Repos, Fire Sales, and Bankruptcy Policy

Transcription

1 Repos, Fire Sales, an Bankruptcy Policy Gaetano Antinolfi Francesca Carapella Charles Kahn Antoine Martin Davi Mills E Nosal Preliminary an Incomplete May 25, 2012 Abstract The events from the financial crisis have raise concerns that the failure of a large financial institution coul lea to estabilizing fire sales of assets. The risk of fire sales is irectly relate to the exemptions from the bankruptcy stay enjoye by many financial contracts, notably repos. In this paper, we construct a moel of repo transactions an consier the effects of changing bankruptcy rule on the activity in the repo market an in the market where a efaulte borrowers collateral can potentially be resol. We fin that exempting repos from the automatic stay of bankruptcy is beneficial for creitors that hol the borrowers assets. The exemption may increase the size of repo activity, which can either be goo or ba for welfare. Finally, the exemption can also lea to amaging fire sales in the resale market, which reuces welfare via its effect on investment activity. The parameters of the moel influence the size of these effects an, thus, the nature of the optimal bankruptcy rule. 1 Introuction An institution that suffers large losses may be force to sell assets at istresse or fire-sale prices. If other institutions revalue their assets at these temporarily low market values, then the initial sale can set off a cascae of fire sales that inflicts losses on many institutions, (French et al., (2010)). A number of commentators have ientifie fires sales as epleting the balance sheets of financial institutions an aggravating the fragility of the financial system in the recent financial crisis, (Shleifer an Vishny, 2011). Therefore, via efaults an fire sales, one trouble institution can amage another an, as a result, reuce the financial system s capacity effi ciently allocate resources. Many commentators have ientifie the freezing of the repurchase an agree sale, repo, market as key contributor to the recent financial crisis. This paper evelop a moel a repo market. We examine the implications of ifferent policy rules on the activity in this market, as well as in the market where the collateral of a efaulte borrowers can be resol. An important 1

2 feature of the repo market is that the repo lener (or creitor) hols collateral of the borrower. Uner current bankruptcy rules, the lener can liquiate the collateral if the borrower efaults. In effect, the repo contract is exempt from a stanar bankruptcy proceure, a stay, which prevents creitors from liquiating the collateral of the efaulting party. This exemption from the stay, which is affore to many financial contracts, has raise concerns that the efault of a large financial institution coul trigger estabilizing fire sales of assets. Such fears are base on the failure, or near failure of Bear Stearns an Lehman brothers in In our moel, the possibility of borrower efault motivates leners to request collateral from buyers. The insurance function of the collateral is imperfect. If bankruptcy rules impose a stay on the liquiation of the borrowers collateral, then the lener receives a low value from the collateral. This low value can represent the cost to the lener of not having access to its funs uring the liquiation process, as well as the risk that the collateral coul lose value. If, instea, bankruptcy rules allow the lener to liquiate the borrower s collateral, then that collateral can be sol a seconary market. In that seconary market, leners an investors compete with each other to obtain goos from a market traer. Investors have access to a prouctive technology that prouces consumption goos, using the market traer s goo as input. Hence, if their ability to obtain the market goo ecreases, the economy as a whole suffers a loss of prouctivity, as economy-wie investment ecreases. This assumption is important an captures the fact that fire sales affect the economy through an externality. Absent this assumption, leners woul internalize all the effects of their sales of assets on the economy an make effi cient investment ecisions in the repo market. With this assumption, leners o not take into account the effect they have on investors when choosing their activity in the repo market. The externality creates a trae-off when consiering bankruptcy rules. On the one han, exempting repo transactions from the bankruptcy stay is esirable since it increases the value of the borrower s collateral to the lener since it can be liquiate in case of efault. In this sense, exemption from the bankruptcy stay makes the repo market more liqui. On the other han, this increase activity results in more pronounce fire sales in case of borrower efault, which reuces the investment activity, which is not esirable. The relative size of these two effects epen on the parameter values of the moel an, thus, so oes the optimal bankruptcy rule. Our paper focuses on the trae-off between the liquiity of the repo market an the potential for fire sales relate to the exemption from the stay. These effects are iscusse in Roe (2011). Duffi e an Skeel (2012) also iscuss a number of aspects of bankruptcy stay relate to our paper. The paper that is most closely relate to our is Acharya, Anshuman, an Viswanathan (2012) which focuses on a trae-off similar to ours. Our paper oes not aress some of the 1 It is worth noting that these fears were, for the most part, not realize in part because Bear Stearns was purchase by JP Morgan Chase, an the US broker ealer unit of Lehman i not eclare bankruptcy. 2

3 benefits of the exemption from the stay associate with close-out netting. An analysis of these benefits is provie in McAnrews an Robers (2003). [We shoul also reference Bolton & Oehmke] The paper is organize as follows. The basic moel, without efault, is presente in the next section. The basic moel is generalize to allow for efaults in section 3. Section 4 examines the nature of a government policy intervention an carefully analyzes the trae-offs that the government faces. Section 5 offers some concluing comments. 2 The Basic Moel The economy has 3 ates 1, 2, an 3 an 2 goos a an c. Goo c is perishable an goo a is urable. One might want to think of goo a as being asset. There are 4 types of agents: leners, L, borrowers, B, investors, I, an traers, T. The number or measure of agents is n i, where i {L, B, I, T }. Leners an borrowers are born at the beginning of ate 1. Borrowers live at ates 1 an 2, an leners live at ates 1, 2 an 3. Investors an traers are born at the beginning of ate 3 an live only at ate 3. All agents can commit to their promises. Leners want to consume goos a an c at ates 2 an 3. They like goo c more than goo a. Leners can prouce goo c only at ate 1, where one unit of costly effort prouces one unit of goo c. The preferences of a lener, U L, are given by U L = u (c 2 ) + c 3 + γa 2 + γa 3 c 1, where u is increasing an strictly concave, an 0 < γ < 1. Subscripts inicate when the goos are prouce or consume. Borrowers like to consume goo a at ate 2, an can prouce goo c only at ate 2. In aition, they possess a technology that can costlessly an instantaneously convert one-for-one goo c into goo a. The preferences of a borrower, U B, are U B = a 2 c 2. Traers are enowe with c units of goo c at ate 3. They like to consume goos a an c at ate 3, an their preferences, U T, are given by U T = a 3 + c 3. Investors are enowe with ā units of goo a at ate 3, where ā < c. 2 like to consume goos a an c, an their preferences, U I, are They U I = a 3 + c 3. Investors have access to a technology, f, that prouces goo c using goo c as an input. f is increasing, strictly concave an f (ā) > 1. The last assumption implies that f is a prouctive investment in the sense that if the investor coul 2 We consier the case where ā > c in footnotes. 3

4 exchange his enowment of goo a for ā units of goo c, then marginal return is strictly greater than one for all levels of input c (0, ā]. Agents trae in pairs; that is, they are bilaterally matche. Agents are matche at the beginning of ate 1 an at the beginning of ate 3. The ate 1 an ate 3 matching processes of inepenent of one another. Since investors an traers are not alive at ate 1, only leners an borrowers enter the matching process at that time. There are gains from trae. For example, borrowers an leners benefit from traing goo c at ate 1 for goo c at ate 2, an traers an investors benefit from traing goo a for goo c at ate 3. Trae between borrowers an leners is straight forwar since agents can commit. A matche lener prouces goo c at ate 1 an gives it to the borrower, who converts it into goo a. In return, the borrower promises to prouce goo c for the lener at ate 2. This traing arrangement can be compactly represente as (c 1, c 2 ). Note that since goo c is converte to goo a one-for-one, a 1 = c 1 ; an since goo a is urable, a 2 = a 1. Trae between a matche investor an traer at ate 3 is qui-pro-quo. The investor exchanges his enowment of goo a for the traer s enowment of goo c. The traing arrangement between investors an traers can be represente as (a 3, c 3 ), i.e., the investor gives up a 3 units of his enowment of goo a an receives c 3 units of the traers enowment of goo c. The terms of trae in a ate-1 match between a lener an borrower, (c 1, c 2 ), are etermine by bargaining. Given the contract offer (c 1, c 2 ), the lener s payoff is u (c 2 ) c 1 an the borrower s is a 2 c 2. Since, technology implies that c 1 = a 1 = a 2, the total payoff (or surplus) in a borrower-lener match is S BL = u (c 2 ) c 2. The borrower accepts contract (c 1, c 2 ) only if a 2 c 2, an the lener accepts only if u (c 2 ) c 1. For simplicity assume that the lener has all of the bargaining power, an makes a take-it-or-leave-it offer to the borrower. This bargaining protocol implies that the lener will choose will choose c 2 = a 2, i.e., the lener receives the entire match surplus. The lener will offer the contract (c 1, c 2 = c 1) to the borrower, where u (c 2) = 1 since this offer maximizes match surplus. The borrower will accept this offer. Let m ij represent the probability that agent i is matche with agent j at ate 1, an let m represent the measure of prouctive ate 1 matches. An example of a matching technology, which we will use later on, is a Leontief matching technology that take the form m = min { n L, n B}, m LB = m/n L an m BL = m/n B. For this matching technology, agents irect their search to a prouctive partner, an the short sie of the market etermines the number of matches. Note that leners have nothing to offer any potential traing partner at ate 3, an this is inepenent of being matche or not at ate 1. Therefore, the expecte payoff to a lener, which is measure before agents are matche at ate 1, is m LB [u (c 2) c 2]. The bargaining assumption implies that the expecte payoff to the borrower is zero. 4

5 Only investors an traers enter the ate 3 matching process (since leners have no strict incentive to o so). The terms of trae in an investor-traer match are etermine by bargaining, where we assume that the investor has all of the bargaining power. In an investor-traer match, the investor offers the terms of trae (a 3, c 3 ), i.e., the investor offers to give up a 3 units of his enowment for c 3 units of the traer s enowment. The investor s payoff is f (c 3 ) + ā a 3 (an his surplus is f (c 3 ) + (ā a 3 ) ā), an the traer s surplus is a 3 + ( c c 3 ) c (an his surplus is ( c c 3 ) + a 3 c). Hence the match surplus is S IT = f (c 3 ) c 3. The traer will accept the offer (a 3, c 3 ) only if a 3 c 3. Since the investor has all of the bargaining power, he will offer (a 3, c 3) to the traer, where a 3 = c 3 = min {ā, c} = ā, which implies that the match surplus is f (ā) ā. Let M ij represent the probability that agent i is matche with agent j at ate 3, an M represent the measure of ate-3 prouctive matches. For the Leontief matching function, M = min { n I, n } T, M IT = M/n I an M T I = M/n T. The expecte payoff to the investor is M IT (f (ā) ā) + ā, an the expecte payoff to the traer is c. 3 Let p a represent the value to an investor of having an aitional unit of goo a, measure in terms of goo c, at the beginning of ate 3 before matching takes place. Then p a = M IT f (ā) + ( 1 M IT ), i.e., the investor is inifferent between receiving p a units of goo c for sure, an receiving an aitional unit of goo a. 4 Finally, note that a planner, whose objective is to maximize total social surplus, cannot o any better than the equilibrium outcome. 5 Total social surplus S is S = m (u (c 2 ) c 1 ) + m (a 2 c 2 ) + M [( c c 3 ) + a 3 c] + M [(ā a 3 ) + f (c 3 ) ā] = m (u (c 2 ) c 2 ) + M (f (c 3 ) c 3 ), since c 1 = a 2. Assuming that the planner must respect agent participation constraints, total social surplus will be maximize at c 2 = c 2 an c 3 = c 3, 3 If ā c, then the investor will offer ( a 3, 3) c, where a 3 = c 3 = min {ā, c} = c. In this case the expecte payoff to the investor is M IT (f ( c) c) + ā, an the expecte payoff to the traer is c. 4 If ā > c, then the price of goo ā is 1 since if the investor is given an aitional unit goo a he will simply consume it. The average price of goo a, however, is IT f (ā) ( ) M + 1 M IT > 1. ā We woul argue that in this case, the average price is the relevant statistic when thinking about gains from trae. 5 The planner takes as given the matching techologies an the bargaining protocol of the agents. 5

6 where c 2 an c 3 correspon to the take-it-or-leave-it offers mae by the lener an investor, respectively; see above. The planner can implement this surplus as long as u (c 2 ) a 2 c 2 an f (c 3 ) a 3 c 3, i.e., agent participation constraints are satisfie. Although the planner can reistribute surplus between the borrower an investor by increasing a 2 from c 2 an between the investor an traer by increasing a 3 from c 3 compare to the equilibrium outcome, he cannot increase total surplus compare to the equilibrium outcome. 6 3 Borrower Default an Repo Contracts We exten the basic moel by introucing the possibility of exogenous efault by borrower. Default is moele by having the borrower probabilistically ie between ates 1 an 2. With probability δ a fraction of borrowers ie, an with probability 1 δ no one ies. 7 We will refer to the former outcome as the efault state, an the latter as the no-efault state. From a ate 1 perspective, the probability that a borrower ies is δ. We use two parameters to escribe efault so that we can moel a rare event, a small δ, such as a major financial meltown, a big. When a borrower ies, any goo that he hols, e.g., goo a, will be elivere to his creitor at ate 3. However, the goo is elivere after the ate-3 matching process is conclue. The lener will, therefore, want to rethink the mechanics of his traing relationship with a borrower. Although there is little the lener can o about a borrower s broken promise to supply goo c at ate 2 the borrower is ea after all the lener can prevent the borrower from holing goo a between ates 1 an 2 by entering into a repo-type contract. In a repo contract the lener prouces goo c at ate 1 an gives it to a borrower; the borrower then converts goo c into goo a, an gives goo a back to the lener to hol as collateral. The collateral goo a is then transferre back to the borrower at ate 2 if he prouces goo c for the lener. A repo contract partially protects the lener against a borrower efault: If the borrower ies, the lener owns the collateral, an he has the option to enter the perio 3 ranom matching process. A repo contract can be represente compactly as ( c 1, c 2 ). We let the tile enote the (optimal) ecisions mae by the lener. If the borrower oes not efault, then the lener receives c 2 units of goo c from the borrower, an the lener transfers the collateral, c 1 units of goo a, to the borrower. If the borrower efaults, then, at ate 2 the lener owns the repo collateral, which is c 1 units of goo a. Suppose a matche borrower ies. The lener can always consume the collateral at ate 2 or 3, an his payoff is γ c 1. If, however, the lener enters the 6 If ā > c,... 7 We can assume that with probability 1 δ, a finite number, i.e., a set of measure zero, of borrowers ie. This way there can be efaults even in goo times, but these efaults are essentially unimportant for the economy. This woul correspon to situations (in the real worl) where there are fails or efaults an these have no significant implications for asset prices or economic activity. 6

7 ate 3 matching process an is matche with a traer, then there are gains from trae, since the lener s valuation of goo a to goo c is γ an the traer s is 1, an the lener s payoff can be greater than γ c 1. 8 Terms of trae in a lenertraer match is represente by (ã 3, c 3 ), where the lener gives ã 3 units of goo a to the traer in exchange for c 3 units of goo c. The payoff to the lener associate with (ã 3, c 3 ) is γ (ã 2 ã 3 ) + c 3, where ã 2 represents the amount of goo a that the lener brings into the match, an the payoff to the traer is ã 3 + c c 3. Total surplus in a lener-traer match, S, is S = (1 γ) ã 3. Assume the lener has all of the bargaining power, an makes a take-it-orleave-it offer to the traer. The traer will accept the lener s offer if ã 3 c 3, an the take-it-or-leave-it assumption implies that ã 3 = c 3. Hence, the payoff to a matche lener with collateral equal to a 2 is { ã γ (ã 2 ã 3 ) + c 3 = 2 if ã 2 c γ (ã 2 c) + c if ã 2 > c, an the payoff to the traer is c. Since the lener s expecte payoff associate with entering the ate 3 matching process is strictly greater than γã 2 = γ c 1, he will always enter the ate 3 matching process when his borrower efaults. The repo contract ( c 1, c 2 ) that the lener offers the borrower in a ate 1 match is affecte by the possibility that his borrower efaults. Since the lener has all of the bargaining power in the ate 1 match, as above, it will be the case that c 1 = c 2 = ã 1 = ã 2. Denote the probability that the lener is matche with a traer in the event that his borrower ies as M an M an the number (measure) of matches between traers an either leners or investors at ate 3 in the efault state. For the Leontief matching technology, M = min { n I + m, n } T an M M = m + n I. For this matching technology, leners irect their search to traers, as o investors. Consier the following two maximization problems. This first problem escribes a situation where a efaulte lener s collateral which is c 1 1 units of goo a is less than the traer s enowment. In this situation the lener will be able to exchange all of his collateral for goo c at ate 3 if he is matche with a traer. The lener s ate-1 problem, which is to choose the amount of goo c to prouce at ate 1, enote c 1 1, is max c 1 c (1 δ ) u ( c 1 ) [ 1 + δ M c ( 1 M ) ] γc 1 1, (1) i.e., the lener prouces c 1 1 units of goo c 1, which the borrower converts into c 1 1 units of goo a. If the borrower oes not ie, then the lener consumes c 1 1 units 8 If the lener is matche with an investor, there are no gains from trae since both agents have goo a an his payoff will be γ c 1. 7

8 of goo c at ate 2. If the borrower ies, the lener enters the ate-3 matching process; he consumes c 1 1 units of goo c if he is matche an c 1 1 units of goo a if he is not. The secon problem escribes the situation where the lener s collateral which is c 2 1 units of goo a is greater than the traer s enowment. In this is situation, the lener will only be able to exchange part of his collateral for goo c at ate 3 if his is matche with a traer. The lener s ate-1 problem is to choose the amount of goo c to prouce at ate 1, enote c 2 1, max c 2 1 c 2 1+(1 δ ) u ( c 2 ) [ 1 +δ M ( γ ( c 2 1 c ) + c ) + ( 1 M ) γc 2 1 ] In both problems, if the borrower efaults, the lener enters the ate-3 matching process, an is matche with a traer with probability M. The first-orer conitions, with equality, for each of these problems are (1 δ ) u ( c 1 ) ( 1 + δ M + ( 1 M ) ) γ = 1 an (1 δ ) u ( c 2 1) + δ γ = 1, respectively. Note that the first-orer conition associate with problem (2) oes not epen on matching the ate-3 matching probability. Since γ < 1, note that c 1 1 > c 2 1. Note that if c 1 1 > c, then problem (1) oes not make economic sense since it assumes that the lener can trae c 1 1 units of goo a for c 1 1 units of goo c if he is matche with a traer at ate 3. But this outcome is impossible. Similarly, if c 2 1 < c, then problem (2) oes not make economic sense since it assumes that the lener can only trae a fraction of the c 2 1 units of goo a for goo c if he is matche with a traer at ate 3. But the lener will be able to trae all of his units of goo a. Hence, if, c 1 1 < c, then problem (1) is the solution to the lener s ate-1 contract offer to the borrower, i.e., c 1 = c 2 = c This solution to the problem is consistent with the lener traing all of the collateral for goo c at ate 3 if he is matche with a traer. An, if c 2 1 > c, then problem (2) is the solution to the lener s ate-1 contract offer to the borrower, i.e., c 1 = c 2 = c This solution is consistent with the lener only traing a fraction of the collateral for goo c. Finally, if c 1 1 > c an c 2 1 < c, then neither problem solves the lener s ate-1 contract offer. Problem (1) assumes that if the lener is matche with a traer he is able to exchange all of his holings of goo a for goo c; but this is inconsistent with c 1 1 > c. Problem (2) assumes that the lener is able to exchange only part of his holings of goo a for goo c; but this is inconsistent with c 2 1 < c. There is a iscrete ecrease in the marginal benefit associate with having an aitional unit of goo c at c = c; the expecte marginal benefit falls from δ [ M + ( 1 M ) γ ] to δ γ. Since 9 If c 1 1 < c, problem (2) is not a relevant economic problem since c2 1 < c. 10 If c 2 1 > c, then c1 1 > c, an problem (1) is not a relevant economic problem. (2) 8

9 (1 δ ) u ( c)+δ [ M + ( 1 M ) γ ] > 1 an (1 δ ) u ( c + ε)+δ γ < 1 for any ε > 0, the lener s ate-1 contract offer to the borrower is c 1 = c 2 = c when c 1 1 > c an c 2 1 < c. (Note that, inepenent of the problem that etermines the value of c 1, c 1 > c ) Suppose that the efault state is realize an, as a result, n B borrowers ie in between ates 1 an 2. Then, at ate 3, traers, investors an leners will enter the matching process. Denote the probability that an investor is matche with a traer in the efault state as M IT IT, where M = M. The terms of trae between a matche investor an traer is not a function of the matching probability M. Hence, the investor exchanges min {ā, c} = ā units of goo a for ā units of goo c with the traer. From the investor s ate 3 perspective, the price of goo a, measure before agents are matche at ate 3, p δ a, is p δ a = M IT f (ā) + ( 1 M IT It is important to emphasize that p δ a p a, since M IT M IT.12 When M IT > M IT, p a > p δ a, an the lower price in the efault state will be referre to as a fire sale of asset a. The value of asset a ecreases to investors because leners enter the ate-3 matching process to sell the collateral of their efaulting borrowers, which reuces the probability that they, the investors, are matche with traers. (In the no-efault state, an event that occurs with probability 1 δ, the price of asset a is p a.) There are real affects associate with the fire sale since the total amount of real investment falls, compare to the situation where borrowers o not efault. In the basic moel, where borrowers o not efault, a government is unable to increase total social surplus, compare to the equilibrium allocation. When borrowers can efault, however, a government may be able to increase total social surplus, compare to the equilibrium allocation, by mitigating negative effect that fire sale has on investment. For example, suppose that the government can limit the flow of leners that enter the ate-3 matching process. In practice, the government can limit the flow of leners by imposing a bankruptcy stay on the assets of a efaulting party; in this situation, leners who are subject to the bankruptcy say are essentially prohibite from ate-3 matching process, an, therefore, will consume their collateral holings at ate-3. One can interpret the situation where leners are able to, without restriction, enter the ate-3 matching process with the collateral of the efaulte borrower as the government exempting a bankruptcy stay on the assets of the efaulting party. We now investigate how government policies that limits the leners of efaulte borrowers participation in the ate 3 matching process can affect total social surplus. 11 This is important an shoul be state somewhere. 12 As above, M IT represents the probability that an investor is matche with a traer in the no-efault state. In the no-efault, leners o not enter the ate-3 matching process. ). 9

10 4 Government policy The government s policy instrument, θ, controls the flow of leners into the ate-3 matching process in the efault state. Let θ represent the fraction of leners who hol collateral at ate 3 that are allowe to participate in the ate- 3 matching process. One can interpret θ = 1 as an exemption from bankruptcy stay or a superpriority rule for repo collateral for all leners; θ = 0 as a bankruptcy stay on all repo collateral; an θ (0, 1) as a partial exemption from a bankruptcy stay in the sense that some leners, θ m of them, are exempt from a bankruptcy say an others, (1 θ) m of them, are not. If a lener is subject to a bankruptcy stay he is unable to participate in the ate-3 matching process, an consumes his collateral holings at ate-3. Government policy instrument, θ, may affect the payoffs an behavior of the various agents in the economy. The expecte payoff to a borrower, W B, is W B = m BL (1 δ ) (ã 1 c 2 ), where m BL is the probability that a borrower is matche with a lener at ate 1. Since the lener has all of the bargaining power, c 2 = ã 1, which implies that W B = 0. Hence, the payoff to an behavior of the borrower are unaffecte by government policy θ. The expecte payoff to a traer, W T, is W T = (1 δ) [ M T I (â 3 ĉ 3 + c) + ( 1 M T I) c ] + δ [ M T I (â 3 ĉ 3 + c) + M T L (ã 3 c 3 + c) + ( 1 M T I ] (â3 ĉ 3 ) + δm T L (ã 3 c 3 ) + c, = [ (1 δ) M T I + δm T I M T L ) ] c where the hat over the a 3 an c 3 represents (optimal) offers mae by the investor to the traer, M T L = M θm n T θm + n I an M T I = M n T n I θm + n I. Since investors an leners have all of the bargaining power in their matches with traers, â 3 = ĉ 3 an ã 3 = c 3, which implies that W T = c. Hence, the payoff to an the behavior of the traer is unaffecte by government policy θ. In fact, ĉ 3 = c 3 = min { c, ā} = ā, which is the trae allocation in a traer-investor match in a worl without efault. The payoff to the investor, W I, is W I = (1 δ) [ M IT (f (ĉ 3 ) â 3 + ā) + ( 1 M IT ) ā ] + δ [ M IT (f (ĉ 3 ) â 3 + ā) + ( 1 M IT ) ā] = [ (1 δ) M IT + δm IT ] (f (ĉ3 ) â 3 ) + ā. 10

11 Although the behavior of the investor is unaffecte by government policy since â 3 = ĉ 3 = c 3 = min {ā, c} = ā his payoff is affecte since the matching probability M IT is a function of θ. Finally, the payoff to the lener, W L, W L = m LB { ã 1 + (1 δ ) u ( c 2 ) + δ[ θ(m ( c 3 + γã 2 γã 3 ) + ( ) 1 M γã2 )] + (1 θ) γã 2 }. Clearly, government policy may affect both payoff to an behavior of the lener. In orer to evaluate the best policy, the government must unerstan how the behavior of a lener which is simply his choice of c 1 is influence by changes in θ. As in section 3, consier two maximization problems, which generalize problems (1) an (2) to take account of government policy, θ. The problems are, max c 1 c (1 δ ) u ( c an ) + δ { θ [ M c ( 1 M ) ] } γc (1 θ) γc 1 1 (G1) max c 2 1 c (1 δ ) u ( c 2 ) ( ( 1 + δ {[M γ c 2 1 c ) + c ) (G2) + ( 1 M ) γc 2 1 ] + (1 θ) γc 2 1}. The first-orer conitions for these problems are, (1 δ ) u ( c 1 ) ( ) 1 + δ γ + (1 γ) θm = 1 (3) an (1 δ ) u ( c 2 1) + δ γ = 1, (4) respectively. As in section 3, c 1 1 > c 2 1. Qualitatively speaking, the characterization of the lener s behavior regaring his choice of c 1 is ientical to that in section 3. In particular, if c 1 1 < c, then the lener s choice of his ate-1 prouction of goo c, c 1, is given by the solution to (3); if c 2 1 > c, then c 1 = c 2 1, where c 2 1 is given by the solution to (4); an if c 1 1 > c an c 2 1 < c, then c 1 = c. It shoul be note that, compare to a worl with no efault, ate-1 prouction by the lener, c 1, is lower in a worl with efault, i.e., c 1 < c 1. Proposition 1 c 1 is weakly increasing in θ. Proof. If c 1 1 < c, then c 1 = c 1 1. From (3), we have ( ) c 1 θ = c1 1 (1 γ) θm / θ = δ θ (1 δ ) u (c 1 1 ). (5) Since θm = { θ if θm + n I < n T θn T θm+n I if θm + n I > n T, 11

12 an ( ) { θm = θ 1 if θm + n I < n T n I n T ( θm+n I ) 2 if θm + n I > n T, (6) we get that c 1 1/ θ > 0. If c 2 1 > c, then c 1 = c 2 1 an, from (4), c 1 / θ = c 1 1/ θ = 0. Finally, if c 1 1 > c an c 2 1 < c, then c 1 = c an c 1 / θ = 0. The intuition behin this proposition is straightforwar. Having access to the ate-3 matching process is valuable for the lener. Since, for any given θ, the lener s ate-1 prouction of goo c is less than the effi cient amount, c 1, the lener may increase his ate-1 prouction if he is given better access to the ate-3 matching process. The lener will increase his ate-1 prouction if the current prouction level is less than c. In this situation, the lener will value the increase prouction one-for-one if he is matche with a traer at ate 3. If current prouction is greater than or equal to c, the lener will not increase ate-1 prouction with increase access to the ate-3 matching process since an aitional unit prouction will only be value at γ < 1 at ate 3, even the if lener is matche with a traer. The government seeks to maximize total social surplus, S, which is given by S = n B W B + n L W L + n I (W I ā) + n T (W T c). Given the assume bargaining conventions, the expression for total social surplus can be simplifie to S = n L W L + n I (W I ā), which means we only nee to focus on the behavior of an payoffs to leners an investors. Since the surplus to investors is W I ā = [ (1 δ) M IT + δm IT ] (f (ā) ā), (7) where ĉ 3 = c 3 = min {ā, c} = ā, the government policy θ affects the investor s surplus only through the matching probability, M IT. Proposition 2 The investor s payoff is weakly ecreasing in θ. Proof. Note that M IT θ = { 0 if θm + n I n T mnt ( θm+n I ) 2 if θm + n I < n T an, since M IT = min { n I, n } T /n I, M IT / θ = 0. Therefore, W I / θ = δ ( M IT / θ) (f (ā) ā) or { W I θ = 0 if θm + n I n T mn δ T (f (ā) ā) if θm + n I < n T. (8) ( θm+n I ) 2 12

13 This proposition accors with intuition. If the number (measure) of traers is relatively large in the sense that θm + n I > n T then increasing access to the ate-3 matching process for leners has no effect on the investors surpluses since investors are matche with probability one at ate 3. If, however, the number of traers is not relatively large in the sense that θm + n I < n T then increasing access to the ate-3 matching process to leners will reuce the probability that investors are matche with traers an, hence, reuces the payoffs to leners. Turning to leners, since, c 1 = c 2 = ã 2 = ã 1 an ã 3 = min { c 1, c}, the surplus function for a lener can be simplifie to W L = m LB { c 1 + (1 δ ) u ( c 1 ) + δ γ c 1 + δ θm ã 3 (1 γ) }. (9) To assess the its policy, the government must unerstan how W L is affecte by a change in θ. Proposition 3 The lener s payoff is strictly increasing in θ. Proof. The erivative of (9) with respect to θ is W L θ { = m LB c1 θ [ 1 + δ γ + (1 δ ) u ( c 1 )] + m LB δ (1 γ) ã } 3 θm + θ { ( ) } θm m LB δ (1 γ) ã 3 + (10) θ The first line of (10) is equal to zero. When c 1 < c, this is implie by (3), recognizing that c 1 / θ = ã 3 / θ. When c 1 c, (4) implies that c 1 / θ = ã 3 / θ = 0. Therefore W L θ = m LB δ (1 γ) = m LB δ (1 γ) > 0 { ( ) θm θ { ã 3 } ã 3 if θm + n I < n T n I n T ã ( θm+n I ) 2 3 if θm + n I > n T The intuition behin proposition 2 is straightforwar. Holing c 1 constant, an increase in θ increases the chances that the lener will be able to participate in the ate-3 matching process. This unambiguously increases the surplus of the lener because in the event of a borrower efault, the value of either part or all of the lener s collateral a increases from γa to a. As well, if c 1 < c, then, holing the ate-3 matching probability constant, an increase in θ optimally increases c 1 an, by construction, the lener s collateral holings. Since it is optimal to increase c 1, the lener s surplus must also increase. Propositions 2 an 3 ientify the trae-off that the government faces. An increase in θ (weakly) lowers the probability that an investor will be matche 13

14 with a traer an, hence, (weakly) lowers investment. But an increase in θ strictly increases the probability that a lener will be matche with a traer, in the event of a borrower efault, an this enhances the liquiity of a lener s collateral. To assess a government policy that changes the value of θ, one simply has to compare the investment effect with the liquiity effect. Generally speaking, the net effect can go either way, since the magnitues of the effects epen upon moel parameters. Consier first the situation where m + n I < n T. One can interpret this as a liqui ate-3 market for both investors an leners since they are always matche with probability one. It shoul be obvious that the optimal government policy in this situation is θ = 1, i.e., there shoul be an exemption for a bankruptcy stay for all leners, since allowing efaulte leners to participate in the ate-3 matching process oes not isplace any investment activity. To see this formally, note that, using (8) an (10), W θ = W nl L θ + W ni I θ = mδ (1 γ) ã 3 > 0, i.e., the government shoul choose θ as high as possible. Consier now the interesting case where the ate-3 market is quite illiqui from the investor s perspective in that n I > n T. For this case, again using (8) an (10), W θ = 1 ( θm + n I ) 2 [(1 γ) ã 3 (θ) (f (ā) ā)]. (11) It turns out that optimal government policy can be etermine by comparing the value of f (ā) ā with that of (1 γ) ã 3 (θ) for various values of θ. For example, Proposition 4 Suppose n I > n T. If (1 γ) ã 3 (0) > (f (ā) ā), then the optimal government policy provies an exemption from a bankruptcy stay of all leners, i.e., θ = 1. If (1 γ) ã 3 (1) < (f (ā) ā), then is the optimal government policy requires a bankruptcy stay for all leners, i.e., θ = 0. Proof. Since ã 3 = min { c 1, c}, Proposition 1 implies that ã 3 / θ 0. If (1 γ) ã 3 (0) > (f (ā) ā), then from (11) W/ θ > 0 for all θ [0, 1], an setting θ = 1 is optimal. If (1 γ) ã 3 (1) < (f (ā) ā), then from (11) W/ θ < 0 for all θ [0, 1], an setting θ = 0 is optimal. This proposition is quite interesting. Even though the ate-3 market is quite illiqui from the perspective of investors even if leners are not permitte to participate it may be optimal for the government to exempt efaulte leners 14

15 from a bankruptcy stay. This may happen when, intuitively speaking, the liquiity value of allowing leners to have access to traers is greater than the investment value associate with investor-traer matches. It is true that when θ = 1, leners will isplace economy-wie investment when n I > n T. However, the value of the liquiity, (1 γ) ã 3, generate by leners excees that of the isplace investment. The next proposition fully characterizes optimal government policy when n I > n T. Proposition 5 When n I > n T, an optimal government policy either provies an exemption from a bankruptcy stay for all leners, θ = 1, or imposes a bankruptcy stay on all leners, θ = 0. Proof. When W (θ) is strictly monotonic in θ, the optimal θ is either θ = 0 or θ = 1, see Proposition 4. Suppose that W (θ) is weakly monotonic in θ. There are two cases to consier. If (1 γ) ã 3 (0) = (f (ā) ā) an ( ã 3 (θ) / θ) θ=0 = 0, then θ = 1 an θ = 0 are optimal policies, (an, in fact, any θ [0, 1] is an optimal policy). If (1 γ) ] ã 3 (0) < (f (ā) ā), ( ã 3 (θ) / θ) θ=0 > 0, an ã 3 (θ) / θ = 0 for all θ [ˆθ, 1, ˆθ > 0, then the unique optimal policy is θ = 0. Suppose that W (θ) is not monotonic in θ. ) W (θ) is not monotonic only if: (i) (1 γ) ã 3 (0) < (f (ā) ā); (ii) (1 γ) ã 3 (ˆθ = (f (ā) ā), for some ˆθ < 1; an (iii) ã 3 (θ) = c 1 (θ) < c at θ = ˆθ. In this situation, a global maximum of W (θ) will occur at either θ = 0 or θ = 1 (or both). That is, a global maximum cannot occur at θ (0, 1). The optimal government policy is unique if an only if W (0) W (1). When n I > n T, the optimal government policy is either a bankruptcy stay for all leners or an exception from a bankruptcy stay for all leners. Although a partial exemption, i.e., θ (0, 1), is permitte, it is never optimal, except for the knife ege case where W (θ) = 0 for all θ [0, 1]. But even in this knifeege case, θ = 1 or θ = 0 is an optimal policy. In spite of the relative illiquiity of the ate-3 market, i.e., n I > n T, an exemption from a bankruptcy stay for all leners is optimal when the liquiity value associate with proviing access to the ate 3 market for leners is greater than the isplacement of real investment opportunities. The final case in terms of ate-3 market liquiity to consier is when n I < n T an n L + n I > n T. Here, there exists a θ, say θ, such that θ n L + n I = n T. Clearly, it woul never be optimal for the government to choose a θ < θ. Optimal government policy here somewhat mirrors the case where n I > n T, except now the lower boun of optimal government policy is θ instea of θ = 0. In particular Proposition 6 When n I < n T an n L + n I > n T an optimal government policy either provies an exemption from a bankruptcy stay for all leners, θ = 1, or imposes a bankruptcy stay on fraction 1 θ of leners. Proof. The proof follows those of Propositions 4 an 5. 15

16 5 Final Remarks To be complete at a later ate. 6 (Some) References Acharya, Viral V., V. Ravi Anshuman, an S. Viswanathan Bankruptcy Exemption of Repo Markets: Too Much Toay for Too Little Tomorrow? Manuscript. Bolton, Patrick an Martin Oehmke Shoul Derivatives be Privilege in Bankruptcy? NBER working paper Duffi e, Darell, an Davi Skeel A Dialogue on the Costs an Benefits of Automatic Stays for Derivatives an Repurchase Agreements. Rock Center for Corporate Governance at Stanfor University Working Paper No McAnrews, James J., an William Robers Payment Intermeiation an the Origins of Banking Manuscript. Roe, Mark The Derivatives Players Payment Priorities as Financial Crisis Accelerator. S tanfor Law Review