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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
OPTIMAL CONSUMPTION WITH INTERTEMPORAL
SUBSTITUTION I: THE CASE OF CERTAINTY
Ay man Hindy and Chi-fu Huang
March 1990
WP#3158-90-EFA
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
v»Stf>v
1990
j
OPTIMAL CONSUMPTION WITH INTERTEMPORAL
SUBSTITUTION I THE CASE OF CERTAINTY
:
Ayman Hindy and Chi-fu Huang
March 1990
WP#3158-90-EFA
M.I.T. LIBRARit;:
JUN 2 2 1990
REUiJVtU
OPTIMAL CONSUMPTION WITH INTERTEMPORAL
SUBSTITUTION THE CASE OF CERTAINTY
I:
Ayman Hindy and
Chi-fu Huang'
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
MA
02139
October 1989
Revised March 1990
Abstract
We
study the problem of optimal consumption choice
a class of
utility functions that
perfect substitutes.
The
class
we
consider excludes
additive utility functions used in the bterature.
for
in
continuous time under certainty for
capture the notion that consumptions at nearby dates are almost
all
We
time-additive and almost
all
the non time-
provide necessary and sufficient conditions
a consumption policy to be optimal. Furthermore, we demonstrate our general theory by
solving in a closed form the optimal consumption policy for a particular felicity function.
optimal policy in our solution consists of a (possible)
initial
"gulp" of consumption, or an
The
initial
period of no consumption, followed by consumption at the rate that mountains a constant ratio
of wealth to average past consumption.
*We
are grateful to
Hua He
and Thaleia Zariphopoulou
for pointing out
an error
for helpful discussion.
in
an earlier version, and to John Heaton, Jean-Luc Vila
1
Introduction
The
choice of an optimal consumption plan for an individual under certainty
problem
in economics; see, for
is
a
classical
example, Ramsey (1928) and Modigliani and Brumberg (1954).
In a large part of the literature that addresses this problem, the preferences of an individual are
represented by a time-additive utility function.
utility function
It
has long been recognized that a time-additive
does not incorporate the intuitively appealing notion that past consumption can
contribute to an individual's future satisfaction. Nevertheless, this class of preferences has been
because of
utilized
of time periods
is
its tractability.
In discrete time models,
one can argue that
if
the length
chosen appropriately, then past consumption would have negligible effect on
current satisfaction. Such arguments, however, no longer apply
when we consider a continuous
time model.
Recently, a class of non-time-additive utility functions, due originally to
and Uzawa (1968), has gained increasing significance
Koopmans
(1960)
in the literature of continuous time equi-
librium models; see, for example, Epstein (1987). Also, non-time-additive utility functions that
exhibit habit formation are objects of current research interest in continuous time financial asset
pricing models; see, for example, Constantinides (1988), Heaton (1988), and Sundaresan (1989).
A common
feature of these utility functions
satisfaction at
any time,
that the
felicity,
or the index of instantaneous
a function of the current consumption and a certain functional of
is
A
the past consumption.
is
high level of past consumption can serve either to depress current
appetite as in the case of the
Uzawa type
utility functions or to increase current
appetite as in
the case of the habit formation models.
These non-time-additive
utility functions offer intuitively
appealing notions of interactions
between past and current consumption. They are also mathematically tractable. Optimal con-
sumption
policies
forms of the
can be characterized using dynamic programming. For particular functional
felicity function,
one can even have closed form expressions for these
Unfortunately, these utility functions
much
fail
policies.
to exhibit one key economic property that underlies
of our intuition about the behavior of asset prices over time in the absence of arbitrage
possibiUties.
A common
hypothesis maintained by
have predictable jumps.
agents
knew that the
moment.
If
To support
many
financial economists is that asset prices cannot
this hypothesis, the
price of an asset were to increase
argument goes as
follows:
by a discrete amount
suppose
at a specific
an agent could borrow funds for a very short period of time over which the interest
2
1
he would have to pay
is
negligible, then he
would be able to
access to a frictionless market, he would simply borrow
price were supposed to occur,
"position.
The demand
buy the
some funds
just before the
jump
in the
and then liquidate
asset, wait for its price to increase,
when the jump were supposed
be eliminated. The validity of
consumption
Having
realize an "arbitrage profit".
his
of agents exploiting this arbitrage opportunity would push the price of
the asset up before the time
prices for
Introduction
this
argument depends
to appear and hence the
crucially
on the hidden assumption that
at nearly adjacent dates are almost equcd
agent pays on funds borrowed over a short time interval
There are several economic
tions at nearly adjacent dates.
forces that will bring
The most
primitive
jump would
and thus the
interest that the
negligible.
is
about the closeness of prices for consump-
among
these forces
is
the hypothesis that
agents treat consumptions at nearly adjacent dates as almost perfect substitutes and thus ensure that almost equal prices are established for them.
under certainty, and Hindy and Huang (1989),
ences that exhibit this property.
explicitly
They show
on current consumption and
is
in
Huang and Kreps
(1989), in a model
a model under uncertainty, study prefer-
in particular that if the felicity function
strictly
concave in
it,
depends
then consumptions at nearly
adjacent dates cannot be treated as almost perfect substitutes. Since the functional forms of
Uzawa type and
the felicity function used both in the
in the habit
formation type preferences
are strictly concave, agents with those preferences do not view consumptions at nearby dates
as close substitutes.^ Therefore, such models will reach economic conclusions that violate
some
of our basic intuition about price behavior over time.
The purpose
of this paper
is
to investigate the optimal
for a class of utility functions that treat
key feature of this class of
consumption problem under certainty
consumptions at nearby dates as close substitutes. The
utility functions is that the felicity
function at any time depends only
upon an exponentially weighted average of past consumption - one derives
past consumption.
We
will
demonstrate the tractabihty of
programming can be used to
when one
the optimal consumption policy
we
solve in closed
in
may
our solution
form the optimal consumption
specify periods of
calls for
consumption mixed with periods of
this property.
a (possible)
initial
In particular, the optimal
"gulp" of consumption, or an
'Heaton (1990) studies preferences in discrete lime whose continuous time limits satisfy the notion
Huang and Kreps (1989) and Hindy and Huang (1989).
substitution studied by
policy.
derives current satisfaction only from past consumption,
no consumption. Our closed form solution exhibits
consumption policy
by showing how dynamic
give sufficient conditions for optimality. In particular, for a specific
functional form of the felicity function,
Intuition suggests that
this family
satisfaction only from
of local
period of no consumption, followed by consumption at the rate that maintains a constant
initial
The consumption
ratio of wealth to average past consumption.
rate
determined by the
is
interplay between the effect of past consumption, the current return on investment and the
impatience of the agent.
The
rest of this
under certainty
paper
organized as follows. Section 2 sets up the consumption problem
is
in continuous time.
Section 3 heuristically derives necessary conditions for a
consumption policy to be optimal. Section 4 provides
section 5,
we
sufficient conditions for optimality.
In
solve the optimal consumption problem in closed form for a particular felicity
function. Section 6 contains concluding remarks.
The Setup
2
who
Consider an economic agent
is
from time
lives
t
=
to
<
= T
from consumption
refrain
pattern over his
life
moment, and can consume
for
any period as he sees
at finite rates over intervals.
We
fit.
span by a positive^, increasing, right continuous function C:[0,T]
possible kind of a discontinuity of
consumption
X,
is
C
function
is
an example;
set of the agent, A'+, is the space of all
Note also that a
right-continuity, for
=
The
=
0.
zero for almost
is
all
The
p. 48).
the space of right-continuous and finite variation
function x on [0,T] has a finite left-limit at any
!(<*")
=
x{t),
i
any
t
6 [0,T') denoted by x{f^).
£ [0,T). For convenience, we
Since left-limits exist for any
a;
G
A",
a
jump
By
will use
of of x at
t is
x{t)-x{i-).
points of discontinuity of a consumption pattern
decides to
consume a "gulp", and the
there can be only at most a countable
size of the
number
continuous function of time. Let these
jump
is strict,
we
will explicitly state so.
is
C
are the
moments when the agent
amount of that "gulp". Note that
the
of times of "gulps" since
moments be
use weak relations in our discussion. Hence increasing
a relation
??+,
such functions and the commodity space,
finite right-limit at
any i G A' we have
the convention that x(0~)
is
finite variation
e (0,r] denoted by x{t~) and a
We
X
—
to have a "singular"
example, Royden (1968,
see, for
also
Note that the only
t.
C
a jump. Note also that we allow
the linear span of A'+. Note that
functions.
Ax{t)
is
a continuous increasing function whose derivative
The famous Cantor
t.
t
is
He can
represent the agent's consumption
with C{i) denoting the accumulated consumption from time zero to time
component, that
world where there
and T. The agent can consume
a single consumption good available at any time between
"gulps" of the good at any
in a certain
ri,r2,.
is
.
.,
and
let
C
is
the
an increasing right
jump
at time
equivalent to nondecreasing, for example.
r,
be
When
,
The Setup
2
4
AC(t,). Moreover,
rates
C
is
absolutely continuous on the intervals on which the agent consumes at
and the derivative of C on these
When
this derivative
is
C
zero,
is
intervals,
denoted
the "rate" of consumption there.
c(t), is
constant on the interval, and the agent
is
not consuming at
all.
The agent
or
all his
t,
The
a continuous function of time.
is
dW{t)
=
W{t)r{t)dt-dC{t)
W{t^)
=
W(T,)-ACiT,)
is
a
left
t
r(<).
he can invest part of
We
assume that the
agent's wealth before consumption at
feels,
h«LS
fort e (r„r,+i)
for
alH 6
is
however, that consumption at one time
[0,r].
is
T] and for final wealth W{T).^
[0,
a substitute for consumption at other, near-
by times. Therefore, delaying or advancing consumption
and Hindy and Huang (1989),
i
just the budget constraint over time.
preferences for consumption over the period
very small effect on his total satisfaction.
for all
foraUr..
continuous function and (1)
impose the condition that W{t) >
The agent
He
and at any time
denoted W{t)y changes according to the following dynamics:
Note that W{t)
We
Vl^(O),
weaJth in a riskless asset with instantaneous rate of return
interest rate r(<)
time
with endowment
starts at time
time has a
for a very small period of
Huang and Kreps
(1989), for the case of certainty,
for the case of uncertainty, formalize the idea that
consumption
preferences can be similar to those expressed by the agent. In particular, the authors advance
a family of topologies on the space of consumption patterns over time that capture the notion
that consumptions at nearly adjacent dates are almost perfect substitutes. In this paper,
use one
member
commodity space
of this family given as follows.
X
Fix p
>
1.
Define the topology
T
we
on the
to be the topology generated by the norm:^
\\C\\=[J^\C{tWdt + \CiT)\^]''\
(2)
Moreover, Huang and Kreps (1989) and Hindy and Huang (1989) show that there are
intuitively appealing functional forms of utility
intertemporal substitution properties.
C
which express preferences for consumption with
The agent has one
of them. Fix any consumption pattern
and a corresponding terminal wealth W{T). At any time
^The astute reader might have observed
tinuous function of time.
justification
solution
*The
is
To
W(T) =
T
define the exponentially weighted
that the final wealth should be W(T'''), since wealth
simplify notation,
that consumption gulps at
t,
we
will use the slightly
inaccurate
W{T)
is
a left con-
for final wealth.
add nothing to the satisfaction of the agent. Hence,
Our
at the optimal
U'(r+).
special treatment of
on [0,T) but disagree on
T
C(T)
in the
norm accounts
could not be distinguished
if
for the fact that
the
two consumption patterns which agree
norm were without the term involving C{T).
average of past consumption,
z{t), by:
2(0
where z{0~) >
2(0- )e-^'
a constant and
is
the integral in (3)
=
is
/?
is
+p
C
e-^('-) dC{s)
(3)
,
Note that the lower limit of
a weighting factor.
0~, to account for the possible
jump
right continuous function. Moreover, z has a
component when
f
No-
jump
exactly
C
at
when
C
of
does. Also observe that higher values of
/?
<
=
0,
and that
z{t) is a
does and z has a singular
imply higher emphasis on the
recent past and less emphasis on consumption in the distant past. Note also that the average
past consumption z{i) changes according to the following dynamics:
dzit)
=
fi[dCit)
z{Ti)
=
2(r-)
Given 2(0"), the agent's
+
-
for< e(r,,r,+i)
z{t)dt]
/3AC(r.)
utility for the
for alii,
for all r,.
consumption pattern
C
and
final
wealth
W{T)
is
given by:
U{z{0-),0;{C,W{T)))= f
u{z{t),t)dt
+ V{WiT)),
(5)
Jo
where both
u: 3R+
x [0,r]
—
>
3?
U {— oo} and V: §?+—§(? U {— oo}
V is strictly concave and
the possibility that u or V
functions. Furthermore,
that
we have allowed
V(W^) = ^W°, where a <
0.
The
function u
is,
u
is strictly
are continuous
concave in
can take the value
in the
its first
— oo
and increasing
argument. Note
at zero. For example,
terminology of Arrow and Kurz (1970),
the felicity function that assigns the level of satisfaction derived from past consumption and
V
is
the bequeath function.
topology generated by
T
Preferences given by (3) and (5) are continuous in the product
and the Euclidean topology on
Jf+; see
Hindy and Huang (1989),
C
determines, via the budget
proposition 17.
Starting from a given level of wealth, each consumption plan
constraint, the value of terminal wealth
W{T).
It
W{t), and the average past consumption z{t~).
average past consumption up tiU
and z{t~) the state variables
consumption decision
at
t.
at
t,
t,
also determines for every time
Note that at any
faces the
1^(0) and given 2(0").
We
A'^.
is
will call
the
W{t)
making
his
problem of choosing a consumption pattern C*
- the admissible controls - to maximize his
U{z{0~),0;{C,W{T))), subject to the dynamics given
is
t.
since they represent the status of the agent before
The agent
the wealth
the value z{t~)
excluding the possible consumption at
- the optimal control - from the space
wealth
t,
t,
in (1)
and
(4),
and given that
utility,
his initial
3
6
Formally,
wealth
final
problem
is
let
A{W (0),0) C X+ x JJ+
W{T)
Heuristic Derivation of Necessary Conditions
be the space of
all
pairs of
C
consumption pattern
that satisfy the budget constraint of (1) starting with H^(0).
The
and
agent's
to find:
sup
UiziO-),0;{C,W{T))).
(6)
(c,w(r))e>(W(o),o)
A
solution to (6) exists
the
if
supremum
is finite
and
is
some {C',W'{T)) G
attained by
AiWiO),0).
At times, we
[0,<~]}
will also consider
and W{t) determined by
a sub-problem of (6) at some
(1) (the
C
is
t
with the convention that C^{s)
denotes the consumption pattern C' and
on
[t,T]
with an
initial
of
clear that
C
in (7)
finaJ
wealth
z{t-)e-'^^'-'^
+
W{T)
=
for all
t,
where A{W{t),t)
ioTs>t.
(8)
once W{t) and z{t~) are known, {C{s);s G [0,i~]} has no impact on the choices
and
this justifies the notation
U{z{t-),t-C^,W{T)). Moreover,
{C\s) = C'is) - C'(t~);5 G
W{t) corresponding
{C\W'{T)).
to
[t,T]}
is
if
{C',W'{T))
is
a solution to (7) with z{t-) and
Heuristic Derivation of Necessary Conditions
We now
use heuristic arguments to derive a set of necessary conditions for a solution to (6).
These conditions
will
imply that an optimal consumption pattern can have gulps only at
Suppose (C, W'{T))
is
an optimal solution to (6) and
let
and starting with wealth W{t) and average past consumption z{t~),
J{W{t),z(i~),t) be the
[t,T]. In
other words,
maximum
<
=
0.
the corresponding wealth and
average past consumption, or the state variables, be {W'(t),z'{t~)) for
t
<
.s
that satisfy the budget constraint
r e-^^'-^UC\0
a solution to (6), then
3
(7)
value W(t), where
=
z{s)
It is
+ V(WiT)),
an increasing and right-continuous function on {t,T] representing accumulated
consumption starting from
(1)
uiz{s),s)ds
Jt
{C',W(T))eA{W{t),t)
where
G [0,r): given {C(s);5 G
budget constraint), solve
U(z{r),t;C',WiT))= f
sup
t
t
G [O,^]. At any time
let
the value function
possible attainable satisfaction over the remaining period
let:
rT
JiW{t),z{t-),t)=
sup
[/ u{z{s),s)ds + ViW{T))],
(C'W(T))eA(W(t),i)'-Jt
'
(9)
.
where {z{s);s G [^T]}
Assume
that
Jw{W,z,t) =
t,
J
is
—
is
defined according to (8).
finite
an'^'
1
and
differentiable in
measures the changes
from a very small change
results
W and
The marginal
2.
maximum
in the
wealth when everything else
in
marginal value of the average past consumption z at time
t,
is
value of wealth at time
attainable satisfaction that
kept constant. Similarly, the
=
Jz{W,z,t)
—
g^^'^'
,
measures
the change in the value function for infinitesimal changes in 2, ceteris paribus.
Along the optimal solution, J should
1
satisfy the following conditions:
- Bellman Optimality Principle
The Bellman optimality
all
times
t
principle; see, for example,
and r € [0,T] such that
<
JiW'{t),z'{t-),t)
That
an optimal policy on
is:
<
=
Fleming and Rishel (1975), implies that
r,
f^
u{z'{s),s)ds
+ J{W'{T\z'{r-),r)
T] remains optimal over any subperiod
[t,
for
principle implies that along the optimal solution, the value function
[r,
(10)
T]. In particular, this
J{W',
z',t)
is
a continuous
function of time even at times of jumps. Right-continuity follows from
J{W'{t),z'{t-),t)
= ]imJiW'{T + A),z'{t- + A),T + A) = J{W'{t+),z'{t),t)
for all r.,
(11)
and
left-continuity
is
easy to see.
For brevity of notation, we will use J{t) to denote J{W'{t),z'{t~),t) and similarly use
Jwit) and Jz{t) to denote the
partial derivatives of J{t) with respect to
2 - Continuous Marginal Value of
Along the optimal
time
t
G
t
+ At
That
[0,7")
policy, given the interest rates, the
A< >
or else
some changes
7w{t)€
the
A<
same
Jwit)
marginal value of
in
C
eeJ'
will
^^''
*
e
units of wealth at
units of wealth at time
be called for to change
W*
and
z'.
we must have
is,
Letting
2, respectively.
W and z
must be equal to the marginal value of
for small
W and
is
| 0,
we
line of
get
= lw{t +
At)€ef<^^''^'^'^'.
Jvv(0 = Jwif^)- Thus Jw{t) must be right-continuous on
(12)
[0,7').
Along
arguments, we show that Jw{i) must be left-continuous on (0,r] and thus
continuous on [0,T].
3
8
Similarly,
Heuristic Derivation of Necessary Conditions
by the Bellman optimality principle, a marginal value of
must be equal to
its
function at time
t
marginal contribution to the
+
At. That
felicity
of
J z{i)
is
+
_
to the value
+ At)(e-^^\
J,{t
we conclude that J^CO must be right-continuous on
0,
>
t
is,*
u,{z'is),s)(e-^^'-'Us
—
increase in z{t~) at
+ A/] and
function on {t,t
/t+At
Letting A<
e
(13)
The
[0,T).
left-continuity
established by noting that
Jt-At
—
and letting A<
3-
>
0.
Dynamics of Marginal Values
By
the hypothesis that r{t)
is
continuous in
<,
— —=
(12) implies, as
At —>
that
0,
-T{t)Jw{t).
j^
Next note that
if
z'{i)
is
continuous on some time interval
{t
- A<,t
-I-
A<), then
u,{z'it),t)
At
and
Hence,
in
a time interval over which C" has no "gulps", (13) implies, as At
^Ml
Also note that at
derivative of
V
consumed
t
at
=
-u,{z'{t),t)
+
=
»
0,
that
p7,{t).
T, the marginal value of wealth,
evaluated at the terminal wealth W'{T).
Jw{T)
is
exactly equjil to the
In addition, any unit of the good
T, with no change in wealth, has no effect on the agent's total satisfaction
and hence J z{T) =
4-
<
=
—
0.
Appropriate Times for "Gulps"
Applying Bellman's optimality principle
*We
use u,(r,
t)
to denote
^=^.
in (10)
on an interval
(r,,
r,+i)
when our
solution
prescribes consumption at rates,
we
get the following Bellman equation:
max{u{z'it),t) + Jw[rit)W'it)-c{t)]-i-J,(}[c{t)-z'it)]
>
c(<)
Note that the Bellman equation
tion rate that satisfies (15)
is
linear
+ 7t}J =
m the consumption rate c{t).
=
c(Oe[0,oo]
c{t)
=
oo
when
Jw{t) - Plz{t) >
,
when
Jw{t) - Plzit) =
,
when
Jw{t) - Plzit) <
Bellman equation, therefore, suggests that over the periods
cannot be the case that
W.
(/?
times) the marginal value of z
also suggests that a "gulp" of
It
Jwi''')
~ PJzi''') £
negative. Since the marginaJ values of
will
W
and
+
e) for
G (r —
e,
r
jump
only be a countable
T,
on
5-
No "Gulps"
Now
for all points
number
t
t
JwiT) -
calls for
we cannot have two
successive
W and
Jwi''')
—
/^<^z(t)
>
e
0.
is
strictly
Jwii) — PJzi^)
Hence our candidate policy
moment
0.
A
z,
at
at time r
>
0.
Note that by right
any one moment. This, together with
implies that there are two strictly positive
€i,C2 such that:
Jw{t) - PJzit) =
Jwii)- (3Jz{t)>0
Assuming that
is
=
I3Jz{t)
jumps
Jwit)-0zit)>O
PJzit)
when
any moment r when
at
Since this cannot happen as there can
e).
a "gulp" of size
the continiiity of the marginal values of
numbers
+
at rates,
greater than the marginal value of
at r
some
is
=
suppose that our policy
continuity of C,
which consumption
of jumps, our policy might prescribe a "gulp" only at a
this particular interval, such that
after
is
in
z are continuous, the function
be strictly negative on an interval (t - e,T
should prescribe a
Hence, the consump-
.
consumption might be prescribed
Suppose that we prescribe a "gulp"
0-
G (r.,r,+i). (15)
given by:
is
c{t)
it
for all/
Jw
and Jz are
decreasing just to the
left
(r-ei,r),
on
when
on
<
=
r
(r,r
,
+
e2).
differentiable functions of time, this implies that
of r and increasing just to the right of r. This condition,
however, cannot hold for any size of jump A. From the dynamics of
|[7iv(r+) - /37.(T+)] -
Jw{t) —
jpwir) -
/37,(r)]
=
/3
\u,{z'(t)
+
Jw
I3A,t)
and J^, we get that:
-
t..(z*(r), r)]
.
(16)
10
But by
if
concavity of the
strict
^{Jw — 0J z]
felicity
function in 2, this last quantity
were negative just before the jump,
jump, thus violating the condition that
our analysis
at
Heuristic Derivation of Necessary Conditions
3
=
<
not valid for
is
and
=
t
T.
A
<
=
and
gulp at
t
=
t
Jw — PJz
=
we should
<
=
0,
strictly negative just after the
be increasing just after a "gulp". Note that
is
clearly sub-optimal since the agent cannot
and the
it
wealth
final
All the above considerations lead us to conclude that
"gulp" at
negative. Hence,
T, thus C* does not have any gulps, except possibly
T, however,
derive any consumption satisfaction from
would be
it
is strictly
is
also reduced.
C should take the form
of a possible
followed by periods of consumption mixed with periods of no consumption.
prescribe a
jump AC*(0)
at
<
=
0,
then the size of the
That
vcdue function immediately after the jump.
the
is,
jump
jump should maximize
size
AC*(0) should
If
the
solve the
following problem
max J(W^(0) From the
first
initial possible
order condition,
jump
at
<
=
0,
e,z{0-)
we then know
that
+
/96,0+).
Jw{0^) =
f^Jzi^'^)-
Moreover, after the
the following condition should be satisfied for
all t
G (0,T]; at
times of no consumption:
u(z'{t),t)
+ Jwr{t)W'it) + l,fiz'{t) +
7,
=
(17)
0,
together with
/?7.(0-7h'(0<0,
and
when consumption occurs
at the times
at rates,
we must have
iJwit)-l3'Jz{i)]c'{i)
More generally, consumption only occurs when Jw{t) —
of no consumption
is
should at
all all
=
Q.
(19)
PJz{i)- During these times the strategy
sub-optimal and hence
u{z'{t),t) -\-Jwrit)W'it)
The necessary
(18)
conditions can be expressed
+
J,0z'ii)
+7t <
0.
(20)
more compactly by stating that the value function
times satisfy:
mnxiu{z,t)
In the following section,
+ JwWr{t)-J^Pz +
we provide conditions
Jt,
0, -Jw]
sufficient for
=
0.
(21)
a candidate solution to be optimal.
11
4
Sufficiency
The necessary
conditions in (18) and (19) simply say that the agent should adopt a consumption
policy of a possible initial "gulp" followed by periods of consumption
consumption.
the agent starts with a consumption "gulp", then immediately after
If
marginal value of wealth
z.
Otherwise,
mixed with periods of no
W should be equal to
(/?
During the periods
which the agent consumes at
in
that the marginal value of wealth
always equal to
is
0,
the
(/?
value of any additional unit of the good at any time
is
initial "gulp".
such a manner
rates, he does this in
times) the marginal value of average
In other words, the agent chooses his
past consumption.
=
times) the marginal value of past consumption
would benefit the agent to change the value of the
it
t
consumption so that the marginal
equalized between the two competing
uses available to him, namely investing to increase his wealth and consuming to increase his
average past consumption.
wiU
If this
find it attractive to deviate
condition
is
not satisfied in a particular policy, then the agent
from that policy by reallocating the consumption good in the
activity with the higher marginal value.
When
the agent
least as large as
(/?
is
not consuming,
it
must be the case that the marginal
utility of
wealth
is
at
times) the marginal utility of the average past consumption. In other words,
during these periods the agent gains more in overall satisfaction by investing
the riskless asset and by temporarily refraining from consumption.
The
all his
wealth in
increased wealth will
provide more consumption and bequeath value in the future than immediate consumption. Of
course,
start
when the marginal
consuming again.
such that at any
strictly greater
values of the good in both uses
is
again equalized, the agent
Finally, the necessary conditions instruct the agent to follow a policy
moment
after
t
—
0,
the marginal value of the good
than the marginal value of the good
if
invested,
if
consumed, ^J z,
at "rates" or
whether
later that the closed
it
is
never
Jw-
Note that, by the necessary conditions, when consumption occurs, there
it is
may
is
no
telling
whether
includes singular components. However, the reader will find out
form solution we construct
in the following section for a particular felicity
function shows that the optimal consumption policy contains no singular component.
We now
two
provide sufficient conditions for a consumption policy to be optimal.
steps. First,
we show that there
is
a value function J'{W,z,t) which
the satisfaction over [t,T] that an agent can obtain starting at
past consumption
this
z.
t
is
We
argue in
an upper bound on
with wealth
W and an average
Second, we give conditions under which a consumption policy achieves
upper bound and hence
it is
indeed the optimal policy.
»
.
12
4
Let the agent at any time
t
Sufficiency
have wealth W{t) and average past consumption z{t~). Assume
that the agent has the opportunity to exchange his wealth and average past consumption for a
level of satisfaction J'{W{t),z{t~),t). In other
J'{W,z,t) defined
for all levels of
that provides the agent at time
words, there
a "utility equivalent" function
is
wealth and average past consumption and for
t
all
times in [0,T]
with lump-sum satisfaction for the period [t,T]. Therefore
the agent has the option of choosing any budget feasible consumption plan
till
any time
and
t,
then exchanging his wealth and average past consumption at that time for a lump-sum payoff
in units of utility.
We
will
show that
if
J'{W,z,t)
satisfies certain conditions,
M^(0) and z{0~) at time zero for J'{W{0),z{0~),0)
feasible
consumption policy
till
any time
t
is
then exchanging
better for the agent than adopting any
and then receiving a lump-sum
utility at that time.
Consider the situation in which the agent decides to terminate his consumption at time
t
G [0,T], and exchange his state variables at
t,
W{t) and
z{t~), for J'{W{t),z{t~),t). In such
a case, the agent's total satisfaction will be given by:
r uizis),s)ds + r{W{t),z{r),t)
(22)
Jo
To ensure
that the value in (22)
is
the correct value of utility for
all
possible plans,
we impose
the
following boundary conditions on J'. First, note that the agent might decide not to terminate a
given consumption plan at
be given by
will
(5).
We
in (22) is the utility for
all.
At time T, he would receive V{W{T)) and
require that at T, J*(iy(r),z(T-),r)
and then
"live
given by (22)
The
utility
U
{
time T. Second, note that
his wealth strictly before
time T,
the corresponding utility from such a plan,
= J^uiz€-^^'-^\s)ds +
we impose the boundary condition
V{0).
following theorem shows that the agent prefers receiving J'{W{0),z{0~),0) to the
gained from any consumption plan terminated at ant time
Theorem
J?
all
till
thus the value
on" the satisfaction derived from past consumption. To ensure that the value
is
that J'{0,z,t)
= V(M'(r)), and
a consumption plan followed completely
the agent might follow a consumption plan that exhausts
his total satisfaction
1
Assume
— oo}, concave
that there exists a differentiable function
in
W
and
z,
t
6 (0,T].
J'{W,z,t):^+ X
5f+ X [0,T]
—
whose partial derivatives are continuous functions of time,
that satisfies the following differential equation:
m^{u{z,t)-\-Jw^Vrit)-j;Pz + J^,pj;-j;v} =
together with the boundary conditions:
J'{W,z,T)
=
V{W)
and
V<e[0,T],
(23)
13
=
J'{0,z,t)
/
u{ze-^^'-*\s)ds
+
V{0),
then
''
uiz(s),s)ds
+
yC
J'{W{t),z{r),t) < r(WiO),z{0-),0)
£ AiW{0),0),
j:
where {z{s);0
^
s
<
t
and W{t) are
}
respectively, associated with
the average past
consumption and time
t,
and
let
the points of discontinuity of
C
of notation,
we
Recall that
W
is
C
£ A(W{0),0) that terminates
C
is
{H^(5);0
left-continuous and z
is
<
t)
and
right-continuous.
process and
{2(5);
<
z(s)
+
J'{W{t),z{t-),t)
u{z{s),s)ds
+
J'(M'(0),z(0-),0)
+ E.=o f,r
{j'w{s)rWis)ds - Jty:s)dCis)
+
We
J:{s)l3{dC{s)
<
For convenience
at times denote J''{W{t),z{t~),t), J^{W{t),z(i-),t), J^{W{t),z{t-),t),
f^uizis),s)ds
/o'
<
s
J^{W(t),z{t-),t) by J*(0, Jwit)^ Jzit)i and j;(t), respectively.
=
The wealth
on (0,0 be ti,T2,
the average past consumption associated with
t}, respectively.
wealth,
C.
Proof. Assume that the agent adopts a consumption plan
at
t
and
have
-
zis)ds)
+
J:{s)ds)
(24)
where the equality follows from fundamental theorem of calculus and we have
Note that the boundary conditions
for J* ensure that the left-hand expression
is
set tq
—
0.
the correct
expression of utility even for consumption plans that do not terminate before T, or those plans
for
which the wealth reaches zero
By
the hypothesis that J'
is
strictly before T.
concave
two arguments, we know that
in its first
<
Jwir^Wirt) -
=
Jw(r,){Cir-) - C(r.))
W{T,))
+
+
J;(r.)(z(r.)
j;(r.)/?(C(r.)
- z(r-))
-
C(7;-)).
Substituting this into (24) gives
fu{zis),s)ds
+
J'{W{t),z{t-\t)
Jo
<
J-(H^(0),2(0-),0)
+
f\u{z{s),s)
Jo
+
j;yis)Wis)r{s)
- J:{s)0z{s) +
J:{s)]ds
,
14
4
+
r [/?j;(s) -
Sufficiency
Jw{s)]dCis)
Jo-
<
J-(Vy(0),z(0-),0),
where the second inequality follows from the hypothesis that the integrands are negative and
C
is
increasing.
Now,
if
I
we can show
that there exists an actual policy {C'',W'(T)) 6
A{W{0),0) such that
U{ziO'),0;{C-,W'iT))) = J'{WiO),ziO-),0),
then {C',W'(T))
In fact,
is
the optimal policy.
we will prove more than
C,
of a budget feasible policy
that. In the following theorem,
starting from
guarantees that the associated value function
theorem
U
{
is
t
and any state W(t),z{t~), which
equal to the function J'{W,z,t) defined in
1.
Theorem
3?
any time
we provide a characterization
2 Suppose that there exists a differentiable function J'{W,z,t):^+ X 5?+ X [0,T]
— oo}, concave
in
W
and
z,
—
whose partial derivatives are continuous functions of time,
which solves the differential equation:
Tnax[u{z,
t)
+ JwWr{t) -
J^jSz
+
j;, fij;
- J^] =
Vi G [O.T]
(25)
together with the boundary conditions:
=
J'iW,z,T)
V{W)
and
rT
=
J'(0,z,t)
Starting from any time t 6 [0,T], with
C
6 A{W(t),t), which
is
I
u(2e-^('-'\s)d5.
W{t) and
z{t~), assume that there exists a policy
continuous on {t,T], with a possible
associated state variables are W'(t),z'{t~), such that for all
t
jump AC'{t), and whose
G {t,T]:
j;
=
and
[j;y{W',z',t)-f3j;{W',z',t)]dC'{i)
=
for
u{z',t)
+ j;vW'r{t)-j;Pz' +
(26)
all
e>0,
(27)
i:
ft
and
r(W{T),z{T-),T) = J'iWiT) - AC'{T),zir-) + 0AC'iT),T)
then U{z{t-),t-{C',W'{T)))
=
J'{W{t),z{t-),t) for
In particular, U{z{0-),0;{C',W'{T)))
that
C
attains the
maximum
all
,
(28)
{W{t),zit-),t) e §?+ x Sf+ x [0,T].
= J'{W{0),z{0-),0).
It
then follows from theorem
of U{z{0-),0;{C,W{T))) over the set A{W{0),0).
1
15
Recall from the definition of U{z{t-),t;{C' ,W'{T))) that for r
Proof.
<
t
<
s
< T, we
have:
Uiz'ir),t;iC',W'{T))) =
Now
J'
u{z-iO,Od^+U{z'{s-),s;{C',W'{T))).
(29)
consider the following:
=
J'
=
u{z-{0,0 d^ + J'iW'it'' ), ^'(0,0 + /' (Jw
J'{W{t^),z'it),t)
dw + j; dz- + J^ di)
+ I'{uiz'i0,0 + Jw^"{OriO - j:p^'io +
Jl) d^
+ J\(3J:{W',z\0- Jw{W',z',0)dC'{0
=
J'{W'it+),z'{t)J)
=
J'{W'{t),z'{t-),t),
where the
first
(30)
equality follows from the fundamental theorem of calculus.
foUows from the dynamics of
W
and
z'
and the assumption that
and the third equality follows from equations (26) and
continuity of
C
for all
From equations
t
G (TyT], and from condition
(29) and (30),
we conclude that
(27).
(28),
The
C
last equality follows
t
=
t.
<
<
5
< T,
t
equality
continuous on {t,T],
is
when
for all r
The second
from the
U{z'{r),t;iC',W'{T)))-U{z'{s-),s;{C',W'{T)))
=
J'iW'{tlz\t-),t)-J'iW'{s),z'{s-),s)
m
U(z'it-),t;{C',W'{T)))- J'iW'{t),z'{t-),t) =
where
m
is
a constant independent of
have used the convention that
if
the
t.
Now,
minimum
let
T =
or
for all
te
[t,T],
min{t G [0,T]:Vy*(<)
does not exist, we set the
=
minimum
0},
where we
to be T. Note
from the boundary condition on J' that: J'{W'{t),z'{t-),t) = U{z-{t-),f;{C',W'{T))).
Thus we conclude that
5
We
m
=
0,
and the proof
is
now complete.
I
Optimal Consumption Policy
provide a closed form solution for the optimal consumption problem formulated in section 2
with infinite horizon in a world of constant interest rate
r.
The
felicity
function
is
u{z,t)
=
16
Optima] Consumption Policy
5
-e~^^z° where q <
,
1,
and where the discount factor
^
>
captures the agent's impatience. In
other words, the agent seeks to majcimize
r -e-^'z{tydi,
Q
l/(r(0-),0,C)=
(31)
Jo
given
W(0) and 2(0") and where
problem, which
proof
is
is
z{t)
The
given by equation (3).
is
a modified version of theorem
sufficiency
theorem
for this
given in the following corollary, whose
2, is
omitted.
Corollary
Suppose that there
1
J*(H', z):3?+ X Jf+
—
5f
exists a concave, continuously differentiahle function
U { — oo}, which
max{— +
solves the differential equation:
J^yWr - J'Jz -
6J', pj;
- J^] =
(32)
,
together with the boundary conditions:
roo
J'(0,z)=
1
e-^'-{ze-^^'-'^)°ds
/
and
(33)
Q
Ji
lim e-^'^J'iWiT), ziT))
=
(34)
,
TToc
W{t) and
z{t~), assume that
continuous, with a possible
jump AC'{t), and
for all feasible policies. Starting from any time t £ [0,oo), with
there exists a policy
C € A{W{t),t),
which
is
whose associated state variables are W'{t),z'{t~), such
— + JwWr
t
G {t,oo):
- j;0z* -6J'
=
and
\j;y(W',z\t)-PJ;{W',z',t)]dC\t)
=
forall
a
I
that for all
(35)
€>0,
(36)
and
J'{WiT),z{T-)) = J'{W{t) - AC*(r),z(r-)
then U(z{t-),t;C')
=
J'{W{t),z{t-)) for
In particular, U{z{0-),0;C')
the
maximum
We
will
all
(Wit),z{t-)) €
= J'{W{0),z{Q-)).
It
+ /?AC-(r)),
(37)
R+ X ^+.
then follows from theorem
1 that
C
attains
of U{z{0-),0;C) over the set A{W{0),0).
show that the key feature of the solution
average past consumption
z.
then the optimal behavior
increases and z declines
till
is
If
is
a critical ratio
A:*
>
of wealth
li''
to
the agent starts at time zero with ^,^1 strictly less than k',
to invest
the ratio
^
all
the wealth in the riskless asset and wait while
reaches k'
.
From then
W
on, the agent consumes at the
.
17
^
rate which keeps the ratio
^(Q-\ strictly greater
reducing
equal to k' forever.
If,
on the other hand, the agent starts with
than k', then the optimal behavior
W and increasing
to bring
z,
^
to take a "gulp" of consumption,
is
immediately to k*
^
consumption occurs at the rate that keeps the ratio
Following this gulp, the optimal
.
equal to k' forever.
The
critical ratio
k' depends on the interest rate r, the impatience of the agent captured by 6, the concavity of
the felicity function captured by q, and the rate of decay of past consumption captured by
Our construction
First,
is
of the critical ratio k' and hence the optimal solution
we examine candidate
the policy of keeping
^
period of no consumption
any k-ratio
policy,
if
^,^1
<
denoted j''{W,z),
k.
We
J'{W,z),
that there
two
steps.
the A:-ratio policy
if ^,^_\
>
k, or after
a
show that the value function associated with
will
/3
J*
—
if
<k
and
z
=
—>k.
if
a unique value k' >
is
0,
in
satisfies:
4, we show
>
A;
equal to k forever, after an initial "gulp"
—a + rWJl^ - PzJ^ -SJ'^ =
Second,
For any
solutions of the k-ratio form.
is
/?.
such that the associated value fu:iction,
satisfies the differential inequality
maxi
— + j;^-rW -
j;i3z
-
- J^
6J', (3j;
=
i
(38)
,
together with the boundary condition that
1
It
then follows from theorem
for the agent's problem.
Assumption
1
We
^00
= -
JiO,z)
1,
a
\
/
e-^'(ze-^')
1
'
dt=
that the ratio policy associated with k'
(39)
is
the optimal solution
record our solution in the following statements.
The parameters of
6
the problem satisfy
> QT
and
(1
—
q)/3
>
^
—
This assumption ensures the existence of a solution to the
out the case when the agent
z°.
,,
is
r
infinite
so patient that he prefers to wait
longest possible time. This behavior leads to infinite utility.
horizon program.
and accumulate wealth
Now
fix
a k-ratio policy.
It
rules
for the
18
Optimal Consumption Policy
5
Lemma
t>
1
=
Suppose that ^,^1
k.
The consumption
rate required to keep
-^^ =
k for
all
is
'» = Bm'^^'^
and
the corresponding utility is
z{o-r
"•^-"/^IStt)
Proof. Using
the dynamics of
W and
z,
and equating -^i^) to zero, we can compute
Another simple computation produces the associated
total utility is strictly positive.
Lemma
k,
2 Suppose that ^,J_
and then consuming
utility.
Note that by assumption
c.
the
1,
I
<
k.
Consider the policy of no consumption
to keep the ratio -^tM
=
for
k,
t
>
t'
.
The
till t'
total utility
when
from
^A'./
=
this policy
IS
zio-r
o{6 + al3)
Proof.
a
The reader can
lemma.
establishes the
Proposition
^^(o-r[H^(o)i^
1
\.kz{0-)i
-
]^s
=
easily verify that t'
aP{^)
^j^J
log
S
(40)
+ a0
r+0
.
A
direct
computation
I
The value function j''(W,z) associated with any
j''{W,z)=
<
a{6+Q0)
+
^
Has
^a [k:w
A
A;-ratio policy is
given by:
w
iff<k
w
l(i±M.YB
where
A =
and
^-"/^(fel)
B =
Furthermore,
j''
is
*
+ "^
^-"/?(^)
continuous, concave, has continuous first derivatives
^ + rWJl^ - PzJ^ -6J''
=
if
Ji^-PJ^ =
if
^<k,
^>k,
and
satisfies
(41)
.
19
together with the boundary condition
—
= —r-^
•^(0,2)
Proof.
w >
Suppose that -^
immediately to k
The
'^^
is
.
W -kz
and we define j''{W,z)
1,
a
size of the initial "gulp" required to bring the ratio
l
assumption
ttz
and the
= J^{W - A,z +
+ 0k
'
/3A). Concavity of J* in
rest of the proposition
can be
ezisily verified
both
W and z follows from
by direct computations.
This proposition shows that any ratio policy has a value function which
satisfies
I
the differ-
domain. The value function of the optimal
ential equations (41) in the relevant parts of the
solution, however, satisfies the differential inequality (38) over the whole domain.
There
is
exactly one value k' that produces the optimcd policy.
Proposition 2 Let
k'
=
^ +—(l-a)
6
and note
is
that k'
>
by assumption
1.
ar
The value function J' associated with
the k' -ratio policy
continuous, concave, has continuous first derivatives and satisfies the differential inequality:
maxJ
— + J^rrW - J'^z - 6J',
pj; -
J^
I
=
,
together with the boundary condition that
J{0,z)
1
=
q(q/?
It
then follows from theorem 1, that the
A;'
+
<5)
-ratio policy is the optimal solution for the agent's
problem.
Remark
1
k'
is
the unique ratio with the property that the associated value function J' is
twice continuously differentiate over the whole
domain {W,z).
Proof.
to prove that
In light of proposition 1,
we only need
z°
—
+ Jw'tW a
J'(3z
-6J' <0
pj; -
Jw <
if
if
W
—>k'
z
W<
—
fc*
and
20
Optimal Consumption Policy
5
To prove the
to Figure 1, let
= {W,z)
any point x
inequality, consider
first
such that
W
be the point on the intersection of the line
fi
passing through the point x with slope
^^ = - 4.
W = k*z, to which one would jump
one starts at
if
In other words,
By
x.
=
g
is
W
>
Referring
k'z.
and the straight
k' z
line
the point on the boundary
construction, J'{x)
=
J'{q)-
Let
f{W,z)=—^J'wTW-J'Jz,
a
and note that /(a) — 6J'{a) =
straight line connecting a
and
Applying the fundamental theorem of caJculus along the
0.
i,
we obtain
fUwdW + f^dz).
/(x)-/(a)=
Ja
Noting that along the
line connecting a
direction from a to i,
it
then follows that
—a + Jw^^ Computing fw - Pfz
and
W
in the region
x,
fw —
<
(ifz
that
dW
>
in the
to conclude that
is sufficient
-Sr <0
J'zP^
>
we have dz = -jSdW, and
—>k'.
if
z
k'z, the reader can easily verify that
fw -
Pfz <
O— QT
To prove the second
We
inequality, consider the function
J^-PJ^
in the region
where
W < k'z.
can write
.k'z
z'^-'gi''^)
,
r^-HJ: =
where
+P
where
A is given
hence g{y)
j
in proposition
00 as y
Computing the
|
1.
Note that 5(1)
>
We
1.
It
r
=
0,
'•+^
y
+
aP +
/3
and that
- ^^^ >
1
S
by assumption
0,
1,
00.
derivative of p,
da
AU
dy
r
->r
we
get
P{8 - ar)
aP) _<±££
+P
"
Substituting k' in the expression for
for y
I
then follows that J^,
\k'
^, we
- PJI >
k'{r
+ P))
conclude that j^
0, for all
y{P
=
+
for
r)
j/
=
1,
points (W^z) such that
have thus shown that the nature of the optimal consumption policy
over the ratio of wealth to average past consumption to keep
it
policy of keeping the ratio at or below a fixed constant satisfies
and that
^
W < k'z.
is
I
exerting control
at or below a critical level.
many
>
Any
of the sufficient conditions
'
.
21
However, the
for optimality.
satisfied
by only one
differential inequality (38),
critical ratio k*
We
.
Proposition 3 The optimal solution
gulp of size
A =
(ir^ffTf^
period of no consumption of length
compare
interesting to
It is
starts with the
same
initial
summarize the solution
consume
is to
at rate c*{t)
- ar)W{0) + (i^ +
[(^
t*
this
wealth
=
which
log
(1
_
r+e
k'z(0-
''
y^,L>
,
the crucial condition,
is
is
in the following proposition.
=
\'_^J
q))^(0-)]
if
,
W(t),
^^
<
after
k'
,
an
initial
or after a
-fWiox > k'
if ^,J_(
consumption policy to that adopted by a consumer who
W{0) and
same past consumption experience ^(0") who
the
seeks to maximize:
/
e-^*c{t)''dt,
Jo
where we assume that
consumption rate
is
The propensity
preferences.
=
c*(t)
to
il^'^
consume
is
=
— a >
1
It
can be easily verified that his optimal
^a (0the
same
for agents
=
An
r.
with time- additive and non-time-additive
A
will
be a constant equal to his
consumer with non-time additive preferences, however,
suming a "gulp" of
size
-
—^-jzP—K
if
differ-
agent with time-additive preferences will consume
rW^{t), and hence his wealth at any time
see figure 2.
0.
However, the quantity of consumption and the trajectory of wealth are
Consider the case when S
ent.
c*
> ar and
6
Ws wealth
initial
wealth,
will either start
by con-
too high relative to z{0~), and then keep
is
the level of his wealth constant forever, or he will wait for
t'
= ;^
log
^^p^
,
if
his wealth is
too low relative to ^(0"), and then start consuming the interest on his wealth forever.
The parameter
0, which controls the rate of "decay" of previous consumption, determines
the size of the initial "gulp" or the length of the
of the subsequent constant wealth.
inital gulp, if
The
initial
waiting period, and hence the level
higher the value of
/3,
the smaller the size of the
the solution calls for a "gulp", and the shorter the
case the solution entails a waiting period. This
is
initial
waiting period, in
an intuitive result since a higher
/3
implies
that past consumption experience has "smaller" effect on current, and hence total, satisfaction.
Therefore,
effect of
reaches
when the optimal
solution entails an initial waiting period to accumulate wealth, the
^(0") decays faster with higher
its critical value. Similarly,
means that the
effect of
/?
leading to shorter waiting period before the ratio
when the optimal
—
solution calls for an initial "gulp", higher
the "gulp" wiU decay faster and hence the "gulp" contributes less to
future satisfaction. Therefore, agents with higher
/?
choose smaller
initial "gulps".
22
References
7
Concluding Remarks
6
In this paper,
we have provided
sufficient conditions for
a consumption policy to be optimal for
a class of time-nonseparable preferences that treat consumptions at nearly adjacent dates to be
almost perfect substitutes.
We
demonstrated our general theory by explicitly solving
form the optimal consumption policy
The
in closed
for a particular felicity function.
heuristically derived necessary conditions for optimality in Section 3 can be justified
We
rigorously.
refer the reader to Blaquiere (1985) for details.
We
also
remark that our
conclusion that the optimal consumption policy does not involve any "gulps" after
depends crucially on the assumption of continuous interest
rates.
If
the interest rate
t
is
=
not
a continuous function of time, a different consumption policy with intermediate gulps might be
optimal.
Finally,
we remark
that the dynamics of
-
we report
Observing that the objective functional
differently.
h': 3?+
that the closed form solution
is
in section 5
can be developed
homogeneous of degree a
in z,
and
W and z are linear, we can write J'{W,z) = z°'h'{^), for some function
K. Substituting this form of J* into inequality
(
32),
we obtain an ordinary
differential
inequality that can be solved and the criticaJ ratio k' can be determined by imposing the
condition that the solution of the differential inequality,
function.
We
/i*, is
a twice continuously differentiable
elected to present our step-by-step construction of the candidate solutions and
the optimal policy to illustrate more clearly the nature of the control problem.
References
7
1.
K. Arrow and M. Kurz, Public Investment, the Rate of Return, and Optimal Fiscal Policy,
Johns Hopkins Press, Baltimore, 1970.
2.
A. Blaquiere, Impulsive Optimal Control with Finite or Infinite Time Horizon, Journal
of Optimization Theory and Applications,
3.
G. Constantinides, Habit Formation:
A
Volume 46, Number
4,
Resolution of the Equity
August 1985.
Premium
Puzzle, un-
published manuscript. University of Chicago, 1988.
4.
L. Epstein,
A
Simple Dynamic General Equilibrium Model, Journal of Economic Theory
41, pp.68-95, 1987.
23
5.
W. Fleming and
R. Rishel, Deterministic and Stochastic Optimal Control, Springer-
Verlag, 1975.
6.
J.
Heaton,
An
Empirical Investigation of Asset Pricing With Temporally Dependent Pref-
erence Specifications, niimeo, Sloan School of Management,
7.
J.
MIT,
April 1990.
Heaton, The Interaction between Time-Nonseparable Preferences and Time Aggrega-
tion, unpublished manuscript, University of Chicago, 1988.
8.
A. Hindy and C. Huang,
On
Time
Intertemporal Preferences in Continuous
of Uncertainty, Working Paper No. 2105-89, Sloan School of
II:
The Case
Management, MIT, March
1989.
9.
C.
Huang and D. Kreps, On Intertemporal
of Certainty, Working Paper No.
Preferences in Continuous
Time
2037-88, Sloan School of Management,
I:
The Case
MIT,
revised
March 1989.
10. T.
Koopmans, Stationary Ordinal
Utility
and Impatience, Econometrica 28, pp. 287-309,
1960.
11. F.
Modigliani and R. Brumberg, Utility Analysis and the Consumption Function:
An
Interpretation of Cross Section Data, in Post Keynesian Economics (K. Kurihara Ed.),
Rutgers University Press, 1954.
12. F.
13.
Ramsey,
A
Mathematical Theory of Saving, Economics Journal 38, pp. 543-559, 1928.
H. Royden, Real Analysis, Macmillan, London, 1968.
14. S.
Sundaresan, Intertemporally Dependent Preferences and the VolatiUty of Consumption
and Wealth, Review of Financia.1 Economics
15.
H. Uzawa,
Time
in Vaiue, Capital
Preferences, the
Consumption Function and Optimum Asset Holdings,
and Growth: Papers
Aldine, Chicago, 1968.
2, pp. 73-89, 1989.
in
Honour of Sir John Hicks,
(J,
N. Wolfe, Ed.),
Restricted Region
R
w
Boundary B
Admissible Region
O
The State Space showing the Boundary
B.
times after t=0, the optimal policy
restricts the state to the admissible region O.
For
all
Figure
1
Wealth
7
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