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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
OPTIMAL COMSUMPTION AND PORTFOLIO CHOICE
WITH BORROWING CONSTRAINTS
Jean-Luc Vila*
and
Thaleia Zariphopoulou^
WP#3650-94
January 1994
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
OPTIMAL COMSUMPTION AND PORTFOLIO CHOICE
WITH BORROWING CONSTRAINTS
Jean-Luc Vila*
and
Thaleia Zariphopoulou**
WP#3650-94
January 1994
MA
*
Sloan School of Management, Massachusetts Institute of Technology, Cambridge,
**
Department of Mathematics and Business School. University of Wisconsin. Madison
OPTIMAL CONSUMPTION AND PORTFOLIO CHOICE
WITH BORROWING CONSTRAINTS
Jean-Luc Vila
Sloan School of Management, Massachusetts Institute of Technology
and
Thaleia Zariphopoulou
Department of Mathematics and Business School, University of Wisconsin, Madison
First Draft:
August 1989
Current Version: January 1994
ABSTRACT
In this paper,
we
use stochastic dynamic programming to study the intertemporal consumption and
portfolio choice of an infinitely lived agent
constraint.
exist
We show that,
when
We
the agent's
faces a constant opportunity set
under general assumptions on the agent's
and can be expressed
details,
who
as
utility
feedback functions of current wealth.
utility
We
and
a
borrowing
function, optimal policies
describe these policies in
function exhibits constant relative risk aversion.
wish to thank Minh Chau, Neil Pearson, the associate editor and an anonymous referee for
comments. Jean-Luc Vila wishes to acknowledge financial support from the International
Financial Services Research Center at the Sloan School of Management. Thaleia Zariphopoulou
acknowledges financial support from the National Science Foundation Grant DMS 9204866. Errors
*
their
are ours.
I.
INTRODUCTION
we
In this paper,
consider the Portfolio-Qansumption problem of an infinitely lived agent
in
the
presence of a constant opportunity set and borrowing constraints. Using the method of Dynamic
Programming, we show
policies
do
exist
and can be expressed
this existence result,
we
investment decisions
when
Stochastic
to the
under general assumptions on the agent's
that,
as
feedback functions of the investor's current wealth. Given
are able to describe
how borrowing
the agent's relative risk aversion
dynamic control has
first
constraints affect the consumption and
is
constant.
been used by Merton (1969, 1971)
Portfolio-Consumption problem
function, optimal
utility
when
to obtain an explicit solution
the investment opportunity set
HARA class and when trading
unrestricted.
is
constant, the agent's
More
recently, Karatzas
utility
function belongs to the
et
(1986) generalized Merton's results and obtained closed form solutions for general
al.
is
utility
functions.
Instead of using stochastic control methods, the so-called martingale approach has been alternatively
used by Pliska (1986), Cox and
consumption and portfolio
dynamic programming
consumption
problem
in
Huang (1989
policies
literature.
infinite
1991), Karatzas et
when markets
al.
(1987) to study intertemporal
are complete, which was also the case in the earlier
The martingale technology
set by a single intertemporal
an
&
consists in describing the feasible
budget equation and then solving the
static
dimension Arrow-Debreu economy. The martingale approach
economists for two reasons.
First,
it
is
consumption
appealing to
can be used to solve for the asset demand under very general
assumptions about the stochastic investment opportunity
set.
Second, and consequently,
it
can be
applied in a general equilibrium setting to solve for the equilibrium investment opportunity set (see
Duffie and
Huang
(1985),
Huang
(1987)).
4
Unfortunately, in the presence of market imperfections such as market incompleteness, short sale
constraints
or
transaction
approach
martingale
the
costs,
much
looses
of
its
tractability'.
Consequently, many authors used dynamic programming to analyze the impact of these imperfections
on
asset
demand
(see for example Constantinides (1986), Duffie and Zariphopoulou (1993),
Grossman and Laroque
(1988),
Grossman and Vila
(1992), Fleming et
al.
(1989), Fleming and
Zariphopoulou (1991), Zariphopoulou (1993)).
The present paper extends
who
faces
this line
two constraints. The
of investing
constraint
first
in a risky asset, i.e, the
of research to the dynamic problem of an infinitely lived agent
his
we
his
this
are non-negative constants.
all
on
times,
in
for the purpose
the risky asset, X,, must be less
paper
we
The second
i.e.,
borrow
concentrate on the case
constraint
is
the requirement
W,^0. Using the dynamic programming
associate to the stochastic control problem a nonlinear partial differential equation,
namely the Bellman equation.
We
show
that
if
the
utility
function satisfies
conditions, the Bellman equation has a unique solution which
Using a verification theorem,
value function) which
consumption,
his ability to
investments
wealth W^. In
that the investor's wealth stays non-negative at
methodology,
a limitation
market value of
than a exogenous function X(W,) of
X(W,)=k(W,+L) where k and L
is
is
this solution turns
is
some general
regularity
twice continuously differentiable.
out to be the indirect
utility
function (the so called
therefore a smooth function of the current wealth, W,. Moreover, the optimal
Q and the optimal investment X, are obtained in feedback form through the
first
order
conditions from the Bellman equation. Because of the smoothness of the value function, the optimal
policies are respectively continuously differentiable
In the second part of the paper
we
(Q) and continuous
(X,) functions of the wealth.
study the particular case of a Constant Relative Risk Aversion
He and Pearson (1991) use a duality approach to apply the martingale technology to incomplete markets with short
Although their methodology carries a lot of insight, they do not solve explicitly for the optimal policies.
sale constraints.
5
agent. In particular,
we
describe in detail the optimal policies and
we compare them
to the ones of
the unconstrained problem. In the absence of the borrowing constraint, the optimal investment
X,=(^/Ao') W, where
volatility
show
\i
is
the excess rate of return
of the rate of return on the risky asset and
even
at points
a similar result in
where the constraint
is
smaller (greater) than
In the third part,
is
we
will
the coefficient of relative risk aversion.
We
risky asset
more conservative
(i.e.
rate,
to invest less in the
not binding. This result has to be contrasted with
make him more
that, if the
(less)
agent consumes only his
conservative
if
final
the relative risk aversion,
1.
present an application of our analysis to an optimal growth problem
Robinson Crusoe economy with two
linear technologies,
one
riskless
the investment in the riskless technology must be non-negative.
the average rate of return
is
o
the
Grossman and Vila (1992) who show
wealth the borrowing constraint
is
A
over the risk-free
is
that the borrowing constraint causes the agent to be
risky asset)
A,
on the
is
on
capital in the
economy
to fall
We
and one
show
risky.
We
assume
in a
that
that this constraint causes
even during periods when the constraint
not binding.
The purpose of the paper
is
twofold. First,
we want
examine how borrowing constraints
to
consumption and portfolio decisions. Second, we want
to illustrate
be used rigorously to obtain qualitative properties of optimal
policies
affect
how dynamic programming can
even when
explicit solutions fail
to exist.
The powerful theory of
characterized
as
the
viscosity solutions
unique
(constrained)
is
used
viscosity
in
this
solution
characterization of the value function as a viscosity solution
Bellman equation, which turns out
and such equations do not have
in
to
be
fully nonlinear,
paper.
is
The
of the
value function
is
first
Bellman equation. The
imperative because the associated
might be degenerate (due to the constraints)
general smooth solutions.
The unique
characterization together
6
with the stability properties of viscosity solutions enable us to approximate the value function by a
sequence of smooth solutions of the regularized Bellman equation and
identify the
smooth
limit-
function with the value function. Finally, the characterization of the value function as a constrained
viscosity solution
a.e.
is
natural
due
to the presence of the (state) constraint of non-negative wealth
(W,iO
Vt^O).
The methodology employed herein can be
applied to several extensions and variations of the infinite
horizon problem. In particular the case of finite -horizon can be analyzed very similarly as
discussed in Section V. Also,
if
although more complicated, can
In a general setting the
we
allow investing to
still
more than one
in the
in
imperfect
markets,
to
Bellman equation,
analyze a very wide range of
More
presence of market imperfections.
from the theory of viscosity solutions can be used to provide
of problems
is
be treated with the above method.
above methodology can be used
consumption/investment problems
stock, the
it
for
example,
(see
(i)
precisely, results
analytic results for the value function
Fleming and
Zariphopoulou
(1991),
Zariphopoulou (1993)) Duffie and Zariphopoulou (1993), Duffie, Fleming and Zariphopoulou ( 1993),
as well as
policies
(ii)
convergence of a large class of numerical schemes for the value function and the optimal
when
closed form solutions cannot be obtained (see, for example, Fitzpatrick and Fleming
(1990), Tourin and Zariphopoulou (1993)).
The paper
is
organized as follows.
The general model
the existence result. In Section IV,
we
is
presented
analyze the case of a
application to an optimal growth problem. Section
in
Section
II.
CRRA investor.
Section
Section
VI contains concluding remarks.
V
III
deals with
presents the
II.
THE MODEL
2.1.
A
We
consider the investment-consumption problem of an infinitely lived agent
Consumption-Portfolio Choice Problem
expectation of a time-additive
One
asset
is
riskless
utility
function.
with rate of return
The agent can
r (r2:0).
The other
distribute his funds
asset
is
who maximizes
between two
a stock with value
P;.
We
the
assets.
assume
that the stock price obeys the equation
-^' = ir*\i)dt +
adb
(2.1)
P.
where the excess
rate of return
\i
and the
volatility
a are positive constants. The process
standard Brownian Motion on the underlying probability space (Q,,^,CP).
bt is
a
We assume that there are no
transaction costs involved in buying or selling these financial assets.
The assumption
is,
that the opportunity set
is
constant,
however, possible to allow the market coefficients
is
necessary for the tractability of the model.
r, \i
and o
to be stochastic processes themselves.
Unfortunately, this would increase dramatically the dimensionality of the problem. For example,
r„
\ii,
and
o,
are diffusion processes, the value function will
It
depend on four
state variables
W,
r, \i
if
and
o.
The
controls of the investor are the dollar
amount X, invested
in the risky asset
and the consumption
rate C,. His total current wealth evolves according to the state equation
dW, =
The agent
^ee
faces
Dytjvig and
two
(rW,-C,)dt
-(-
iiX,dt
-(-
oX,db„
constraints. First, his wealth
Huang (1988) on
for t^O
and
must stay non-negative
the importance of this non-negativity constraint.
Wo = W.
at
(2.2)
any trading time.^ Second,
8
the optimal
amount X, must never exceed the (exogenously given) amount X(W,).
In this paper,
we
will
This formulation
consider the case
X(W) = k(W+L) where
a certain fraction
the risk free rate
f
methodology employed here can be used
X
non-negative constants.
X(W,)=W,+L.
If
is
if
the
the investor needs only to put
of his stock purchases and can borrow for the remaining fraction
then X(W,) = (l/f)W,. This case
r,
L are
general enough to encompass several interesting examples. For instance,
is
investor has access to a fixed credit line L, then
down
k and
(1-f) at
treated only for the sake of exposition since the
for the general case of trading constraints X, ^
X(W,) with
being a concave function of wealth (see Zariphopoulou (1993).)
The
i)
control (X,, C,)
(X„ Q)
is
admissible
if
a ^-progressively measurable process
is
generated by the Brownian Motion
ii)
ill)
We
The
the o-algebra
b,.
(Xt,Cj) satisfy the integrability conditions
W^^O
a.e.
where W,
v)
is
Q^Oa.e. (Vt^O)
JXMs<+oo
iv)
where ^,=o(b,; O^s^t)
is
A
J
C
ds
<
+<»
,
a.e.
(Vt^O)
(Vt^O)
(2.3)
the wealth trajectory given by (2.2)
X,^k(W,-l-L)
denote with
and
a.e.
when
(X,,
Q)
is
used and
(2.4)
(Vt^O).
the set of admissible policies.
objective of the investor
is
entails solving the optimization
to
maximize the expectation of a time-additive
utility
function, which
problem
sup
E[
re-'"u(C)dt]
(^-^^
9
where u
the agent's
is
function u
utility
function and
a strictly increasing
is
p>0
is
We
a discount factor.
assume that the
utility
CXO, + <») function' with
lim u'(C) = +00
and
C-O
lim u'(C) =0.
C--
The Value Function
2.2.
The
value function J
We
controls.
defined as the supremura of the expected
is
denote by
J"
2.1:
(i)
Proof: Part
(i)
J*, J
and
(W^O).
^/'(W+L),
(W^O).
A of admissible
follows from the fact that the set
Remark
finite.
2.1:
A
than an investor with wealth
The above
Indeed, J(W)
is
The
J(W)
bounded from below by ju(rW)
(1986). For the purpose of this paper,
shall
in
A function
Karatzas
is
et al. (1986).
called C^(Q)
^Technically speaking, from
if its first
Lemma
They
2.1.,
increasing in L. Part
W>0
(ii)
provided that J"(W)
since the control (X,=0;C,=rWj)
is
is
admissible
of J"(W) can be found in Karatzas
utility
et al.
function and the market coefficients, can
are also discussed in Part
k-derivatives exist
is
assume that J"(W) < +oo\ VW^iO.
necessary conditions on the discount factor, the
be found
finite for all
is
explicit derivation
we
controls
W and constraint X(W)=k;(W + L).
properties ensure us that
and bounded from above by J'(W). The
L=0
W+L and constraint X(W) = k;W will have a larger
follows from the fact that an investor with wealth
feasible set
lines
J".
J"(W) ^J(W) ^J-(W),
(ii)J(W)
over the set of admissible
and J" the value functions associated, respectively, with credit
and L=a>. The following lemma compares
Lemma
utility
and are continuous functions
assuming that J*(W)
is
finite for
every
W
is
IV of
this
paper.
in Q.
enough
to guarantee the fmiteness of J(W).
10
The
following
Lemma
describes elementary properties of J(-).
lemma
2.2:
(i)
J(-)
(ii)
J(
•)
is strictly
is strictly
(Hi) J(-) is
(iv)
concave.
increasing.
continuous on
\imJ'{W)
[0, »)
with J(0)
=
-ju(0).
= +0O.
Proof: See Proposition 2.1 of Zariphopoulou (1993).
2J. The Bellman Equation
In this section,
is
we
completely characterize the value function and
done using dynamic programming which leads
we
derive the optimal policies. This
to a fully nonlinear, second order differential
equation (2.6) below, known as the Hamilton-Jacobi-Bellman (HJB) equation. In Theorem
show
that the value function
is
2.1,
we
the unique C^(0, + oo) solution of the (HJB) equation. This will enable
us to find the optimal policies from the
first
order conditions in the (HJB) equation. They turn out
to be feedback functions of wealth and their optimality
is
established via a verification theorem (see
Fleming and Rishel (1975)).
Theorem
2.1:
Theorem
max
=
^/(»0
with
The value function J
is
the unique C{0,
[-a'X'J"{W)*^J'iW)]
^
+ oc)nC[0, »)
solution of the
Bellman equation:
max [uiC)-CJ'iW)] ^rWJ'iW),
(W>0)
/(0)=!ii2).
2.1
is
the central result of our paper.
The proof of
this result
is
presented in some details in section
III.
Next, using the
first
order conditions in (2.6) and the regularity of the value function,
optimal policies in a feedback form.
we can
derive the
11
Theorem
2.2:
The optimal policy (C',^])
is
given in the feedback
form C',=C'(W',) and X'=X'(W]) where
C'( '), X'(') are given by
C'(W)
where W,
We
this section
viscosity solutions
2.3:
X'(W)
and
=
mm
is
{
C* and
^^
H
^
cr
-J"
X",
\k{W^L)}
(2-7)
(W)
being used.
by stating a result which will play a crucial role in the sequel. For the definition of
and their
stability properties see
The value function J
of the (HJB) equation
The proof
(u'yUJ'(W))
the (optimal) wealth trajectory given by (2.2) with
is
conclude
Theorem
=
Appendix A.
the unique (constrained) viscosity solution in the class of concave functions,
is
(2.6).
rather lengthy and technical and
(1993) and, for the sake of presentation,
it is
it
follows along the lines of
Theorem
3.1 in
Zariphopoulou
omitted. General uniqueness results can be found in Ishii-Lions
(1991) although they cannot be directly applied here because the control set
not compact.
is
overview of existence and uniqueness results we refer the reader to 'User's Guide" by Crandall,
(1992) and to the book 'Controlled
III.
SMOOTH
In this section
main
ideas.
(C*)
we
SOLUTIONS OF THE
present the proof of
The (HJB) equation
degeneracy comes from the
not uniformly
elliptic^
and we know
Our
(2.6)
is
Viscosity Solutions' by
goal
^ o^X^ for the Bellman equation
constants depending only on [a,b].
is
Ishii
and Lions
Fleming and Soner (1993).
EQUATION
2.1.
Before
second order
we
fully
start the details of the proof,
we
discuss the
nonlinear and (possibly) degenerate. The
second order term '/2a'X^J"(W) may become zero. Therefore
that degenerate equations
one-dimensional differential equation
derivative
(HJB)
Theorem
fact that the
for example, Krylov (1987)).
A
Markov Processes and
For a general
is
first
do not have,
in general,
said to be uniformly elliptic (non degenerate)
^ o^X^SCjCl^.''])
for
smooth solutions
showing that the optimal
to exclude this possibility by
(2.6) satisfies 0<Ci([a,b])<
(2.6)
if
is
(see,
X
is
the coefficient of the second-order
any interval
[a,b]
where C, and C, are
12
bounded away from zero
in (0,+oo).
We can
then use the results of Krylov (1987) for the regularity of solutions
of uniformly elliptic equations.
We
next consider an arbitrary interval [a,b] with a>0. Since the value function
derivative exist almost everywhere.
Without
We
X
want
k(W+L)
show
to
that the optimal
Therefore,
is,
it
we
suffices to find a lower
first
we want
non-increasing and
is
we may assume
bounded away from zero
or (/i/a^)(-J'(W)/J"(W)). In the second case
(/i/CT^)(-J'(W)/J"(W)). Since J'(W)
J(«)
is
loss of generality,
bound
for
J"(W)
in [a,b].
concave,
is
that J (a)
it is
in the interval [a,b].
approximate !(•) by a sequence of smooth functions
J'
and
and second
J'(b) exist.
Formally, the optimal
to get a positive lower
strictly positive,
its first
X
is
either
bound of
bounded from below by J'(b)>0.
Since
we do
not
know how
which are solutions of
regular
a suitably
regularized equation.
To
this
The
end,
we
consider the following regularized problem: Let b| be a Brownian Motion independent of
policy (Xt.Q)
admissible
is
(Xt,Q)
ii)
q>0
a.e.
iu)
(Xt,q)
satisfy
is
if:
a ^^-progressively measurable process
i)
b,.
where .^ =
a(bj; 0<s<t);
(Vt>0);
t
r(X,')Ms <
iv)
the
with
We
O
initial
=
rCtds<+co
and
amount of wealth W' >
dW:
(i)
+CO
< X; < k(Wt+L)
W'o=W
a.e.
;
given by
(rWf +/iX;-Cf)dt
condition
(Vt<0)
+
satisfies
a€X:db,
W^
a.e.
+
(^'^^
aeWfdb,'
(Vt>0) and
(Vt>0).
denote by A' the set of admissible policies and define the value function J*(«) to be
J'(W)
The following two lemmas provide
= sup
E
[
re-'"u(Ct)dt]
(^^^
.
regularity properties for the value function
J'.
13
Lemma
3.1:
Proof: See
Lemma
Theorem
3.2:
^'(W)
The value function
J' is strictly
max
strictly
increasing on [0, +«).
of Zariphopoulou (1993).
5.1
The value function J'(W}
=
concave and
is
unique smooth solution of the regularized Bellman equation
the
[-(/{X^^e'\^)y' iW)+^iXJ*\]V)]
*
max[u{C)-Cr\lV)] *rWy{W)
IV>0
,
2
OiXik(W.L)
(3.3)
/-(O) =
with
Proof.
The
Note
Lemma
.
that equation (3.3)
lemma
next
li^
3.3: J'
says that the
is
uniformly
elliptic
above sequence
J'
converges to J locally uniformly as
and see Krylov (1987).
converges to
e
goes to
J,
as « goes to zero.
0.
Proof: See appendix B.
We
next prove that the
first
and second derivatives of
X
implies in particular that the optimal
Lemma
of
(.,
3.4: In
such
that,
any interval
in
[a,b], there exists
J*
equation (3.3)
are
is
bounded away from zero uniformly
bounded away from
in e.
This
zero.
two positive constants R,=R,([a,b]) and R2=R2([a,bJ) independent
for lVe[a,b],
J"(W)>R,
(i)
\J'"{W}
(ii)
\
(3.4)
<R2
(3.5)
.
Proof: See appendix B.
We now
conclude as follows:
max
fiV{lV) =
iifli
We
first
consider the boundary value problem
{-cr(X^*e^li^)V"
-
2
{lV)->-tJLXV
(W)}
+
max {u{c)-cV (^W)} -^rWyi^W)
WG[a,b
ciO
aR2
V{a)=J'(a),
V{b)=J*{b)
.
(3.6)
14
Applying
classical results
we
get that (3.6) has a
by
lemma
of (3.6) as
smooth solution
elliptic differential
equations (see, for example, Krylov (1987)),
V which by uniqueness and lemma 3.4 actually coincides with
3.3 J'(=V*) converges locally uniformly to
J,
J will
J'.
Since
be a viscosity solution of the limiting equation
goes to zero, namely
e
{la^X^V {W)>^iXV{W)}
max
fiV(]V) =
*
2
-
itRi
max{u{c)-cV{W)}
*
rWV{W)
We[a,b],
cio
^^'^^
V(b)=J(b).
V(a)=J(a),
To
from the theory of
pass to the limit,
we used
the fact that J'(W)->J(W) and the stability properties of viscosity solutions (see
We
Appendix A, Theorem A.1).
smooth solution
next observe that (3.7)
V*
V*. This solution
is
is
uniformly
elliptic
and therefore has a unique
of course a solution in the viscosity sense. Finally, using that
unique (constrained) viscosity solution of the Bellman equation
(2.6),
we
get that V*
J
is
= J and therefore
the
J
is
smooth.
Remark
3.1:
Having analyzed the (HJB) equation
does not change
excess returns
/9J(W)
much
if
we allow
/i=(/ii,..,/iN)
=
max
N
Y,
and
[
investing in
volatility matrix
(2.6) for the case of
more than one
E
1 tr(X'EE'X') J"
,
one
we can
stock. In particular,
if
see that the analysis
we have N
(W)
+
(^,X)J'(W)
]
+
max [u(C) -CJ'(W)]
s
all
rWJ'(W)
'k(W.L)
(3.8)
follows along the
same
lines as before
once the
recovered. But, these constants can be obtained as in
separately
+
C
" •'
The proof now
stocks with
the associated Bellman equation becomes
2
K
stock,
local ellipticity constants
Theorem
5.1 of
R, and R, can be
Zariphopoulou (1993) by solving
possible cases of maximization with respect to the X^'s in (3.8).
15
rv.
THE CONSTANT RELATIVE RISK AVERSION (CRRA) CASE
we
In this section,
consider the case of an agent endowed with a
u(C)
where
The
A
is
^_
1-A
C'--^,
utility
A>0. A^l,
("^-l)
the coefficient of relative risk aversion.
case where the borrowing constraint
Merton
=
CRRA
(1971).
Merton proved
X<k(W+L)
is
not present or not binding, has been studied by
that the optimal investment strategy consists of investing a fixed proportion
of wealth in the risky asset. This proportion ^/{Aa^) increases
free rate, ^, increases,
and when the
volatility
a or the
when
the excess return on stocks over the risk
relative risk aversion
A
decreases.
The
following
proposition summarizes the results of Merton.
Proposition 4.1: (Merton 1971)
(i)
If there
is
no borrowing
an optimal control
constraint (formally L=<x:),
exists
provided that the following
inequality holds:
\'
(4.2a)
> o
with A" being defined as follows:
A- =
^
A
A
7
(ii)
The value function J'(*
)
and
{4.2b)
[^-(l-/l)(r + 2)];
=
Jil
la"
(4.2c)
.
the optimal controls (X( '),€(*)) are given respectively by
r(^W)
=
X'iW)
(X-yuiW),
('^^^
JLW,
(4.4)
=
Aa^
(4.5)
C'i^W) = X'W.
When
[^/(Aa')]<k, the borrowing constraint
solution to the unconstrained problem
(i.e.
X<k(W+L)
(4.2)-(4.5)).
We
is
not binding and the solution to (2.5)
assume
in the sequel that:
is
the
16
M>k(Aa^),
that
is
stock in excess of the risk free rate
is
The
the borrowing constraint will be binding*.
fi
is
large,
if
(4.6)
inequality (4.6) will hold
the volatility a
is
low or
if
if
the expected return on the
the risk aversion of the investor
small.
In the presence of
myopic
strategy
borrowing constraints, one natural candidate for an optimal investment strategy
where the investor
invests the
minimum
myopic investment strategy
the
of two quantities: (a) what he would have invested
maximum
without the borrowing constraint and (b) the
is
level of investment
k(W+L).
In other words, the
is
XMw'(W)
=
min[_^ W, k(W+L)].
(4-7)
Act
When
We
the investor follows the myopic strategy, he
will
show
constraint
is
that the myopic strategy
not binding,
is
is
is
not aware of the borrowing constraint until he meets
not optimal in general:
affected by the fact that the borrowing constraint
However, as shown below, when L=0, the myopic strategy
Proposition 4.2:
IfL=0,
The optimal investment, while
the value function /'(•)
and
f(W)
=
may be binding
it.
the borrowing
in the future.
indeed optimal.
is
the optimal control (X('),C(')) are given by
{X'>)^u{W),
('^^^
(4.9)
X^{W)
=
kW,
C^(W)
=
x'w,
("^-^^^
where
XO-'lP-H-Anr.^k-^)].
A
Proof: J* solves the Bellman equation and the optimal policies can be
If
L>
0,
then, to our knowledge, a closed form solution to
to obtain qualitative properties of the optimal controls.
T^rom (4.4), we expect the borrowing constraint
means trivial (see proposition 4.4).
(4-11)
2
problem
More
computed
explicitly
(2.5) fails to exist.
It is
from
possible however
precisely, in Propositions 4.3, 4.4
to be binding for high levels of wealth.
However, the proof of
(2.7).
this
and
4.5 below.
statement
Is
by no
17
we
are able to
compare the optimal controls
(C*,X') with the optimal controls (C',X") and (C^.X") which
provide interesting benchmarks.
The
next proposition shows that in the presence of the borrowing constraint (2.4) the consumer-investor will
always invest
less in
the risky asset than according to the myopic investment strategy (4.7).
Proposition 4.3: The optimal investment strategy, X'(W),
X'^^(IV). Furthermore, for small values of W, X'(W)
at
is
most equal
is strictly less
to the
myopic investment
strategy,
than X^*>^(W).
Proof. See appendix C.
Given the borrowing constraint
_2_ (J')^^
(2.4),
the Bellman equation (2.6) can be rewritten as:
+
7^—4
pi = _2_(J')''^ + rWJ'
+
/ik(W+L)J'
=
1
+
W
when
rWJ'
^i
+
-A
belongs to
i.crk^(W+L)-J"
(4.12a)
U,
W
when
belongs to B,
(4.12b)
2
where we use the following notation and vocabulary:
U
=
{
W>0
—
such that
(T
B
=
{
W>
<
The
meaning
}
>k(W+L)}
such that
Given the optimal unconstrained
interval,
k(W+L)
that the agent
=
unconstrained domain,
=
constrained domain
-i"
strategy,
it
seems
intuitive to expect the unconstrained
next proposition provides a sufficient condition for
Proposition 4.4: If
p + -y >
r
+ k^/2
a positive number
There
ii)
The optimal investment
Proof. See appendix C.
U
to be actually
an
interval.
(recall that 7=/i*7(2cr)) then:
i)
exists
W greater than a
meets the borrowing constraint for every
W*
strategy,
such that
X'(W),
is
U=(0,W) and
greater than
kW.
B=(W',oo);
domain
to be
critical level
an
W*.
18
The
next proposition compares the optimal consumption policy C*(W), to
and C"(W) are defined
and
in proposition (4.1)
Proposition 4.5:
IfA<l,
i)
C-(W),
{
C'(W)
Hi)
£S^
Lim
1
< C-{W) < C'>i\V)[^:llkf
}
(4-13a)
.
satisfies
Min
{
C\W*L); C-{W)[^*^\^
^'^^^^^
}
;
satisfies
= A«
For
iv)
~"
-A
C\W) f-iL]^
<C\W)<
C\\V)
C^CW) where C"(W)
C'(IV) satisfies
IfA>l, C'(W)
ii)
to
(4.2).
\
Max
C"(W) and
(4-l^<=^
;
Win
C-(W) > C"(W)
[0,W], C'(IV)
satisfies
(4.13d)
.
Proof: See appendix C.
Proposition 4.5 provides bounds for the optimal consumption policy. These bounds are best illustrated in
figures
and
1
2. It is
interesting to note that
the relative risk aversion coefficient
to
show
that in this case
monotonic with respect
C'(W)
is
A
is
C*(W) may
equal to
1
lie
(i.e.
outside the interval [C"(W),C"(W)]. In
u(c)=log
not equal to C*'(W)=C"(W).
(c)),
Hence
C*(W) and C"(w) are
fact,
equal.
when
It is
the optimal consumption policy
is
easy
not
to the credit line L, even in the constant relative risk aversion case. This point
illustrates the fact that the
presence of market imperfections significantly modifies optimal policies and that
a priori bounds are very difficult to obtain.
From
proposition 4.5
more and
invests less
(iv)
due
and proposition
it
follows that in the interval [O.W], the investor
consumes
follows that, the wealth of an optimizing
consumer
to the borrowing constraint.
more
will
grow
will
meet the borrowing constraint
(in average)
4.3
It
slowly than the wealth of a myopic investor.
later.
As
a result, an optimizing consumer
19
V.
THE FINITE HORIZON PROBLEM
In this section
we
model and we
discuss the finite-horizon
about the value function and the
state results
optimal policies.
The
investor starts at time te[0,T), with
X, amount of money
invests
The
in stocks.
as in the infinite-horizon case.
The
T>0, having an endowment W, consumes
prices of the riskless asset
same
investor faces the
and the stock
for se(t,T], at rate C,
satisfy the
and
same equations
constraints as before. In other words, the wealth
and the consumption rate must stay non-negative and must meet borrowing constraints X,<k(W,+L)
a.e. for
t<s<T.
The
objective
is
to
maximize the
entails to solve the optimization
expected
total
utility
coming from consumption and terminal wealth which
problem with value function
V
T
V(W,t)
=
supE[ fu.CC
A
where
Ui
is
the utility of consumption and u,
increasing and smooth.
section
The
set
,s)ds + Uj(W.^,T)]
J,
A
is
is
the bequest function. Both u, and u, are assumed to be concave,
the set of admissible policies which
is
defined in a similar way as in
II.
We now
state the
main theorems. Since the proofs are modifications of the ones given
in previous sections
they are omitted.
Theorem
5.1:
The value function
V is
the unique C"{[0,<x>)\[0,T])
and
C^'((0,<x>)x(t,T)) solution
of the Bellman
equation
V
+
max
[l-a'X'V^)
*
nXV^]
+
max [u,(C,f) -CV^\
*
rWV^
=
V(W,T) =u^{W,T).
Theorem
5.2:
X',=X '(W
,
,
The optimal policy
s)
(C',,X',) (t<s<T),
where the Junctions C'( • ) and X'( •
is
)
given in the feedback
are given by
form C* = C'(W],s) and
20
C-(W,t)
where W]
VI.
is
In this section,
The
we
resources
first
^
min
[
ii
J
k(W^L)]
the optimal wealth trajectory.
APPLICATION TO A
initial
X-(W.t)
and
(u;)-'(V^(W,t))
=
GROWTH PROBLEM: SUBSISTENCE LEVEL AND RISK TAKING
consider a single good Robinson Crusoe economy.
is
risk free with rate of return r while the
rate of return ^i+y (/i>0)
and instantaneous
do not assume
amount
that the
representative agent
is
representative agent
is
endowed with
has access to two linear technologies (see for instance Cox, Ingersoll and Ross(1985)).
Wo and
technology
The
to
volatility
second technology
is
risky with instantaneous
a (a>0). By contrast with Cox, Ingersoll and Ross, we
invested in these technologies can be negative.
The
objective of the
maximize
sup
subject to the dynamics of wealth
(i.e.
E
[
(^-^^
re-'"u(C,)dt]
the stock of capital)
dW^ =(rW^-q)dt *MX,dt *aX,db^,
'w„
where the amount X, invested
^6 2)
w
=
in the risky
technology
is
constrained by
(6-3)
< X, < W^
We
assume
minimum amount C
that the agent requires a
u(C)
^
For the agent to
assume that
survive, a
this initial
minimum
Y,
With these
that
we
definitions, the optimal
studied in section IV.
is
wealth
is
satisfied.
= W,-W
required, namely,
We
is:
(6-4)
W^W with W=^C.
Henceforth we
define net wealth Y, and net consumption Z^ by
(6.5)
(6.1)-(6.3) reduces to the portfolio-consumption
problem
analysis of the portfolio-consumption problem, the following results
are straightforward
L The amount
function
and Z, = Q-C.
growth problem
From our
utility
^_
(C-C)'-'^, A>0, A^l,
1-A
=
initial
wealth condition
of consumption so that his
invested in the risky technology
is
smaller than the myopic
amount
21
min
{
W^;
JL(W,-W)}.
Act
2.
on
Hence
constraint (6.3) (non-negativity of the riskless investment) causes the average rate of return
capital (t+^i
^
)
to
be lower even during periods when
this constraint
is
not binding. Therefore,
neglecting such a constraint yields a systematic overprediction of the average rate of return on capital.
VII.
CONCLUDING REMARKS
In this paper,
we have used
with borrowing constraints.
about the agent's
when an
A
stochastic control
We
to study
an optimal consumption-investment problem
have shown that optimal feedback controls
utility function.
exist
under general assumptions
This has enabled us to obtain qualitative properties of these controls even
explicit solution fails to exist.
further quantitative analysis can be
equation, although fully nonlinear,
(see for
methods
is
done by developing numerical methods. Fortunately, the Bellman
'well
behaved' (Theorem
example Fitzpatrick and Fleming (1990)). In
2.1),
and a wide
their paper, Fitzpatrick
class of algorithms
can be used
and Fleming examine a Markov
chain parametrization of a similar investment-consumption problem. They show that the value function and
the optimal policies for the approximate discrete problem converge, respectively, to the value function and the
optimal policies of the continuous problem as the discretization mesh size goes to zero. While the
discretization relies
rely
on
on Kushner's Markov chain
viscosity solution techniques
ideas, the proofs are not of probabilistic nature; rather, they
introduced by Souganidis (1985) and Barles and Souganidis (1988).
The methodology presented herein can be extended
alternative investment opportunity sets
the case where the market parameters
and various
r, /i
to a wide variety of consumption-investment problems with
constraints'.
and a evolve
See Zariphopoulou (1993), Fleming and Zariphopoulou (1991).
in a
An
interesting extension
non-deterministic way,
i.e.,
would be
to consider
where the investment
22
opportunity
introducing
is
stochastic. In this case,
new
we would need
to increase the dimensionality of the
state variables to represent the stochastic
market parameters. In general, multidimensional
problems are hard to solve mainly because the associated
solutions might not exist.
As
maximized
utility as
HBJ
equation
is
a result feedback formulae for the optimal control
the methodology used in the present paper could
(weak) solution of the
HJB
still
problem by
often degenerate and smooth
may
fail
to exist. Nevertheless,
be applied for the unique characterization of the
equation. This characterization will guarantee the convergence
of numerical schemes for the optimal policies despite the lack of smoothness of the value function.
Therefore,
it is
our belief that viscosity techniques are the most appropriate tool to analyze continuous-time
consumption/investment problems
explicit solution fails to exists
in
the presence of market frictions. This
and when,
is
as a result, numerical approximations
true in particular
must be used.
when an
should be
It
noted however that each new case will require a specific analysis since to date no general framework
exists.
Indeed, as opposed to the perfect capital markets literature which has developed a general model (the diffusion
model) and a general tool (the martingale technology), the literature on market
Hence, for each different type of constraints (transaction
specific treatment
*See
Tuckman and
is
required.
Vila (1992).
costs,
frictions lacks
such unity.
holding costs*, borrowing constraints
...)
a
23
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Absence of Arbitrage, and Feasible Consumption
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25
APPENDIX A
VISCOSITY SOLUTIONS
The notion of viscosity solutions was
first
introduced by M.G. Crandall and P.L. Lions (see Crandall and Lions
(1983)).
Definition A.l: Let F:
function
R* - R
J:
is
R*xIbcRxR->R be continuous and non-increasing
and only
From
if
argument.
its last
A
continuous
a viscosity solution of the equation:
F(WJJ',J")
if
in
for every
smooth
then F(WJ(W^),(f,'{'W^),4>"(W^)) <
0.
F(W^J(Wo),0'(Wo),0"(Wo)) >
0.
if
Wo>0
is
a local
maximum
of
(ii)
if
Wo>0
is
a local
minimum
of J-0 then
it
follows that a
(A.1)
R* - R the following propositions hold
(C^) function 0:
(i)
the above definition,
=
J-4>
smooth
viscosity solution of (A.1)
is
also a solution of (A.1) in
the classical sense and therefore the concept of viscosity solution extends the concept of classical ones.
Moreover, viscosity solutions exhibit very good
Theorem A.1
states that
stability properties
under some mild conditions the
as the
Theorem below demonstrates.
limit of viscosity solutions
is
also a viscosity solution
of the limit equation (see Lions (1983) for a proof).
Theorem A.l: Let e>0, F' a continuous function from R* x Rx Rx
F'(WJ'J''J'")=Oin
[0,a).
let J'
be viscosity solutions of
We assume that F' converges locally uniformly on R* x Rx Rx R to some function F and
that J' converges locally uniformly
on
R to R and
on
[0,<b) to
some function J. Then J is a
viscosity solution
ofF(W,J,J',J")=0
[0,co).
In the case of equation (2.6), the function
F(W,J,J',J")
=
^J
-
F
max
OiXsk(W-L)
takes the following form:
{
IcrX^ J" +mXJ'}
2
-
rWJ'
-
max
CiO
{
u(C) -CJ'
}
(A.2)
.
26
APPENDIX B
This appendix presents the proofs of lemmas 3.3 and
Lemma
Proof:
2.1.
J' converges to
3.3:
We
first
observe that
J
locally uniformly as e goes to
J' is locally
J*' is
P" converges towards v
locally
utility
funtion u(«). Moreover, because
bounded. Therefore, there
locally uniformly in (0,a>).
lim
Using Theorem Al, we get that v
v(W)
=
it is
locally
and a function v such
that
almost any interval
independent of
e,
such
that,
.
^
also a viscosity solution of (2.6). Therefore,
3.4: In
^
the unique viscosity solution of (2.6).
is
have the same limit which coincides with
Lemma
concave,
Lemma
and therefore
e
is
J*"
J' is
defined as in
=^-
w-o
function J
subsequence
exists a
is
Moreover,
lim/-(RO
uniformly in
0.
uniformly bounded. Indeed J'<J<J" where J"
This follows from the concavity of the
Lipschitz and hence
3.4.
we conclude
On
the other hand, the value
that all the converging subsequences
J.
[a,b], there exist
two positive constants R,=Ri([a,b]) and R2=R2([a,b])
for We.[a,b],
J"(W)>R,
(i)
\J'"(^)
(ii)
(3.4)
<i?2
(3-5)
I
Proof
(i)
We
first
show
that
lim(J')'(W„)
=J'(WJ
c-O
at
any point
Wp
such that
J'
exists.
Let
Wq
be such a point. Since
J'(W) < J'(Wo)
Since the
(J*')'s
are
bounded
J' is
+ (W-Wo)J' '(Wo).
locally uniformly, there exists a
concave,
we have
(B.l)
subsequence (J'")'(Wo) and a number p such that
27
Urn
Using that
J'n
converges towards
J,
r"(W„)
= p
.
inequality (B.l) yields
J(W) < J(Wo) + p(W-Wo).
But
J
is
concave and differentiable
at
(B.2)
W^, therefore p=J'(Wo) and
Hm(-r'')'(WJ =J'(WJ.
Since
J'' is
non-increasing,
we conclude
that in any interval [a,b] such that J'(b) exists, there exists a constant
C,=Q([a,b]) with
J"(W) > C, on
(ii)
The
The bound
for J'"
interested reader
APPENDIX
C:
more
is
difficult to obtain.
The proof
The case
Lemma
C.l
Proof: Let
i) ii)
of
and
CRRA
iii).
J(W)/(W+L)'-''
ii)
2.1 (ii),
is
and
4.5 iv)
non-increasing and hence
ii)
is
J(W,L)AV''^
=
show
that J(«,«)
et al. (1989)).
we omit
it.
=
J(t,l-t)
that J(t,l-t)
is
To
appears at the end of
f(W)W<
is
(2.2)
credit line
we
ease the presentation,
this
first
prove
appendix.
(l-A)J(W).
is
> (l-A)J(W).
L.
and the constraints
homogenous of degree (1-A)
(2.3)
and
(2.4), a
(see for instance
standard
Grossman and
Then
J(1,LAV) which
J(W,L)/(W-i-L)'"'^
one can show
dynamics
utility function, the
to
4.5.
non-decreasing and hence f(W)(W-i-L)
J(W,L) denote the value function J(W) when the
Laroque (1989), Heming
i)
4.3, 4.4
The proof of Proposition
J(IV)/W'-*
argument can be used
extremely technical and therefore
utility
i)
Given the form of the
is
referred to Zariphopoulou (1993) for details.
is
This appendix presents the proofs of Propositions
Proposition 4.5
(B.3)
[a,b].
is
non-increasing in
W.
where t= W/(W-(-L). Now, by an argument similar
non decreasing
in
t.
to the
one
in
Lemma
28
Proposition 4.5:
IfA<l, C'(W)
i)
satisfies
\
Max
{
IfA>l,
ii)
—
-A
^^]~^
C-{W), C>{W)[
}
< C-{W) <
1
C>{W)[^^]^
.
i'f-13a)
C'(IV) satisfies
1
C^(W) < C'iW) < Min
C'(W)
Hi)
w..
For
Win
Z!^
W
[0,W], C'(W)
= A«
i)
If
A<1,
then from
Lemmas
('^l^b)
;
2.1
and C.l
(i),
we
m^c)
.
satisfies
C-(W) > C-(W)
Proof:
}
satisfies
Lim
iv)
C^Cff+L); C' (W) [^^LlL]^
{
(4.13d)
.
have:
J'(W)W < J(W)(1-A) < J"(W)(1-A).
Recalling that J'(W)=(C')-^
it
follows from Proposition 4.1 that
C'(W) > C"(W). Similarly
J'(W)W < J(W)(1-A) < J*(W+L)(1-A).
Hence
C'(W) > C*(W) [W/(W+L)](1-A)/A.
Also, from
Lemmas
2.1
and
C.l(i),
we have
J'(W)(W+L) > (l-A)J(W) >
(1-A)J*'(W)
and therefore
C*(W) < C<'(W)[(W+L)/W]'^^
ii)
The proof
iii) It
iv)
If
is
similar to that of
follows immediately from
(i)
(i)
and therefore
and
is
omitted,
(ii).
See lemma C.4.
the utility function exhibits constant relative risk aversion, then the Bellman equation does not admit a
closed form solution
on
(O,™).
However,
it is
possible, by following Karatzas et
equation in the unconstrained domain U. For
this
purpose,
we
express
al.
(1986) to solve the Bellman
W as a function of C by inverting the
29
equation:
c^
(c.i)
= -AC^'AV'CC).
(C.2)
j'(W(C))
Differentiating (C.I) with respect to
C
=
yields
J"(W(C))
Hence, the Bellman equation can be rewritten as
^J(W(C))
=
^C'--^
1
C yields
Differentiating equation (C.3) with respect to
(see Karatzas et
al.
W"(C)C
+
solution to (C.4)
is
u's are the roots of the
(i=0,l) are constants
K,'s
—
+
A
(C-3)
.
-
i^rW(C)C-'
7
7
=
(C-4)
= 0.
(C-5)
i.[C-K^C"*-K^C''"l
second order equation below
-(T--f-p)o)
J' *
The
A
the following linear second order differential equation
^(r+7i-:^-;9)W'(C)
W(C)
where the
Ic'-'^W'CC)
+
(1986) for a general presentation of this approach)
7
The general
rW(C)C-^
+
""A
when
w*
> 1>
W
is
in
— =0
-
> w"
and
w*
(C-6)
with
>
any connected subset
(C.7)
A
(a,b) of the
unconstrained domain U.
On
such subset the optimal policy X(W(C)) and the value function J(W(C)) are given by the equations
X(W(C))
=
^—
cr
-J"
=
_ii_W'(C)C
(C-8)
i._^JC-K,a;*C"*-K^cx;-C''"]
=
Act
A"
Act
and
J(W(C))
=
f-^dW
J
The
next
Lemma
lemma shows
C.2: There exist
dW
rc-'^W'(C)dC
=
=
A"
J
that the borrowing constraint,
W>0 such
1 [S—
1
-
-A
"'
(0,W)
is
is
'^'
K,C""-^].
(C.9)
w"-A
w'-A
X<k( W+L),
that the open interval
K,C"'-^not met
included in
when
W
U and such
is
small enough.
that
X'(W)=k(W+L).
Proof: Per absurdum, suppose that there exists a sequence
(4.12b) and the definition of the constrained
domain
B,
W„ it
such that X'(W„) = k(W„+L).
follows that
From equation
30
^J(WJ > _^(J'(W„))"^
If
A< 1,
the
then from equation (C.IO)
minimum
But by
it
follows that J(W„)
of the right hand side of (C.IO) for
Lemma
is
*
bounded away from
all positive
(C-10)
k^LJ'(WJ.
by a positive constant, namely
values of J'(W„).
2.1,
limJ(WJ < limJ-(WJ =0
,
n-«
n-"
which yields a contradiction.
If
A>1, then from equation
minimum
(C.IO)
it
follows that J(W„)
of the right hand side of (C.IO) for
all
bounded away from
is
positive values of J'(W„).
limJ(WJ < limJ-(WJ
=
-«
(-<=)
But by
by a constant, namely the
Lemma
2.1,
,
n--
n-»
which yields a contradiction.
Furthermore, from equation
(0,oo).
The
The proof
next
Lemma
lemma
C.3:
is
(4.6), the
borrowing constraint must be binding and hence
U cannot be equal
to
therefore complete.
describes the behavior of
i)
LimX'{W)
=
X*(W) and C*(W)
as
W goes to zero.
;
w.o
u)
Proof:
i)
The argument
ii)
It
Lim C'{W)
is
=
similar to the
0.
one
in
Lemma
C2.
follows from equation (4.12a) that
1
-A
^^-^^^
^J(W) > ^^(J'(W))"^.
1 -A
Hence, since
lim
J(W)
=
(
respectively -<»
w-o
when
A<1
(respectively
A>1),
it
follows that
)
31
Urn
J'(W)
= +C0
.
w-o
Since
1
C-(W)
the proof
Using
next
tlie
is
[J'(W)]"'^
,
complete.
two preceding lemmas, we conclude that
lemma
Lemma
=
describes the behavior of
in the interval (0,W*), the constant
C*(W) and X*(W)
Ki must be zero. The
in the interval (0,W*).
C.4: In the interval (0,W), the following inequalities hold:
K,>Q,
i)
kW
ii)
< X-{IV) <
JLW,
Act
C-(W) > C-(W)
Hi)
;
lim*^'^^^
=
1
,
w-oC-{W)
d "A"
s
„
<
iv)
,
dlV'v)
dW
.,
>
and
dX- ^
/i
dW
Proof:
i)
In the interval (0,W*),
Ao"
we have
J(W(C))<r(W) =(l)^u(W)
=
lu(C-K,C--)< l[u(C)-u'(C)K„C''*]
A-
A-
where the
and
last inequality
holds because u(
after rearranging terms,
we
get that
concave. Using equation (C.9) to get the expression of J(W(C))
• ) is
K^, is
A-
non-negative.
If
Ko=0, then J(W)=J"(W) which contradicts the
fact that the constraint is binding.
ii)
From equation
(C.5) and (C.8)
we
X(W(C))
get that:
-
JLw(C)
Aa^
=
JL1c"X(i-'^*)
^
Acr^ A"
and
X(W(C)) -kW(C) =^(C)
=1
A"
[(Ji,-k)C -K,(_^,a;--k)C"*]
Act
Act
.
(Cll)
32
From
it
follows that ip{')
is
a concave function of C. Furthermore from
Lemma
C.2,
V(C)<kL
if
C
[0,C'(W)) and ^6(C'(W)) = kL. Hence, V(C) must be positive (because Vi(c(0))=^(0)=0) and non
in
is
(C.ll),
decreasing which yields that
X>kWanddX/dW>k.
From equation
iii)
(C.5) and
W(C)
for
We[0,W]
K,=0,
it
follows that
J-fC-K^C"*]
=
with Ko>0. Therefore
C"(W) > A'W =C-(W)
This completes the proof of proposition
']£ ^
iv)
v)
It
^
C(l-a,-) ^
Q
W^
W^
follows from equation (C.6) and the fact that
=^ -JL
^
dW
vi)
W^
We
4.5.
X,,W^-X^W^^
dW^
.
next
Lemma
Proof:
show
C.5:
For every
From Lemma
W>0,
C.4
ii), it
^-^('^')'C"'
Act
that, actually, the
K,=0 and Ko>0.
<
C-K„u.-C-*
second part of property
X'(W) <
j^
Act"
(ii)
above holds
for every value of
W.
JLw
follows that
W>
L
lJL-1
kAo^
Hence,
if
W>W*
X-(W) <
k(W + L)
<
JLw
Act
The
next
lemma shows
the risky asset.
that
when
W
is
large enough, the agent will invest the
maximum
allowed
k(W+L)
in
33
Lemma
C.6: There exists a
number IV such
absurdum suppose
Proof: Per
that there exists a
<
From
that (W,<o)
the Bellman equation (4.12a),
it
divide each side of (C.12) by
1/(1-A).
From
Proposition
4.5,
k(W +L)
<
A^
1-A
.
C.7: If
-y
+p>r+
k ^
,
then
is,
W < W, < W,
From Lemma
C.l,
(C.12)
.
J(W„)AV„J'(W„) goes to
get
(C-13)
.
0< \
k(kAa^-At) which contradicts condition (4.6).
U= (0, W).
]W„W,[ c
Within the interval (Wj.W,), there
«.
^(W„.L)J'(WJ
.
2
suppose that there
;
-
Hence we
interval ]0,W*[, the unconstrained
Proof Per absurdum, suppose that besides the original
another interval. That
n
let
to A".
:^
.
r
rW„J'(WJ
Using equation (4.11) for the expression of A" we get
Lemma
such that
.
.
W„J'(WJ. Then we
1-A
W„ - «
follows that
C'(W„)/W„ goes
^
a subset of the constrained domain B.
sequence
^J(WJ < _^J'(WJC-(WJ
We
is
U
exist
exists
two numbers
and X*(W,)-kW,
two constants
W, and W,
such
= X"(W,)-kW, =
K^, K, such that
domain, U, contains
that:
(C.14)
kl.
W(C), X(C) and J(C) are given by
equations (C.5),(C.8) and (C.9). Let Ci=C(Wi); i=l,2. Let V(C) given by (C.16) below:
i>{C) =
V-(C)
^^ '
X-(C)
-
kW(C)
^^-^^^
,
(C.16)
=l[M-k][C-lC,^^'-'^C"^-tC^'""-^C"-]
^M-k
A-
^M-k
'
with
M
We
=
recall that
for every
C
JL
(C.17)
.
w*> l>0>(j', cj*>A and
in the interval (Ci,C2),
If
Ko<0 and Ki<0, then
If
Ko>0 and
K,<0, then
that
M>
k.
Given the definition of
ip{-),
it
follows that V'(C)<kL,
with equality at C, and Cj.
V'(*)
is
increasing which
)/'(•) is strictly
is
impossible since ^(C,)=V'(C2).
concave which
is
impossible since V(C)<kL=^^(C,)=^(C2).
34
From
the two observations above,
X'(W)<MW,
C
follows that K,>0.
Funhermore, we know from Lemma C.5
that
hence
K,((j--l)C-' >
Let
it
be
the
point
K,(l-u)-)C-"
where
Kj(k-Mw-)(-w-)(l
for
reaches
V(*)
-a)-)C""
,
>
(^•^^)
C € (C,,C,).
its
minimum
in
K,(M(j*-k)w-(cj--l)C-*,
]C„C;[.
for
C=C
Since
xl>'{C)>0,
(^-1^)
.
Multiplying (C.18) and (C.19) and rearranging terms yields
(^-^O)
> (w--cj-)[(w-+a;-)M-k]
From equation
(C.6),
follows that
it
w*+w = y
(7+^-r), and finally that
^>r>5*7.
- ^
2
(C.21)
'
If (/ik)/2
+
r
<
/9
+
7,
then
we have
a contradiction
and therefore U=(0,W).
35
36
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