Using Elasticities to Derive Optimal Bankruptcy Exemptions

Transcription

1 Using Elasticities to erive Optial Bankruptcy Exeptions Eduardo ávila YU Stern January 25 Abstract This paper characterizes the optial bankruptcy exeption for risk averse borrowers who use unsecured contracts but have the possibility of defaulting. It provides a novel general forula which holds in a wide variety of environents for the optial exeption as a function of a few observable sufficient statistics. Knowledge of borrowers leverage, the sensitivity of the interest rate schedule faced by borrowers with respect to the level of the bankruptcy exeption, the probability of bankruptcy and the change in consuption by bankrupt borrowers is sufficient to deterine the optial bankruptcy exeption. When calibrated to US data, the optial bankruptcy exeption iplied by the odel ($,) is larger than the average exeption in the US ($7,), but of the sae order of agnitude. JEL nubers: K35, E2, 4 Keywords: bankruptcy, default, unsecured credit, general equilibriu, sufficient statistics Contact: edavila@stern.nyu.edu. I would like to especially thank John Capbell, Eanuel Farhi, Oliver Hart and Alp Sisek for any helpful coents, as well as Itay Goldstein (discussant). I a also grateful to Philippe Aghion, Adrien Auclert, Raj Chetty, Gita Gopinath, athan Hendren, Kyle Herkenhoff, Ben Iverson, Jonathan Parker, Adriano Rapini, José Víctor Ríos Rull, Andrei Shleifer, Jerey Stein, Motohiro Yogo, Mark Wright and participants in Harvard Macro lunch, 24 BER SI Corporate Finance eeting and YU Stern. Financial support fro Rafael del Pino Foundation is gratefully acknowledged. A previous version of this paper circulated as "Using Elasticities to erive Optial Bankruptcy Policies".

2 Introduction Motivation Should we ake bankruptcy procedures harsher or ore lenient for borrowers who decide not to repay their debts? What is the socially optial level of bankruptcy exeptions? These are iportant norative questions see, for instance, White (2), who recently docuents the large increase in consuer bankruptcy filings in the US in the last decades that are still subject to debate. This paper shows that a few observable variables are sufficient to answer such questions under a broad array of circustances. This paper addresses the proble of optial bankruptcy design by analytically characterizing the welfare axiizing bankruptcy exeption for risk averse borrowers who use unsecured contracts but can choose to default. The welfare axiizing bankruptcy exeption optially trades off reduced access to credit ex-ante with iproved insurance ex-post. Main results I derive the ain results of the paper in a baseline two period odel with risk averse borrowers who only have access to a contract that proises a flat repayent; I subsequently extend the results in any diensions. Throughout the paper, the bankruptcy exeption is defined as the dollar aount that a borrower who declares bankruptcy is allowed to keep. The ain contribution of this paper is to characterize a) the welfare change induced by a arginal change in and b) the optial bankruptcy exeption, as a function of a few observable sufficient statistics. This characterization provides a clear interpretation of the forces that deterine the optial exeption for a broad range of priitives and directly links the theoretical tradeoffs to a sall set of observable variables. Most of the intuition behind the results can be seen in the case with logarithic utility borrowers who always clai the full bankruptcy exeption. In that case, the optial exeption (easured in dollars) ust satisfy the following equation: = βπ, () Λε r, where β is the borrowers rate of tie preference, π is the probability of default in equilibriu, Λ is a easure of borrowers leverage and ε r, is the sensitivity of the interest rate schedule faced by borrowers with respect to the bankruptcy exeption easured in dollars. Iportantly, all four variables in the forula for have direct epirical counterparts. Equation () captures the key tradeoff regarding the optial deterination of the bankruptcy exeption. On the one hand, if borrowing rates rise quickly with the level of the bankruptcy exeption (high ε r, ), it is optial to set a low exeption level, especially when borrowers leverage is high (high Λ): a low exeption facilitates the access to credit ex-ante in that case. On the other hand, if default is very frequent in equilibriu (high π), especially when borrowers value consuption relatively ore in the terinal period (high β), it is optial to set a high exeption level, allowing borrowers to consue ore when bankrupt. Equation () trades off these forces optially. The decisions of how uch to borrow and when to default do not affect the assessent of arginal interventions nor the forula for directly: the fact that borrowers borrow and default optially guarantees that the effect on these variables induced by a change in the bankruptcy exeption vanishes fro the optial exeption forula. This fact greatly siplifies the characterization of the optial exeption. This paper iposes no restrictions on the shape of the contracts used, but takes the set and shape of the contracts available as given: this is the key friction that the optial exeption addresses. When the set of available contracts All variables in equation () are endogenous. The logic behind this characterization is identical to the one behind the CAPM, in which the beta of an asset (an endogenous object) becoes a sufficient statistic to deterine expected returns, or consuption based asset pricing odels, in which the consuption process, independently of how it is generated, is a sufficient statistic for pricing assets.

3 does not allow agents to reach the first-best allocation, which iplies perfect insurance for the risk averse borrowers, an optially deterined bankruptcy exeption iproves welfare. Hence, the optial exeption proble solved in this paper is the second-best proble solved by borrowers who use a given set of contracts but can choose the level of bankruptcy exeptions ex-ante. Calibration I show the applicability of the theoretical results by calibrating the forula for the optial exeption to US bankruptcy data and assessing the agnitude of the welfare gains generated by adjusting the bankruptcy exeption. The preferred calibration to US data iplies that the optial bankruptcy exeption should be in the range of, dollars, an aount slightly larger than the average exeption in the US (of approxiately 7, dollars). For the sae calibration, ipleenting the optial exeption achieves welfare gains on the order of approxiately.3% of ex-ante consuption. This calibration exercise is not eant to settle the debate on the level of exeptions. Only further work on the easureent of the key required variables will help us refine the optial prescription for the level of exeptions. Extensions I explore several extensions to understand which departures fro the baseline odel affect the optial exeption forula. First, allowing for additional argins of adjustent, like labor supply or effort choices, does not odify the optial exeption forula as long as these choices enter separably into borrowers utility function. Second, allowing for ore general utility specifications for consuption like Epstein-Zin preferences or state dependent utility only affects the optial exeption forula through changes in the borrowers stochastic discount factor. Third, allowing for ultiple traded contracts with arbitrary payoffs is straightforward: a leverage-weighted average of interest rate sensitivities captures then the arginal cost of increased leniency. Fourth, when borrowers are ex-ante heterogeneous but exeptions cannot be individual specific, the optial exeption becoes a function of a weighted average across borrowers of the arginal effects present in the baseline case. Fifth, whether borrowers are price takers or not deterines whether the equilibriu interest rate sensitivity or the interest rate schedule sensitivity is the right easure of the arginal cost of changing exeptions. Sixth, exeptions contingent on aggregate shocks can iprove welfare by targeting the sae set of sufficient statics identified in the baseline odel for each aggregate state. Seventh, the optial exeption forula in econoies with non-zero output gaps accounts for the equilibriu effect of exeptions on aggregate deand. Finally, I show that, in a dynaic context, the optial exeption is given by an average across periods/states of benefits and costs of the bankruptcy exeption, all weighted according to the borrowers own stochastic discount factor. These extensions yield two robust insights. First, the precise deterination of the region(s) in which borrowers decide to default does not odify the optial exeption forula, because borrowers default optially. Hence, evaluating the welfare iplications of changing exeptions does not require explicit odeling of the ultiple factors that ay influence default decisions. 2 Second, only easures of consuption of bankrupt borrowers are required to assess the welfare benefits of varying exeptions. Preference paraeters, like risk aversion, deterine how to translate easures of consuption into welfare, but the responses of labor supply and other endogenous choice variables are irrelevant once borrowers consuption is known, as long as utility of consuption is separable and other econoic choices are not subject to frictions. 2 For instance, default is jointly deterined with other endogenous choice variables. Siilarly, forward looking borrowers internalize the option value of waiting for uncertainty to be realized before defaulting. Both these considerations, which prevent a siple characterization of default regions, only affect the optial exeption through the sufficient statistics. 2

4 Related Literature This paper relates to several literatures in econoics. 3 First, this paper is ost directly related to the literature on general equilibriu with incoplete arkets, which has studied the possibility of default in very general environents in which arkets are exogenously or endogenously incoplete. Zae (993) and ubey, Geanakoplos and Shubik (25) are the first to theoretically analyze the core tradeoff present in this paper. They show that default ay be welfare iproving in a odel with incoplete arkets, since it enhances insurance opportunities by introducing new contingencies into contracts. These papers take default penalties as exogenous and do not characterize optial penalties, which is the approach I adopt in this paper. 4 It is well known that default is only beneficial when arkets are initially incoplete. Allowing for default when agents can write fully state contingent contracts, as in Kehoe and Levine (993), Alvarez and Jerann (2), or Chien and Lustig (2), only restricts the contracting space, reducing welfare unequivocally. Second, this paper is also related to the quantitative literature on default and bankruptcy. On the one hand, the papers by S. Chatterjee,. Corbae, M. akajia and J.V. Ríos-Rull (27) or Livshits, MacGee and Tertilt (27), aong several others, provide a careful quantitative structural analysis of unsecured credit and default fro a acroeconoic perspective. ue to their dynaic nature and their rich general equilibriu features, these papers ust rely on nuerical ethods to evaluate the welfare iplications of bankruptcy policies. This paper, by contrast, focuses on providing analytical insights into this question. Livshits (24) provides a recent survey of this growing literature, which has not reached a consensus on the optial level of exeptions. The work by Gropp, Scholz and White (997), Gross and Souleles (22), Fay, Hurst and White (22), Mahoney (22), Iverson (23), obbie and Song (23), and Severino, Brown and Coates (23), aong others, uses icroeconoetric ethodology to understand the iplications of actual bankruptcy policies. See White (27, 2) for recent surveys of this body of work. Third, this paper presents a novel application of the sufficient statistic approach to the proble of bankruptcy and security design. As stated by Chetty (29), the central concept of the sufficient statistic approach is to derive forulas for the welfare consequences of policies that are functions of high-level elasticities rather than deep priitives. The optial policies characterized under this approach are robust to a broad range of environents. Soe recent successful applications of this approach are iaond (998) and Saez (2) hence the title analogy on incoe taxation, Shier and Werning (27) and Chetty (28) on uneployent insurance, Arkolakis, Costinot and Rodríguez-Clare (22) on the welfare gains fro trade liberalizations and Basu et al. (23) on productivity and capital accuulation. Fourth, this paper is related to the literature on optial contracting and corporate finance, which studies borrower-lender relationships, bankruptcy, and default. 5 Costly state verification, liited coitent or enforceent, oral hazard, adverse selection, and secret cash-flow anipulation are features studied in static and dynaic environents within this extensive literature. Instead of fully optiizing on the shape of the contracts given priitives, in this paper I take the set of contracts available as given, while optiizing on the level of the bankruptcy exeption. Fifth, the results of this paper also relate to the work in security design otivated by risk sharing, copactly 3 I do not refer to the vast legal literature on bankruptcy; see Heralin, Katz and Craswell (27) for a survey. 4 It is surprising that, despite acknowledging see section 7 of their paper that the optial penalty associated with default is neither zero nor infinite, ubey, Geanakoplos and Shubik (25) do not characterize it forally. 5 Townsend (979), iaond (984), Gale and Hellwig (985), Bolton and Scharfstein (99), Bizer and emarzo (992), Hart and Moore (994, 998, 995), Krasa and Villail (2), Albuquerque and Hopenhayn (24), Jappelli, Pagano and Bianco (25), Rapini (25), Bisin and Rapini (26), Hopenhayn and Werning (28), Grochulski (2), Gale and Gottardi (2) are exaples of this vast literature. Bolton and ewatripont (25), Freixas and Rochet (28) and Tirole (25) are textbook references. 3

5 suarized in Allen and Gale (994) and uffie and Rahi (995). This literature starts by allowing for soe for of arket incopleteness and then asks the question of which securities should be introduced in the arket to iprove risk sharing and welfare. The optial bankruptcy exeption in this paper solves a restricted optial security design proble, under a set of particular behavioral (e.g., risk averse borrowers and risk neutral lenders) and environental (e.g., restricted set of contracts traded) constraints. Finally, a dynaic version of the environent used in this paper following Eaton and Gersovitz (98), has becoe the workhorse odel to understand sovereign default. If the ex-post penalties iposed on sovereigns that default were enforceable, the results of this paper would also apply to the international context with inor odifications. Forally closer to this paper, Bolton and Jeanne (27, 29) use a two-period odel of defaultable credit to study the optial nuber of creditors and the ability to refinance. See Aguiar and Aador (23) for a recent survey of the sovereign default literature. Outline Section 2 lays out the baseline odel and analyzes its positive iplications and section 3 executes the norative analysis, solving for the optial bankruptcy exeption. Section 4 calibrates the theoretical forulas to US data and section 5 analyzes ultiple extensions. Section 6 concludes. All proofs and derivations are in the appendix. 2 Baseline odel This section presents the ain results in a stylized odel. Section 5 extends the results in any diensions. First, I characterize the equilibriu behavior of borrowers and lenders for a given level of the bankruptcy exeption. Subsequently, I characterize the welfare axiizing bankruptcy exeption fro an ex-ante perspective. 2. Environent Tie is discrete, there are two dates t =, and there is a unit easure of borrowers and a unit easure of lenders. Because borrowers are risk averse, the results of this paper are ore applicable to household borrowing, rather than to corporate borrowing. 6 Borrowers Borrowers are risk averse and axiize expected utility of consuption with a rate of tie preference β >. Their flow utility U (C) satisfies standard regularity conditions: U ( ) >, U ( ) < and li C U (C) =. Unlike any papers in the optial contracting literature, which assue risk neutrality to siplify the contract design process, the results of this paper exploit interior optiality conditions. Hence, borrowers axiize: ax C,{C } y,b,{ξ } y U (C ) + βe[v (C )], where V (C ) = ax ξ {,} { ξu ( C ) + ( ξ )U ( C )} and ξ is an indicator for default for every realization y. If a borrower decides to repay, ξ =, while if he decides to default, ξ =. C denotes consuption for a borrower who defaults and C denotes consuption for a borrower who repays. There is a single consuption good (dollar) in this econoy, which serves as nueraire. 7 Every borrower is endowed with y units of the consuption good at t =. At t =, every borrower receives a stochastic endowent of y units of the consuption good, whose distribution follows a cdf F ( ) with support in [ y,y ], where y and 6 There is scope to adapt the results of this paper to the corporate context, in which firs often assued risk neutral engage in risk anageent with iperfect instruents and becoe endogenously risk averse, along the lines of Froot, Scharfstein and Stein (993) and Rapini and Viswanathan (2). 7 Allowing for positive inflation is straightforward 4

6 y could be infinite. The realizations of y are iid aong borrowers. Therefore, under a law of large nubers, there is no aggregate risk in this econoy. All variables are observable by the parties. Borrowers can trade a single noncontingent contract, i.e., a debt contract. That is, borrowers issue debt with face value B, due at t =, and receive fro lenders q (B,)B units of the consuption good at t =. Hence, the gross interest rate faced by borrowers can be defined as +r q. When needed, I denote by r the logarithic interest rate, i.e., r log( + r). In the baseline odel, borrowers take into account that the interest rate at which they are able to borrow depends on the aount of debt they take. Hence, the budget constraint at t = for borrowers is: C = y + q (B,)B, where the unit price of debt taken q (B,) is a function of B and, as described below. At t =, once y is realized, borrowers can decide to repay the aount owed B or to default. 8 If they default, they consue C = in{y,}, that is, they keep the bankruptcy exeption of units of the consuption good, unless is larger than y, in which case they only keep y units. Any positive reainder y is seized fro borrowers and transferred to lenders, although lenders only receive a fraction δ [, ) of the transferred resources. This loss captures the resource costs associated to the bankruptcy procedure. I assue throughout that there is no ex-post renegotiation of the ters of the contract. The exeption takes a value in the interval [, ]. Because the bankruptcy procedure cannot rely on external funds, y. To siplify the exposition, I further restrict to be greater than the lowest realization of y, that is, > y. 9 Therefore, for a given realization of y, the budget constraints at t = when a borrower chooses to repay and when it chooses to default are, respectively: C = y B C = in{y,} I assue that borrowers rate of tie preference β, initial endowent y and distribution of future endowents F ( ) are such that borrowers borrow in equilibriu, that is, B >. condition. Lenders The appendix provides an exact sufficient Given the exeption level and borrowers default behavior, I assue that lenders supply credit to borrowers according to an interest rate schedule q (B,), which depends on the face value of the debt B and the exeption level. As long as lenders ake zero profit and q (B,) is well behaved, the ain results of this paper are valid without further specifying the lenders behavior. That said, to be concrete, I use as a running exaple the case in which lenders are risk neutral, perfectly copetitive and require a given rate of return + r, which can differ fro the borrowers rate of tie preference β. In that case, q (B,) takes the for: q (B,) = δ ax{y,} B df (y ) + df (y ) + r, where represents the default region and the no default region, which are deterined in equilibriu. See the appendix for how to handle risk averse copetitive lenders. 8 I use the words bankruptcy and default as synonys in this paper. See White (2) and Herkenhoff (22) for how borrowers ay default on their obligations without entering in bankruptcy. Allowing borrowers not to repay without declaring bankruptcy does not change the optial exeption forula, as long as they behave optially. See the discussion in section 5. 9 The results extend naturally to the case in which < y, but the analysis becoes ore tedious, since ultiple cases have to be analyzed then. Those results are available under request. 5

7 Equilibriu definition/regularity conditions An equilibriu, for a given level of exeption, is defined as a set of consuption allocations C, {C } y, default decisions {ξ } y, aount of debt issued B and price q such that borrowers default and borrow optially internalizing that their choices affect the price of the debt and lenders offer an interest rate schedule q (B,) while aking zero profit. As usual in probles with continuous distributions of shocks, convexity is in general not guaranteed. To ease the exposition throughout, I work under the assuption that the borrowers proble is convex, so first-order conditions are necessary and sufficient to characterize the optiu. I further assue that borrowers indirect utility W () defined in equation (7) is also convex in. The appendix provides sufficient conditions for convexity and the online appendix shows nuerically that the odel is in practice well behaved. After understanding how the level of the bankruptcy exeption affects ex-ante welfare, I solve for the optial ex-ante exeption. The optial ex-ante exeption ust be interpreted as an optial coitent device chosen by borrowers given the set of contracts available to provide insurance and enforce ex-post repayents, which allows the to borrow ex-ante. The following rearks highlight the two key features of the econoic environent. Reark. (Given set of contracts/assets) The key friction in this paper is that the set of contracts/assets available to borrowers is given: borrowers do not choose the shape of the traded contracts optially. This paper does not take a stand on why borrowers do not use a contract or a set of contracts that deliver the first-best outcoe of perfect insurance. Transaction costs, hysteresis in contracting, inforational frictions or bounded rationality are plausible explanations. For instance, the rich literature on financial contracting referenced above contains elaborated justifications for why debt contracts are prevalent. Taking as given the set of traded contracts/assets, as in the classic research in general equilibriu theory, this paper deterines the optial degree of leniency in bankruptcy. The possibility of bankruptcy introduces new contingencies which iprove risk sharing, a echanis originally pointed out by Zae (993) and ubey, Geanakoplos and Shubik (25). Reark. (Constant exeption level) In the baseline odel, an exeption level that does not depend on the realization of y is optial even when a nonlinear bankruptcy schee that depends on y is feasible, because the first-best outcoe in this econoy involves risk neutral lenders providing a flat consuption profile (full insurance) to risk averse borrowers at t =. A constant exeption level is not constrained optial in soe of the extensions: I ll point that out whenever that is the case. In those situations, I a solving a second best proble with iperfect instruents. Then, I justify the choice of a constant exeption with the fact that it is the one used in practice. 2.2 Equilibriu characterization First, I characterize the optial ex-post default decision by borrowers and then solve for the optial ex-ante choice of B. Finally, I characterize the equilibriu pricing schedules offered by copetitive lenders. Borrowers default decision At t =, given his ex-ante choice of B, a borrower solves the proble: { ( ) ( )} ax ξu C + ( ξ )U C ξ {,} { Because flow utility is strictly onotonic, this proble is equivalent to ax {ξ } C,C }. The optial default decision is given by a threshold on the realization of y. When y is high, it is optial not to default, but when y is sufficiently low, it is preferable to default than to repay the loan. Figure shows graphically the default proble at t =. The upper envelope of the default and repayent options deterines the optial consuption choice given B. The 45 degree line is shown for reference. 6

8 C C = y B C = in{y, } 45 y B + B y y Forced efault Strategic efault Figure : Optial default decision given B Forally, the optial default decision is: ξ =, if y < + B efault ξ =, if y + B o efault The default threshold is deterined by the indifference condition between the aount to be repaid B and the aount transferred to lenders y. I assue that an indifferent borrower decides not to default. Given the default decision, the fraction of borrowers that defaults in equilibriu is deterinistic and given by F ( + B ). This odel incorporates forced default, which occurs when a borrower does not have enough resources to fully pay back its debt (it occurs when B > y ), and strategic default, which happens when borrowers have enough resources to fully repay but they decide not to do it (it occurs when + B > y > B ). This distinction, which often plays a proinent role in discussions about bankruptcy exeptions, is not relevant for the results of this paper. Borrowers optial choice of B Given their optial ex-post default decision and taking into account the interest rate schedule offered by lenders, borrowers optially choose how uch to borrow. The proble solved by borrowers at t =, for a given exeption, is: [ˆ axu (y + q (B,)B ) + β U (y )df (y ) + B y ˆ +B ˆ ] y U ()df (y ) + U (y B )df (y ) +B Under the assued regularity conditions, the proble solved by borrowers is convex in B, so the following first-order condition fully characterizes the solution to (2): [ U (C ) q (B,) + q ] (B,) B = β B ˆ y (2) +B U (y B )df (y ) (3) Intuitively, the left hand side of (3) represents the arginal benefit at t = of increasing the face value of the debt by a dollar. This arginal benefit is given by q, the aount raised at t = per dollar proised at t =, corrected by how the induced interest rate increase affects the total aount borrowed B. Borrowers value this change at their arginal 7

9 Repayent B ax {y, } 45 y + B y y Figure 2: Repayent to lenders given B utility U (C ). The right hand side of (3) represents the arginal cost of repaying the debt, given by the arginal utility when the payent is due. This cost is only paid in those states in which borrowers do not default; debt iposes no effective costs on borrowers in those states in which it does not have to be repaid. Equation (3) allows to characterize analytically how the total aount of credit changes in equilibriu with the level of the bankruptcy exeption. Unsurprisingly, the sign of db d is abiguous. Forally, db d has the following sign: ( ) ( db sign = sign U (C ) q [ d B q + q ] [ B +U q (C ) B + 2 ] ) q B B + βu () f ( + B ) There are three distinct effects that deterine the sign of db d. First, all else equal, an increase in reduces q, which reduces household consuption C and increases the value of U (C ); this effect induces borrowers to borrow ore. Second, all else equal, an increase in varies the unit aount that can be raised at t =, given by the direct price effect q (B,) and the change in the derivative of the interest rate schedule 2 q (B,) B B ; this effect is in general abiguous. Third, all else equal, an increase in reduces the arginal cost of borrowing because the default region widens, which reduces the likelihood of having to pay back the debt. This effect induces borrowers to borrow ore. This ter captures the idea that borrowers decide to borrow ore ex-ante anticipating not having to pay back their debts, an effect that is often described as oral hazard. The epirical literature on this issue finds that high exeptions are often associated with high levels of borrowing. uerical solutions of the odel with standard paraetrizations find that B can increase or decrease with. That said, it is not necessary to take a stance on whether borrowers borrow ore or less when exeptions change to understand the effects on welfare of varying bankruptcy exeptions, as long as they choose how uch to borrow optially. Lenders interest rate schedule When lenders are risk neutral and perfectly copetitive, given borrowers default decision, they offer the following interest rate schedule for given levels of B and : q (B,) = δ +B y B df (y ) + y +B df (y ) + r (4) 8

10 Figure 2 graphically shows the repayent to lenders. The upper envelope between ax{y,} and B represents the effective repayent to lenders. The credit spread is positive, that is r r >, to account for the possibility of default, and approxiately equal to the expected unit loss for lenders, because of risk neutrality. Two properties of the interest rate schedule are iportant for the analysis. First, the interest rate schedule decreases with the aount of debt B. Second, the interest rate schedule decreases with the level of the bankruptcy exeption. Forally (the exact expressions are in the appendix): q (B,) B < and q (B,) < (5) Intuitively, for a given level of, the required spread increases with the aount of credit issued. This occurs for two reasons. First, the per unit fraction of liabilities recovered by lenders in default states decreases with the total aount of credit. Second, because the default region widens, ore resources are lost as bankruptcy costs. Also, for a given level of B, the required spread increases with the level of the bankruptcy exeption. There are again two reasons for this. First, the recovery rate for lenders in default states decreases with the level of the exeption, since borrowers get to keep a higher exeption. Second, because the default region widens, ore resources are lost as bankruptcy costs. We can forally express the equilibriu changes in interest rates induced by changing the bankruptcy exeption in the following way: dq (B,) d = q (B,) db B d + q (B,) The last ter in equation (6) is negative, but depending on the sign of db d, dq (B,) d can take any sign. As long as borrowing increases with, that is, db d >, observed interest rates increase in equilibriu, that is, dq (B,) d <. 3 Optial bankruptcy exeption After characterizing the equilibriu for a given exeption, I now study the proble of deterining the welfare axiizing exeption. Because lenders ake zero profit in equilibriu, axiizing borrowers indirect utility is equivalent to axiizing social welfare in this econoy. First, I study how varying the bankruptcy exeption affects ex-ante welfare. Then, I solve for the optial exeption. I denote the indirect utility of borrowers, as a function of, by W (): (6) W () = U (y + q (B (),)B ())+ [ +β y U (y )df (y ) + +B () U ()df (y ) + ] y +B () U (y B ())df (y ), (7) where B () is given by the solution to equation (3) and q (B (),) is given by: q (B (),) = δ +B () Propositions and 2 present the ain results of this paper. Proposition. (Marginal effect of varying on welfare) y B () df (y ) + y +B () df (y ) + r a) The change in welfare induced by a arginal change in the bankruptcy exeption is given by: d = U (C ) q (B (),) B + ˆ +B βu ( C ) df (y ) (8) 9

11 b) The change in welfare induced by a arginal change in the bankruptcy exeption, expressed as a fraction of t = consuption, is given by: d U = Λε r, + { } Π C, (9) (C )C C q (B (),) q = where Λ q B y +q B is a easure of borrowers leverage, ε r, log(+r) denotes the sei-elasticity of the interest rate schedule offered by lenders with respect to the level of the exeption and Π {C } +B C βu (C ) C U (C ) df (y ) is the price-consuption ratio fro the borrowers perspective of a clai that pays the arginal value of increasing the bankruptcy exeption by one unit. Proposition characterizes the effect on social welfare of a arginal change in the bankruptcy exeption. The derivation of equation (8) crucially exploits the fact that borrowers borrow and decide when to default optially. On the one hand, a arginal increase in the exeption akes borrowing ore expensive through a reduction in the price on the debt issued q, which affects the total aount aount of debt outstanding B. This change is valued by borrowers according to their t = arginal utility U (C ). This increase in borrowing costs is the arginal cost of a ore lenient bankruptcy procedure. On the other hand, a arginal increase in the exeption increases the resources that borrowers can keep when they default while claiing the full exeption. Averaging over the pertinent realizations of y and weighting this gain by the arginal utility βu ( C ) in those states, the arginal welfare gain of a ore lenient bankruptcy procedure becoes β +B U ( C ) df (y ). Equation (9) expresses the change in welfare as a oney-etric dividing by U (C ) before noralizing by initial consuption C. The ter Λ easures borrowers leverage. The ter ε r, denotes the partial derivative of the interest rate schedule with respect to the bankruptcy exeption. Equation (5) guarantees that ε r, is strictly positive. Π { C } is the price fro the borrower s perspective at t = of a clai that pays borrowers consuption only in default states in which borrowers clai the full exeption those in which y >. Π {C } C C express this price in relative ters to current consuption C : this is a easure of the arginal benefit for borrowers of increased leniency. The ratio Π {C } C, which I refer to as the price-consuption ratio, is deterined by the product of two ters. First, it can be high when the ratio of arginal utilities U (C ) U (C ) is high. Second, it can be high when consuption growth in those states C C is also high. Both ters are in general related, and tightly linked when utility is CRRA or Epstein-Zin cases, as discussed below. The optial bankruptcy exeption can be found as: = argax W () Under the assued regularity conditions, the optial bankruptcy exeption ust be a solution to the equation d =. Proposition 2. (Optial bankruptcy exeption) The optial exeption expressed in units of the consuption good, i.e., dollars is characterized by: where Λ q B y +q B = is a easure of borrowers leverage, ε r, log(+r) = Π {C } C Λε r,, () q (B,) q denotes the sei-elasticity of the interest rate schedule offered by lenders with respect to the level of the exeption and Π {C } C ebt-to-equity ratios, that is, L q B y are frequently used as easures of leverage. The variable Λ is a onotonic transforation of L: Λ = L +.

12 +B C βu (C ) C U (C ) df (y ) is the price-consuption ratio fro the borrowers perspective of a clai that pays the arginal value of increasing the exeption by a unit. The expression for optially trades off the arginal benefit of increasing consuption in default states (nuerator) against the arginal cost of restricting access to credit (denoinator). When is high, borrowers face higher interest rates, which akes borrowing less profitable, at the cost of iproved insurance when declaring bankruptcy. A low akes ex-ante borrowing less costly while aking bankruptcy ore painful. A high value for is optial when Λ and ε r, are large. Intuitively, if interest rates schedules are very sensitive to increasing the bankruptcy exeption, aking default ore attractive by increasing is very costly in ters of curtailed access to credit; this effect is odulated by the aount borrowed Λ. A low value for is optial when Π {C } C, the noralized welfare gain of a arginally higher exeption is large. Although equation () ust hold at the optiu, it does not provide a characterization of as a function of priitives, because all right hand side variables are endogenous. CRRA utility To build further intuition, let s assue that borrowers have constant relative risk aversion utility (CRRA) preferences, that is, U (C) = C γ γ, where γ C U (C) U (C). Assuing a particular utility specification only affects directly the price-consuption ratio Π {C } C. The arginal cost of increased leniency Λε r, does not depend directly on the utility function, only through the effects on B and q. We can thus write: Π { C } C Therefore, an increase in leniency increases C = β ˆ +B ( C C ) γ df (y ) and has both discount rate and cash flow effects. When the CRRA coefficient γ is greater than one, the discount rate effect ( C C ) γ doinates this is the standard paraetrization, see Capbell (23) but, if γ is less than one the consuption growth ter C C doinates. With logarithic utility (γ = ) both effects exactly cancel out. We expect t = consuption to be higher than the exeption level, that is C C <, so the arginal loss generated by a higher exeption is increasing in the risk aversion paraeter γ. The forces that deterine the price-consuption ratio are the sae that deterine the price-dividend (consuption-wealth) ratio in standard consuption based asset pricing odels see Capbell (23) for a review. In the classic Lucas (978) odel, price-dividend ratios are constant and equal to the rate of tie preference β for investors with logarithic utility. That sae result applies here with one odification: instead of β, the relevant price-consuption ratio becoes βπ, because we are interested in the price-dividend ratio of a security that only pays in default states with positive recovery by lenders. Logarithic utility is an often used benchark specification for preferences, the price-consuption ratio can be written as: Π { C } C = βπ, where π +B df (y ) is the unconditional probability that a borrower consues the bankruptcy exeption. It can alternatively be written as: π = π π, the unconditional probability of default π ties the conditional probability that a borrower clais the full exeption π. Hence, when borrowers have log utility and lenders always clai the full exeption, only the probability of default is needed to assess the arginal benefit of increasing the bankruptcy exeption. When borrowers risk aversion is larger than unity and < C, the logarithic utility forula provides a lower bound for the optial exeption that does not require to easure consuption. See the appendix for a foral arguent.

13 Hence, when borrowers have logarithic utility, the optial bankruptcy exeption can be written as: = βπ Λε r, In this baseline odel, assuing that lenders are risk neutral, we can further rewrite the equation that characterizes the optial exeption as: ( β ( + r ) ) γ = C, () ϒ + δ ( ϒ) where ϒ = f ( +B )B F( +B ) F( ) is a easure of curvature of the distribution F ( ) that can take values in [, ]. When F ( ) is a unifor, ϒ =. Intuitively, the optial exeption seeks to equalize consuption at t = in default states with consuption at t =, although with a correction that depends on the ter β(+r ) δ+( δ)ϒ. Intuitively, all else constant, when β ( + r ) >, it is optial to consue ore at period, which calls for lower exeptions. Siilarly, when δ + ( δ)ϒ, which easures log(+r), is low, high exeptions are optial. All else constant, there is no clear coparative static on the bankruptcy cost paraeter δ. Two effects copete. On the one hand, a high δ (low bankruptcy costs) aplifies the sensitivity of interest rate schedules to exeption changes because ore resources flow fro borrowers to lenders in bankruptcy. On the other hand, a high δ dapens the sensitivity of interest rate schedules to exeption changes, because there is no loss associated to defaulting in ore states. All these effects are odulated by borrowers risk aversion: high risk aversion γ pushes towards C. Although equation () is helpful to provide intuition behind the sufficient statistics, I focus throughout this paper on equations like (9) and (), since those hold ore generally. I conclude this section with two rearks. Reark. (Sufficient statistics) Three observable variables: leverage, the sensitivity of interest rate schedules with respect to the exeption level and the price-consuption ratio, suffice to deterine the optial exeption, independently of the rest of the structure of the odel. For log utility borrowers, the price-consuption ratio siplifies to the probability of default when borrowers clai always the full exeption. For instance, the distribution of incoe shocks or the level of interest rates only affect through these sufficient statistics. The logic behind these sufficient statistics is siilar to the one behind the CAPM, in which the beta of an asset becoes sufficient to deterine expected returns. It is also siilar to the logic behind consuption based asset pricing odels, in which the consuption process, independently of how it is generated, is sufficient to deterine asset prices and expected returns. Reark. (Security design interpretation) The key tradeoff in this paper can be interpreted as a security design proble. Assue that borrowers can choose between two securities, priced by the sae set of lenders. They can issue either a noncontingent bond, or a bundle of that noncontingent bond with a put option (whose strike is deterined by the optial default decision). Given their incoe process, borrowers will prefer one of these two securities; the choice of adjusts paraetrically between both contracts and selects the optial traded security. 4 Calibration Although the ain contribution of this paper is theoretical, the upshot of the approach followed is that the key theoretical tradeoffs can be quantified by easuring a few observables. To show the applicability of y results in practice, I first calibrate the optial exeption to US data and then study the agnitude of the welfare gains generated by arginal changes in the bankruptcy exeption. 2

14 Optial exeption Because it entails inial inforational requireents, I calibrate the optial exeption for the baseline odel assuing that borrowers have CRRA utility. 2 borrowers is given by: I have just showed that the optial exeption for CRRA utility = βπ ( C C ) γ Λε r, (2) Therefore, four observable variables and two preference paraeters, the rate of tie preference β and the coefficient of relative risk aversion γ, need to be calibrated. I use a yearly calibration. Calibrating Λ, β, and γ is straightforward. To calibrate Λ, I target the sae average ratio of unsecured debt to personal disposable incoe as Livshits, MacGee and Tertilt (27), which is q B y = 8.4%. This value iplies that Λ = I use β =.96 as the annual rate of tie preference: this is a standard choice. I adopt γ = for the baseline calibration for risk aversion, but report the results for other levels of risk aversion. Finding appropriate values for π, C /C, and ε r, requires further work. First, π can be written as the product of the unconditional probability of default π with the probability of claiing the full bankruptcy exeption conditional on defaulting π. To deterine π, I use the average probability of filling for chapter 7 bankruptcy, also fro Livshits, MacGee and Tertilt (27), which is.8%. This choice should raise no concerns. I deterine π by using the fact that roughly 9% of bankruptcies are filed as no-asset bankruptcies see Lupica (22). This iplies that % of bankrupt borrowers fully clai their exeption. Second, I use C /C =.9 for the change in consuption by bankrupt borrowers. A % reduction in consuption ay be considered as a large change, but it is within the range of variation docuented in Filer and Fisher (25) using PSI data on changes in food consuption for bankrupt individuals. I purposefully pick a relatively large nuber for C /C to ake sensitive to varying risk aversion; otherwise, the results when C /C are nearly identical to those in the log utility case as long as γ is within a reasonable range. Finally, calibrating the sensitivity of the interest rate schedule with respect to the exeption ε r, is not easy. Ideally, we would observe how interest rate schedules vary with changes in exeptions for the sae borrower. o paper has been able to follow that approach, precisely because the required pricing data is sensitive for lenders. Instead, I use the average estiate fro Gropp, Scholz and White (997), who estiate the (equilibriu) effect on interest rates of changes in bankruptcy exeptions over tie and across states. A value of ε r, = is within the reasonable range for the average response of interest rates in the population. This choice iplies that increasing the bankruptcy exeption by a hundred thousand dollars increases the equilibriu interest rate by 25 basis points. dq q q Conceptually, this value corresponds to d, as defined in equation (6), and not to q, since it does not control for the change in interest rate caused by changes in the level of borrowing db d, which is found to be positive in the data. Therefore, the chosen value for ε r, ay be upward biased, which eans that the optial exeptions in table 2 could be seen as lower bounds for. Alternatively, we can think that we are calibrating the odel in which borrowers are 2 As shown in section 5, allowing for Epstein-Zin preferences and assuing that the certainty equivalent of t = consuption equals C yields identical results. 3 In actual econoies, a substantial fraction of borrowing is collateralized. Under the assuption that collateralized borrowing is fully secured, which iplies that the interest rate charged is not sensitive to, equation (8) derived below iplies that the denoinator of the optial exeption only has to account for the fraction of unsecured credit. If collateralized lending is not fully secured, the optial exeption found in this section should be adjusted downwards. Also, the bulk of exeptions in the US are hoestead exeptions. Hence, I iplicitly assuing that borrowers endowent at t = is in the for of a house. 3

15 price takers. As shown in section 5, the equilibriu seielasticity is the correct one in that case. 4 Table suarizes the choices of paraeters and variables used in the baseline calibration. Paraeter/Variable Value Paraeter/Variable Value β Rate of tie preference.96 π Probability of default.8 C /C Consuption change.9 π Probability of no-asset bankruptcy. Λ Leverage.775 ε r, Interest rate sensitivity Table : Calibrated variables γ. Using equation (2), table 2 presents the optial exeption for different values of the risk aversion coefficient γ = (log) γ = 5 γ = γ = 2 γ = 5 39, 638 6, 46 2, , 435 6, 922, 79 Table 2: Optial exeption (easured in dollars) This analysis concludes that the actual average bankruptcy exeption across US states, of roughly 7, dollars, is very close to the optial value iplied by the odel. The preferred calibration suggests that the optial exeption should approxiately be, dollars, although the level of risk aversion has a draatic effect on. Given that ε r, could be even lower, the quantitative results of this paper suggest that there ay be scope to increase welfare by raising exeptions. In general, this is because the existent epirical research has found low values for interest rate sensitivities. On the size of welfare gains The results regarding show that the optial exeption should be slightly larger than the average exeption across US states. Given that result, a natural question is how large are the welfare gains fro changing at the current level of exeptions. To answer this question, I focus again in the CRRA case and define σ () d U (C )C as the welfare gain, expressed as a fraction of initial consuption, of increasing the bankruptcy exeption starting fro a level. The value of σ (), as shown in section 3, is given by: σ () = Λε r, + ( ) C γ βπ C For the purposes of this exercise, I freely use σ () to assess changes of any agnitude, given the difficulty of calibrating d for all values of and then integrating over those to actually copute welfare changes. Ideally, if we could obtain epirical counterparts of the different sufficient statistics for all values of, we could calculate the exact welfare gain or loss induced by a nonarginal change in by integrating d over the relevant range of exeptions. Siilarly, we could find the fixed point that yields. Hence, although equation (8) is exact for all values of the bankruptcy exeption, the calibration becoes an approxiation because I ipose constant values for the sufficient statistics. Hence, the results are ore precise for sall interventions and should be interpreted otherwise 4 It is hard to ake a case for whether borrowers take interest rates as given or not for unsecured borrowing. A previous version of this paper used price taking as the baseline case. 4

16 with caution. The value of σ () still deterines whether a sall change in exeptions is welfare iproving or not. If σ () is positive, this paper guarantees that a sall increase in exeptions is welfare iproving and viceversa. Using the sae set of paraeters described in table 2, and starting fro the average exeption of = 7, dollars, I now show in table 3 the welfare gains associated to a, dollars increase in. γ = (log) γ = 5 γ = γ = 2 γ = 5.84%.27%.89%.62%.897% Table 3: Welfare gain/loss (, dollars change) easured as a fraction of initial consuption Therefore, table 3 provides the welfare gains, easured as a percent of initial consuption C, fro increasing the bankruptcy exeption by, dollars when = 7,, for the sae paraetrization used to derive. As expected, the agnitude of the welfare gain grows with the value of, when > 7, and viceversa. For instance, for the benchark calibration, a, dollars increase in increases welfare by.89%. Increasing by 3, dollars, which would ove the econoy fro the average 7, until the optiu of,, yields a welfare gain of roughly.3%; this easures the total welfare loss incurred by not having the optial exeption. 5 Extensions The goal of this section is to understand how departures fro the baseline odel affect the expression for the optial exeption. In order to do that, I extend the baseline odel in ultiple diensions. I address several natural extensions forally and discuss soe others before concluding. To ease the exposition, I analyze every extension separately and oit equilibriu definitions and regularity conditions. 5. Endogenous incoe: elastic labor supply and effort choice in frictionless arkets In the baseline odel, borrowers incoe is exogenously deterined. However, it is often argued that creditor friendly bankruptcy procedures reduce borrowers welfare by distorting labor supply decisions. It is true that seizing a fraction of borrowers labor incoe has a negative effect on labor supply. Siilarly, large exeptions ay distort ex-ante effort choices by borrowers. However, the optial exeption forula does not change when effort and labor supply are affected by changes in exeptions, as long as those decisions are ade optially. To capture these concerns, I odify the baseline odel in two diensions. First, I allow borrowers to choose labor supply in both periods. Second, I assue that the distribution of incoe F (Y ;a) is a differentiable function of a noncontractible effort choice a, ade by borrowers at t =. This effort choice creates another for of oral hazard. Borrowers can work at given wages w and w in the background, perfectly copetitive/constant returns to scale firs provide labor deand curves. Borrowers flow utility is now given by a well behaved function U (C,;a), where denotes hours worked and a is the quantity of effort exerted. For now, I ake no assuptions on the separability between consuption, leisure and effort. For siplicity, I abstract fro liits on wage garnishents and assue that all labor incoe is transferred fro borrowers to lenders in bankruptcy. Borrowers now solve: ax U (C, ;a) + βe a [V (C, )] C,{C } y,b,{ξ } y,,{ } y,a s.t. C = y + w + q B ; C = y + w B ; C = in{y,}, 5