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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
OPTIMAL CONSUMPTION AND PORTFOLIO RULES
WITH DURABILITY AND LOCAL SUBSTITUTION*
Ayman Hindy and
Chi-fu
Huang
November 1989
Revised November 1991
WP#
3367-92-EFA
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
OPTIMAL CONSUMPTION AND PORTFOLIO RULES
WITH DURABILITY AND LOCAL SUBSTITUTION*
Ayman Hindy and
Chi-fu
Huang
November 1989
Revised November 1991
WP#
3367-92-EFA
OPTIMAL CONSUMPTION AND PORTFOLIO RULES
WITH DURABILITY AND LOCAL SUBSTITUTION*
Ayman Hindy
Graduate School of Business
Stanford University
Stanford,
CA
94305
and
Huang
Sloan School of Management
Chi-fu
Massachusetts Institute of Technology
Cambridge,
MA
02139
November 1989
Revised November 1991
Abstract
We
study a model of optimal consumption and portfolio choice which captures,
ferent interpretations, the notions of local substitution
goods.
The
class of preferences
we consider excludes
all
and
in
two
irreversible purchases of durable
nonlinear time-additive and nearly
the non-time-additive utility functions used in the literature.
dif-
We
all
discuss heuristically necessary
conditions and provide sufficient conditions for a consumption and portfolio policy to be opti-
we demonstrate our general theory by solving in a closed form the optimal
consumption and portfolio policy for a particular felicity function when the prices of the assets
follow a geometric Brownian motion process. The optimal consumption policy in our solution
mal. Furthermore,
consists of a possible initial "gulp" of consumption, or a period of no consumption, followed
by a process of accumulated consumption with singular sample paths. In almost
all
states of
nature, the agent consumes periodically and invests
more in the risky assets than an agent with
time-additive utility whose felicity function has the same curvature and the same time-discount
parameter. We compute the equilibrium risk premium in a representative investor economy
with a single physical production technology whose rate of return follows a Brownian motion.
In addition, we provide some simulation results that demonstrate the properties of the purchase
series for
durable goods with different half-lives.
*We are grateful to Hua He for pointing out an error in an earlier version, and to Darrell Duffie, John Heaton,
Jean-Luc Vila Thaleia Zariphopoulou and anonymous referees for helpful comments. We acknowledge financial
support from National Science Foundation under grants NSF-SES 9022937 and NSF SES-9022939
,
We
Summary
Introduction and
1
study the problem of optimal consumption and portfolio choice for an agent whose prefer-
ences over consumption patterns are given by:
E [^
u{z{t)J)dt ^V{W{T),z{T))\
(1)
,
/o
where
z{t)
and where
C( t
)
z{i)
is
+
r(0-)e-'^'
/?
e-^C-) dC{s)
/'
(2)
,
the process of average past consumption derived from the consumption process
which denotes the
factor, z{Q~)
=
>
total
amount
a constant, and
is
of consumption
W{T)
is
the
till
time
t.
The parameter
random wealth
are continuous and increcising functions. Furthermore,
V
is
left at
a weighting
/J is
time T. Both u and
concave and u
is
concave
V
in its first
argument.
Two interesting
specified in (1)
economic notions are captured
and
In
(2).
local substitution, that
is
in
two
different interpretations of the
model
one interpretation, preferences given by (1) embody the idea of
consumptions at nearby dates are almost perfect substitutes.
In a
second interpretation, the model represents preferences over the service flows from irreversible
purchases of a durable good that decays over time.
Local substitution in continuous time, studied by Hindy,
and Huang (1991),
is
the notion that consumption at a point in time depresses marginal utility
of consumption at nearby times.
phenomenon that
Huang and Kreps (1991) and Hindy
Suppression of appetite following a large meal
leads to periodic consumption.
Local substitution
which are widely used
is
consuming
in the finance
Epstein and Zin (1989),
who study
tinides (1990),
on satisfaction.
at adjacent dates as very similar alternatives.
and economics
utilities; see, for
Uzawa
little effect
absent in the nonlinear time-additive representations of preferences
used non-time-additive
(1960) and
a natural
In addition, local substitution implies that
delaying or advancing consumption for a short period of time has
In other words, agents regard
is
literature.
Many
researchers have recently
example, Epstein (1987), Duffie and Epstein (1989),
generalizations of the utility form introduced by
Koopmans
(1968) in the context of continuous time equilibrium models, and Constan-
Detemple and Zapatero (1990), Heaton (1990), and Sundaresan (1989) who
study habit formation models.
The
utility representation in (1) differs
preferences literature in one aspect which
is
from most specifications
in the
non-time-additive
very crucial in capturing the notion of local sub-
2
1
stitution. In
Summary
most of the currently used non-time-additive models, the index of instantaneous
satisfaction, or felicity, at
Hindy,
Introduction and
any time depends explicitly on the consumption rate
Huang and Kreps (1991) and Hindy and Huang (1991) show
that
at that time.
when the
felicity
function depends explicitly on current consumption rates, the resulting preferences do not exhibit local substitution. In essence, the marginal utility for
consumption
at
any moment
is
too
consumption rate and hence the agent would not tolerate delaying or
sensitive to the current
advancing consumption even for a very small period of time. The
other hand, has the key feature that the
utility function in (1),
on the
function at any time depends only upon an
felicity
exponentially weighted average of past consumption - one derives satisfaction only from past
consumption. This feature
is
and those studied by other researchers, except
In the durable
good
till
market
time
for the
t.
we study
the main difference between the utility function
for
here
Heaton (1989, 1990).
good version of the model, we interpret C{t) as the
total purchase of a durable
Purchases of the good are irreversible either because there
is
no secondary
good, or because of prohibitively high selling costs. The good provides a flow of
services z{t) at time
Absent any new purchjises, the service flow declines over time because
t.
Discrete time versions of this model
of the deterioration in the stock of the durable good.
have been studied by
Dunn and
Singleton (1986), Eichenbaum, Hansen and Singleton (1988)
and Hotz, Kydland and Sedlacek (1988), among others. The model we study here can
be viewed as a limiting case of the model of Grossman and Laroque (1990)
purchasing cost and infinite selling cost.
We
will
if
we
also
specify zero
henceforth use the terms "consumption" and
"durable purchase" interchangeably.
The
specification of utility in (1) leads to an economically interesting
consumption behavior
and also to a technically chedlenging optimization problem. The optimal consumption behavior
for the agent with preferences given
as
most models predict but only
from past consumption.
When
by
(1)
is
periodic.
periodically.
The agent does not consume
Most of the time the agent derives
the marginal value of average past consumption,
all
the time
satisfaction
z, is
higher
than the marginal value of wealth, a situation that arises only periodically, the agent consumes.
Otherwise, the agent refrains from consumption. In the durable good interpretation, the agent's
purchases are periodic.
Between purchases, the agent
is
content to use the service flow from
the existing stock of the durable.
From the
technical point of view, the standard
dynamic programming equations
of the
Bellman-Jacobi type do not apply since a heuristic derivation of these equations would suggest
a bang-bang control policy, but there
is
no natural upper bound on consumption.
We
follow a
different
We
approach to establish optimality.
provide necessary and sufficient conditions for a consumption-portfolio policy to be
We
optimal.
show that
it
is
necessary and sufficient that the value function, or the
maximum
attainable utility starting from any state, satisfies a differential inequality which can be viewed
as
an application of the Bellman optimality principle. In particular, the conditions require that
at all times the
marginal value of wealth
is
at least as large as
times) the marginal vaJue
(/?
Furthermore, consumption should only occur when these
of the average past consumption.
marginal values are equal.
We also construct
class of felicity
a closed form solution of the optimal consumption problem for a particular
and bequeath functions
a geometric Brownian motion.
than a
critical
consumption
number, say
is
The
in the case
—
ratio
of the return on the risky assets.
is
subject to
it
=
k'.
to keep the ratio
—
less
This requirement implies that
The quantity
bounces back and forth to
random shocks because of the randomness
Furthermore, the sample path properties of the Brownian
motion lead to peculiar behavior of the quantity
k'.
—
is
Because of uncertainty about asset returns, however, an interesting
periodic.
number
prices of the risky assets follow
idea of the optimcil solution
and consume only when
k',
phenomenon appears. The
the critical
when the
hit it
^
^
during the times when
fluctuates so fast that
whenever
it
it is
very close to
hits the value
fc*
again uncountably infinite number of times. This results in
a process of optimal cumulative consumption with nontrivial increasing sample paths in which
consumption occurs only periodically.
We
provide a complete solution for the optimal consumption and portfolio rules in this
example and show how the solution changes with changes
In particular,
utility
whose
We show
in
is
we compare the behavior
felicity
function has the
in the
parameters of the problem.
of our agent to that of an agent with a time-additive
same curvature and the same time-discount parameter.
that the agent with local substitution preferences behaves in a less risk averse manner,
the sense that he invests higher proportions of his wealth in the risky assets. This behavior
reinforced by lower riskless rates, lower time-discount rate
(lower
and higher
effects of substitution
/?).
It is
worthwhile to mention that one obtains periodic consumption behavior
models when the marginal
felicity
of consumption evaluated at zero
is finite.
agent with such preferences will refrain from consumption once his wealth
possibly time varying, level; see
consumption
is
periodic, even
Cox and Huang
when the
felicity
in
time additive
In such a case,
falls
below a
an
critical,
(1989). In contrast, in our analysis, the optimal
function has infinite slope at zero. In addition.
4
in
Introduction and
1
choosing consumption, the agent considers both current wealth and past consumption expe-
rience.
Local substitution and durability produce important effects of past consumption and
purchases on current
We
utility.
study the implication of the agent's behavior on the equilibrium in a^set markets and
on the determination of the
We
premium.
risk
show that the agent's
two-fund separation and, hence, the Capital Asset Pricing Model holds.
Consumption Capital Asset Pricing Model
is
Summary
no longer connected to the marginal
fails
On
the other hand, the
to hold since the marginal utility of
utility of wealth.
Bergman (1985) and Grossman and Laroque
portfolio choice implies
(1990).
We
consumption
Similar observations were
also
compute the
risk
made by
premium
in
equilibrium in the presence of a production technology with constant stochastic returns to scale.
We
show that the equilibrium
risk
premium
will
be lower in the case of local substitution, or
durability, than in the case of time-additive utility. This result
is
consistent with the less risk
averse behavior of the agents with local substitution or the case of durable goods.
We
also study the properties of
We
measured consumption and purchases over some fixed pe-
construct temporal aggregates of purchases for a consumer
who
faces investment opportunities similar to the historical record of the U.S. stock market.
For
riod using simulation.
durable goods with different half-lives, we analyze the autocorrelations of the changes
in
mea-
consumption
We find that all autocorrelations are negative reflecting the durability of the
good. We also find that, for a fixed half-life, the longer the measurement period is,
the smaller
the absolute value of the autocorrelation. This results from temporal aggregation.
sured purchases.
is
In addition,
we
good, the higher
find that, for
is
any fixed measurement period, the longer
is
to purchase small quantities frequently in contrast with the
optimal policy for long-lived durables which
calls for
also suggests that the time-series of purchases of
The
for a
the half-life of the
the absolute value of autocorrelation. This reflects the fact that the optimal
policy for short-lived durables
than that
is
good with long
half-life.
sporadic and large purchases. This finding
good with short
The simulation
results
half-life will
be "smoother"
we report confirm
this intuition.
analysis of this paper suggests that local substitution, or durability, alone lead to
lower risk
premium on
equity.
Many
shown that habit formation leads
researchers,
to higher
most notably Constantinides (1990), have
nsk premium on
equity. Habit formation, which
is
complementarity over longer periods, can be combined with local substitution, or durability, to
produce interesting and
of
rich
behavior and asset price dynamics.
The
recent empirical evidence
Eichenbaum and Hansen (1990), Gallant and Tauchen (1989), and Heaton (1989, 1990)
suggest that there
is
substitution over short periods.
In particular,
Heaton (1990) showed.
after correcting for the
problem of temporal aggregation, that habit formation aJone does not
provide significant explanation power for asset pricing over the time-additive models; while
local substitution, or durability, does. In addition, given local substitution, or durability, habit
formation over long horizons becomes more significant in
its
explanatory power. These results
imply that a model which combines local substitution and habit formation
In fact, this
The
good.
its
is
variables.
purchases of durable goods and relate
series of
More work needs
to be done to achieve this objective.
Caballero (1991), and the references therein, for more on this important
refer the reader to
topic.
focused on the decision of an Individual to purchase a durable
important to construct an aggregate
dynamics to macroeconomic
We
worth pursuing.
the next step in this research program.
is
analysis of this paper
It is
is
The contribution
of this analysis to the macroeconomics literature on durable goods
is
providing a closed form solution to the problem of an individual. This can serve as a concrete
basis for further aggregation results.
The
rest of this
paper
is
problem under uncertainty
organized as follows. Section 2 sets up the consumption and portfolio
continuous time. Section 3 provides a heuristic derivation of the
in
necessary conditions for optimality, whereas Section 4 shows that these, together with some
other regularity conditions, are sufficient.
infinite horizon
in closed
form
program. In Section
for
a particular
6,
felicity
of geometric Brownian motion.
we
Section 5 provides a verification theorem for an
solve the optimal
when the
function
Section 7 provides
consumption and portfolio problem
prices of risky cissets follow a process
some comparative
implications and section 8 presents computations of the risk premium.
measured, temporally aggregated, purchases appear
statics
and equilibrium
The simulation
in section 9. Section 10 contains
results of
concluding
remarks.
2
Formulation
Consider an agent
is
who
lives
from time
t
—
to
= T
t
in
a.
world of uncertainty where there
and T. The agent has the
a single good available for consumption at any time between
opportunity to invest
in
a frictionless securities market with
tinuously traded, and indexed by n
pays dividends at rate Pn{t), and
as pniS(t),t),
=
0, 1,2,.
.
.
,
A'^.
sells for 5„(<), at
TV
+
1
Security n, where n
time
t.
We
denotes transpose.
=
1,2,
.
., A'',
.
is
risky,
assume that p„(<) can be written
where we have used S{t) to denote the column vector
'The superscript
long lived securities, con-
[5i(t), 52(<)i
•
•
•>'S'n(0]
'•
6
Formulation
2
Security
at
time
is
where r{S,t)
t,
pay dividend, and
locally riskless, does not
sells for
B{t)
=
exp{/o r(5(5),s)<i5}
the instantaneous riskless interest rate at time
is
and
t
r(-,-) is Borel
measurable.
We
fi.
will use the following notation: If
In addition,
The
if ct is
a matrix,
a vector in S?^,
/i is
denote
let \a\^
tr(crcr^),
where
let |^|
tr is
be the Euclidean norm of
the trace of a square matrix.
price process'^ for the risky securities follows a diffusion process given by:
S{t)+ f p{S{s),s)ds^ S{0)+ f ti{S{s),s)ds+ f (T{S{s),s)dB{s)
Jq
where
B
an M-dimensional,
is
M
> N, standard Brownian motion
probability space {Cl,J^,P) and the notation
probability one.
Assume
that for each integer
we
of notation
will
m
0, there exists
sometimes use
a.s.,
(3)
defined on a complete
denotes statements which are true with
a.s.
that this diffusion process
>
Vi G [0,T]
Jo
Jo
is
strictly positive
a constant c^ so that
and
n{t), p{t), <t(<),
with probability one and
<
£'[|5(t)|^'"]
r(t) to
For brevity
e'^'"'."'
denote fi{S{t),t), p{S{t),t),
a{S(t),t) and r(S{t),t), respectively.
We
assume that the agent has only access to the information contained
We
prices of the risky securities.
where
!Fi
is
assume that
!Ft
contains
the probability zero sets of J^, or
all
All processes to be discussed will be adapted to
The agent can consume
almost
Let
all
is
F
s
<
t}
Is
complete.
is
also refrain
from consumption altogether
t
for
at
some
can have a singular
a continuous nontrivial increasing function whose derivative
is
zero for
t.
X+
continuous.
be the space of
all
processes x whose sample paths are positive, increasing and right-
Recall that an increasing function x{u,.) has a finite left-limit at any
denoted by x{u),t~).
for the
<
moment, and can consume
time. Moreover, the sample path of cumulative consumption at any time
component, that
€ [0,r]},
{!Ft;t
F.''
the single good at "gulps" at any
The agent can
over intervals.
finite rates
F =
the smallest sub-sigma-field of /" with respect to which {5(5);
We
measurable.
denote this information structure by
in the historical
We
will use the
convention that i(w,0~)
sample paths of any i G X+, a jump of
i(a;,.) at r is
=
€ (0,T]
a.s. Since left limits exist
Ax{u,t) =
—
Y
t
x{uj,t)
T®
-
x{u!,t~).
B{[0,T]), the product
^A process
is a mapping Y:U x [0,7^
S? that is measurable with respect to
sample
9? is a
sigma-field generated by /" and the Borel sigma-fieid of [0,T]. For each ui e Q,Y(u,\ .).[0,T]
path and for each
(
€
[0,
T], y(.,
(): fJ
—
5f is
a
random
''The latter can be ensured by a growth condition on
*The process
Y
is
said to be adapted to
F
if for
—
variable.
(/x
each
information constraint-, the value of the process at time
t
—
t
p)
€
and on
cr;
see
Friedman (1975, theorem 5.2.3).
This is a natural
[0,T], Y(t) is .Ft-measurable.
cannot depend on information yet to be revealed.
The
C
stochastic process
X+
€
is
a consumption pattern available to the agent with C(u;,t)
denoting the cumulative consumption from time
points of discontinuity of C(w,<) are the
to time
moments when
For any
in state u.
t
€
a;
fi,
the
the agent consumes a "gulp". Moreover,
C{^,t) has an absolutely continuous component over the intervals during which the agent
consuming
at rates C(u;,t),
Finally, C(u>,<)
may have a
where C'(u;,<) denotes the consumption rate at time
t
in state
is
u>.
singular part. Singular components of consumption processes will
play an important role in our analysis of the optimal consumption policy.
An investment
strategy
is
an TV-dimensional process
A —
{A{t)
=
{Ai{t),.
.,AN{t}};t G
.
[0,T]}, where Anit) denotes the proportion of wealth invested in the n-th risky security at time
t
before consumption and trading.
where
1 is
strategy
A
a vector of
I's.
The proportion
A consumption
C
plan
invested in the riskless security
€
X+
is
is 1
— j4(()^l,
said to be financed by an investment
if
\V{t)
= W{0) +
+
where W{t)
is
[W{s)r{s)
/J
+
W{s)A^(s)Is-^{s){fiis)
J^\Vis)A'^{s)Is-i{s)(r{s)dB(s)
the wealth at time
t
V<G[0,r]
-
r{s)S{s))) ds
- C(r)
a.s.,
before consumption and /5-i(0
is
an
A'^
x
TV
diagonal matrix
with the n-th element on the diagonal being equal to S(t)~^. Note that the wealth process has
left-continuous sample paths and that W(<'*")
The agent
given by (1).
= W{t) —
AC(t).
derives satisfaction from past consumption and from final wealth and utility
The
felicity
function at time
t
is
defined over an exponentially weighted average
is
of past consumption
r{t)
=
z{0-)e-'^'
+
(3
e-^^'-'UC{s)
f
a.s.,
(5)
Jo-
where ^(0~) >
by path
is
a constant,
/? is
a weighting factor, and the integral of (5)
in the Lebesgue-Stieltjes sense.
to account for the possible
jumps whenever
higher values of
C
/3
jump
of
C
at
Note that the lower limit of the integral
<
=
0,
and that
does. Moreover, z has a singular
z
is
has been shown by Hindy and
defined path
in (5)
is
0~,
a right continuous process which
component whenever
imply higher emphasis on the recent pa^t and
in the distant past. It
is
less
Huang (1991)
C
does. Observe that
emphasis on consumption
that this kind of preferences
views consumption at nearby dates as close substitutes: delaying or advancing consumption
for
a very small period of time, without violating the condition that consumption at
only on information available up to
t,
t
depends
has a very small effect on the agent's total satisfaction.
8
Formulation
2
A
consumption plan
missible
(4)
if
is
C
A
and the investment strategy
and
well-defined^,
m
for all integers
>
that finances
0,
it
there exists
are said to be ad-
km
so that, for
all
t,
(6)
<
E[\W{t)\'^"']
Denote by C and
A
€<""'.
the space of admissible consumption plans and trading strategies, respec-
tively.
Formally, the agent manages wealth dynamically to solve the following program:
supegc
s.t.
E
C
is
a.nd
where
Wq
is
«(^(0> t)dt
[lo
financed by
+ V(WiT+), z{T))\
A 6 ^ with W{0) = Wq,
AC {t)>0
Wit) -
the initial wealth of the agent, u{-,t)
is
(7)
Wt e [0,T],
the felicity function at time
t,
V
and
is
the
function for final wealth and terminal stock of the durable good. Note that the second
felicity
constraint of (7)
is
a positive wealth constraint - wealth after consumption at any time must
be positive.
One
pattern
objective of this paper
C
is
€ C financed by some
to derive necessary
A
£
A
and
to he optimal.
To achieve
the utility maximization problem of the agent at time
exponentially weighted past consumption
is
sufficient conditions for a
t
>
we
this end,
consumption
will also consider
given that wealth
is
W
and the
both before consumption at that time.
z,
We
define^
J{W,z,S,t) = supcec,
+ V{W{T+),z{T))]
C is financed hy A £ At with W{t) = W,
V5 € [t, T],
and W(s) - AC{s) >
Et [f^u{zis),s)ds
s.t.
where
from
t
t
Ct
and At are the spaces of admissible consumption plans and trading strategies starting
with the convention that Ct-
replaced by
5, for all 5
G
[t,T],
z(s)
'For this we
mean both
=
for all
C
€
Ct,
the admissibility
is
W
defined by (6) with
and
=
2e-^('-''
+
/3
r e-^^''-''>dC{v).
the Lebesgue integral and the Ito integral are well-defined.
control depending on (W{i), z{t~),S(t),t) and C(t) depends on the history of {VV, 5),
solution
(8)
to the stochastic differential equation (4).
E, denotes expectation conditional on F,.
When
A{t)
is
a feedback
we mean there
exists a
Note that J{W{t),z{t~),S{t),t)
is
the agent's optimal expected
consumption purchases at that time and J{W{t'^),
utility after the
Note that
if
consumption purchases
C
at time
z(t), S{t),t) is
we
Often,
t.
a solution to (7) financed by A, then
is
solution to (8) financed by
A
life
time
=
time
the optimal expected
will refer to
d
utility at
J
t
before
life
time
as the value function.
{C{s) - C{t~);s 6 [t,T]}
is
a
restricted to [t,T].
Heuristic Derivation of Necessary Conditions
3
We
will use
that
it
Bellman's optimality principle to derive necessary conditions about J assuming
has certain smoothness properties.
usual Bellman equation in
One
difficulty that arises in this
dynamic programming
plans are purely at rates and are feedback controls.
and may have singular parts. Moreover,
in gulps
components,
it
cannot be expressed
in
if
is
that the
when the admissible consumption
derived
is
endeavor
Here, cumulative consumption can be
the cumulative consumption has singular
feedback form. Thus, our derivation of the necessary
conditions will be heuristic in nature.
First,
z{t~
=
)
we observe that
if
a consumption gulp of size
is
z, 5,
= J{W - A,
2
+
/?A, S,
t).
(9)
so because both quantities are equal to
u{z{s), s)ds
E,[J^
where {z{s);s €
size of the
[i,
jT]}
Jw
+ V(W{T+),
z(T))],
and W{T'^) are defined along the optimal path on
[t,T].
Moreover, the
gulp should be chosen to maximize J. Thus, we must have
Jw{W where
W{t) = W,
and S(t) = S, we must have
2,
JiW,
This
A is prescribed at timet when
and J^ denote the
A,
z
first
+
/3A, 5,
t)
= I3J,{W - A,
partial derivatives of
2
+
/3A, 5,
(10)
<),
J with respect to
its first
and second
arguments, respectively.
We now
show that
(9)
and (10) imply that
where a gulp of consumption
that
A
is
is
=
W and
z defined
jSJ^ at
[-Jw{W - A,z +
JwiW
any {\\\z,S,t]
W while noticing
through (10) gives
l3A,S,t)
+ 0J,{W - A,z +
+JwiW-A,z+f3A,S,t)
=
be equal to
prescribed. Differentiating (9) with respect to
an implicit function of
JwiW,z,S,t)
Jw must
- A,z + l3A,S,t),
f3A,S,t)]
—
10
3
Heuristic Derivation of Necessary Conditions
where we have used (10) to get the second equality. Similarly, we have Jz{^,
A,z +
/3A, S,
z, S, t)
— J^iW —
Given (10), we then have
t).
JwiW,z,S,t) = (iJ,{W,z,S,t)
at
(W,z,S,t) where a consumption gulp
Second, assume that J
any time
<T,
t
lemma imply
=
to apply.^ For
programming and the generalized
Ito's
+ Jl^^[V^J{s) + J,{s)\ ds
- Jw{s)]dC{s) + EiiJirt) - Jir.)] - E:[Jw{t,)^W{t,)
maxdc,^ Et [//+^ u{z{s),s)ds
dC
on
generalized Ito
[t,t
+
(12)
a.s.,
denotes the consumption plan on the time interval
prescribed by
The
lemma
that
+Jj(r,)Aa(r,)]]
dC
for the generalized Ito's
the principle of optimality in dynamic
+ j;+^[0J,{s)
where
prescribed.
is
smooth
sufficiently
is
(11)
A), J{s) and
lemma we
will use
[t, t
+
A),
r, is
jump
the i-th
point
derivatives are evaluated at (H^(5),z(5~), 5(5),5),
its
throughout can be found
Krylov (1980, theorem 2.10.1) and
in
Dellacherie and Meyer (1982, VIII. 27): Fix a consumption policy financed by a trading strategy and define the
proportion invested in risky securities A(t) accordingly. Let S{t),W(t),z{t) follow the dynamics in (3), (4) and
SR be once continuously differentiable on 9?" + ^ x [0,7^ in
(5), respectively. Let f{W,z,S,t):di'^'^^ x [0, T]
—
all
of its
arguments and twice continuously differentiable on
5?''+^
I <
A^
x [0,r]:<)i,(W,2,5,
+
M.
smooth manifold
possibly on a
3, is
t)
=
0,
/
(A smooth manifold
=
=
"''^
x [0,7^
in S?""''^
in its first
x ["i^
all its
arguments.) Then, for
/(W(0),z(0-),5(0),0)+
/
all
N+
[0,
T]
—
JJ
optional times
[p^/(3)
2
arguments, except
=
given by: A<
's
ij, where (t>i(W,z,S,t):U'^*'^ x
1,2
continuously differentiable in
f(W{t^),z(t),S{t),t)
M
3?
+
,
/
<
=
<
1,
<
2
(W,z,S,t) £
I, with
T, we have:
/,(^)]ds
Jo
+ /
Jo
[fw{s)W{s)A{s)'^ Is-^{s)c{s)
+ fs{sf a{s)]dB[s)
+ / iPMs)-fw{s))dC{s)
Jo-
+ '^Af(T,) -J2[fw(r,)AW{T,) +
t
where
ri, T2,
.,
are the
jump
(W(t),z{t~),S(t),t), where
points of
V* f
is
C
on
f,(T,)^z(T.)]a.s.,
I
[0, t],
where
all
the partial derivatives are evaluated at the points
/ associated with the trading strategy and is
the differential generator of
given by:
D-^/
=
fw[Wr + WA''ls->{ti-rS)]-f.0z + fJin-p)
+2fws(W<Ta^ Is-,A) +
and where A/(r,)
=
tr{fss<Ta''
)]
/(iy(r.+ ),2(r,),5(r.),r.)-/(H'(r.),2(r.-),5(r.),r,).
.
12
SufRciencv
4
\imJ(W,z,S,t)
=
\imJiW,z,S,t)
WIO
=
V{W,z),
(15)
rT
The
first
boundary condition
involve no gulps at time
T
u(ze-^^'-'\s)ds
/
+
V{0,ze-'^^^-'^).
(16)
Jt
implied by the fact that any optimal consumption plan must
is
since such a gulp cannot contribute to the agent's satisfaction from
increases in z and, furthermore,
reduces the terminal wealth.
it
The second condition
is
implied
by the constraint that wealth at any time cannot become negative. Thus, whenever wealth
zero, the only fe^isible policy afterwards
4
is
is
no consumption.
Sufficiency
we provide a
In this section
two
steps.
First,
theorem
verification
we show that
if
for
optimal controls.
there exists a solution
with the two boundary conditions (15) and (16), and J
then J{W,z,S,t)
> J{W,z,S,t)
for all
{W,z,S,t).
J
We
will
proceed with
to the differential inequality (14)
satisfies
some
regularity conditions,
Second, we then give conditions so that
attained by a candidate feasible investment and consumption policy.
J{W,z,S,t)
is
follows that
J{W,z,S,t) = J{W,z,S,t) and the candidate investment and consumption policy
is
It
then
the optimal policy.
Proposition
in its first
1
Let there be a differentiable Junction J{Y,t):{Q,oo)^'^^ x [0,T)
two arguments, which
continuously differentiable in
is
continuously differentiable in
its first
N+2
A'2
and
(16).
In addition,
let
of
its
>
3i,
concave
arguments, and twice
arguments, except possibly on a smooth manifold
satisfying the differential inequality (14) in the interior of
(15)
all
—
its
M,^
domain with boundary conditions
u and J satisfy the growth conditions: there exist
7v'i
>
and
>
Then, for
E,
J
all
t
\u{x,t)\
<
A-i(l
+
\x\f^
\J{Y,t)\
<
A'i(i
+
|F|)^'^
Vi e
3?+,
<
e [0,T],
vy 6 <+',
t
e [0,T],
e [0,T],
uizis),s)ds
+ V{WiT+),z{T)) <JiW{t),z{t-),S{t),t)<J(W{t),zit-),S{t),t),
(18)
For the definition of a smooth manifold see footnote
7.
13
for all
C
E Ct financed by some
consumption and wealth,
C" G
t
z'
and
The
is
consump-
^
u{z'{s),s)ds^V{W'{T^),z'{T))
denote the weighted average past consumption and the wealth associated with
{C',A'), respectively, then
and (C, ^')
z{t~), S{t)^t),b. feasible
by A' G At so that
Ct financed
W
and A.
< T and given {W{t),
J{W{t),z{t-),S{t),t)=Et
where
C
G [t,T], are the average past
s
I
there exists, for any time
if
tion plan
G At, where z{s) and W{s),
respectively, associated with
Proof. See appendix.
Now,
A
must be the case that J{W{t),
it
z{t~), S{t),t)
-
J{W{t), z{t-), S{t),
the optimal consumption and investment policy starting from the {W{t), z{t~),S{t),t).
following
is
the main theorem of this section, which provides conditions for a candidate
consumption and investment policy to be optimal.
Theorem
at
1
any time
t
Let J satisfy the conditions of Proposition
Assume furthermore
1.
G [0,T], with W{t),z(t~),S(t), there exists a consumption policy
by A' G At with the associated state variables
W'is) =
W
and
z'
,
such
that,
that, starting
CG
putting g
—
Ct
financed
inf{5
>
t
:
0},
Js
=
W a.e.se
r[Jwi^V',z',S,v)-PJ,{W',z',S,v)]dC'{v)
=
V5G
u{z',s)
+ V'^'J +
it,g) a.s.,
(i,p]a.5.,
(19)
(20)
and
J{W{t,),z{t-),S{t,),t,)
where
ti's
- JiW{t,) - AC'{ti),z{t-) +
are the times of gulps prescribed by
J{W{t),z{t~),S{t),t) and {C',A')
at
is
C
on
0AC'{t,),Sit,),t,)
[t,g).
=
a.s.,
(21)
Then J{W{t),z(t~),S{t),t) —
an optimal consumption and investment policy starting
t.
Proof. See appendix.
As a
I
recipe for decision making, theorem
follow as long as his wealth
is
is
1
outlines general principles that the agent should
strictly positive. In particular, the
to use a control policy that keeps the pair
(19)
t),
satisfied. In addition, the
agent
iW,z)
in the region
may consume
only
theorem instructs the agent
where the
differential equation
when the marginal
utility of
wealth
14
is
A
5
equal to
Verification
Theorem
for the Infinite
Horizon Program
times) the marginal utility of the service flow. Finally, in the contingency that
{(3
the optimal consumption policy calls for a "gulp", the size of the gulp should be chosen such
that the value function immediately before
is
equal to the value function immediately after the
gulp.
The optimal consumption and
all
5
the necessary conditions of Section
A
The necessary and
program of
of course satisfies
Horizon Program
for tfie Infinite
sufficient conditions for optimality presented in the
for the finite horizon
1
3.
Theorem
Verification
Theorem
portfolio policy characterized in
(7). In this section,
replace the admissibility conditions of (6) for
C
we
will let
financed by
A
T =
above two sections are
oo and
V =
in (7)
and
by:
/•oo
E[
where
for
z is associated
\u{z{s),s)\ds]
with C, and for
all
integers
m
>
<oo,
and
(22)
T
G 5?+, there exists
fc^
so that,
alU < r,
E[\C{t)\'^^]
<e*"',
E[\Wit)\^"']
<e*m«.
(23)
Similarly let
A
C and
policies starting at
spaces starting at
=
<
denote the space of admissible consumption policies and investment
0,
respectively. In addition, Ct
and At are the corresponding admissible
t.
Consider the infinite horizon program:
J{W, z, S,
t)
=
supeec,
s.t.
E [/,°° u{z{sl s)ds\Tt]
C is financed hy A E At
with W{t)
and W{t) - ACis) >
We
= W,
(24)
Ws>t.
record below a verification theorem for this infinite horizon program:
Theorem
2 Let
J
differentiable over
in its first
N+2
inequality of
(
:
J?^"'""'
KiJ:"*"''
—
5?+ be positive, concave in its first two arguments, continuously
in all of its arguments,
and twice continuously
arguments, except possibly on a smooth manifold
M,
I4) with the boundary condition
/•oo
limJ(W,z,S,t)=
W[0
^
/
Jt
uize-^'^'-^\s)ds
<
00.
differentiable over K^"*"
satisfying the differential
15
In addition,
>
A'2
J
satisfies the
growth condition: for every
Assume furthermore
consumption policy
such
,
6
KJ >
there exists
Jf+,
and
so that
\J{Y,t)\
z'
T
\Y\)^^
<+^
\A^ €
any time
t
<
t
€ [0,T].
(25)
00 with W{t),z{t~),S{t), there exists a
€ Ct financed by A' £ At, with the associated state variables
putting g
that,
Af (1 +
starting at
that,
C
<
—
inf{s
>
t
:
W'{s) =
=
y a.e.se{t,g)a.s.
r[Jw(W\z',S,v)- 0J,(W\z\S,v)]dC'iv)
=
"i
+
\im Et[J{W'{t
s—•oo
s),z'{t
+
+
V'^'J
+ s)J +
s-),S{t
and
0},
J,
+
u{z',s)
W
s)]
=
0,
(26)
se{t,g]a.s.
(27)
a.s.,
(28)
and
J{W{t,),zit-),S{t,)J,)- J{W{t,)- AC'{t,),z{t-)
where
t, 's
are the times of gulps prescribed by
C
on
(3AC'{t,),S{t,),t,)
program, two more conditions are added.
latter condition ensures that the agent exhibits
wealth indefinitely without consumption
technical convenience
is
and we can replace
6
A
(29)
is
an
First,
J
t
is
positive. Second,
increases to infinity.
enough impatience so that accumulating
not optimal.
We
imposed the former condition
for
by a stronger condition which requires that (28)
it
holds not just for the optimal policy but for
a.s.,
t.
the expected value of J{t) along the optimal path must converge to zero as
The
=
Then J — J and (C',A')
[t,g).
optimal consumption and investment policy starting at
In the infinite horizon
+
all
feasible plans.
Closed Form Solution for an Infinite Horizon Program
In this section
we provide a
closed form solution for a particular example of the optimal con-
sumption and portfolio problem formulated
this analysis,
we use a
felicity
in section 5, in
function u{z,t)
= -e~
z"',
which the horizon
where q <
1,
is
infinite.
In
and where the discou nt
factor S expresses the agent's impatience.
In addition,
motion given
we assume that the
prices of the risky securities follow a geometric
Brownian
by:
rt
S(t)+
f p(s)ds =
Jo
S{0)+ f'Is{s)^lds+
Jo
['
Jo
Is{s)adB(s),
(30)
16
A
6
where Is{t)
where
/i is
is
a diagonaJ
N
N
x
Closed Form Solution for an Infinite Horizon Program
an TV-dimensional vector of constants, and a
instantaneous interest rate r
We
a constant.
is
assume that aa^
W,
function J depends only on wealth
=
We
interest rate are constant, the value
the average past consumption
is
z,
and
t.
Also,
henceforth denote this function simply by J{W,z).
The
easy
it is
time separable and with a
our attention on the function J{W,
will therefore focus
The
nonsingular.
is
e~*' J{VK, 2,0), since the felicity function
constant impatience factor.
l,...,N,
M matrix of constants.
an A^ x
is
Note that since the asset price parameters and the
to see that J{W,z,t)
—
matrix whose diagonal elements are Sn(t),n
and
z, 0)
becomes
differential inequality of (14)
-or
max{— + max[V'^ J]a
The boundary condition when
W—
The optimal investment
the TV risky assets at
ratio of wealth
W
states of nature.
all
to average past
If
/^_l
=
-j^
k' the
is
strictly less
consumption policy
^
is
is
equal to k'
if
see, the erratic
z less
is
than^ a
critical
<
to keep the
k' in almost
initial
all
consumption
than k', then the optimal consumption policy
Starting from that
to keep
number
W{0) and
^
<
z declines until the
(random)
moment, the agent consumes
k' forever in
to take a "gulp" of consumption, reducing
^
amount required
the
all
the agent starts at time zero with ^,^_l
immediately to k'
that required to keep
.
ratio
to invest constant fractions of wealth in
is
W increases on average and
amount required
the other hand,
the ratio
(32)
o)
the agent starts at time zero with wealth
time r when the ratio
^
decision
consumption
consume nothing and wait while
On
+
The optimal consumption
times.
experience -:(0~) such that
only when
(31)
show that the optimal solution to the agent's problem takes the form of a
will
barrier policy.
to
0.
,^.^^c. ^"-
a[af}
is
- Jw} =
is
/(O.^)-
We
J,
S J,
A
states of nature.
>
k'
,
then the optimal
W and increasing
z,
to bring
Following this gulp, the optimal amount of consumption
.
k' forever,
and consumption occurs only when
^
=
k'.
As we
is
shall
behavior of the price of the risky assets implies that the optimal consumption
pattern will have "singular" sample paths. In almost
a non-trivial amount at infinitely
many
all
states of nature, the agent
points of time. However, the times
consumes
when consumption
occurs are a set of Lebesgue measure zero.
In essence,
when the wealth
into consumption.
'Recall that
Conversely,
we use weak
relations
in relation to z
when the wealth
and hence x
less
is
too high, the agent will convert wealth
relative to z
than y means that i
is
is
low, the agent will refrain
strictly less
than or equal to
y.
17
from consumption altogether to accumulate her wealth and
mean
time, will enjoy the
from her past consumption purchases. The singularity of the sample paths
satisfaction derived
of the optimal consumption pattern
This
in the
caused by the erratic behavior of a Brownian motion.
is
a feature of the modeling choice in which the returns on risky securities follow a Brownian
is
motion. The economic content of the solution
the periodicity of consumption, which
is
in
is
part due to the fact that the agent derives satisfaction from past consumption.
We
will
employ
and portfolio
tion
First,
We
policies.
N
is
>
the expected
denoted
6amer
in all states of
life-
time
the policy of investing constant fractions
consumption policy of keeping the ratio
nature and consuming only when
with any
utility associated
if
—
—
—
This
k.
the initial conditions are such that
if
jtqA <
/:-ratio barrier policy
k.
and
We
will
show that
{W,z),
initial state
j'^(iy, 2), satisfies
V^j''-Sj''] =
J^ Second, we
together with
and hence
will
its
show that there
is
(3J^
=
if
W < kz
if
W>kz.
a unique value k'
make two
observations.
{W,
initial state variables
consumption pattern
=
such that the /c'-ratio policy
all
First, let
z) for a fc-ratio policy.
and the average past consumption, given
>
and
the conditions of
Theorem
2
the optimal policy.
is
pattern with
of wealth
>
associated value function, J'{W,z), satisfy
Before proceeding, we
j''(W,z)
is
a (random) period of no consumption
max\— +
scalar a
policy
followed after an initial "gulp" of consumption
k, or after
For
portfolio policies of the k-ratio barrier form.
risky assets together with a
than or equal to k forever
,qJ.
present the logic behind the optimal solution in two steps.
the corresponding k-ratio
0,
of wealth in the
policy
methods to compute the function J and the optimal consump-
we analyze general consumption and
any k >
less
probabilistic
0.
for the
same
Thus j''{aW,az) =
z°'j''{W/z,l).
fc-ratio policy
a°'j''{W,z).
Second,
must be time independent and
is
it
is
in
with
That
(4)
C
be the optimal consumption
By
the linearity of the dynamics
and
(5), respectively,
initial state variables
is,
J^
is
aC
{aW,az)
homogeneous of degree q
is
the
for
any
in that
easily seen that the optimal investment strategy
a function of
W and
z.
A'
Direct computation shows that
7*
^'(^.^) = -7777r-r'
(33)
'WW
where F
=
{aa'^)(fi
-
rl) and 1
is
an TV-vector of
I's.
The investment
policy
is
always propor-
18
A
6
Closed Form Solution for an Infinite Horizon Program
tional to the vector T.^°
The
object of control under a fc— ratio policy
keep this ratio
less
than
k.
is
the ratio
— and the purpose of control
is
to
^
<
k,
Suppose that the agent starts from a state {W,
and suppose that she invests constant proportions of her wealth
by
^l*^,
and consumes nothing
until the first
minimum amount
T on, the agent consumes the
t
>
us
T.
The wealth dynamics
compute the expected
From
of this policy
utility
=
Given that
j'^
J{W{t),z(t))^
is
=
such that
risky assets, denoted
Also suppose that from
k.
required to keep the ratio
—7^ <
k for
well-defined by Lions and Sznitman (1984). Let
is
we can
write, for
iW,z) such that
—
<
k:
E[r e-^'—e-'''^'ds + e-^'j''{W{T),ze-^^)'\
homogeneous
We
^(r)"J(^-,l).
of degree q, for a point on the
know
thus
boundary {W{t),z{t)),
that
j''[w{T),ziT))=akz{Tr,
for
some constant
a^.
Hence, we can rewrite j''{W,z)
•^'^^'
'">
=
+
-^^^
a(Q/3 +
from a state
—
motion with
drift fik
7
=
[/x
—
<
k,
the logarithm of the ratio
=
r
rl]^{aa'^)~^[fi
—
+
f3
rl].
-\-
bkf
-\-
for
—
—
,
before
some
scalar 6^-
Then
starting
reaching k, follows a Brownian
a standard deviation Uk
(1985, proposition 3.23),
^ ^-a.dogi-log^) ^
(35)
.
J
I-
A^ = bkT
— 6^7/2 and
From Harrison
E[^-(a/3+5)xj
as:
(
a(a[i
that the constant investment policy
(34)
^ [e-(-^+*)^]
l^!.y
0)
-
t°^
(?)
Assume
all
j''{W,z) obtained from such a policy.
the definition of the utility,
J*(ir,2)
time r when -jJ^
N
in the
z)
=
bk^/j, where
we conclude that
{^^^
•kz'
(gg)
where
ak^\\{pi\^2al{aii^b))\-iik\.
Substituting from (36) into (35), we get, when
^'"^•^' =
a(Q/3 + ^) +
;;?iT7)
—
<
fc.
i°'-s?^i-'"y°'q(q/3 +
^
(37)
<5)
Here we note the two-fund separation property implied by (33); see Merton (1971).
(^^'
19
Now, consider the
A = YTW
situation
such that
Jy^,
=
f3J^
Jw =
when
PJz
for all
^ = k.
From
(a/?
To summarize, the
A^ - bkT and a
where a^
is
A =
egy
>
The
k.
fc-barrier policy prescribes
+
—
>
A;.
We
+
A:-ratio
consumption policy
in deriving (41)
and
we
is
is
(
we
38),
get
given by
given by (37).
started by assuming a constant proportions investment strat-
j'^
.
If
J^ were the optimal value function, the optimal investment strategy
the optimal investment strategy would be equal to r/(l
to the system of equations
make
is
/3fc)-a/3fc
would be determined through (33). In particular, since the
to
of size
bkT. This investment strategy together with the fc-ratio consumption policy implies
the functional form
1
jump
then choose the value of the constant a^
this condition
<5)[ait(l
utility
a
value function associated with a constant proportions investment strategy
given in (40) and a^
Note that
^
boundary and the corresponding
*° ^ point on the
Notice that (39) implies
when
j''
composed of (37) and
6jt
=
—
1/(1
A;-ratio policy
q^). Thus,
- Ok) and
investment behavior of
generates the functional form of J*.
We
this solution
j'^ in
is less
Assumption
1
<
k,
than
that the implied
need to make enough
as-
exist.
We
whose proof uses
ele-
sumptions on the parameters of the problem to ensure that a desired solution of q^
assume throughout
^
there exists a solution
if
a concave function, then we have produced a consistent
j''
would keep
this section that:
The parameters of the problem
S
> ar
-ii
—
2(1
-q)
satisfy
and
(I
—
q)(3
>
S
-
Given these restrictions, we can easily establish the following
mentary algebra and
is left
for the reader.
r
result,
20
A
6
Lemma
1/(1
There exist a unique
1
1
that solves the
system of equations of (37) and
bj^
—
— Ok) and
^
'"'-
+
[7/2
Moreover, a^
(r
+
+
0)
> a and a^
As a consequence
We
<
qj.
Closed Form Solution for an Infinite Horizon Program
will henceforth
(q/?
is
+
6)]
of q^ being independent of k, A''
strictly concave,
=
6)]^
The
-
a*).
It is
Lemma
+
4(r
/j)(
J^JTT]
also independent of k.
is
1
and by A' the constant
also understood that the
Ok
following proposition gives properties of
The function J^ of (4I) with a^ replaced by a*
and
-
T/{1 - Q^)
denote by a* the solution characterized in
in the definition of Uk of (40).
2
+(r + f3) + {a0 +
2(r + /3)
v/[7/2
independent of k.
proportions investment strategy r/(l
Proposition
-
is
is
j''
replaced by a'
defined in (41).
continuous,
increasing,
twice differentiate, has continuous first derivatives, has continuous second
derivatives except possibly on the smooth manifold
Aik =
{(W', z)
:
W
=
kz], and satisfies,
together with A'
—a + V'^'j''-8J'=^0
if
\V<kz
=0
if
W >kz,
(43)
=0,
(45)
Jwand
the
PJz
(42)
boundary conditions
a(ap +
0)
UmE\e-''j''{Wkit),Zkit))\
where
and
W^ and
z^ are the wealth
and
the average past
consumption policy and the investment strategy A'.
consumption associated with
In addition,
e~'*'j'^
the k-ratio
satisfies the
growth
condition of (25).
Proof.
All the assertions except for the last can be verified by direct
the fact that q
<
>
0,
< q' <
1.
and hence j''{Wk{t),
computations using
For the last boundary condition note that Wk{t)
z^it)) is less
than a^z^it), where we
=
z[Q-)e-'^'
+
(i
f e-^('-'UCk{s)
Jo
<
z{O-)e-^'
+
[0Ck{t)],
kz^it) for
all
recall that a^ is defined in (40).
Note that
Zk{t)
<
21
where Ck denotes the
see that Ckit)
portfolio rule
Ar-ratio
< W{t)
a.s.,
consumption
where W{t)
From the budget
policy.
is
the wealth realized at time
A' with no consumption withdrawal before
t.
we can
constraint of (4),
when
t
following the
Using the dynamics of
W,
one can
easily verify that
VimEle-^' J{Wk{t),Zkit))\ < limE[ake-^'z^{t)] < lim E[afce-*'[2(0-)e-'"
ttoo
L
t]oo
J
+
/?W^(t)]"
fjoo
But
[z(0)e-^'
+
/JH^CO]"
^
[/SW^CO]"
uniformly in
w
as
«
j
oo
hence
/?Vr(i))i
=
VimEle-^' J{VVk{t),Zk{t))]
<
limEfe-*'(2(0)e-^'
(too
+
and thus
VimE\e-^'fi"W''{t)\.
(Too
J
I
l\m E\e-^' 13° W°{T)]
J
1-
But
where the
first
inequality follows from the property that
the definition of a'.
Hence E[e-*'H^°(<)] <
q < q', and the
W{0)e^^°'-°''^^.
last inequality
Noting that a - q' <
from
0,
the
required result follows.
The
last assertion is obvious.
Now, we show that a
the function
j'' is
I
^--ratio policy
financed by A'
is
an admissible policy
in (24).
Moreover,
indeed the expected utility that the agent gets from following this
A:-ratio
policy.
Proposition 3 The wealth process and
rier
the
consumption pattern associated with a k-ratio
consumption policy financed by the investment strategy A'
function e~
J {W\z) gives
satisfy (23).
bar-
In addition, the
the expected utility of following these policies at time
t.
22
A
6
Proof.
Let
W^, Ck, and
denote the wealth, the consumption, and the average past
z^
consumption associated with the
Closed Form Solution for an Infinite Horizon Program
A:-ratio
proof of Proposition 2 that Ck(t) < W{t)
time
at
t.
when
t
consumption policy financed by A'. Recall from the
a.s.,
where we
recall that
W(t)
the wealth realized
is
following the investment strategy A' with no consumption withdrawal before
< W{t)
Similarly, Wk{t)
a.s. It is easily verified
Brownian motion and thus W{t)
that
W
log-normally distributed.
is
a geometric
satisfies (23) since it is
The
first
assertion follows.
For the second assertion, identical arguments used to prove Theorem 2 show that
E[J'e-'"'-^^dv +
where we have used Proposition
by letting
From
5
—
oo.
The
2.
assertion follows from
we know that any
A'
satisfies
fail
to satisfy the differential inequality (31).
(31)
is
e-''j''(Wk{t),Zkit)),
monotone convergence theorem
I
the above discussions,
many
=
e-''j''{Wk{s),Zk{s))\
A:-ratio
consumption policy together with
of the sufficient conditions for optimality. All policies, except one, however,
the optimal solution, which
is
The unique
barrier policy that satisfies inequality
recorded in the following proposition.
Proposition 4 Let
f3{a'
Then
fc'
>
and
- a)
the solution to (24) consists of the investment strategy
A' and
the k' -ratio
barrier consumption policy.
by lemma
Proof.
First k'
Next
the value function associated with the A;'-ratio barrier policy be J'
let
Proposition
2,
>
we only need
to
1.
show that J*
Jw -
start with the first inequality.
region
W
<
In view of
satisfies:
0Jz <
—a + V'^'J'-6J'<0
We
.
if
W <k'z
if
W>k'z.
and
Using the definition of J', we can write J^Y - /3J'
k' z as:
J'w-0J;
=
«^""'5(-j;^)
g{y)
=
/3(a-
-
a)2/'*'
where
+
^j/(-'-''
^
al3
+
S'
in the
23
where a
is
a constant. Note that g(l)
Computing the
derivative of g,
—
=Q
ay
we
=
and that since a* >
0,
J/
>
1.
It
-q)2/
/?(q
'
+
y
-r,
k-
Now, we
then follows that J^y
-
>
/3J'
which one would jump
if
— -q-
oo as y
By
x.
for y
=
j oo.
is
1,
and that j^ >
W < k'z.
2) such that W
z) such that
=
(H^,
W = k'z and the straight
In other words, a
one starts at
{W,
0, for all points
Let a be the point on the intersection of the line
^jj-
|
.
=
Consider any point i
verify the second inequality.
the point x with slope
g{y)
get
Substituting k' in the expression for j^, we conclude that j^
for
0,
line passing
the point on the boundary
construction, J'{x)
=
>
k'z.
through
W = k'z, to
J*(a).
Let
-a + J^rW
- J'J^
z)=JwrW - ^'
-^7
ZJ,
fiW,
,
'WW
and note that /(a) -
6 J' {a)
straight line connecting a
and
=
0.
i,
Applying the fundamental theorem of calculus along the
we obtain
/(£)-/(«)=
r fwdW + f,dz.
Ja
Noting that along the
line
direction from a to x,
it
connecting a and x, we have d z
then follows that
/w —
z"
—
+ JwtW - :piT—l ^
/3/j
<
I'^
WW
Computing fw — Pfz
easily verify that
The
at
t
=
in the region
fw -
Pfz
<0
W
for k'
>
=
J'zPz
-SJ'
= -fSdW, and
dW
>
in the
to conclude that
is sufficient
<Q
that
W
if
—>k'.
^
k'z,
and using the properties of q', the reader can
1
pf^ri^-
following proposition records the optimal consumption and investment policy starting
0.
The optimal
policy from any
t
>
can be constructed similarly.
Proposition 5 Let
—
r
A'it)^—
— a*
for all
t,
1
and
lei
the budget feasible
consumption process
C
,
which has continuous sample paths almost
surely, be given by:
C'(t)
= AC'(0) + J^r^-^dl(s)
P-a.s.
(46)
24
A
6
Closed Form Solution for an Infinite Horizon Program
where
an<f
'
AC-(O)
=
l{t)
=
sup [log
W{i)
=
(vyo-AC-(0))e('^^^^^^"^'rT^^'"l^<'»
i(0
=
[zo
maxjO,
+
\^ ^^.
^5/^-
(47)
).
P-a.s.,
log fc']^
P-
0AC'iQ))e-'^'
(48)
P-a.s.,
(49)
a.s.
(50)
where W*{s) and z'{s) are the state variables associated with
C.
C
The strategy A' and
defined above are the unique optimal solution for the agent's problem.
Proof.
To prove
this proposition, all
The
ratio barrier strategy associated with k'.
if
>
.^-\
k'.
The
size of the
from which we compute
Let us proceed
policy
is
now
A
to
A
jump
we need
fc*
to
show
is
that the above strategy
ratio barrier policy calls for a
calculated to achieve the condition
is
jump
at
i
a
^,
^
7^
the
is
t
=
—
0,
^'^
as in (47).
compute the consumption process
after
t
=
0.
The
i-'-ratio barrier
equivalent to the condition that
W'(t]
<
k'
<
logk'
'it,P-a.s.
or
(51)
z'it)
^
W'(t)
log
Now
yt,P-a.s.
(52)
consider the "unregulated" process {W,z), which gives the wealth of the agent and her
past average consumption under the assumption that no consumption takes place after the
initial
jump
at
i
=
0.
The
values of
will fail to satisfy condition (51) for
the solution
is
some periods and
(50).
The
ratio -j4y
some sample paths, and the idea
C
of
to ensure that at the
this regulation, define the process
r
W(s)
log—
0<3<(^
let
for
and
is satisfied.
l{t)= sup
and
z{t) are given in (49)
to "regulate" this ratio using the consumption process
optimal solution condition (51)
To achieve
W{t) and
the "regulated" process log
log
—^
1
log
+
P-a.s.
'
2(5)
be given by:
—— =
;
^ z'{t)
For each state u, the sample path
^-
/(u--, .)
log
^ z{t}
lit)'
P-a.s.
^
has the following properties:
(53)
^
'
25
•
/(w,
•
log
•
l(u), .)
.) is
increasing and continuous with /(a;,0)
^^ <
log
for all
A:*
>
<
when
increases only
=
log ^./J^a'
variables associated with the optimal
is
)
almost
all
u.
logfc*.
a candidate for the logarithm of the ratio of the state
consumption plan, since from the above construction, we
can easily conclude that condition (51)
is
satisfied.
The question now becomes whether
consumption process C* that could enforce the relationship
exists a feasible
both sides of (53) using
lemma, we
Ito's
+
Witl
_
i(0
+
H-(0)-AC'(0)
^°^
+
z(O-)
;
/
Jo (1
7(1
/3AC-(0)
-
Expanding
2(1-Q-)2J'"
+ /?AC-(0)^yo^
-
/;
in (53).
there
get
^
(T^r-'^.^(.) {[^ ^]
^/'.^^^
^
^
+
^°^2(0-)
z'{t)
'°^
0, for
0, P-a.s.
This "regulated" process (log ^.Ay
^
=
7o
2(1
^
-2a')
- Q-)2
P-
.C'(.)
a.s.
""^
J
P-a.s.
-T'^adB(s)
Q*)
Thus, we conclude that the consumption process given by:
satisfies the
/
increases for
all
condition (27) of
Now
let
sample paths.
Theorem
2
is
Therefore,
=
C
increases only
when
increases only
^.xJ
y{0)l{0)
+
=
0w{l)l}.(t\
-
From
Ito's
f y{s) dlis) + Jo/' lis) dyis)
lemma, we
P-
a.s.
Jo
Noting that
/(O)
- 0,P -
a.s.,
and that dl{s) - ^^r^, we
yit)l{t)
= C-(0 +
find that
r lis) dyis) P - a.s.
Jo
Integrating the second term in the right-hand side by parts,
r.,,s
=
r
w'is)z'is)
^^'^ =
Jo0W'is) +
z'is)''^'^
we have that
^
a.s.
when
k' and hence
satisfied.
the process y be given by y{t)
y{t)lit)
C
Note from the above equation that
condition in (53).
get:
26
Comparative Statics
7
But from the properties of
/,
we know that
increases only
/
when
-j^fpf
=
k'
.
We
thus conclude
that
a.s.
The
solution
we constructed has the following
agent consumes the
greater than
(/?
minimum
amount required
possible
times) the marginal value of
K
marginal values are equalized.
features. After the initial (possible) gulp, the
to keep the marginal value of wealth
Consumption takes place only when these
z.
the marginal value of z drops below
(/?
times) the marginal
value of wealth, the agent stops consuming. These rules result in a consumption process with
singular sample paths.
set of all times
Consumption occurs
uncountably
at
infinite
when consumption occurs has Lebesgue measure
paths. This result occurs because of the
number of
times, but the
zero, for almost all
sample
unbounded variation property of the Brownian motion
sample paths.
7
Comparative Statics
It is
interesting to
compare the behavior described
an agent with a time-additive
in the previous section to the behavior of
utility function defined
on the subspace of absolutely continuous
consumption patterns and given by:
€
where
c{t} is the
consumption rate
at
time
e-^^'SlLdt],
t.
(55)
Note that the
of the utility function analyzed in section 6 as
(i
]
utility function in (55)
is
the limit
oo.
Consider the consumption-investment problem of an agent whose preferences are given by
(55) and
who
faces the investment opportunities given by (30)
Merton (1971) shows that the agent's optimal
to a constant fraction of total wealth,
policy
is
to
and a constant
consume continuously
riskless rate.
at a rate equal
and to invest constant proportions j4^r of
his
wealth
in the risky assets.
Now compare
the investment policies of the agent
we studied
Merton's time-additive agent, assuming that the corresponding
in this
felicity
lemma
1,
Hence the agent whose preferences exhibit
we know that a' >
same
functions have the
curvature q, and that the agents have the same time preference parameter
investment opportunities. From
paper with those of
6,
and
face the
a. Furthermore, lim^jooQ'
local intertemporal substitution invests
more
same
=
in
oi-
the
27
and appears to be
risky assets
substitution (higher
/3),
Furthermore, the weaker the
less risk averse.
effect of local
the less that the agent invests in risky assets, ceteris paribus.
One
could explain such a behavior by arguing that the fact that past consumption affects current
and future
utility leads the
agent to tolerate higher probabilities of reduced future level of
wealth and hence take more risky positions.
The reader can
easily verify that
^- <
0,
and
^-
<
0.
In other words, higher values
of riskless rate and higher discount factors lead to less investment in the risky assets, ceteris
paribus.
Another interesting property of the solution
state distribution
for the non-time-additive utility is the steady
and the expected value of the the ratio
^.
Following the computations in
Harrison (1985, §5.6), we conclude that, from any starting point {Wo,zo), as
pjm<A^lif.f^
l-
J
z{t)
\
1
t
—> oo:
ifx<r
if
I
>
(56)
r
and
where Pq and Eq are the probability distribution and expectation, respectively, conditional on
starting at (1^(0), 2(0)) and where
and where 7
=
[/x
—
^*
=
a
=
rl\^{aa'^)
Denote limt_oo ^0(777^] by
One can
^[fi
^
+^+
r^- 2(l-a')^
'
v/7
(1-a-)'
—
(t-Joo-
rl].
The reader ca n
easily verify that:
intuitively understand these relationships as follows.
past consumption eventually settles
down
to (t-Joo-
Wealth
is
The
ratio of wealth to average
the investment in risky assets
which the agent holds for purposes of future consumption, wherejis z
is
an indication of
past consumption experience, which also contributes to his future satisfaction.
his
The agent
decides on an optimal division between these two sources of future satisfaction. This division
is
naturally dependent on the strength of the local substitution effect and the outside investment
28
Equilibrium Risk Premium
8
When
opportunities.
relatively
more
the local substitution effect
satisfaction from z,
high (low
is
and hence [^]oo
investment opportunities are more attractive (high
low.
is
r or
the agent tends to derive
/?),
On
low a),
[
the other hand,
—
]oo is
when the
high, reflecting
more
emphasis on deriving future satisfaction from wealth.
Premium
Equilibrium Risk
8
In this section,
we
on asset prices
briefly state
some of the implications
in equilibrium.
We
Cox, Lngersoll and Ross (1985a).
of the form of utility studied in section 6
adopt the representative agent equilibrium framework of
In the general
framework of section
4,
we note
that the
optimal investment policy implies the mutual fund separation results of Breeden (1979) and
Merton (1971). Thus the Intertemporcd Capital Asset Pricing Model of Merton (1973) holds
parameter production processes, the constant
In the special case of constant
in equilibrium.
proportion optimal policy given in (33) implies two-fund separation.
Therefore, the Capital
Asset Pricing Model holds. However, the Consumption Capital Asset Pricing Model of Breeden
(1979)
fails
to hold because the marginal utility of instantaneous consumption
to the marginal utility of wealth.
For similar results, see
is
no longer equal
Bergman (1985) and Grossman and
Laroque (1990).
Next, we consider the determination of the risk premium. Suppose, for simplicity'^ that
there
is
one risky production technology whose rates of return follow a (/i.(7)-Brownian with ^
and a being
strictly positive scalar
given in section
such that a^i
agent with preferences as
Using arguments similar to those of Cox, lngersoll and Ross (1985b), and
6.
using the computations of Harrison (1985, §3.2),
r, is
< 6}^ Consider an
we conclude that the equilibrium
riskless rate,
given by:
r
=
a
=
In contrast,
curvature
if
a and
Given that a^ <
fi
-
{I
- a)a^
where
^ [^(M + 0- ctV2)2 +
2(T2(a/3
+
6)
-
{fi
- aV2)]
+
.
the agent has a time additive utility function whose felicity has the same
the
6,
same time discount parameter
6,
the reader can easily verify that
the interest rate,
is
given by /i-(l -a)a^.
a > a and hence the
risk
premium, which
This result can be easily generalized to the case of many production technologies.
'^More specifically, V{t), the level of risky capital at time t starting from one unit continuously reinvested,
evolves according to the following equation; dV{t) = fiV{t)di + <tV {i) d B (t) where B is a one-dimensional
standard Brownian motion. Note that we use /i and <7 in a different meaning from that in section 6, and 6 is the
parameter of time discount.
,
29
is
the excess of fj. over the riskless rate,
This
is
lower in the case of preferences with local substitution.
consistent with our earlier observations that the agent
is
substitution behaves in a less risk averse
In addition, as local substitution decreases
utility.
9
manner compared
(/?
whose preferences exhibit
local
to an agent with time-additive
increases), the risk
premium
increases.
Temporal Aggregation and Properties of Consumption
Series
Recorded purchases are thus
Purchcises of durable goods are typically measured periodically.
a time aggregate of purchases during the measurement period.
It
useful to
is
examine the
properties of the process of measured purchases as predicted by the model developed in this
paper.
The
properties of recorded purchases for an individual will depend mainly on two factors:
—
the wealth relative to the stock of the durable good, or the ratio
measurement period and the length of the measurement period
durable good. For example,
half-life for
an individual
if
,
at the beginning of the
relative to the half-life of the
one measures the daily purchases of a durable good with a long
who
starts with a ratio
—
much lower than
the critical ratio k',
then one will record zero purchases with very high probability.
On
measures the quarterly purchases of a durable good with a short
half-life for
starts with a ratio -^ very close to the
probability,
The
boundary value
k', then
one
the durable good, and the initial value of
—
measurement period,
We
an individual who
will observe,
with high
whose
price follows a geometric
return
/i
relative to the half-life of
assume an economy with two
investment opportunities: a riskless asset with constant interest rate
r
= 6%, and
is
given by^'' u{x,t)
=
simulate the daily behavior of an individual
We
a risky asset
Brownian motion process with instantaneous expected rate of
14.8%, and instantaneous standard deviation of the rate of return a
individual's felicity function
in section 6.
= 22%. The
T^^e~^'^'x~''^^^.
who
follows the optimal solution prescribed
than aggregate his purchases of the durable good over the period of measure-
ment, which we take to be one week, one month or one quarter.
of changes in the purchases in two successive
We
report the autocorrelation
measurement periods. To study the dependency
'''The astute reader will observe that this utility function does not satisfy the nonnegativity condition
in
one
can best be illustrated by simulating the process
of purchases of the durable good for a typical individual.
We
if
"smooth" consumption over successive quarters.
interplay between the length of the
=
the other extreme,
proving the verification theorem.
problem having an optimal solution.
We
note that this utility function gives
rise to
we used
a well posed optimization
Temporal Aggregation and Properties of Consumption
9
30
we
of this statistic on the initial value,
start the
measurement from the twenty
that correspond to the twenty values {1.00,0.95,0.90,-
function reported in (56). Finally,
We
week, one month and two years.
the standard error on
Figures
half-life
1
all
we report
•
•
Series
different states
,0.05} on the steady-state distribution
these statistics for durable goods of half-life one
use 1000 data points to estimate each statistic and hence
reported autocorrelations
is
0.0316.
to 3 report the autocorrelation of changes in the purcha.ses of a durable
good with
one week, one month and two years, respectively. Each figure reports the autocorrela-
tions for weekly,
monthly and quarterly measurements. Note that
all
measured autocorrelations
are negative. This reflects the periodic nature of purchases of durables. Above-average changes
of purchases are
more
likely to
be followed by below-average changes.
We
also observe a clear
pattern in the value of the autocorrelations. First, the closer the starting state to the critical
boundary
is,
vation period
the lower, in absolute value,
is,
the lower
is
is
the autocorrelation. Second, the longer the obser-
the absolute value of the autocorrelation. This
is
a reflection of
the effect of time aggregation. As more of the purchases are lumped together in one number,
the variability of recorded changes in purchases within periods of measurement
is
reduced.
Figure 4 reports the autocorrelation of changes in the quarterly purchases of the durable
good. The figure displays the autocorrelations for durable goods with
month and two
is
years.
The
figure
shows that the shorter the
the absolute value of the autocorrelations.
for short-lived durables
is
This
of the good
reflects the fact that the
one week, one
is,
the smaller
optimal policy
to purchase small quantities frequently in contrast with the optimal
policy for long-lived durables which calls for sporadic
The
half-life
half-life
and large purchases.
autocorrelation of changes in the purchases of the durable
cant way, across starting states.
The
differ, in statistically signifi-
following table reports the average, across initial positions,
of autocorrelations of simulated changes in purchases for different durable goods
measurement periods.
Half
Life
and
different
31
purchases,
we would
like to illustrate the "typical"
sample path behavior of purchases over
some long horizon. Figures 5 and 6 show simulated sample paths
durable goods with
half-life
of quarterly purchases of
one month and two years, respectively. In addition, both figures
show the sample path of the agent's wealth. The contrast between
clearly the effects of durability
A
wealth.
on the
haJf-life will
be purchased
amounts, relative to wealth, than a durable good with a shorter
good with short
and 6
illustrates
variability of changes in purchases relative to changes in
durable good with a longer
of a durable
figure 5
half-life will
less frequently
half-life.
As a
and
result,
in bigger
purchases
be "smoother" than purchases of a good with long
half-life.
Finally,
individual.
we remind the reader that
More work needs
to Caballero (1991),
of these results apply only to the purchases of an
to be done to derive the aggregate purchases.
and the references
therein, for
more on
this
We
refer the reader
important topic.
Concluding Remarks
10
In this paper,
for
aJl
we
heuristically discussed necessary conditions
and provided
sufficient conditions
a consumption and portfolio policy to be optimal for a class of time-nonseparable preferences
that consider consumptions at nearby dates to be almost perfect substitutes.
Our model
also
admits an alternative interpretation as a model of irreversible purchases of a durable good
whose service flow deteriorates as the stock of the good
ages.
We
demonstrated our general
theory by explicitly solving in closed form the optimal consumption and portfolio policy for
a particular felicity function when the prices of the risky assets follow a geometric Brownian
motion process.
We
have explicitly solved
for the equilibrium interest rate in a
Cox, Ingersoll, Ross (1985)
type economy with a physical production technology whose rate of return follows a Brownian
motion. In addition, we provided some simulation results that demonstrate the properties of
the consumption/ purchase series for durable goods with different half-lives.
Two
avenues of future research deserve attention. First, we can incorporate the notion of
habit formation into our model while preserving the feature that consumptions at nearby dates
are almost perfect substitutes. This can be accomplished by defining
past consumption,
Zi
and
Z2,
where
z^
two weighted averages of
has heavier weights on more recent past consumption
than Z2 and 22 has heavier weights on more distant past consumption than
functions defined on
21
and
Z2,
^i.
Consider
denoted by u{zi,Z2,t), with the property that U12 >
0,
felicity
where
32
11
Ui2 denotes the cross partial derivative of the
higher the distant past consumption
of
more
is,
first
two arguments of
u. In this
References
formulation, the
the higher the marginal felicity for an additional unit
we can
recent past consumption will be. Therefore,
interpret Z2 as a living standard.
This formulation treats consumption at nearby dates as close substitutes and incorporates the
notion that "habits" are formed not over a short period of time but rather over longer horizons.
Second, the interest rate derived in Section 8
yield curve.
explicit
One needs
of limited interest since
is
it
implies a
flat
to explore possible specifications of the production technology to derive
and interesting term structure of
interest rates.
References
1 1
1.
K. Arrow and M. Kurz, Public Investment, the Rate of Return, and Optimal Fiscal Policy,
Johns Hopkins Press, Baltimore, 1970.
2.
Y. Bergman,
Time Preference and Capital Asset
Pricing Model, Journal of Financial
Economics 14, 145-160, 1985.
3.
D. Breeden,
An
Intertemporal Asset Pricing Model with Stochastic Consumption and
Investment Opportunities, Journal of Financial Economics 7, pp. 265-296, 1979.
4.
R. Caballero, Durable Goods:
An
Explanation for Their Slow Adjustment, mimeo, Columbia
University, 1991.
5.
G. Constantinides, Habit Formation:
of Political
6.
J.
Economy
A
Resolution of the Equity
J.
Cox and C. Huang, Optimal Consumption and
Cox,
Prices,
8.
J.
Cox,
J. Ingersoll,
and
and
metrica 53, pp. 385-407
9.
S.
Ross,
An
Econometrica 53, pp. 363-384
J. Ingersoll,
Puzzle, Journal
98, pp. 519-543, 1990.
follow a Diffusion Process, Journal of Economic
7.
Premium
S.
,
Ross,
Portfolio Policies
when Asset
Prices
Theory 49, pp. 33-83, 1989.
Intertemporal General Equilibrium Model of Asset
,
1985a.
A Theory
of
Term Structure
of Interest Rates, Econo-
1985b.
C. Dellacherie and P. Meyer, Probabilities
Holland Publishing Company,
New
and Potential B: Theory of Martingales, North-
York, 1982.
33
10.
J.
Detemple and
tion,
11.
F. Zapatero, Asset Prices in
Econometrica 59, pp. 1633-1658
,
An Exchange Economy
with Habit Forma-
1991.
D. Duffie and L. Epstein, Stochastic Differential Utility and Asset Pricing, Econometrica,
1991, forthcoming.
12.
K.
Dunn and
K. Singleton, Modeling the
separability of Preferences
Term
Structure of Interest Rates Under Non-
and Durability of Goods, Journal of Financial Economics 17,
pp. 27-55, 1986.
13.
M. Eichenbaum and
L.
Hansen, Estimating Models with Intertemporal Substitution using
Aggregate Time-Series Data, Journal of Business and Economic Statistics
14.
M. Eichenbaum,
L.
Hansen and K. Singleton,
A Time
8,
530069, 1990.
Series Analysis of Representative
Agent Models of Consumption and Leisure, Quarteriy Journal of Economics CIII, pp.
51-78, 1988.
15.
L.
Epstein,
A
Simple Dynamic General Equilibrium Model, Journal of Economic Theory
41, pp. 68-95, 1987.
16.
L.
Epstein and
S. Zin,
Substitution, Risk Aversion, and the Temporal Behavior of Con-
sumption and Asset Returns:
17.
W. Fleming and
A
Theoretical Framework, Econometrica 57, 1989.
R. Rishel, Deterministic and Stochastic Optimal Control, Springer-
Verlag, 1975.
18.
A. Gallant and G. Tauchen, Seminonparametric Estimation of Conditionally Constrained
Heterogeneous Processes: Asset Pricing Applications, Econometrica 57, 1091-1120, 1989.
19. S. J,
Grossman and G. Laroque, Asset Pricing and Optimal
Portfolio Choice in the Pres-
ence of Illiquid Durable Consumption Goods, Econometrica 58, pp. 25-51, 1990.
20. J.M. Harrison,
Brownian Motion and Stochastic Flow Systems, John-Wiley
&
Sons, Inc.,
1985.
21. J.
Heaton,
An
Empirical Investigation of Asset Pricing with Temporally Dependent Pref-
erence Specifications, unpublished manuscript, Sloan School of Management,
22. J.
MIT,
1990.
Heaton, The Interaction between Time-Nonseparable Preferences and Time Aggrega-
tion,
unpublished manuscript. University of Chicago, 1988.
34
11
23. A.
Hindy and C. Huang, On Intertemporal Preferences
for
References
Uncertain Consumption:
A
Continuous Time Approach, Econometrica, 1991, forthcoming.
24. A. Hindy, C.
The Case
Huang and D. Kreps, On Intertemporal
Preferences in Continuous
Time
I:
of Certainty, Journal of Mathematical Economics, 1991, forthcoming.
25. J. Hotz, F.
Kydland and G. Sedlacek, Intertemporal Preferences and Labor Supply, Econo-
metrica 56, pp. 335-360, 1988.
26. T.
Koopmans, Stationary Ordinal
and Impatience, Econometrica 28, pp. 287-309,
Utility
1960.
27. N. Krylov, Controlled Diffusion Processes, Springer- Verlag,
28. P.L. Lions
and A. Sznitman, Stochastic
New
York, 1980.
Differential Equations with Reflecting
Boundary
Conditions, Communications on Pure and Applied Mathematics 37, pp. 511-537, 1984.
29. R.
Merton,
Optimum Consumption and
Portfolio Rules in a Continuous
Time Model,
J.
Econ. Theory 3, pp. 373-413, 1971.
30. S.
Sundaresan, Intertemporally Dependent Preferences and the Volatility of Consumption
and Wealth, Review of Financial Economics
31.
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.),
35
Appendix
A
Proofs
Proof of proposition
W{s) and z{s~) be
1
C
Let
Ct financed
€
hy
A
€ At he
a,
feasible policy for (8).
Let
the wealth and average pa^t consumption at time 5 6 [t,T] associated with
(C, A). Since wealth after consumption must be nonnegative, whenever the wealth reaches zero,
the only feasible policy
is
to
consume and
invest zero afterwards.
W has left-continuous and that z has right-continuous sample paths. For simdenote J and
partial derivatives wit h respect to W ,z ,S and
of notation, we
Recall that
plicity
will
its
time, evaluated at (H^(5), 2(5"), 5(s),s), by J{s), Jwis), Jz{s), Js{s), Js(s), respectively.
Note that, given the admissibility condition
(6)
and the growth condition on u given
in (17),
we know
Et
Define g
—
ini{s
does not exist,
we
stochastic interval
>
set
[t,
t
:
it
W{s) =
to be
0},
T}^
r
<
00.
(59)
where we have used the convention that
if
the infimum
Also, let the points of discontinuity of C{u),s) on the
g) be ti{u), t2{u>),
Fix r G {t,T). Generalized Ito's
\u{z{s),s)\dt
J
.
.
..
lemma
implies that
qAt
u{z{s),s)ds+ J{W{gAT),z{{gA t)-),S{qA T),g A r)
=
u{z{s), s)ds
J''^^
+
+ JiW{t}, 2(r ), Sit), t) + j'^\v^J(s) +
[Jwis)W{s)Aisyis-^ {s)<t{s) +
J
Jsis)^(T{s)] dBis)
+ 'Z[Jitt)-J(U)]-Yl[Jw{U)AW{U) +
where g
we know
At
denotes min[^,r].
is
a.s.,
By
the hypothesis that J
<
JwiU)AW{U) + Uu)Az{t,)
=
-JwiU)AC{U)-\-J^{t,)0AC{t,)
g Q:
q{u>)
is
[(5J,is)
J
concave
an optionaJ time
in that, {ui
<
t]
e Tt ^t &
[0,7^.
-
Jn-{s)] dC{s)
(60)
in its first
that,
J{tt)-JiU)
'e
liU)Az{t,)]
+
Js{s)] ds
V^
a.s.
two arguments,
A
36
Proofs
Thus,
f<>^'u{z{s),s)ds+J{W{gAT),ziigAT)-),S{gAT),gAT)
(61)
< J{W{t), z{r), 5(0,
where we have used the
Now
fact that
J
Js(s)^a{s)] dB{s)
the differential inequality and
satisfies
C
is
increasing.
define
Qn
From
+ If^'[Jw{s)W{s)A(sfls-i is)ais) +
=
\n{{s
<gAT:
J'[Jw{s)W{s)A{syis-i{s)ais) + Jsisf a(s}]'^ds =
the application of the generalized Ito's
r
lemma
in (60),
we know that
\jw{s)W{s)A{s)'^ Is-iis)a{s) + JsS{s)^a{s)]'^ds < oo
and, hence, we conclude that gn
]
g
At
On
a.s.
n}.
the stochastic interval
[t,
a.s.,
p„],
J'[Jwiv)W{v)A{vfls-i{v)aiv) + Jsivfa{v)]dB(v)
is
a martingale; see, for example, Liptser and Shiryayev (1977, chapter 4).
Replacing
Et
£>
J
A r of (61) by g^ and taking expectation gives
u{z{s), s) ds
Note that (59) and the
Et
+
J{W{gr,), z{g-}, 5(£>„), q„)
fact that gn —> g
/
A
u{z{s),s)ds
as
71
—
-
(62)
t a.s. imply
—> Et
We
by Lebesgue dominated convergence theorem.
Et[J{W{g,,), z{g-), 5(e„), p„)]
<JiW{t),z(r),S{t),t).
u{z{s),s)ds
/
want to show that
Et[J{W{g A
zHg A
r),
oo. In our current context, this conclusion follows
r)"), S{g
Ar),gA
r)]
from Lebesgue theorem by the
left-
continuity of the sample paths of {W{s),z{s~),S{s)), the growth condition on J in (17) and
the
moment
conditions on
W
and on
C
in (6); see, for
example, Fleming and Rishel (1975,
theorem V.5.1).
The
assertion then follows from letting t
W ^0, and
from the
fact that
V
is
]
T, from boundary conditions of J at
increasing and
W{T+) < W{T).
I
T
and
at
37
Proof of theorem
1.
The
E(
uiz'is), s)ds
+
u{z'(s), s) ds
+ JiW'{t+),
[J'
=
E,
+
=
where the
first
J"
J
allows us to write
JiW'iQ), z'{q- ), SiQ),
z'(t),
[Jwis)W'{s)A'isyis-.is)ais)
g)]
5(0,
+ Jsisfa{s)]dB{s}]
JiW'it+),z'{t),S{t),t)^j{W'it),z'{r),S{t)j),
equality follows from (19), (20) and (21), the second equality follows from the
arguments used to prove Proposition
identical
lemma
generalized Ito's
1,
and the third equality follows from
the boundary conditions of (15) and (16) allows us to conclude that
is
an optimal policy.
Using
J — J and thus {C',A')
I
Proof of theorem
we
(21).
2.
For any
finite
T, using identical arguments as in proving
Theorem
1
get,
rT
J{W(t),z{r),S{t),t)>Et
Since J
and
is
still
positive,
u{z{s), s)ds
+ J{W{T), z{T-), S{T), T)
we can drop the second term on the right-hand
maintain the inequality. Since the inequality holds for
and conclude that J >
Then
j
all
side of the
T >
t.
We
above relation
can
let
T —
>
oo
1
in
J.
the assertion follows from arguments identical to those used to prove
addition to using condition (28).
I
Theorem
Durable Good with One
Week
Half Life
Autocorrelation of Change in Purchases
1.00
"
0.90
'
0.80
OJO
'
'
0.60
starting point
Measurement frequency
'
0.50
0.40
"
0.30
0.20
0.10
on distribution function
weekly
Figure
'
1
monthly
x
quarterly
Durable Good with One Month Half Life
Autocorrelation of Change in Purchases
1.00
0.90
0.80
0.70
Measurement frequency
0.50
0.60
starting point
0.40
0.30
0.20
0.10
on distribution function
weekly
Figure
2
monthly
x
quarterly
Two Year Half Life
Durable Good with
Autocorrelation of Change in Purchases
-0.1-0.2-0.3-
•1
-I
1.00
0.90
0.80
0.70
0.60
starting point
Measurement frequency
0.50
on
0.30
0.20
0.10
distribution function
weekly
Figure
0.40
3
monthly
x
quarterly
r
Quarterly Purchases of the Durable
Good
Autocorrelation of Change in Purchases
1.00
"
0.90
'
0.80
"
0.70
'
starting point
Half Life
"
0.60
1
on
week
Figure
0.50
0.40
"
'
O.bo
0.20
distribution function
1
4
'
month
>«
2 years
0.10
>
c
Purchases and Wealth/lOO
120
M
O
o
en
O
o
160
200
240
280
320
360
O
>
c
Purchases and Wealth/100
40
o
80
120
160
200
240
280
320
360
Date Due ^-^-f^
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