Optdef JEDC Final

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Contents lists available at ScienceDirect

Journal

of

Economic

Dynamics

&

Control

journal homepage: www.elsevier.com/locate/jedc

Optimal

bankruptcy

code:

A

fresh

start

for

some

R

Grey

Gordon

IndianaUniversity,DepartmentofEconomics,100S.WoodlawnAve.,Bloomington,IN47405,UnitedStates

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received14August2017 Revised9October2017 Accepted22October2017 Availableonline27October2017

JELclassiﬁcation:

D14 D52 D91 E21 K35

Keywords:

Bankruptcy Life-cyclemodels Incompletemarkets

a

b

s

t

r

a

c

t

Whatistheoptimal consumerbankruptcylaw?Ianswer thisquestionusingan incom-pletemarketslife-cyclemodelwithaplannerwhocanchoosestate-contingentbankruptcy costs.Ideveloptwomaintheoreticalcharacterizations. First,wheneverdebtdischargeis allowed,it shouldoccur withoutcost.Second,bankruptcyshouldalwaysbeallowedfor highly-indebtedhouseholds.Quantitatively,theoptimalpolicycangenerateawelfaregain aslargeas11.6%.However,attractiveinformaldefault,asymmetricinformation,andmoral hazardcanreducethewelfaregaintoaslittleas0.7%.

1. Introduction

Bankruptcypolicyvariesgreatlybytimeandlocation.InmanyEuropeancountries,thereislittletonodebtforgiveness. BankruptcylawsintheUnitedStates,ontheotherhand,arewidelyconsideredpro-debtor.Moreover,viewsontheproper amountofdebtforgivenesshavechanged dramaticallyoverthelast two hundredyears.Inthe U.S.,debtors’prisons have beenreplacedwitharelativelyswiftbankruptcyprocess,which,untilrecently,offeredanear-completedischargetoalmost everyone.In2005, theBankruptcyAbusePreventionandConsumerProtection Actrestrictedthisnear-completedischarge toonlythosewithbelow-medianincome,forcingabove-medianincomehouseholdstopayalltheirdisposable income(in thespecificsensedefinedbythelaw)forfiveyears.1 _Which_of_the_many_possible_bankruptcy_{laws—ranging}_from_complete dischargeforalltonodischargeatall—isbest?

Toanswer this question, Iuse an incomplete markets life-cycle model of bankruptcy andallow a planner to choose state-contingentbankruptcypenalties.The plannerspeciﬁeswhethera householdmayﬁleforbankruptcy, anyassociated

R _I_thank_Kartik_Athreya,_Satyajit_Chatterjee,_Daphne_Chen,_Hal_Cole,_Bulent_Guler,_Aaron_Hedlund,_Juan_Carlos_Hatchondo,_Aubhik_Khan,_Dirk_Krueger,

AmandaMichaud,VictorRíos-Rull,Pierre-DanielSarte,MichèleTertilt,JuliaThomas,andEricYoung,aswellasanonymousreferees.Ialsothankconference participantsfromtheEuropeanMeetingoftheEconometricSociety2014,ComputinginEconomicsandFinance2014,MidwestMacro2013and2014,and aUIUCmini-conferencein2014,aswellasseminarparticipantsatFloridaStateUniversity,theOhioStateUniversity,andPurdueUniversity.

E-mailaddress:[email protected]

1_Robe_et_al.₍₂₀₀₆₎_document_both_the_historical_and_geographical_variation_in_bankruptcy_laws,_and_Coleman₍₁₉₇₄₎_documents_the_variation_in_U.S. bankruptcylawsovertime.AsWangandWhite(2000)summarize,“theUnitedStatesisextremelyunusualinhavingveryprodebtorbankruptcylaws” (p. 255).FordetailsontheBankruptcyAbusePreventionandConsumerProtectionAct,seeWhite(2007).

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ﬁlingcostsandearningsrestrictions,andhowlongabankruptcyremainsonahousehold’screditrecord.Inchoosingthese penalties,theplannerfacesatrade-off betweenimprovingcreditandaddingstate-contingencytodebtrepayment.

Ianalyticallycharacterizetheplanner’soptimalpolicyforabroadclassofutilityfunctions,socialwelfarefunctions,and labor efficiencyprocesses under full information.I find two main results.First, whenever the optimal policy givesdebt forgiveness,itisgivenwithoutcost,a“freshstart.” Thefreshstartintheoptimalpolicydeviatessubstantiallyfromcurrent U.S.policyinthattherearenofilingcostsandabankruptcyfilingneverimpactsahousehold’screditrecord.Second,Ifind that bankruptcyshould always be allowed insome states.In particular,ifa householdcannot repaytheir debtor would otherwiseprefertoinformallydefault,bankruptcyshouldbeallowed.Asacorollaryofthisresult,anaturalborrowinglimit economy—aneconomywherehouseholdshavemaximalcommitmenttorepaytheirdebt—issuboptimal.

Quantitatively,Iﬁndtheoptimalpolicyproducesaconsumptionequivalentwelfaregainof11.6%relativetoaneconomy calibrated to match the U.S. when informal default is unattractive and there is full information. It doesso by typically allowingbankruptcyforthosewithbadpersistentshocksandhighdebtlevelsbutforbiddingitotherwise.Undertheoptimal policy,householdsaccumulatedebtequaltohalfofaverageearningsbytheirmid-30sandonlybeginsavingforretirement intheirmid-40s.Theoptimalpolicyresultsinasmallincreaseindefaultrates,aﬁfteen-foldincreaseindebt,and50%lower interestratesandcharge-off rates.

Whilethegainsfromtheoptimalbankruptcyarequantitativelylarge,thenumbersabovewerecalculatedassumingcostly informaldefault,fullinformation,andfullabilitytounderstandacomplicatedbankruptcylaw.Thelastoftheseassumptions seems tomatterquantitatively little:Asimplerulegeneratesawelfaregain of10.6%,onlyonepercentagepointlessthan the11.6%fromtheoptimalrule.Theassumptionoffullinformationmattersmuchmore.Speciﬁcally,whenhouseholdsare allowed to reduce their earnings ina wayunobservableto bankruptcycourts, accessto bankruptcyis severelyrestricted and the welfare gain obtained by the optimalpolicy fallsto 5.5%. This gain is not much more than the 5.4% produced bya naturalborrowinglimiteconomy. Whileasymmetricinformationwithmoral hazardsigniﬁcantly impairstheoptimal policy,limitedcommitment intheform ofan attractive outsideoptioncan be evenworse. Inparticular, whenthe value ofinformal defaultis madeaslargeasthe theory allows,the optimalpolicy producesa welfaregain of only0.7% under both fullandasymmetricinformation.Thislast resultsuggeststhatoptimalpolicy—whenit hascontrolover thevalueof informaldefault—shouldmakeitasunattractiveaspossible,aclaimIproveunderfullinformation.

1.1. Relatedliterature

Thispaperispartofa largequantitative literature,surveyedby Livshits(2015),thatinvestigates consumerbankruptcy lawandits implicationsforconsumption,credit,andwelfare.Policy evaluationshavebeendone bymodeling speciﬁc re-formsorconsideringmoretheoreticalexercises,suchaseliminatingbankruptcy.2 _Most,_though_not_all,_of_these_experiments havebeenconductedundertheassumptionsoffullcommitment(inthesenseofbankruptcybeingtheonlydefaultoption) andfullinformation.Theoverridingthemeofthisliteratureisthatharsherbankruptcylawsalmostalwaysimprovewelfare, eventothepointthateliminatingbankruptcygreatlyimproveswelfareintheabsenceofshockstohouseholdexpenditures. The presentpaperhelpstointerpret theseresults bytheoretically andquantitatively characterizingthe optimalpolicy. Thecalibratedmodelreproducestheﬁndings oftheliteraturethateliminating bankruptcyproduces alarge(5.4%)welfare gain relativeto theU.S.when thereisfull commitmentandfull information.However,the theoreticalresultsshow there isalwaysabetterpolicyallowingbankruptcyinsome states,suchasstateswhererepaymentisimpossible.Quantitatively, the optimalpolicy underfull commitment andfull information produces a welfare gain of 11.6%. This showsthat while thegainfromeliminatingbankruptcyislarge(astheliteraturehasrepeatedlyfound),thereisafarbetteroptionallowing bankruptcy. Moreover,themeans by whichthislarge gainis achieved—defaultconcentrated amongthosewiththe worst persistentshocksandcostlessbankruptcyconditionalondefault—suggestpathsforpolicygoingforward.

Byanalyzing the optimalpolicy inthe face oflimitedcommitment,the paperalso contributesto andhelpsinterpret a literature thathasallowed formultiple defaultoptions.3 _The _most_closely_related_is_Chatterjee_and_Gordon ₍₂₀₁₂₎ _that quantitatively shows(1) eliminating bankruptcyimproves welfare evenwhen households mayinformally defaultand(2) making theinformal defaultoptionascostly aspossible improveswelfare.The presentpapershowsthat even whenthe firstfinding holds,theoreticallyit isalways optimaltoallowsome bankruptcy.In fact,allowing bankruptcyifandonly if householdswouldotherwiseinformallydefaultachievesthesamewelfare(quantitatively) asthefull optimalpolicywhen informaldefaultisveryattractive.Asfortheirsecondfinding,theresultsconfirmittheoreticallyandquantitatively: Theo-retically,theplannershouldmakeinformaldefaultasunattractiveaspossible;quantitatively,theoptimalpolicycanachieve awelfaregainofonly0.7%ifinformaldefaultisveryattractive.

This paper is also connected to a theoretical literature that can be divided into two branches. The ﬁrst, beginning with Kehoe and Levine (1993) andcontinued by Kehoe and Levine (2001; 2006), Kocherlakota(1996), and Alvarez and Jermann(2000) has focused oncomplete markets withlimited commitmentinthe formof a participationconstraint. In

2_This _approach _has_been _followed _by _Athreya ₍₂₀₀₂₎_, _Athreya _et_al. _(2009b_), _Athreya _et_al. _(2009a_),_Li_and _Sarte ₍₂₀₀₆₎_, _Livshits _et_al. ₍₂₀₀₇₎_, Chatterjeeetal.(2007),ChenandCorbae(2011),ChatterjeeandGordon(2012)andGordon(2015)andothers.

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theseeconomies,equilibriaareParetoeﬃcient.The secondbranchhaslookedatasymmetricinformationwithmoral haz-ard.Inthisliterature,eﬃciencytypicallyrequiresthataninverseEulerequationholds,aconditionthatisoftenimpossible whenhouseholdshaveunimpededaccesstosavingandborrowingatarisk-freeprice.4_Bizer_and_DeMarzo₍₁₉₉₉₎_,_Bisin_and

Rampini(2006),andGrochulski(2010) allshow thatbankruptcycan beusedto maketheinverseEulerequation holdby impedingthisfree-ﬂowofcredit.

Akey distinction betweenthispaperand thetheoretical literature is thetype of debtcontractsused. The theoretical literaturehasemployed Dubeyetal.(2005)(DGS)-styledebtcontractswhereborrowinglimitsare takenasexogenousby agentsandinterest ratesbear aconstant riskpremium.Incontrast,thispaperandthe quantitativeliterature havealmost exclusivelyworkedwithEatonandGersovitz(1981)(EG)contracts.WithEGcontracts,eachlevelofborrowinghasa poten-tiallydistinctpriceandthereisno“borrowinglimit” intheconventionalsensebutinsteadaLaffercurve.

As it turns out, DGS contracts are essential forthe theoretical literature’s results.For instance, a feature common to

KehoeandLevine(2006)andGrochulski(2010)isthat,forthedecentralizedequilibriumtoachievetheeﬃcientallocation, theexogenous(toborrowers)borrowinglimitsmusttakeonveryspeciﬁcvalues.5_{Additionally,}_these_exogenous_limits_must, atleastinGrochulski(2010),bedecreasinginwealth(p.366–367).ForbankruptcypolicythatisMarkov,thistypeofpricing requiresDGScontractsratherthanEGcontracts.

AsIestablish inAppendixC, EGcontractshaveatleast fourconceptualandtheoretic advantagesover DGS-style con-tracts.First, DGS contractstypically have multiple equilibria corresponding to various exogenous borrowinglimits while EGcontractsdonot.6 _Second,_DGS _contracts_are _not_necessarily_competitive _in_the_face_of_free_entry._In_contrast,_EG con-tractsarepricedtoexactlypreventthistype ofentry.Third,evenfixingtheexogenousborrowinglimit,DGScontractscan havemultipleequilibria,someofwhichcanhaverisk-freeratesandnoequilibriumdefaultandotherswithriskyratesand equilibriumdefault.Fourth, whentherearemultipleequilibriaofthissort, itisoftenpossibleforfirmstoenterandmake strictlypositiveprofits.Again,EGcontractsarenotsubjecttotheseissues.

The cost of usingEG contracts is the non-lineardebt pricing that makes budget constraints much more complicated andtheoretical results harderto obtain. Moreover, itlikely makes implementing a constrained eﬃcientallocation of the typeinGrochulski(2010) impossible.7 _So_instead_this_paper_proceeds_in_the_other _direction,_starting_from_a_{decentralized} equilibriumandworkingtowardseﬃciency.

2. Model

Thelife-cyclebankruptcymodelisessentiallyLivshitsetal.(2007)augmentedtohaveelasticlaborsupply(withinelastic asaspecialcase)butwithoutexpenditureshocks,i.e.,directshockstoahouseholdsnetworth.8

2.1. Modelsetup

TheeconomyispopulatedwithacontinuumofhouseholdswholiveforatmostTperiods.Thehouseholdlaboreﬃciency

eisstrictlypositiveandfollowsaﬁnite-stateMarkovchain

πee

|t thatis age-dependent.Newbornhouseholdsdrawefrom adistribution

πe

.

Marketsare incompletewithhouseholdsonlyhavingaccesstoabond a∈Awitha<0beingunsecureddebt(negative networth)anda≥0beingsavings.Aisaﬁnitesetcontainingnegativeandpositiveelements,aswellaszero.Forsimplicity, Iassumehouseholdsmaynotdefaultonsavingsa≥0.Newbornhouseholdshavea=0.

Householdshave a bankruptcy flag hindicating whether a bankruptcy is on their credit record, h=1, ornot, h=0. All households beginlife withh=0. When a householdfiles forbankruptcy or has a bankruptcy record, it is assumed thatthey maynot borrow. Thisrestrictionis fortractability,andit meansthat onlyone bankruptcylawisneeded(since householdswillnotdefaultona_≥0)ratherthantwo.Thisassumptioniscommonintheliteratureandcapturesanempirical observationduetoMusto(2004)thatcreditaccessislimitedwhilearecordofbankruptcyremainsandincreasesassoonas itisremoved.However,nothinglegallypreventscreditorsfromlendingtobankrupthouseholds,andthelackofcreditmay reflectthereactionofcreditorstoinformationcontainedinabankruptcyrecord.Iwillrevisitthispointinthediscussionof

Proposition3.

4_In_commonly_used_notation,_eﬃciency_dictates₍_u₍_c₎₎−1₌₍_β₍₁₊_r₎₎−1_E_[₍_u₍_c₎₎−1_]_(a_result_due_to_Rogerson,₁₉₈₅_)._Jensen’s_inequality_then_implies

u(c) =β(1+r) Eu(c) .

5_See_Proposition₃_on_p.₁₃_in_Kehoe_and_Levine₍₂₀₀₆₎_and_the_proof_of_Proposition₂_on_p.₃₆₃_in_Grochulski₍₂₀₁₀₎_.

6_In_this_paper’s_framework,_the_equilibrium_is_unique_up_to_a_tie-breaking_rule._In_{inﬁnite-horizon}_problems,_Auclert_and_Rognlie₍₂₀₁₄₎_have_shown uniquenessundertheassumptionofi.i.d.shocksorpermanentexclusionfromcreditmarketsfollowingdefault.

7_The_credit_limits_in_Grochulski₍₂₀₁₀₎_induce_a_very_speciﬁc_amount_of_borrowing_(see_equation₃₅_on_p._363)._In_an_EG_framework_with_discrete shocks,debtpricesarestepfunctionsandsothereareimplicitrestrictionsonborrowingquantities.

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Preferencesareadditivelytime-separableoverconsumption candlabornwitha discountfactor

β

_>0.Laborischosen fromaﬁnitesetN⊂R++_,_and_I_assume _the_period_utility_function_u₍_c_,_n₎ _is_{well-deﬁned}_for_any_c_>₀_and_any_n_∈_N_._The periodutility uiscontinuousandstrictlyincreasing inc—butnot necessarilyconcaveordifferentiable—anddecreasingin

n.Additionally,itisunboundedbelowwithlim_c_↓₀u

(

c_,n

)

₌_−∞foralln.Havingutilityunboundedbelowisimportantfor thetheoretical results.However,itdoesmeanu(0,n) isnotdeﬁned, andsoIrequirehouseholdschoosec>0(thechoice setisstill compactbecauseais restrictedto lieina ﬁniteset).Additionally,Iallowforpsychiccostsofdefault,

κ

(e)≥0, thatdependonhouseholdeﬃciency.Thesecostsapplyifeitherformaldefault(bankruptcy)orinformaldefault(theoutside optiondiscussedbelow)occurs,andtheyareassumedinvarianttoplannerpolicy.

Ahousehold’sstateis(a,e,t,h). Thepriceofadiscountbondwithfacevalue a isqt(a,e). Forsavings,a≥0,default isnotallowed, andsothepriceforsuch abondissimplyequaltotherisk-freerateoftransferring resourcesacrosstime,

¯

q<1,whichItaketobeexogenousfortractability.Fordebt,a<0(whichimpliesh=0),creditorsexpectarepaymentrate

pt(a,e).Consequently,ano-arbitrageconditionhas

q t

(

a , e

)

=q ¯p t

(

a , e

)

. (1)

Whentherepaymentratept islow,soisthepriceqt,whichimpliesahighinterestrate1/qt−1.Equilibriumwillrequire that therepaymentrateis consistentwithhouseholddefaultdecisions.Forconvenience, Ideﬁne pt

(

a≥0,e

)

=1 sothat

qt

(

a,e

)

=q¯pt

(

a,e

)

foralla,e.

Bankruptcypolicy isdeﬁnedby theinstrumentsavailabletotheplanner.Speciﬁcally,theplannercandoallofthe fol-lowing:

1. Specifywhetherahouseholdisallowedtoﬁleforbankruptcy,Dt

(

a,e,n

)

=

{

0,1

}

,ornot,Dt

(

a,e,n

)

=

{

0

}

. 2. Chargeabankruptcyﬁlingcost

ζ

t(a,e,n)≥0.

3. Retainabadcreditrecordwithprobability

λt

_∈[0,1].

Theplannermusteitherforgiveallofahousehold’sdebtornoneofit.However,thisassumptionisnotasstrongasit seemsbecausepartialforgivenessmaybeincorporatedviaalottery.9 _The_policies_available_to_the_planner_cover,_as_special cases,manyofthetypesofpenaltiesthattheliteraturehasusedtomodelbankruptcy.

Householdsmayalways informallydefault by choosingan outside optiondeliveringVO

t

(

a,e;

γ

)

inlifetimediscounted utility terms where

γ

isa policy instrument ofthe planner.10 _The _outside _option_is _meant _to _represent_the _best_option consumers havefor dealingwith debtother than repaying it orobtaining a discharge, andthe planner mayhave some controlover howattractivetheoutsideoptionis.Forexample,debtcollectionmethodsareextensivelyregulatedby law— includingwhen andhowoftendebt collectorscan makephonecalls, informationdebtcollectors mustprovide,andeven whatmaybewrittenonenvelopes—andabankruptcyreformcouldbepairedwitharevisionoftheselaws.11_However,_the planner’sinﬂuencemaybelimited.Technology,whichDrozdandSerrano-Padial(2017)haveforcefullyarguedisofessential importance inthedebt collectionindustry,isone potentiallylimitingfactor. Anotherisa household’sabilitytoleave the country.12_To_ﬂexibly_capture_the_planner’s_ability_or_inability_to_control_the_outside_option’s_{attractiveness,}_I_assume

γ

_must bechosenfromaset

,whichItaketobeﬁnite.Notethatif

isasingleton,theplannertakesVO _as_given_and_can_only reformbankruptcypolicy.

Imakethreekeyassumptionsregardingtheoutsideoption.First,whenitischosen,thecreditorgetsnothing.Whilenot withoutlossofgenerality,recoveryratesondefaulteddebtcanbeaslowas12–14%netofcollectioncosts(Chatterjeeand Gordon,2012).Second,VO

t

(

a,e;

γ

)

islessthanorequaltoan“autarky” valueVtA

(

a,e

)

forall

γ

∈

.Ideviatefromtheusual deﬁnition of autarky to allow for savings at the risk-free rate andapply a psychic cost of default in the ﬁrst period of autarky.Thatis,VO

t

(

a,e

)

≤VtA

(

a,e

)

:=Xt

(

0,e

)

−

κ

(

e

)

where

X t

(

a, e

)

=maxa∈A,n∈Nu

(

c, n

)

+

β

e

π

ee|tX t+1

(

a , e

)

s.t. c +q ¯a =en +a, c > 0, a ≥0 (2)

with XT

(

a,e

)

:=maxn∈N,en+a>0u

(

en+a,n

)

. Third, VtO

(

a,e;

γ

)

is invariant to the bankruptcy policy instruments Dt,

ζ

t,

λ

t. Whilethismightseemrestrictive,itneednotbe.Forinstance,ifchoosingtheoutsideoptionresultsinbeingpermanently barredfromcreditmarkets,thentheoutsideoptionisindependentofbankruptcypolicy.

2.2. Householdproblem

Fortheremainder ofthepaper, Iwillsuppressthedependenceofvalue functions, policies,andpricesonthe planner policiesexceptwhennecessaryforclarity.Giventhepolicyinstrumentsandprices,thevaluefunctionofahouseholdwith

9_For_instance,_suppose_one_had_ei._∼i.d.

U[1,1+σ]discretized.Forσ≈0,theearningsriskisnegligible,butacutoff ¯e∈[1,1+σ]suchthatDt(a,e,n) =

{0,1}ifandonlyife≤e¯causesafraction ( e¯−1) /σofthedebttobeforgiven.Thisargumentcanbeextendedtoanyeﬃciencyprocessbyincorporating asmalli.i.d.shock.

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a,e,tthathasnorecordofabankruptcyis13

V t

(

a, e, h =0

)

=maxo,doV tO

(

a, e

)

+dV tD

(

a, e

)

+

(

1−o

)(

1−d

)

V tR

(

a, e

)

s.t. o ∈

{

0, 1

}

, d ∈n∈ND t

(

a, e, n

)

, od =0

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wherethevalueofrepayingdebtis

V R

t

(

a, e

)

=maxa_∈A,n∈Nu

(

c, n

)

+

β

e

π

ee|tV t+1

(

a , e , 0

)

s.t. c +q t

(

a , e

)

a = en +a, c > 0

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andthevalueofdefaultingviabankruptcyis

V D

t

(

a, e

)

=maxa∈A,n∈Nu

(

c, n

)

−

κ

(

e

)

+

β

e

π

ee_|t

(

1−

λ

t

)

V t+1

(

a , e , 0

)

+

λ

tV t+1

(

a , e , 1

)

s.t. c +q ¯a +

ζ

t

(

a, e, n

)

= en, c > 0, a ≥0

D t

(

a, e, n

)

=

{

0, 1

}

.

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Notethat since bankruptcy policy conditionson labor, to obtain a dischargethe householdmust chooselabor such that

Dt

(

a,e,n

)

=

{

0,1

}

.Ifahousehold’sbudgetconstraintwouldbeemptyuponchoosingtorepay,theymustchooseeitherthe outsideoptionor,iftheplannerletsthem,bankruptcy.

Thevalueofhavingabankruptcyrecord,h₌1_,is

V t

(

a, e, h =1

)

=maxa∈A,n∈Nu

(

c, n

)

+

β

e

π

ee|t

(

1−

λ

t

)

V t+1

(

a , e , 0

)

+

λ

tV t+1

(

a , e , 1

)

c +q ¯a = en +a, c > 0, a ≥0. (6)

NotethatIdonotallow householdsinbadstandingaccessto anoutsideoption.Thisiswithoutlossofgenerality(wlog) becauseVO

t

(

a,e

)

islessthanthevalueofautarkyandVt

(

a,e,h=1

)

isgreaterthanit.

2.3.Equilibrium

Equilibriumforagivensetofpolicyinstruments,Dt,

λ

t,

ζ

t,

γ

andarisk-freepriceq¯isa setofpolicies,ct,nt,at,dt,ot, valuefunctionsVt,pricesqt,andrepaymentratesptsuchthat

1.Householdsoptimizetakingpricesasgiven.

2.Thepricescheduleqtisgivenbynoarbitrage:qt

(

a,e

)

=q¯pt

(

a,e

)

. 3.Repaymentratespt areconsistent:

p t

(

a , e

)

=

e

π

ee|t

(

1−max

{

d t+1

(

a , e

)

, o t+1

(

a , e

)

}

)

. (7)

2.4.Planner’sproblem

Theplannersolves

max_ζ_,_λ_,D,γa,e,t

α

t

(

a, e

)

V t

(

a, e, h =0

)

s.t. D t

(

a, e, n

)

∈

{{

0, 1

}

,

{

0

}}

,

λ

t∈[0, 1],

ζ

t

(

a, e, n

)

≥0,

γ

∈

∀

a, e, n, t

(8)

where

αt

(a,e)_≥0istheweightplacedontype

(

a_,e_,t_,h₌0

)

.14

3. Theoreticalresults

Thissectioncharacterizes theoptimalpolicytheoreticallyunderfullinformation.Tosimplifytheproofs,householdsare assumedtorepaywhenindifferentbetweenbankruptcyandrepayment.Ifindifferentbetweenbankruptcyandtheoutside option,theyareassumedtoﬁleforbankruptcyifitisanoption.AllproofsarerelegatedtoAppendixA.

Beforecharacterizingtheoptimalpolicy,itisessentialtoestablishthatequilibriumexistsgivenplannerinstruments.The onlywaythiswouldnotbethecaseisifasolutiontothehouseholdproblemdidnotexist.However,sincehouseholdsonly haveaﬁnitenumberofchoices, andtheyalwayshaveatleastonefeasiblechoice(speciﬁcally,neverborrowingorsaving), thisisguaranteed.

Proposition1. Foranypolicychoiceoftheplanner,anequilibriumexists.

13 _To_reduce_notation,_I_treat_bankruptcy_and_repayment_as_feasible_options._If_bankruptcy_is_not_feasible,_then_the_value_function_should_be_reformulated withoutreferencetoVD_and_similarly_for_VR_._{Additionally,}_default_options_are_only_available_if_a_<_0._To_handle_the_case_of_t₌_T_,_take_V

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Proposition 2 shows that the outsideoption can be madeirrelevant: If VO _is _suﬃciently _negative, _the _outside_option will neverbe chosen inequilibrium. Consequently, the plannercan completelyeliminate equilibriumdefault whenVO _is suﬃcientlylow:ByspecifyingDt

(

a,e,n

)

=

{

0

}

foralla,e,n,t,householdsalwayschoosetorepaytheirdebtandthemodel isequivalenttoastandardAiyagari(1994)model.The keytotheproofisthat alowerboundonlifetimeutilitywouldbe violatediftheoutsideoptionwerechoseninequilibriumandwassuﬃcientlyunattractive.

Proposition2. Thereexistsa

δ

suchthatVO

t

(

a,e

)

<

δ

foralla,e,timpliestheoutsideoptionisneverchoseninequilibrium.

Aprincipaldiﬃculty incharacterizingbankruptcypolicyisthatimprovementsinVtalsoimproveVtR−1andVtD−1 through

continuationutility.DependingonhowVR

t−1−VtD−1 changes,bankruptcymaybecomemoreattractive,worseningcreditand

potentiallywelfareforaget−2households.However,Lemma1showstheincreaseincontinuationutilityintheVD t−1

prob-lem canbe undone through acommensurate increase in the

ζt

₋1 ﬁlingcost. This causesthe spreadVtR−1−VtD−1 tonever

decrease,whicheffectivelyallowsVt increasestotrickledowntoyoungerhouseholds.

Lemma 1. Fixsome (a, e,t) with t_<Tand supposeVD

t

(

a,e

)

is well-deﬁned in thatithas atleast one feasible choice.If the

continuation utility oftheVD

t

(

a,e

)

problem increasesdue to some policy change,VtD

(

a,e

)

can be held constantby increasing

ζ

t(a,e,n)foralln.

Almost as a corollary of thisresult is that

λ

t>0, i.e., retaining a bankruptcy ﬂag, is unnecessary.Speciﬁcally, lower-ing

λt

to zero increases continuation utility in the VD

t problem, but that effect can be undone via higher ﬁling costs.

Proposition 3 formally establishes that

λ

t=0 isweakly optimal. This result standsin sharp contrast to the U.S. system whereabankruptcyﬁlingstaysonahousehold’screditreportfortenyears.

Proposition3. Foranypolicywith

λ

t>0,thereexistsanotherpolicywith

λ

t=0generatingthesameplannerutility.

Becauseof theassumptionthat abankruptcyrecord preventsborrowing, setting

λ

t=0both removesinformationand enablesbankrupthouseholdstobeginborrowingagainassoonaspossible.AsmentionedinSection2,nothinglegally pre-ventslendingtobankrupthouseholds,butneverthelesstheyseemtohavelittleaccesstocredit.Ifthisisbecausebankruptcy signalsalackofcreditworthiness,thenretainingthisrecordwouldpresumablybe optimal.Inotherwords,

λt

=0—inthe sense ofremoving information—would likelynot be optimalunderasymmetric information.However, thelogic ofsetting

λ

t=0—inthesense ofallowing bankrupthouseholds tobeginborrowingquickly—would stillseemto apply.Additionally, whiletheinformation-basedexplanationofreducedcreditpost-bankruptcy isveryintuitive, Athreyaetal.(2012b)argue— basedonresultsfroma modelthat allowsborrowingpost-bankruptcyandasymmetricinformation—thata moreplausible explanationisafull-informationresponsetothepersistentshocksthattriggerbankruptcy.15_{Consequently,}_the_optimal

λ

t, bothintermsofdroppinginformationandallowingborrowing,maybeclosetozeroevenwithasymmetricinformation.

Fromthispointon,Iwillwlogconsideronlypoliciesthathave

λ

t=0forallt.Aconvenientpropertyofthisisthatthere isnolongeraneedtocarryaroundabankruptcyﬂag(ashouseholdsarebornwithh=0andnevertransitiontoh=1).So, IwriteVt(a,e)inplaceofVt

(

a,e,h=0

)

andsimilarlyforthepolicyfunctions,neverreferringtoVt

(

a,e,h=1

)

oritspolicy functions. Anotherconvenientpropertyisthat thecontinuationutilities oftherepaymentandbankruptcyﬁlingproblems arenowthesame.

Lemma2establishesakeymethodfortheplannertoimproveonanexistingpolicy.Speciﬁcally,supposetheplannercan improveVt whileloweringmax{dt,ot}.Inthiscase,two indirectbeneﬁtsaccrue toaget−1households.First,they have improvedcontinuationutility.Second,theyhaveimproveddebtpricing.TheseeffectselseequalincreaseVt−1,VtR−1,andVtD−1.

Byincreasingt−1ﬁlingcoststoholdVD

t−1ﬁxed(Lemma1),Vt−1 increasesandmax

{

dt−1,ot−1

}

decreases.Thatis,improved

welfareandcreditforagethouseholdscanbetranslatedintoimprovedwelfareandcreditforaget₋1households.Using backwardsinduction,theseimprovementscanbeappliedtoallyoungerhouseholds.

Lemma2. Consideranarbitrarypolicy(

ζ

,D,

γ

).Let

(

ζ

˜,D˜,

γ

)

beapolicythatforsomethasVt˜

(

a,e

)

higher—relativeto(

ζ

,D,

γ

)—foralla,e,t˜≥t andmax{dt(a,e),ot(a,e)}lowerforalla,e.Thenthereexistsapolicy(

ζ

∗,D∗,

γ

)satisfying 1.

ζ

_t_˜∗=

ζ

˜_˜

t andD∗t˜=D˜˜t forallt˜≥t 2. D∗_t_˜=D˜t forallt˜<t.

thathas higherVt˜

(

a,e

)

—relative to (

ζ

,D,

γ

)—for all a,e,t˜.Moreover, forall a,e,t˜≥t the policies (

ζ

∗,D∗,

γ

) and

(

ζ

˜,D˜,

γ

)

inducethesameV˜t

(

a,e

)

.

Proposition 4showsthatthe plannershould not tiedebtforgiveness tolabor supply.The intuitivereasonis thatonly allowing bankruptcyforsomeamounts oflabor (or,equivalently,earnings)distortsthehousehold’sdecisionandinflicts a dead-weightloss.Specifically,iftheplannerwantsahouseholdtofileinsomestatea,e,t,thenchoosingDt

(

a,e,n

)

=

{

0,1

}

(7)

foronlysomevaluesofnconstrainsthehousehold’slaborchoiceandreducesVD

t relativetoapolicywithDt

(

a,e,n

)

=

{

0,1

}

foralln.Hence,usingDt

(

a,e,n

)

=

{

0,1

}

forallnimprovesVt(a,e)withoutchangingmax

{

dt

(

a,e

)

,ot

(

a,e

)

}

=1.Ontheother hand,iftheplannerdoesnotwantahouseholdtoﬁle,hemaysimplysetDt

(

a,e,n

)

=

{

0

}

forallnwithoutaffectingwelfare orcredit.Consequently,thereisnoreasontotiedebtforgivenesstohoursworked.

Proposition4. Forany policy(

ζ

,D,

γ

) specifyingDt(a,e,n)varyingin nforsome a, e,t,thereis a policy(

ζ

∗,D∗,

γ

) with

D∗_t_˜

(

a˜,e˜,n

)

invarianttonandVt˜

(

a˜,e˜

)

higherforalla˜,e˜,t˜.

Wlog,InowrestricttheplannertochoosingpolicieswithDt(a,e,n)invariantton.

Anotherdead-weightlossis inducedby havingstrictlypositive filingcosts.Specifically, iftheplannerdoesnot wanta householdtofile,hemaysimplynotletthembysettingDt

(

a,e,n

)

=

{

0

}

.Ontheother hand,ifhedoeswantthemtoﬁle, thenthe damage tocreditors,max{dt(a,e), ot(a,e)},is not mitigatedby having

ζ

t(a, e, n)>0. Hence,in eithercaseit is optimaltoset

ζ

t

(

a,e,n

)

=0.Proposition5establishesthisimportantresult.

Proposition5. Considera policy(

ζ

,D,

γ

) anddeﬁnet₌max

{

0

}

_∪

{

t

|

∃

a_,e_,nwith

ζt

(

a_,e_,n

)

_>0

}

. Ift_>0_,thenthere isa policy (

ζ

∗,D∗,

γ

)—identical to (

ζ

, D,

γ

) fort>t—with

ζ

∗ equal to zero thathas higherVt(a, e) for alla,e, t.If under the

originalpolicy

ζt

(

a,e,n_t

(

a,e

))

>0andd_t

(

a,e

)

=1,thenV_t

(

a,e

)

isstrictlyhigherunderthenewpolicy.

Propositions3–5 provide the first main resultof the paper: The planner should eitherallow a household to file for bankruptcy,makingthemaswellofaspossible,orcompletelypreventthemfromfiling.16 _The_optimal_policy_thus _differs fromU.S.bankruptcypolicy intwokeyways.First,bankruptcyintheU.S.hasbothdirectcostsintheformoffilingcosts andindirectcosts inthe formofexclusion.The optimalpolicy hasneither andallows—inthe truestsense—a freshstart. Second,U.S.lawwithfewexceptionsallowseveryhouseholdtofile.Intheoptimalpolicy,accesstobankruptcyisrestricted. Becausetheplanner’sproblemreducestochoosingDt(a,e,n)∈{{0},{0,1}}foralla,e,t(andanarbitraryn)and

γ

∈

, theplanner’s choice setis ﬁnite.Moreover, every choice isfeasible asProposition 1 showsan equilibriumexists forany plannerpolicy.Consequently,anoptimalpolicyexists,andthisisestablishedformallyinProposition6.

Proposition6. Anoptimalpolicywith

ζt

(

a_,e_,n

)

₌

λt

₌0foralla,e,n,tandwithDt(a,e,n)invarianttonforalla,e,texists.

Intheabsenceofﬁlingcostsandlabor distortions,bankruptcyisalwaysbetterthanautarky,whichitselfisbetterthan theoutsideoption by assumption.Hence, formal defaultvia bankruptcyisalways preferredin welfareterms toinformal defaultvia the outside option: VD_≥_VO_. _{Additionally,}_creditors _receive _nothing_when _a _household _defaults _irrespective _of whetherthedefaultisformalorinformal.Consequently,theplannershouldalwaysallowbankruptcywhenevertheoutside optionwouldotherwisebepreferable.Proposition7formallyestablishesthissecondmainresult.

Proposition7. Supposea policy (

ζ

,D,

γ

) has

ζ

equalto zero.If Dt

(

a,e,n

)

=

{

0

}

forsome a,e,t whereVtO

(

a,e

)

>VtR

(

a,e

)

orVR

t

(

a,e

)

is undeﬁned, then there is another policy (

ζ

∗, D∗,

γ

) with

ζ

∗ equalto zero that (1)speciﬁes D∗t˜

(

a˜,e˜,n˜

)

=

{

0,1

}

wheneverVO

˜

t

(

a˜,e˜

)

>V R

˜

t

(

a˜,e˜

)

orV R

˜

t

(

a˜,e˜

)

isundeﬁnedand(2)hashigherVt˜

(

a˜,e˜

)

foralla˜,e˜,n˜,t˜.

AtrivialconsequenceofProposition7isthattheplannershould notsetD=

{

0

}

forallstates:BecauseVR _is_undeﬁned ifdebtislargeenough, D=

{

0,1

}

isalways optimalforhighlyindebtedstates.Onereasonthat isinteresting isanatural borrowinglimiteconomy,formallystatedbelow,willneverbeoptimal.

Deﬁnition. AnaturalborrowinglimiteconomyisaneconomyhavingDt

(

a,e,n

)

=

{

0

}

foralla,e,n,twith

γ

havingVtO

(

a,e

)

lowenough,inthesenseofProposition2,tomaketheoutsideoptionirrelevant.

Thisdeﬁnition preciselycaptures the notionof anatural borrowinglimit economy inAiyagari (1994).Speciﬁcally, (1) householdsalways chooseto repaytheir debt;(2)they can borrow atarisk-free rateup tothe netpresentvalue ofthe worstpossibleearningsstream;and(3)theyneverborrowmorethanthisamount.(Whilelaboriselasticallysuppliedhere, one can have inelasticlabor supply, like inAiyagari, 1994, by making N a singleton.)Because a naturalborrowing limit requiresD=

{

0

}

forallstates,evenforstateswheredebtisextremelylargeandhencerepaymentisnotfeasible,itmustbe suboptimal.ThisisformallystatedinCorollary1.

Corollary1. Anaturalborrowinglimiteconomyisweakly inferiorto onethatallowsbankruptcywheneverVO

t

(

a,e

)

>VtR

(

a,e

)

orVR

t

(

a,e

)

isundeﬁned.

This result sheds light on an important question of whether the natural borrowing limit is optimal. This question hasbeen investigatedquantitatively by numerousauthors who, almostwithout exception, ﬁnda naturalborrowinglimit

16 _The_bang-bang_nature_of_this_result _is_driven_by_the _assumptions_that_the _planner_cannot_partially_forgive _debt_and _cannot_transfer_resources_to creditors.Whilethelogicofeliminatingdead-weightcostswouldseemtocarryovertoamoregeneralsettingthatrelaxestheseassumptions(sothat,e.g.,

λt=0wouldstillbeoptimal),thebang-bangresultwouldlikelynotsurvive.Asdiscussedinfootnote9,thetheorycanallowforpartialdebtforgiveness,in

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significantlyimproveswelfarerelativetoeconomiescalibratedtomatchU.S.statistics.17_However,_this_result_shows_that_for awideclassofutilityfunctionsandlaborefficiencyprocesses—evenefficiencyprocesseswherethenaturalborrowinglimit isverylarge—itisalwaysbettertoallowbankruptcyinsomestates.

An important caveat to this and the other results is the absence of general equilibrium (GE) effects, which Li and Sarte (2006) have shown can be importantin large-scale bankruptcy reforms.In the partial equilibriumcontext of this paper,improving value functions(e.g.,by allowing bankruptcyinhighly-indebted states) orincreasing borrowingoptions alwaysimproveswelfare.InGE,thisneednotbethecase:Makinghighly-indebtedstatesunattractiveorlimitingborrowing mayincrease theaggregatecapitalstockandleadtohigherwages,potentiallyimprovingwelfare.Onewaytointerpretthe resultsisby consideringthemasareformcoordinatedwithmonetary policyinordertoachieve aﬁxed realinterest rate andrealwage.

AnotherconsequenceofProposition7isstatedinCorollary2:Optimally,thepolicyresultsintheoutsideoptionnever beingchosen.ThisstandsincontrasttotheU.S.whereinformaldefaultisaregular occurrence.Thebasicintuitionisthat formaldefaultcanreplicateorimproveonwhateverinformaldefaultaccomplishesbothforhouseholdsandcreditors.

Corollary2. Withoutlossofgenerality,theoptimalpolicyhastheoutsideoptionneverchosen.

Whiletheoutsideoptionisneverchosen,itstillplaysacriticalrole.Speciﬁcally,itpinsdowninwhichstatestheplanner “must” allowa householdtoﬁle forbankruptcy.The plannerallows bankruptcywheneverVR_<_VO_._If_VO _is_very _low,_this onlyoccursforlargelevelsofdebt.IfVO _equals_the_value_of_autarky,_this_occurs_for_small_amounts_of_debt.

Infact,Proposition8showsthatiftheoutsideoptionvalueisequaltotheautarkyvalueandtherearenopsychiccosts, the planner can do no better than a zero borrowinglimit economy. The result obtains because bankruptcy is optimally allowed whenevertheoutsideoptionisbetter thanrepayment.Fort=T,there isnoborrowingsothe valueofrepaying debtisstrictlylessthanthevalueofautarky.Consequently,repaymentisworsethantheoutsideoptionandsotheplanner allows bankruptcy for all indebted households.This then causes qT−1=0, resulting inall householdsaged T−1having

value of repaying lessthan the value ofautarky andthe outsideoption. Thisprocess repeats resulting in a full collapse ofthe creditmarket. The resultisqualitatively similarto that inBulowandRogoff (1987) where reputationalone isnot enoughtosustaindebt.

Proposition 8. Suppose there are no psychic costs of default, i.e.,

κ

(

e

)

=0. Further, suppose thatfor all

γ

∈

and all a, e,

t,VO

t

(

a,e

)

=VtA

(

a,e

)

. Then, an optimum features qt

(

a,e

)

=0 for alla<0and all e,t. Thatis, an optimum features a zero

borrowinglimit.Moreover,if

α

t(a,e)>0foralla,e,t,theneverypolicyproduces Vt(a,e) identicaltothatofazeroborrowing

limiteconomyforalla,e,t.

Proposition 8 shows that if default outside the bankruptcy system is too attractive, nothing can be gained from bankruptcy. Proposition 9complements itby showing that informal defaultshould optimallybe madeasunattractive as possible.The resultholds because(1)bankruptcyisoptimallyusedinplace oftheoutsideoption,soloweringVO _has_no directeffectonwelfareand(2)aworseoutsideoptionprovidestheplannerwithadditionallyﬂexibility aboutwhendebt shouldbeforgiven.

Proposition9. Consideranyoptimalpolicy(

ζ

,D,

γ

).Ifthereisa

γ

∗∈

withVO

t

(

a,e;

γ

∗

)

≤VtO

(

a,e;

γ

)

foralla,e,t,then

γ

∗

ispartofanoptimalpolicy(

ζ

∗,D∗,

γ

∗)having

ζ

∗₌0.

4. Quantitativeresults

The modelisnow broughttothe datatobetter characterizethe optimalpolicy.Thissection assumesfullinformation andfullcommitment(the latterallowingVO _to_be_arbitrarily _negative)_and_so_gives_an _upper_bound_on_what_bankruptcy policycanaccomplish.Thenextsectionwillassesshowmuchandinwhatwaysconstraintsontheoptimalpolicymatter.

As inLivshitsetal. (2007), amodel periodis threeyears.Households age1(real age24) to R−1 (real age63) have one labor eﬃciencyprocess, representing workingagerisk, and householdsaged R(real age 66) toT (real age 84) have another,representingguaranteedretirementincome.TheworkingageprocessisestimatedfromthePanelStudyofIncome Dynamics (PSID) data.As istypical inthe literature, theretirement process hasno risk, anditis calibratedto matchan averagereplacementrate.

4.1. Estimation

ThePSIDdataIuseisfromHeathcoteetal.(2010)andrangesfrom1967to2002.Ithasbeencleanedandprocessedin anumberofways,suchasextrapolationoftop-codedvaluesassumingaParetodistributionanddroppingobservationswith

(9)

Table1

PSIDestimation.

First-stageregression GMM

ν1 −0.959 (0.080) ρ 0.969 (0.003) ν2 0.237 (0.020) ση2 0.027 (0.002)

ν3 −0.017 (0.002) ση,21 0.233 (0.004)

N 142,595 σ2

ε 0.065 (0.002)

R2 _0.030

implausiblelevelsofconsumption.Inadditiontotheseassumptions,Irestrictthesampletoindividualswithagesbetween 24and63and—fortheeﬃciencyprocess estimation—requiretheheadand,ifpresent,the“wife” (inthePSIDsense)work acombinedamountofatleast260h.Iassumetheprocesshastheform

loge i,t =

ν

0+

ν

1 h +

ν

2h 2+

ν

3h 3+u i,t

u i,t = z i,t+

ε

i,t

z i,t =

ρ

z i,t−1+

η

i,t

z i,1∼N

(

0,

σ

_η2_,1

)

,

η

i,t∼N

(

0,

σ

_η2

)

,

ε

it∼N

(

0,

σ

_ε2

)

(9)

wherehisageover10,t isageminus 23,andallshocksarei.i.d.Allowingforaseparate varianceofthepersistentshock early in life allows, to some extent, for features that are missingfrom the model such as collegeattainment, race, and gender.Imeasureei,tusingequivalizedtotallaborearnings(theheadandwife’sjointlaborearnings)dividedbytotalhours (headandwife’sjointhoursworked).Conceptually,ei,tcapturestheextraearningsaccruingtoanindividualifhoursworked arescaledinproportiontothecurrentdivision oflaborinahousehold. Ifthe householdhassizeone, itisjusttheusual deﬁnitionofthewage.

Thecoefficientsare identified followinga procedure similar toHeathcote etal.(2010). Theage profilecoefficients

ν

1,

ν

2,

ν

3 areobtainedfromanOLSregressionthatcontrolsfortimeeffects(

ν

0 isalsoobtainedfromOLS,butitisusedasa

normalizationconstantinthemodel).Theshockprocessparameters

ρ

_,

σ

2

η,1,

σ

η2,

σ

ε 2areidentiﬁedusingthevariancesEt

(

u2_i_,_t

)

andthesecond-orderautocovariancesEt

(

ui,tui,t+2

)

foreachage,year,andcohort.AsinHeathcoteetal.(2010),second-order

autocovariancesareusedbecauseobservationsonlyoccureverytwo-yearsafter1995.Intotal,292momentsidentifythe4 shockprocessparameters,andIusetheoptimalGMMestimator.

Table1displaystheparameterestimates.Thefirst-stageregressioncoefficientsimplymeanlabor efficiencydeclines3% fromage24to30(recallequivalizedwagesareused)andthengrowssteadilyfromage30to64foratotalincreaseof36%. TheestimatedshockprocessishighlypersistentwithparametersremarkablysimilartothoseinStoreslettenetal.(2004). Thisisdespitealargenumberofdifferences,foremostbeingadifferentdependentvariable.

4.2.Calibration

Tousetheestimatedefficiencyprocessparameters,Ispecifythemodelefficiencyprocessanalogouslyto(9)andconvert theannualestimatestothree-yearestimates.Specifically,Itake

ρ

3 _as_the_model_persistence,

(

₁₊

ρ

2₊

ρ

4

)

σ

2

η asthemodel persistentshockinnovationvariance,

σ

2

η,1 asthemodelvariance forz1,and

σ

_ε2 asthemodel varianceforthei.i.d.shock.

Thisconversiongivesthatthethree-yearaheadforecastmeansandvariancesarethesameinthemodelanddataandthat theinitialvariancesarethesame.Exceptfor

ν

0whichisusedasanormalizationconstant,theage-proﬁleparameterscarry

overdirectly.Forretiredhouseholds,

loge i,t=

ν

0+

ν

1h R−1+

ν

2h 2R−1+

ν

3h 3R−1+z i,R−1+log

(

. 5

)

(10)

wherehR−1=63/10.The log(.5)termgeneratesan exogenous 50%declineinlabor eﬃciencyandismeant tocapture

re-ducednon-assetincomeinretirement.

Consistentwiththetheory,thepersistentandtransitoryshocksarebothdiscretized.18_The_transitory_shock_is_discretized using3pointsat0and ± 1standarddeviations.The persistentshockisdiscretized using7linearly-spacedpointsonan age-dependentgridthatalways covers3standarddeviations(forretirees,thegridisthesameasforR₋1).Thetransition probabilitiesarecomputedusingTauchen(1986)’smethod.

Thelaborgridistakentobearbitrarilyﬁnebetween0.001and0.999(onecanthinkofthestepsizebetweengridpoints asbeingmachineepsilon).Theassetgridcannotbeasﬁnesinceincreasingthegridalsoincreasesthesearchspaceforthe planner’soptimalpolicy.Iuse50strictlynegativepointsand250pointsoverall.

Theutilityfunctionischosentobe

(

cθ

(

1−n

)

1−θ

)

1−σ_/

(

₁₋

σ

)

_._The_parameter

σ

_is_set_to₁₊₁_/

θ

_so_that_constant_relative riskaversion,1−

θ

(

1−

σ

)

,equals2foranycalibratedvalueof

θ

.Therisk-freepriceq¯issettogivea2%annualrealinterest rate.

18 _The_grids_used_in_the_computation_are_such_that₍_ε_,_z₎_can_always_be_recovered _from_e_. _That_is,_e₌_f₍_ε_,_z₎_is_invertible,_which_is_required_to_be perfectlyconsistentwiththetheory.Althoughonecanconstructexampleswherefisnotinvertible,thereisalwaysanarbitrarilysmallperturbationoff

(10)

Forwelfarecomparisons,Ispecifyanexante welfarefunctionwhere

α

1

(

a=0,e

)

=

πe

forall eand

αt

(

a,e

)

=0forall

othera,e,t(andsotheplannerseekstomaximizewelfareofnewbornhouseholds).Thishasbeenthemostcommonchoice intheliteratureforlife-cyclemodelsfeaturingdefault.19_This_welfare_measure_accounts_for_the_welfare_of_older_households becauseV1

(

a=0,e

)

implicitlyvaluesutilityatallfuturestatesreachedwithpositiveprobability.

Forcomparisonwiththeoptimalpolicyanddeterminingtheremaining parametervalues,Iattempttocapturethe cur-rent U.S. bankruptcy system andthe value of informal default. In the U.S., any householdin good standing can ﬁle for bankruptcy,soIsetDt

(

a,e,n

)

=

{

0,1

}

foralla,e,n,t.Sinceabankruptcystaysoncreditreportsfor10years,Iset

λt

=0.7 (for all t) to match thisduration. Filing costs in the U.S. arecomprised of two main components,official filing fees and attorneycosts.OfficialChapter 7bankruptcyfilingfeesin2016 were$335dollarsbutcanbe waivedfordebtorsearnings lessthan $150belowthe povertylevel (U.S.Courts,2016).Attorney’s fees,whichpresumably areincreasing inlabor effi-ciency,aresubstantial.White(2007)statesthata“typical” debtor’scostoffilingisbetween$1800and$2800(p.192).So,I parameterize

ζ

t(a,e,n) toallowforprogressivecosts

ζ

¯e.Theslopecoeﬃcient

ζ

¯isthenchosentogiveaverageﬁlingcosts equalto1% ofaverageearnings.Thisisbased onan$1800 ﬁlingcost asa fractionof3 yearsofearningsat$60,000.The psychiccosts

κ

(e)areparameterizedasmax

{

0_,

κ

0+

κ

1zi,t+

κ

2

ε

i,t

}

.

Informaldefaultismodeledasautarkywithanadditionalpsychiccost

κ

O_≥₀_in_the_ﬁrst_period,_which_makes_VO t

(

a,e

)

=

VA

t

(

a,e

)

−

κ

O.20 Thecost

κ

O ismeanttocaptureallcostsassociatedwithinformaldefaultsuch asphonecallsfrom credi-torsandwagegarnishments(withthemaintainedassumptionthatallcollectioneffortshaveazeronetrecoveryrate).For approximatingthecurrentU.S.system,Itreat

κ

O_as_a_parameter_and_calibrate_it._Later_in_this_section_I_will_treat

κ

O_as_the planner’spolicyinstrument

γ

(recall

γ

inﬂuencestheoutsideoptionvalue)andassumetheplannercanmake

κ

O_arbitrarily large.Inthenextsection,Iwillassumetheplannercanonlychoose

κ

O₌₀_.

The8parameters

(

β

,

θ

,

ν

0,

κ

0,

κ

1,

κ

2,

κ

O,

ζ

)

are usedtomatch8momentsfromthedata.The discountfactor

β

is

pri-marilyusedtomatchawealth-incomeratioof1basedonanannualratioof3.TheCobb–Douglasutilityweight

θ

isused tomatchthefractionofhours spentworking.InthePSIDsamplewithagerestrictionsbutnorestrictiononhoursworked, thefractionofpotential worktimespent workingis27.6%.(Thisestimateassumes16hofworktimeadaywith365work daysavailable peradult.)The earningsprocess parameter

ν

0 sets averageearnings to1,anormalization. Thepsychiccost

parameters

κ

0,

κ

1,

κ

2areusedtomatchadebt-incomeratioof0.028,aﬁlingrateof1.20%,andanannualizedinterestrate

of12.7%.ThefirstnumberisatriennialconversionoftheannualnumberinLivshitsetal.(2007),thefilingrateisbasedon Chapter7and13personal bankruptcyfilingsrelative totheworkingagepopulationin2015convertedtoa triennialrate, andthelastisfromGordon(2015).Theproportionalfilingcostparameter

ζ

isusedtomatchaﬁlingcost-earningsratioof 0.01.Thinkingoftheoutsideoptionasanabsorbingstatewithdebtsinpermanentcollection,thepsychiccostofinformal default

κ

O_is_used_to_target_the_percent_of_consumers_with_a_third-party_collection._In_the_past_decade_this_has_varied_from 12to15%(FRBNY,2017),andIuse13%asatarget.ThecalibratedparametersandmomentsaregiveninTable2.

Themodeldeliverseverytargetedmomentwithmixedperformancefortheuntargetedmoments.Thepopulationindebt isintherangeof6.7%(Chatterjeeetal.,2007)and17.6%(Wolff,2010),theformermeasuringstrictlynegativedebtpositions andthelatterweaklynegative.Theannualizedcharge-off rateisabouttwicewhatitshouldbe,buttheimpliedmodelrateis closelylinkedtotheinterestratebecausetherearenotransactioncosts.21_As_is_typically_the_case_for_a_normally_distributed eﬃciencyprocess,themodelunder-predictsearningsandwealthinequality.

Themodel’simpliedpsychiccostsareincreasinginbothpersistentandtransitoryearningswithamedianvalueof1.06. Thismedianvalueisverylarge,around16%intermsofconsumptionequivalentvariationforanewborn,andconsequently householdswithabove-medianearningsdefaultveryinfrequently.However,thepsychiccostsquicklygotozerofor house-holdswithnegativeshocks.e.g.,foratransitoryshockone-standarddeviationbelowitsmean,

ε

it=−0.25,thepsychiccost iszeroifzi,t iszero.

Fig.1 plotslife-cycleprofiles forconsumption,earnings, assetholdings,anddefaultrates(bothbankruptcy filingrates andoutsideoptiontake-uprates).Inthedata,bankruptciesaremostfrequentamong30–47yearolds(Livshitsetal.,2007, Figure1,p.404).Themodelcapturesthisbutistooextremewithessentiallynoonefilingforbankruptcyafterage45.Older householdsprefertheoutsideoptionbecausetheyhavelittleneedtoborrowinretirement,whichmakesgoing toautarky (withtheadditional

κ

O_cost)_relatively_less_costly.

4.3. Baselinecases

Before analyzing thefull optimal policy,it is usefulto considersome baseline cases.The ﬁrst baseline is the current U.S. system (referred to as US). The second is a zero borrowinglimit (ZBL) economy where qt

(

a,e

)

=0 for all a<0, e,

t, whichprovides a lower bound onutility.A thirdbaseline case isa naturalborrowinglimit (NBL)economy, which has

Dt

(

a,e,n

)

=

{

0

}

for all states.The last baseline must,by Corollary 1, improveon the NBL economy. Thispolicy speciﬁes

19_For_instance,_Livshits_et_al.₍₂₀₀₇₎_,_Athreya_et_al._(2009b₎_and_Gordon₍₂₀₁₅₎_all_use_this._While_testing_other_welfare_functions_is_trivial,_this_has_not beendoneforbrevity.

20_To_be_precise,_once_a_household_chooses_the_outside_option,_their_policies_are_as_if_in_autarky._That_is,_their_policies_are_the_optimal_policies correspond-ingto(2)witha=0intheﬁrstperiodandwhateverisimpliedbytheautarkypoliciesandshocksthereafter.Consequently,VO

t(a,e) =Xt(0,e) −κ(e) −κO

fora<0.Bydeﬁnition,VA

t(a,e) :=Xt(0,e) −κ(e) fora<0,soVtO(a,e) =VtA(a,e) −κO.

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Table2

Calibration.

Statistic Target Model Param. Value

Independentlydetermined

Annualrisk-freerealinterestrate 2% 2% q¯ 0.942

Badcreditrecordduration(years) 10 10 λ 0.7

Relativeriskaversion 2 2 σ 1+1/θ

Jointlydetermined

Pct.ofhouseholdsﬁling 1.20 1.28 κ0 1.06

Pct.ofhouseholdseverchoosingoutsideoption 13.00 13.10 κO _0.008

Debt-earningsratio× 100 2.80 2.69 κ1 1.23

Annualizedinterestrate 0.127 0.132 κ2 5.72

Wealth-incomeratio 1.00 1.01 β1/3 _0.945

Filingcost-earningsratio 0.010 0.010 _ζ¯ 0.008

Laborsupply 0.28 0.27 θ 0.36

Earnings(normalization) 1.00 1.01 ν 2.03

Untargetedmoments

Populationindebt∗ 6.7%–17.6% 11.6%

Annualizedcharge-off rate 0.05 0.10

Debt-incomeofﬁlers 0.54 0.90

Filerswithbelow-medianincome 0.69 1.00

Earningsgini 0.61 0.50

Wealthgini 0.80 0.72

Mean-medianearnings 1.57 1.51

Mean-medianwealth 4.03 2.46

Meaneﬃciency(normalization) – 2.92

Note:Modelincomeismeasuredasen+(1/q¯−1) aand“indebt” isa<0.

Table3

Welfareandallocationsfromdifferingdefaultpolicies.

Statistic ZBL US NBL D{} _D∗ _Simple _Naive _D∗

asym

WelfaregainrelativetoUS −1.18 0.00 5.42 7.43 11.6 10.6 3.47 5.54

Filingrate(%) 0.00 1.28 0.00 1.48 2.91 2.12 1.02 1.26

Defaultrate(%) 0.00 2.33 0.00 1.48 2.91 2.12 1.02 1.26

Totaldebt 0.00 0.03 0.12 0.29 0.46 0.60 0.07 0.13

Pop.indebt(%) 0.00 11.6 30.0 30.6 40.1 39.6 24.0 29.6

Interestrate(%) – 13.2 2.00 6.15 6.19 4.64 4.04 3.86

Charge-off rate (%) – 10.4 0.00 8.22 4.40 3.55 2.22 2.60

Totalassets 1.17 1.03 0.93 0.80 0.50 0.30 1.02 0.92

Totalconsumption 1.07 1.00 1.07 1.06 1.06 1.05 1.07 1.07

Totalearnings 1.00 1.01 1.01 1.02 1.03 1.04 1.01 1.01

Misreporting(%) – 0.00 – 0.00 0.00 0.00 23.6 25.8

Totaloutsideoption(%) 0.00 13.1 0.00 0.00 0.00 0.00 0.00 0.00

Note:Interestandcharge-off rateshavebeenannualized;Misreportingistherateconditionalonﬁling; Totalassetsaremeasuredasa(1−max{d,o}) dμ(capitalingeneralequilibrium).

Dt

(

a,e,n

)

=

{

0

}

forallstateshavingVtR

(

a,e

)

<VtO

(

a,e

)

orVtR

(

a,e

)

undeﬁned.IrefertothisastheD{}economy.Forallthe policiesexceptUS,Itake

κ

O_arbitrarily_negative_to_focus_on_what_optimal_bankruptcy_policy_can_achieve_if_informal_default canbemadeverycostly(

κ

O₌₀_will_be_taken_up_in_Section₅_).

TheresultsforZBL,NBL,US,andD{}_are_summarized_in_Table₃_._The_reported_welfare_gain_is_the_consumption_equivalent

welfaremeasurerelativetoUS.ZBLisworsethanUSwithawelfarelossof1.2%.NBLontheotherhanddoesmuchbetter witha welfaregain of5.4%.ComparingtheamountsofdebtinUSandNBL,debtisroughly 4timeslargerinNBL.These ﬁndingsagreewithalargeliterature thathasfoundtheNBLdoesmuchbetterthanmodeleconomiescalibratedtomatch U.S.datamomentsbecauseimprovedcreditallowshouseholdstoself-insure well.Fig.1showsNBLincreasesconsumption ofyounghouseholdsrelativetoUS,reducingsavingsearlyinlife.Front-loadingconsumptionimproveswelfaresince

β

_/q¯is signiﬁcantlylessthan1.

D{} _generates_a_7.4%_welfare_gain_relative_to_US,₂_percentage_points_larger_than_NBL’s_gain._While_Corollary₁_shows_D{}

mustweaklyoutperformNBL,quantitativelythereisasizabledifference.ThisisdespiteD{}_and_NBL_differing_only_in_states

wherehouseholdscannotrepay,inwhichcaseD{} _provides_a_fresh_start_and_NBL_does_not._While_implementing_NBL_would

beverycostly(probablyinvolvingareturntodebtors’prisons),implementingD{}_would_be_much_less_costly:_Courts_would

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30 40 50 60 70 80 0.8

0.9 1 1.1 1.2 1.3 1.4

Consumption

30 40 50 60 70 80

0.4 0.6 0.8 1 1.2 1.4 1.6

Earnings

D*

NBL

US

30 40 50 60 70 80

Age

0 1 2

3

Assets

30 40 50 60 70 80

Age

0 0.02 0.04 0.06 0.08

Default Rates

Bankruptcy

Outside Option

Fig.1. Life-cycleproﬁlesforD∗_,_NBL,_and_US.

4.4. Theoptimalbankruptcyrule

Thefulloptimalpolicy,labeledD∗,isnowconsidered.Itiscomputedusingageneticalgorithmandmultigridonasuper computerwiththehouseholdproblemsolvedsixmilliontimes.InterestedreadersmayconsultAppendixBformoredetails. AscanbeseeninTable3,D∗generatesa11.6%welfaregainrelativetoUS,morethandoublingNBL’s5.4%gain.Tosee whythewelfaregain isso large,firstnote thatrelative toUS,D∗ generates15timesmore debtand,despitethis, similar defaultrateswith50% lowerinterest rates.Thisdebtisevidenceofimprovedconsumptionsmoothingthat isalsoevident inthelife-cycleprofilesinFig.1.UnderD∗,anaveragehouseholdaccumulatesdebtequaltoroughlyhalfofmeanearnings by age40.It isnot until themid-40swhere deleveragingoccursandmeanassetsgrow inpreparationforretirement.In contrast,USandNBL havepositiveaverage assetpositionsthroughoutthelife-cycle.D∗’s largerdebtamounts inmid-life increase labor through a negative wealth effecton leisure, which results in households workingmore when their labor efficiencyishighest.Asaresult,averageearningsare2%largerinD∗thaninUS(ascanbeseeninTable3).

Theoptimalpolicyimprovesconsumptionsmoothingbyallowingbankruptcyonlyforthosewhobeneﬁtthemostfrom it. Thiscan be seen inFig. 2, whichplots the optimalpolicy forselect ages.A blackdot means that the state is visited in equilibriumwithprobability greater than 10−5_._A _blue_plus _sign _means_that _the _household_ﬁles_for _bankruptcy_if_the

plannerletsthem,i.e.,dt

(

a,e

)

=maxDt

(

a,e,n

)

.AredcirclemeansDt

(

a,e,n

)

=

{

0,1

}

,andsothehouseholddefaults.These parts ofthepolicy arethe mostmeaningfulbecausechanging thepolicy from{0,1}to{0}hasa directeffecton default decisions andthe planner’sobjectivefunction (byvirtueof exantewelfare andthe probability beinggreaterthan 10−5_).

Thehorizontalaxisisdebt(recallaverageearningsareroughly1),andtheverticalaxisgivesthestandarddeviationsfrom thepersistentshockmean.Thetopandbottompanelspresentthepolicyforthemedianandworsttransitoryshockvalues, respectively.

The optimalpolicytends to allowbankruptcywhen, conditionalon alevel ofdebt,householdsreceivethe worst per-sistent shock occurringwithsome non-negligible probability. Inthis sense, theplanner onlyallows default forthe most unlucky households. Restricting defaultto only a small fraction of households is important because default rates effec-tively act as a borrowing tax: Borrowing interest rates relative to risk-free rates are 1/

(

1−Ee|e,tdt+1

(

a,e

))

or roughly

Ee|e,tdt+1

(

a,e

)

forsmalldefaultrates;hence,ax%defaultratetranslates(roughly)intoaborrowingtaxofx%.Theoptimal

policymitigatesthisdistortionbyrestrictingdefaulttothosewhobeneﬁtthemostfromit.

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0 0.5 1 1.5 2 2.5 -3

-2 -1 0 1 2 3

Stdev from persistent mean

Median transitory shock, Age 30

Pr>.00001

& d=maxD

& d=1

0 1 2 3 4 5

-3 -2 -1 0 1 2

3

Median transitory shock, Age 51

0 0.5 1 1.5 2 2.5

Debt

-3 -2 -1 0 1 2 3

Stdev from persistent mean

Worst transitory shock, Age 30

0 1 2 3 4 5

Debt

-3 -2 -1 0 1 2

3

Worst transitory shock, Age 51

Fig.2. Theoptimalpolicybyage.

peoplewish to ﬁle inresponse tothe negativetransitory shock, theplannergenerallydoesnot let them(i.e., anegative transitoryshockcreatesmoreblueplussignsbutnotmanymoreredcircles).

Theoptimalpolicyrespondsmoretopersistentshocksthantransitoryshocksbecausethelatterareeasilyinsuredusing creditwhiletheformerarenot. Becausetransitoryshockshavenoimpactonfutureearnings, anegativeshockhittingan aget householdhasno impact onqt(a,e). Hence, householdscan readilyborrow to smooth out thissmallreduction in lifetimeincome.Incontrast,persistentshocksare muchhardertoself-insure via credit.Notonlydo theyreduce lifetime incomebyasubstantialamount,theyalsoreduceqt(a,e),whichmakesborrowingcostlypreciselywhenhouseholdswant toborrow. So,the planneris more likelyto offer bankruptcyin response to anegative persistentshock thana negative transitoryone.

Livshitsetal.(2007),Athreyaetal.(2009a),andothershaveshownthathigh-bankruptcy-costregimesaremore prefer-ablewhen shocks arelesspersistent andconverselyforlow-cost regimes. Theoptimal policy—by treating persistentand transitoryshocksdifferently—canhavethebestofbothworlds,requiringhouseholdstoself-insureagainsttransitoryshocks butallowingbankruptcyforinsuranceagainstpersistentones.

5. Constraintsontheoptimalpolicy

Theoptimalbankruptcypolicy vastlyimproveswelfare.However, the11.6%welfaregain isanupperbound becauseof three constraints on optimal policy.The first constraint is that the policy should be simple enough forhouseholds and creditors to understand. The second constraintis asymmetric informationwith moral hazard:By tying debtforgiveness toparticularefficiencylevels,theoptimalpolicy underasymmetricinformationincentivizeshouseholdstomisreporttheir efficiencystatus.Thethirdconstraintislimitedcommitmentinducedbyanattractiveoutsideoption.Thissectionexplores howmucheachoftheseconstraintsaffectstheoptimalpolicy.

5.1. Simpleimplementation

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Fig.3. AveragemarginaleffectonPr(maxD|).

Ofcourse,theparameterizationshould beabletocapturemuch oftheoptimalpolicy’svariationintherelevantregion ofthestate space, whichistheset

=

{

(

a,e,t

)

|

VD

t

(

a,e

)

>VtR

(

a,e

)

orVtR

(

a,e

)

infeasible

}

.Foranystate in

,households choosedt

(

a,e

)

=maxDt

(

a,e,n

)

andsochangingDt(a,e,n)from{0,1}to{0}orviceversaleadstoadirecteffectondefault decisions.Sucha changewillalsohaveadirecteffectonplannerwelfaretotheextentthestate isvisitedinequilibrium. Aftersomeexperimentation,thelogitregression

P r

(

maxD i,t=1

)

=1/

(

1+exp

(

−

δ

i,t

))

,

δ

i,t = b 0+b 11[age i,t≥40]+b 2log

(

−a i,t

)

+b 3z ˜i,t+b 4