The Review of Economic Studies, Ltd.
Optimal Lending Contracts and Firm Dynamics
Author(s): Rui Albuquerque and Hugo A. Hopenhayn
Source: The Review of Economic Studies, Vol. 71, No. 2 (Apr., 2004), pp. 285-315
Published by: Oxford University Press
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ReviewofEconomicStudies(2004) 71, 285-315
) 2004 The ReviewofEconomicStudiesLimited
and
OptimalLendingContracts
FirmDynamics
RUI ALBUQUERQUE
University
ofRochester
and
HUGO A. HOPENHAYN
TorcuatoDi Tella
and Universitat
University
ofRochester
2003 (Eds.)
FirstversionreceivedMay 2002; finalversionacceptedJanuary
constraints.
We developa generalmodel of lendingin thepresenceof endogenousborrowing
cannotbe
arise because borrowersface limitedliabilityand debt repayment
Borrowingconstraints
ofborrowing
ofdebtarecloselylinkedwiththedynamics
In themodel,thedynamics
enforced.
perfectly
thevalueof
In fact,borrowing
mustsatisfya dynamicconsistency
constraints.
constraints
requirement:
access to
access to short-term
current
debtrestricts
byfuture
capital,butis itselfdetermined
outstanding
where
in modelsofexogenousborrowing
is notguaranteed
credit.Thisdynamicconsistency
constraints,
of debt.We characterize
function
theabilityto raiseshort-term
capitalis limitedby someprespecified
at all histories-andderive
constraints
theoptimaldefault-free
contract-whichminimizesborrowing
consistent
The modelis qualitatively
forfirmgrowth,
survival,
leverageand debtmaturity.
implications
and
staticswithrespecttotechnology
andsurvival
withstylizedfactsonthegrowth
offirms.
Comparative
defaultconstraints
arederived.
1. INTRODUCTION
of firmgrowthand survival.1Such condeterminant
constraints
are an important
Borrowing
facedby firmsor
investment
of
to
straints
arise
in
connection
the
opportunities
financing
may
This
recession.
a
to
survive
as
those
paperdevelopsa
required
temporary
liquidityneeds,such
firm
for
its
studies
and
of
constraints
dynamics.In
implications
theory endogenousborrowing
A lending
to
default.
and
limited
the
firm's
ourmodel,debtis constrained
liability option
by
schedule.The choice
and a repayment
contractspecifiesan initialloan size, futurefinancing,
thefirm'sfuture
of thesevariablesin turndetermines
futuregrowth,
capacity,and
borrowing
and firmdynamicsarejointly
constraints
itsabilityandwillingness
to repay.Hence,borrowing
determined.
We studythisdynamicdesignproblem.
At
Ourmodelbuildson Thomasand Worral's(1994) modelof foreigndirectinvestment.
fixed
a
which
a
has
or
timezero a riskneutralborrower
requires
project
(firm entrepreneur)
initialset-upcost.Everyperiodtheprojectyieldsrevenuesthatincreasewiththeamountof
of firm
determinants
are important
constraints
1. Thereis considerableevidencesuggestingthatfinancing
firms
mayexplainwhysmallmanufacturing
dynamics.Gertlerand Gilchrist(1994) arguethatliquidityconstraints
and Timmermann
firms.Perez-Quiros
of monetary
policythando largermanufacturing
respondmoreto a tightening
as measured
conditions
ofcreditmarket
totheworsening
aremoresensitive
(2000) showthatinrecessionssmallerfirms
areessentialin
constraints
ratesanddefault
(1989) showthattheliquidity
premia.EvansandJovanovic
byhigherinterest
constraints
viewfinancial
thedecisiontobecomean entrepreneur.
Fazzari,HubbardandPetersen(1988), amongothers,
and Cabral and Mata (1996) are able to fit
as an explanationforthe dynamicbehaviourof aggregateinvestment,
a simplemodelof financing
firmsby estimating
of Portuguesemanufacturing
reasonablywell the size distribution
constraints.
Forsurveyssee Hubbard(1998) andStein(2003).
285
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286
REVIEW OF ECONOMIC STUDIES
capitalinputanda revenueshock,whichfollowsa Markovprocess.A riskneutrallender(bank)
financestheinitialinvestment
and providesliquidityto supportthefirm'sgrowthprocess.At
to and
anypointin timetheprojectmaybe liquidated.A lendingcontractspecifiestransfers
fromtheborrower
on all pastshocks.The firm,
anda liquidation
decision,contingent
payments
has limitedcommitment
andcan chooseto defaultat anytime.Defaultgivesthefirman outside
value whichincreaseswiththeamountof capitalfinancedand thecurrent
revenueshock.We
and limitedliability
thatmaximizestotalfirmvaluesubjectto theno-default
studythecontract
constraints.
The optimalcontract
betweenthevalueforthelender(whichwe
definesa Paretofrontier
call long-term
debt)and thevalue fortheentrepreneur
(whichwe call equity).By defaulting,
theentrepreneur
obtainsan outsidevaluebutloses itsequity.Thus,thefirm'sabilityto expand
is constrained
entitlement.
by theentrepreneurs
Equitygrowsovertimeas thefirmpays off
thelong-term
debt.This weakensborrowing
as theincreasedequityprovidesthe
constraints,
to
of
accumulate
amounts
bondingnecessary
capital.
increasing
debtequal totheinitialset-upcost.
an initiallong-term
Competition
bylendersdetermines
Theequilibrium
contract
maximizestheentrepreneur's
value)subject
equityvalue(andtotalfirm
to expectedrepayment
of thisset-upcost.A uniquedebtmaturity
structure
attainsthisinitial
eitherleadstodefaultora lowerinitialfirmvalue.
equityvalue.Anyotherdebtmaturity
In theoptimallendingcontract
equitygrowsat themaximumpossiblerate(theinterest
a
are no longerbinding.Along
constraints
level
at
which
rate),eventually
reaching
borrowing
thispath,dividendsarezero.As equitygrows,so does thesize ofthefirmanditsprobability
of
survival.
in
Ourmodelis thusconsistent
size
effects
found
the
literature
on
withthefirm
and
age
firm
In addition,
ofthe
determinant
itimpliesthatthecapitalstructure
is animportant
dynamics.2
firm'sgrowth
in Zingales(1998).
andexitdecisions,in accordancewiththeevidencepresented
firmrespondsto Tobin'sQ as
of a financially
constrained
Moreover,we showthatinvestment
well as thecurrent
levelof cash flows.Finally,we showthatfirms
withhighermarket-to-book
ratioof assetsdisplaya lowerratioof long-term
debtto short-term
debt,conditionalon the
revenueshock(e.g. Barclayand Smith,1995). This property
on the
arisesbecauseconditional
revenueshock,a firm
withhighermarket-to-book
withhigherequity
valueofassetsis also a firm
weakerborrowing
debt.
constraints
andlowerlong-term
entitlement,
The growthin thefirm'sequityis statecontingent,
musttrade-off
as theoptimalcontract
constraints
across
even
the
different
states.
As
a
borrowing
though processforfirm
consequence,
shocksis first-order
a more
andvalueexhibits
for
firm
Markov,theresulting
size,
profits
process
histories
structure.
As
an
firms
better
of
even
when
are
with
shocks
i.i.d.,
complexlag
example,
shockswillhavehigherequityandtotalvalue.Thisis nottheresultoflack of insurance,
as the
contract
we consideris fullystatecontingent.
offirmequityandsize on itspast
The dependence
is analysedin thispaper.
history
The optimallendingcontract
has someappealingcomparative
statics.Projectswithlower
sunkcosts,betterprospectsor growthopportunities
can sustaina largerinitialdebtand size,
exhibithighersurvivalprobability,
debtis fasterand borrowing
therepayment
of long-term
constraints
are eliminated
or
sooner.A lowervalueof default(e.g. betteroutsideenforcement
creditrating)implieslargerfirmsize,leverage,and,consistent
withBarclayand Smith(1995),
morelong-term
debt.Firmswithhigherrevenuesare also predicted
to havemoreleverageand
debt.Consistent
withthisprediction,
Titmanand Wessels(1988) presentevidence
long-term
thatfirmswithgreatersales displayhigherdebtto assetratios.Higherinterest
rateslead to a
smallerinitialsustainable
debtandfirm
size.Thoughtherelationship
betweenriskandborrowing
2. See Evans(1987) andHall (1987) forevidenceon growth
offirms
properties
byage andDunne,Robertsand
Samuelson(1989a,b)forevidenceon entry
andexit.
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ALBUQUERQUE & HOPENHAYN
OPTIMAL LENDING CONTRACTS
287
constraints
is less clear,ouranalysisindicatesthatriskierprojectscouldfacetighter
constraints.
Theseandothercomparative
models
staticsquestionsobviouslycannotbe addressedbyexisting
offirm
whichassumeexogenousborrowing
constraints.
dynamics
A theoryof endogenousborrowingconstraints
must addressthe followingdynamic
access
current
access to capital,butis itselfdetermined
consistency:
by future
equityrestricts
to credit.This dynamicconsistencyis not guaranteedin models of exogenousborrowing
wherethe abilityto raise short-term
constraints,
capital is limitedby some pre-specified
function
of equity.Of all dynamically
consistent
lendingplans,the optimalcontractis the
one thatmaximizesaccess to short-term
capitalandtheexpectedrateof decreaseoflong-term
debt.
of theoptimalcontract
versionof
in thedeterministic
is straightforward
Implementation
ourmodeland can be achievedby an initiallong-term
debtwitha specificmaturity
structure.
In everyperiodtheentrepreneur
makespayments
to thelenderand expandsits capacitywith
retained
case the
The outstanding
debtdecreasesovertime.In thestochastic
earnings.
long-term
firm'searnings
withhighlyproductive
tofinanceitsexpansionwhenconfronted
maynotsuffice
debtis necessarilystatecontingent.
In parallelto thedeterministic
states,so long-term
case,
averagedebtdecreasesovertime.
The paperthatis mostrelatedto ours is Thomas and Worral(1994). In theirmodel,
shocksarei.i.d.,thereis no liquidation
valueandoutsidevalueis givenbythefirm'srevenues.
Our extensionsare important
forthe analysisof firm
forseveralreasons.As a framework
thepossibility
of liquidation/exit
shocksare veryrelevant.Secondly,
and persistent
dynamics,
of the
a generaloutsidevaluefunction,
we are able to examinetherobustness
by considering
This
results.Finally,in contrast
to thecontract.
to theirmodel,thelenderhas fullcommitment
turnsout to simplify
theanalysisconsiderably
by allowingtheuse of dynamicprogramming
methodsandthusproviding
a moreextensive
oftheoptimalcontract.
characterization
Ourtheoryof debtis relatedto HartandMoore(1994, 1998).3In HartandMoore(1994)
thethreatof repudiation
setsa lowerboundon thepresentvalue obtained
by theentrepreneur
debtpayments
to an upperboundon thevalueofdebt.In addition,
byhim,whichis equivalent
are subjectto a cash flowconstraint.
Whilein their
These are also ourtwomainassumptions.
and
set-updebtis eitherraisedor notto fundtheproject,we let leveragebe statecontingent
timevarying.
and we allowforliquidationof
we letrevenuesbe statedependent
Furthermore,
thefirm.These featuresgiveus addedmarginsto discussthedynamicsof real and financial
choicesof firms.In our set-up,financialconstraints
giveriseto threetypesof inefficiencies:
feasibleinitially,
as in Hartand Moore; (ii) firmsmaybe
(i) projectsmaynotbe financially
creditconstrained
and producebelowtheoptimallevel,as in Thomasand Worral(1994); (iii)
be
too soon.
projectsmay terminated
As in
and Rosenthal(1990), ourmodelimpliesa maximumsustainablelongFernmindez
termdebt.In theirpaper,defaultconstraints
schedulesand in some
puta limiton repayment
cases makeit infeasiblefortheborrower
theloansreceived.
back
to
to credibly
commit paying
In suchcases, thelendermustforgivea certainfraction
of theinitialdebt.Noticethatif the
unlessthe
initialinvestment
exceedsthisborrowing
limit,theprojectwill notbe undertaken
contributes
of a projectthusdependson the
withits own funds.The feasibility
entrepreneur
the
natureof defaultconstraints.
Bulow and Rogoff(1989b) show thatif upon defaulting,
borrower
cannotbe excludedfromsavingatthemarket
rateandhas accessto actuarially
interest
total
fairinsurance,
inanyfeasiblecontract,
thereis nofinancially
feasiblecontract.
Furthermore,
debtis limitedbythecostsbornebytheborrower
upondefault.
3. Fora surveyon thecorporate
see HarrisandRaviv(1991).
literature
capitalstructure
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288
REVIEW OF ECONOMIC STUDIES
withincomplete
This paper is also relatedto the literature
on optimaldebt financing
contracts.4
We brieflyreferto the workthatis mostrelatedto ours. A dynamicmodel of
and lendingwithno defaultconstraints
introduced
was first
by Eatonand Gersovitz
borrowing
(1981), in the contextof international
lending.Kehoe and Levine (1993) presenta general
constraints
underno default(orparticipation)
Theseparticipation
constraints.
equilibrium
theory
to the
(2000) applythisframework
generateendogenousdebtlimits.Alvarezand Jermann
work
the
Also
is
analysisof risksharingand assetpricingwithlimitedcommitment. related
can
be
by Marcetand Marimon(1992). Their simulationssuggestthateconomicgrowth
Bulow
and
Rogoff(1989a) study
substantially
bythepresenceoflimitedenforcement.
impaired
a modelof international
wherethelendercan punishthe
commnitment,
lendingwithimperfect
borrower
andcontracts
can be renegotiated.
bymeansofdirectsanctions
thatgivesriseto a holdenforcement
is
a
sourceof contractual
Imperfect
incompleteness
is
An
obvious
of
this
up problem.
bonding.In ourmodel,the
way dealingwith problem through
is
builds
borrower
resolved
over
as
the
time
hold-upproblem gradually
upthisbondbyincreasing
itsclaimstofuture
of
contracts
A
in
the
context
similar
arises
situation
profits.
repeatedinsurance
whenagentscannotcommitnottotakeoutsideoffers
inthefuture.
HarrisandHolmstrom
(1982)
use thismechanism
becomes
to explainan increasing
whentheabilityofworkers
wageprofile,
knownovertime.Anotherexampleis Phelan(1995), thatconsidersa repeatedmoralhazard
modelwhereagentscan recontract
withoutsideprincipals,
increasingprofilesof
generating
consumption.
Diamond(1989) studiesreputation
buildingin a modelwithbothadverseselection(in
where
and
moral
hazard
(in projectchoice).Thereis a sequentialequilibrium
projectriskiness)
rate
interest
ratesdecreaseovertimeas theprobability
of defaultdecreases.The fallin interest
increasesthevalue of maintaining
and thusreducestheincentivesto take
a good reputation
andthelender.
excessiverisk,mitigating
theconflict
ofinterest
betweentheborrower
Thispaperis organizedas follows.Section2 providesa simpleexampleofourframework.
Section3 introduces
theoptimalcontract
themodel.In Section4 we characterize
alongseveral
more
financial
dimensions.
standard
Wealsodiscusshowtoimplement
theoptimalcontract
using
instruments.
As an alternative
Section5 presentstheproblemunderstudyas a
formulation,
results
constrained
growth
problem.Section6 concludes.We leavetheproofsandothertechnical
to theAppendix.
2. AN EXAMPLE
Considera projectthatrequiresan initialinvestment
Io andgivesrevenuesR (kt) in everyperiod,
wherekt is theworking
Netprofits
in
capitalemployed periodt. Cash flowsaredeterministic.
are givenby R(kt) - (1 + r)kt,whichare maximizedat theoptimallevelof workingcapital
k*. SupposethatR(k*) - (1 + r)k* > rlo, so investing
in thisprojectis profitable.
However,
theentrepreneur
the
would
has no wealth.In absenceof enforcement
problems, entrepreneur
and
k*
from
its
revenues
initiallyraisetotaldebtof Do = I0 + k*,reinvesting
everyperiod
to thedebtholder.The Modigliani-Miller
theoremappliesto thisset-up,so
makingpayments
thespecificpayments
tobe madeareundetermined.
The totalvalueoftheprojectis
R (k*) - (1 + r)k*
r
whichbytheabovecondition
exceedstheinitialinvestment.
4. AghionandBolton's(1992) seminalpaperdevelopsa theoryofcapitalstructure
based on wealthconstraints
on thepartoftheentrepreneur
Foran excellentsurveyof
andon theinability
contracts.
ofthepartiestowritecontingent
theliterature,
see Hart(1995).
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ALBUQUERQUE & HOPENHAYN
OPTIMAL LENDING CONTRACTS
289
Now supposethattheentrepreneur
withvalue kt
has an alternative
outsideopportunity
and unlesstheprojectgrantshimthisvalue,he would choose to default.Accordingto this
outsideopportunity
theentrepreneur
can stealtheworking
capital.It is usedhereforsimplicity
ofexposition
therestofthe
andis a specialcase ofthemoregeneralframework
usedthroughout
andthat
paper.Supposealso thatthisoutsideoptionis notcontractible
(1)
rio < R(k*) - (1 + r)k* < r(lo + k*).
As we willnowshow,thiscondition
to starttheprojectat
impliesthateventhoughitis efficient
fullscale,a default-free
i.e. Io < Do <
contract
mustnecessarily
involveborrowing
constraints,
Io + k*,so thatko < k*.To see this,noticethatiftheprojectwerecarriedoutin fullscale from
thestart,
thevaluetotheentrepreneur
wouldbe
Vo= W- o
R (k*) - (1 + r)k*
r
whichfromequation(1) impliesthatVo < k* andthustheentrepreneur
wouldchoosetodefault.
Thisexamplealso illustrates
thatborrowing
wouldariseevenifIo = 0.
constraints
We nowderivetheoptimalno-default
Let { Dt} and {Vt} denotethedebt
lendingcontract.
andequityvaluesin thiscontract.
Giventheoutsideopportunity,
itfollowsthat
kt < Vti,
withequalitywhenVt< k*,so as theequityofthefirm
increasesovertime,ktwillalso increase.
It is obviousthatwhileVt < k*,no dividendswillbe distributed,
so kt = Vt= (1 + r) Vt-1
This
in
turn
a payment
time
receive
that
at
t
the
lender
must
+
(1
implies
r)kt-1.
rt = R(kt-1) - kt = R(kt-1) - (1 + r)kt-1.
(2)
Whenk*is reached,theborrower
receivesrk*everyperiodas dividendsandthelendergetsthe
cash
flows.
be
remaining
LettingDo theinitialdebtandko = Do - 0o,itfollowsthatktwillgrow
atrater untiltheoptimallevelk*is reached.LettingW(ko) denotethetotalvalueoftheprojectif
theinitialsizeis ko,itfollowsthatthemaximum
initialdebtsatisfiesVo= W(ko)- lo = ko.The
initialtotaldebtDo = Io+ kotogether
withthematurity
defined
structure
byequation(2) define
theoptimallendingcontract.
structure
and theinitialdebtarejointly
Noticethatthematurity
determined:
a different
unlesstheinitialdebt
constraint
structure
violatestheno-default
maturity
is smaller.
In an abstractsense,theoptimaldebtcontract
policyforthefirmand a
specifiesa growth
a
of
cash
One
described
flows.
of
contract
was
this
above,involving
sequence
implementation
to
is
consider
initial
loan
and
a
an
An
alternative
single
Io + ko
repayment
plan.
implementation
initiallong-term
loans
loanskt.Short-term
debtBo = 10,together
witha sequenceofshort-term
arerepaidin fullattheendoftheperiod.All remaining
loan
to
the
are
long-term
profits applied
untilequityreachesthepointV = k*.Atthatpoint,thefirmis able toraisetheoptimalamount
of short-term
in greater
capital.Section4.5.1 in whatfollowsdiscussestheseimplementations
detailandgenerality.
Havingreachedk* does notimplythattheentrepreneur
getsridof thebank.Therecould
stillbe positivedebtatthispoint.Whatis trueis thatbeyondthislevelifequitykeepsincreasing,
a pointcan be reachedwheretheentrepreneur
does notneedthebank:whenequityis sufficient
to financetheproject.Whenuncertainty
is addedlateron,thisstatement
needstobe qualifiedto
say:underall contingencies.
Thisexampleshowshowborrowing
binds.
arisewhentheno-default
constraint
constraints
It also pointsto an important
difference
betweenthistheory
andstandard
modelson firm
growth
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REVIEW OF ECONOMIC STUDIES
290
Slope of-1
*--------------------k
V(B)
TheParetoFrontier
ko
-----------------Vmin
Bo Bmax
B
FIGURE 1
The Paretofrontier
withlimitedborrowing.
The lattermodelsconsiderthe firm'sdynamiccapitalaccumulation
In our set-up,theforward
to
constraint.
some
problemsubject
exogenouslygivenborrowing
debtlimitsandthe
between
current
link
nature
of
the
establishes
a
no-default
constraint
looking
the
structure
offuture
In
model
considers
this
our
sense,
optimaldesignoflong-term
repayments.
debtcontracts
andborrowing
constraints.
debtdecreases.
in theequityvalue Vtoccursat thesametimeas thelong-term
The growth
Thisis no coincidence;as we establishlater,theoptimalcontract
definesa pathalongthePareto
frontier
defined
ofthelenderandthefirm,
Figure1
Bt, Vtrespectively.
bythevalueentitlements
debt
level
of
the
Pareto
V
In
is
the
frontier,
1,
Bmax
(Bt).
long-term thatcan
plots
highest
Figure
This
be financed.
be crediblyrepaid.Anyinvestment
than
cannot
more
Bmax
projectrequiring
firm
from
to
the
defines
a
level
that
is
of
defaulting
implicitly
just enough keep
equity,Vmin,
wereBmaxgranted.
is loosenedand V increasesby
constraint
As B decreases,theborrowing
is thattheincreasein equitythat
morethanone-for-one
(moreon thisbelow). The intuition
resultsfromreducingdebt,improvesentrepreneur's
incentives
leadingto an increase
thereby
=
further
decreasesin
in firmvalue.As theunconstrained
V
is
i.e.
k*,
reached,
any
optimum
debtresultin one-for-one
increasesinequity,sincethetotalvalueoftheprojectis not
long-term
changed.
of debt?In ourexample,shortWhatare theimplications
of ourcontract
to thematurity
termcapitalis constrained
a negativerelationship
by kt < Vt = V(Bt). Thisdefinesimplicitly
betweenshort-term
debt.Faced withthisshort-term
constraint,
borrowing
capitalandlong-term
the firmwould choose to repayits long-term
debt as fastas possibleuntilthisshort-term
constraint
does notbind.
borrowing
of the
notindependent
The limiton totaldebtDt = kt + Bt < V(Bt) + Bt is typically
thecase that
betweenshortand longterm.Indeed,as we showlaterit is generally
composition
is: debt
thattotaldebtcan be highertheloweritslongtermcomponent
V'(Bt) < -1 implying
in Figure1.
structure
matters.
This is illustrated
in thecurvedportionof theParetofrontier
no
structure
whereV'(B) = -1 the maturity
Clearly,in the regionof the Paretofrontier
are
forfirmvalue.Thisdiscussionsuggestshowspecialaremodelswherefirms
longermatters
tohave
witha fixedtotalborrowing
confronted
limit.Moreover,
itmaynotbe desirableforfirms
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ALBUQUERQUE & HOPENHAYN
OPTIMAL LENDING CONTRACTS
291
thefreedom
and
tochoosehowmuchtorepayeachperiodatwill.Indeed,ifgivensuchfreedom
and
theinitialloan I0 + ko,thefirmwouldchoosenotto repay,keeptherevenues,
maintaining
exitin thefollowing
the
periodwithan outsidevalueR(ko) > (1 + r)ko= Vi. To avoiddefault,
initialloanwouldthenhavetobe lower.
In therestofthepaper,ourexampleis generalized
intwomaindimensions.
First,a general
classofno-default
shockstorevenues
areconsidered.
constraints
Second,we introduce
persistent
and considerthepossibility
We deriveoptimal
of liquidationas partof thelendingcontract.
andstudytheirimplications
forfirmgrowth
andsurvival.
lendingcontracts
3. THE MODEL
Timeis discreteandthetimehorizonis infinite.
Attimezeroan entrepreneur
by
maystarta firm
a projectwhichrequiresa fixedinitialinvestment
pursuing
Io > 0. The projectgivesa random
streamofrevenuesR(k,s) eachperiod,wherek is thecapitalinputands E S C 9 is a revenue
shock.The revenueshocks followsa Markovprocesswithconditional
cumulative
distribution
function
F(s', s). F(.) is jointlycontinuous.
The timingof eventswithina periodis as follows.First,theshocks is observed.After
observingtherevenueshock,thefirmcan eitherbe liquidated,at a value L(s), or continuein
Ifthefirm
in operation,
continues
salestakeplace,andrevenues
operation.
inputsarepurchased,
R(k,s) arecollectedattheendoftheperiod.Theserevenuesarean increasing
ofboth,
function
k ands. The revenueshocks is publiclyknown,so thereis no asymmetry
ofinformation.
Theentrepreneur
haslimitedliability.
It starts
withzerowealthandthusrequiresa lenderto
financetheinitialinvestment
andtheadvancements
ofcapitaleveryperiod.5Both,entrepreneur
andlender,discountflowsusingthesamediscountrater > 0.
have limited
Lenderscommitto long-term
contractswiththe firm.However,contracts
as theborrower
can choose to default.As in Hartand Moore (1994), onlythe
enforceability
or not,the
borrower
has the abilityto runthefirm.If thematchis endedeithervoluntarily
in
residualvaluefortheborrower
discussed
more
detail
is givenbya function
O (k,s), whichis
below.
A long-term
contractspecifiesa contingent
liquidationpolicy et (et = 1 if exit is
recommended
and et = 0 otherwise),
that
advancements
kt fromthelenderto thefirm
capital
takeplace at thebeginning
a
of
dividend
ofeachperiod,anda cashflowdistribution
consisting
flowdt and payments
to thelenderR(kt,st) - dt whichtakesplace at theend of theperiod.
dividendsand
Because the firmhas no additionalfundsdt
O0.The capitaladvancement,
=
at
time
are
on
the
t,
liquidationpolicy any
{kr-1, dr-1, er-1, Sr}r=l
historyht
contingent
ofprevioustransfers
andall shocks,including
st.6 Let H be thesetofall possiblehistories.
1. A feasiblecontract
is a mappingC : H -+
Definition
2 x {0, 1} suchthatforall
ht E H and (kt,dt, et) = C(ht), dt > 0, and et = 1 if er-1 = 1,forsome r < t.
Thetiming
ofeventsis as follows.Attimezero,a competitive
setoflendersoffer
long-term
contracts
to thefirm.
If thefirmacceptsa contract,
thelenderpaysfortheinitialinvestment
Io
andcarriesthecontract
as stipulated,
theagenthas notdeviatedfromthecorresponding
provided
5. If thefirmstartswithwealthw < Io,thentheprojectonlyneedsfinancing
of I' = I0 - w. If w > 10,then
thereis no needforexternal
lending.
6. To clarify
thenotation,
we shalluse letters
without
todenotecurrent
periodvaluesandwitha prime
subscripts
todenotenextperiod'svalue,exceptwhenan explicitreference
tolongerhorizonstakesplaceinwhichcase we shalluse
t,t + 1,....
subscripts
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292
REVIEW OF ECONOMIC STUDIES
and the firm
repayment
plan or defaulted.If the agentdeviates,the contractis terminated
tobe implemented.
continues
theplandefined
liquidated.Otherwise,
bythecontract
3.1. Contracts
withperfectenforceability
cancommittotheabovecontract
In theabsenceofenforcement
thelenderandthefirm
problems,
Since flowsare discountedat the same rate,the optimal
withoutany additionalconstraints.
forthematch.
contract
maximizestotalexpecteddiscounted
profits
function.
The following
Let 7r(s)= maxkR(k,s) - (1 + r)k denotetheprofit
assumptions
a
solution
to
this
maximization
guarantee
problem.
profit
R has thefollowing
properties:
Assumption1. Thefunction
(1) R(k,s) is continuous.
(2) Foreach s, R(k,s) - (1 + r)k is quasiconcaveink and has a maximum.
(3) Thereexistssomeb < oo suchthatforall s and k, -b < R(k,s) - (1 + r)k < b.
thefollowing
ThetotalsurplusofthematchW(s) satisfies
equation:
programming
dynamic
(3)
= max L(s), 1 Ar
W(s')F(ds', s) .
[R(s) +
W,(s)
If W(s) = L(s) forall s E S, thefirmwouldnotbe viableand wouldbe immediately
closed.The survivalset,S = {s : W(s) > L(s)}, is thesetof statesat whichthefirmwould
continuein theindustry.
Assumption2. ThesurvivalsetS is non-empty.
Withperfect
theModiglianiandMiller(1958) theorem
appliesandthecapital
enforceability
fromthis.First,survivaland
of thefirmis indeterminate.
Therearetwoimplications
structure
of
of its capitalstructure.
Second,thereis a multiplicity
growthof thefirmare independent
values.
the
same
have
that
debt
for
debt
shortand
present
long-run
optimal
repayment
plans
3.2. Contracts
withlimited
enforceability
contract
thelong-term
of default,
As illustrated
in ourexample,whenfirms
havethepossibility
to
firm's
the
about
details
now
ability default
give
proposesa uniquedebtrepayment
plan.We
andtheconstruction
ofthelong-term
contract.
3.2.1. Entrepreneur'soutsideopportunities. If thefirmchoosesto defaultit will do
a firmobtainsa
so priorto makinganypayments
to thelender.We assumethatby defaulting
ofthemodeland
is one oftheprimitives
totalvaluegivenby a function
O(k, s). Thisfunction
whichis common
facedbythefirm,
thevalueoftheoutsideinvestment
summarizes
opportunities
can collectrevenuesand disappear,
knowledgeto bothparties.For example,if theborrower
thenO(k, s) = R(k,s), as in Thomas
itselfas a new firm,
without
beingable to re-establish
butis excludedfromborrowing,
and Worral(1994). If thefirmcan continueoperations
saving
and insurance(as in Manuelli(1985), Marcetand Marimon(1992)), thenO(k, s) is thevalue
a firm
obtainedbythefirm
maybe excludedfrom
Alternatively,
optimalself-financing.
through
as inBulowandRogoff(1989b). O (k,s)
butnotfromsavingorpurchasing
insurance,
borrowing
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ALBUQUERQUE & HOPENHAYN
OPTIMAL LENDING CONTRACTS
293
willbe thevaluefunction
thusobtained.Another
exampleis obtainedifthefirmcan establisha
newcontract
witha bank,afterpayinga costforbreachofcontract.
If at thebeginning
thenthelatter
of someperiodthebankdecidesto liquidatethefirm,
ofthefirm's
obtainsa valueO (0, s). Thisvaluerepresents
an inalienablecomponent
capital;itis
theresidualvaluethatcannotbe takenawayfromtheentrepreneur
suchas hisopportunity
cost.
The difference
L (s) - 0 (0, s) thusrepresents
thecomponent
of theliquidationvaluethatcan
be appropriated
thanL(s) - 0 (0, s)
greater
bythebank.As inHartandMoore(1994) anything
couldbe renegotiated
We assumethat0 (0, s) = 0, forall s. Thisis not
downbytheborrower.
without
theexposition
of thepaper.In AppendixA we showwhat
loss,butit greatlysimplifies
on O.
modifications
arerequiredto generalizetheresults.We makethefollowing
assumptions
Assumption3. Thefunction0 has thefollowing
properties:
(1)
(2)
(3)
(4)
(5)
0(0, s) = 0,forall s.
O(k, s) > 0.
O(k, s)- k < L(s).
O is a continuous
function.
inbotharguments.
O is non-decreasing
Part2 is inlinewithourlimitedliability
Part3 saysthatinvoluntary
separations
assumption.
areless efficient
thanliquidation.
in an abstract
3.2.2. Long-termdebtcontracts. In thissectionwe formulate
thecontract
are discussedin Section5. A contract
form;alternative
specifiesa liquidation
implementations
advancesofcapitalkt,anda dividenddistribution
dt.This
policy,history-dependent
contingent
contract
debtlevelorvalueto
definesan equityvalueforthefirmVtandthelong-term
implicitly
thelenderBt. Theequityvalueforthefirm
dividends
whereas
sumoffuture
givesthediscounted
thelong-term
debtorvaluetothelendergivesthediscounted
cashflowsto thelender.Thus,the
totalassetvalueafterhistory
by Wt= Vt+ Bt.
htis defined
The totalvalue of debt includesdebtoriginatedat possiblydifferent
periodsof time.
of
all
debt
are
However,becausethereis onlyone lender,thesedifferent
homogeneous
vintages
andthereareno seniority
debtand Bt as longclaims.In spiteofthis,we label ktas short-term
termdebt.
ofperiodt + 1 after
LettingVt+1(s') denotethecontinuation
equityvalueatthebeginning
=
in
the
firm
not
willchoose to default periodt providedthatthe
(ht, kt,dt, et,s'),
history
ht+l
in thematch:
valueofoutsideopportunities
is lowerthanthevalueofitsentitlement
bystaying
O(kt,st) <
dt' +
Vt+l(s')F(ds', st)).
(4)
Sincethelendercan alwaysincludein thecontract
to liquidatethefirmand
a recommendation
sinceliquidation
or enforcement
is moreefficient
thandefault,
we requirethattheparticipation
constraint
be
satisfied
at
all
times.
equation(4)
A feasiblecontractis enforceableif, afterany historyht, the triplet(kt,dt, Vt+l(s'))
satisfiesequation(4). It is easy to see thatafteranyhistory,
is also
contract
thecontinuation
an elementof thesetof enforceable
be
the
of
values
set
contracts.
C
(V, B)
Letting2 (s)
t2
suchthatthereexistsan enforceable
contract
withinitialvaluesVo = V and Bo = B andinitial
states, itfollowsthat(Vt, Bt) E -2(st) forall t. The setofoptimalcontracts
givesvaluesthatare
in theParetofrontier
havetheproperty
that
of a2(s). Moreover,
as seenbelowoptimalcontracts
forall t, (Vt,
arein theParetofrontier
of? (st).
Bt)
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294
REVIEW OF ECONOMIC STUDIES
In that
was depictedin Figure1 foran examplewithout
The Paretofrontier
uncertainty.
firmvaluesubjectto total
solvestheprimalproblemofmaximizing
exampletheParetofrontier
thedualproblemofmaximizing
is through
debt.An alternative
toconstructing
theParetofrontier
the debtvalue subjectto the firm'sequityvalue.7In the moregeneralscenarioof random
can be characterized
thisfrontier
B(V, s) whichgivesthemaximum
bya function
productivity,
forV.
debtthatis enforceable
fora givenlevelofequityV and states anda domainrestriction
contract
alternative
In fact,any
debtcontract.
Thus,underlying
B(V, s) is theoptimallong-term
will givea lowerfrontier,
so thatforthesame levelof V and shocks, thefirmwill be more
constrained.
financially
statecontingent
3.2.3. Equilibriumcontracts. We cannowdescribethemaximum
longlenders
of
The
existence
0.
termdebtthatcan be crediblyrepaidfromtime
manycompetitive
debt
withthisinitiallong-term
receivethehighestsharevalueconsistent
requiresthatborrowers
level.
and givesthehighest
2. An equilibrium
contract
C(.) is feasible,enforceable,
Definition
lender
withthe
consistent
breakingeven: Vo = sup{V :
possibleinitialvalueto theborrower
>
B(V, so) Io} whentheinitialshockis so.
of an initialinvestment
back to Figure1, noticethatfinancing
Io requiresthat
Referring
debtin excess of Bmax
value requiringlong-term
Bo < Bmax be raised.Any investment
This is because,theprojectdoes notgenerateenoughcash flowsto repay
cannotbe financed.
thebondholders
and stillgrantenoughfuturedividendsto thefirm,thatwouldkeep it from
defaulting.
to
If Io cannotbe financed,
no contract
is possibleunlessthefirmhas fundsto contribute
For
at
least
thisinitialinvestment.
More specifically,
thefirmmustcontribute
Io B(Vo,so).
defined
structure,
byhighervaluesO (k,s), willreducethetotal
example,a weakerenforcement
bythefirm.
surplusoftheprojectandthusrequirea higherinitialinvestment
a
credible
In general,financing
of Io is feasibleonly whenever
contingent
repayment
of the
on characteristics
schedulecan be agreeduponby bothparties.This dependscritically
of
the
consider
As
and
outside-value
function. an example,Bulow
feasibility
Rogoff(1989b)
commitment.
one-sided
with
a
between
and
debtcontracts
a lending
country
borrowing
long-term
cannotbe excludedfromcontingent
savings(whattheycall cashTheyshowthatiftheborrower
value of thepenalties
discounted
the
then
is
in-advancecontracts), debt restricted
by
present
In thiscase, thispresentdiscountedvalue is givenby Bmax,and
frombreachingthecontract.
whenthereareno penaltiesthefrontier
collapsesandthereis no feasiblelending.
4. THE OPTIMAL CONTRACT
We now construct
the dynamicprogramming
problemthatsolves for the Paretofrontier.
The values Vt+1(s') providea summarystatisticforthe futurecontractand togetherwith
constraint.
or participation
to verify
thisnon-default
Using Vtas a
(kt,dt, et,st) are sufficient
be
can
the
contract
statevariable,following
specifiedin recursive
Spearand Srivastava(1987),
=
=
s
and
and
st
form.Everyperiod,giveninitialvalues Vt V
assumingliquidationis not
values V(s'). In turn,the
continuation
and
contract
the
recommended,
specifiesa pair (k,d)
anddividendactions.Thisformulation
investment
continuation
valuesV (s') willdictatefuture
7. Thisalternative
formulation
theanalysisconsiderably.
simplifies
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ALBUQUERQUE & HOPENHAYN
OPTIMAL LENDING CONTRACTS
295
is consistent
withreallifebondcontracts
in whichcovenants
arewritten
restrictions
establishing
in thefirm'sinvestment,
dividendandfinancing
policies.
Sincethecurrent
valuefortheentrepreneur
is thesumofthedividends
paidoutthisperiod
plus the discountedvalue of the streamof futuredividends,the followingequitycash flow
condition
mustbe satisfied
ifthefirm
is notliquidated:
d
lV(=
+r
1V
+fV(s')F(ds',s).
(5)
,
Thecontinuation
continuation
contract.
equityvaluesV (s') mustbe supported
byan enforceable
value V(s') > 0 is feasible,since it can be
By Assumption
3(1), any positivecontinuation
obtainedby givingthefirma transfer
thefirm(or committing
t < V(s') and liquidating
to
no future
value V (s') < 0 is notfeasible.Hence,an equity
advancements).
Anycontinuation
value V (s') can be supported
contract
if,andonlyif,V (s') > 0. This is the
by an enforceable
domainrestriction
indicatedabove.
constraint
to
Usingequation(5), theenforcement
equation(4) simplifies
O(k, s) < V.
(6)
condition
d > 0 andtheequitycashflowequationsimplify
to8
Finally,thelimitedliability
1
f
V(s')F(ds', s)
<
V.
(7)
We now writethe problem'sBellmanequation.We assumethatthe lenderhas access
to perfectcapitalmarkets;a negativeperiodpay-off
resultsin increasedlending.The lender
maximizesthe debtlevel B(V, s) by choosingan enforceablecontractthatgives a current
whenthecurrent
revenueshockis s. Recallingthatthetotalsurplus
expectedvalueV tothefirm
ofthematchis givenby W(V, s) = B(V, s) + V, andthatthelenderhas fullcommitment
to the
itis immediate
to see thattheoptimaldebtcontract
also maximizesW(V, s) givenV.
contract,
The function
thefollowing
W(V, s) satisfies
equation:
dynamicprogramming
W(V, s)
=
max{L(s),
maxk,
V(s')>o
+
+
r R(k,s) -(1 r)k
fW(V(s'),s')F(ds'; s)
(8)
subjectto equations(6) and (7). Standardresultsin dynamicprogramming
implythatthereis a
uniquesolutionW(-) to thisproblem.9
It is interesting
to comparethisprogramme
withtheone obtainedforthecase of perfect
constraint
enforcement,
equation(3). Notice thatif the no-default
equation(6) were never
thenk wouldbe chosenso thatR(k, s) -(1 +r)k = n7(s)andthesolution
toequation(8)
binding,
wouldgiveW(V, s) = W(s). Also,iftherewas unlimited
V wouldgrowwithout
bound
liability,
to achievetheefficient
levelofcapitalnextperiodforall statesin S.
In theoptimalcontract,
thelenderdecidesto terminate
and liquidatethefirm
thecontract
whenever
itsvaluereachesB(V, s) = L(s)- O (0, s) = L(s). In sucha state,thelendergivesthe
lowestfeasiblevalueofequitytothefirm,
O (0, s) = 0. Becauseoflimitedliability
equation(7)
8. Strictly
does notrequirethatdt is non-negative
at anypointintime.In principle
the
speakinglimitedliability
couldaccumulatesomeformof precautionary
rateto cover
entrepreneur
savingand investthemat themarketinterest
futurefinancial
needs.In theoptimalcontract
See the
thelenderis implicitly
doingthesesavingsfortheentrepreneur.
ofthecontract
discussedinSection4.5.1 foran example.
implementation
9. Thepossibility
ofliquidation
introduces
a non-convexity
intheabovedecisionproblem.Thoughnotexplicitly
statedin theaboveformulation,
in ouranalysiswe considerthepossibility
on theliquidation
ofrandomization
decision.
Thisis explicitly
addressedin AppendixC.
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296
REVIEW OF ECONOMIC STUDIES
thelenderdoes not"waste"equityassignments
thehighest
inthesestates.SinceL(s) represents
valuethatcanbe appropriated
ofas standard
itcanalso be thought
bythelenderuponliquidation,
collateral.
In thenextsubsections
forthe
we will discussin turntheproperties
of thedebtcontract
frontier-astateafterwhichtheborrowing
theefficient
constraints,
designofoptimalborrowing
will neverbind-the growth
constraint
of firms,
and thecapitalstructure
and survivalpatterns
policy.
4.1. Short-run
and equity
constraint
borrowing
As seenin theexample,theleveloflong-term
advancements
debtsetsa limiton theshort-term
of working
short-run
capital.A directconsequenceof this,is thata staticproblemdetermining
ofk canbe separated
definetheindirect
fromthedynamicchoiceofV(s'). Inparticular,
financing
function
profit
1(V, s) = maxkR(k,s) - (1 + r)k
(9)
subjectto O(k, s) < V.
The solutionto thisproblemis simple.Let K(s) = inf{k: R(k,s) - (1 + r)k = 7r(s)}, and
defineVU(s) = O (K(s), s). This is the smallestcontinuation
value forthe firmthat,once
withstaticprofit
maximization.
reached,is compatible
Thus,if V > Vu(s), k is chosenso that
cannotbe enforced
maximization
and
R(k,s) - (1 + r)k = 7r(s).If V < VU(s), current
profit
k is chosenso thatO(k, s) = V. Hence,R(k,s) - (1 + r)k < 7r(s),ifandonlyif V < VU(s).
Theseresultsfollowdirectly
1 and3.
fromAssumptions
The frictions
betweentotaldebt,equityandfinancial
constraints
seenearlierinourexample
else
are alreadyapparent
constraint
here.Fromtheshort-run
financing
equation(6), everything
thehigherthedebtlevel B (lowerV), theless capitalthefirmis able to borrowfor
constant,
This negativerelationship
debtand short-term
betweenlong-term
capitalis at the
production.
heartofthefinancing
constraint.
We makethefollowing
function:
on theindirect
profit
assumptions
Assumption4. ThefunctionI has thefollowing
properties:
ins, strictly
increasbounded,increasing
(1) H is twicecontinuously
uniformly
differentiable,
in
<
V
V
Vu
(s).
ing
for
concaveifV < Vu(s).
(2) H is concavein V, and strictly
(3) ThereexistsM < oc, suchthatVu (s) < M foreverys.
Lemma2 in AppendixB givessufficient
forAssumptions
conditions
4(1) and4(2) tohold.
in s, requiresthattherevenuefunction
that11is increasing
Loosely speaking,theassumption
increasesby morethanthe outside-value
functionwhen shocksincrease.Assumption4(2)
withrespectto k be greaterthan
requiresthatthedegreeof concavityof therevenuefunction
thedegreeof concavity
of theO function.
One examplethatsatisfiesall of theserequirements
is O(k, s) = R(k,s) (thisis theassumption
made in Thomasand Worral(1994)). Another
interesting
exampleariseswhen shocksare projectspecific,i.e. the outsidevalue does not
respondto changesin shocks(O(k, s) = k).
4.2. Firmvalueand theefficient
frontier
The deterministic
in Section2 showsthatthereis a minimumV
examplewe constructed
levelofdebt)thatis consistent
Forhigherdebtlevels,thetotalvalue
withefficiency.
(maximum
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ALBUQUERQUE & HOPENHAYN
OPTIMAL LENDING CONTRACTS
297
of thefirmis lower.In thissectionwe extendthisresultto thegeneralcase, wherethisupper
boundon debtis definedforeach state.For debtvalues above thatlimit,an increasein the
Provideddebtis belowthis
matters.
raisestotalfirmvalue,so capitalstructure
equityofthefirm
is notrelevantand thetotalvalueof thefirmcoincideswiththeperfect
limit,capitalstructure
levelofequity-the
minimum
enforcement
case. Thus,we call thislimit-anditscorresponding
efficient
frontier.
From(9) itfollowsthatifthecurrent
high,V > Vu(s), theperiod
equitylevelis sufficiently
return
willbe identicalto theone obtainedin theunconstrained
problem,i.e. 1-(V, s) = 7r(s).
totalvalue W(V, s) = W(s), sincethecontract
However,thismaynotensurethatthecurrent
mustalso guaranteethatthe enforcement
will not bind in any futureperiod.For
constraint
example,if VU(s) < V < -i+r VU(s')F(ds', s), thenitmustbe thecase thatV(s') < VU(s')
withpositiveprobability
on a subsetof thesurvivalsetS, andthusnextperiod'sunconstrained
maximum
cannot
be guaranteed.However,if V is highenough,it may be possible
profit
to guaranteethattheenforcement
will notbindin any futureperiodand thusthe
constraint
unconstrained
solution
will
be
attained.
optimal
initialvalueforthefirm
let Vn(s) be theminimum
levelof current
Abusingthenotation,
thatis neededto guarantee
willnotbindforat leastn periods,
thattheenforcement
constraint
thecurrent
one,whenthestateis s. ThenVn(s), n > 1,can be defined
recursively
by
including
V(s) = max(Vu(s),1
(10)
f Vn-1(s)F(ds', s)
fors E S, withVo(s) = 0. Let
V(s) = limn,,
Vn(s).
Since Vn(s) is an increasing
bounded,this
4(3) is uniformly
sequence,whichby Assumption
follows
thatV (s)
limitexists.Furthermore,
it
dominated
theorem,
usingLebesgue's
convergence
is a solutionto
V(s) = max(V(s),
V(s')F(ds', s)
s
S.
(11)
verified.
Thissolutionis unique,as Blackwell'ssufficient
canbe immediately
conditions
The functions
debtB(s) = W(s) - Vf(s)
V'(s) and W*(s)definethemaximumlong-term
sincefor
consistent
withunconstrained
We call thefunction
frontier,
V(s) theefficient
financing.
thetotalsurplus
cannotimprove
equityvaluesabove V(s) (i.e. debtvaluesbelowB(s)) thefirm
debtlevel.Thisinterpretation
is derivedfromLemma1.
bymanipulating-its
Lemma 1. Lets be in thesurvivalsetS. Then:
in V.
(1) W(V, s) is weaklyincreasing
(2) Forall V > V(s), W(s) = W(V, s).
(3) Forall V < V(s), W(s) > W(V, s).
Proof See AppendixD.
I
of
An immediate
and survivalare a function
consequenceofLemma1 is thatfirmgrowth
thecapitalstructure
belowtheefficient
frontier
only.These age and patheffectsare analysed
in whatfollows.
extensively
constrained
firms
inthemodel.Short-run
Lemma1 helpstoidentify
constrained
financially
maximization.
firmsare thosethatare unableto borrowenoughcapitalto achievestaticprofit
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298
REVIEW OF ECONOMIC STUDIES
W(s1)
.
----W(V,s1)
W(V,
s2)
W(s2)
V(S2)
V(s1)
V
FIGURE2
The valuefunction
fortworevenueshocks
firms?No.
These firmshave current
excess marginalreturns.
Are thesetheonlyconstrained
As discussedabove,firmswithequityvalues VU(s) < V < V(s), facea positiveprobability
of reachingstateswherethe marginalreturnis positive.All thesefirmshave totalsurplus
excessmarginal
returns.
W(V, s) < W(s) andfuture
2
illustrates
the
function
W fortwopossiblerevenueshocksst > s2. This figure
Figure
in
it strengthens
the statement
of thevalue function.
In particular,
depictsseveralproperties
Lemma1 thatthevaluefunction
monotone
is weaklymonotone
in V forfixeds, to strictly
(see
AppendixC fora technicaldiscussion).We leavethediscussionof otherproperties
impliedby
Figure2 forlater.
A directimplication
of strictmonotonicity
of thevalue functionW, is thattheoptimal
contract
frontier
and thatall
recommends
thatno dividendsbe distributed
belowtheefficient
be
constraint
allocated
to
the
of
debt
(i.e.
earnings
equation(7) holds
repayment long-term
as an equality).This is truedespitethefactthattheborrower
and lenderare riskneutraland
allowsforfaster
discountflowsatthesamerate.Thereasonis thatdelayingdividend
distribution
and
of
the
incentives
to default
debt.
reduces
More
equitygrowth repayment long-term
equity
and relaxesthe short-term
firm
increases
as
does the
value
total
constraint.
Thus,
borrowing
maximum
debt
at
time
can
be
that
zero
long-term
repaid.Formally:
credibly
Proposition1. If V < 1r f V(s')F(ds', s) theoptimalcontractrequiresthatV =
are distributed.
l+r f V(s')F(ds', s), so no dividends
The relaxationof the borrowing
forthe case of
has a simpleinterpretation
constraint
deterministic
revenues.The contractgivesthe firma certainvalue, equal to the discounted
valueof itsshareofprofits.
Thisvalue,i.e. theanticipation
ofthefirm'sshareoffuture
profits,
is preciselywhatholdsthefirmfromdefaulting.
to defaultincreasewiththe
Since incentives
amountofcapitaladvanced,thehighertheequityshare,themorecapitalcan thusbe advanced.
As theoutstanding
andlong-term
debtis repaid,
retained
equitygrowsovertimethrough
earnings
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ALBUQUERQUE & HOPENHAYN
OPTIMAL LENDING CONTRACTS
299
thecapitaladvancesincreasetowardstheunconstrained
mixis
level.Clearly,thedebt-equity
in determining
theborrowing
important
capacityof thefirm:theincreasedequityprovidesthe
frontier
is reached
ofcapital.Once theefficient
bondingnecessaryto raiseincreasing
quantities
thisroleforequitydisappearsas thelong-term
ofthecurrent
revenue
contract
becomesa function
shockonly.Also, onlyaftertheefficient
frontier
is reachedwill dividendsbe distributed.
This
featureof theoptimaldividendpolicyis presentin otherpapersas well.In Spence(1979), in
orderto meeta cash flowconstraint,
firms
distribute
dividendsonlyonceitscapitalreachesthe
size.
long-run
4.3. Firmgrowth
and survival:age effects
thatfirms
ratesattheirearlystages
Manystudieshavedocumented
experience
verylargegrowth
of lifecoupledwithdramatic
ofthe
turnover
rates.In thissection,we discusstheimplications
forfirmgrowth
andsurvival.
optimaldebtcontract
Thomasand Worral(1994) showthatcapitaladvancements-andthusfirmsize-grow
overtime.In thissectionwe provea similarresultforour generalset-up.We
monotonically
have alreadyestablishedthaton averageV increases(and thus,B decreases).However,given
thatrevenuesare stochasticand debtis contingent,
values
choiceof continuation
a non-trivial
mustbe made,tradingoffborrowing
an
future
We
constraints
different
provide
paths.
along
characterization
oftheoptimalstatecontingent
debt
debt.Foranyhistory,
elementary
contingent
decreasesmonotonically
overtimeconditional
on therevenueshock.
Theresultspresented
ofthevalue
inwhatfollowsandinthenextsectionuse strict
concavity
function
W(., s) forL(s) < W(V, s) < W(s). Allowingforrandomliquidation,
AppendixC
showsthisholdsunderAssumption
4(2).10
Take Vt < -41 f V(st+l)F(dst+l, st), so theconstraint
in equation(7) binds.It is easy
to see thatat an interiorsolutionforthe optimalcontract,V(st+l) will be chosen so that
Noticethat
W1i(V(st+l),st) is equalizedforall thosestateswerethefirmis notliquidated.11
thisderivative
servesas an indexforcontingent
a lowerderivative
equityanddebt.By concavity,
to lowervaluesofcontingent
debtforall states.By theenvelopetheorem,12
corresponds
1
=
(12)
Wi(Vt, st)
Ol,(Vt,
st) + W1(V(st+l), st+l).
1+r
Giventhat I1 > 0, thisimpliesthatthelevelofcontingent
debtdecreasesovertime,conditional
on therevenueshock.Moreover,it will strictly
decreaseif and onlyif 1- > 0, thatis when
debt(forall
constraints
bind.Noticethatthisimpliesthatstatecontingent
short-run
borrowing
transits
a statewhereitfacesno short-term
states)willnotchangeaftera firm
borrowing
through
constraints
anditwillstrictly
decreaseotherwise.
thisimpliesthatevenifthefirm
Interestingly,
transits
a periodwithnegativecash flowsits contingent
debtwill notgrowand may
through
indeeddecrease.
Giventhenegativerelationship
for
betweenk and long-term
debt,thefollowing
property
firmsize andprofits
follows:
Proposition2. Conditionalon the revenuestate of thefirms, firmsize and profits
increasewithage.
10. In particular,
ourconditions
is concavewhenO (k, s) = R(k, s) andthereis no
implythatthevaluefunction
value,as in ThomasandWorral(1994).
liquidation
11. Concavity
thatthevaluefunction
is differentiable
almosteverywhere.
guarantees
12. Pointwisedifferentiation
of the objectivefunctionwithrespectto V(s') yieldsthe first-order
condition
on thelimitedliabilityconstraint
W1(V(st+l), st+1) = Xt whereXt is theLagrangianmultiplier
equation(7). The
resultnowfollowsfromusingtheenvelopeconditionW1(Vt,st) =
1(Vt, st) + xt.
-4L.
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300
REVIEW OF ECONOMIC STUDIES
Another
Whilethefirmis in a
ofourdiscussionconcernsfirmgrowth.
important
corollary
setwhereV < V(s), firms
willgrowwithage. Once a regionis reachedwhereV = V(s), age
willhaveno further
effect.
Thisimpliesthat:
growth
Proposition3. Younger
firmswill,on average,growfaster.
Thisresultdoes notrequiremeanreversion
as inmodels
shocksorlearning,
ofproductivity
ofpurestochastic
and
likeJovanovic
evolution,
(1992).
(1982)
Hopenhayn
The factthatconditionalon s, the equityvalue is an increasingsequencealso has an
offirms,
exhibitverylargeturnover
firm
exit.Mostindustries
important
consequenceconcerning
are
lower
and
for
small
rates
for
smaller
and
survival
firms.
predominantly
young
Empirically,
foryounger
firms.
We addressthisevidencenow.
Let S*(V) be the set of values s forwhicha firmwithequityV remainsactive.It is
immediateto see thatthe optimalexit rule satisfiesthe followingproperties:
(i) V2 > V1
of W(.) withrespectto V);
impliesthatS*(V2) Q S*(Vi) (followsfromthemonotonicity
(ii) S*(V) C S; (iii) if s e S, thens e S*(V(s)). This showsthattheexit set is weakly
in V. Intuitively,
haveloweroptionvalueofstaying.
decreasing
highlyindebted(lowerV) firms
matters
fortheexitbehaviourof firmsin thatit inducesinefficient
Thus,thecapitalstructure
thathighdebtlevels
liquidation.Zingales(1998) presentsevidenceforthetrucking
industry
affect
thesurvivaloffirms
decisions.
theirinvestment
byconstraining
on s), ifa firmdoes notexitat timet in states, then
Because,Vtis increasing
(conditional
itwillnotexitanytime
in thefuture
ifitreturns
tothesamestate.Thus,accordingto ourtheory,
limitedenforcement
contributes
to a positiverelationship
betweenfirmsurvivalandage.
Proposition4. Conditionalon the revenuestate of thefirms, survivalprobability
increaseswithage.
Theseresultsare a reflection
of financial
constraints
beingrelaxedwithage. Fazzariet al.
find
in
that
addition
to
flows
area significant
variableto predict
the
firm's
cash
(1988)
q-value
firms'investment.
In ourmodel,theq-valueofa firm(as measuredbytheratioof W tothebook
valueofassetsIo) is a sufficient
hasreachedtheefficient
statistic
oncetheequityvalueofthefirm
frontier.
Priorto thatpoint,thesize andgrowth
bothbyitsequityvalue
of a firmis determined
V andrevenueshocks. In thisrange,a highq-valuemayreflect
eithergoodrevenueshocksor
lowerborrowing
constraints.
Moreover,sincein thisregionthereis a one-to-one
mapbetween
thepairs(V, s) and(W, F), cashflowsandthefirm's
and
determine
its
decisions
q-valuejointly
Our
model
relevant
thus
the
firm's
cash
flows
are
a
in
to
addition
growth.
q-value,
suggeststhat,
variablebutonlyforyoungfirms.
4.4. Firmgrowth
and survival:patheffects
In theprevioussubsection,
we established
conditional
thegeneralprincipleof age and growth:
on a givenshocks, Vtis a non-decreasing
time.In
i.e.
firm
value
tends
to
increase
over
process,
thissubsection,
we considertheeffect
ofproductivity
shocks.The mainquestion
of thehistory
we addressis: do better
histories
lead tolowerborrowing
constraints,
largerfirmsize andhigher
survivalrates?The answeris a qualifiedyes.Thisquestionis particularly
becauseit
important
matters.
constraints:
also
impliesthatage alonedoes notexplainfinancial
history
To makeourstatements
twosequencesofshocks:
precise,considerthefollowing
{so,sl ...,
{SO, S ...
ST-1,sT},
,_ST , sT,
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(13)
ALBUQUERQUE & HOPENHAYN
OPTIMAL LENDING CONTRACTS
301
wherebothsequencesstartandendwiththesameshocksandst > s. Let {Vo,V1,..., VT) and
valuesequences.As we willsee,ingeneralV' #VT,
{Vo, , . . ., VT} denotethecorresponding
so history
ifVt> Vt/,
t = 1,..., T.
matters.
We saythatthecontract
leadstopositivepersistence
Positivepersistence
forthecurrent
shock,firmsize and
impliesthat,controlling
productivity
value increasewithpastproductivity
shocks.Our originalquestioncan be restatedas: When
does theoptimallendingcontract
lead to positivepersistence?
To motivate
themechanism
forhistory
considerthecase ofi.i.d.shocks.Under
dependence,
whatconditions
willthecontract
thatareidenticalup
be path independent?
Taketwohistories
to timet and differ
onlyin periodt, wherest > s. Let Vt+l(st+l) and V/+1(st+l) denotethe
continuation
valuesforthetwohistories
at timet + 1. Usingequation(12) it is
corresponding
easyto see thatif Vt+1(st+l) = V'+1?(st+1)then
Hnl(v,
st,)=nl(v',s).
Thisholdsforall st ands onlyif IH12= 0, i.e. whenthemarginaleffectof V is independent
of theshocks. This is obviouslya veryexceptionalcase. In general,pathdependencewillbe
affected
In thecase of i.i.d. shocks,positivepersistence
by thesignof thiscrosspartialeffect.
ariseswhenH12 > 0 whilenegativepersistence
arisesin theoppositecase. Whenshocksare
notindependent,
butnot
correlation
forpositivepersistence,
of
shocks
is
a
positive
keycondition
sufficient.
Giventheseremarks,
we
make
the
fortheremaining
section
in
this
following
analysis
twoassumptions:
Assumption5. Forall s', F(s', s) is non-increasing.
Assumption6.
V < Vu(s).
H12(V,s) > 0 whenever
The first
increaseswiththe
distribution
assumption
requiresthattheconditional
probability
current
shockin thefirst-order
stochastic
dominancesense.Thisis a verystandard
assumption,
whichis satisfiedby manystochasticprocesses,in particularby first-order
auto-regressive
R and
on thefunctions
less thanone.A sufficient
condition
processeswithabsolutepersistence
for
the
B.
second
in
in
is
of
(of supermodularityHI (V, s)) given Appendix This
assumption
O
condition
relatesthemagnitudes
of thecrosspartialsof bothR and 0. Considerthefollowing
extreme
cases. If R12(k,s) > 0 and O(k, s) = O(k), thenH12 > 0 forV < VU(s). However,
if R(k, s) = R(k) and 012(k,s) > 0 thenF12 < 0 forV < VU(s) and valuesof k suchthat
to shocks
revenuebe moreresponsive
R1(k,s) > 1+ r. Thus F12 > 0 requiresthatthemarginal
thanthemarginal
outsideoption.
Thefollowing
statesan important
oftheseassumptions,
namelythat
proposition
implication
continuation
values
in
are
equity
increasing st+1.
V(st+1)
concave in V for V < V(s). At an interior
Proposition5. LetW(V, s) be strictly
solutionW12(V,s) > 0 so anyoptimalcontinuation
policyV(s') mustbe non-decreasing.
Proof See AppendixD.
II
In words,Proposition
5 saysthatrevenuestateswheremarginal
returns
tocapitalarehigher
areassignedhigherequityvalues.
Does monotonicity
in theequityvalueassignment
Forthecase
implypositivepersistence?
of i.i.d.shocks,thisis easyto prove.Indeed,takest >
andlet Vt > Vt'be thecorresponding
st
frontier
Assumethesevaluesaresuchthatat leastforV'/theefficient
equityvalueassignments.
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REVIEW OF ECONOMIC STUDIES
302
is notreachedin thefollowing
period.Now giventhat
EVt+l = Vt(1+ r) > V(1 + r) = EVt+1
it is straightforward
to showthatVt+l > Vt+1. Thus,evenwhenshocksarei.i.d. themodelis
able to deliverpersistence.
in equityvaluesneed notoccur.
Whenshocksare seriallycorrelated,
positivepersistence
The difficulty
arisesbecausehighershocksin one periodincreasethelikelihoodofhighshocks
in thefollowing
period.A higherinitialequityvaluemaynotbe enoughto sustainsignificantly
example.
largerexpectedequityneedsforthefollowing
period.Considerthefollowing
are identicalunderso and si,
Example1. Thereare threestates:so < sl < s2. Profits
i.e. F(V, so) = H(V, st). However,HI(V, si) < HI(V, s2). Moreover,
supposeP(s2 I Sl)=
P(so so) = 1 - e, P(so, Si) = P(so I s1) = P(s2, so) = E, for e small, where P(si, sj) =
= si I = sj). Now considerthe paths
st
P(St+l = si, st = sj) and P(si I sj) = P(=t+l
{so, S1, s2),
Iso,so,s21.
areequatedacrossstatesforperiod1:
Fromtheoptimalcontract
attimet = 0, derivatives
W1(Vo,so) = Wi(VI,s1).
Forperiod2 we get
W1(Vo,so) = "I (Vo, so) + Wi(V02,S2),
W1(VI, s1) = Hi(Vi, Sl) + W1(V12,S2).
at si andmovingintostate
The valuesVij aretheoptimalcontinuation
valuesfora firmstarting
> V02,so thatthebetter
leads to higher
of
a
that
contradiction
V12
history
way
Suppose
by
sj.
mustsimilarly
entitlement
atperiod2. Then,byconcavity,
theoptimalcontract
assignV0o> Voo
so)). Thus
(becauseW1(V10,
so) = W1(V12,
S2) < W1(Vo2,
s2) = W1(Voo,
(1 + r)VI = rVIo + (1 - E)V12
>e Voo+ (1 E)V02
= (1 +r)Vo-
83
(1 -2r)Voo- -Vol+
2
1 -- 28 V02
> (1 + r)Vo,
wherethe firstinequalitycomes fromthepreviousdiscussionand the last inequalitycomes
fromV02 > Vol > Voowhichfollowsfromsupermodularity.
But, VI > Vo,impliesthat
<
<
with
S2)
so).
si)
n1(Vo,
Together
Hi(V1,
W1(V12,
W1(V02,
s2) we obtainthecontradictionthatW1(VI, sl) < W1(Vo,so).
shocksso thatpositive
in productivity
It is possibleto restrict
thenatureof persistence
in equityobtains.Forthatpurposewe makethenextassumption.
persistence
ins, thenH1(
Assumption7. If (V(s), s) is non-decreasing
is non-decreasing
inso. HI
fV(s)F(ds,
so), so)
7. Thentheoptimalcontract
Assumption
Proposition6. Supposethat1 and F satisfy
displayspositivepersistence.
Proof See AppendixD.
I
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ALBUQUERQUE & HOPENHAYN
OPTIMAL LENDING CONTRACTS
303
Remark1. Thisresultcan be strengthened
iftheR.H.S. of
to strictly
positivepersistence
7 is strictly
Assumption
increasing.
The following
in someusualparametrizations.
7 is verified
exampleshowsthatAssumption
be R(k, s) = sk', with0 < a < 1, andtheoutside
Example2. Let therevenuefunction
=
function
be O (k,s) k. It followsthat
opportunity
HI(V,s) = sVa - (1 + r)V
Vi(V, s) = asVa-1 - (1 + r),
whenV < Vu(s). It can be shown(see AppendixD) thatif1-1(V(s), s) is non-decreasing
in s
fV(s)
V(s)
(s,so)s
2(s,so)so ds.
F(ds,so),so)
Ids 1 +r
dso
s
Whatconditions
makethelastintegral
condition
is thatthetermin
A
sufficient
positive? strong
curvebracketsbe positive.An alternative
is to assumethetermin curvebracketsis increasing
in s andhas positiveexpectation.
Whens = pso + e and8 is
Two specialcases areofinterest.
distributed
withcdfG, thenF(s, so) = G(s - pso). The termin brackets
has thesamesignas
A
in
sufficient
In
the
condition
this
case
is
e
has
that
specialcase where
F.
positivesupport.
withcdfG, thenF(s, so) = G(lns - p Inso) andtheterm
ns = p Inso + E ande is distributed
in brackets
has thesamesignas IS, whichis alwayspositive.
NoticethatProposition
6 impliesthatmarginal
willbe higherforhigher
returns
productivity
shocks.Using thesemarginalreturns
it followsthat
as a measureof borrowing
constraints,
forhighershocks.
constraints
aretighter
borrowing
Do good historiesincreasethe probability
The
of survivalunderpositivepersistence?
answeris again a qualifiedyes. As discussedabove,thesurvivalset S(V) increaseswithV.
increasethesurvivalset.However,a goodhistory
also changesthe
Consequently,
goodhistories
conditional
distribution
of shocks.It is conceivable-yetunlikely-thatthelattereffectcould
contribute
tosurvival.
Thiscouldhappenifthedifference
betweenthevalueofstaying
negatively
and thevalueof liquidation
is notmonotonic
takescareof this
in s. The following
proposition
anomaly.
ins.
Proposition7.
(i) W(V, -) is weaklyincreasing
Thenoptimalliquidationis determined
(ii) L(s) is constant.13
by an increasingthreshold
s*(V).
Proof See AppendixD.
II
Giventheassumptions
forpositivepersistence,
good histories
implybothan increasein V
and an increasein theconditional
of part(ii) of the
distribution
fors. Undertheassumptions
aboveproposition,
survivalprobabilities
also increase.
4.5. Thestructure
and composition
ofdebt
In previoussectionswe focusedon thereal side,considering
theimplications
of theoptimal
contract
forfirm-size
Thissectionturnstothefinancial
thefinancial
sideandconsiders
dynamics.
13. The conditionthatL(s) is constantcan be replacedby
s) +
--TT[I-[(V,
in s. Detailsareavailablefromtheauthorsuponrequest.
decreasing
f L(s')F(ds',
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s)] - L(s) is non-
304
REVIEW OF ECONOMIC STUDIES
debtcanbe decomposedin a long-run
structure
anditsdynamics.
As indicatedbefore,totalfirm
Bt = Wt- Vt andtheshort-term
component
capitaladvancement
kt.Mostofourdiscussionis
centred
on thelong-run
component.
no dividends
are
theefficient
4.5.1. Debt repayment
schedule. Priortoreaching
frontier,
increasewiththevalueofthe
distributed
andall profits
arepaidto thelender.Sincetotalprofits
if liquidationdoes not
fora fixedstates debtrepayments
increaseovertime.Moreover,
firm,
occur:14
r
Bt - EBt+i = lit - 1
EBt+1,
and sinceFit increasesand EBt+I decreasesovertime,it followsthatfora givens, debtfalls
in expectation
of an increasing
rate.In addition,as theequityof thefirmgrows,itsdebtmatudebtdecreases.These
increasewhilelong-term
ritychanges:short-term
capitaladvancements
in Barclayand Smith(1995). Theyshowthat,conwiththefindings
areconsistent
predictions
ratioW/Io (or Tobin'sQ, as
ditionalon therevenueshock,firmswithhighermarket-to-book
debtB/k.'5 The intuto
short-term
debt
defined)displaya lowerratioof long-term
previously
value
with
itionforthisresultinoursetting
firm
a
on s,
is thatconditional
highermarket-to-book
and
hence
of assetsW/o10,
to theentrepreneur
is also a firmwithhigherequityentitlement
V,
debt.In BarclayandSmith(1995)
weakershort-term
andlowerlong-term
constraints
borrowing
thisresultis explainedbyappealingtoMyers(1977). InMyers(1977) theexistenceofriskyfixed
claimscreatesan underinvestment
mayrejectpositive
problemaccordingto whichstockholders
ifthesefixed
netpresentvalueinvestment
problemis eliminated
options.Thisunderinvestment
Because
thevalue
claimsmaturebeforeanyopportunity
to exercisetherealinvestment
options.
of theseoptionscan be measuredbythefirm'smarket-to-book
ratio(SmithandWatts,1992),a
debt.
ratioshouldbe associatedwithshorter-maturity
highermarket-to-book
debtcan be obtainedforthe
A sharpercharacterization
of the dynamicsof long-term
of theexamplein Section2 by allowing
case. Here we providea generalization
deterministic
fora generaloutside-value
continuation
Let
be
the
function
equityvalueforthefirm
Vt
O (kt).
=
at timet. The optimalcontract
that
+
implies
Vt+l min((1 r)Vt,VU).
Since Bt =
+ Bt+1] decreasesovertime,it followsthatI-I(Vt) > rBt. In
to
H (Vo) > rB0 = rIo > 0, withstrictinequality
ifthelendercontributes
at time0,T1-[HI(Vt)
particular
11
will
be
so
an initialinvestment
>
0.
increases
over
Since
time, (Vt)
positiveeveryperiod,
Vt
Io
thecash flowof thecontract
willbe positiveeveryperiod.Each periodtheborrower
paysback
=
thecapitaladvancedwithinterest
thequantity
and contributes
H-(Vt) R(kt) (1 + r)ktto
therepayment
frontier
oflong-term
debt.Once theefficient
VUhas beenreached,theborrower
andpaystheconstant
amountR(k*) - (1 + r)k*- rVu, wherek*is the
keepsrVu as dividends
loan.
amount
of
as
interest
on theoutstanding
long-term
profit
maximizing
capital,
In theabove implementation
thecash flowis controlled
of theoptimalcontract,
by the
0.
An
initial
loan
R
lender.An alternative
when
is
(kt) kt+1>
implementationpossible
Io + k0
theneedof shortterm
without
allowstheentrepreneur
to self-finance
thecapitalaccumulation
R(kt)- kt+ 1,keeping
makesrepayments
advancements.
Attheendofeveryperiodt theborrower
the
borrower
thequantity
frontier
is
for
Once
the
efficient
reached,
paysevery
kt+1 production.
is eithernotdefaultcontract
debt
the
lender
k*
other
to
R(k*)
period
rVU.16Any
long-term
schedule
prooforleadstoa lowerinitialdebtlevel.Thisis obvious,becauseanyotherrepayment
14. Whenliquidation
occursBt = L(st).
similarevidence.In thesepapershowever,
15. GuedesandOpler(1996) andStohsandMauer(1996) also present
of all debt,
debt maturity
is measuredas term-to-maturity
of new debt issues or weightedaveragedebtmaturity
debtB is in dollars.
Bothmeasuresarein years,whereasourmeasureoflong-term
respectively.
as
16. It is straightforward
to checkthatthiscontract
givesthesamevalueat time0 tobothlenderandborrower
thecontract
thatcombineslong-term
andshort-term
debt.
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ALBUQUERQUE & HOPENHAYN
OPTIMAL LENDING CONTRACTS
305
resultsin a lowerV0,andconsequently
is theone thatat
a lowerko.Thus,theoptimalcontract
eachpointmaximizesthetotaldebtthatcan be sustained.
4.5.2. Statecontingent
debt. Letus nowturntoa characterization
ofthestatecontingent
natureofdebt.The questionwe addressin thissectionis whether
debtis
theleveloflong-term
intheproductivity
monotonic
ifthelendingcontract
shock.Thiswouldbe thecase,forinstance,
wereimplemented
statecontingent
debt
Attheefficient
frontier,
byan equity-share
arrangement.
thefollowing
B(s) satisfies
equation:
B (s)=
I+
1 rr(s) - d(s) +
B (s)F(ds', s)
.
The firsttermb(s) = 7r(s) - d(s) corresponds
to debtpayments.
Using standarddynamic
andfirst-order
stochastic
dominanceof F(s', s), itis easyto showthat
programming
arguments
ifb(s) is non-decreasing
in s, thenB (s) willbe a non-decreasing
function.
Whathappensbeforetheefficient
frontier
is reached?On theone hand,moreproductive
statescan sustainmoredebt.On theotherhand,itis advantageous
to increasetheequitystake
in good statesas well. Dependingon whateffectdominatesB(V (s), s) maybe increasingor
thatimplythatstate
conditions
decreasingon s. The followingproposition
presentssufficient
is
debt
monotone
even
outsidetheefficient
frontier.
contingent
Proposition8. Let V(s') denotetheoptimalpolicyfunction
starting
from(V, s). Assume
thatFI(V (s), s) - (1 +r) V(s) is increasing
is
constant.
ins whenever
(s),
s)
(V
Supposealso
I-11
thatthecontract
displayspositivepersistence.
Ifs2 > Sl thenB(V(s2), S2) > B(V(sl), st).
Proof See AppendixD.
II
Let us return
under
to debtpayments
at theefficient
and analysetheconditions
frontier
whichb(s) is non-decreasing
are
in s. At the efficient
debt
frontier, payments obviously
in therangewheredividendsare notdistributed,
i.e. d(s) = 0. Thisoccursin those
increasing
stateswhereV(s) > VU(s). In contrast,
forstateswhereV(s) = VU(s),debtpayments
are
b(s) = 7r(s) - (1 + r)V(s) +
JV(s')F(ds', s).
The lasttermis non-decreasing,
Theremaining
term
dominance.
ofstochastic
bytheassumption
7r(s) - (1 + r)Vu(s), is determined
function:
bytherevenueandoutside-value
d((s)
-
(1 + r)V(s))=
R2(k(s),s) - (1 + r) [02(k(s) s) -
1(k(s), s) R12(k(s),s)
RiI(k(s),s)
If theoutside-value
function
O is notverysensitiveto changesin s and 02 is small,or R12is
willbe increasing.
As an example,if O (k,s) = XkandR(k,s) = ska, this
small,debtpayments
derivative
willbe positiveifandonlyif 1 - a > Xa. Thisprovidesbothan examplewhereB (s)
willbe increasing
that1 - a > Xa
as well as a counter-example.17
thecondition
Furthermore,
also impliesthatn(V(s), s) - (1 +?r)V(s) is increasing
ins whenever
HI (V(s), s) is constant,
in s evenoutsidethe
meansthatB(V(.), .) is increasing
which,ifthereis positivepersistence,
efficient
frontier.
ds
17. Foran exampleofmonotonicity
a
take1 - a > ka. Fortheconverse,
considerthecase wherethe1 - a
<, if
and shocksare i.i.d.For anotherexample,takeO = XR and R = ska, wherethederivative
is positiveif and only
1 - a > (1 + r)X.
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306
REVIEW OF ECONOMIC STUDIES
4.5.3. A sequenceof short-term
contracts. It is easyto see thattheoptimalcontingent
debtand growthpolicycan also be implemented
contracts.
This
by a sequenceof short-term
is accomplished
by rollingoverthedebtfromone lenderto anotherwhilehavingall lenders
limitimplicit
coordinate
ona short-term
limit.Letk(B, s) be theshort-term
borrowing
borrowing
in theoptimalcontract
andlet17r(k(B,
s) - (1 + r)k'] be themaximal
s), s) = maxk'<k(B,s)[R(k',
thatcanbe achievedsubjecttotheborrowing
limitk(B, s). Thentheentrepreneur's
profit
optimal
borrowing
problem:
policycanbe derivedfromthefollowing
dynamicprogramming
V(B, s) =
to:
subject
1
maxr,B(,')
[r(k(B, s), s) - r +
V(B(s'), s')F(ds', s)
r < nr(k(B,
s), s)
f
B(s')F(ds', s) = B(1 + r) - r.
Thisis a standard
given
dynamicinvestment
facingan exogenously
problemforan entrepreneur
constraint
This
themaximalshort-term
debt
constraint
k(B, s).
borrowing
represents
borrowing
theentrepreneur
limitis the
can raise.It is important
to noticethatimplicitin thisborrowing
factthatcurrent
short-term
lendersanticipate
thatfuturelenderswill also determine
financing
this
rule.
on
lenders'
sideregarding
This
the
coordination
using
equilibrium
obviouslyrequires
theborrowing
iflenders
limit.Thisis clearlynottheonlyequilibrium
contracts:
withshort-term
in
the
that
the
short-term
will
raise
all
not
be
able
to
future,
anticipate
capital
entrepreneur
any
breaks
down.
financing
4.6. Somecomparative
statics
4.6.1. The role of collateral. One interesting
staticsexerciseregardsthe
comparative
roleof collateralin thedesignof theoptimallong-term
debtcontract
and thedynamicsof the
constraints.
Consider
two
the
same
initial
investment
firms
with
lo, one witha larger
borrowing
shareofthisinvestment
This
be modelledas a
firms
are
can
sunk.
the
identical.
Otherwise
being
lowerliquidation
withlower
valueforall revenuestates,whichmeanslowercollateral.The firm
value
has
a
set
from
of
survival
states
and
thus,
liquidation
larger
equation(11), higherV(s). It
also has lowertotalfirmvalueforgivens. Thus,lowercollateral(or greater
sunkcosts)leadsto
lowerdebtandleverageforgivens.
Lowerliquidation
debt
also reducesthetotalvalueW atanystate.Sincetheinitiallong-term
is thesame,thestarting
lower
as
smaller.
entitlement
must
be
V0
equity
Interpreting liquidation
lowercollateral,
initial
themodelimpliesthatprojectswithhighersunkcostswillfacestronger
constraints
andwillstartsmaller,
borrowing
takinga longertimeto mature.
4.6.2. Firmsize,betterprojectsand betterenforcement.A projectwithhigher
returns,
i.e. higherR(k, s) valuesor betterenforcement,
i.e. lowerO(k, s), will havea higherindirect
profitfunction-I(V,s). This impliesthattotalvalue W(V, s) will be higher.An immediate
consequenceis thattheprojectcan sustaina largerinitialdebtand size,exhibithighersurvival
are eliminated
of long-term
constraints
debtis fasterand borrowing
probability,
repayment
G
sooner.Alternatively,
a projectcan be moreattractive
if it facesa betterinitialdistribution
or conditionaldistribution
F forrevenueshocks.In bothcases, it is easy to show18thatthe
initialvalueoftheprojectis higherandconsequently
initialequityandtotaldebtwillincrease.
firmswill startlargerand face a higherprobability
of survival.In additionto
Consequently,
18. Theprooffollowsa similarargument
as theone usedforProposition
7 andis availablefromtheauthors
upon
request.
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OPTIMAL LENDING CONTRACTS
ALBUQUERQUE & HOPENHAYN
307
theseimplications
andas expected,
tohavegreater
allowsfirms
a better
enforcement
technology
leverage.
Theprediction
thatlargerfirms
(withhigherR(k,s)) havemoredebtis supported
byTitman
andWessels(1988) whopresent
salesalso displayhigherdebtto
evidencethatfirms
withgreater
assetratios.Also,interpreting
a low O (k,s) as highcreditrating,
themodelpredictsthatfirms
withhighercreditratinghavemorelong-term
debt.Thisis supported
in
byevidencepresented
BarclayandSmith(1995).
4.6.3. More riskyprojects. We do nothaveanygeneralresults.However,thefollowing
can lead to
increaseinthespreadofa project'sreturns
examplesuggeststhata meanpreserving
initial
constraints.
tighter
borrowing
Let R(k,s) = sl-aka for0 < a < 1, O(k, s) = )k, theliquidationvalue L = 0, and
shocksbe i.i.d.It followsthat
7r(s)--s (1
VU(s)=X=s
-
a)I
1+r
r
(jr
Hencea meanpreserving
increasein spreadof r(s) resultsalso in a meanpreserving
increasein
of
Since
spread Vu(s).
V(s) = max(Vu(s), 1r V) whereV is defined
by
implicitly
V = Emax (VU(s),
l
,
it followsimmediately
that V'(s) increases.Thus, once the projectreaches maturity
the
entitlement
to the lendermustbe smalleron average.Now suppose thatunderthe initial
distribution
of returns,
W(so) - V = I, whereV(so) = V. Thisimpliesthattheprojectstarts
unconstrained.
Yetwitha meanpreserving
V increasesso the
increasein thespreadofreturns,
constrained.
projectis initially
5. THE OPTIMAL DEBT CONTRACT AS A CONSTRAINED GROWTHPROBLEM
Itis possibletorewrite
ourmodelinsucha waythatitdoesnotexplicitly
modeltheaccumulation
of financial
assets(ortherepayment
ofdebt).The focusis on constrained
capitalaccumulation.
Thisis helpfultostresstheconnections
betweenourmodelandothermodelsoffirm
with
growth
constraints.
Redefine:
exogenousborrowing
1-(k,s) = max{R(x,s) - (1 + r)x Ix < k},
and assumethatforeverys E S and V > 0 thereexistsa uniquek suchthatO(k, s) = V.
Givens, thisestablishesa one-onemapbetweenequityvaluesV and capitaladvancements
k.
Thisallowsus torewrite
problemequation(8) as:
W(k,s)
=
max L(s); maxks') 1+ r[(k
s)+
W(k(s'), s')F(ds', s)
wheretheinnermaximization
is subjectto
O(k,
s) -l+r r
O(k(s'), s')F(ds', s).
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(14)
308
REVIEW OF ECONOMIC STUDIES
on the
This set-uphas the structure
of a capital accumulationproblemwitha constraint
investment
rate.An equilibrium
choosesko so thatW(ko,so) - O(ko,so) > o. This
contract
is equivalent
constraint
to setting
theinitiallong-term
debtandequitylevelsB(Vo,so) > lo from
before.
An interesting
our assumptionsis the case O(k, s) = k, where
example satisfying
rate.
equation(14) establishesthattheexpectedgrowthof capitalcannotexceedthe interest
=
steal
the
can
the
Anotherspecial case resultswhen O(k, s)
R(k, s), where
entrepreneur
rate
bound
on
the
In
an
revenuesof thefirm. thiscase equation(14) represents upper
growth
ofrevenues.
We can use thedeterministic
case to showthatourmodelspecializesto Spence (1979).
If O(k) = R(k),
as
Interpret
kt thefirm'scapitalstockand assumethereis no depreciation.
thentheentrepreneur
can keep therevenue,butnotthecapital.Everyperiodthefirmmakes
at timet,
an investment
decision.Followingthenotationin Spence,let mtbe theinvestment
i.e. R(k) = k0. Then,
so kt = mt + kt-1.Specializeto thecase in whichR is homogeneous,
betweeninvestment
kt= (1 + r)l/akt_landmt= ((1 + r)1/a- l)kt-1,whichis theconstraint
whichin Spence'sanalysisis takenas an
andthefirm'scapitalusedin Spence.Thiscoefficient,
is notindependent
rate.
oftheinterest
exogenousparameter
6. FINAL REMARKS
basedon
constraints
In thispaperwe havedevelopeda generalmodelofendogenous
borrowing
constraints
In ourmodel,borrowing
oflimitedenforcement
theassumptions
andlimitedliability.
and
ofborrowing
ariseas partoftheoptimaldebtcontract.
Ourmodelextendsprevioustheories
for
a
Moore
Hart
and
such
as
Thomas
and
Worral
and
(1994), allowing
general
(1994)
lending,
of exit.The modelhas
forfirm'sshocks,capitalaccumulation
and thepossibility
specification
tendto growfasterand
thatyoungerfirms
forfirmgrowth
and survival,
implying
implications
The model
withempiricalregularities.
havelowersurvivalrates.Bothproperties
areconsistent
and
ofthefirm'sinvestment
determinant
also suggeststhatthecapitalstructure
is an important
assetratios,greater
debthavehighermarket-to-book
exitdecisions.Firmswithhigherlong-term
bettercreditratingsandcollateral.
revenues,
levelinordertodescribethegeneralproperties
abstract
Wehavekeptouranalysisata fairly
and
of a class of modelsof borrowing
and lendingbased on theidea of limitedcommitment
is veryflexibleand could be used to
limitedliability.We certainly
believethatour structure
theprojectas a new
developmorespecializedmodels.Forexample,thoughwe haveidentified
loan to
thereis no reasonwhyto do so. An establishedfirmmaylook fora long-term
firm,
financea newinvestment.
ofborrowing
feature
an important
Ourmodellingapproachhighlights
dynamicconsistency
future
tocredit.
access
itself
limited
but
is
restricts
current
constraints:
access
to
by
equity
capital,
access
one
that
maximizes
is
the
Ofall dynamically
the
contract
consistent
lendingplans, optimal
debt.But thereare others,
to short-term
capitaland theexpectedrateof decreaseof long-term
thanarejustified
constraints
whichcan be suboptimal
andthuslead to tighter
bythe
borrowing
This suggeststheimportance
of our modellingapproach.Models of exogenous
environment.
that
therearebettercontracts
constraints
borrowing
alwaysleave openthequestionof whether
couldimplyweakerconstraints.
is limitedby a
assumethatborrowing
Some modelsof exogenousborrowing
constraints
firm'sassets.Ourmodelsuggeststhattherelevant
variableis notassetsbutequity.Kiyotakiand
in a modelwhere
Moore (1997), studythemacroeconomic
of limitedborrowing
implications
firms'collateralis subjecttoendogenous
In thepresenceofsunkcostsofinvestment
fluctuations.
or whensomeassets-such as humancapital-are notfullyappropriable,
loanscannotbe fully
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ALBUQUERQUE & HOPENHAYN
OPTIMAL LENDING CONTRACTS
309
collateralized
withassets.In suchcases,morelendingcan be sustained--asin ourmodel-with
thethreat
ofdepriving
theborrower
fromitsequity.
Thereareseveralinteresting
One ofthemis toexplorethegeneral
extensions
ofthetheory.
In thispaperwe treat
considered.
ofthetypeofborrowing
constraints
equilibrium
implications
mostexisting
thevalueof defaultas exogenous,withfairlygeneralproperties
to accommodate
models.In an equilibrium
thevalue of defaultwill notbe exogenousand should
framework,
in turnbe influenced
Thereare manyalternative
ways of closinga
by theoptimalcontract.
modelof thesortdevelopedhere,someof whichhave been alreadyexploredin theliterature
(see, forinstance,Ghoshand Ray, 1996, 2001). As an example,considera situationwhere,
withotherlenders,at some
contracts
borrowers
can enterintonewlong-term
upondefaulting,
additionalcost. An equilibrium
contractwouldrequiretakingintoaccountthisendogeneity.
This is obviouslya veryinteresting
area of research.Thoughourpaperdoes notaddressthis
thegeneralresultsobtainedherecouldstillproveveryusefulindeveloping
equilibrium
problem,
thatresearch
programme.
whichare worth
constraints
Thereare obviouslyalternative
waysof modellingborrowing
In
in
are
an alternative
informational
absent thismodel,
exploring. particular,
asymmetries,
therelationconsiders
sourceoffrictions
togenerate
creditconstraints.
Diamond(1989, 1991a,b)
structhe
and
the
between
unobservable
of
maturity/seniority
ship
quality theborrower's
project
tureof debt.Green(1987) analysesdebtcontracts
whenagentshave
in a repeatedenvironment
unobserved
endowments.
Atkeson(1991) and Marcetand Marimon(1992) studytheeffectof
moralhazardongrowth
inthecontext
ofinternational
growth
lending.Developingmodelsoffirm
and survivalbased on suchfoundations
of researchin thisarea.
direction
is also an important
SomerecentworkincludeDeMarzoandFishman(2001) andClementiandHopenhayn
(2002).
APPENDIX A. GENERALIZINGTHE OUTSIDE OPPORTUNITYFUNCTION
Thisappendixshowshowtomodify
themodeltoallowfor0(0, s) > 0. Since O (0, s) > 0 is theinalienablecomponent
oftheentrepreneur's
offuture
share,feasibility
equityvaluesmustbe adjustedso thatV(s') > O (0, s'). Thismeansthat
the
uponliquidationV = 0(0, s), andthelender'svalueis B(V, s) = L(s) - 0(0, s). It also meansthatin designing
efficient
frontier
thelendertakesintoaccountthepayments
madeto thefirmin theliquidationstates.Thatis, thenew
efficient
is givenby
frontier
V(s) =
max(Vu(s),1i'
0(0, s),
fV(s')F(ds',
s))
s i
else.
All proofsremainunchanged,
oftheefficient
frontier.
exceptfortheconstruction
APPENDIX B. PROPERTIES OF THE INDIRECT PROFIT FUNCTION
Lemma2 givesconditions
listedin thetextin
underwhichtheindirect
functionl (V, s) displaystheproperties
profit
4.
Assumption
Lemma 2.
in V for V < Vu(s).
bounded,strictly
(i) 1Fis continuous,
increasing
uniformly
(ii) If R and 0 are continuously
differentiable
and 01(k, s) > 0, thenF1(V, s) = I (ks)-+r)
and
(iii) IfR and O are twicecontinuously
difgerentiable
Rll (k, s)Ol (k, s) - Oll0(k, s)(R1 (k, s) - (1 + r)) < 0(< 0)
fork < K(s), thenIll(V, s) < 0(< 0)for V < VU(s).
(iv) If R and O are twice continuouslydifferentiableand
R12(k,s)01 (k,s) - 012(k,s)(Rl(k,s)
- (1 + r)) > 0(> 012(k,s))
for allk < K(s), then H21(V, s) > 0(> O)for V < VU(s).
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310
REVIEW OF ECONOMIC STUDIES
(v) If R and 0 are continuously
differentiable,
01(k, s) > 0 and
Rl(k, s) (1 + r)
R2(k,s) >
02(k, s),
01(k,s)
thenI- is increasing
ins.
Proof (i) is an immediateconsequenceof the maximumtheoremand the properties
givenfor R and O in
theorem;(iii) and (iv) followfrom
Assumptions1 and 3, and; (ii) is a directapplicationof the implicitfunction
theexpression
theprofit
function. II
differentiating
giveninpart(ii). Part(v) followsfromdifferentiating
APPENDIX C. CONCAVITY AND MONOTONICITY OF THE VALUE FUNCTION
The nextlemmasdiscussconcavity
andstrict
ofthesurplusfunction
W. Thepossibility
ofexitintroduces
monotonicity
a potential
inthetotalsurplusfunction
underwhichthefunction
lemmagivesconditions
non-concavity
W(., s). The first
W is concavein V without
usingrandomizations.
Lemma 3. SupposeR(k,s) - (1 +r)k > L(s) is satisfied
and Fl(V, s) is concavein V foreach s. Thenthetotal
concaveforV < Vu(s), then
surplusfunction
W(V, s) willbe concavein V forall s. Furthermore,
ifn (V, s) is strictly
W(., s) is strictly
concavein V whenever
V < V(s).
ofW(., s) followsfromtheconcavity
of I (V, s) byTheorem9.8 in Stokey,
Lucas andPrescott
Proof Concavity
on n that
(1989). Now supposethatV < V(s). Thenthereexistsan n suchthatV < Vn(s). We willshowbyinduction
thisimpliesthatW(., s) is strictly
concavein a neighbourhood
of V. Forn = 1, V < VU(s). Since in thisregionthe
return
function
concave.Supposenowthatthe
1nis strictly
concave,thenthevaluefunction
W(-,s) willalso be strictly
resultis trueforall s and V < Vn-1(s) andthatVu(s) < V < Vn(s). LettingV(s') be theoptimalcontinuation
values,
thenV(s') < Vn-1(s') on a subsetof S withpositivemeasuregivens. Now,strictconcavity
followsby theinduction
hypothesis. II
thatR(k,s) - (1 +r)k > L(s) forall (k,s) eliminates
thepossibility
ofexit,makingthisassumption
too
Assuming
is to allowforrandomizations.
In particular,
randomizations
on theexitdecision
strongforourpurposes.An alternative
forsomevaluesof (V, s) aretobe expected.The secondlemmashowsthattheserandomizations
arealso sufficient.
Suppose thatat the beginningof the period,afterobservinga shocks and whenthe initialvalue is V, a
randomization
is usedanddefinethenewvaluefunction
Wrby
Wr (V, s) = maxxe[0,1],
V21>0, V,1
>0 XW(V2,s) + (1 - k)L(s)
subjecttoXV2+ (1 - X)V1 = V,
wherethefunction
W is definedas in equation(8), butreplacingWr forW undertheintegral
sign.
Lemma 4.
Thevaluefunction
Wr(., s) is concaveforall s e S.
Proof The prooffollowsthelinesoftheone givenforLemma3.
II
1 thatthevaluefunction
is strictly
monotone.
andProposition
Finally,itis an immediate
consequenceofconcavity
Lemma 5. SupposeW(V, s) is concavein V forall s. ThenifW(V1,s) > L(s) and V1 < V(s) itfollowsthat
W(V2,s) < W(Vl , s) forall V2 < V1.
ofthevaluefunction
itfollowsthat
Proof If W(V2,s) = W(Vl, s) forsomeV2 < V1 < V(s), thenbyconcavity
Lemma1. II
W(V2,s) = W(V(s), s), contradicting
APPENDIX D. OTHER PROOFS
of W(., s) followsimmediately
fromthemonotonicity
of (., s)
ProofofLemma1. The weak monotonicity
We nowestablish2. Firstnoticethatfors 4 S, W(V, s) = L(s),
applyingstandard
dynamicprogramming
arguments.
foranyV. Now,if V = V (s), thenitis possibleto chooseas continuation
valuesV(s') = V(s'). We willshowthatthe
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OPTIMAL LENDING CONTRACTS
ALBUQUERQUE & HOPENHAYN
311
theproperty
definedby 2. Supposethenthat
dynamicprogramming
equationgivenbyproblemequation(8) preserves
W(V, s') = W(s') forV > V(s'). Then,lettingV(s') = V(s'), itfollowsimmediately
(8) that
from-equation
W(V,
s)=
--r
=l+r
s')F(ds',
s)
[(s) + W(F(s'),
[(s)
+
f
(s')F(ds', s)] = W(s).
Sincethesetoffunctions
theuniquesolution
W(.) suchthatW(V, s) = W(s) fors E S is closedin thenormtopology,
to thedynamic
thisproperty.
programming
equation(8) also satisfies
We now establish3. NotethatW(s) > L(s), as s e S. AssumethatW(V, s) > L(s), otherwise
theresultis
trivial.If V < V (s), thenthereexistssome n suchthatV < Vn(s). We now proveby inductionthatthisimplies
forn = 1, forin thiscase V < VU(s) and lettingV(s') be theoptimal
W(V, s) < W(s). The resultis immediate
continuation
valuesstarting
from(V, s),
W(V,s) <
7
(s)
+fW(V(s'),s')F(ds', s)
(s) +
< -+r
s) = W(s).
WV(s')F(ds',
To continue
withtheinduction
argument,
supposethatV < Vn-1 (s) impliesthatW(V, s) < Wf(s)forall s. ThenifV <
as forthecase
Vn(s), eitherV < VU(s) or V <
f Vn-1(s')F(ds', s). In thefirstcase,usingthesameargument
1_ s) < W(s). In thelattercase, V(s') < Vn-1(s') fora subsetofS withpositive
n = 1,itfollowsimmediately
thatW(V,
measure.Usingtheinduction
it followsimmediately
thatW(V, s) < ?r[nr(s) + f T(s')F(ds', s)] =
hypothesis,
W(s).
II
5. We first
concavein V forV < V'(s) and W12> 0, thenany
ProofofProposition
provethatif W is strictly
Considertheproblem
optimalcontinuation
policyV(s') mustbe non-decreasing.
f
g(V,s) = max W(V(s'),s')F(ds', s)
to
subject
fV(s')F(ds', s) < V.
(D.1)
Let s2 > sl. We now showthatforanypair (V, s) the solutionto equation(D.1) is V(s2) > V(sl). Since W is
concavein V, it is almosteverywhere
of problemequation(D.1) implythat
differentiable.
The first-order
conditions
and because
Otherwise,
W1(V(s2), s2) = W1(V(sI), sl) at all pointswherethesolutionis interior.
by strictconcavity
that
W12> 0, itfollowsimmediately
W1(V(sl), s)
< W1(V(s2),
s1)
_ W1(V(s2),s2).
Since V(s') is an increasingfunction,
f V(s')F(ds', s2) > f V(s')F(ds', sl). LettingV (s') obtainedfrominitial
values(V, si), itfollowsthatV2(s') < V1(s') forsomesetofpositivemeasure.Butthen,bythestrict
of W,
concavity
thisinequality
mustholdforall s'.
We now provethatif W(V, s) is strictly
concavein V forV < V(s) and FI12(V,s) > 0 and concave,then
withpositivecrosspartialderivatives
W12(V,s) > 0. We willshowthattheBellmanequationmapsthesetoffunctions
intoitself.So supposethatwe startwitha function
concaveforV < V(s'). It follows
W12> 0 and,as assumed,strictly
thatthefunction
itfollows
concavein V, forall V < V(s). Usingtheenvelopetheorem,
immediately
g willbe strictly
thatg,(V, si) = -L- W1(V (s'), s'), whereVi is theoptimalsolutionstarting
fromsi. Let s2 > sl and assumethat
V < V(sl). By theresultabove it followsthatVl(s') > V2(s'), so gl(V, sl)
gl(V, s2), and thusg12 0. The
>
<
function
W satisfies
thefollowing
Bellmanequation
,
W(V,s) = max L(s),
+r [I(V, s) + g(V,s)]
wherefunction
to dependon W as definedin equation(D.1). Since 112 > 0 and gl2 > 0 as wasjust
g is understood
proved,itfollowsthatthefunction
W12> 0 as well. II
6.
ProofofProposition
We provethisproposition
in a sequenceofclaims.
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312
REVIEW OF ECONOMIC STUDIES
Claim 1.
7. Then
SupposethatI and F satisfy
Assumption
V (s) F (ds, so) s
,
ins ==n +rr
W1(V (s) , s) constant
1
is non-decreasing
in so.
Corollary1. In theoptimalcontract,I 1(V (s), s) is weaklyincreasing.
thats1 > so and I1(V(sl), sl) <
Proof. Suppose,by way of contradiction,
continuation
derivative
fromsi andsince
starting
l(V(so), so). Lettingci be the
n1I1(V(s1),s1) + c1 = -I(V(so), so) + co,
itfollowsthatcl > co. Let V1(s) denotethecontinuation
atsl and Vo(s) thecontinuation
valuesstarting
valuesstarting
fromso. By concavity
of W,itfollowsthatV1(s) < Vo(s). Hence
1ll(V(sl),sl)
=
+rj
1
>
(I
rf1
fV(s)F(ds,s)',Sl)
Vo(s)F(ds, sl), s
.
(D.2)
ins, byClaim 1 it followsthat
Finally,sinceW1(VO(s),s) is constant
11 (
l (l
Vo(s)F(ds,sl), Sl >i
(D.3)
Vo(s)F(ds,so),so
= 1I(V (so),so).
Combiningequations(D.2) and (D.3), it followsthat
1Ii(V(sl),sl)
tion. II
> nlI(V(so), so), thusgenerating
a contradic-
Now we proveClaim 1.
in dynamicprogramming.
induction
Supposethatwe startwith
ProofofClaim 1. The prooffollowsthestandard
Thenwe need to showthatT W
a function
W thatsatisfiesthisproperty.
thenew value function.
Let T W represent
in s. Considers1 > so. ThenT W1(V(sl), sl) =
satisfies
thisproperty.
Take V(s) suchthatT W1(V(s), s) is constant
in s, so
in theproofofCorollary1, itfollowsthatIl1 (V(s), s) is non-decreasing
TW1(V(so), so). Usingtheargument
7
usingAssumption
1-I1
1
V(s)F(ds,
sl),sl>
T1r1
1
V(s)F(ds, so), so
II
Claim 2. Take (V1,sl) and (Vo,so) such thatsl > so, and W1(V1,sl) < W1(V0,so). Let V1(s) denotethe
from(VO,so). ThenV1(s) > Vo(s) and
optimalcontinuation
policyfrom(V1,sl) and Vo(s) theoptimalcontinuation
< W1(Vo(s), s).
W1(V1(s), s)
let V1 satisfyW1(V1,sl) = W1(V0,so). By concavityof W in V it follows
Proof To provethisstatement,
of thepolicy
thatV1 < V1. Let V1(s) denotetheoptimalcontinuation
from(VI, sl). Fromthemonotonicity
starting
it followsthatV1(s)
function,
V1(s). FromClaim 1 it followsthat I1(V1,sl) > l1(Vo,so) and consequently
<
theinequalities,
of Win V itfollowsthatV1(s) > Vo(s). Combining
W1(V1(s), s) < W1(Vo(s), s). Usingtheconcavity
>
itfollowsthatV1(s) Vo(s) andthusW1(V1(s),s) < W1(VO(s),s).
II
Combiningtheseresultswe can now proveProposition6. Considertwo histories(so,sl, sT-1, sT) and
....
fort = 1. Let ct andEtdenotethecorresponding
derivatives
inequality
ST-l, sT) wherest > st withstrict
theclaim,
of the....
valuefunction
at t. Obviouslyco =
FromClaim 1 it followsthatcl i ~1. Applyinginductively
0g.
it followsthatct
ft forall t > 1. LettingVT(s) and VT(s) denotethecorresponding
policiesat T followingthe
<
fromtheconcavity
of W in V itfollowsthatVT(s) > VT(s).
histories,
respective
II
(so, s
ProofofExample2.
Since 1(V(s), s) is non-decreasing
in s, itfollowsthat
V'(s)
V(s)
(1 - a)s
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(D.4)
OPTIMAL LENDING CONTRACTS
ALBUQUERQUE & HOPENHAYN
d +1r
d +r
) dso
Now consider
1
V(s)F (ds,so),so
f+r
dsoo
asro
=
V(s)F(ds,so)
V(s)F(ds, so) - (1 - a)so
sf
313
V (s)F(ds, so).
we get:
andrearranging
Integrating
bypartsthesecondintegral
fV(s)F(ds,so) - (1- a)so
Va( sV ) +/ V'(s)(1-
F(s, so))ds
s0)ds
V(s)F(ds,so) + (1 - a)soJ
V'(s)F2(s,
=
s
f
Fi(s, so)s + F2(s,so)so ds.
V(s)(
The inequality
followsfromequation(D.4) and F2(s, so) < 0 (i.e. F(s', .) is non-increasing). II
in V. Hereitis shownthatW(V, .) is
7. In Lemma5 we showthatW is weaklyincreasing
ProofofProposition
also weaklyincreasing
in s.
Assumefirst
thatwe startwitha weaklyincreasing
function
W on theR.H.S. of theBellmanequation.Suppose
valuesfrom(V, si). Let ti, i = 1,2 denotetheprobability
distribus2 sl andlet Vi (s) be theoptimalcontinuation
>corresponding
function
tion
to F(., si). By Assumption
dominatest1, so thereexistsa transition
5, A2 stochastically
measureon
P(s, s') withsupporton theset {(s, s') e S x S : s' > s} suchthat#2= Pjq. Let vl be a probability
S x 91+ withsupport
on thegraphofthisfunction
V1(.) suchthatvl (ds, V1(ds)) = Al (ds). It followsbyconstruction
thatf W(VI(s), s)F(ds, sj) = f W(V, s)vl (dV, ds). Definea measurev2 on S x 91+ by liftingvl withP, i.e. for
marginal/r1
anyrectangleset A x B let v2(A x B) = f P(s, A)v1(ds, B). It is easy to verifythatsincevl has first
function
thereexistsa transition
and A2 =
thenv2 willhavefirst
theorem,
marginal
I[2. By theRadon-Nikodym
PptI,
measureon 91+ foreachs, whichmaybe interpreted
Q : S x N+ - [0, 1] suchthatv2 = -Q2. Q(s, .) is a probability
as a randomized
on continuation
valuesV, giventhatstates. Since W is concavein V, theoptimalcontinuation
strategy
i.e.
no lowerthantheone obtainedfromthisrandomized
strategy,
policyV2(s), given(s2, V) mustgivea pay-off
f
W(V2(s),s)F(ds, s2)
fW(V,s)Q(s, V)F(ds,sl)
s)v2(dV,ds).
=f W(V,
(D.5)
Finally,comparing
v2 andv1,
fW(V,s)v2(dV,ds) =
Sf
W(V,s)P(s',s)vl(dV,ds)
W(V, s')v (dV, ds'),
(D.6)
wherethelastinequality
followsfromthefactthatP(s', .) has supporton valuess > s' and theinduction
assumption
thatW is weaklyincreasing
in s. Combining
equations(D.5) and (D.6), itfollowsthattheBellmanequationmapsthe
setofnon-decreasing
functions
intoitself,andthusW is non-decreasing.
Now lete(V) = sup{s I -I [Fl(V, s) + g(V, s)] < L}, wherefunction
g is as definedin theproofofLemma5
is weaklydecreasingand
to checkthatthisfunction
above,andifthissetis emptysetit equal to infS. It is immediate
givestheexitthresholds. II
from(V, s), and Vi(s')
8. Let s2 > sl. DefineV(s') as theoptimalpolicyfunction
starting
ProofofProposition
as theoptimalpolicystarting
from(V(si), si), fori = 1,2. We wantto showthatB(V(s2), s2) > B(V(sl), sl). Using
thefirst-order
condition
forV (s'):
B1(V(s2), s2) =
(V(s2), s2) + B1(V2(s'), s')
1+r 7l1
1
i l(V(sl), sl) + B1(VI(s'), s')
l+r
-= Bl(V(sl), sl).
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314
REVIEW OF ECONOMIC STUDIES
Positivepersistence
and concavity
implythatB1(V2(s'), s') < B1(V1(s'), s'). Thus,111(V(s2),s2) > F1(V(sl), sl).
ofFI(., s) thereexistsV2 > V(s2) suchthat 11
also implies
By concavity
'(V2, s2) = 1(V(s1), s1). Positivepersistence
that
f
W(V2(s'),s')F(ds', s2) >
ins if
Hence,B(V(s), s) is increasing
s')F(ds',
sl).
W(VI(s'),
FlT(V(s2),s2) - (1 + r)V(s2) > FI(V(sl), si) - (1 + r)V(sl).
The assumption
on theproposition
impliesthat
s2) - (1 + r)V2
fl(V(sl), s1) - (1 + r)V(sl) <I(V2,
<
< Fl(V(s2), s2) - (1 + r)V(s2),
wherethesecondinequality
followsfromI-(V, s) - (1 + r) V beingdecreasing
in V, forfixeds.
II
Thispaperhascirculated
withImperfect
underthetitle"OptimalDynamicLendingContracts
Acknowledgements.
We appreciatethehelpfulcomments
of Dilip Abreu,V. V. Chari,Raquel Fernandez,AndyNewman,
Enforceability".
David PearceandNancyStokey.We also thankseminarparticipants
ofChicago,
atLondonBusinessSchool,University
ofRochester,
theNBER SummerConference,
Northwestern
UCLA, University
Yale,Universidade
University,
Cat61ica
theSED 1999Conference
andtheSimonSchoolofBusiness,andthe2001 AFA Meetingsin New Orleans.
Portuguesa,
the financialsupportfromtheDoctoralScholarshipProgramof the Banco de Portugal.
Albuquerqueacknowledges
financial
The usualdisclaimer
fromtheNationalScienceFoundation.
Hopenhayn
acknowledges
applies.
support
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