A Simple Model of Bankruptcy in General Equilibrium

Transcription

1 A Simple Model of Bankruptcy in General Equilibrium Mattew Hoelle University of ennsylvania Department of Economics 160 McNeil Building, 3718 Locust Walk, iladelpia, A May 23, 2009 Abstract In tis paper, I introduce a simple model of ouseold bankruptcy into a dynamic general equilibrium model. Te basic idea is tat ouseolds can trade in assets in te initial state and ten coose not to ful ll teir portfolio commitments in future states. Wen tis occurs, te returns to creditors must be diluted in order for markets to clear. Under tis formulation, I prove tat a bankruptcy equilibrium exists and even sow ow te result generalizes to a model wit real costs of bankruptcy. I also study ow a production model can be analyzed using te exact same framework were te production model is suc tat (i) rms nance production using te bond market and (ii) bot rms and ouseolds can declare bankruptcy. 1 Introduction In te development and study of general nancial models, two fundamental features ave been commitment and perfect foresigt. Wit tese two foundations, ouseolds are modeled as making all economic decisions for te entire nite lengt of te model at te initial date-event. At tis node, ouseolds not only directly trade in te commodity and asset markets currently open, but also make contingent transactions in all future commodity and asset markets. Wile tis simple framework does wonders wen coming to grips wit nancial markets, te reality is tat no mecanism exists to freely and perfectly enforce commitment. Te autor wises to tank is advisor, rof. Felix Kübler at te Swiss Banking Institute, University of üric. Te autor furter acknowledges te support received wile visiting te Swiss Banking Institute. Additionally, valuable comments from seminar participants in üric (ISB) and Toulouse (TSE) and conference participants at te 5t Annual CARESS Cowles Conference were implemented trougout tis work. Comments are welcome at moelle@econ.upenn.edu. 1

2 In tis work, I relax te assumption tat ouseolds must ful ll teir commitments in te nancial markets. Simply put, ouseolds are permitted to sort sell assets in te initial date-event, reap te bene t (increase in wealt) at tat time, and ten decide not to repay te debt owed at a future date-event. A similar idea was developed by Dubey, Geanakoplos, and Subik (2005) and termed "default". In teir paper, ouseolds own a portfolio of assets. At te time wen te debt from olding a sort position on an individual asset comes due, te ouseold can coose wat fraction (if any) of tis debt to repay. Te cost of less tan complete repayment is utility loss, wit greater loss for a greater amount of debt left outstanding. My model of bankruptcy di ers noticeably from Dubey, Geanakoplos, and Subik (2005). As above, ouseolds own a portfolio of assets. At any future date-event, ouseolds identify te value of teir entire portfolio. If te value is su ciently negative, ouseolds ave a binary decision: eiter repay te full amount owed or default on te entire portfolio. Te idea of defaulting on an entire portfolio and not making good on any commitments is wat I term "bankruptcy". Declaring bankruptcy is not costless of course, but unlike Dubey et. al., te cost is a nancial one a ecting te budget set, rater tan canging te ouseold preferences. Concerning bankruptcy and not default, te two seminal works are Sabarwal (2003) and Araujo and ascoa (2002). In bot setups, a ouseold declaring bankruptcy will forfeit some parameterized sare of its endowments. Wit investment constraints, tis con scation of endowments or wage garnisment can be any fraction independent of te size of bankruptcy debt. Te idea of bankruptcy exemptions resembling tose tat are exaustively detailed in te bankruptcy code is taken up by Sabarwal. In is setup, te exempt items tat a ouseold maintains possession of after bankruptcy are endowments. In my setup, te exempt items are assets meaning tat a bankrupt ouseold may still collect te returns from te previous purcase of exempt assets. I believe tat tis distinction is merely a question of ow one interprets te mytical endowments and assets of nancial models and does not take away from eiter result. Wile investment constraints are certainly observed in reality, a bankruptcy model is best served by not including suc exogenous constraints. Witout a clear understanding of ow te constraints are set in reality, te resulting equilibria in a model wit suc constraints will be arbitrary. Furter, te optimal ouseold portfolio decisions may be more a product of te exogenous constraints rater tan te endogenous decisions made in te face of bankruptcy. Wit te focus entirely on te bankruptcy mecanism, I seek to endogenize te constraints placed on ouseold portfolio decisions. Tis task was rst undertaken in te second model introduced in Araujo and ascoa (2002). Te two most direct metods to bound assets in a bankruptcy model are to eiter (i) ave te bankruptcy cost be some xed proportion of te total bankruptcy debt or (ii) ave te bankruptcy cost be wage garnisments tat approac complete garnisment as te debt becomes unbounded. Araujo and ascoa took te second approac. Implicitly, tis requires tat te cost 2

3 of bankruptcy depends on a ouseold s debt at te time of bankruptcy. Tis does not t te US legal framework, wic I will now discuss. Accounting for over 70% of te individual bankruptcies, 1 capter 7 is te most common form of bankruptcy and also te most interesting to model. Brie y, capter 7 allows a bankrupt ouseold to completely discarge nancial debt. 2 Tus, te cost of capter 7 bankruptcy is not tied at all to te bankruptcy debt. Capter 13 bankruptcy composes all but te remaining 0:1% of individual bankruptcies. 3 Capter 13 involves a 3 5 year repayment plan between te debtor ouseold and its creditors. 4 Suc a dynamic cannot be modeled in tis 2 period model and even if it could, a ouseold is still responsible (eventually) for its debt under capter 13. Tus, te potential for abuse of capter 7 privileges presents a more interesting policy to model. Te obvious abuse is tat ouseolds wo anticipate a negative return for teir portfolio consisting of bot long positions and sort positions would seek to increase teir sort positions to increase consumption in te initial state. Suc a ouseold would attempt to borrow as muc as possible knowing tat te debt would be erased in te following period. A recent 2005 clari cation of te bankruptcy law sougt to reduce te number of abuses. It states tat any ouseold applying for bankruptcy must pass a "means test". Simply put, if a ouseold s average income over te previous 6 monts is above te state median, ten (unless a large number of statutorily allowed expenses can be deducted) te ouseold s capter 7 claims are transferred to capter In tis instance, a ouseold would not be able to discarge te entire debt and would be subject to te 5 year repayment plan as regulated by capter 13. Terefore, it appears obvious tat if a ouseold will fail te "means test" in some state next period, ten te ouseold will not run up debt beyond its means to repay in tat state. Catterjee et. al. (2007) sow tat in a properly calibrated macroeconomic model, tis policy is welfare improving. Returning to te work of Araujo and ascoa (2002), existence is only guaranteed witout investment constraints wen te asset payouts for creditors are strictly concave functions of te assets. Suc an assumption is clearly required in te bankruptcy model as te assets may not be linearly independent for bankrupt ouseolds. Tis allows ouseolds te possibility to purcase an unbounded amount of one asset, sell an unbounded amount of a second asset, and still collect bounded nancial payouts. Te problem wit tis assumption is tat it contradicts te US bankruptcy code. Te code reads tat creditors wit a claim in a bankruptcy case are divided into 6 classes. 6 Repayment for a lower class will only occur once te classes above ave been repaid in full. Tus, trougout tis work, I will assume tat all creditors belong to te same 1 ttp://en.wikipedia.org/wiki/capter_7_bankruptcy 2 11 U.S.C. 7 (moderate court fees apply) 3 ttp://en.wikipedia.org/wiki/capter_7_bankruptcy 4 11 U.S.C. 1322(d) 5 11 U.S.C. 707(b)(1) and te Bankruptcy Abuse revention and Consumer rotection Act of U.S.C

4 class (te model is speci ed wit only ouseolds and witout government or nancial institutions) and are repaid equally. Tus, in eac state of uncertainty, tere are two classes of ouseolds (and two subclasses of te second class) and eac as its own linear asset payout. A bankrupt ouseold as a nancial payout tat is a negative fraction of its asset purcases only. Tis bankruptcy cost is independent of a ouseold s debt at te time of bankruptcy, but is rater only a function of te relative makeup of a ouseold s portfolio. Witin te class of solvent ouseolds, eac ouseold wit an asset sale must ful ll its commitment entirely and repay te entire parameterized payout. For asset purcases, all solvent ouseolds are treated equally and receive some positive fraction of te original asset payout. Tis fraction is endogenously determined based on te number of bankruptcy declarations trougout te economy. Tis idea of proportional repayment to creditors captures te idea of large, anonymous asset pools tat are used by Dubey et. al. and appear to be te best way to model lack of commitment in a general equilibrium framework. An additional feature of te model is te possibility for a cain reaction of bankruptcy as in Dubey et. al. Te principle is tat a bankruptcy declaration by one ouseold will dilute te payouts to all nonbankrupt ouseolds olding long positions in assets. As tese payouts are diluted, te total value of ouseolds portfolios will decrease and tis may cause additional ouseolds (ouseolds tat would ave been solvent ad tey received teir committed - nancial payout) to declare bankruptcy. Tis rst e ect is identical to te cain reaction of default and an example is discussed in Dubey et. al. In tis work, I provide an example of my own since a second e ect exists tat may extend te cain reaction. Wen payouts are diluted, te seized assets of bankrupt ouseolds ave less value. Tus, fewer total funds are returned to te creditors and tis depresses teir payouts even furter. Te bankruptcy code allows for certain possessions (assets) of a bankrupt ouseold to be exempt from liquidation (depending on te state in wic te ouseold resides). Te most common (and controversial) exemption is a ouse and te exemption is termed te "omestead exemption". Wile it is clear tat exempting more items will increase te incentive to declare bankruptcy by reducing te cost, te welfare e ects of suc a policy are unclear. To model tis realistic feature of bankruptcy, I only require tat te size of te exemption is bounded above. Wit tis requirement, te existence of bankruptcy equilibria is guaranteed and te model can be used to evaluate exemption policies. Witout an upper bound on exemptions, a ouseold would be able to sell an unbounded number of assets, use tese funds to purcase te exempt items, and ten declare bankruptcy and walk away wit an arbitrage pro t (te value of te exempt items). Suppose tat bankruptcy imposes real costs on te economy. Suc costs would be te resources lost in te process wereby te bankruptcy court liquidates bankrupt ouseolds assets and ten distributes tese funds to te creditors. Toug I do not ave a clear idea about ow large tese real costs may be, I do want te model to be able to quantify te bankruptcy e ects over 4

5 a range of possible real costs. I dedicate a brief section to explaining ow te real costs are incorporated and ow te proof of existence generalizes. As it turns out, te only signi cant di erence (and an interesting implication) is tat asset payouts are diluted more wen real costs are present. Te addition of rms to a pure excange model is often of secondary importance in general equilibrium. Tis is because te primitives of rms can be appended to a pure excange model in suc a manner as to leave all te results virtually intact. Wile production adds an element of "reality" to te model, teorists in general equilibrium understand tat te framework for adding production as been well-establised and te pursuit is only wortwile if an interesting question about production as been posed. In my situation, te questions of rm bankruptcy and rm nancing are interesting enoug to warrant formal analysis. I am interested in te e ects of rm bankruptcy on rm production and ouseold consumption. I introduce a production model in wic bot rms and ouseolds can declare bankruptcy and rms use te nancial markets to nance production investment. As sareolders in tese rms, ouseolds play a key role in tat teir preferences determine te rms optimizing decisions, wic ten directly a ect te ouseolds nancial decisions (troug te equity contracts tat te rms issue). I am able to view te production model wit rm bankruptcy as a simple extension of te pure excange model wit ouseold bankruptcy. I prove existence of a rm bankruptcy equilibrium and discuss some of te equilibrium implications. Tis paper is organized into ve remaining sections and an appendix. Section 2 introduces te model and proves te existence of a bankruptcy equilibrium, te fundamental concept in tis paper. Section 3 discusses ow to incorporate bankruptcy exemptions into te setup. Section 4 generalizes te model to include real costs of bankruptcy. Section 5 introduces rm production and proves te existence of a rm bankruptcy equilibrium. Section 6 concludes. Te appendix contains te proofs of te key results. 2 Te Model I consider a 2-period general equilibrium model wit uncertainty. Te nancial instruments are numeraire assets in unit net supply. Let tere be S states of uncertainty in te second time period and denote te rst time period as state s = 0, so tat te states belong to te nite set s 2 S = f0; :::; Sg: In eac state, tere are L pysical commodities and te commodity l = L will be te numeraire (meaning tat all oter commodities in tat state are priced relative to l = L). I will denote commodities wit a subscript and states in parenteses. Let te set of ouseolds be denoted by 2 H were H~[0; 1] and R 2H d = 1: I will denote ouseold variables and parameters wit a superscript. De ne te Borel-measurable function (A) : H![0; 1] as (A) = R 2AH d: 5

6 Tere are a nite number of distinct types of ouseolds denoted by f 2 F = f1; :::; F g: De ne H f = f 2 H : is of type fg : I explicity assume (toug te assumption is witout loss of generality) tat (H f ) = 1 F 8f: Furter, to be of te same type, I mean tat 8; 0 2 H f for some f; ten X = X 0 = X f (consumption set), u () = u 0 () = u f () (utility function), and e ; z (0) = e 0 ; z 0 (0) = e f ; z f (0) (initial endowments). Tese ouseold primitives will be introduced sortly. Trougout tis paper, I will refer to bot ouseolds and types of ouseolds as simply ouseolds. No confusion sould arise, but ouseold will be linked wit te equilibrium variables (since ouseolds of te same type may make di erent optimizing decisions), wile types will be linked wit parameters (as all ouseolds of te same type ave identical parameters). De ne G = L(S 1) and ten denote ouseold consumption as x 2 R G. Concerning notation, x (s) 2 R L is te vector of consumption by ouseold of all commodities in state s; and x l (s) 2 R is te scalar denoting te consumption by ouseold of good (s; l), or te l t pysical commodity in state s: Te primitives for te ouseolds are te consumption set X f, te utility function u f : X f! R, and te endowments of bot pysical commodities e f 2 R G and initial assets z f (0) 2 R J : To caracterize and compute equilibria, I assume tat te model satis es standard smoot assumptions: A.1 X f = R G : A.2 u f is C 2 ; di erentiably stricting increasing (i.e., 8x f 2 X f ; Du f (x f ) >> 0); di erentiably strictly quasi-concave (i.e., 8x f 2 X f s.t. Du f (x f ) f = 0 and f 6= 0, ten T f D2 u f (x f ) f < 0); and satis es te boundary condition (i.e., clu f (x f ) X f were U f (x f ) = fx 0 2 X f : u(x 0 ) u(x f )g): A.3 e f >> 0: Let tere by J S independent assets wic are traded at state s = 0 and return strictly positive payouts r j (s) in states s > 0: As a simpli cation of a more general asset structure wit stocks (real assets) and bonds (eiter nominal or numeraire assets), I limit te analysis to only numeraire assets, but numeraire assets in unit net supply. Wit unit net supply, I can endow all ouseolds wit a strictly positive initial asset vector so tat no ouseold enters te model close to a bankruptcy position. I collect te asset payouts into te S J yields matrix 2 Y = 4 r 3 1(1) ::: r J (1) : : 5 r 1 (S) ::: r J (S) wit te payouts in terms of te real pysical commodity l = L: Denote te portfolio of ouseold assets as z 2 R J were zj 2 R is te scalar denoting te amount of asset j eld by ouseold : Te assets are in unit net supply, 6

7 so I must endow all ouseolds wit te initial assets z f (0) 2 R J : te assumptions placed on te parameters Y and z f (0) : I specify A.4 Y is a strictly positive matrix wit full column rank and is in general position. 7 A.5 z f (0) > 0 and 1 F f2f zf j (0) = 1 8j:8 Let te equilibrium commodity prices be denoted by p 2 R G nf0g: Under (A:2), ten p >> 0: Since commodity l = L is te numeraire, I normalize p L (s) = 1 8s 0: Te assets pay out in tis numeraire commodity, so Y as real units. To consider te nominal value of te asset payouts, I will make te identity transformation 2 p L (1) r 1 (1) ::: 3 r J (1) Y = 4 0 ::: : : 5 : 0 0 p L (S) r 1 (S) ::: r J (S) Tus, I view te yields matrix Y as a matrix wit payouts in terms of te unit of account. Let te equilibrium asset prices be denoted by q 2 R J : Te asset prices can be tougt of as te nominal returns of te assets in state s = 0: De ne te (S 1) J returns matrix as q R = ; Y were R (as wit Y ) pays out in te unit of account. Te yields matrix Y is te exogenous parameter, but its elements will only be te asset returns in states s > 0 if tere is no bankruptcy (reduction to te standard GEI model). Oterwise (as I will discuss sortly), te asset payouts will be di erent depending on if a ouseold does or does not declare bankruptcy and does or does not old a long position in tat particular asset. Tis paper will only consider te e ects of bankruptcy under capter 7 for two important reasons. First, capter 7 makes up more tan 70% of te individual bankruptcy cases. 9 Second, capter 13 bankruptcy requires a 3 5 year repayment plan, a dynamic tat cannot be modeled in 2 periods. Narrowing te focus, I consider only te unsecured credit market. Unsecured credit is tat wic is obtained wit no collateral riding on a ouseold s decision to declare bankruptcy or not. Eiter credit market (secured or unsecured) is certainly worty of study, but te secured credit market leaves open a very important question: were do te collateral requirements come from? Certainly, te ability of ouseolds to default on credit would make lenders want 7 By general position, I mean tat any J rows of Y will be linearly independent. 8 Te reason I use assets in unit net supply is tat assumption (A:5) endows te ouseolds wit assets suc tat it is not close to a bankruptcy position initially. A more realistic model is contained in section 5 were bot bonds and stocks exist, wit bonds being te only asset tat will be diluted as a result of bankruptcy. 9 ttp://en.wikipedia.org/wiki/capter_7_bankruptcy 7

8 to impose a collateral aquisition plan tat is contingent on default/bankruptcy. However, suc collateral requirements would seem to be an endogenous feature of te model. Tus, I study only te unsecured credit market (tis better ts te anonymous interaction paradigm of general nancial markets anyway). Wit tis bankruptcy problem, tere are two fundamental problems tat need to be overcome to guarantee existence: boundedness and convexity. As stressed above, I do not arbitrarily restrict ouseold decisions wit constraints. Terefore, te bankruptcy decision must be incorporated in my model in suc a way so tat te asset coices of ouseolds are endogenously bounded. Concerning convexity, it is clear tat te budget set written in terms of te single asset variable z 2 R J is not convex. For some given state s > 0; te payout function for a bankrupt ouseold is concave and te payout function for a solvent ouseold is concave, but over its entire domain (and allowing a ouseold to be eiter bankrupt or solvent depending on market conditions), te payout function is not concave. Tis is akin to te problem contained in te Dubey et. al. model wit default. In tat case, te tresold between being a debtor and not being a debtor was exactly set at zero asset oldings. Tus, by de ning 2 R J as asset purcases and 2 R J as asset sales, ten it is unambiguous (and state independent) tat a debtor is a ouseold wit > 0: By de ning two asset variables, Dubey et. al. are able to maintain convexity. 10 In my model, te tresold between being bankrupt and solvent depends not only on all te assets in te portfolio, but also on te state-speci c asset payouts. Terefore, in order to employ te Dubey et. al. sceme to convexify te budget set, I would need to introduce an additional S asset variables tat would denote te portfolio eld by a bankrupt ouseold in eac state s > 0: For obvious reasons, I coose not to do tis. Moreover, te bankruptcy problem in reality is a nonconvex one; eiter a ouseold is or is not bankrupt. In order to obtain existence, I will need a continuum of eac of te nite type of ouseolds f 2 F (te measure of eac type is identical). 11 Wit te continuum, ten te weigted budget set for eac type f 2 F will be convex. Furter, wile te demand correspondence for eac individual ouseold will be uc, but not convex-valued, te demand correspondence (wit te continuum of ouseolds of te same type) for eac type f 2 F will be convex-valued. 12 I will now discuss ow capter 7 bankruptcy is introduced into te model. 10 By de ning an additional asset variable, Dubey et. al. may ave doubled te number of independent assets. Consider te statement from ame (93): "by separating purcases from sales, I ave allowed for te possibility tat agents go long and sort in (i.e. buy and sell) te same security. I ave implicitly contemplated tis possibility in te [GEI] model, but wen default is not possible, suc an action is irrelevant. However, wen default is possible, suc an action may not be irrelevant; it may bene t a trader to go long and sort in te same security if e does not intend to meet all is obligations." 11 Tis was rst recognized in te Araujo and ascoa (2002) paper on bankruptcy. 12 Toug not explicitly speci ed, te assumption of perfect competition is only truly meaningful in a model wit nite ouseolds if it is actually te di erent types of ouseolds tat are nite, wile te number of ouseolds belongs to a continuum. 8

9 In te United States, under capter 7 bankruptcy law, a bankrupt ouseold is completely discarged from te debt (no future wage garnisments). 13 Tis does not capture te entire e ect of bankruptcy owever. First, unmeasurable psycological costs seem to be important. 14 Second, te law states tat ouseolds cannot declare bankruptcy again for 6 years after te beginning of a capter 13 case and 8 years after a capter 7 discarge. Tird, te bankruptcy court may retroactively void asset transactions if it suspects tat assets are being idden. In a simple 2-period model, it is di cult to account for te psycological costs and te 6 year or 8 year e ect. Tus, I ignore tese two factors. In several paragraps, I will discuss te tird factor at lengt. In terms of assets, te discarge works as follows. A bankrupt ouseold is in debt on its nancial assets (a necessary condition) and cooses to forfeit its portfolio of assets. Tus, te ouseold loses te entire positive value of te long positions it may old on any of its assets, but also is not responsible for any debt incurred wit its sort positions. Te assets of bankrupt ouseolds wit positive value are con scated, liquidated, and ten tese funds are divided among te vector of asset pools. Wile tere are in nite ways to make suc a division, I employ te "distribution rule" described in assumption (A:7) below. So a bankrupt ouseold can walk away from its nancial portfolio and its debt wit one important caveat. Te person in carge of liquidating a bankrupt ouseold s assets to repay te creditors is called te case trustee. Tis person s job is to return as muc value as possible to te creditors. As suc, te case trustee is endowed wit "avoiding powers" including te power to undo any transfer of assets or property tat occurred in te 90 days leading up to te bankruptcy ling. So, te spirit of te capter 7 bankruptcy law proclaims tat te penalty for bankruptcy is only te loss of currently eld nancial assets and not any wage-garnisment or con scation of endowments, but tis idea of "avoiding powers" presents a new reality. For nancially poor ouseolds wit sort positions in all assets, tere is no role for te trustee s "avoiding powers" as tese ouseolds would not ave ad any assets to selter from te state in te past 90 days. However, for ouseolds currently olding long positions in assets, any transfer of assets in te previous 90 days could be voided. As a result, te proceeds of suc a nancial transfer (aving been consumed in previous time periods) must be o set by a loss of endowments in te current time period. Anoter reasonable assumption is tat ouseolds declaring bankruptcy wit long positions in assets are more likely to be investigated for fraud and will ave to fund teir defense wit current period income. In eac situation, te cost of bankruptcy is proportional to te size of long positions tat a bankrupt ouseold maintains. Tis cost is given by j jr j (s)(zj ) 15 (2.1) were >> 0 is te parameter tat approximates te "avoiding powers" of te U.S.C. 701, Weter or not we care about tese in a serious economic model remains an open question. 9

10 case trustee. 16 Te fact tat is strictly positive is crucial in te proof of existence, but is also important as it allows tis relation between bankruptcy and te relative makeup of a ouseold s portfolio (a relevant feature 17 ) to be included in te model. If a ouseold declares bankruptcy in some state s > 0; ten its total wealt in tat state is given by p(s)e (s) j jr j (s)(zj ) : Tis expression is explicit about ow te bankrupt ouseold must pay te costs of bankruptcy out of its endowments in tat state. In 2005, te bankruptcy law was clari ed to prevent abuse of capter 7 bankruptcy. 18 Eac applicant for capter 7 must pass a "means test". If te average montly income for a ouseold over te past 6 monts is above te state median, ten (unless a large number of statutorily allowed expenses can be deducted) a ouseold s capter 7 claim will be converted to capter 13 (or in my model witout capter 13, te ouseold will be liable for te entire nancial commitment). By making te following assumption, I am making te equivalent assumption tat, for some state s > 0; a ouseold will fail te "means test" and not be able to declare capter 7 bankruptcy: 19 A.6 8f 2 F; 9s > 0 s.t. e f l (s) > 1 F f2f e f l (s) 8l: Implicit in tis assumption is tat te mean ouseold income is arbitrarily close to te median. De ne Sf = fs > 0 : ef l (s) > 1 F f2f e f l (s) 8lg as te set of states at wic ouseold f cannot declare bankruptcy by law. 20 Te "means test" togeter wit te cost of bankruptcy in (2:1) serve to bound te assets of bankrupt ouseolds. 16 Tis is fundamentally di erent tan te wage garnisments of A. Bankrupt ouseolds only pay a cost of bankruptcy if tey possess long positions and tese long positions are a ouseold coice. Wit wage garnisments, all bankrupt ouseolds lose endowments as all ouseolds are endowed wit strictly positive endowments (not cosen). Furter, in te main model of A, te cost of bankruptcy strictly increases wit te size of te bankruptcy debt. In my setup, te cost only increases if te ouseold cooses to increase te long positions of assets. 17 Te relevance of tis supposed connection between te cost of bankruptcy and te size of asset purcases will depend on empirical work (not considered ere). 18 Bankruptcy Abuse revention and Consumer rotection Act of Anoter assumption wit te same e ect is to assume tat wit some small probability, a ouseold does not le its paperwork correctly (or onestly) and is not given a capter 7 discarge. Excluding cases tat are dismissed or converted to capter 13, a debtor receives a discarge in 99% of te capter 7 cases, but not all. 20 If I were to introduce spot prices into te de nition of Sf ; ten a small price cange may switc s 2 Sf to s =2 S f or vice-versa. As a result, te budget correspondence would not be continuous. Rater tan (A:6); I could ave assumed tat te ouseold income is actually only te labor income and tis is measured in terms of commodity l = 1 in every state. Ten te inequality in (A:6) would read e f 1 (s) > 1 F f2f e f 1 (s): 10

11 Recall tat te asset payouts in Y are assumed to be strictly positive. Suppose a ouseold declares bankruptcy in some state s = 1: If te model allowed for Arrow securities, ten it would be optimal for tis bankrupt ouseold to sort te Arrow security paying out in state s = 1 to 1 and receive an arbitrage pro t (iger payout in te initial state witout reducing payout in any oter state). In an attempt to acieve suc an aim, a ouseold could replicate te desired Arrow securities wit te portfolio z 0 = 2 4 r 1(1) ::: r J (1) : : r 1 (J) ::: r J (J) ! 0 : Tis replicated portfolio z 0 must contain some assets eld in long position. Bankrupt ouseolds will receive a strictly negative payout as a result of olding long positions and tis negative payout increases wit te size of te long positions. As a result, te cost of bankruptcy wit >> 0 prevents te replication of Arrow securities for te purpose of creating arbitrage pro t. Claim 1 For bankrupt ouseolds, te assets z are bounded. roof. If a ouseold does declare bankruptcy in some state (at least one suc state exists), ten its total wealt is j jr j (s)(z j ) p(s)e (s) (2.2) in tose states in wic it does declare bankruptcy. Its total wealt is j r j (s)zj p(s)e (s) (2.3) for tose states (at least one suc state exists) in wic eiter te ouseold cannot declare bankruptcy (s 2 Sf 6= ; for 2 H f ) or te ouseold cooses not to declare bankruptcy. Te payout rj (s) is te asset payout for nonbankrupt ouseolds and takes into account te dilution of payouts caused by bankruptcy. Having to repay its full commitment, ten rj (s) = r j(s) > 0 for zj < 0: Suppose tat tere exists an optimal sequence of asset coices zj suc tat z j! 1 for some j: From (2:2) and wit >> 0; te ouseold would be consuming outside te consumption set (contradiction). Wit z bounded above, suppose tere exists an optimal sequence of asset coices zk suc tat z k! 1 for some j: Ten from (2:3) and wit r (s) >> 0; te ouseold would be consuming outside of te consumption set (contradiction). A ouseold will determine te value of its portfolio at any state s > 0, tat value given by j r j (s)z j ; and determine if tis provides a iger payo or if declaring bankruptcy provides a iger payo. Te bankruptcy payo is j jr j (s)(zj ) : So a ouseold will declare bankruptcy in some state s > 0 if s =2 Sf wit 2 H f and j r j (s)z j < j jr j (s)(z j ) (2.4) and is indi erent towards te prospect of bankruptcy if (2:4) olds wit equality. As a casual observation, notice tat (2:4) olds only if j r j (s)z j < 0: 11

12 De ne Hs 0 = f : s =2 Sf for 2 H f and (2:4) oldsg as te set of bankrupt ouseolds in state s: Wen ouseolds declare bankruptcy, te returns to creditors must be diluted in order for markets to clear. As in Dubey et. al., te assets are pooled, so all creditors will receive te identical, lower returns given by: r j (s) = j (s)r j (s) if z j 0 and r j (s) = r j (s) if z j < 0 8j were j (s) is te dilution factor. Te overall SJ vector is an endogenous price variable. For tese bankrupt ouseolds, ten I need to ave a "distribution rule" for ow te positive value of a bankrupt ouseold s assets: j j (s) j rj (s)(zj ) (2.5) are divided among te asset pools. Looking at (2:5); a bankrupt ouseold forfeits its long positions in assets and teir value j j(s)r j (s)(zj ) : In addition, te bankrupt ouseold must pay te cost j jr j (s)(zj ) : Tis total value will be returned to te creditors. Since te set of creditors di ers for every asset, te following assumption is needed to dictate wat value is returned to wic asset pool. A.7 If 2 Hs; 0 ten (s) r j (s) zj te variable (s) is de ned as (s) = is returned to te asset pool j were j ( j(s) j )r j (s) zj k r : (2.6) k(s) zk Wit (A:7); te positive value of te bankrupt ouseold 2 Hs 0 is used to pay o te creditors in assets j tat as sorted. Since 2 Hs 0 implies j jr j (s)(zj ) > j j(s)r j (s) zj j ( j(s) j )r j (s)(zj ) > j r j(s) zj ; j r j(s) zj ten (s) < 1: Of course in te extreme case were a bankrupt ouseold does not old any long positions, te assumption (A:7) plays no role. I will now explicitly write down te budget set of te ouseolds. B (p; q; ) = f(x; z) 2 X R J : p(0)(e f (0) x(0)) qz f (0) qz 0; p(s)(e f (s) x(s)) max n j r j (s)z j ; j jr j (s)(z j ) o 0 8s =2 S f ; p(s)(e f (s) x(s)) j r j (s)z j 0 8s 2 S f g for 2 H f : I de ne a bankruptcy equilibrium as (x ; z ) 2H ; p; q; s.t. 12

13 8 2 H, given (p; q; ) (x ; z ) 2 arg max u (x): subj to (x;z)2b (p;q;) zj = 1 8j: 2H 1 F 1 F j (s) 2H X f2f ef l 2H (s) = x l (s) 8(l; s) =2 f(l; 1); :::; (L; S)g: X f2f ef l (s) j r j(s) = x L (s) 8s > 0: zj =2H 0 s zj 2H 2H 0 s (s) zj (H) (M) = 1 8j; s > 0: (AC) Te nal set of equations are te aggregate consistency (AC) equations tat allow me to determine te equilibrium value of te dilution factor j (s): First, I sow ow (AC) plays a role in te proof of existence and ten I analyze some of te equilibrium properties implied by te equations. Claim 2 Wit (AC); ten j r j (s)zj =2H 0 s 2H 0 s j jr j (s)(z j ) = j r j(s): (2.7) roof. Te nancial payout for 2 H 0 s can be rewritten as j j (s) j rj (s)(z j ) j j(s)r j (s)(z j ) : Using te de nition of 2H (s) = j ( j(s) j )r j (s) zj k r ; k(s) zk ten te left-and side of (2:7) is given by: j j(s)r j (s) zj j r j(s) zj =2H 0 s 2H 0 s (s) k r k(s) z k : Since (AC) implies tat j (s) r j (s) zj 2H =2H 0 s r j (s) zj 2H 0 s (s)r j (s) zj = r j (s); ten summing over all j yields (2:7): Rearranging (AC) to solve for j (s) yields: 1 zj (s) zj =2H j (s) = 0 s 2H 0 s : (2.8) 2H 13 z j

14 As bot te numerator and denominator are strictly positive, ten j (s) > 0 and bounded. Suc a result is in contrast to Dubey et. al. In teir formulation, pessimistic beliefs o te equilibrium pat can be self-ful lling in equilibrium. Tus, I do not need to introduce a re ned equilibrium to eliminate suc undesirable equilibria. Using te market clearing condition 2H z j = 1 and te accounting equality 2H z j = 2H z j =2H z 0 s j 2H z 0 s j ; ten (2:8) is equivalently written as: 1 (s) zj 2H j (s) = 1 0 s : zj 2H Recalling tat 0 (s) < 1; te following intuitive results old: 1. If z j H0 s (or H 0 s = ;), ten j (s) = 1: 2. If z j < 0 for some 2 H0 s; ten j (s) < 1: Te rst statement says tat even under bankruptcy, so long as te bankrupt ouseolds do not abandon a sort position on a certain asset j; ten tat asset payout r j (s) will not be diluted. Te second statement says tat even wit te "distribution rule" and assumption (A:7); te creditors in asset pools suc tat zj < 0 for some 2 H0 s can never recover enoug resources to return te original payout r j (s): As a wole, if bankruptcy occurs in some state s > 0; ten j (s) < 1 for some asset j and j r j (s)z j < r(s)z for some solvent ouseold. As a quick digression, consider te rst order condition wit respect to z for all ouseolds: ^R = 0: Te matrix ^R is te ouseold-speci c (S 1) J payo matrix q ^R = ^Y were ^Y is te S J matrix wit terms ^r j (s) de ned as ^r j (s) = r j (s) for =2 Hs 0 and ^r j (s) = (s)r j (s) for 2 Hs: 0 Te term (s) is a nonnegative fraction of wat te bankrupt ouseold actually pays j jr j (s)(zj ) compared to wat is owed j r j (s)z j < 0 : (s) = j jr j (s)(zj ) : j r j (s)z j 14

15 By de nition, (s) 2 [0; 1]: Ten ^Y is a nonnegative matrix for all ouseolds. Te transformation from Y to ^Y is not rank-preserving. Tus, any regularity properties would not old for te bankruptcy equilibria. 21 Taking any j; since ^r j (s) > 0 for some s > 0; ten q j > 0 in equilibrium. Claim 1 in te previous section sowed ow te bankruptcy setup is able to bound te assets z for bankrupt ouseolds. Te next claim will bound te assets z for entirely solvent ouseolds (ouseolds tat do not declare bankruptcy in any state s > 0). Claim 3 For entirely solvent ouseolds of a certain type f 2 F; te set 2H f z is bounded. roof. Suppose tat z is not bounded for all 2 H f 2H f suc tat =2 [ s>0 Hs: 0 Divide te assets into subsets s.t. 1. If zj! 1; ten j 2 J : 2H f 2. If! 1; ten k 2 J : 2H f z k Under te assumption tat z is not bounded (and since ^Y z and 2H f ence ^Y z is bounded from te budget constraints), ten J 6= ;; J 6= 2H f ;; and for any #J #J states s 2 S 1 : det r j (s) j r j2j ;s2s1 k (s) i k2j ;s2s 1! 0: By te de nition of te ouseold-speci c asset payouts: rj (s) = j (s)r j (s) as! 1 8j 2 J ; 8s 2 S 1 rk(s) = r j (s) 8k 2 J ; 8s 2 S 1 : For j 2 J ; 21 By taking te inaccurate view tat te only assets wic matter in te (F OCz) for bankrupt ouseolds are tose s.t. zj 0 since tese are te only ones entering te budget constraint, ten one arrives at te follow contradiction. Using te inaccurate view, ten ^r j (s) = jr j (s)1fzj 0g: Seeing tat olding onto positive assets is costly in bankruptcy, te ouseold will let (zj )! 0: Using (F OCz) and olding everyting else constant (te Lagrange multipliers do not cange), ten (zj )! 0 implies tat ^r j (s) is increasing and ten q j is increasing. However, for market clearing to old, if one ouseold reduces te amount of asset eld ((zj )! 0), ten te price q j can only decrease in order to entice oter ouseolds to buy up tis slack. Tis contradiction justi es my formulation of te ouseold-speci c asset payouts. 15

16 1. by te de nition of j (s) 8s > 0 1 (s) j (s) = 1 2H 0 s 2H zj zj ; and 2. since zj is bounded for bankrupt ouseolds as sown in claim 1, 2H 0 s ten j (s)! 1 8s > 0: Terefore det r j (s) j2j ;s2s1 j r k (s) k2j ;s2s 1 i! det (r j (s)) j2j ;s2s 1 j (r k (s)) k2j ;s2s1 and from above det r j (s) j2j ;s2s1 j r k (s) k2j ;s2s 1 i! 0: Tis contradicts te assumptions of (A:4) tat Y as full column rank and is in general position (any subset of rows s 2 S 1 is linearly independent). Teorem 1 states te general existence of a bankruptcy equilibrium wit te proof in te appendix. Teorem 1 Given assumptions (A:1) equilibrium (x ; z ) 2H ; p; q; exists. (A:7) and >> 0; ten a bankruptcy A feature of bankruptcy equilibria is te possibility for a cain reaction of bankruptcy (as in Dubey et. al.), a concept tat is termed contagion in nance. Te basic idea is tat bankruptcy by one ouseold may cause oter ouseolds (wo would ave remained solvent ad tey received te entire payout for teir asset purcases) to declare bankruptcy. Tis e ect can snowball. As te asset payouts become more diluted, ten te value of bankrupt ouseolds seized assets fall. As a result, te state cannot return as muc value to te asset pools and te payouts are diluted even furter. In te following example, for a simple speci cation of parameters, I give an example of a bankruptcy equilibrium wit tis cain reaction. Toug markets are complete, te presence of bankruptcy is an externality tat prevents te areto optimal allocation from being acieved. In te bankruptcy equilibrium, te total sortfall of bankruptcy is equal to te sum across ouseolds of te di erence between wat a bankrupt ouseold pays j jr j (s)(zj ) and wat a bankrupt ouseold owes j r j (s)z j : I can ten compute te GEI equilibrium and determine wat te initial sortfall of bankruptcy is (wen none of te asset payouts are diluted). Te total sortfall of bankruptcy is larger tan tis initial sortfall because of te cain reaction mecanism described above. Te ratio of tese two sortfalls (total divided by initial) will be termed te "bankruptcy multiplier". 16

17 Example 1 Consider a model wit J = S = 2; F = 3; and L = 1: Let u f (x ) = a f 0 ln(x 0) a f 1 ln(x 1) a f 2 ln(x 2) by te utility of te ouseolds. Let 1 1 te payouts for te numeraire assets be Y = : Eac type of ouseold 1 2 as initial assets z f (0) = 1 3 ; 3 1 : In tis setup, it turns out tat all ouseolds 2 H of te same type H f will optimize wit te same vector x ; z = x 0 ; z 0 8; 0 2 H f (and teir bankruptcy decisions will be identical as well). Te cost of bankruptcy is parameterized by = (0:1; 0:1) : Using te "means test" and assumption (A:6); I will speci y endowments suc tat ouseolds 1 and 2 can declare bankruptcy in state s = 1; but not s = 2: Houseold = 3 cannot declare bankruptcy in eiter. Tese endowments are given by: e 1 (0) = 2 e 1 (1) = 1 e 1 (2) = 3 e 2 (0) = 2 e 2 (1) = 1 e 2 (2) = 3 e 3 (0) = 8 e 3 (1) = 4 e 3 (2) = 3 r(0) = 12 r(1) = 8 r(2) = 12 were r is te vector of total resources (including te payouts of te numeraire assets) in eac state. Tere is noting special about te values of te endowments cosen. I merely wanted to give ouseolds 1 and 2 an incentive for bankruptcy by giving tem bot a low endowment in s = 1: Te preference parameters are given by: a 1 0 = 37=12 a 1 1 = 285=56 a 1 2 = 3=2 a 2 0 = 10=3 a 2 1 = 153=140 a 2 2 = 1 a 3 0 = 67=12 a 3 1 = 21=4 a 3 2 = 203=82 In te model witout bankruptcy, te GEI allocation is identical to te areto optimal allocation. Wit tis allocation (and subsequent assets), te initial bankruptcy sortfall is given by = 2 in state s = 1 : j jr j (s)(zj 2) = 0:01714 j r2 j (s)z2 j = 0:10277 Initial sortfall = 0:08563 : Te bankruptcy equilibrium as asset prices q = (1; 1:5) and dilution variables (1) = (53=70; 65=70) for te only state (s = 1) in wic bankruptcy is declared by ouseolds. In s = 1; bot ouseolds tat can declare bankruptcy (ouseolds 1 and 2) do declare bankruptcy. Te portfolios of tese two ouseolds are given by: z 1 = 1; 1 2 z 2 = (1; 1) : Terefore, in te bankruptcy equilibrium, te total sortfall of bankruptcy is given by: j jr j (s)(zj 1) = 0:05 j r1 j (s)z1 j = 75=140 j jr j (s)(zj 2 ) = 0:1 j r2 j (s)z2 j = 17=70 Total sortfall = 0:62857 : : 17

18 If te initial sortfall calculated above were 0; ten te bankruptcy equilibrium would be identical to te GEI equilibrium. As it is, te small initial sortfall starts te cain reaction wic lowers te payouts of assets in state s = 1 from (1) =! 1 to (53=70; 65=70) : 22 Wen te instantaneous cain reaction is complete, te resulting bankruptcy multiplier is given by: Bankruptcy multiplier = 7:34: Given te bankruptcy equilibrium, te consumption equivalent ce is te % of consumption tat eac type of ouseold would be willing to forfeit in return for te government forbidding bankruptcy declarations (leading to te areto optimal allocation). If te consumption equivalent is positive, ten te ouseold as iger utility wit te GEI allocation: ce 1 = 30:28% ce 2 = 9:44% ce 3 = 2:69% : Toug not explicitly considered in tis work, wat is implicit in te above example is tat if some policy existed to make a transfer of size 0:08563 to ouseold = 2 in state s = 1; ten te sortfall of magnitude 0:62857 could be preempted. 3 Extension I: Bankruptcy Exemptions Te two most straigtforward (and awed) metods to model bankruptcy witout exogenously imposed investment constraints are to assume tat te bankrupt ouseolds eiter (i) pay a cost proportional to te total bankruptcy debt or (ii) su er a garnisment of wages (as in Araujo and ascoa). Bot modeling coices contradict te legal framework for bankruptcy (capter 7) in te US. Obviously, if unbounded bankruptcy imposes a complete garnisment of current period wages (and ouseold preferences satisfy te boundary condition), ten no ouseold would ever rack up an unbounded bankruptcy debt. In my model, te cost of bankruptcy is only paid on te asset purcases (zj ) : For te common situation wit poor ouseolds declaring bankruptcy (poor meaning no asset purcases), ten my model allows for a complete discarge of te debt and no costs of bankruptcy. To sow tat I am not being deceptive about te distinction between ouseold endowments e f (s) and asset purcases (zj ) ; I will extend te analysis to a specialized example wit typical bankruptcy exemptions. Consider te ctitious state of Hamilton (an almagam of several US states). In Hamilton, te following items are exempt under capter 7 bankruptcy (meaning tat te ouseold will maintain possession of tese items under bankruptcy): te rst $100; 000 in value of te primary residence (partial omestead exemption) 22 Helping tis process along of course is te fact tat ouseolds = 1; 2 old a mixture of bot long and sort positions. 18

19 te primary ouseold automobile te college savings accounts of any dependents any jewelry or artwork wit total value less tan $100; 000: Te bankruptcy statutes of course include a long laundry list of exemptions, but for illustrative purposes, te above list is kept sort. Houseold endowments will certainly include all commodities tat unequivocally belong to a certain ouseold and cannot be owned by a di erent ouseold. Tis will include a ouseold s labor income and some (but not all) of te ouseold s durable goods. Te assets will include te traditional nancial instruments (stocks, bonds) as well as a ouseold s liquid personal assets on wic nancial contracts can be written to spread ownersip among ouseolds. Wen uncertainty remains about de ning an asset, a secondary property will de ne assets as tose ouseold possessions tat would be seized in a bankruptcy case. One of te most unconventional assets will be a ouseold s primary residence. As most Americans do not own teir omes entirely, tere will be two assets associated wit a ouse: te mortgage eld in a sort positions and te actual ouse eld in a long position. As banks will take a ouseold mortgage, repackage it, and sell it o to investors, ten all ouseolds in te economy can tecnically own an individual ome (even toug just one ouseold will reside in it). 23 Tus, a ome can be tougt of as an asset wit risky return depending on te ousing prices. Consider a ouseold wit 2 automobiles, a ouse wort $300; 000, college savings accounts, and jewelry and artwork totalling $200; 000 in value. Te ouseold olds a portfolio of bonds (including te ome mortgage) suc tat it is contemplating bankruptcy in some state s > 0: Te primary automobile will be a ouseold endowment of tat particular commodity. De ne J 0 as te set of assets tat are not exempt and terefore subject to liquidation during bankruptcy. For tis example, ten J 0 = fbonds (including ome mortgage), second auto, value of ome beyond $100; 000, and value of artwork and jewelry beyond $100; 000g: De ne J 1 as te set of assets wic are exempt during a ouseold bankruptcy. For tis example, ten J 1 = fcollege savings accounts, rst $100; 000 of te value of te ome, and rst $100; 000 of te value of te artwork and jewelryg: 23 Tis is akin to te situation in te movie "It s a Wondeful Life" in wic Jimmy Stewart s caracter does not ave te funds to return all te townspeople s savings during te bank run because teir money is invested in te omes of te oters. 19

20 To obtain existence, I will need to place an additional assumption on te assets in J 1 : A.8 For all assets j 2 J 1 ; and 8 2 H; tere exists an upper bound K j 2 R suc tat z j K j: It is clear tat te exempt assets in te example (college savings accounts are implicitly bounded above) all ave suc a bound preventing arbitrarily large long positions. I can now write down te bankruptcy decision: 8 9 < X X max rj (s)zj ; : j r j (s) z X j rj (s) z = j (3.1) ; j2j 0[J 1 j2j 0 j2j 1 Notice tat tere is no cost of bankruptcy for te exempt assets j 2 J 1 : I can now de ne te corresponding distribution rule: j (s) j rj (s) z j j2j 0 (s) = r j (s) zj j2j 0[J 1 Notice tat 0 (s) < 1 as in section 2. Consider te two polar cases. If J 1 = ;; ten none of te assets are exempt and te analysis reduces to tat considered in section 2. If J 0 = ;; ten all ouseolds tat can (all suc tat 2 H f and s =2 Sf ) will declare bankruptcy. It is immediate tat increasing te bankruptcy exemptions will increase te incentive for bankruptcy and reduce j (s) for some (j; s): Weter or not suc a policy move is areto improving is not possible to determine because te bankrupt ouseolds may ave a iger payout (consider (3:1)), but it may now be more costly to transfer wealt between states since te asset payouts are diluted more. Wile all-or-noting normative improvements are unlikely over a range of parameters, tis setup does allow for policy analysis of te impact of bankruptcy exemptions on prices and allocation. 4 Extension II: Real Costs of Bankruptcy Now suppose tat bankruptcy imposes real costs on te economy and tat none of te assets are exempt under bankruptcy (returning to te original setup of section 2). In section 2, I ad implicitly assumed tat te entire positive value tat a bankrupt ouseold forfeits j j (s) j rj (s) zj is made entirely available to pay o creditors. Instead, suppose tat only j j(s)r j (s) zj 20

21 wit 2 [0; 1] is made available to te creditors. j (1 ) j (s) j rj (s) zj Te di erence is irrevocably lost to te economy. As a result, te market clearing conditions are appropriately adjusted: zj = 1 8j: 2H X 1 F f2f ef l 2H (s) = x l (s) 8(l; s) =2 f(l; 1); :::; (L; S)g: X 1 F f2f ef l (s) j r j(s) j (1 ) j (s) j rj (s) z j = x L (s) 8s > 0: 2H 0 s 2H (M) As a result of te real costs, eac bankrupt ouseold 2 H 0 s as assets wit positive value j j (s) j rj (s)(z j ) ; but due to ine ciencies in te bankruptcy process, only j j(s)r j (s)(z j ) is made available for te various asset pools. Using te "distribution rule" akin to (2:7); ten de ne RC(s) as RC(s) = j j(s)r j (s) zj k r (4.1) k(s) zk were RC stands for "real costs". Since j (s) < j (s) j ; ten RC(s) (s) (wit strict inequality for (s) 6= 0) and tis olds 8 2 Hs 0 and 8s > 0: Te resulting (AC) conditions are given by: j;rc (s) zj z j RC(s) z j = 1 8j; s > 0: (AC) 2H =2H 0 s 2H 0 s Te next claim sows tat te (AC) conditions are su cient for existence. Claim 4 Wit (AC); ten j r j (s)zj =2H 0 s j r j(s) 2H 0 s 2H 0 s j jr j (s)(z j ) = (4.2) j (1 ) j (s) j rj (s)(z j ) : 21