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Working Paper Series
Collateralized Debt as the Optimal
Contract
WP 90-03(R2)
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Jeffrey M. Lacker
Federal Reserve Bank of Richmond
Working
CoIJXCERALIZED
DEBT
Jeffrey
Paper
90-3R2
AS THE
OPTIMU
CONTRACT
M. Lacker
Research Department,
Federal Reserve Bank of Richmond
P.O. BOX 27622, Richmond, VA 23261, 804-697-8279
November 3, 1989
Revised, February 1, 1991
This material has been published in the Review of Economic Dynamics, the only definitive repository of the content
that has been certified and accepted after peer review. Copyright and all rights therein are retained by Academic
Press. This material may not be copied or reposted without explicit permission.
IDEAL (International Digital Electronic Access Library)
Helpful comments were received from Stacey Schreft, John Weinberg, Doug
Diamond, Kevin Reffett, Monica Hargraves,
Charles Kahn, James Bullard, John
Caskey, William Gale, George Fenn, two anonymous reviewers,
and at seminars
at Purdue University,
the Federal Reserve Bank of Richmond, Florida State
University,
The University
of Virginia, the 1990 NBER Summer Institute,
and
at the Econometric
Society 1990 North American Meetings,
although all errors
remain the author's responsibility.
This is a revised version of Federal
Reserve Bank of Richmond Working Papers 90-3 and 90-3R.
The views expressed
are author's and do not necessarily
reflect the views of the Federal Reserve
Bank of Richmond or the Federal Reserve System.
In a two-agent,
ABSTRACT :
conditions
environment,
contract
feasible
of one good
The
second
contract5
amount
thus
in order
are that
in the
sense
available
to the
lender
to a wide
aversion
good
that
range
set of all resource
for extraneous
usually
and some of the
and is essential
constraints.
of the borrower
contracts.
of financial
good
amount
is
are paid.
in nontrivial
The critical
conditions
in a particular
way,
is nonincreasing.
The
to the borrower
an upper
and
a fixed
good
second
debt
randomization.
pays
first
are heterogeneous
it impose5
risk-sharing
a collateralized
but if the
good,
available
in feasible
the
the borrower
incentive
preferences
risk
collateral
good
information,
which
allowing
as collateral,
to satisfy
contracts
applications
first
serves
the absolute
of the
second
under
within
even
contract,
of the
all of the
good
for optimality
and that
debt
and none
insufficient,
allocation
allocations,
In a collateralized
private
are described
is the optimal
incentive
two-good,
limit
can sharply
on the compensation
The re5ults
arrangements.
constrain
suggest
-l-
Why
agent's
is there
payment
is noncontingent
because
general
to the
the treatment
equilibrium
be fully
agent,
smaller
theory
then
seems
than
payments
answer
with
under
contract."
The
first
good,
which
second
good
and this
feature
studied
of one of the goods
and nonverifiable;
for simplicity
"contract"
in this
setting
each
agreed
good,
earlier,
such
"Arrow-Debreu
of agents'
the
upon
expected
incentive
is derived
here,
as in Townsend
of Arrow-s.
environment.
of one agent
endowments
implied
payment
contract
value
debt
payment
of
(the
is private
information.
A
schedules,
is the
one
for
given
solution
the weighted
only by resource
by private
The
Conditions
are nonrandom.
1371, maximizing
constrained
The
endogenously.
of arbitrary
An optimal
[2],
paradox
success.
insight
but the realized
all other
this
for insufficient
the endowment
is random
is private
is a "collateralized
as collateral
in general
is a noncontingent
limited
sharing
in
occasionally
to resolve
on the
events
noncontingent
of payments
only
while
is
will
as a quid pro quo for some consideration
utilities
conditions
arrangment
is a pair
as a loan advance.
program,"
directly
an
This
information
arrangement
a two-good risk
serves
payments
Efforts
have met with
builds
observed
be completely
the possibility
the optimal
In the environment
"borrower")
paper
is made.
some relevant
the optimal
default
is to examine
are found
the
in which
in this
key innovation
contractual
as in a "default."
occasional
proposed
with
publicly
will
arrangements
of circumstances,
or no payment
that
contractual
otherwise,
range
of uncertain,
suggests
inconsistent
contractual
a wide
If instead
and find environments
payment
over
[I, 111.
contingent
but this
observed
as in a '*default," little
occassionally,
puzzling
In 50 many
debt?
to an
average
feasibility
Contract5
of
and
-2-
maximum
generality
Prescott
are allowed,
and Townsend
random
good
all of the
random
transfered
only
is
determined
functions
the
payment
The main
and most
needed.
The
to the random
rates
Edgeworth
here
of this
control
does
R, and pays
nonrandom
up for"
payment
R,
amount,
good
R; the exact
is
schedule
good
the gap between
R.
This
feature
An interesting
is not to compensate
is a set of conditions
for each
play
the
lender
realized
roles
utility
random
for contracts
Ex post verification
plays
the
environment.
of preferences
is
good
less,
relative
in the
that
their
marginal
sense
apart.
never
state
Essentially,
be tangent
(see Figure
3 below).
marginal
displays
nonincreasing
The results
which
are
no role.
the
inside
are that
good.
which
(collateral)
bounded
must
under
in this
of diversity
nonrandom
agents
minor
the borrower's
to the
type
allocation
the borrower,
of the two
and allow
the
fixed
a fixed
nonrandom
constraints.
its role
are everywhere
that
and the
a certain
than
drawn
The
of it "make
is the optimal
good,
theory,
variation.
paper
contract
respect
as in
of the borrower.
like the
and that
with
is that
R.
Thus
incentive
must
Boxes
aversion
good
lender
assumptions
finite,
transfers
random
critically,
curves
lotteries
at least
is less than
constraints.
from the
of substitution
indifference
other
debt
endowment
pays
is
endowment
it is less than
in that
the honesty
result
extraneous
the borrower
realized
random
of the
directly
collateralized
the
incentive
of collateral
but to ensure
contract
whenever
the
by the
quite
property
First,
good
as collateral
actual
follows
debt
whenever
when
involving
[28].
In a collateralized
of the
possibly
utilities
rely
functions
the
Two
are
absolute
on optimal
of bounded
risk
-3-
The persuasiveness
depends
on the plausibility
particular
the way
prefences
minimize
would
implies
seem that
some of the
will
repay
good
corresponds
property.
are unique
shortfall
The
limited
has,the
made
It is easy
farmer
to a similar
rise
to collateralized
might
more
be a plot
productive
previously
pay as much
contract
possible.
utility
a fixed
surrender
"reduced
debt
than
economic
structure
else,
by using
it to some other
naturally
calls
either
lender.
because
is worth
use.
for the borrower
that
retaining
with
might
any
also give
and 50 might
The collateral
which
of a learning
to the borrower,
chattels
chattels.
of a stock
Nobody
portable,
So the optimal
farmer's
good with
who
if chattels
out of the harvest,
contract.
it, or because
agent's
but
the borrower's
environments
capital
hand
The
durable,
of preferences,
as the optimal
anyone
as the machine
amount
harvest.
as well,
of some of the
related
form"
to the
It
the borrower
farmer:
property
direct
of land or a durable
acquired
transferring
possesses
of the
receives.
a loan to a farmer
from the next
to an individual,
to imagine
rise
proceeds
in
in any collateralized
have
Imagine
suited
repay
up by the
.
state?'
to the chattels
lender
and specially
be of only
contract
(private)
The difference
lender
the contract
in
ceteris paribus, to
attempts,
the
here
on preferences,
has to be involved
in every
proposed
are heterogeneous.
good that
why wouldn't
collateral
out of the
collateral
like this
contracts
conditions
contract
of the collateral
otherwise
over
preferences
the optimal
something
contract;
personal
that
of debt
of the required
in which
the amount
debt
might
of the explanation
good
the borrower
is
of knowledge
or setup
else will
give
cost to
be willing
to
and so the optimal
the machine
as often
as
-4-
The
that
collateral
without
good
held
that
it imposes
under
is
available
of
loan.
but perhaps
intertemporal
rates
amount
formation
because
distortions
capital,
in the
serve
suggest
as
all,
it is rarely
other
than
those
other
future
implicit
legal
bankruptcy
beyond
the
the
for
simple
examples
might
not be able
of
implications
marginal
distortions
limit
an upper
in the rate
are not
can
collateral
of the
associated
because
lender
in
in
on the
of capital
fully
funded
some types,
[41], and
such
as human
[15].
case that
type
with
the potential
complex
models
a borrower
promised
extensions
lending.
can be viewed
that
we have,
income
and these
Indeed,
often
noted
with
of financial
no resources
streams
serve
a creditor's
collateralizes
associated
50 far,
literally
as a contingent
implicitly
phenomena
has
in payment;
for example,
"unsecured"
repayment,
the
inputs
along
are available,
proceedings
from the promised
Although
here,
of the exact
collateral
debt.
of capital
the
in which
consumption,
projects
collateral
a generalization
to equalize
distortions
efficient
provides
agents,
feasible
perhaps all debt contracts are implicitly collateralized.
After
dates
failure
sense
allocations,
the borrower
Briefly,
across
collateral
reported
that
the
of the
that
to situations
in the
incentive
amount
utility
paper
of borrowers'
can obtain,
choice
poorly
The results
above,
include
otherwise
this
and
attainable
As a result,
of substitution
a borrower
constrain
constraints"
allocation
the
on the expected
In a sense
contract
are resource
In fact,
insufficient.
constraints
loan
bound
"borrowing
intertemporal
that
sharply
contracts.
of collateral
the
can
an upper
a desired
to the optimal
contracts.
borrower
feasible
the phenomenon
contracts
constant
by the
to obtain
is essential
it the only
are the trivial
obtain
good
claim,
at
as
rights
distinct
an unsecured
bankruptcy
contracts,
are far
including
in
-5-
the present
unsecured
one,
and
it is worth
(explicitly)
The economic
problem
under
that
which
discussion
The
finds
of collateral
of the
optimal
debt
version
[22].)
on Townsend's
costly
randomized
dominate
in the present
randomization
that
states
and
the possibility
states"
it is well
give
based
known
schedules
can
rise to debt
randomized
condition
paper
are those
payment
arbitrary
and a simple
cannot
the crucial
of simulating
and Moore
the return
that
rationing,"
is found
Bester
be faked while
the return
This
seems
asymmetry
question
"default."
contracts
under
which
and Hellwig
to the borrower
implausible
of why borrowers
In the present
[8]
in most
do not exploit
paper
the borrower
in any state.
[16] and Kahn
assume
and lender,
of "credit
is unobserved.
it evades
any amount
renegotiation,
borrower
paper,
of a model
"default
in all other
Hart
in the working
and do not in general
5.
to derive
contracts
and randomized
A
in Section
attempts
[36, 40, 141, but
In contrast,
contract.
is contained
debt
conditions
is unnecessary.
In the context
can hide
schedules
(38, 24, 21, 28, 381.
are allowed
contexts,
policies
debt
2 describes
the programming
results;
appears
of optimal
verification
verification
deterministic
contracts
assume
state
models
Section
and displays
previous
discussion
between
one.
1.
the main
contracts
discusses
(A fuller
known
contracts
is a collateralized
introduction
the best
in Section
4 contains
constrained
contracts.
Probably
that
contract
the distinction
debt may be a subtie
optimal
Section
them.
that
is described
3 defines
the optimal
remainder
secured
environment
Section
contracts.
considering
and Huberman
the borrower's
but are not verifiable
[19], in models
resources
focused
are observed
by a third
party
such
on
by both
as a court,
-6-
"enforceable"
and thus
ascertain
litigants'
contracts
wealth
are ex ante contingent.
obligation
be a "sum
been
certain,"
difficult
since
to reconcile
widespread
presents
[13],
penalties
borrower,
penalties
are
in loan
assymetric
other
contract
In addition,
paper
attention
described
provides
restriction:
but
one
can costlessly
and the other
is that
the
essential
hazard
contract
to contracts
two alternative
to
of default.
If
by the
the borrower
produce
evidence
and ex ante
debt
reguires
is an
risk
and utilities
of effort
in which
cannot
of larger
can costlessly
and
similarity.
technological
is that
lender
committing
uncollateralized
and uniqueness
be restricted
loan model,
of the one presented
distributions
property
the
contingent.
of collateral
ex ante moral
on probability
with
to be forfeited
their
of the debt
system
lender
case
have
It seems
in the event
the treatment
in which
profits,
are genuinely
on the
a special
and highlights
ratio
must
Innes
this
return
unifies
legal
investment
good
at the
partnership
[29, 261.
that
relies
as a second
environments
likelihood
debt,
that
an admissible
is calculable
reckoning
in the
one-good,
that
The optimality
is nondecreasing.
justify
than
is virtually
and restrictions
century
on the borrower
interpreted
contracts,
contract.
requires
court5
contracts
is that
and besides,
an imperfection
[17, 181 recently
as the monotone
return;
12th
penalties"
information
neutrality
realized
the
the present
Innes
that
at least
his environment
Thus
optimal
claim
debt
"nonpecuniary
here.
the
that
But
enforce
principle
an amount
settling
such
contingent.
and often
legal
meaning
in a risk-neutral,
an optimal
impose
regularly,
[34, pp. 59-701,
use of contracts,
Diamond
these
in which
enforceable
quite
be made
A longstanding
time the suit is brought
arrangements,
cannot
hide
such
choice.
the payment
assumptions
hide
than
the
actual
the return.
-7-
These
results
essential,
seem
because
arrangement
plausibility
cannot
literatures
but
that
except
available
worth
in a number
noting
collateral
"worth
on
information
adverse
selection
collateral
the borrower
collateral
the
rationing
which
debt
"credit
that
would
assumptions
contracts
that
rationing"
state
Such
question
sharing
even
with
the
of nonexistent
that
the
lender
is also
that
selection
depend
on them,
as given.
the
It is
the role
of
the collateral
with
is
the present
in the
(private
large
ex ante
[5, 6, 7, 8, 35, 391,
In general,
the
on the use of contracts
that,
over
if it were
possible,
all of the promised
not be incentive
Thus the possibility
be quite
to arise.
based
important
returns)
would
in various
investigated
consistent
to hand
paper.
might
taken
assumed
in the sense
contracts
are assumed
contracts
is imposed.
results
prefer
selection
evidence
and others
have
future
contract
of the present
due to adverse
above
due to adverse
is insufficient
[5, 6, 7, 81.
one might
are generally
concerning
in every
is not,
of debt
Collateral
debt
risk
contract
to the borrower,
rationing"
a collateralized
some'general
or of assuming
it is generally
[3, 4, lo]).
"credit
cited
of papers
than
of the borrower
again,
under
lender
(for example,
literature
with
to the
the effects
to agents
in loan contracts,
less"
paper
but
that
is quite
from the borrower.
for the works
form of contracts
neutrality
can produce
resources,
resources
examine
known,
Second,
a borrower
existing
risk
and the debt
contracts.
hide
existing
First,
it, as is well
to monotone
is able to hide
settings,
limited.
in his setup
of assuming
but
Vast
without
is optimal
the restriction
resources
somewhat
sensitive
compatible
of credit
to the way
in
-8-
1. Environment
There
goods,
are two
called
referred
agents,
a and b, and
named
1 and 2, and
indexed
to as "the borrower,"
by i.
indexed
Agent
a will
h of good i is ehir h=a,b, i-1,2.
Endowments
known
nonrandom,
nonnegative
The endowment,
1 is random,
and will
as collateral,
will
50 it will
be restricted
endowment
of the
Agent
yl of good
by 8.
Good
to as the
in which
occasionally
e,2,
eblr
ealr
2 will
turn
"collateral
eb2 = 0, and the
by
Cal
=
0
Cbl
=
ebl
In a previous
-
(possibly
the goods
Ca2
Ylr
period,
a some consideration--a
compensate
place
Ylt
+
contingent,
before
and ub(chlrcb2)
utility
functions
Written
Uh(Chlrch2)
=
the endowments
loan
utility
and eb2,
of agent
of
are
a of
out to function
good."
lender
possibly
are consumed.
cb2
advance,
b for the earlier
derive
be
The endowment
1 and y2 of good 2 to agent b after the endowments
take
are two
Attention
has no
good.2
a is to make transfers
given
ua(cal,ca2)
case
collateral
the transfers
Agents
be refered
to the
After
then
constants.
be denoted
There
b as "the lender."
and agent
agent
good
by h.
from
=
eb2
Consumptiona
(1.1)
Y2*
are realized,
for example.
are
Y2I
+
consumptions
agent
b gives
The transfers
according
to the
a and b respectively.
to be additively
of
are received.
from
agent
a to b
consideration.
for agent5
are assumed
-
ea2
negative)
separable,
= Uhl(chl) + Uh2(ch2), for h=a,b. 3
functions
Both
agents'
and 50 can be
-9-
The realized
information
the good
and pretend
that
actually
The
need
lower
assumption
only
than
and higher
do not exist
contract.
pretending
be checked
assumption
must
and Weinberg
and zero
that
of the
than
both
evidence
collateralized
hiding
assumptions
summarized
about
debt
of costly
1:
(a)
19, 8 random.
(b)
density
8, where
strictly
increasing;
derivitives.
than
is strictly
of our environment
are
1 and make
actually
positive
It is perfectly
it seem that
realized.
e,l =
continuous
on n, the
(c)
differentiable;
and ual, ua2,
it is prohibitively
e&S = 0,
support
of
For all feasible,
Uhi( ), h=a,b, i=1,2, are continuous,
continuously
(d)
ebl 1 0,
of 0 is absolutely
0 I 00 c Bl < +a.
consumption8
twice
e,2 > 0,
The distribution
f( ) that
concave,
but
Endowments:
n = [00,0,],
nonnegative
good
elements
as follows:
Assumption
with
the primitive
0 is
costly
ual,
and Ub2 have
costless
smaller
to make
for
than
ua2,
and Ubl are
finite
agent
a to hide
actually
it seem that
lower
of goods
falsification.
The
it
the
for both
against
produce
[23] for an analysis
is.
information
be checked
for the equilibrium
units
is larger
report
private
one cannot
it actually
incentive
the alternative
constraints
a to hide
to falsify
amount
are symmetric
that
than
is that
The usual
costs
not bind
costly
private
for agent
is smaller
the realized
against
The constraint
See Lacker
amount
realized.
incentive
does
is essentially
it is costless
that
of this
falsification
reports.
good
to be prohibitively
actually
and thus
random
the realized
implication
is that
and faking,
that
that
it is assumed
is.
was
of the
To be specific,
do not exist,
constraints
state
a.
to agent
In contrast,
good
endowment
realized,
0 is larger
- 10 -
Note
that
Ub2 need
good
might
surrender
agent
leave
some
not be increasing,
agent
of good
b, or may
50 that
b indifferent
2 might
or even
be a pure
in fact be costly.
"acquisition"
worse
off.
punishment
of the
Thus
that
collateral
having
provides
a
agent
no gain
to
4
2. Contracts
Agents
agree
meet
to a repayment
advance
will
random
was
Agent
value
some
maxiumum
these
smaller
place
contract
e’.
following
amount
that
Bore1
on the
Then
it seem that
to in the
For a given
is a family
some time
loan
later
nonrandom
the
by agent
and Townsend
specifies
The
allows
realized
a, transfers
contract.
a probability
of probability
set of feasible
equal
are
For
[28] and Townsend
display,
the
value
8', is not necessarily
displayed
sets of the
focus
of 0 and either
value,
to be random.
below.
the other
value
agreed
Prescott
A contract
f&Q, on B, the
realized
The displayed
z(dyl,dy218'),
(y~,yz).~
with
advance,
detail
we may
repayment.
b, or makes
the
for a loan
in more
so that
along
to a schedule
are allowed
in exchange
governing
the
Given
according
generality,
is a measure,
for each
explicitly,
a observes
0.
and,
to be described
B is realized,
value
value,
tranfers
transfers
period
to be seen by agent
to the true
to take
of the
endowment
endowments.
true
contract
not be treated
characteristics
the
at an initial
[37),
B', the result
distribution
measures,
over
z'( , Iti)
tranfers
[-eblrelx[0tea21*
An application
Townsend
[38]),
self-selection
of the well-known
allows
us to restrict
constraint.
Revelation
attention
Specifically,
Principle
(see Myerson
to allocations
for a given
contract
that
[25],
satisfy
a
s, and a given
- 11 -
state
8, agent
utility,
a chooses
given
B’E[Bo,B]
an announcement
to
maximize
expected
by
rr [u,l(e-yl)
+Ua2(ea2-Y2)1
n(dYlrdY2(e’)
9
Define
e.
B*(B)
Since
given
as the
both
agents
contract,
both
both
h'lrdy2)~)~
and the realized
effective
announced
are aware
agents
agents
state
schedule
state
by agent
of the choice
can calculate
know
will
chosen
that
has the property
problem
the actual
the true
facing
Therefore,
B*(B).
be z(dyj,dy218)
a when
(2.1)
schedule
agent
state
a for any
for a contract
relating
E i(dyl,dy2lfl*(t9)).
transfers
This
that
rr [ual(e-yl)
+Ua2(ea2-Y2)1
x(dYltdY2Ie)
1 rr [u,l(e-Y1)
+ua2(ea2-Y2) 1 A(dYlrdY2Ifl’ 1
5.t.
v(e,e’ jam,
A contract
the
is incentive
B*(B)
fact that
(IF' 1, agent
contract,
an identical
attention
maximizes
a always
there
feasible
exists
allocation.
to contracts
reveals
a contract
Thus
which
Under
which
(IF’).
satisfy
a contract
the actual
satisfies
is no 1055
there
the
(IF’ 1
e-e.
if it satisfies
(2.1).
truthfully
(IF’
z that
state. 6
) follows
satisfies
For any given
(IF') and which
in generality
incentive
feasibility
from
results
feasible
z is resource
transfers,
and thus
feasible
satisfies
if it is a probability
in
in restricting
condition
(IF’).
A contract
is
measure
over
- 12 -
R:BXBXW[O,~],
rj- ~(dy~&zle>
=
A deterministic
functions
resource
of the
state,
feasible
A deterministic
deterministic
contract
is one
yl(O)
(RF’ 1
vea
1,
in which
and yp(0).
contract
version
is incentive
of
A collateralized
where
debt
incentive
=
MIN[&R],
Y;(e,R)
=
Q
=
contract,
feasible
Y;wv
is
if it satisfies
the
following
1
ualV-ylW))
+ ua2(ea2-Y2(e’))
f3.t. e-e.
or debt
contract,
contract
(IFI
for short,
is a
(y;(B,R),y;(B,R)), satisfying:
(2.2)
vea,
VoEIRrB1lr
1.
constant
is plotted
in Figure
Figure
For realizations
in (i?o,O1).
The corresponding
of B that
are
(2.3)
veqeo,R).
-uHl(Q)luH2(ea2-Y;(e,R) 1
R is an arbitrary
2.
contract
(IF')
+ Ua2(ea2-Y2(e))
Y;p (e,R)
A deterministic
feasible
v(e,e’ pmn,
and
are deterministic
if it satisfies
u,lWnUW
resource
transfers
An example
consumption
large
enough,
(2.4)
of a debt
schedules
agent
contract
are shown
a transfers
in
a
-13-
constant
is not
R, of good 1, and none of good 2.
amount,
sufficient
to allow
1, and transfers
some
incentive
of good
2.
feasibility.
for &[80,R].
decreasing
satisfies
resource
The
largest
and
Condition
with
the contract
that
y;(eD,i)
R.
= e,2,
for small
available.
debt
may be
and the debt
of all of the
collateralized
(2.4) makes
of the collateral
There
collateral
contract
8; the contract
verified
good
ever
can be termed
some value
contract
good
cannot
would
with
all of good
the endpoint
R, and will
y&J,R)
that
of 0
strictly
the debt
contract
feasibility.
is y;(fIo,R), and this
contract
a transfers
along
(2.4),
realization
y:(O,R) for 0 below
Condition
incentive
If the
R, then agent
It can be readily
amount
debt
transfer
the payment
y;(B,R) = 0 for B=R, determines
condition
ensure
'
of R below
corresponding
in the
more
under
the collateral
lowest
be constructed
require
transferred
collateral
associated
01, call
it R, such
to R requires
For R > R, a
state.
since
the
(2.4) is undefined
than
a has
agent
i is the value of R for which the collateral constraint yz(8) I
e,2 just binds, and no debt contract with R > R is feasible.
3. Optimal
Contract5
An optimal
for which
that makes
without
appropriate
is one that
is no alternative
one agent
making
solutions
merely
there
contract
better
the other
to a particular
for this
be an extension
off
agent
is resource
resource
(in the
worse
private
environment,
of a result
and
incentive
and
feasible
contract
sense of ex ante expected
utility)
off.
and incentive
feasible,
Optimal
information
"Arrow-Debreu
as in Townsend
of Prescott
contracts
[37].
can be found
program"
Proof
and Townsend
of this
[28] to a
that
as
is
would
- 14 -
continuous
measures
Mm
over
&
5.t.
and
transfers
subject
A(
, 16) for each
(RF')
and
(PI)
a solution
three
1, imply
assumption
permits
Assumption
constraints,
deterministic
1:
Debt
sufficient
enter
constraints.
linearly
utilities,
The program
and Townsend
constraint
l(a) and
expected
contract
[28] and
in both
the
set is convex.
l(b) guarantee,
If the
then
we
Contracts
for the optimality
conditions
the debt
restricting
Proposition
the
as Assumptions
further
that
2r
variables
In Pl the
exists.
1 is not
Assumption
agents'
as in Prescott
choice
of Collateralized
section
weights.
incentive-compatibility
the
and the
Pareto
sum of the two
as possible,
Because
Assumption
In this
and
set is nonempty,
4. Optimality
nonnegative
the weighted
general
function
that
BUI, to
(IF')
to maximize
[37].
constraint
probability
j-Jr [Ubl(ebl+YI) + Ub2(Yz)l~(dYl~dY21e)fodB
Pl is as fully
objective
is to choose
+ u a2(ea2-y2)lA(dylldy2)B)f(e)de
to feasibility
Townsend
The program
is omitted.
Aa and xb are arbitrary
is chosen
know
space,
ual(fl-Yl)
XaSSS[
+
where
state
-ui>(ca3)/u&(cal)
Let Assumptions
contract
are described
contract
attention
solves
of the debt
which,
is optimal.
The
to deterministic
1 and 2 hold.
Pl.
Then
together
following
contracts.
is nonincreasing.
a
contract.
with
- 15 -
Proofs
are
a with
of agent
risk
aversion
keep
in mind
respect
allows
that
contract
some
insight
that
is resource
this
utility
relax
the
different.
utility
for that
For
right
randomness
side of
does
violations
increasing
not
absolute
with
the addition
might
Assumption
be enough
Proposition
1.
is implicit
in the
incentive
outlined
If faking
above
would
"slack"
is ual(B'-yl)
private
fail because
than
If instead
side of
a's
could
side of
(IF') is
satisfied
for 8 =
(IF') may be
is no more
for ualr
risk
the
averse
with
+ ua2(ea2-y2),
before.
Thus
and may
and
adding
in fact
ual displayed
(IF') would
be made
for 8 > 8 and 6" = 8, and the slack
both
agents
falsification
nonexistant
left
into any constraints,
the right
to compensate
side of
agent
is unchanged.
are still
no smaller
feasibility.
are required
=
8, and add randomness
is nonincreasing
+ ua2(ea2-y2)
of randomness
standard
constraints
aversion
aversion,
l(d) concerning
but
deterministic
a way that
The
constraints
(IF') is certainly
risk
Fix
truthtelling)
of y1 and y2 than
of incentive
[27]),
ual(cal)
a given
feasible.
0' = 8 the right
introduce
Thus
aversion
absolute
(see Pratt
be finite.
constraints?
incentive
u,l(t?-B'+B'-yl)
SO the
functions
1, consider
(assuming
risk
risk
Nonincreasing
good.
s( , Ia), in such
feasibility
absolute
to the distribution
obtained
state
B > $ and
Because
function
smaller
state,
absolute
if 6 > 0.
and incentive
for that
regard
cause
only
must
into Proposition
incentive
0' < 8.
utility
utilities
for 0 = 8, 50 the
unchanged
8, and
of range
is allowed
to the allocation
expected
a wide
is the
1, the random
to good
marginal
(l-a)-l(6+caJ)l-o
For
-u~j(cal)/u;l(cal)
in the Appendix.
goods
costs
for the extraneous
plays
is possible
information
for 8 < 0'.
in
and is costless,
assumption,
In this
a role
risk.
then
additional
case the proof
for 0 < 8 and 8' = 8 the right
as
side of
-
(IF')
is no smaller,
decreasing.
that
constraints
the
is strictly
the proof
However,
in private
reasonable
and
information
implications
risk-sharing
is that
1 allows
to deterministic
aversion
1 makes
environments
could
to prove
no use
problem
of the
of this
Assumption
a version
The problem
is strictly
for the optimal
relax
us to rewrite
contracts.
if risk
do not bind
one
of optimality
Proposition
larger
of Proposition
for B < 0' generally
conjecture
16 -
type
incentive
contract.
A
l(d) but use
Proposition
Pl with
fact
1.
attention
is now to choose
some of
restricted
payment
schedules
Ylr and y2 to
MAX
Xar [u,l(e-Yl(e))
+ua2(ea2-y2(e))lf(e)de
+ xb l [ubl(ebl+Yl(e)) + ub2(Y2(mfwfl
5.t.
In P2 the
expected
that
utilities,
(IF)
to maximize
chosen
subject
constraints
to the
some difficulties.
(IF) involves
In order
to be reduced
approach
replace
almost
a continuum
to make
to a managable
always
taken
when
a weaker
first
form of a control
problem,
because
y1 and y2 and their
control
theory
derivatives
requires
that
The
the state
(IF) with
order
space
the constraints
we restrict
The
tractable
point
(RF) and
central
one
problem
is
these
constraints
here
(in fact the
The problem
are in terms
state
(IF).
for each
is continuous)
in the
attention
taken
agents'
incentive-
contracts,
approach
condition.
at each
and
of constraints,
the problem
form.
sum of the two
feasibility
presents
for each
et < e.
simpler
to deterministic
P2 still
0,
the weighted
relevant
The program
alternative
need
is
contract
compatibility
and
(RF)
(=I
is to
then
the
of function5
space.
to functions
takes
The
of bounded
- 17 -
7
variation.
The derivative
This
everywhere.
fact can be used
incentive-compatible
Proposition
Let Assumption
-
- me)uide-Yl(e))
nondecreasing
Condition
(4.1) states
function
(4.1) does
contract
payment
condition
of the
bounded
variation.
condition,
(IF)
for
then
variation,
property
almost
of
Thus
imply
equality
0
a. e.
8, agent
(4-l)
a's utility
feasibility.
functions,
for a contract
but this
(4.1) is weaker
the debt
is optimal
under
under
the
If the
then
(IF)
of
the debt
if (4.1)
is optimal
to 8.
for functions
Because
by construction,
is a
to satisfy
is not true
(IF).
than
contract
contract
2
continuous
and sufficient
of utilities,
(IF) with
then
incentive
are absolutely
(yl,yz)
B', for t?' very close
(IF) and the debt
is
this
stronger
weaker
condition
l
An immediate
that
exists
If the contract
for any given
not by itself
concavity
satisfies
substituted
1 hold.
of the announcement,
schedules
because
variation
a convenient
yp(e)u&(ea2-y2(e))
that
(4.1) is necessary
contract
to derive
(IF) and is of bounded
Condition
of bounded
contracts.
2:
satisfies
of a function
agent
implication
(4.1) is that
a can shed via an incentive
the ex ante utility
Proposition
of
of agent
2 implies
a, va(8)
feasible
there
is
contract.
= u,l(e-yl(e))
a limit
to the risk
Define
va(8)
+ ua2(ea2-Y2(e))-
as
Then
- 18 -
Corollary
1:
satisfies
(IF) and
v;(e)
L
Let Assumption
in a sense
slope
of the borrower's
concave,
value
opportunity
variation,
contract
the borrower
bears
"most"
ex post utility
a smaller
with
payment
of u&(8-yl(8))
for risk
(yl,y2)
then
a.e.
+ ua2(ea2-y2(B'))
strictly
smaller
is of bounded
If the
u;l(e-ww
Thus,
ual(e-yl(e’))
1 hold.
sharing
of the risk,
is the partial
respect
yl(e)
in that
to
This
v&(O),
that
state
affords
the
the minimum
derivitive
Note
in a given
state.
by reducing
e.
in that
slope
of
if ual
e implies
is
a
an indirect
of agent
a's ex
post utility.
Some
additional
notation
will
be helpful.
Define
v,(B,R) and Vb(e,R)
by:
va(etR) = ual(e-y;(flJW
vb(&R)
These
are the
contract
=
Ubl(ebl+Y;(erR))
ex post utilities
debt
+ Ub2(Yi(erR))
of the two
agents
in state
8, under
the debt
R.
We can now describe
contract
+ ua2(ea2-yi(etR))
R is the unique
contract
satisfy
conditions
optimal
under
contract.
which
The
the collateralized
first
condition
debt
is that
the
- 19 -
Condition
1:
X,E [6v,:;“)]
1
P
+ xg
P(R
0,
(Derivations
appear
the collateral
If /J > 0, then
is that
sense
second,
their
Condition
risk
0.
~1 is the multiplier
(Recall
I e,2.
does
the value
constraint
associated
y;(BO,R)
not bind,
of xa/xb
condition
with
= e,2 by
50 that
for which
R < R, then
R is optimal.
R = R, and Condition
for the optimality
rates
of substitution
For all 6' in the
_
interior
u;l(e-y;(e,R))
of agent
a with respect
of debt
are sufficiently
1
contracts
different
are bounded
in the
apart:
of 0,
fm
ui2(ea2-Y2(epR
F(e)
1
ub2(Y;(frR))
p(B,R) = -uHj(e-y;(e,R))/uHj(0-y;ce,R))
aversion
that
does bind,
of the two agents
marginal
ubl(ebPYhR))
where
=
Aa and xb.
and crucial
2:
/,
constraint
1 determines
the preferences
that
yz(BO)
the collateral
p given
=
in the Appendix.)
constraint
P = 0, and Condition
The
- R)
If the collateral
definition.)
determines
[“vb;;.“‘]
to good
is the
1.
-
coefficient
The term
of absolute
in braces
on the
- 20 -
right
side
The main
of Condition
result
3:
be discussed
Let Assumptions
R satisfies
and
suppose
P = 0, so that
the collateral constraint does not bind.
from
risk-sharing
this
case
Condition
collateral
shown
good
does
and assume
tangent,
the
and the
lender
edge
that
for
lender.
state
a
1 and 2 with
steps.
now that
R
Then
ual
First,
abstract
is linear.
In
to
Indifference
corn,
collateralized
lies entirely
expected
giving
can,
debt
curves
the
that
collateral
satisfy
state
in Figure
the borrower
more
ceteris paribus, make both
contract
on the boundary
consumption
values
for an arbitrary
Box
the northwest,
more
8, the two agents'
and the borrower
for 8 > R, and the western
lender's
in two
2 is equivalent
in the Edgeworth
The
result
for any given
towards
contract
the
that
a move
off.
Conditions
that
all ea,
are never
than
case
the northern
minimizes
For
curves
better
feasible
3:
highly
In this
this
= 0 and Condition
3 are
below.
contract.
to understand
3 is simply
indifference
more
optimal
considerations
p&R)
Condition
Condition
contract
1 and 2 hold,
debt
It is easiest
agents
and will
collateralized
is the unique
3.
negative,
can now be stated:
Proposition
good
2 is always
edge
is the
incentive
of the Edgeworth
for 8 < R.
of the collateral
This
good.
Boxes;
contract
- 21 -
A very
that
simple
all Utility
and ub2(c)
is yi(B)
example
function5
+ y2(B)
following
MAX
problem,
equivalent
[ebl
s
P2'
lower
constraint.
effect
thus
just
user
ual
this
is not
lower,
(evaluated
using
Consider
is nondecreasing.
the
slope
.in the borrower's
now marginally
1
utility
to minimizing
debt
lender.
E[(l-q)yz(e)],
subject
contract
the borrower's
constraints.
to
value
(RF),
It is easy to
the expected
user
105s
(the borrower)
(IF), and the &,
is the unique
born
8,
by agent
feasible
incentive
less risky,
a.
and this
resource
contract
Imagine
allows
contracts
increase
in
in yz(6')
is now
in the total
consumption
I, and
decrease
u;l(e-yl(e))
a reduction
a gain
feasible
a marginal
feasibility.
The borrower's
has a direct
via Corollary
by a marginal
allowing
ex post utility.
of corn
ex post utility
accompanied
can be lower,
consumption
Alternative
of the borrower's
state
1
Gb
for the
from the higher
(the lender),
to maintain
and so v;(B)
(P2’
W2mfwde
+
reservation
linear,
the risk
enough
the
loss.
incentive
for some given
large
feasibility
(IF)
and
The collateralized
can affect
yl(e)
payment
= c,
to P2:
of collateral
Value
on the
can affect
good)
= ubl(c)
Incentive
to q c 1.
Suppose
here.
= ua2(c)
Ual(C)
the total
words,
Ylw
is equivalent
minimizes
When
+
arbitrary
due to the transfer
that
3 is equivalent
at work
I v - we) + ea2- Y2(e)mew
Vb is some
to the
and that
of the collateral
(RF)
show that
are linear,
L 0; in other
5.t.
where
the principle
Condition
= qc.
a's valuation
agent
illustrates
variation
schedule
in ex ante expected
is
utility
- 22 -
that
can be shared
measure
of the value
compared
with
yl(B)
to the
the wedge
measured
between
agents'
that
For
a bit more
insight
=
xb
In the
above
would
utilities
are equated
be zero
be equated
8’
I B of imperfect
absolute
aversion,
and more
emphasizing
the marginal
state,
is less
lender.
2, note
that
1
that
6Y;mR)
6R
and
risk-sharing,
term
C&m)
6R
3
constraints,
since
weighted
the
marginal
the bracketed
is a measure
due,
is scaled
here measures
'
average marginal
Instead,
negative,
p(B,R), which
a "second-order
corn
to the
of Condition
As a result,
1.
The entire
u;l(fJ-yl(fl)) of a perturbation
literally
good
in
Box.
in risk-sharing
without incentive
state-by-state.
2 is always
imperfection.
It bears
changes
of the Edgeworth
collateral
side
is
2, associated
the marginal
+ ~bUb2(Yhm))
for every
in Condition
informational
risk
curves;
improvement
of the
The gain
of Condition
interior
into the right
environment
of Condition
in states
side
2 is a
6R
same
would
cost
the
-X&2(ea2-yi(etR))
utilities
side
left
-Xau~l(e-y;(e,R)) + $&(ebl+Y;(bR))
+
expression
in risk-sharing.
of the
more
side of Condition
6Vb(&R)
1 6v,(fitR) +
a
6R
right
the value
of giving
right
indifference
toward
disutility
8.
by the
are movements
2 states
The
improvement
the
for all
agents.
of this
1058,
and y2(8)
Condition
than
by the two
term
of the utility
of course,
by the
on the
to the
coefficient
the effect
of
on
in va(B).
the right
effect."
utility
side
of Condition
2 is quite
By giving
the borrower
less
of corn
for the borrower
collateral
is reduced,
and
- 23 -
this
relaxes
the
If p&R)
risk.
incentive
is small
in risk-sharing
between
the
side of Condition
side of Condition
2.
in u&
the value
2 implies
curves
the borrower
is small
and
to bear
little
less
improvement
Condition 2 can be thought of as an upper
Thus
Alternatively,
indifference
and allows
the change
is acheived.
bound on p&R).
by the right
constraint
of indirect
a minimum
risk-sharing
value
measured
for the wedge
of the two agents , as measured
by the
left
Condition 2 can be thought of as an upper bound on
Thus
the lender*8 valuation of the collateral good.
Some
interesting
commenting
"1055,"
on.
aspects
the collateral
ex post utility
evaluated
the
good
lender's
called
to take
verification
In one
(see
as the costly
is fully
lender
point
but because
of view
of time
the debt
is
when
as long as the
(U&I > 0),
unlike
the
it is strictly
also that
Thus the contract
[21] for a discussion
a
is compensated
collateralized
Note
case,
takes
states,
of the collateral
in this
are worth
as it is to the borrower,
the lender's
possession
lender
for B < R; in fact
to the
contract.
The
lender
in these
lender
of view.
value
the
setting
it will
be in
for nonpayment,
is fully
as
time.
the costly
consistency
in the
environment).
important
differ
From
point
and renegotiation-proof
setup
lower
the contract
has a positive
interest
verification
setting
though
for in the original
consistent
costly
range.
from the borrower's
collateral
to the
is always
this
undercollateralized,
contract
of some collateral
is not as valuable
in 0 over
debt
in this
1, of R - B for 0 < R.
of good
(if ~62 1 0) by the transfer
increasing
contracts
In the collateralized
in terms
lender's
of debt
respect
significantly
verification
optimal
from debt
environment.
collateralized
contracts
debt
contracts
in some other
It is easy to establish
in this
settings,
that
such
when
- 24 -
E[Vb(e,R)]
ub2 1 0,
debt
is strictly
R and R' > R, the ex post
contracts,
every
state
expected
always
under
utility
prefers
environment
in R, because
increasing
the
R'.
contract
has no interior
there
[20]),
However,
as the
next
borrower
can constrain
maximum
one.
which
section
depends
explores,
contracts
credit
implies
such
might
of the collateral
good
lender
in this
(see Williamson
an interior
that
in
lender's
that
rationing"
the availability
in a way
the
to R, and the
respect
This
on just
fact,
feasible
b is greater
of agent
of this
with
be no "Williamson
will
(401 and Lacker
Because
R to a smaller
a larger
utility
for any two
maximum.
of collateral
to the
be thought
of as "credit
can
a sharp
rationing."
5. Collateral
Constrained
The borrower's
constraint
than
borrower's
longer
collateralized
contract
or incentive
so that
As mentioned
of collateral.
debt
the
the multiplier
sufficient,
and
earlier,
debt
there
contract
It is impossible
for values
of R greater
impose
may
utilizes
all of the
to construct
than
R (less
be a value
a
R without
violating
feasibility.
3 covers
Proposition
binding
the
endowment
collateralized
resource
endowment
on contracts.
el) for which
Contracts
case
in which
p is zero.
so we need
a more
the collateral
When
general
constraint
p > 0, Condition
result
to cover
is not
2 is no
this
case.
- 25 -
Condition
4:
For all
B in the
-
4:
optimal
1 and 2 hold,
binds
(so that
1 and 4 with
p L 0.
and suppose
that
R = RI, and that R
Then R is the unique
contract.
4 differs
from Condition
~1 on the right
to a further
above,
constraint
Conditions
Condition
containing
f(e)
uh(ea2-y2(e,R))
1F(e)
ui~(y;(&R))
u;l(e-y;(B,R))
Let Assumptions
the collateral
satisfies
of n,
P(fltR)
‘,ibF(e)’
+
Proposition
interior
hurdle
side.
A binding
for the optimality
it is impossible
2 by the presence
to constuct
collateral
of the debt
a debt
of a positive
constraint
contract.
contract
that
gives
As was
provides
term
rise
noted
greater
expected
utility
contract.
more
to the
However,
expected
interior
utility
described
earlier.
incentive
constraint
reducing
PWW,
borrower's
utility.
enough,
exceed
"risk
lender
for the
By giving
the risk
premium"
the value
the cost
lender
that
born
with
more
corn
1 can be relaxed
by the borrower.
can be used
enough
do not resemble
via the indirect
the borrower
to Aa,
constrained
debt
the
(at the rate
lender's
in the
expected
50 that R = i and jb is large
agent b with more expected
the lender's
effect
and less collateral,
slightly
the
debt
can provide
risk-sharing
The reduction
to increase
relative
of providing
associated
does R, the collateral
contracts
in Corollary
If xb is large
then
than
lower
valuation
utility
of the
will
- 26 -
collateral
Once
good.
P(fitR)I or an upper
that
the
again,
bound
this
on the
The
collateral
constraint
the
constraint
is independent
contract.
project,
If agent
he may
,constraint.
present
value"
information
More
could
in the
occur
sense
either
valuation
be quite
severe.
the project
agent
utility
due to the
a could
on
good.
for example,
of agent
a under
an investment
collateral
"having
obtain
bound
collateral
Notice,
considering
financing
an upper
of the
of the expected
despite
that
a positive
financing
net
in a perfect
environment.
generally,
the
collateral
marginal
rates
Such
derived
directly
a wedge,
have
dynamic
could
to obtain
intertemporal
could
lender's
a is an entreprenuer
be unable
This
is essentially
quite
models
far-reaching
is that
constraint
of substitution
will
implications,
between
and the
a key property
of substitution
the
lender.
of the environment
since
rates
a wedge
of the borrower
from the primitives
intertemporal
drive
here,
of many
are equated
across
agents.
6. Concluding
Remarks
An explanation
analysis
of the ubiquity
necessarily
has been
environment.
Whether
this
one
"match"
between
finds
the
situations
depends
in which
on whether
as suggested
should
perhaps
understanding
in the
people
more
carried
explanation
actually
Introduction,
of financial
out
as only
contracts
in the
depends
find themselves.
analogous
a small
on how
step towards
The
attractive
and the
The plausibility
can be found
results.
proposed.
possible
and Conditions
environments
contracts.
has been
simplest
is plausible
the Assumptions
"realistic"
be viewed
of debt
which
Consequently,
an improved
also
deliver,
this
- 27 -
Appendix
Proposition
certainty
Suppose
18
equivalent
a contract
contract
R solves
(yl(8),y2(0))
Replace
Pl.
defined
it with
the
by
ual(e-y#W
= rrual(e-Yl)~(dYl,dY2(e)r
ua2(ea2-Y2(e)
1 = rrua2(eaz-yz)~(dyl~dyz18).
Because
ual and ua2 are concave,
nonnegative.
function
satisfied,
variable
Because
ubI and uh2 are concave,
is not reduced
50 we only
the risk premium,
by this
need
substitution.
to check
incentive
yi(e)
the value
- E,[yile],
is
of the objective
Feasibility
is obviously
compatibility.
Define
a
yl(e,O' ) by
u,lWy1uW
1) = srual(e-yl)“(dYlrdy2)e’)i
yl(B,B')
announced
is the
state
certainty
equivalent
is 8' but the true
By Assumption
2, and Pratt
for any given
8'.
Using
of the random
state
[27], Theorem
this
and
(IF'):
variable
is t9. Note
2,
yl(e,e*)
that
y1 if the
y1(8,8')
= yl(B).
is nonincreasing
in
e
-
28 -
ual(e-yl(0)
+ ua2(ea2-Y2(e))
= rr [ual(e-yl)
+ua2(ed2-yz)lr(dylrdY21e)
and thus
1
JJ
=
u,l(e-yl(W
1
ualuJ-ylW))
(yl,y~)
satisfies
Proposition
(IC).
[u,l(e-yl)
2:
1)
+
+
Ua2 (ea2-Y2 (0’ 1)
ua2(ea2-Y2(e’
) 11
(IC).#
Suppose
(yl,y2)
is of bounded
variation
and satisfies
Then
Ua2 (%2-Y2 (0 1)
2
-
Ua2 (ea2’Y2 (0’))
u,l(e-yl(e))
v(e’
This
+ Ua2(ea2-Y2)l"(dYlrdY21e')
implies
that
where
D-yl(B)u;2(ea2-y2(8))
ual(e-yl(e’ 11,
,epa2xn,
5.t.
e2e.
left derivatives
D-yl(t?) and D-y2(8)
I D-yI(B)u&(B-yI(B)).
exist
almost
Propositions
3 and
and y2 both
both
-
everywhere,
Because
where
they
exist,
the derivatives
do exist
they
must
of yl
satisfy
(4.1).#
ua2(ea2-Y2(e)
1 t and define
now
an arc x:WX2.
X2,
and
recover
rewritten
inverse
yl(e)
=
Define
set of arcs
of ual
= u,l(e-yl(e)
x(B)
of bounded
and y2(0
1 and q(e)
= (xI(6'),x2(8))'.
set of absolutely
and 42 as the
e - ##q(O),
as follows:
x,(e)
the vector
Let A be the
let B be the
41 as the
4:
continuous
variation
inverse
of ua2.
=
A contract
arcs
from n to R2.
Given
= ea2 - +2(x2(0).
is
from n to
Define
i, we can
P2 can now
- 29 -
Choose
an arc XEB to
MAX Aa j- [xl(e)
+
(P3)
+ xz(~)lf(@)d~
Xb J- [$l(B,Xl(B))
s.t.
+ $2(X2(8))]f(B)dB
a.e.
X’ uwwxw)
ve-,
www)
where
x(8)
u,2(0),
x2
= ~~lrX1(e)lx~~r~21r
= ua2(eal+ea21r
ubl(ebl+e-d1(xl)),
and
+2(x2)
measure
unambiguously
endpoints,
only
dx
of bounded
exist;
define
at discontinuities
of x.
differ
equivalent.
arcs will
by removable
An arc of bounded
be refered
The jump
exists
only
almost
absolutely
to as functions
everywhere.
continuous
measurable
respect
to the Lebesgue
Define
the
The measure
measure
a Bore1
respectively.
At the
as x(81).
function
x'(B)dB
and dr(8)
measure.
Lagrangian
function
as
The
role
of dx occur
at 8 and x,(B)
would
is a measure
Two arcs x
class,
into
((8)dr(8),
that
but
result.
The derivitive
dx can be decomposed
and a measure
=
at
below.
an equivalence
x,(e).
of x
are said to be
no ambiguity
-
atoms
Discontinuities
a crucial
is thus
when
x at e is Ax(B) = x+(e)
in
limits
discontinuities
variation
that
2.
is a discontinuity
and they play
=
left and right
is said to be removable.
are not removable,
and i that
and x+(B)
If there
=
rise to a ¶Z2-valued regular
the
and x+(01)
2
+l(e,xl)
Note
Assumption
gives
BEintn,
x,(B)
as x(00)
x+(e) I the discontinuity
endpoints
each
call them
x,(BD)
using
variation
For
on n.
= Ual(e+eblb
= ub2(eb2+ea2-42(x2)).
is convex,
An arc x:*X2
hu
= ~~dt~12~+2~t~h~(C~(~~)))r
wdo
{(x,=)E1R4(=~2(e,x),x~x(e)}
Bore1
~1 = Ual(O)t
x'(e)
an
where
is singular
((4)
with
is
- 30 -
L(e,x,q
=
-x,09
+ x2)fv)
-
wbl(edw
+ 4b2(x2)ifw
if xEx(B)
=
Define
rL(e,z)
=
our problem
the
function
1i.m v4,x0,z0+w
a++,=
rL is well-defined
Define
otherwise.
as the recession
rL(e,Z)
For
+a0
and
independent
rL(B,z)
of L (see
a]):
- W,xo,20)1/a
of
(xO,zO)
= 0 if zl + 22 10,
as long
as L(B,xO,zO)
and rL(B,z)
s
01
=
< +a.
= +Q if 21 + 22 < 0.
81
L(e,x(e),x'(e))de
+
where
BO
<(8)dr(8)
If xEA
then
wwWww)
s
80
x'(e)de.
[30, sec.
functional
JL(x)
for JEB,
and x'EZ(B,x)
is any representation
the
second
term
of the singular
in JL vanishes.
measure
Our problem
dx -
can be
restated:
Choose
J&B
We will
apply
is, problems
Lagrange:
with
to
MIN
results
fixed
JL(x)
(P4)
of Rockafellar's
endpoints.
Consider
[32] for Lagrange
the
following
problems;
Problem
of
that
- 31 -
Choose
xEB
to
MIN
x(eO)
s.t.
where
2
x"EX(Bo)
right
endpoint,
JL(x)
< +a, there
i(SO)
Proof:
= x0,
=
endpoints.
left endpoint
Define
x0 = (JC~,Z~) and
and the
largest
respectively.
with
exists
ii
x(BO)EX(BO),
an arc &ED,
x(Bl)EX(f?l),
equivalent
= Xl I and JL(;)
wl)=(el),
+
JL(X)
rL(eOrx+(eOk~o)
other
x1
=
and
to x on intn,
= JL(x).
For all such x and ;, we have:
JL(;f)
This
feasible
For any arc EB
1:
x(el)
and
are arbitrary
smallest
Lemma
with
but
the
(P5)
x0
=
and xlEX(81)
= G1(e1)ri2)r
feasible
JL(x)
rLu4p+(e0)-x0)
,iAx,(e,))
= we1
+
rL(eJ-x+(el)).
= 0, for all x(BO)Ex(BO)
and
SO ~~6 1 = JL(x )-#
implies
arc kB
that
any feasible
such that
G(e)
= 0.
equivalent
arc that
solves
equivalent
to the
Define
x(e)
V&intn,
For any arc that
rL(el,~(el)-x+(el))
fixed
=
arc XEB
P5 for endpoints
endpoint
problem
is endpoint-equivalent
to any
rL(eO,x+(eO)-ii(eO))
=
and
solves
5' and -1
x .
P5, with
the Hamiltonian
Wedbp)
=
sup,
{P'Z - Lub,z)
P4, there
1za2),
is an endpoint-
Therefore
endpoints
P4 is
5' and -1
x .
- 32 -
where
e2
problem
Define
are multipliers.
P(8)
P(8)
as (pE9121H(#,x,p)<+a}.
i.e. the
= (@2(plIC,p2=~1)r
normal
cone
Then
to X(B,x).
for our
Then
we
have:
H(B,x,p)
=
+a
if Few)
P1wu;l(mw
where
we have
used
the
PEA, the Hamiltonian
where
Arcs
if
aH(&x(ti),p(B))
is the
and pEB are
P-(eww),
along
dX
E and 'IIare Bore1
(i) R(B)
is normal
to P(B)
the
Lemma
2:
satisfy
-
Any pair
(EHC).
for F(B).
For arcs XEA
and
Extended
following:
(a)
of H(B,*,*)
Hamiltonian
veEz),
at
(x(O),p(O)).
conditions
x,(e)qe),
(EHC),
x+(e)Ex(e),
for any representation
X’(e)de,
=
www)
measurable
at x,(B)
and p+(B)
(HC)
a.e.
set of subgradients
w
to X(B)
at p,(B)
= p2(8)
can be written:
said to satisfy
with
=
PI(B)
if pa(e)
w,x,z)
aH(e,x(e),p(0)
and P+vmw;
e(e)d7(6)
normal
conditions
E
(HC) holds
(where
fact that
(-~'(e),xW))
XEB
-
dp
-
p’(e)de
and dr(8)
is nonnegative)
and x+(B)
[7]-a.e.,
and
it is true
(ii) ((0)
[r]-a.e.
of arcs XUL and pur
that
satisfy
(HC) also
is
that
- 33 -
See Rockafellar
Proof:
1321, Proposition
3, p. 168.X
Now define
w,p,s)
Define
where
=
the
SUPXIZ
{p-z + 8.X -
L(e,x,z))xa2,za2j.
functional
rM(8,s)
is the recession
It is readily
verified
function
that
of M(O,*,*).
L is a Lebesgue-normal
integrand;
that
is,
the epigraph
L(e,-,a)
epi
is closed
proper
=
and depends
in the
sense
correspondences
can then
apply
Theorem:
integrand,
that
oppositely
L is not equal
and P(B)
following
Suppose
and P(B)
finite.
L is convex
to +a everywhere.
and upper
convex,
are closed-valued
Let XEB
Then x is optimal
and is
Furthermore,
semicontinuous.
Lesbegue-normal
and upper
and pEB be a pair
(EHC) and such that JL(x)
infinite.
Jpl(p) are both
Also,
theorem:
correspondences.
satisfy
on 8.
are closed-valued
L is a proper,
and X(B)
semicontinuous
arcs
Lebesgue-measurably
that
X(B)
the
{(x,z,a)~51a>_L(e,x,z))
and Jh(p)
of
are not
for P5 and JL(x)
and
the
We
- 34 -
This
Proof:
is a simplified
version
of Rockafellar's
Theorem
2
[32, p. 1711.
We will
conditions
now
show
of the Theorem
-1
x ) and thus
is optimal
R and construct
the
seeking
a solution
instead
of the
but
with
and derive
posited
xi,
satisfies
contract
constraint
convenience
we will
RE(-ebld1)*
for P4.
is optimal
First,
debt
limits.
Thus
at the endpoints.
collateral
ambiguity
and thus
debt
take
results,
Note
The Hamiltonian
R.
the
(EHC).
an arbitrary
contract;
not to bind,
suppress
and write
that
Here
we treat
of xa/xb
this
is found
as assumed
x* and p.
if R = R, then
xi.
that
= 22.
(HC) can be written:
we are
that,
as a free parameter,
must
with
exceed
3.
For
of xi and pA on R where
xi(OC)
for
on intn,
is consistent
restrict
and
to of0 and -1
x
in Proposition
We will
0
value
an arc pAa
xa/xb
x
Since
continuous
X,/Xb
that
the
feasible
of xi equal
we construct
the dependence
conditions
call
Next
A,
satisfies
(with endpoints
xf; is absolutely
for the value
A value,
contract
for P5
to P5, we set the endpoints
an expression
debt
a collateralized
corresponding
relevant
discontinuous
along
that
attention
no
to
the
for the
-
p’(e)
E
wme)
~'(0
E
G,(e)
=
L&(e)
+
=
ff &e)E(xl&e)),
[t-b
if
x;(e)=&(e),
if
x;(e)=el,
+ ~bYl(fl))f(h+m)
if x&e)~(~,x2b
[(-Aa + ~bwwf(fl),+c)
if
&e)=x2,
(-ODt(-xa + ~bwv)f(e)l
if
&e)=s,
debt
contract,
x~mE(xlr~lm)
for ewwlir
eqR,el].
If x;(eo)
for
Using
these
me)
facts
=
demonstrate
while
= 3,
x;(B)
-
= ICI for &[Bo,R],
x;(B)E(s,x2)
the contract
and
for &(t9o,R) and x&B)
is "collateral
=
constrained."
we have:
(-1, + $we))f(e)
em(e)
We will
+ ~bvl(e))f(e)l
('Aa + bv2(e))f(e)
In the collateralized
X2
Lip)
(-Aa + ~bvl(e))fw
(-Qr(-Xa
G2m
35 -
later
+
that
(-Aa
if
+ bwe))f(e)
eE(eO,R)
(A.21
if 8E(R,01).
- 36 -
+ (-Aa+ bJ9(e))f(e)
mpw
so that
the
formula
The normality
for p'(e)
of n(6)
in
and
(x;+(8),x;+(B)).
given
in (A.2)
(EHC)
implies
ia achieved
suP~AP(e)(xl+x2)j(xl,x2)~x(e)}
(-Aa+ ~b”2(e))f(h
2
is consistent
that
if
with
[32].
The normality
=
(Axhu + ~xhmP(e)
=
(G(e) + A&mm)
=
(Ax;(e) + &om+(o
(A.1).
Ap(t9) # 0 then
by (x;(8),xg(B)),
See
(A-3)
of c(e) implies
(x;-(0),x:-(e)),
that
if Ax*(e)
f 0, then
rL(e,ax*(e))
where
rL is the recession
conditions
can be shown
R > 'ebl
=$ Pm
R < i
p(B0)
a
p+(eg) < 0
&intn
Solving
P-(h)
3
3
function
to imply
= 4ae1)
= Ap(60)
the
above.
following
= p-(h)
= p+(eO)
Together,
endpoint
these
two
relations:
= 0
= 0
R = i
Ap(e)
=
0
the differential
= 0, yields:
defined
equation
(A.2), with
endpoints
p+(BO)
I 0 and
- 37 -
01
P = xaS -u~l(e-y~(e,R))f(e)de
+
e0
uh(e'&etR))
f(e)de
1
ui2(ea2-Y2(etR) 1
xb J-" Ub2(YhR)
80
+
xb s
Bl
U~l(ebl+Y;(e,R))f(e)de
R
(A-4)
where
Jo E -p+(BD)Q(O).
The
increasing
in absolute
decreasing
in R.
For xa/xb
and
Solving
eE(eo,el),
P(e)u;1(8-y;(B,R))
+
(A.4) and the
implies
that
the second
side
term
is negative
is positive
and
and
by
(A.4) with
fi = 0.
If &,/xb
R = i
< A, then
JJ.
p(e),
for
on the right
& by
1 A, R is determined
(A.4) determines
term
in R, while
value
Define
first
yields:
=
P+(eO)U;l (0)
x E &(kR)
a
fact that
p(B)
(A.5) to substitute
(A-5)
6R
< 0 in
Xa6va(B,R)/6R + xb6Vb(e,R)/6R
(A.5).
for p(B)
It remains
is increasing
to show that
in (A.3), we obtain
in
(A.3) holds.
e
Using
- 38 -
-
Ubl(ebl+Yhw
f(e)
Ukt(earY2(~tR
F(e)
1
-
ub2(Y;(f’R))
U;l(e-y;(fl,R))
_ p(eIR)P(e)u~l(e-Y;(e,R))
&F(e)
1
U&l (0)
NLR)p+(kdXbF(e~
When
R < R, p+(Bo) = 0 and
then
(A.6)
JL(x*)
Thus
is implied
is finite,
the
is implied
by Condition
and thus
conditions
(A.6)
(A-6)
4.
JL(x*)
Since
and JM(P)
of the Theorem
by Condition
2.
x* is feasible
When
< 0,
by construction,
are not oppositely
are fulfilled
p+(eol
infinite.
and x* is optimal
for P4 and
P5.
TO establish.the
an optimal
Obviously
debt
otherwise
contract
2 must
we can assume
they
uniqueness
as above,
be feasible,
that
of the optimal
and
i and x* differ
would
belong
to the
and that
so JL(x)
contract,
2 is some
other
< +a~. Without
on a set of positive
same equivalence
suppose
optimal
x*
is
contract.
loss of generality,
Lebesgue
class.
that
Then
measure,
we have
for
- 39 -
JL(%
JL(x*)
01
J00
=
-
rw&w
L(e,x*(e))lde
81
1
s
rL:,(etx*v))(~l(e)
- x;(e))
-
L$2te,x*(e))(G2(e)
- x2*(e))lde
00
I
1
e1
[p'(e)
- p(e)p(e)l(Xl(e)
- x;(o)
+
pwwX2(o
- +u)ld~
80
e1
I
-
J
p(e)&(e)
+ iii(e) + p(e)&(e)
- x;(e))
- X;‘(B)
- Isolde
60
P-cmflce)
J
volw
The
>
0.
first
two
conditions,
equality
final
J&l
latter
and the
uses
Proposition
The
inequalities
the
follow
fact that
L*
Xl
*
> Lx2
from Assumption
formula
1, p. 161, and the fact that
inequality
uses:
implies
+
of
from the convexity
"integration-by-parts"
p(Bo)
p(B) 5 0, VB; El(B)
= +m); and the convexity
G>(e)
+ t2ww
of u&(dl(xl))
L, the Hamiltonian
3.
in Rockafellar
= Ap(B0)
+ &(c9)
in x1
= p(B1)
10
2
u~1(41G1(o))
1
u~1(41(x~(e)))
=
X;P (8)
- x;P (8)
-
m(+e)
-
p(e)&(e)
subsequent
(321,
= Ap(t91) = 0.
Vf9 (otherwise
(from Assumption
that
i&(e)
The
-
x;(o)
-
x;(o).
2).
The
- 40 -
Because
x*'(e)
JL(x)
-p(B)
a.e.
> 0 VBEintn,
and t(e)
> JL(x*)
= 0.
the
last
But
if this
and x* is the unique
inequality
is true
optimal
is strict
unless
G'(8)
; and i* are equivalent.
contract.
=
Thus
Figure 1. A Collaterallized Debt Contract.
Figure 2.
Consumption Schedules under
A Collateralized Debt Contract.
0
I
I
I/
I-
O %
R
I
I
/
L
f
I
Figure 3. Indifference Curves That Satisfy
Condition 2, for a fixed 8
- 41 -
NOTES
it is often
1. Indeed,
contracts
are distinguished
partnership
there
With
latter).
as an optimal
the
two
(see Lacker
alterations,
the model
first,
contracting
date;
related
and Weinberg
and de Roover
former,
arrangement:
initial
to find environments
from the closely
and pawnbroking
concerning
at the
difficult
in which
arrangements
of
[23] and the
citations
[12] and Caskey
presented
the collateral
and there
here
good
loan
(91 concerning
the
delivers
pawnbroking
is durable,
and exists
is a possibility
of the borrower
absconding.
2. It might
setting
where
patterns.
agents
the
costless
inconsequential,
Assumption
1.
5. This
some
of goods
"nonpecuniary
presumes
sort that
faking
that
can punish
agent
with
= Ca2r
endowment
eb2 = 0.
with,
and it
it for u, as well,
relaxing
The
a special
Assumption
latter
case
is
of
- ua2(ea2-y2).
to an enforcement
a for withholding
in a
ebl = eb2 = 0; 00 =
and then
are ua2(ea2)
access
here
cumbersome.
by setting
is effectively
have
choose
that do not exist.
penalties"
agents
these
be dispensed
more
can be obtained
= CalI UaZ(Ca2)
would
dispensing
be considerably
specified
determine
lender
prevents
so his environment
The
the
of ub can easily
(13) setup
E 0, Ual(Cal)
l(d) to allow
that
in principle
would
the environment
ex ante which
actions
conjecture
analysis
4. Diamond's
0, Ub2(cb2)
to embed
separability
as if nothing
although
take
One might
3. Additive
seems
be fruitful
stipulated
facility
transfers,
of
but
-
is not capable
of overcoming
the
42 -
informational
imperfection
of costless
falsification.
6.
(IF')
holds
7. A real-valued
is the difference
function
if 8*(B)
function
[33, pp.
variation
98-1001.
is not unique
on an interval
of two monotone
of bounded
See Royden
even
real-valued
can have
or if agent
is of bounded
functions
a countable
a randomizes.
variation
on the
number
if it
interval.
A
of discontinuities.
- 43 -
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@FedFRASER