Incentive-compatibility, limited liability and costly liquidation in financial contracting

Games and Economic Behavior 118 (2019) 412–433

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Incentive-compatibility, limited liability and costly liquidation in financial contracting ✩ Zhengqing Gui a,∗ , Ernst-Ludwig von Thadden b,c , Xiaojian Zhao d,e a

Economics and Management School, Wuhan University, China University of Mannheim, Germany CEPR, United Kingdom of Great Britain and Northern Ireland d Monash University, Australia e Chinese University of Hong Kong, Shenzhen, China b c

a r t i c l e

i n f o

Article history: Received 3 March 2017 Available online 30 September 2019 JEL classification: D86 G33 Keywords: Financial contracting Incentive-compatibility Limited liability Indivisible collateral Costly liquidation

a b s t r a c t This paper studies a financial contracting problem where a firm privately observes its cash flow and faces a limited liability constraint. The firm’s collateral is piecemeal divisible and can only be liquidated continuously by resorting to the service of a costly third party, typically associated with bankruptcy. In this situation, multi-class collateralized debt is optimal, in which the firm makes several debt-like promises with a seniority structure. The decision over continuous and piecemeal liquidation depends on both the cost of introducing the third party and the firm’s funding need. Allowing the firm to refinance ex-post through surreptitious liquidation may reduce the firm’s ex-ante payoff, consistent with covenants in debt contracts prohibiting the sale of assets. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Information asymmetry is one of the most important impediments to efficient interactions in financial markets. In corporate finance, firms usually have an informational advantage over outside investors, because it is often difficult for investors to observe the outcomes of firms’ activities directly. Similar problems are prevalent in consumer finance, in particular the housing market, where fluctuations in the incomes of borrowers are not easily observed by lenders. Such information asymmetry in the post-contracting stage is one of the main concerns in the theory of financial contracting. In a typical corporate financial contracting environment with asymmetric information, a firm borrows from an investor to invest in a risky project from which it will receive returns that are only observable to itself. Moreover, ex post the firm has limited liability, so it

✩ We thank the Editor, Marco Battaglini, the Advisory Editor, and two referees for their constructive suggestions. We also thank Yeon-Koo Che, YiChun Chen, Kim-Sau Chung, Yuk-Fai Fong, Martin Hellwig, Susheng Wang, participants to conferences and seminars at Chinese University of Hong Kong (Shenzhen), Henan University, Hong Kong Baptist University, Hong Kong University of Science and Technology, Max Planck Institute for Research on Collective Goods, Nankai University, Nanyang Technological University, National University of Singapore, Sun Yat-Sen University, University of Mannheim, University of Queensland, Wuhan University, the 2016 CUHK Workshop on Game Theory and Applications, the 2017 HKUST Workshop on Industrial Organization, CMES 2016, AMES 2017, NASMES 2018, and EEA-ESEM 2018 for their valuable comments. Gui and von Thadden acknowledge financial support from the German Science Foundation (DFG) through CRC TR 224 (Project C9). Gui also acknowledges financial support from the Youth Scholar Team in Social Sciences of Wuhan University: “New perspectives for development economics research”. Junjie Ren provided excellent research assistance. Corresponding author. E-mail addresses: [email protected] (Z. Gui), [email protected] (E.-L. von Thadden), [email protected] (X. Zhao).

*

https://doi.org/10.1016/j.geb.2019.09.011 0899-8256/© 2019 Elsevier Inc. All rights reserved.

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cannot be expected to make arbitrary repayments even if it wanted to. In response to such restrictions, the parties ex-ante use instruments such as collateral or threats of bankruptcy agreed upon in an incentive-compatible contract to secure the truthful execution of their agreement ex-post. Most papers in this literature assume that the threats and promises underlying these contracts can be arbitrarily finely adjusted and costlessly executed. This can be because the firm’s collateral is infinitely divisible (among the many examples, see, e.g., Gale and Hellwig, 1985; Faure-Grimaud, 2000; Povel and Raith, 2004b), or the penalties imposed by the investor can be applied without cost (e.g., Bolton and Scharfstein, 1990; Lacker, 2001). In this paper, we move away from these assumptions, and ask what happens if the firm’s assets are not perfectly divisible, or the ex-post execution of threats is costly. We do this by means of a simple model in the tradition of Faure-Grimaud (2000) that assumes that collateral is not perfectly divisible. In practice, most collateral comes in indivisible units, and even when these units (such as small production units) can eventually be divided into smaller pieces, this requires technical or legal expertise to disentangle the multiple links that make the units cohere in the first place. Alternatively, if the contracting parties agree to play a lottery in order to achieve a division in expected terms, they may disagree on the credibility of the randomization procedure and thus again need an independent third party, if they use it at all. Whether for legal, technical, or implementation reasons, such third-party procedures are costly, which limits their use.1 In this paper, we study the use of such costly procedures in the execution of financial contracts when the firm privately observes its cash flow, is ex-post cash constrained, and its collateral is imperfectly divisible. We solve a financial contracting problem with ex-post information asymmetry, where collateral must be liquidated piecemeal, if the parties do not use a costly third party who has technical or legal expertise in dividing the collateral or implementing a randomization scheme. Using a third party is costly, but it is the only way to avoid the transfer of collateral as a whole. Hence, in particular, the use of randomization or piecemeal liquidation is endogenous in our model, and we show that they can co-exist in an optimal contract. The optimal contract derived in our model is essentially a multi-class collateralized debt (MCD) contract, which takes a piecewise structure involving both piecemeal and continuous liquidation. Such a contract is akin to a combination of several debt contracts with different priority in case of default. There is an endogenous face value of overall debt, and if the firm defaults (i.e. does not service all of its debt), the available cash is allocated to the different debt classes in decreasing order of seniority. Low debt tranches do not receive cash, but some assets either piecemeal or through costly liquidation. In an alternative interpretation, for each piece of its collateral, the firm has to make a fixed cash repayment to prevent it from being liquidated. We introduce the notion of the MCD contract, show that it is indeed optimal, and discuss the structure and comparative statics of the optimal MCD. The intuition for the MCD structure to be optimal results from a tradeoff between the third-party liquidation cost and the deadweight loss from liquidation, and is most simple to understand in the special case of one single indivisible asset. When the firm’s cash flow is sufficiently high, it is required to repay a fixed amount of cash to the investor without any liquidation. When the firm’s cash flow falls short, it has incentive to make a full cash repayments to the investor so as to minimize asset liquidation. Thus, liquidation has to be continuous, which necessitates the introduction of a third party. However, when the firm’s cash flow is sufficiently low, it has to liquidate a large fraction of the collateral even with a full cash repayment and continuous liquidation. Therefore, the firm would rather liquidate the collateral piecemeal without involving a third party and save the liquidation cost. This exhibits a potential logic for the firm’s decision on how to deal with its assets in the face of bankruptcy: If it is optimal to keep all assets intact, then the parties settle the contract without third-party help. If it is optimal to divide assets into pieces, a more expensive process is required. It is only worth paying the fixed cost to split up an asset if the allocation is substantially different from not splitting it up. The comparative statics of optimal MCD contracts is straightforward. When the liquidation cost is sufficiently high, the firm would never want to introduce a third party to perform continuous liquidation, so it will liquidate the collateral piecemeal whenever it has insufficient cash. Our optimal contract thus resembles the one derived in Lacker (2001). Similarly, when the liquidation cost is sufficiently small, the firm would prefer continuous liquidation when it is insolvent. In this case, our optimal contract resembles the debt contract in Faure-Grimaud (2000). We also discuss some important features of optimal MCD. First, in our model, the incentive-compatibility constraint may not bind in an optimal MCD contract. The firm loses its collateral when its cash flow is sufficiently low, but its valuation of the collateral could be higher than the fixed cash repayment when there is no liquidation. In other words, the firm is punished more severely than incentive-compatibility would require when it is insolvent. The reason is that the firm cannot falsely claim a state that necessitates a cash repayment higher than its realized cash flow, i.e., the firm cannot exaggerate its cash holdings. That means our incentive constraint is specified consistently with the firm’s limited liability. This is noteworthy because other papers in this literature, such as (Faure-Grimaud, 2000; Povel and Raith, 2004b; Hege and Hennessy, 2010), have assumed that the firm can repay any amount of cash out-of-equilibrium irrespective of its realized cash flow, implying that the firm must be punished uniformly across the state space, i.e., the firm is indifferent between repaying the loan (when it can afford to) or relinquishing the collateral. However, when the incentive constraint is correctly specified as in our model, the firm strictly prefers the fixed cash repayment, i.e., a repayment that is strictly lower than the firm’s loss from liquidating the collateral. If the incentive constraint is misspecified this cannot happen because it requires

1

See Lacker (2001) for a discussion of some of the problems of randomization as a contracting tool.

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the firm to have no incentive to spend some extra money out of its empty pocket to avoid liquidation. It allows the firm to make infeasible deviations that violate limited liability and may sometimes lead to suboptimal contracts. Second, we discuss a variant of the model where the firm can surreptitiously liquidate its collateral after the realization of its cash flow, and before the execution of the contractual transfers. This assumption relaxes the firm’s limited liability and enables it to exaggerate its cash flow ex-post. We show that in this variant of the model the optimal contract still exhibits a piecewise structure when surreptitious liquidation is less efficient than liquidation by the original investor. When private surreptitious liquidation is sufficiently efficient, the optimal MCD contract we derive is a debt contract without bankruptcy, i.e., the firm always has enough cash to repay the debt, and avoid any liquidation by the incumbent investor. Moreover, we show that allowing for surreptitious liquidation may hurt the firm, as it makes the incentive constraint stronger by relaxing the firm’s limited liability. This paper can be viewed as a synthesized framework of financial contracting with ex-post information asymmetry and indivisible collateral. When the cost of continuous liquidation is small, our model becomes similar to the classical model of Costly State Verification (CSV) established by Townsend (1979) and Gale and Hellwig (1985). When the cost of continuous liquidation is infinity, our model resembles that of Hart and Moore (1998) in the sense that the firm’s cash flow cannot be verified by a third party. Moreover, this paper can also be interpreted as a theory of bankruptcy, if one views the bankruptcy court as a trusted third party. When the firm is insolvent, it can either turn to a bankruptcy court which verifies its cash repayment, divides its collateral, and transfers part of the assets to the investors, or it can keep its cash and leave the assets wholesale to the investor. This latter option resembles the strategy often chosen by defaulting homeowners in the real estate market. Our model thus provides a benchmark for studying the firm’s tradeoff between these two choices and shows that they can endogenously arise within the same contract. It also offers an explanation for the existence of bankruptcy courts. If it is optimal to keep all assets intact, then there is no need to turn to a third party. If it is optimal to divide assets into pieces, then a more expensive process is required and bankruptcy courts may have to play a role. Related literature Our paper builds on the theory of financial contracting with ex-post asymmetric information about borrower returns, as developed, i.e., by Townsend (1979), Diamond (1984), Gale and Hellwig (1985), Bolton and Scharfstein (1990), and Faure-Grimaud (2000), and extends it to problems of indivisible or imperfectly divisible collateral.2 Under perfect divisibility of collateral, most of the previous literature along the lines mentioned above has found optimal contracts that can be interpreted as debt. Border and Sobel (1987) and Mookherjee and Png (1989) showed that debt may be Pareto dominated when random auditing is allowed, and Krasa and Villamil (2000) argued that whether the optimal contract is stochastic depends on the informed party’s commitment power. Lacker (2001) argued that stochastic contracts become desirable if collateral is indivisible and provides a condition for a debt contract to be optimal in a model of indivisible assets with stochastic liquidation. That paper also assumed that the contracting parties cannot credibly pre-commit to randomize their future actions, so randomization mechanisms can only be used if they are ex-post optimal. As a consequence, Lacker showed that with indivisible collateral the optimal contract takes a bang-bang structure: The firm is obligated to repay a certain amount of cash to the investor. If it fails to repay this amount, its collateral will be seized as a whole, and the firm must make a smaller repayment. Our paper is distinct from Lacker (2001) in several aspects. The key difference is that in our model the third party can do something more than a randomization device: He may also inspect the firm’s cash repayment, which provides the basis of state-contingent continuous liquidation. In this sense, our model builds a bridge between several different approaches in the field of financial contracting, such as the CSV model (Townsend, 1979; Gale and Hellwig, 1985) and the Hart-Moore model (Hart and Moore, 1998). Our approach is less drastic than Lacker (2001), as we make a stronger assumption that allows the firm to commit to introducing the third party, but possibly at a cost. If the cost of relying on a third party is high, we recover some of his predictions as a special case of our model. Moreover, Lacker (2001) is concerned with the question when debt is optimal, while in our paper the question is “What is an optimal contract”. Finally, in our model, assets can be partially divisible, and can also be surreptitiously liquidated by the firm; these topics are not covered in Lacker (2001). There are also other papers working with the assumption of indivisible collateral, but they do not focus on how the indivisibility affects the structure of the optimal contract (Yao and Zhang, 2005; Gorton and Ordonez, 2014). Our paper is further related to the literature on optimal bankruptcy design (Aghion et al., 1992; Berkovitch and Israel, 1999; Bris et al., 2006; von Thadden et al., 2010; Gennaioli and Rossi, 2013). When facing financial distress, the firm can either go through the bankruptcy procedure and have its assets liquidated by the decision of the court or reach an agreement with the investor that includes a transfer of asset ownership. Our paper provides a new perspective on the tradeoff between going through the bankruptcy procedure and reaching an out-of-court settlement, and how this choice varies with the realized states. The rest of the paper is organized as follows. Section 2 introduces the model. Section 3 presents our main results. Section 4 provides the example of one single indivisible asset to be used as collateral. Section 5 discusses the incentive constraint in the model and modifies our benchmark model to allow for surreptitious liquidation. Section 6 concludes. All the proofs are relegated to the Appendix.

2 There is another stream of literature that studies the problem of hidden actions due to the separation between ownership and control. See, e.g., Innes (1990) and, more generally, Tirole (2010).

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2. Model Consider a financial contracting environment between a firm and an investor. The firm is endowed with I 0 ≥ 0 units of cash (numeraire) and seeks funding from the investor who is assumed to have deep pockets. The firm has the opportunity to undertake a risky project. The project needs I > I 0 units of cash to initiate and will generate some cash flow ω and fixed assets with future value V to the firm and a market value of W < V . For notational convenience, we denote W / V = α < 1. Hence, as in Hart and Moore (1998), assets should optimally remain in the ¯ ].4 Moreover, possession of the firm; liquidation is socially inefficient.3 ω is stochastic with c.d.f. F and p.d.f. f over [0, ω the distribution of ω is common knowledge, while the realization of ω can be only observed freely by the firm. Thus ω is the firm’s private information, to which we will sometimes refer as its (ex-post) type. Assume that f (ω) ∈ [ f , f ] for ¯ ], and I < E (ω), so the investment is profitable. We let B denote the firm’s initial borrowing, and allow it to any ω ∈ [0, ω borrow more than its funding need, i.e., B ≥ I − I 0 . After the realization of ω , the firm can use the cash or liquidate part of its assets to repay the investor. Due to the information asymmetry between the two parties, cash repayments and the asset ˆ. liquidation can only be contingent on the firm’s report of the state, which is denoted by ω We suppose that the firm’s cash repayments cannot exceed its ex-post cash holdings, i.e., the firm is protected by limited ˆ ) the firm’s cash repayment when it reports ωˆ , then limited liability requires that liability. If we denote by R (ω

R (ω) ≤ I 0 + B − I + ω

¯ ]. for any ω ∈ [0, ω

(LL)

By assuming limited liability, we rule out the firm’s ability to borrow after the realization of ω . In a dynamic model the firm may be able to obtain additional finance at this stage in order to prevent assets from being liquidated, as the private continuation value of its assets (V ) is larger than their liquidation value (W ). In our static model, this is impossible because the firm has nothing to pledge in the future. In Section 5 we discuss a variation of our model in which ex-post borrowing is allowed and takes the form of surreptitious liquidation. ˆ ) the fraction of assets to be liquidated when the firm reports ωˆ . Since the investor and Furthermore, we denote by X (ω other market participants have the same valuation of the firm’s assets, which is lower than that of the firm, it does not matter whether the firm liquidates the assets and pays the liquidation value to the investor or whether the firm transfers the assets to the investor. Different from ω , we assume that assets are observable and verifiable. We assume that the firm’s assets consist of finitely many indivisible units and that, without further action, each such unit can only be transferred as a whole. Hence, the firm’s assets are what we call piecemeal divisible and thus can be divided into finitely many parts. In other words, in the absence of third-party intervention (see below), the liquidation decision ˆ ) must be taken from a finite set, denoted by X = {x0 , x1 , . . . , xn }, with n ≥ 1 and 0 = x0 < x1 < · · · < xn = 1, implying X (ω that no liquidation and full liquidation can always be chosen. We analyze the general case n ≥ 1 in Section 3; the case n = 1 of a single indivisible asset (think of a house as collateral for a mortgage) will serve as a leading example in Section 4. While the firm’s asset base as such is only piecemeal divisible, it can be continuously liquidated if the parties bring in a third party. This costs c ≥ 0 and makes it possible to liquidate the assets in any desired way. This may either be through technical and legal expertise with respect to the assets or by using and certifying an appropriate stochastic mechanism.5 The third party can be interpreted as a court, which supervises the continuous liquidation procedure or which implements stochastic liquidation, but since we do not model institutional details here, it can also be interpreted as a private out-of-court ˆ mediator.6 Liquidation supported by the third party can take any value in the closed interval [0, 1]. Let ψ be a function of ω ˆ : ψ(ωˆ ) = 1 if yes, otherwise ψ(ωˆ ) = 0. that indicates whether the third party should be introduced when the firm reports ω ˆ ) is affected by ψ(ωˆ ). We will formally refer to this constraint as the feasibility Therefore, the liquidation decision X (ω constraint, i.e.,

If ψ(ω) = 0 then X (ω) ∈ X . If ψ(ω) = 1 then X (ω) ∈ [0, 1].

(FC)

Varying values of c correspond to different cases in the literature. If c = 0, the firm’s assets are perfectly divisible as in models of Faure-Grimaud (2000) and others. If c is sufficiently large, the asset is completely indivisible, such as in Lacker (2001). If c is positive but small, this is as in the models of costly state verification of Townsend-Gale-Hellwig. For simplicity, we assume the cost c is monetary and borne by the investor. A contract now specifies: (1) the firm’s initial borrowing B; (2) the cash repayment R from the firm to the investor; (3) the fraction of assets to be liquidated X ; and (4) a function ψ indicating whether the third party is called upon. (2), (3), ˆ . In the following analysis, we will denote the contract by , i.e.,  = ( B , R , X , ψ). and (4) are all functions of ω

3 In Hart and Moore (1998), the cash flow is observable by both parties but not verifiable by outsiders, in order to simplify the problem of ex-post renegotiation. 4 Our analysis will go through if ω is distributed over [0, +∞). 5 Randomization achieves continuous liquidation in expected value, which is all that matters in our risk-neutral setting. Stochastic liquidation has been discussed, e.g., by Mookherjee and Png (1989). 6 See, e.g., Aghion et al. (1992) or von Thadden et al. (2010) for fuller models of bankruptcy and bargaining.

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In addition, we assume that both contracting parties are risk-neutral. Given the realized cash flow ˆ , the ex-post utility functions are report ω

ω and the firm’s

ˆ ) = − B + R (ωˆ ) + X (ωˆ ) W − ψ(ωˆ )c u I (, ω, ω for the investor, and

ˆ ) = I 0 + B − I + ω − R (ωˆ ) + [1 − X (ωˆ )] V u F (, ω, ω for the firm. We assume that the parties can commit to the ex-post execution of the contract ex-ante. Therefore, by the revelation principle, we can without loss of generality focus on direct mechanisms. In these mechanisms, the firm must have incentives to truthfully report its cash flow. That leads to the incentive-compatibility constraint

ˆ ) for any ω, ωˆ such that R (ωˆ ) ≤ I 0 + B − I + ω. u F (, ω, ω) ≥ u F (, ω, ω

(IC)

Note that we only require that each (ex-post) type of the firm has no incentive to choose the repayment/liquidation combinations designed for other types among those it can afford in state ω . Hence, the set of deviations possible for each type depends on the (endogenous) contract. This complication of the incentive-compatibility constraint has been overlooked in the literature up to now7 and makes the analysis more difficult than under the standard approach (which ignores the qualification of feasible deviations). However, such qualification is essential in a model built on the very notion of limited liability. Finally, the investor should at least break even if she accepts the contract, which implies her individual rationality constraint:

E ω u I (, ω, ω) ≥ 0.

(IR)

Hence, a full statement of the contracting problem is

max 

subject to

E ω u F (, ω, ω), (LL), (FC), (IC) and (IR).

We say that a contract  is optimal if it is a solution to this problem. 3. Analysis

ˆ , we say that  is weakly dominated by We start with some concepts and notations. First, for any two contracts,  and 

ˆ if the following two inequalities hold:

ˆ , ω, ω), E ω u I (, ω, ω) ≤ E ω u I ( ˆ , ω, ω). E ω u F (, ω, ω) ≤ E ω u F ( Second, since

E ω u F (, ω, ω) = I 0 + B − I + E ω ω + V − E ω [ R (ω) + V X (ω)], the problem is equivalent to maximizing B − E ω (ω), where

(ω) = R (ω) + V X (ω) is the firm’s valuation of its total ex-post payout. By (IC), the firm can always understate its cash flow, so (ω) must be nonincreasing. Lastly, we introduce a specific form of debt contract that uses the piecemeal structure of the firm’s assets. In such a contract, the firm makes a debt-like promise against each piece of its asset base. For each such piece, the firm has to pay a fixed amount of cash to prevent this piece from being liquidated, either partially (at cost c) or as a whole (at cost 0). These promises have a priority structure in the sense that the most junior promise is broken first if the firm’s total cash is insufficient to repay its debt in full, then the second-lowest promise, etc. Definition 1. Recall that X = {x0 , x1 , . . . , xn }. Then  is a multi-class collateralized debt (MCD) contract if there exists a p sequence {(ω j , ωcj , φ cj )} j =0,1,...,n , such that:

7

See, e.g., Faure-Grimaud (2000), Povel and Raith (2004b), and Hege and Hennessy (2010) and our discussion in Section 5.

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Fig. 1. The structure of an MCD contract (n = 1).

Fig. 2. The structure of an MCD contract (n = 2). p

p

p

c ¯ = ω− (a) 0 = ωn ≤ ωnc −1 ≤ ωn−1 ≤ · · · ≤ ω0c ≤ ω0 ≤ ω 1; p 0,

p 0

(b) If ω ≥ ω the firm repays ω and there is no liquidation; (c) For any j = 1, . . . , n, p (c.1) if ω ∈ [ω j , ωcj−1 ), then liquidation is piecemeal with a constant cash repayment, i.e. p

ψ(ω) = 0, X (ω) = x j , and R (ω) = I 0 + B − I + ω j ; p (c.2) if ω ∈ [ωcj , ω j ), then liquidation is continuous with a compensating cash repayment, i.e. φ c −( I 0 + B − I )−ω

ψ(ω) = 1, X (ω) = j , R (ω) = I 0 + B − I + ω and constant total payout φ cj ; V (d) The total payout (ω) is weakly decreasing in ω . Fig. 1 and 2 provide graphical illustrations of MCD contracts with n = 1 and n = 2, correspondingly. With MCD, the firm p p p issues n claims with a total face value of ω0 . If ω ≥ ω0 , the firm repays in full and there is no liquidation. If ω < ω0 the firm defaults. Depending on the firm’s cash position in default, its outstanding collateralized claims are dealt with in order p p of seniority. If ω ∈ [ω1 , ω0 ), the most junior claim is not repaid in full and all the other claims are repaid in full. The most junior claim is secured against a piece of assets worth 1 − x1 , which can be either liquidated in full, with no cash payout p (if the cash flow attributed to this claim is relatively low: ω ∈ [ω1 , ω1c )) or liquidated continuously with compensating cash p payouts (if the cash flow attributed to this claim is relatively high: ω ∈ [ω0c , ω0 )). On both sub-intervals, the value of the p p firm’s total payout for the firm, (ω), is constant. If ω ∈ [ω2 , ω1 ), also the second most junior claim is not repaid in full, and so on all the way up the priority ladder to the most senior claim n. Here the superscript p stands for “piecemeal p liquidation”, i.e., ω j is the left endpoint of an interval with piecemeal liquidation (in which the whole unit is liquidated).

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ωcj is the left endpoint of an interval with continuous liquidation. Finally, note that, for any j ≥ 0, R (ω) may have discontinuities at ωcj , while X (ω) may have discontinuities at p both ωcj and ω j . We shall show below that optimal contracts will indeed exhibit such jumps, which result from switches Similarly, the superscript c stands for “continuous liquidation”, i.e.,

to and from regimes with the bankruptcy cost.8 In what follows, we will show that any MCD contract satisfies (LL), (FC) and (IC); moreover, any contract that satisfies (LL), (FC), (IC) and (IR) is weakly dominated by an MCD contract. Hence, if the contracting problem has a solution, one solution is an MCD. Proposition 1. Any MCD contract satisfies (LL), (FC) and (IC). The proof of Proposition 1 is centered around verifying the incentive constraint. Since, by (d) of Definition 1, the firm’s p payout (ω) is nonincreasing, it is not profitable for the firm to understate its cash flow. Moreover, when ω ∈ [ωcj , ω j ) for some j, the firm cannot exaggerate its cash flow, because higher cash flow leads to higher cash repayments, which must p exceed its realized cash flow ω . When ω ∈ [ω j , ωcj−1 ) for some j, the firm can exaggerate its cash flow, but its payout is then the same as that with a truthful report. Proposition 2. Any contract that satisfies (LL), (FC), (IC) and (IR) is weakly dominated by an MCD contract with B = I − I 0 . As a consequence of Proposition 2 we shall from now on restrict attention to MCD contracts with maximum equity participation (B = I − I 0 ), the latter in the spirit of Gale and Hellwig (1985). This is partially for tractability, partially because MCD contracts have a natural institutional interpretation, as discussed above.9 The proof of Proposition 2 involves several steps. First, for any contract that satisfies (LL), (FC) and (IC), whenever there is an ω j such that the firm liquidates x j piecemeal at ω j , we identify the smallest possible state that yields the cash p p repayment R (ω j ), and define it as ω j . It can be shown that all the states between ω j and ω j yield the same total payout p

value, i.e., (ω) is constant on [ω j , ω j ]. Second, consider an alternative contract that is defined as follows: When p , j

ω is

p , j

sufficiently close to ω and ω ≥ ω the firm liquidates x j piecemeal, otherwise, the firm liquidates as little as it can with continuous liquidation. At the same time, we make the firm’s expected payout identical between the two contracts. The key point in this step is to find the cutoff that determines whether the firm in state ω should use continuous liquidation. Roughly speaking, the benefit of using continuous liquidation is that it enables the firm to minimize its liquidation for a given ω , therefore such benefit decreases with x j − X (ω). The cost of using continuous liquidation is the constant c, which is ultimately borne by the firm. The cutoff is then pinned down by resolving this tradeoff. Finally, we can verify that this new contract is an MCD contract that weakly dominates the initial contract. Note that while any contract that satisfies (LL), (FC) and (IC) is weakly dominated by an MCD contract, it does not imply that MCD is the only possible structure for optimal contracts. To see this, suppose that  is an optimal MCD contract. By p Definition 1, an MCD contract has (ω) = φ cj for any ω ∈ [ωcj , ω j ), implying that X (ω) has a slope of −1/ V , as shown in

Fig. 2. Consider another decreasing function Xˆ (ω) defined for any and satisfy the following:

ω ∈ [ωcj , ω pj ). Let Xˆ (ω) have a slope smaller than −1/ V

E ω [ X (ω)|ωcj ≤ ω < ω j ] = E ω [ Xˆ (ω)|ωcj ≤ ω < ω j ]. p

p

This transformation requires that we increase X (ωcj ) and decrease limω→ω p − X (ω) without changing the integral of X (ω) j p

on the interval [ωcj , ω j ), which is possible because of the discontinuities. Then, suppose that

 p ( B , R , Xˆ , ψ) if ω ∈ [ωcj , ω j ), ˆ =  otherwise.

ˆ is also optimal, although it is not an MCD contract. In other words, ˆ generates the same expected We can prove that  payoff for two contracting parties as , but has a steeper liquidation function on one of its intervals with continuous ˆ incentive-compatible. liquidation; this feature still makes  8 Both R (ω) and X (ω) have discontinuities at ωcj because the firm switches from piecemeal liquidation to continuous liquidation at ωcj . Under piecemeal liquidation the firm does not use all the realized cash to repay its debt, while under continuous liquidation the contract can have a full cash repayment. R (ω) jumps from a partial cash repayment to a full cash repayment, and X (ω) jumps from x j to something between x j and x j +1 , so they are all disconp p p tinuous at ωcj . With optimal MCD, X (ω) may have discontinuities at ω j , because the firm faces a tradeoff at ω j : On the one hand, increasing ω j reduces p

the expected liquidation on the interval (ω j ) and shifts the X function downward, which may reduce the firm’s loss from liquidation. On the other hand, increasing

p j

c j,

ω enlarges the interval (ω ω

p j)

on which liquidation is continuous, thus increasing the probability of incurring the bankruptcy cost c; this is p

bad for the firm. Whether the firm wants to reduce ω j and make X (ω) continuous at occur. 9 All our results hold for more general MCD contracts with B ≥ I − I 0 .

ω pj depends on these two opposing forces, both possibilities can

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Based on Proposition 2, we know that the firm’s problem can be solved by finding an optimal MCD contract, a probc ¯ lem that can be solved using the standard Lagrangian method. Therefore, remembering the convention that ω− 1 = ω , the problem of finding an optimal MCD contract can be rewritten as n n −1   p p p (ω j + x j V )[ F (ωcj−1 ) − F (ω j )] + φ cj [ F (ω j ) − F (ωcj )]

min

p

{(ω j ,ωcj ,φ cj )} j=0,1,...,n

j =0

j =0

subject to the constraints that are imposed by the MCD structure: p

p

p

¯, 0 = ωn ≤ ωnc −1 ≤ ωn−1 ≤ · · · ≤ ω0c ≤ ω0 ≤ ω

ω0p ≤ φ0c ≤ ω1p + x1 V ≤ φ1c · · · ≤ φnc−1 ≤ V , and the investor’s participation constraint:

−( I − I 0 ) +

n  j =0

ω pj

p

p

(ω j + x j W )[ F (ωcj−1 ) − F (ω j )] +

n −1   j =0

[ω + α (φ cj − ω) − c ]dF (ω) ≥ 0.

ωcj

Proposition 3 now shows that the resulting MCD has a potentially rich structure, depending on the size of the liquidation cost c and the firm’s funding need I − I 0 . Proposition 3. There exist two cutoffs, c and c, and two functions, I (c ) and I (c ), such that: (a) If I − I 0 < I (c ), then any optimal MCD contract with B = I − I 0 has only continuous liquidation; moreover, when c > c, I (c ) = 0; (b) If I − I 0 < I (c ), then any optimal MCD contract with B = I − I 0 has only piecemeal liquidation; moreover, when c < c, I (c ) = 0. Proposition 3 is proved by exploiting the Lagrangian. Let L be the Lagrangian for the firm’s problem, and λ be the multiplier for the investor’s participation constraint. We first assume that there is piecemeal liquidation in an optimal MCD contract. By first-order necessary conditions, we can compute a lower bound for λ. Then, we show that a larger λ is associated with a larger I − I 0 , so there is an upper bound for I − I 0 , which in turn serves as a sufficient condition for continuous liquidation, i.e., the result stated in (a) of Proposition 3. Following the same procedure, we assume that there is continuous liquidation in an optimal MCD contract to get the sufficient condition in (b) of Proposition 3. According to Proposition 3, whether the firm chooses piecemeal or continuous liquidation in bankruptcy depends largely on the relationship between the funding need I − I 0 and the liquidation cost c. When c is small, we have I (c ) = 0, and thus any optimal MCD contract has continuous liquidation, as continuous liquidation is not very costly to implement. In particular, when I − I 0 is also sufficiently small such that I − I 0 < I (c ), any optimal MCD contract has only continuous liquidation. On the contrary, when c is large, we have I (c ) = 0, and thus any optimal MCD contract has piecemeal liquidation as it is too costly to introduce the third party. In particular, when I − I 0 is also sufficiently small such that I − I 0 < I (c ), any optimal MCD contract has only piecemeal liquidation. Proposition 4 characterizes optimal MCD with only piecemeal liquidation. It provides a necessary condition that deterp p p mines ω j based on ω j −1 and ω j +1 . Proposition 4. For any optimal MCD contract with only piecemeal liquidation, there exists μ ≤ α such that for any j = 0, 1, . . . , n − 1, p p p if (ω j −1 ) < (ω j ) < (ω j +1 ), then p

ω pj − ω pj+1 − μ(x j+1 − x j ) V =

p

F (ω j −1 ) − F (ω j ) p

f (ω j )

(1)

.

To understand the intuition of (1), consider the problem of choosing the optimal p j

p j +1

ω pj on the interval [ω pj+1 , ω pj−1 ]. By

ω has to repay ω units of cash, and transfer a fraction x j +1 of its assets p p p p to the investor, while the firm with a type higher than ω j has to repay ω j and transfer x j . Note that ω j +1 < ω j , and definition, the firm whose type is lower than

x j +1 > x j . Therefore, it is as if the investor is selling x j +1 − x j of the assets to the firm whose ability to pay is distributed p p over [ω j +1 , ω j −1 ]. Standard theory of optimal pricing tells us that the price net of the cost should be equal to the inverse of the conditional hazard rate (Tirole, 1988), i.e., p

ω pj − ω pj+1 − μ(x j+1 − x j ) V = 



Price







Shadow cost



p

F (ω j −1 ) − F (ω j ) p



f (ω j )





Inverse of the conditional hazard rate

.

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Fig. 3. Optimal MCD contract with only piecemeal liquidation (n = 4).

ω pj repays extra ω pj − ω pj+1 units of cash, and retains a fraction p p the investor actually sets a “price” that equals ω j − ω j +1 , and her “shadow

In other words, the firm whose ex-post type is higher than p , j

x j +1 − x j of its assets. Thus, by choosing ω cost” is x j +1 − x j multiplied by μ V ≤ W , which indexes the efficiency loss from liquidation. The investor’s marginal revenue is determined by the inverse of the conditional hazard rate. Hence, (1) comes from setting the price equal to the marginal revenue. Moreover, since (1) holds for any j, the multiplier before the “shadow cost”, which is μ, should be constant for all j in an optimal contract. This suggests that the marginal efficiency loss from liquidation is constant for all j at the optimum. Fig. 3 depicts the optimal MCD contract characterized in Proposition 4 with n = 4. This contract can be implemented by p p a sequential repurchase agreement: For any j, the contracting parties set a price ω j − ω j +1 for a fraction x j +1 − x j of the assets; the firm can repurchase its assets if it can afford the corresponding price after the realization of ω . Moreover, the firm must repurchase in order of seniority: It must pay for x j +1 − x j before x j − x j −1 . We finally turn to the question how much piecemeal liquidation is used in an optimal MCD contract? In other words, what is the ex-ante probability for whole parts of the asset base to be transferred without outside intervention? Note that p with MCD, piecemeal liquidation is implemented only when ω ∈ [ω j , ωcj−1 ). Proposition 5 tells us that the length of these intervals is bounded above by a linear function of c. Proposition 5. For any optimal MCD contract and any j = 0, 1, . . . , n − 1,

ωcj − ω pj+1 ≤

c 1−α

(2)

. p

To prove Proposition 5, observe that if ωcj > ω j +1 , then ∂ L /∂ ωcj must be nonpositive, otherwise one can reduce ωcj to increase the value of L. Proposition 5 implies that with optimal MCD, the intervals implementing piecemeal liquidation shrink to zero as c converges to zero. This is the case of several papers in the literature, such as Diamond (1984) and Faure-Grimaud (2000). Corollary 1. If c = 0, then the optimal MCD contract has the following feature: There exists a cutoff p p p R (ω) = ω0 , X (ω) = 0; when ω < ω0 , R (ω) = ω , X (ω) = (ω0 − ω)/ V .

ω0p such that when ω ≥ ω0p ,

Proposition 4 and Corollary 1 are more elaborate discussions of the two extreme cases considered in Proposition 3: Proposition 4 speaks to the case where c is sufficiently large, while Corollary 1 talks about the case where c = 0. 4. An illustrative example In this section, we assume that assets are completely indivisible, i.e., n = 1, and show how the structure of the optimal MCD contract varies with the bankruptcy cost c and the funding need I − I 0 . According to Proposition 2, when n = 1, there p is only one class of debt, and the optimal MCD contract is pinned down by three parameters: The face value of debt ω0 , c c the threshold for wholesale surrender of the collateral ω0 , and the firm’s total loss under continuous liquidation φ0 . By Proposition 3, if c and I − I 0 are sufficiently small, liquidation in any optimal MCD contract is fully continuous, which is shown in Fig. 4.

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Fig. 4. Optimal MCD contract with indivisible assets (c is small).

Fig. 5. Valuation of contractual transfers with indivisible assets (c is small).

It is also instructive to study the valuation of the contractual transfers from both parties’ perspectives. As a complement to Fig. 4, the total transfers from the firm as valued by the firm, R (ω) + X (ω) V , and as valued by the investor, R (ω) + X (ω) W , are depicted in Fig. 5. The contract in Fig. 4 differs from the optimal contract in Faure-Grimaud (2000) in an important aspect: The value of the p p firm’s expected payout conditional on [0, ω0 ) is determined by φ0c , which exceeds the face value ω0 . In other words, while p the firm needs to pay ω0 to settle its debt, it will be punished by more than this amount if it defaults. According to Gale and Hellwig (1985), a standard debt contract is defined by the following three properties: (1) the firm pays a fixed amount of cash when it is solvent; (2) the firm is declared bankrupt if this fixed cash repayment cannot be met; (3) in bankruptcy, the investor recoups all of the firm’s available assets. As Fig. 5 shows, an optimal MCD contract satisfies p the first two properties but not the third one. When the firm fails to repay the face value of debt ω0 , assets liquidated in market value are not enough to make up for the gap between the firm’s cash repayment and the debt’s face value. However, the firm is allowed to keep some of its assets in this situation. In other words, there is a violation of absolute priority, since the rule states that creditors must be served before equity holders. Such violations of absolute priority have long been standard in practice and have been widely documented (see, e.g., Franks and Torous, 1994). Financial theory has usually justified them by invoking the benefits of renegotiating ex-post inefficient contracts.10 Our model of full ex-ante commitment highlights another reason: Due to the ex-post difference in continuation values, investors suffer a loss when the firm is insolvent, even when the debtor loses more in default than

10

This includes, e.g., the above mentioned papers Aghion et al. (1992) or von Thadden et al. (2010).

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Fig. 6. Optimal MCD contract with indivisible assets (c is medium).

Fig. 7. Optimal MCD contract with indivisible assets (c is large).

by repaying in the good states. A standard debt contract would exacerbate this latter punishment, shifting the balance inefficiently towards punishment rather than preservation of value, and is thus not optimal in our framework. Turning to the case of large borrowing needs I − I 0 depicted in Fig. 6, the firm would have to liquidate a large fraction p of assets by using the third party when its cash flow is far below the face value ω0 . In this case, the relative benefit from implementing continuous liquidation becomes small and the firm would rather transfer the assets fully than incur the cost of continuous liquidation. Continuous liquidation and the wholesale transfer of the asset thus coexist in an optimal MCD contract in an interesting way: If the realized cash is small, the firm keeps the cash and walks away from the contract, leaving the asset wholly with the investor. If the cash is intermediate, liquidation is moderate, and the firm prefers to repay what it has and liquidate partially. This behavior resembles actual practice in the housing market and does not seem to have been analyzed in the contracting literature so far. If, on the other hand, c is large, the firm will never implement continuous liquidation. The optimal MCD contract we derive in this case is a pure “walk away contract”, similar to the one derived in Lacker (2001) and is depicted in Fig. 7. Hence, even in the simplest case of assets composed of only one indivisible unit, an optimal MCD contract can take a number of different forms that resolve the underlying tradeoff between the costs of defaulting in an interesting way. 5. Discussion Nonbinding incentive-compatibility constraint. As discussed in Section 1, the literature on financial contracting with ex-post information asymmetry has frequently employed a shortcut with respect to the incentive-compatibility constraint that is inconsistent with the crucial assumption of limited liability. Taking Faure-Grimaud (2000) as an example, in its benchmark model, the incentive constraint is specified as

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ˆ) u F (, ω, ω) ≥ u F (, ω, ω

ˆ. for any ω, ω

423

(IC )

This implies that the firm can deviate to announce any payoff realization, regardless of whether the deviation entails feasible transfers. In other words, the constraint ignores the limited liability constraint off the equilibrium. The correct incentive constraint (IC) is weaker than (IC ), because under the correct constraint a type-ω firm cannot ˆ with R (ωˆ ) > ω . Thus, there are fewer deviations being ruled out, which suggests that (IC) is less restrictive mimic state ω than (IC ). Other papers that have used this restricted approach include Faure-Grimaud and Mariotti (1999), Povel and Raith (2004a), Hege and Hennessy (2010), Khanna and Schroder (2010), Arve (2014), and Tamayo (2015).11 Clearly this approach simplifies the analysis considerably, because (IC ) immediately implies that R (ω) + X (ω) V is constant across ω and that the incentive-compatibility constraint and the investor’s participation constraint therefore bind. If one does not show that the solution under the restricted approach is also optimal in the larger admissible set, the analysis is incomplete. The case c = 0 in our model fills this gap. Our model also shows that, when c > 0, the incentive constraint may cease to be binding, which shows that the restricted approach is not necessarily valid. In fact, whether p (IC) binds in an optimal MCD contract actually depends on the cutoffs {(ω j , ωcj , φ cj )} j =0,1,...,n resulting from the firm’s optimization problem. An extreme example is that when c is sufficiently large, the contract in Fig. 7 is optimal, but (IC) p binds only when ω0 = V . Moreover, even if c is sufficiently small, and an optimal MCD contract only has continuous liquidation, as depicted in Fig. 4, (IC) may not be binding. Proposition 6 provides a necessary condition for a binding (IC) when an optimal MCD contract has the structure displayed in Fig. 4. Proposition 6. If the optimal MCD contract only involves continuous liquidation and has a binding incentive constraint, then p

1 − F (ω0 ) p 0)

f (ω

> c,

p

where ω0 is determined by a binding (IR). The condition stated in Proposition 6 is satisfied by c = 0. When c > 0, this condition depends on the hazard rate of

ω at the face value ω0p , and is not necessarily satisfied by an optimal contract. The key intuition for the nonbinding (IC)

is as follows. When continuous liquidation is costly, the firm’s objective function becomes two-dimensional: It not only wants to minimize the ex-ante probability of being liquidated, but also aims to save the cost incurred by implementing continuous liquidation. In fact, the firm’s objective would be a linear combination of the expectation of two variables, X p and ψ . Therefore, compared to the model with c = 0 and a binding (IC), the firm may reduce ω0 and make the incentive constraint slack in order to reduce the expected liquidation cost. Surreptitious liquidation. In the model discussed in the previous sections, the firm’s limited liability constraint only refers to its cash flow, because we have assumed that assets are observable and verifiable. Alternatively, one may assume that the firm can generate cash by liquidating some of its assets privately and thus mimic types at which the contract specifies higher cash repayments. In this part, we provide a simple example to discuss how the structure of an optimal MCD contract varies if we allow for such private surreptitious liquidation. The model is as in Section 2, with the exception that the firm can liquidate its assets right after the realization of ω and before the execution of contractual transfers. We assume that assets that are liquidated this way generate W s units of cash to the firm, where 0 < W s < V . If W < W s , liquidation by the firm is more efficient than liquidation by the incumbent investor, and vice versa. Here the subscript s stands for “surreptitious liquidation”. To simplify our analysis, we assume that assets are indivisible, i.e., n = 1. Let Y denote the quantity of assets being liquidated secretly, then Y ∈ {0, 1}. We assume that the incumbent investor is aware of the possibility that assets may be liquidated secretly by the firm, but she cannot observe how much of the assets are liquidated.12 In other words, Y is the firm’s private information, and it is a function of ω . Moreover, we rule out continuous liquidation by assuming that c is sufficiently large. This is a relatively strong assumption, but it suffices to make the point. The general case is similar, because the secret realization of ex-post cash is limited and comes at a cost that the contracting parties anticipate and therefore try to avoid in any optimal contract. Finally, we assume maximum equity participation, B = I − I 0 . The firm’s utility function is then:

ˆ ) = ω + Y (ω) W s − R (ωˆ ) + [1 − X (ωˆ ) − Y (ω)] V . u F (, ω, ω 11 See (IC) in Faure-Grimaud and Mariotti (1999), (3) in Faure-Grimaud (2000), (A.1) in Povel and Raith (2004a), (27) in Hege and Hennessy (2010), Section 1.1 in Khanna and Schroder (2010), (11) in Arve (2014), and (P.3) in Tamayo (2015). 12 If the incumbent investor is unaware of the possibility of secrete liquidation, the model becomes similar to the costly state falsification model studied in Lacker and Weinberg (1989) and Crocker and Morgan (1998).

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Fig. 8. Optimal contract with indivisible assets and surreptitious liquidation (c is large).

The limited liability constraint, the feasibility constraint, and the incentive-compatibility constraint are

R (ω) ≤ ω + Y (ω) W s ;

(LLs )

0 ≤ X (ω) ≤ 1 − Y (ω);

(FCs )

ˆ ) for any ω, ωˆ such that u F (, ω, ω) ≥ u F (, ω, ω

(ICs )

ˆ ) ≤ ω + Y (ω) W s and 0 ≤ X (ωˆ ) ≤ 1 − Y (ω). R (ω

The participation constraint (IR) is the same as that of our benchmark model. Note that even if we allow for surreptitious ex-post liquidation, the firm is still subject to an ex-post resource constraint, i.e., the total amount of cash that can be used to repay its loan is bounded. Thus, our main point that (IC) must incorporate the firm’s affordability constraint still holds. Proposition 7, with its graphical illustration in Fig. 8, characterizes an optimal contract. Proposition 7. Suppose that n = 1, c is large, I − I 0 ≤ max{ W , W s }, and surreptitious liquidation is allowed. p

p

p

p

(a) If W s ≤ W , then there exist ω0 , ω1 , and an optimal contract such that: (1) when ω < ω0 , R (ω) = ω1 ≤ 0, X (ω) = 1; when ω ≥ ω0p , R (ω) = ω0p , X (ω) = 0; (2) Y (ω) = 0 for any ω; (3) either ω1p = 0, or ω1p = ω0p − W s , where ω0p is determined by a binding (IR). (b) If W s ≥ W , then there exists an optimal contract such that: (1) R (ω) = I − I 0 , X (ω) = 0 for any ω ; (2) when ω < I − I 0 , Y (ω) = 1; when ω ≥ I − I 0 , Y (ω) = 0. The inequality I − I 0 ≤ max{ W , W s } in Proposition 7 serves as a sufficient condition for the existence of optimal contracts: The firm’s funding need should not exceed the maximum market value of assets. Several remarks regarding Proposition 7 and Fig. 8 may be helpful. First, note that when W s ≤ W , there is an optimal contract characterized in Proposition 7 with no surreptitious liquidation, and when W s ≥ W , there is an optimal contract with no liquidation by the investor. This latter case is trivial, as debt is riskless for the investor. Second, while there is an optimal contract without any surreptitious liquidation when W s ≤ W , it is still different from the optimal contract depicted in Fig. 7 when secret liquidation is not possible. According to Proposition 7, the firm’s p cash repayment ω1 may be negative, i.e., it may receive reimbursements from the investor in bankruptcy. However, from Definition 1, cash repayments with MCD should be nonnegative. This is purely due to the new incentive constraint (ICs ). Since the incentive constraint only applies to deviations that are affordable to the firm, allowing for surreptitious liquidation makes it possible for the firm to exaggerate its cash flow and deviate to another state with a cash repayment higher than ω . p p p Therefore, if ω0 is small, the firm at state ω < ω0 may still be able to repay ω0 by surreptitious liquidation. The incumbent investor may have to subsidize a low-type firm to prevent it from exaggerating its cash flow, which entails a reimbursement p when ω < ω0 . Third, there are still cases in which (ICs ) is slack in an optimal contract. Based on part (a) of Proposition 7, when p p p p W s ≤ W , either ω1 = ω0 − W s , or ω1 = 0 with ω0 determined by a binding (IR). In the former case (ICs ) is slack, while

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in the latter case whether (ICs ) binds depends on I − I 0 . Therefore, even if we allow for surreptitious liquidation, the firm still faces an affordability constraint, because it is impossible for the firm to liquidate as much as it wants without any cost. Hence, the incentive constraint must still be specified consistently with the firm’s limited liability, just as our (ICs ); moreover, the misspecified incentive constraint in the previous part of this section cannot be used as a shortcut. Our final observation is that allowing for surreptitious liquidation may sometimes reduce the firm’s expected utility. This result is formally stated in Proposition 8. Proposition 8. Suppose that n = 1, c is large, I − I 0 ≤ max{ W , W s }, and surreptitious liquidation is allowed. (a) If W s ≤ W , then the firm is weakly worse off compared to the model without surreptitious liquidation; moreover, if I − I 0 < W s + ( W − W s ) F ( W s ), then the firm is strictly worse off. (b) If W s ≥ W and I − I 0 ≥ W , then the firm is weakly better off compared to the model without surreptitious liquidation; moreover, if W s > W , then the firm is strictly better off. When W s ≤ W , the optimal contract derived in part (a) of Proposition 7 has no surreptitious liquidation, so the firm p should be weakly worse off than in our benchmark model. Furthermore, when ω1 < 0, the two optimal contracts are distinct, and thus the firm should be strictly worse off. This is the case when the firm’s funding need is sufficiently small, i.e., when I − I 0 < W s + ( W − W s ) F ( W s ). On the contrary, when W s ≥ W , we need the firm’s funding need to be sufficiently large, i.e., I − I 0 ≥ W , to ensure that introducing surreptitious liquidation makes the firm better off compared with the benchmark case. On the one hand, allowing for surreptitious liquidation relaxes the firm’s liquidity constraint, so it could be beneficial to the firm when such liquidation is not too costly, i.e., when W s is large. On the other hand, since the incentive constraint interacts with the firm’s limited liability, surreptitious liquidation enables the firm to make a cash repayment higher than its realized cash flow. That is, it enlarges the scope to which the incentive constraint applies. Therefore, the revised incentive constraint (ICs ) is stronger than (IC). As a result, when W s is small, the firm may have to be reimbursed in an incentivecompatible contract, which implies that the optimal MCD contract in the model without surreptitious liquidation is ruled out by (ICs ). In this case, the firm is strictly worse off.13 Clearly, not both cases of the previous two propositions are equally reasonable. If W s > W , it would be efficient for the firm to sign the initial contract with the outsider that can pay W s rather than the incumbent investor in our model. If the firm cannot find such an investor ex-ante, it is unikely that the firm finds her ex post. Therefore, W s < W is clearly the more relevant case. In this situation the firm is strictly worse off if surreptitious liquidation is allowed and I − I 0 is sufficiently small. This observation provides a new justification for covenants in debt contracts that prohibit the sale of some assets owned by the firm. 6. Conclusion This paper has reconsidered a classical financial contract model where a borrower generates assets and cash flow which are his private information. We have argued that disentangling and liquidating complex assets is generally costly and requires third-party intervention, which is typically associated with bankruptcy. A cheaper alternative is to transfer individual assets piecemeal, which can be done bilaterally. In such an environment an optimal contract takes the form of collateralized debt with multiple classes of debt which are serviced according to seniority. Moreover, the first type of liquidation, using costly intervention, and piecemeal liquidation of collateral may coexist in an optimal contract, and the structure of such optimal contract varies with the liquidation cost and the borrower’s funding need. The model rationalizes observed behaviors in the real estate and other markets, where borrowers often cede their property wholesale without going through bankruptcy or repaying all their cash. Our work adds to the growing literature in security design that goes beyond the classical emphasis on the optimality of standard debt and identifies richer security structures. With multi-class collateralized debt in our model, the firm will take two different actions, depending on the realized performance of its project. This feature is similar to the option-like feature in papers on optimal securities such as Schmidt (2003), where the security holder takes a state-dependent action. While we focus exclusively on debt, papers such as Fulghieri and Lukin (2001) and Fulghieri et al. (2017) focused on the relative advantages of debt and equity in addressing the impact of asymmetric information on firm value, and showed that funds may be better raised through equity than debt, thus reversing the classical pecking-order theory. Similar to our paper, Stenzel (2018) studied the role of different classes of debt in mitigating informational frictions, but focusses on an environment where the issuer may be required to sell some of these assets in the secondary market. In a different perspective, our paper can be viewed as a contribution to the theory of corporate restructuring. In practice, the firm faces the tradeoff between going through the bankruptcy procedure and reaching an agreement directly with the investor. Our model provides a synthesized framework explaining the coexistence of these two alternatives. We believe that

13 Although our discussion focusses on surreptitious liquidation, a similar intuition applies to other variations of this model that try to relax the limited liability constraint ex-post.

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our approach of making the different costs of resolving ex-post asymmetric information in contracting frameworks explicit can have a wide range of applications in other related fields, like retail contracting,14 auditing, taxation, and divorce. Appendix A. Proofs Proof of Proposition 1. Let  be an MCD contract. By definition, it is straightforward to verify that  satisfies (LL) and (FC). To prove (IC), note that the monotonicity of (ω) ensures that the firm does not want to understate its cash flow, therefore we only need to rule out the firm’s incentive to exaggerate. p p Suppose that ω1 < ω2 , and (ω1 ) > (ω2 ). If ω2 ∈ [ω j , ωcj−1 ) for some j = 1, 2, . . . , n, or ω2 ≥ ω0 for j = 0, then

ω1 < ω pj , otherwise (ω1 ) = (ω2 ). However, ω1 < ω pj implies that p

I 0 + B − I + ω1 < I 0 + B − I + ω j = R (ω2 ). p

If ω2 ∈ [ωcj , ω j ) for some j = 0, 1, . . . , n, then (LL) binds at its inability to repay R (ω2 ). Hence,  satisfies (IC). 2

ω2 , implying that a type-ω1 firm cannot misreport ω2 due to

ˆ that Proof of Proposition 2. Let  be a contract that satisfies (LL), (FC), and (IC). Our goal is to find an MCD contract  p c c satisfies the conditions listed in the proposition and weakly dominates . We will construct {(ω j , ω j , φ j )} j =0,1,...,n and proceed the proof in several steps. p

Step 1. We first construct {(ω j , ωcj )} j =0,1,...,n . For any j = 0, 1, ..., n, if X (ω j ) = x j and ψ(ω j ) = 0 for some

ω j , then

ω pj = R (ω j ) − ( I 0 + B − I ). Moreover, when x j < c /(1 − α ) V ,

ωcj−1 = min{sup{ω : (ω) = (ω j )}, ω pj−1 }; when x j ≥ c /(1 − α ) V ,

ωcj−1 = min{sup{ω : (ω) = (ω j )}, ω pj +

c 1−α

p

, ω j −1 }.

Here ωcj−1 is determined in two different expressions, because when x j is large, it is inefficient for the firm to liquidate x j piecemeal and keep more than c /(1 − α ) units of cash at hand, implying that the firm would rather liquidate continuously p with full cash repayments when ω > ω j + c /(1 − α ). This tradeoff is absent when x j is small. p

p

¯ for Finally, if there does not exist any ω j that satisfies X (ω j ) = x j and ψ(ω j ) = 0, then ω j = ωcj for j ≥ 1, and ω j = ω j = 0. p It can be verified that {ω j } j =0,1,...,n is nonincreasing: Suppose that X (ω j ) = x j , X (ωk ) = xk , and ψ(ω j ) = ψ(ωk ) = 0 for

ω j , ωk , with j < k. From (IC) and x j < xk , we know that R (ω j ) > R (ωk ), so ω pj > ωkp . Also, for ¯ , which implies ω pj ≤ ω¯ . Moreover, note that any ω , R (ω) ≤ I 0 + B − I + ω some j , k ∈ {0, 1, . . . , n} and

p

sup{ω : (ω) = (ω j )} ≥ ω j ≥ R (ω j ) − ( I 0 + B − I ) = ω j . Therefore, for any j = 1, 2, . . . , n, except that

ωnp may be negative.

ω pj ≤ ωcj−1 ≤ ω pj−1 , suggesting that {(ω pj , ωcj )} j=0,1,...,n satisfies condition (a) of Definition 1

Step 2. We then construct {φ cj } j =1,...,n . For any j = 0, 1, . . . , n − 1, if ω pj



ω pj



φ cj dF (ω) = ωcj

ωcj < ω pj , then let φ cj be the solution for

[(ω) − ( I 0 + B − I )]dF (ω). ωcj

ω pj = ωcj , then φ cj is irrelevant to the contract. Since (ω) is nonincreasing because of (IC), we know that p {(ω j , ωcj , φ cj )} j =0,1,...,n satisfies condition (c) of Definition 1. If

14

See, e.g., Chen et al. (2019).

Z. Gui et al. / Games and Economic Behavior 118 (2019) 412–433

ˆ . Let Bˆ = I − I 0 . For any j = 0, 1, . . . , n, if Step 3. Now we are ready to construct  j = 0, then

427

ω ∈ [ω pj , ωcj−1 ) for j ≥ 1, or ω ≥ ω0p for

ˆ ω ) = 0. Rˆ (ω) = ω j , Xˆ (ω) = x j , ψ( p

If

ω ∈ [ωcj , ω pj ), then Rˆ (ω) = ω, Xˆ (ω) =

φ cj − ω V

ˆ ω ) = 1. , ψ(

Note that by the definition of φ cj , p

(ω j ) − ( I 0 + B − I ) ≤ φ cj ≤ ω + V , which implies

ω pj + x j V ≤ φ cj ≤ ω + V . ω ∈ [ωcj , ω pj ), we have Xˆ (ω) ∈ [0, 1]. Finally, if ωnp < 0, one can increase ωnp and reduce ω0p to save the p ˆ is an MCD contract by construction. firm’s liquidation, thus it is without loss to let ωn = 0. Hence,  Therefore when

ˆ . To do this, we need the following lemma. Step 4. Our remaining task is to prove that  is weakly dominated by  p

p

p

Lemma 1. For any j = 0, 1, . . . , n, if ω ∈ [ω j , ωcj−1 ) for j ≥ 1, or ω ≥ ω0 for j = 0, then (ω) = I 0 + B − I + ω j + x j V .

ω ∈ [ω pj , ωcj−1 ) for some j ≥ 1. Then, there exists some ω j such that X (ω j ) = x j and ψ(ω j ) = 0. By our p construction, ω = R (ω j ) − ( I 0 + B − I ) ≤ ω j , implying that the firm at state ω j and state ω j can mimic each other. From p p (IC), (ω j ) = (ω j ). Since (ω) is nonincreasing, it must be constant on [ω j , ωcj−1 ); this again comes from our definition p p c of ω j −1 . Therefore (ω) = (ω j ) = I 0 + B − I + ω j + x j V . Our analysis can be extended to the case when ω ≥ ω0 , so the

Proof. Suppose that p j

lemma is proved.

2

ˆ (ω) + ( I 0 + B − I ) for any ψ( ˆ ω) = 0. Furthermore, for any j = 0, 1, . . . , n − 1, our construction of By Lemma 1, (ω) =  p ˆ (ω) + I 0 + B − I |[ωc , ω p )]. Therefore, φ cj implies that E ω [(ω)|[ωcj , ω j )] = E ω [ j j E ω u F (, ω, ω) = I 0 + B − I + V + E ω [ω − (ω)]

ˆ (ω)] = V + E ω [ω −  = E ω u F (ˆ , ω, ω). The last equality comes from the fact that Bˆ = I − I 0 . In other words, the firm’s expected payout is the same in the two contracts, it is remaining to show that the investor is also weakly better off. p p ˆ ω) = 0. Suppose that ω ∈ [ω j , ωcj−1 ) for some j = 1, 2, . . . , n, or ω ≥ ω0 for j = 0. Then according to our construction, ψ( We have three cases to consider: If ψ(ω) = 0, then the firm must liquidate some x j  with j  ≥ j + 1 in the original contract. Therefore,

R (ω) ≤ I 0 + B − I + ω j = I 0 + B − I + Rˆ (ω); p

R (ω) + X (ω) W = α (ω) + (1 − α ) R (ω)

ˆ (ω)] + (1 − α ) R (ω) = α[I 0 + B − I +  ≤ I 0 + B − I + Rˆ (ω) + Xˆ (ω) W . If ψ(ω) = 1 and x j < c /(1 − α ) V , then p

R (ω) + X (ω) W − c ≤ (ω) − c = I 0 + B − I + ω j + x j V − c p

≤ I0 + B − I + ω j + x j W . If ψ(ω) = 1 and x j ≥ c /(1 − α ) V , recall that our construction of

ωcj−1 implies ωcj−1 ≤ ω pj + c /(1 − α ). Then

428

Z. Gui et al. / Games and Economic Behavior 118 (2019) 412–433

c

p

R (ω) ≤ I 0 + B − I + ω ≤ I 0 + B − I + ω j +

1−α

,

R (ω) + X (ω) W − c = α (ω) + (1 − α ) R (ω) − c p

≤ I0 + B − I + ω j + x j W . From the three cases above, we obtain that for any

ˆ ω) = 0, ω such that ψ(

R (ω) + X (ω) W − ψ(ω)c ≤ I 0 + B − I + Rˆ (ω) + Xˆ (ω) W . Suppose that x j  with j 

ω ∈ [ωcj , ω

p ) j

(3)

ˆ ω) = 1. If ψ(ω) = 0, then the firm must liquidate some for some j = 0, 1, . . . , n − 1. Then ψ( p

≥ j + 1 in the original contract, thus, R (ω) = I 0 + B − I + ω j  , and X (ω) = x j  ≥ c /(1 − α ) V . Put differently, if the newly constructed contract has continuous liquidation at ω , while the initial contract has piecemeal liquidation, it must be that the firm liquidates too much in the initial contract ( X (ω) = x j  ≥ c /(1 − α ) V ). Applying continuous liquidation enables the firm to make full cash repayments and hence increases the efficiency of the contract. However, Lemma 1 and p the monotonicity of (ω) imply that (ω) ≤ I 0 + B − I + ω j +1 + x j +1 V . Therefore, we must have (ω) = I 0 + B − I +

ω pj+1 + x j+1 V , and ω ≥ ω pj+1 + c /(1 − α ). Thus, Rˆ (ω) = ω, Xˆ (ω) =

φ cj − ω V

;

R (ω) + X (ω) W = α (ω) + (1 − α ) R (ω) p

≤ α (ω) + (1 − α )( I 0 + B − I + ω j +1 ) ≤ α (ω) + (1 − α )( I 0 + B − I + ω −

c 1−α

)

= α (ω) + (1 − α )[ I 0 + B − I + Rˆ (ω)] − c .

(4)

If ψ(ω) = 1, then

R (ω) + X (ω) W − c = α (ω) + (1 − α ) R (ω) − c

≤ α (ω) + (1 − α )[ I 0 + B − I + Rˆ (ω)] − c .

(5)

Therefore, from (4)–(5), we have

ˆ ω ) = 1] E ω [ R (ω) + X (ω) W − ψ(ω)c |ψ( ˆ ω ) = 1] ≤ E ω [α (ω) + (1 − α )( I 0 + B − I + Rˆ (ω)) − c |ψ( ˆ (ω) + (1 − α ) Rˆ (ω) − c |ψ( ˆ ω ) = 1] = E ω [I 0 + B − I + α ˆ ω) = 1]. = E ω [ I 0 + B − I + Rˆ (ω) + Xˆ (ω) W − c |ψ( This inequality, together with (3), implies that

E ω u I (, ω, ω) = − B + E ω [ R (ω) + X (ω) W − ψ(ω)c ]

ˆ ω)c ] ≤ − Bˆ + E ω [ Rˆ (ω) + Xˆ (ω) W − ψ( = E ω u I (ˆ , ω, ω). ˆ , and our proof of the proposition is completed. Hence,  is weakly dominated by 

2

p

Proof of Proposition 3. If {(ω j , ωcj , φ cj )} j =0,1,...,n minimize the firm’s expected payout subject to the investor’s participation constraint, then there exists a constant λ such that they also minimize the following Lagrangian:

L = E ω [ R (ω) + X (ω) V ] + λ{ I − I 0 − E ω [ R (ω) + X (ω) W − ψ(ω)c ]} subject to all the constraints listed in the reformulated problem. Moreover, if λ ≤ 1, L becomes a nondecreasing function of E ω R (ω) and E ω X (ω), suggesting that L is minimized when E ω R (ω) = E ω X (ω) = 0, a contradiction to (IR). Hence, we must have λ > 1 and a binding (IR) in any optimal MCD contract. Moreover, we have the following Lemma. Lemma 2. λ is nondecreasing in I − I 0 .

Z. Gui et al. / Games and Economic Behavior 118 (2019) 412–433

429

Proof. For any j = 1, 2, suppose that  j is an optimal MCD contract when I − I 0 = I j . Then for any  j , let R j = E ω R (ω) and X j = E ω X (ω). By optimality, we have

R 1 + X 1 V ≤ R 2 + X 2 V + λ1 ( I 1 − I 2 ); R 2 + X 2 V ≤ R 1 + X 1 V + λ2 ( I 2 − I 1 ). These two inequalities jointly imply that

(λ1 − λ2 )( I 1 − I 2 ) ≥ 0. Therefore, if I 1 > I 2 , then λ1 ≥ λ2 .

2

Proof of part (a). Suppose that  is an optimal MCD contract, and  has piecemeal liquidation. Let x j be the smallest quantity p of assets transferred as a whole, then j ≥ 1, and ω j < ωcj−1 .

ωcj−1 < ω0p , then there is only continuous liquidation when ω ∈ [ωcj−1 , ω0p ). If we denote by L (a, b) the value of the p ¯ ) is Lagrangian conditional on the interval (a, b), then L (ω j , ω If

p

p

p

p

p

p

¯ ) = (ω j + x j V )[ F (ωcj−1 ) − F (ω j )] + φ0c [ F (ω0 ) − F (ωcj−1 )] + ω0 [1 − F (ω0 )] L (ω j , ω p

p

ω0

p

− λ{(ω j + x j W )[ F (ωcj−1 ) − F (ω j )] +

p

p

[ω + α (φ0c − ω) − c ]dF (ω) + ω0 [1 − F (ω0 )]}.

ωcj−1

First-order conditions are

∂L

p

p

p

= [(1 − α λ)(φ0c − ω0 ) + c λ] f (ω0 ) − (λ − 1)[1 − F (ω0 )];

p

∂ ω0 ∂L

p

p

= [(1 − α λ)(ω j + x j V − φ0c ) + λ(1 − α )(ωcj−1 − ω j ) − c λ] f (ωcj−1 );

∂ ωcj−1

∂L p = (1 − α λ)[ F (ω0 ) − F (ωcj−1 )]. ∂φ0c p

When 1 − α λ > 0, we have ∂ L /∂φ0c > 0, thus φ0c = ω0 . Moreover,

ωcj−1 < ω0p implies that ∂ L /∂ ω0p ≤ 0, which implies

p

λ[1 − c

f (ω0 ) p

1 − F (ω0 )

] ≥ 1.

c must be sufficiently small so that the left-hand side is positive. Actually there are two necessary conditions: p

1 − F (ω0 )

c<

p

f (ω0 )

1

≤

f

,

p

1 − F (ω0 )

λ≥ If

p 0 ) − cf (

1 − F (ω

p 0)

ω

≥

1

= λ1 .

1−cf

(6)

ωcj−1 = ω0p , then p

p

p

p

p

p

p

p

p

p

p

¯ ) = (ω j + x j V )[ F (ω0 ) − F (ω j )] + ω0 [1 − F (ω0 )] − λ{(ω j + x j W )[ F (ω0 ) − F (ω j )] + ω0 [1 − F (ω0 )]}. L (ω j , ω The first-order condition is

∂L p

∂ ω0

p

p

p

p

= [(λ − 1)(ω0 − ω j ) + (1 − α λ)x j V ] f (ω0 ) − (λ − 1)[1 − F (ω0 )]. p

Similarly, we have ∂ L /∂ ω0 ≤ 0, which implies

ω0p − ω pj +

1 − αλ

λ−1

A necessary condition is

p

xjV ≤

1 − F (ω0 ) p 0)

f (ω

≤

1 f

.

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Z. Gui et al. / Games and Economic Behavior 118 (2019) 412–433

1 + xjV f

λ≥

1 + αx j V f

= λ2 .

(7)

ω pj < ωcj−1 implies either λ ≥ λ1 , or λ ≥ λ2 . We also know that c < 1/ f is necessary p p Put differently, λ < min{λ1 , λ2 } is sufficient for ω j = ωcj−1 , and c ≥ 1/ f is sufficient for ωcj−1 = ω0 . Hence,

From (6) and (7), we know that p 0.

for ωcj−1 < ω from Lemma 2, there exists a cutoff I such that I − I 0 < I is sufficient for that  has only continuous liquidation. Moreover, when c ≥ 1/ f ,  cannot have continuous liquidation, so it is without loss to make I = 0 when c ≥ 1/ f = c. Proof of part (b). Suppose that  is an optimal MCD contract, and  has continuous liquidation. Further assume temporarily p that c ≥ (1 − α ) V . Let ωcj be the lower bound of the states that induce continuous liquidation, then ωcj < ω j . If

ωcj > 0, then j ≤ n − 1 and ω pj+1 ≥ 0. The Lagrangian conditional on [ω pj+1 , ωcj−1 ) is p

p

p

p

L (ω j +1 , ωcj−1 ) = (ω j +1 + x j +1 V )[ F (ωcj ) − F (ω j +1 )] + φ cj [ F (ω j ) − F (ωcj )] p

p

p

p

+ (ω j + x j V )[ F (ωcj−1 ) − F (ω j )] − λ{(ω j +1 + x j +1 W )[ F (ωcj ) − F (ω j +1 )] ω pj



p

p

[ω + α (φ0c − ω) − c ]dF (ω) + (ω j + x j W )[ F (ωcj−1 ) − F (ω j )]}.

+ ωcj

The first-order conditions are

∂L p p = [(1 − α λ)(ω j +1 + x j +1 V − φ cj ) + λ(1 − α )(ωcj − ω j +1 ) − c λ] f (ωcj ); ∂ ωcj

(8)

∂L p = (1 − α λ)[ F (ω j ) − F (ωcj )]. ∂φ cj

(9) p

When 1 − α λ < 0, we have ∂ L /∂φ cj < 0, suggesting that φ cj = ω j +1 + x j +1 V . Moreover, which implies

c

p

λ(1 − α )(ωcj − ω j +1 −

1−α

ωcj < ω pj implies that ∂ L /∂ ωcj ≥ 0,

) f (ωcj ) ≥ 0.

A necessary condition is p

p

p

c ≤ (1 − α )(ωcj − ω j +1 ) < (1 − α )(ω j − ω j +1 ) ≤ (1 − α )(x j +1 − x j ) V

≤ (1 − α ) V . However, this contradicts our assumption that c ≥ (1 − α ) V . If ωcj = 0, then the Lagrangian conditional on [ωcj , ωcj−1 ) is p

p

p

L (ωcj , ωcj−1 ) = φ cj [ F (ω j ) − F (ωcj )] + (ω j + x j V )[ F (ωcj−1 ) − F (ω j )] ω pj



p

p

[ω + α (φ0c − ω) − c ]dF (ω) + (ω j + x j W )[ F (ωcj−1 ) − F (ω j )]}.

+ ωcj

The first-order conditions are

∂L p

∂ω j

p

p

p

= [(1 − α λ)(φ cj − ω j − x j V ) + c λ] f (ω j ) − (λ − 1)[ F (ωcj−1 ) − F (ω j )];

∂L p = (1 − α λ) F (ω j ). ∂φ cj p

Similarly, when 1 − α λ > 0, we know that φ cj = ω j + x j V . p

λ[1 − c

f (ω j ) p

F (ωcj−1 ) − F (ω j )

Two necessary conditions are

] ≥ 1.

ωcj < ω pj implies that ∂ L /∂ ω pj ≤ 0, which implies

Z. Gui et al. / Games and Economic Behavior 118 (2019) 412–433

p

F (ωcj−1 ) − F (ω j )

c<

p

f (ω j )

1

≤

f

,

p

F (ωcj−1 ) − F (ω j )

λ≥

F (ω

c )− j −1

F (ω

431

p ) − cf ( j

ω

p ) j

≥

1 1−cf

= λ1 .

Hence, there exists a cutoff I such that I − I 0 < I is sufficient for that  has only piecemeal liquidation. Moreover, since all the analysis in this part is conducted under the assumption that c ≥ (1 − α ) V , it is without loss to make I = 0 when c < (1 − α ) V = c. 2 p

p

p

Proof of Proposition 4. For any optimal MCD contract with only piecemeal liquidation, if (ω j −1 ) < (ω j ) < (ω j +1 ),

ω ω pj is then

p j +1

p j

<ω <ω

∂L p

∂ω j

p j −1 .

Following the proof of Proposition 3, the first-order derivative of the Lagrangian L with respect to

p

p

p

p

p

= [(λ − 1)(ω j − ω j +1 ) + (1 − α λ)(x j +1 − x j ) V ] f (ω j ) − (λ − 1)[ F (ω j −1 ) − F (ω j )].

Since (IC) is slack at will give us (1). 2

ω pj and ω pj−1 , we must have ∂ L /∂ ω pj = 0. Simplifying this equality, and letting μ = (α λ − 1)/(λ − 1)

ωcj ∈ (ω pj+1 , ω pj ), then the first-order derivatives of the Lagrangian p L with respect to ωcj and φ cj are given by (8) and (9). By (9), when 1 − α λ < 0, we know that φ cj = ω j +1 + x j +1 V ; when p c 1 − α λ ≥ 0, we have φ j ≤ ω j +1 + x j +1 V . In both cases, we have Proof of Proposition 5. For any optimal MCD contract, if

∂L c p ≥ λ(1 − α )(ωcj − ω j +1 − ) f (ωcj ). ∂ ωcj 1−α Since ∂ L /∂ ωcj ≤ 0, we immediately have (2). A similar analysis will go through if

ωcj = ω pj > ω pj+1 . 2

Proof of Proposition 6. If there exists an optimal MCD contract that has only continuous liquidation, then first-order derivative of the Lagrangian L is

∂L p 0

∂ω

p

ω0c = 0. The

p

= c λ f (ω0 ) − (λ − 1)[1 − F (ω0 )].

Thus, p

1 − F (ω0 ) p 0)

f (ω

=

λc > c. 2 λ−1

Proof of Proposition 7. Let  be a contract that satisfies (LLs ), (FCs ) and (ICs ). Since it is too costly to use continuous liquidation, in contract  the firm only has three possible decisions to deal with its assets: X = Y = 0 (no liquidation), X = 0, Y = 1 (surreptitious liquidation), or X = 1, Y = 0 (full liquidation). By incentive-compatibility, there are at most three distinct cash repayment choices in the contract that correspond to the three decisions on the assets. Let ωxy be a state with X = x ∈ {0, 1} and Y = y ∈ {0, 1}, and r xy be the cash repayment specified for state ωxy . We first characterize the relationship between r10 , r01 and r00 . Lemma 3. r10 < r01 = r00 . Proof. Suppose that r10 ≥ r01 . The firm at state ω10 can misreport state ω01 without any form of liquidation, which is a violation of (ICs ). Thus, r10 < r01 . Suppose that r01 < r00 . Similarly, the firm at state ω00 can misreport state ω01 without surreptitious liquidation, which violates (ICs ). Suppose that r01 > r00 . The firm at state ω01 can mimic state ω00 with surreptitious liquidation, but it would pay less cash by reporting ω00 . This also contradicts (ICs ), so r01 = r00 . 2 From Lemma 3 and (LLs ), we must have r10 ≤ 0. Since any state can deviate to repay r10 with full liquidation, from (ICs ) we also have r00 ≤ r10 + V . Finally, since the firm’s payout function should be nonincreasing in ω , we can prove that  is p p weakly dominated by a contract with the following cutoff structure: There exists ω00 such that when ω < ω00 , R (ω) = r10 , p X (ω) = 1; when ω ≥ ω00 , R (ω) = r00 , X (ω) = 0. Note that the firm would surreptitiously liquidate its assets only when p ω ∈ [ω00 , r00 ). In what follows, we assume that  exhibits such cutoff structure without loss of generality.

432

Z. Gui et al. / Games and Economic Behavior 118 (2019) 412–433 p

p

To prove part (a) of the proposition, assume W s ≤ W and ω00 < r00 . If ω00 > 0, then by (ICs ), r10 ≥ r00 − W s , otherwise the firm at state ω00 will misreport ω10 . Since r10 ≤ 0, we have r00 ≤ W s , so the firm at state ω10 can also misreport ω00 p and repay r00 to the investor. (ICs ) again implies r10 ≤ r00 − W s . Therefore, r10 = r00 − W s . If ω00 = 0, then let r10 = r00 − W s . It can be verified that r10 ≤ 0. ˆ that is identical to  except that for any ω ∈ [ω p , r00 ), we have Rˆ (ω) = r10 , Xˆ (ω) = 1 Consider an alternative contract  00

and Yˆ (ω) = 0. It can be proved that, when

p ω ∈ [ω00 , r00 ),

ˆ , ω, ω) = ω − r10 , u F (, ω, ω) = ω + W s − r00 = u F ( ˆ , ω, ω) = −( I − I 0 ) + W + r10 . u I (, ω, ω) = −( I − I 0 ) + r00 ≤ u I ( ˆ. Thus,  is weakly dominated by  Moreover, if r10 < 0 and r00 > r10 + W s , the firm can further increase r10 and reduce r00 to minimize its probability of ˆ when r10 = 0 and being liquidated. Hence, we have either r10 = 0, or r00 = r10 + W s . Finally, (IR) should be satisfied by  r00 = V , implying that I − I 0 ≤ W F ( V ) + V [1 − F ( V )]. A sufficient condition for this inequality is I − I 0 ≤ W . p To prove part (b) of the proposition, assume W s ≥ W and ω00 > 0. By an argument similar to the proof of part (a), we ˆ that is identical to  except that for any know that r10 = r00 − W s . Consider an alternative contract  Rˆ (ω) = r00 , Xˆ (ω) = 0 and Yˆ (ω) = 1. It can be proved that, when

ω ∈ [0, ω

p 00 ),

p ω ∈ [0, ω00 ), we have

ˆ , ω, ω) = ω + W s − r00 , u F (, ω, ω) = ω − r10 = u F ( ˆ , ω, ω) = −( I − I 0 ) + r00 . u I (, ω, ω) = −( I − I 0 ) + W + r10 ≤ u I ( ˆ . Also, (IR) should be satisfied by ˆ when r00 = W s , implying that I − I 0 ≤ W s . Thus,  is weakly dominated by  After changing some notations, we can get the results stated in Proposition 7. 2 Proof of Proposition 8. When surreptitious liquidation is not allowed, by Proposition 2 and Proposition 3, any optimal MCD p p p p contract has the following form: When ω < ω0 , R (ω) = 0, X (ω) = 1; when ω ≥ ω0 , R (ω) = ω0 , X (ω) = 0, where ω0 is determined by a binding (IR). That is, p

p

p

W F (ω0 ) + ω0 [1 − F (ω0 )] = I − I 0 .

(10)

Therefore the firm’s expected utility is p

ω0 E ω u F (, ω, ω) =

ωdF (ω) +

ω0p

0

ω¯ =

ω¯ p (ω + V − ω0 )dF (ω)

ωdF (ω) + V − ( I − I 0 ) − ( V − W ) F (ω0p ).

(11)

0

When surreptitious liquidation is allowed, and W s ≤ W , there is no surreptitious liquidation in the optimal contract characterized in Proposition 7, so the first statement of (a) is straightforward. To prove the second statement of (a), note p p p that the firm is strictly worse off if and only if ω1 < 0. This is the case when ω1 = ω0 − W s . From a binding (IR), p

p

p

p

( W + ω1 ) F (ω0 ) + ω0 [1 − F (ω0 )] = I − I 0 , p

p

which implies that I − I 0 = ω0 + ( W − W s ) F (ω0 ) < W s + ( W − W s ) F ( W s ). The last inequality comes from the fact that is bounded above by W s . When surreptitious liquidation is allowed, and W s ≥ W , the firm’s expected utility is I− I 0

E ω u F (, ω, ω) =

ω0p

ω¯

[ω + W s − ( I − I 0 )]dF (ω) +

[ω + V − ( I − I 0 )]dF (ω) I −I0

0

ω¯ =

ωdF (ω) + V − ( I − I 0 ) − ( V − W s ) F ( I − I 0 ).

(12)

0 p

If further we have I − I 0 ≥ W , then by (10), ω0 ≥ I − I 0 . We can see that the firm is weakly better off by comparing (12) and (11). Finally, the firm is strictly better off if we have W s > W . 2

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