HD28
.M414
ot^misy
no.
3VI
^^
.^.y^Wrt*-
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Un intertemporal rrererences in Uontinuous Time:
The Case of Certainty*
Ayman Hindy, Chi-fu Huang, and David Kreps
March 1991
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
WP//
3311
On Intertemporal Preferences in Continuous Time:
The Case of Certainty*
Ayman Hindy, Chi-fu Huang, and David Kreps
March 1991
WP# 3311
On
Intertemporal Preferences in Continuous Time:
The Case
Ayman
of Certainty*
Hindyl Chi-fu Huang* and David Kreps^
March 1991
To appear
in
Journal of Mathematical Economics
Abstract
DifTerent topologies on the space of certain consumption patterns in a continuous
time setting are discussed. A family of topologies which give an economically reasonable sense of closeness and have an appropriate intertemporal flavor is suggested. The
topological duals of our suggested topologies are essentially spaces of Lipschitz contin-
uous functions. Any utility functional whose felicity function at any time t depends
on the consumption at that time and is not linear in it is not continuous
in any one of our topologies. A clciss of utility functionals that are continuous in the
suggested topologies and that capture the intuitively appealing notion that consumptions at nearby dates are almost perfect substitutes is provided. We then give necessary
and sufficient conditions for a consumption plan to be optimal for this class of utility
functionals. We demonstrate our general theory by solving in closed form the optimal
explicitly
consumption problem
for a particular utility function.
The optimal
solution consists
of a (possible) initial "gulp" of consumption, or an initial period of no consumption,
followed by consumption at the rate that maintains a constant ratio of wealth to an
index of past consumption experience.
'This paper merges results reported in two earlier papers, a paper with the same title by Huang and
Kreps and a paper titled "OplimaJ Consumption with Intertemporal Substitution I: The Case of Certainty"
by Hindy and Huang. The authors would like to thank Peter Diamond, Darrcll DufTie, Oliver Hart, Hua
He, John Heaton, Larry Jones, Andreu Mas-Colell, Scott Richard, Tong-sheng Sun, Jean-luc Vila, Thaleia
Zariphopoulou, and the anonymous referees for comments. Itzhak Katznelson told us about Orlicz Spaces.
'Graduate School of Business, Stanford University, Stanford, CA 94305
'Sloan School of Management, Massachusetts Institutes of Technology, Cambridge, MA 02139
^Gradu ate School of Business, Stanford University, Stanford, CA 94305
M.I.T.
AUG
LIBRARIES
1
5 1991
RK;fj\/Fn
Summary
Introduction and
1
Introduction and
1
Consider an economic agent
1
Summary
who
from time
lives
consumption commodity consumable
to time
Suppose there
1.
any time between zero and one.
at
We
is
a single
ask the
fol-
lowing questions: First, how might we represent the agent's consumption pattern over his
lifetime? Second,
if
we
A'+ denote the space of possible consumption patterns, what
let
an appropriate topology on X+; that
"closeness" with the appropriate intertemporal flavor and which
mathematically? Third, what form
is
one which gives an economically reasonable sense of
is,
will
equilibrium prices take
tinuous linear functional? Fourth, what
is
agent whose preferences are continuous
in
is
if
relatively well-behaved
they are given by a con-
the optimal consumption-savings behavior of an
such a topology?
There are '"standard" answers to these questions. Consumption patterns are represented
by a function
c:[0, 1]
—
where
3f+,
such consumption patterns
c{t)
u
topology.
In addition, prices
Preferences are represented by U(c)
q.
the felicity function of the individual and y{t)
is
consumption
is
at time
t.
Two
they are close as functions; say, in
c(t) are '"close" if
some L^
the sup-norm topology or in
appropriate value of
c{i) represents the rate of
and
=
come from
Z', for the
f^ u(c{t),y{t)J)dt,
some functional
where
of the consumption
rates. ^
A
This
a generalized version of the Ryder and Heal [1973] preferences analyzed by Magill
is
typical
example
is
y(t)
=
y{t)
+
Jq 0(t,s)c{s)ds for
some functions
y(t)
and
6{t,-).
[1981].
We
at
are dissatisfied with these standard answers.
one time should be something of a substitute
for
Basically
we
consumption
feel
that consumption
at other, near-by times.
Consequently, prices for consumption at close-by times should be approximately the same.
These, however, are not the properties of the above standard answers.
find
answers to the questions of the previous paragraph that
and retain,
at the
will
for
economic analysis.
Readers familiar with the literature on commodity differentiation,
Colell [1975]
and Jones [1981, 1984]
will at this point rightly
of signed measures on
the weak topology.
A
thus set out to
conform to these intuitions,
same time, mathematical properties important
questions are readily available in this literature. Take the
We
example, Mas
for
think that the answers to our
commodity space
to be the space
equipped with the topology of weak convergence, or simply
[0, 1]
consumption pattern
is
represented by a measure on
right-continuous increasing function where the value of this function at time
cumulative consumption from time
as well as gulps are allowed.
to time
Then
t.
[0, 1]
t
or by a
denotes the
In this representation, continuous injestion
as long as the agents' preferences are continuous in
the weak topology and satisfy a strong monotonicity property equilibria exist with prices
that are continuous functions of time.
Preferences continuous in the weak topology treat
nearby consumptions as close substitutes and equilibrium prices are continuous functions
of time,
all
one caveat.
When
functional.
the questions
A
we
raised
above seem to have been answered!. There
merely continuous function
the felicity function
is
is
is,
however,
not, in general, a reasonable representation of
a function only of c and
(,
we have the standard time-additive
utility
Introduction and
1
Summary
2
consumption and asset prices over time.
continuous function can be nowhere differentiable - take, for example, a Brownian
A
motion sample path. In
is
fact,
most continuous functions are nowhere
perhaps not so counter-intuitive when time
is
question.
When
prices for
consumption should be continuous functions of time.
This
differentiable.
not the characteristic of the
commodity
in
the commodities are indexed by time, our intuition certainly suggests that
We
also think that
more
should be true. Prices should not fluctuate over time in a nowhere differentiable fashion.^ In
particular,
model.
we
think that interest rates should exist in a reasonable economic intertemporal
This condition
equivalent to the requirement that prices for consumptions be
is
This additional requirement
absolutely continuous functions of time.
commodity
results of the
We
begin
in
what makes the
is
differentiation literature unsatisfactory for our purposes.
Section 2 with a formulation of the consumption set and consumption space.
For the set of feasible consumption patterns, we use the space of positive, increasing, right
continuous functions on
[0,1],
denoted A'+.^ For given x G A'+ and
the cumulative consumption from time zero to time
t
t
£
[0,1], x{t)
denotes
under consumption pattern
In
x.
order to speak of net trades, the space of consumption bundles, or the commodity space,
the linear span of A'+. Denoted A', this
[0, 1]
is
is
the space of functions of bounded variation on
that are right-continuous.
The space
A'
comes equipped with a number of topologies, any of which could conceiv-
ably serve our purposes. In order to investigate their suitability, we construct in Section 3
a "wish
we
list"
of properties for a suitable topology.
Our wish
mind
give useful mathematical properties, keeping in
do optimization, and conduct equilibrium
desirable economic properties.
The
analysis.
discussion here
list
comes
two
in
that, in the end,
we
parts. First,
will
Second, and more important,
is
want to
we
list
the heart of this paper in terms of
economics; essentially, we are giving our intuition concerning when different consumption
patterns should be "close".
Once we have our wish
done
in Section 4.
We
list,
we can
see
how
well the standard topologies fare. This
is
many
norm
The weak
quickly look at the topologies induced by the total variation
and the Skorohod topology and show that these two
topology has
is
we
of the properties
fail
rather substantially.
desire, and, indeed as
we mentioned above,
a substantial literature on economies where the commodity space
is
there
the space of signed
measures, endowed with this topology; see Mas-Colell [1975] and Jones [1981, 1984]. But,
we mentioned above, the weak topology has too large a topological
see, lacks a certain uniform property that we find appealing.
as
We
turn instead, in Section
will use
for
5, to
dual, and, as
we
will
a family of norm topologies. For this introduction, we
one representative member of this family, given by the norm
I G X; the topology induced by this norm
is
||ar||
=
/q |a-(i)|d<-|-|a:(l)|
denoted hereafter by T.
We
will
show
^Prices for consumption will in general fluctuate in a nowhere differentiable fashion in a model with
uncertainty due to temporal resolution of uncerttiinty. See, for example,
We
weak
nondecreasing, and so
forth.
will use
relations throughout.
Huang
[1985].
For example, negative means nonpositive, increasing means
forth. If the relation is strict,
we
will use strictly negative, strictly increasing,
and so
The Consumption
2
that,
Consumption Space
Set and
on A'+, these norm topologies are
3
weak topology.
topologically equivalent to the
all
However, preferences continuous and uniformly proper, a concept due to MasColell [1986]
discussed in section 2, in any of the proposed topologies exibit a trade-off property which
we
find appealing. This property
is
essantially that agents tolerate delaying or advancing
increasingly larger quantities of consumption for decreasingly shorter periods of time.
contrast, preferences continuous and uniformly proper in
tolerance for this kind of intertemporal substitution.
have
all
the properties
we
the weak topology
The norm
fail
In
to exhibit
we introduce
topologies
from the point of view of economic intuition and,
find desirable
in
addition, they are mathematically very well behaved. For example, preferences continuous
in these topologies
In Section 7,
admit continuous numerical representations.
we
turn to the characterization of the topological dual spaces for our
In particular, the dual to (A', T), denoted, hereafter, as A'*,
topologies.
Lipschitz continuous functions. In this section,
we
also confirm
some
the space of
is
desirable mathematical
properties of the dual spaces, notably that the duals are vector lattices in the order duals
and the order
intervals are weakly compact.
Section 8 gives necessary and sufficient conditions for a consumption pattern to be
optimal for a class of
An
model.
continuous in our topologies
utility functions
optimal consumption plan
in
a consumption-savings
in
genercd will be periodic even
if
the felicity function
has an infinite slope at zero. Whether one should consume at a given time depends on the
relation
between ones wealth and a weighted average of past consumption.
solution for a particular utility function
is
provided in Section 8.3. Finally,
A
all
closed form
proofs are in
the appendix.
There are three companion papers to
this.
Mas-Colell and Richard [1991] does the
hard work of establishing existence of equilibrium for economics based on our, and other,
Hindy and Huang [1990a, 1990b] generalize and extend our analysis to models
topologies.
where there
is
uncertainty and temporal resolution of that uncertainty, and they consider
some of the standard models
the impact of our topology on
The Consumption
2
and Consumption Space
Set
Consider an economic agent who
from time zero to time one. This agent can consume
lives
a consumption commodity at any time during his
the agent
tive
consume only
consumption
positive
is finite.
life-time
consumption from time zero to time
X
is
if
x(<)
increasing,
limits x{t~)
=
it
is
span.
It
is
t.
:
[0,1]
Note that
—
>
5f,
with x{t) denoting the cumulative
this representation
absolutely continuous and in discrete lumps
\ims-\tx(s)
and xif^)
avoid complications,
is
we
natural to require that
consumption pattern can be represented by a
=
permits consumption
if a:{<) is
has the following two properties. First, for each
possible kind of discontinuity
To
life
amounts of the consumption good, and that the cumula-
Thus a
positive increasing real-valued function x
at "rates"
theory of finance.
in the
lim,|tx(5) exist and are
t
6
discontinuous.
[0, 1],
finite.
As
both unilateral
Second, the only
a jump.
shall further
assume that x
is
right-continuous: lim^if x{s)
—
2
x{t), for all
That
6 [0,1).
t
Consumption Space
Set and
The Consumption
is,
if
the agent consumes a nontrivial stocl<
consumption commodity during the time
at
time
To sum
t.*
we
up,
4
he in fact
interval (t~ ,1"^),
The set
[0, 1].
of such functions
denoted by X+.
Throughout the paper, Xr
is,
of the
consumes x{t'^) — x{t~)
will represent the agent's life-time consumption pattern by a
positive increasing right-continuous real-valued function x on
is
amount
for r
G
[0,1] will
denote the indicator function of
Xr represents a unit gulp of consumption at time
That
[r, 1].
Also, scalars will be denoted by
r.
a,b,...
Let
X
denote the linear span of X+, which
functions of bounded variation defined on
X
in the usual way: xi
refer to
is
a
X
>
12
if
x^
We
Our agent
X+
=
will
(x
-
j/)"*"
+
y,
xi
if
use
x+
>
x(0)
X2
if
0,
> and > on
induces orders
xi
-
0:2
G -Y+
consumption
We
\ {0}.
set.
will
Note that
A'
to denote the positive (increasing)
=
then x+(0)
x(0).
Then we can
with inf(x,j/) defined similarly.
be assumed to have preferences over
a complete and transitive binary relation
>:
preferences are increasing: Vxi,X2 G A'+, xi
the direction of xq: Vx G A'+ and Va
special role played by xo here.
>
X+
as the agent's
X, we
also use the convention that
define sup(x,2/)
the space of right continuous real- valued
The cone
X+, and
to
For x G
lattice, in the usual fashion.
[0,1].
X2 £
commodity space and
as the
part of X.
-
is
It
the direction of which preference
>
0,
would
is
x
on X+.
A'+
We
that are given mathematically by
will
assume throughout that these
+ x^ >: ^i; that they are strictly increasing in
+ axo >- x. The reader may be put off by the
suffice to
assume that there
is
some x G
X+ \ {0}
always strictly increasing, although condition
and our uniform properness assumption, to be discussed
shortly,
in
below
(c)
would need to be modified
accordingly. For the analysis of Mas-Colell and Richard [1991], one would wish to use the
social
endowment
Moreover, we
the topology
T
in place of xo-
will
assume that there
relativized to
X+.
is
given a linear topology
T on
A';
with T"^ denoting
Preferences are assumed to be continuous in
uniformly proper with respect to T. Continuity of preferences
standard and needs no justification.
in
a fixed topology
Uniform properness, on the other hand,
standard. This condition, introduced by Mas-Colell [1986],
is
The
merely
recall his definition:
1
(Mas
Colell) Preferences
respect to a linear topology
if
we
will
X+,
v
£
V
,
=
will
and a T-open neighborhood
and scalar a >
0,
x
- ax' + av
assume that preferences are uniformly proper with respect to
particular choice of x*
we
A'+ are said to be uniformly proper with
there exists a vector x' G A'4.
of the origin such that, for every x G
In fact,
T
'^
x.
for the
xo-^
V/ < 0. We will continue to use
*Here we have used the convention that x{t) = i(l) V< > 1 and i{t) =
throughout, so that the value of any i € A' for arguments greater than one or less than zero
always well defined.
*As we mentioned earlier, our use of xo in this special role is purely to ease exposition. Any x' g X+
this convention
is
T
y on
fairly
not so
commodity
reader should consult Mas-Colell [1986] for further commentary. Here,
space.
V
is
is
and
shown there to be intimately
related to the existence of equilibria for economies with an infinite dimensional
Definition
T"*"
A Wish
3
A
3
List for
Wish
5
r
List for
What
properties do
First,
we want
T
T
we want
for the
topology T? The
to be well behaved
of desiderata
list
mathematically. Ideally,
T
comes
two parts.
in
should be a locally convex,
metric topology, and A' should be complete and separable under this metric.
we wish
to invoke uniform properness
wish for
T
and undertake optimization
for
Also, since
our agents, we
will
In order to apply the equilibrium existence results of
to be a linear topology.
Mas-Colell [1986], one would wish that the lattice operations are continuous with respect
to T, or (even) that {A',A'+,T) forms a
much
too
hope
to
Banach
when we come upon economic
for,
and Richard [1991] allow us to weaken
But
lattice.
this, as it will
considerations.
Happily, Mas-Colell
desideratum substantially:
this
turn out, will be
We
want the
will
order intervals to be weakly compact and the topological dual to be a vector lattice in the
order dual.^
The second and more substantial part of our wish list concerns the economics of the
situation. Our intuition suggests that certain consumption streams ought to be close, and
we will look for a topology on A' that accomodates this intuition. Consider the following
requirements, phrased in terms of properties of y.
Two
(a)
patterns of consumption that have almost equal cumulative consumption at every
point in time should be close.
h
Xn
y for
all
n implies x y
We
t.
=
The
idea of this metric
X
any other path y that
e
up and to the
point {t,x{t)). Hence
X
is
within
With
(b)
this
If
e
To do
so,
the Prohorov metric. For x and y in
p{x, y)
moving
n implies y
if
is
inf{f
>
x(<
:
+
e)
depicted in Figure
lives entirely
left
and down
y
within
is
e
y
consumption of a
in
also regarded as insignificant.
known metric on A^,
is
for all
sup(g[o,i] \xn{t)
-
x{t)\
=
+
1;
e
>
the
y{t)
>
0,
then
x.'
consumption between i„ and x must be small
wish to weaken this so that shifts
amounts of time are
limn^oo
if
and y y Xn
y,
In (a), differences in cumulative
time
Formally,
at every
fixed size across small
we begin by introducing a
well
A.)., let
-()-(}.
x(t
neighborhood of a consumption path
e
within a "sleeve" around x that
is
determined by
and to the right (c in each direction) from x at every
of x, then at every time
of the total consumption under y at
t,
the total consumption under
some time no more than
f
away from
t.
we can pose our second requirement:
p(xn,x)
implies y
^
y
as n
—
<
GO, then Xn
y
y for
all
n implies x
y
y,
and y y Xn
for all
n
x.^
1*^0
could be used instead.
^For the definition of a vector lattice and the order dual, see Schaefer (1974).
^The astute reader will notice that this rules out the standard model where
felicity of
consumption
rates, unless felicity
reader will also note that (a)
is
linear.
We
will
could be said more compactly
demonstrate
as:
T
utility is the integral of
this explicitly in section 6.
The
relativized to .\'+, denoted T'^ should be
no stronger than the sup norm topology relativized to A'+.
*Or: T'*' should be no stronger than the weak topology on A'+ generated by the space of continuous
functions on
[0, 1].
A Wish
3
Our
T
List for
6
economic consideration sharpens (b) somewhat. The idea
final
amount
shift of a given
great consequence in terms of preference. Suppose that
increasingly larger
may
amount
of consumption across a small
in (b) is that the
of time should not be of
we imagine
instead the shift of an
amount of consumption over a decreasingly smaller period of
time.
It
help to give a concrete example: Consider comparing the consumption patterns anXi/2
and anX(n+i)/2ni where a„ is a strictly positive scalar. As n goes to infinity, the shift in
time grows small (and so, in particular, p(anXi/2i«nX(n+i)/2n) ~^ 0)i but if a„ grows with
n, the
amount
for a„ to
shifted
go to
grows large.
wish to assert that, a priori,
enough so that the difference
infinity slowly
To be more
negligible uniformly.
We
precise,
it
should be possible
terms of preference becomes
in
we know, by the presumed
strict
monotonicity of
Xo that Xo + anXi/2 ^ a„Xi/2- ^^ wish to assert that the "added utility"
from \o eventually compensates for the shift of On from 1/2 to {n + l)/2n, if a„ goes to
infinity sufficiently slowly. For example, we might insist that, if a„ is o{n) (i.e., an/n -* 0),
>-
in the direction
then the growth in a„
order to
come
relative to the
growth
in the
Or we might
slow enough.
is
require that a„
is
o{n^'^), or o(lnn), in
to the desired conclusion that the shift in consumption through time
amount
amount
of consumption shifted. But there
shifted such that,
if
some
is
the shift in time
is
is
small
"sufficiently slow" rate of
small, the change in utility
is
small.
To measure what
is
meant by a
"sufficiently slow" rate of
are given a concave, continuous, strictly increasing function
/i(0)
p{z)
=
=
and
\n{z
+
—
fJ,{n)
1).
By
a-nlp{n) —* 0, or On
we
=
oo as n
—
/i
growth, we suppose that we
:
[0,oo)
For examples, think of n(z)
oo.
^
=
[0,oo), such that
z^^^ for
p >
"slow growth" of a sequence of numbers a„ relative to p
o{p{n)). Note that linear growth
is
1,
or
meant
the fastest "slow growth" that
is
are willing to contemplate, since p must be concave. In general,
we pose the following
condition.
(c) For fixed
and
as above,
{j/„(l)} are
Xn
n,
p
+ Xoy
if
{xn) and {y„} are two sequences of claims with
both o{p{n)), and
(ii)
p{yn^Xn)
<
l/n, then for
all
(i)
{x„(l)}
sufficiently large
Vn-
Note that the condition depends on the
when p{z) =
We
fixed p. It
becomes stronger the
faster
p grows,
p is given uniformly for all agents
in the economy; it enters as part of the data of the economy under consideration. But we
do not wish to prespecify what rate p is "slow enough." We only suppose that some "slow
so
it is
strongest
enough" rate can be specified
z.
in
are imagining that
a particular economic application.^
'it has been said to us that this condition is not very natural - how would one discover what is the
appropriate function p for a given population of consumers? We suggest the following. First, for a given
consumption path i, let N,{x) denote the set of x' G A'+ that are within f of i in the Prohorov metric.
We
suggest that a natural property for consumer preferences
I "delayed and lowered by
Nt(x)
is
sleeve
drawn around
is
That
is,
in
is:
For each i €
terms of figure
1,
X+,
the >;-worst element of
the worst consumption pattern in the
one admits to this structural
at the end of section 6, if
added to that representation), then we can ask of a given consumer: Take any path
x would be the lower-right boundary of the sleeve.
property (which, we observe,
some impatience
f".
will hold for the sort of representation
If
we suggest
Three Candidate Topologies
4
7
Three Candidate Topologies
4
Several topologies on the space of functions of bounded variation have been extensively
analyzed by mathematicians. In this section, we ask how each of these do
wish
terms of our
in
list.
For the space of functions of bounded variation, a mathematically natural norm
For x £ A', the total variation of x, denoted by TV(a;),
variation:
/J |di(<)lvariation
(The L^ norm on consumption rates in
norm on cumulative consumption.) Note
well that
we
topological vector lattice with A'+ the positive cone,^° then
norm
at least as strong as the total variation
topology.
well mathematically, aside from lack of separability,
economic wish
Xn given by
(a),
In the first place, (a)
list.
= k/n
a;„(<)
for
e {(k
/
we should have x„ converging
norm. In the same way (b)
is
if
- l)/n,k/n] and x
to x, but (of
+
need to have a topology
will
But while
performs
this topology
in
terms of our
consumption patterns
=
given by x(t)
course) we do
|3-(0)|
to have A' be a
performs miserably
it
total
just the total
is
we hope
violated: Consider the
is
TV(x) =
is
the standard models
is
According to
t.
not, in the total variation
when the underlying time space is a
have the sort of economic qualities we desire.
violated. Total variation,
continuum, measures time much too
finely to
Consider next the Skorohod metric originally introduced on the space of functions which
are right-continuous and have finite
Let
A
denote the class of
=
A e A, then A(0)
strictly increasing,
and A(a) =
strictly positive scalars ( for
and sup^gfQ
that (A',
\x{X{t))
ji
-
y(t)\
(RCLL). This metric
left limits
For x,y e
1.
continuous mappings of
X,
which there exists
<
(.
It
is
defined as follows:'^
define S{x,y) to be the
in
A
:
X
X
X
—^
R+
itself.
If
infimum of those
a A such that supj^rg
can be shown that S
onto
[0, 1]
ij
|A(<)
—
t\
<
e
a metric, and
is
a separable metric space.
5") is
Moreover, this metric does possess some of the economic qualities that we desire. Since
two consumption patterns that are
metric, (a)
we
define
i„(0 =
by !„(<) =
a:„
1
Unhappily, (b)
satisfied.
is
for
>
<
(n
close in the sup
+
for
/
<
1/2,
is
not
In particular,
difference x«„
it
=J>
t
G [1/2, (n
The Skorohod topology
Xt does not
approach zero.
frequently used on A'
We
is
still
—
is
+
l)/2n], and
also not very well
Xt in this topology
=
k.
if <„
—
/,
their
continue our search.
the topology of weak convergence denoted by
denoting convergence in this topology. Let C[0,
X € A'+ with z{a)
for
if
(so (b) should apply), but Xn does
l)/2n, then p(x„,Xi/2) -^
not a linear topology. Whereas Xt„
One other topology
%,, with
note, in this regard, that
comes to exploiting the linear structure of the space of consumption claims.
it is
—
are close under the Skorohod
We
i„(0 = 2n{t - 1/2)
not approach Xi/2 in the Skorohod topology.
behaved when
norm
satisfied.
1]
be the space of continuous
Suppose you possess this path. What is the most you are willing to have this path
Now what path x
if in compensation we give you an additional xo up front?
"delayed and lowered" by,
minimizes this amount, over all paths with z(a) = it? (For preferences with representations as at the end of
6, this optimization problem is tractible.) Finally, how does this amount change as k increases? The
section
answer to this question, of course, gives us /i.
'"which we would want if we were to rely on the existence result of Mas Colell [1986]
"a
good reference
is
Billingsley (1968, p???).
A
5
Family of
Norm
Then x^
functions on [0,1].
/eC[0,l]
x if and only
=>
if
-*
Jq_ f(t)dx(t) for
Jq_ f(t)dxa(t)
all
12.13
Recall that the
separable
8
Topologies
weak topology
Holmes
(cf.
is
a linear, Hausdorff, locally convex topology.
[1975, exercise 22c]).
It is
It
is
metrizable on A'+ by precisely the Prohorov
metric (hence has a countable neighborhood base and
sequential), but
is
it is
not metrizable
on X.
point of view of economic desiderata, the
From the
Preferences continuous in the weak topology on
X+
weak topology
obey (a) and
will
quite an attractive candidate, except for being non-metrizable on
all
is
well behaved:
(b). Altogether,
of A'.
It
it is
has the further
virtue of having been thoroughly analyzed in the economics literature, e.g., in Mas-Colell
and Jones [1981,
[1975]
1984].
Preferences that are continuous and uniformly proper in the weak topology need not be
behaved
well
terms of
in
Fix some
preferences:
(RecjJl that /i(oo)
That
A
is,
=
/z,
oo
any fixed function ^, however. This
is
and consider the continuous function f{t)
=
(c), for
is
assumed.) Define linear preferences by x
preferences are given by evaluating consumption
fortiori,
y
=
a:„
is
\/Ji{n)Xo and
=
j fd{Xn + Xo)
Roughly put, "shadow
?/„
=
l
quite an impact,
if
+
prices" for
that
amount
=
—
0.
shadow
^J^)< j fdyn
A
consumption
amount
We
is
is still
at times just after
we norm C[0,
rise
very quickly.
shift
is
should become negligible, but
we have picked
around zero quickly enough so that the prespecified
Norm
Topologies
too strong along the time dimension.
1]
time zero
too much.^''
'^Integrals here are defined from
'"'if
strictly
of consumption from time zero to time \/n has
have discussed three possible topological structures on
total variation
is
increases quickly enough. Condition (c) holds that there
prices (given by /) to rise
Family of
it
= ^fiy)il + 2y/lU^)).
that the slow-enough-increase must be prespecified, and
slow-enough-increase
5
j fdx > f fdy.
by /.
plans at the "prices" given
But, for every n,
a slow enough increase in the amount shifted so that the
is
1.
\/^4")Xi/n- Note that x^ and yn qualify in terms of
l/x//x(n)
a result, shifting an increasing
the point of (c)
i(
+
strictly positive.
(c). In particular, Xn(a)/fi{n)
As
y
2y/l/n{l/t)
continuous and uniformly proper in the weak topology, and
is
increasing because /
Consider
y
so even for linear
0-
to
1,
A'.
The Skorohod topology does
so that /(O)i(O) enters as a
bv the sup norm, then the weak topology
The topology generated by
is
better,
term in the integral.
weak* topology of A' generated
first
just the
byC[0,l].
'*For any continuous function /, and for preferences defined from / as in the example, we can find a rate
^l that grows slowly enough so that (c) would be satisfied for these preferences and for /i. Thus the
order of quantifiers in (c) is all important - /i must be picked first, and then, what we have shown, there
function
are "nice"
for this
ft.
(i.e.,
linear) preferences continuous
and uniformly proper
in
the
weak topology that
violate (c)
A
5
but
Norm
Family of
violates (b)
it
9
Topologies
and
is
The topology
not linear.
of
weak convergence
except that preferences continuous and uniformly proper
we
In this section,
(c).
will
norm
introduce a family of
in
is
quite suitable,
weak topology may
the
violate
topologies which will be free of
all
these defects.
To do
this, the
Musielak [1983]
is
reader must be introduced to the notion of an Orlicz space, for which
an accessible reference. To make this paper close to self-contained, we give
a very quick introduction here, adequate
for,
To begin, consider the case where the
that we defined the following norm on X:
and told
terms of our particular application.
in
=
function n{z)
z^'^, for
some p >
Imagine
1.
i/p
=
Mv
Since our functions x are
all
of
From standard Banach space
logical linear space.
It is
I
+
\x{t)\Ut
\x{\)\^
Jo
bounded
theory,
variation,
it is
clear that all these integrals exist.
it is
clear that (A',
worth noting, perhaps, that
X
||
a separable normed topo-
||p) is
be complete
will not
in this
norm:
Consider, for example, the sequence of right-continuous, bounded variation functions i„
that are indicator functions of [l/2n, l/(2n
-
U [l/(2n -
1))
2),
l/{2n
-
3))
U
...
U
[1/2,
1].
These are Cauchy and converge to the obvious measurable function, but what they converge
to has
unbounded variation and
is
not right-continuous. A'+, however,
This norm has the right "flavor" for
•
II
in
lip
(c) as well, for the case /i(z)
is
=
complete.
Roughly put,
z^/p.
trades off differences in consumption paths in time and in amounts, where differences
amounts are scaled by being
consumption paths
then the difference
What,
circle in
then, do
0(1 /n)
is
in the
power
p.
time dimension and
Hence
if
the difference in two
o{'n}l^) in the
space dimension,
small.*^
is
we do
X, denoted
raised to the
if /i(z) is
not of the form 2^/p? For
/x
define
rj
=
/x"^
and the unit
C,,, as
C„ = {xe
X
:
C v{Ht)\)dt + 7?(|x(l)|) =
1}.
Jo
The convexity
measuring
all
of
r/
ensures that this unit circle
is
appropriately shaped to be a "gauge" for
other vectors x: Define, for x other than i
positive scalar a that satisfies
||a:||,,
is
xja G
C,,. It
well defined. Moreover, for /x(z)
definition of the
norm. With
Proposition
{X,\\
1
[[,,)
is
operations defined in Section
=
=
0, ||x||^ to
be the unique strictly
takes a proof, but because of the continuity of
z^l^, this definition is
r/,
equivalent to the standard
this definition:
a separable
2,
normed
topological linear space.
With the order
order intervals are weakly compact in each of these topolo-
gies.
We
omit the proof
for the first assertion of this proposition as
of the theory of Orlicz spaces; see Musielak [1983, theorem
a proof of the second assertion.
''See Proposition 4 for an exact statement.
1.5].
it is
a simple application
In the appendix,
we provide
A
5
Family of
Norm
10
Topologies
Note that a countable dense
set of (A',
•
||
||^)
the set of piecewise linear functions that
is
change their values at rational time points and have rational right-hand derivatives. These
Thus the space
functions are absolutely continuous.
dense
(AM|
in
norm
the
||r,)
||,,
Fixing n and the corresponding
the
we
topology relativized to A'+
Tjj
the topology induced by
77,
denoted by 7^, and
is
||^
||
denoted by T^+. For the special case where n{z)
is
=
z^l^,
use notation such as Tp.
The
following basic results are established in the appendix.
Proposition 2 For any
is
is
by consumption at rates.
•
||
of absolutely continuous functions
- any consumption pattern can be approximated arbitrarily closely in
/i
and corresponding
weaker than both the Skorohod and
rj,
T\
is
no stronger than
which in turn
Tj,,
total variation topologies.
The first part of this proposition is fairly
more than linearly, large differences in
intuitive. Since
7;
is
convex,
Note that,
the dimension of space.
topologies, the lattice operations are most definitely not continuous
topological vector lattice. For example, Xi/2-X(n+i)/2n
~
tends to accentuate,
it
and
(A',
in
our norm
7^)
is
not a
0- but the corresponding positive
parts (or negative parts, or absolute values) do not converge.
We
will
be looking
complete, transitive, continuous
7^, for
some
on A'+ that are given by a binary relation y that
for preferences
prespecified
We
tf.
T^
the
in
will
topology, and uniformly proper with respect to
continue to assume that
y
increasing as before.
is
following results establish that
we
Proposition 3 Preferences
that are continuous with respect to
Moreover, Xj^
is
>:
separable under
preference relation
y
T^
,
T^
satisfy (a)
and hence every continuous (complete and
omitted.
and
(h).
transitive)
on A'+ admits a continuous numerical representation.
ately implies the satisfaction of (a).
is
The
are looking for the "right" thing given our desiderata.
Satisfaction of (b) follows as a simple corollary to Proposition 5, which in turn
proof
is
The
The
separability of A'+ follows standard lines
existence of a numerical representation
is
immedi-
and the
standard given continuity
and separability - see, for example, Fishburn [1970].
Proposition 4 Fix
ing as above,
fi
and
the corresponding
and 7^ uniformly proper
rj.
Suppose that y
is
for the specific choice x'
T^ continuous,
= \o. Then y
increassatisfies
(c).
In the appendix,
in
Tr,.
Thus
for
we prove one technical
any neighborhood
V
— x„ =
y„
must not stand
This leaves open one
on
X+?
It is
with respect to 7^
But:
is
+ a:„,
in the relation
issue:
clear from the
x„ and
of the origin,
the definition of uniform properness to xo
j/„
fact: If
How do
y
?/„
- Xn
j/„
are as in (c), then
is
-|-
the topologies
Xn-
T^
^
eventually inside V. Apply
to conclude that, eventually, Xo
to \o
y^- x^
That does
+ a;„ - Xo +
it.
compare with the weak topology
example of Section 4 and Proposition
4 that
uniform properness
stronger than uniform properness with respect to the
weak topology.
1
6
On
the Standard Models and the 7^ Topologies
Proposition 5 For any
ft
and
T^
rj,
is
1
weak topology on
topologically equivalent to the
X+.
Hence we reaffirm (the obvious): Uniform properness
is
not just a topological property
on X^.
On
C
the Standard Models and the 7^ Topologies
The standard models
of preferences for consumption through time assume that preferences
are defined over consumption in rates and are given by a numerical representation of either
the time-additive form
U{x)= [
u{x'{t),t)dt,
Jo
or the non-time-additive form
U(x)=
where
x'
of
The
x'.
denotes the
first
f\ix'{t),y{t)j)dt,
(1)
derivative of cumulative consumption x, and y
is
some functional
The non-time-additive form
[1973] by Magill [1981], who used
time-additive form needs no further explanation.
Ryder and Heal
includes, for example, a generalization of
= yit)+ I
yit)
9{t,s)x'{s)ds,
Jo
for
some
real-valued functions y and 6, and a special case of Magill [1981], the so-called
habit-formation model, recently analyzed by Constantinides [1990] and Sundaresan [1989]
when
0{t,s)
=
e~^*'~'' for
all 5
<
<
and zero elsewhere.
Preferences so given are not defined for consumption other than at rates (where x
is
not absolutely continuous), but we might consider the following procedure for extending
preferences to general x: Find a sequence of absolutely continuous Xn that approximate x,
and then define U{x)
=
lim„_oo U{Xn), hoping that the limit
of the approximating sequence
a:„.
We
will
it is
well defined
want preferences to be
sense of approximation had better be no weaker than
remarked that
is
T^
T^
continuous, so the
convergence - and we have already
possible to approximate arbitrarily closely, in terms of
by absolutely continuous x„. Hence
and independent
T^
,
any x G A'+
program does have some hope of working.
this
It will
only work, though, in very special cases:
Proposition 6 Suppose
that preferences
on A'+ are represented by
f/(i)= f\{x'{t),yit),t)dt
the utility function
(2)
Jo
for some (jointly) continuous function u, for
X,
all absolutely
continuous consumption paths
where
y{t)
= yit)+
/
Jt-ki{t)
0ii,s)dx{t-s)
(3)
12
7
Duality
for
some continuous function
y,
preferences are continuous in
any
(for
T^
point of this
explicitly
is
that
=
and
+
a(y,t)z
(i(y,t),
is
is
x'.
If these
where a and
every point
felicity at
/?
are
that converge in 7^ to
a;„
Indeed,
ar„
converges to x
in
time depends
in
in the literature),
not continuous. In the proof of the above proposition,
not linear in
0.
t.
when the marginal
a sequence of consumption patterns
and
consumption paths
on the consumption rate at that time (a case of great interest
the utility function
u{x',y,t)
ki, k2,
relativized to absolutely continuous
,
then u must have the form u{z,y,t)
t]),
jointly continuous functions of y
The
and nontrivial continuous functions
we constructed
while t/(i„) /* U{x)
a:
norm
the sup
if
topology, so any
treatment of preferences for consumption through time that asked for property (a) would not
admit the standard models
felicity at
is
any time
is
in the literature.
The reason should be
clear:
Unless marginal
independent of the consumption rate at that time, consuming steadily
not well approximated in the standard models by consumption at quickly varying rates.
Yet the cumulative consumption paths of such patterns, and thus the corresponding
be made close
What
in
sort of
[1984, p. 525],
y,
can
the sup norm.
model might then be used to replace the standard model? Following Jones
we make the
that the marginal felicity
be one where the
is
following proposal:
too sensitive to
felicity at
time
t
is
The problem with
the standard models
is
current consumption. A more apt model would
derived only from a functional of the recent past
consumption and possibly the nearby future consumption; and the current consumption
contributes to the current felicity only through this functional.
Proposition 7 Let u{z,t)
and continuosly
be jointly continuous
differentiable in its first
argument. Then
W(x)= /
u{y{t),t)dt
Jo
for
all
X+,
X e
and k^ such
from
zero,
is
continuous in
that either k\ is
and some
The form
T^
,
where y(t)
a constant or ki
is
is
defined in (3) for
differentiable function 9, with
bounded derivative.
given in the above proposition does indicate the sort of "intertemporal substi-
tution of consumption at near-by times" that motivated this study.
for this
7
form to represent uniform proper preferences
Other conditions, which we leave
zero.
some functions ki
bounded away
differentiable with derivative
is
for y{t) to
for interested readers,
One
sufficient condition
be bounded away from
can also be generated,.
Duality
For the duration of this section, we assume that some pair
/x
and
77
as in Section 5 has
been
fixed.
Mas-Colell and Richard [1991]
tells
us that
it
will
be possible to establish existence
of equilibrium in economies where preferences are increasing and uniformly proper in our
7
13
Duality
topology
we can
if
establish that the topological dual forms a vector lattice in the order
drawn from the order
dual. Moreover, equilibrium prices for these equilibria will be
Hence our aim
in this section
is
to characterize the topological dual space of (A", 7^), which
we denote by A'^, and to verify that A'^ does form a vector lattice in the
More important, we will show that A'* is composed of absolutely continuous
marks the topological
time. This
As a consequence, X^
is
7^,
the whole of A', %,
order dual.
functions of
and the weak topology. Although
is
weaker than the weak topology.
a proper subset of the space of continuous functions and does not
contain any function that
Our
between
distinction
X+, on
topologically identical on
dual.
not absolutely continuous.
is
analysis here will be simple applications of the duality results developed in Musielak
However, the reader should note that A'
[1983, section 13] for Orlicz spaces.
subspace of an Orlicz space, and thus A'^
is
a much
is
a proper
smaller space than the topological dual
of an Orlicz space.
Before proceeding, a definition
Definition 2
an
is
t]
N -function
is
if
=
lim
= supfrs —
T)*(s)
Theorem
and only
Let
1
if
t]
be
an
r](r)
N
and
:
r
=
lim
ztoo
Z
ziO
Putting
in order.
>
0],
tj*
function and
the conjugate function oft].
is
<^
oo.
Z
A"
:
—
»
3?
a linear functional.
an absolutely continuous function f on
there exists
(l)(x)
= [
f{t)dx{t)
Vi 6
[0, 1]
Then
<j>
E.
X^
if
so that
A',
JOwith
f
£ L^
,
where
Z"*
=
{y: f' V'illvimdt
v'il\y{t)\) --
+
as j
^
0+},
Jo
and
T]'
is
the conjugate ofr]. In addition, in the above representation,
functional
Thus,
if
if
and only f
T/
is
a positive linear
positive.
is
satisfy
derivatives. A'* contains
functions include
ri(z)
=
\\y\\r,'
all
<
oo. Since Lipschitz continuous functions
the Lipschitz continuous functions.
z^ with p
>
continuous functions whose derivatives
if
is
an TV function. A"* can be identified with absolutely continuous functions
whose derivatives y
Now what
4>
r]
is
1.
In these cases. A'*
lie in
L' with
asymptotically linear and thus
course the linear case
show immediately that
r]{z)
=
\z\.
Arguments
is
-
+
-
=
have bounded
Note that examples of
is
1.
not an TV function? This includes of
identical to those used to prove
in the linear case AT* is
N
composed of absolutely
Theorem
1
the space of absolutely continuous function
with bounded derivatives - exactly the space of Lipschitz continuous functions. For
t?
that
is
14
Optimization
8
asymptotically linear, one quickly shows that 7^
The
the space of Lipschitz continuous function.
is
Thus
equivalent to T\.
following theorem,
whose proof
A'^
is
is
also
omitted,
records this fact.
Theorem
2 Let
be asymptotically linear in that
7/
lim
z—>oo
for
some
Then
scalar a.
G A'^
4>
if
(f>{x)
^
and only
=
= a >
0,
z
f
if
f{t)dx{t)
X
£X
Jofor
some
Lipschitz continuous function f
.
In addition,
(j)
is
and only
positive if
if
is
f
positive.
Given that any
it
is
77
must be asymptotically
for all
77.
Note that
and limj^oo vi^) = 00, if 7? is not an N function,
Thus Theorems 1 and 2 have completely characterized X^
convex with
linear.
77(0)
=
equilibrium prices for consumption are representated by an absolutely
if
continuous function of time, f{t), the interest rate exists for almost
-f'{t)/f(t) at time
The
all
t
and
is
equal to
t.
following proposition shows that
Proposition 8 For any
the space
rj,
X^
X*
is
is
a
lattice in the order dual.
a lattice in the order dual.
Optimization
8
we formulate
In this section
the optimal consumption problem for an agent whose prefer-
ences are continuous in our topologies.
This agent also has a preference for
final
wealth.
Formally, the agent's preferences on A'+ x S?+ are represented by the functional
f\iy{t),t)dt
+
V{w{l+)),
Jo
where
y{t)
=
2/(0-)e-^'
+
/' 6-^('-')dx(0
(4)
Jo-
>
and
at time 1.
The
function u satisfies the conditions of Proposition 7 and
in its first
argument, and
with y{0~)
> 0,and where
V
w{l'^) denotes the agent's wealth after consumption
continuous, strictly concave, and increasing.
an exponentially weighted average of past consumption and
of general y's defined in (3). Higher values of
consumption and
The
w{0).
single
less
t,
/?
is
strictly
concave
Note that y
is
a special case of the family
imply higher emphasis on the recent past
emphasis on the distant past consumption.
consumption good
At any time
is
is
the numeraire and the agent starts at time
with wealth
he allocates his wealth, w(t), between consumption and investment
in
,
,
Optimization
8
a
15
with instantaneous rate of return
rislcless asset
any consumption pattern
We
sup
/
6A'+
xex
^0
Note that w(i)
A
straint.
a
is
.
a continuous function of time. For
r{t),
denote the times of jumps of i.
+
V(w(l'^))
dw{t)
-
w{t)r{t)dt
w{t^)
=
i£;(r,)
w{t)
>
0,
-
will
dx(t)
fort G
- Ai(r,)
=
f3[dx(t)
y(n)
=
y(T^
-
+
)
y{t)dt]
/3Ax(r,)
r,+i)
(t,,
for all
t
forallr,-,
for
dy{t)
hrt G
G
t
(5)
[0,1],
(7-,,r,+i)
for
alH
forallr,.
continuous function and (5)
left
clear that at
is
u{y(t),t)dt
consumption pattern x G A'+ that
wealth w{0)
It is
policy
.
investigate the optimal consumption decision of the agent:
s.t.
initial
x, Ti,T2,.
is
just the intertemporal budget con-
satisfies the
budget constraint (5) with the
simply called budget feasibie.
any time
t,
given
u;(i)
and y(t~), the subsequent optimal consumption
independent of past decisions. Thus, {w{t),y{t~)) are, using the terminology of
dynamic programming, the "state variables"
for the
above dynamic choice problem.
thus define the "indirect utility function" or the "value function" J{w(t),y(t~
supremand of the agent's optimization program starting from
w and
dynamics of
t
),t) as
We
the
given {w{t),y{t~)) and the
y described in (5) and (6).
Heuristic Derivation of Necessary Conditions
8.1
We now
use heuristic arguments to derive a set of necessary conditions for a consumption
pattern x to be optimal.^^ These conditions will imply that an optimal consumption pattern
can have gulps only at
The reader
approach
=
0.
After
=
t
0,
consumption occurs only when pjy —
will see that the following derivations deviate
Jw
somewhat from the standard
dynamic programming. This occurs because consumption other than
in
feasible, and,
is
<
more important, the standard Bellman's equation
will
at rates
be linear in the
"control" and hence the standard arguments are unapplicable.
Suppose x'
is
be (w'{t), y'{t~
in
w
w
and
and
(a)
all
y.
an optimal consumption pattern, and
for
))
G
t
[0, 1].
Assume
the corresponding state variables
J
is finite
and differentiable
Let Juj{w,y,t) and Jy{w,y,t) denote the partial derivatives of J with respect to
y, respectively.
Along the optimal solution, J should
Bellman Optimality Principle
times
let
that the value function
<
<
<
r
<
:
The Bellman optimality
principle'^ implies that for
1
J{w'{t),y'{t-),t)
= j\iy'{s),s)ds + Jiw-iT),y'{T-),T).
'*These heuristic arguments can be made formal. The reader
'^See, for example,
satisfy the following conditions:
Fleming and Rishel [1975,
p.??].
is
(7)
refered to Blaquiere [1985] for details.
Optimization
8
Form
this,
16
immediate that J{w'{t),y'(t~
it is
times of the jumps of x'. This
is
so as the
),t) is
jumps of x* contribute
only through their contribution to y' and at the
jump
to the integral of
a continuous function of time even at
moment
to the agent's satisfaction
jump
of a
the contribution of the
felicities is zero.
For brevity of notation,
we
will use J(t) to
denote J(w'{t),y'(t~),t) and similarly use
w
Jw{t) and Jy{t) to denote the partial derivatives oi J{t) with respect to
(b) Continuous Marginal Value of
rates, the marginal value of
r'
+ Ai
y
Along the optimal
:
6
<
y, respectively.
policy, given the interest
must be equal to the marginal
[0, 1)
,
'^'^'
value of eeJ>
in
.
w and
units of wealth at time
€
and
units of wealth at time
"
x' will be called for to change w' and y'.
t
+ At
That
A< >
for small
is,
or else
some changes
we must have
JAt)^ = 7^(< + AOfer^'^(^)'^\
Letting At
Along the same
thus Jw{t)
is
Similarly,
at
t
we
0,
[
Thus Jw{t) must be right-continuous on
we
show
that Jwit) must be left-continuous on (0,
arguments,
get Ju,(t)
line of
continuous on
must be equal to
its
—
-|-
is
(c)
»
'
implies, as
<•
f
if
increase in y(t~)
felicity
function on {t,t
+
+
Jy{t
+
Uy{y'{s),s)ee-^^'-'+^'Us
-f
At] and to
At)€e-^^'.
(9)
[0,1).
The
Jy{t)€e-^^^
(10)
0,
that
:
By the hypothesis that
_
y'(t)
is
r(t) is
continuous in
continuous on some time interval
^^
in
{t
— At,
'
uyiy
t -\-
A/), then, as
At —
(t),t).
a time interval over which x' has no "gulps", (9) implies, as At
^dAll = -Uy(y'{t),t) + 0y{t).
'We
t,
(8)
-
and
Hence,
left-
0.
dJJt)
Next note that
and
is,^*
Uy{y'is),s)ee-f^^'-'1ds
Dynamics of Marginal Values
At —
1]
established by noting that
Jy{t-At)e=
At —
At. That
(
we conclude that Jy{t) must be right-continuous on
0,
continuity o{ Jy{t)
letting
principle, a marginal value of
marginal contribution to the
/
[0,1).
[0, 1].
Jy(t)e=r
and
Ju,(<"'").
by the Bellman optimality
the value function at time
Letting At
=
(8)
use Uy(y,t) to denote the partial derivative of « with respect to
y.
—
0,
that
•
0,
.
Optimization
8
17
Also note that at
=
<
Times
(d) Appropriate
an interval
(r,, t-,+i
)
Jw(a) = V'{w(a)) and Jy(a) =
1,
"Gulps"
for
when our
:
0.
Applying Bellman's optimality principle
solution prescribes consumption at rates,
in (7)
on
we
get the following
for
alK G (r.,r,+i)
Bellman's equation:
m^ix{u{y'{t),t)
+ lw[r{t)w'{t)-x'{t)] + lyf3[x'{t)-y'{t)] + Jt} =
(11)
Note that the Bellman equation
sumption rate that
=
i'(0 6[0,oo]
x'it)
The Bellman equation,
for
(r
some
—
e,r
e
+
>
0.
=
oo
when
l^{t)
-
>
,
when
J^,(t)
- 0Jy{t) =
,
when
7^{t) - 0Jy[i)
By
0.
PJyi''')
<
0.
.
- PJyii)
continuity, Jw{t)
be negative on {t
will
Hence our candidate policy should prescribe a jump
moment
l3Jy(T)
=
on
r,
when
at r
-
(,t
for all points
Since this cannot happen as there can only be a countable
e).
-
<
Suppose that we prescribe a "gulp"
our policy might prescribe a "gulp" only at a
that 7u;(r)
f3ly{t)
therefore, suggests that a "gulp" of consumption might be prescribed
any moment r when JwiT) -
Jw(t) - f3Jy(T) <
Hence, the con-
x'{t).
satisfies (11) is given by:
x'{t)
at
consumption rate
linear in the
is
number
+
e)
t
G
of jumps,
this particular interval, such
0.
at
No "Gulps" after =
time r > 0. Note that by
at
any one moment. This, together with the continuity of the marginal values of
(e)
t
Now
:
suppose that our policy
right continuity of x',
a "gulp" of size
calls for
A
we cannot have two successive jumps
w
and
y,
are differentiable functions of time, this implies that Jw{t)
—
implies that there are two strictly positive numbers fi,f2 such that:
JUi)-0Jy{t)>O
7w{t)-0yit) =
when
O
PJy{t)
is
decreasing just to the
left
t
on
Jw{t)-fiJyit)>0
Assuming that J\v and Jy
(r-ei,r),
on
=
T,
(r,r
+
€2).
of r and increasing just to the right of r. This condition,
however, cannot hold for any size of
jump A.
i,From the dynamics of
Jw
and Jy, we get
that:
|[7.(r+) But by
strict
/?7,(r+
)]
- j^yUr) - PJyir)] =
concavity of the
felicity
(3[uy{y'{T)
+
/3A,r) - «,(j/-(r),r)]
function in y, this last quantity
is
.
(12)
negative. Hence,
if
di[Jw — /3</y] were negative just before the jump, it would be strictly negative just after the
jump, thus violating the condition that Jw - 0Jy be increasing just after a "gulp". Note
that our analysis
possibly at
<
=
is
not valid for
and
i
=
I.
A
<
=
gulp at
and
t
=
1,
/
=
1,
thus x' does not have any gulps, except
however,
cannot derive any consumption satisfaction from
it
is
clearly sub-optimal since the agent
and the
final
wealth
is
aJso reduced.
Optimization
8
18
All the above leads us to conclude that x' should take the form of a possible "gulp" at
t
=
followed by periods of consumption mixed with periods of no consumption.
0,
should prescribe a
jump Ai*{0)
at
<
=
0,
jump should maximize
then the size of the
value function immediately after the jump. That
is,
the
jump
If
size Aa;'(0)
we
the
should solve the
following problem
max J(u;(0) iFrom the
order condition,
first
initial possible
at times of
jump
at
<
=
f,
we then know
+
j/(0~)
/3e,0+).
that J^i^"^)
=
fiJyi^'^)-
Moreover, after the
0, the following condition should be satisfied for
all
t
G
(0, 1];
no consumption:
+ lwr{t)w\t) -ly^y'{t) +
u{y'{t),t)
lt
=
(13)
Q,
together with
A(0-7^(0<0,
and at the times when consumption occurs
we must have
at rates,
[7^(0 - /37,(0]x*'(0 =
More
generally,
(14)
0.
(15)
consumption only occurs when J w{t) - PJy(t).
strategy of no consumption
is
sub-optimal and hence
uiy'(t),t)
The necessary
conditions on
value function should at
all
During these times the
+
Jwr{t)w'(t) - lydy'it)
+ 7, <
0.
(16)
J can be expressed more compactly by
stating that the
times satisfy the following differential inequality:
max|u(j/,0
+
Jyjwr{t)
Besides this differential inequality,
it is
-
Jy(3y
+
clear that
Jt, (iJy
J must
-
J^,]
=
also satisfy
0.
(17)
two boundary condi-
tions:
J{w,y,l)
J(0,j/,0
The first of
The second
=V(w)
and
=f,'u{y€-0i'-'),s)ds-\-V{O).
these boundary conditions follows because at
is
i
=
1
no consumption
is
,
,
^
'
optimal.
a consequence of the positive wealth constraint - when the agent's wealth
reaches zero, he must henceforth refrain from consumption.
We
will
show
in the following section that the differential inequality plus the
ary conditions are almost sufficient for optimality.
readers familiar with dynamic programming.
are allowed, standard results in
We
two bound-
This deserves some commentary for
observe that
if
only consumption rates
dynamic programming show that the Bellman equation,
together with the appropriate boundary conditions, are not only necessary but also
cient.
suffi-
This occurs because the optimal consumption rates would be functions of J. Readers
unfamiliar with dynamic programming are referred to Fleming and Rishel (1975, pp.??),
for
example,
for a readable account. In
our case, however, consumption does not have to be
Optimization
8
19
at rates.
Hence optimal consumption may not be readily determined once we know J
turns out
that the sufficient conditions for optimality include additional conditions
optimal policy beyond those imposed on J
.
It
on the
These additional conditions are obtained from
.
the preceeding heuristic analysis.
Sufficiency
8.2
We
provide two results.
conditions (18),
theorem
in
is
First,
we show
an upper bound
which we show that
if
that the solution to (17), with the boundary
on the value function J
.
Second, we give a verification
a consumption pattern x* satisfies certain conditions,
then the life-time utility derived from x'
is
given by the solution of (17) and (18). Hence
and x'
coincides with the upper
bound on
Proposition 9 Assume
that there exists a differentiable function
[0, 1]
—
Si
'
of time,
U {-oo}, concave
in
that solves (17) with
utility
w and
the
is
J
optimal.
J'{w,y,t):^^ X
J?+
X
whose partial derivatives are continuous functions
y,
Then
boundary conditions (18).
J(ti;(0), j/(0~),0)
<
J-(ti;(0),j/(0-),0).
As we mentioned
almost sufficient
may
J'
is
in that
The problem
above.
before, the differential inequality
is,
they produce a function that bounds the value function J from
of course, that a budget feasible consumption pattern that attains
not exist. Note that this
at rates
is
not a problem in the standard models as consumption
and the consumption pattern that produces J* can be calculated as a function of
J',
and thus J' = J
for
a consumption pattern x' to attain J'.
occur only at
Theorem
and the boundary conditions are
<
=
.
The
following verification theorem gives a set of conditions sufficient
In particular, they stipulate that gulps can
and afterwards, consumption occurs only
3 Let J' satisfy the conditions of Proposition
a budget feasible consumption pattern x'
,
which
is
9.
at times
where J^
(0, 1], with a possible
Ai'(O), and whose associated state variables are w'(t),y'{t~), such that for
f^\jl{w\y\t) -
(ij;{w\y\t)]dx'{t)
^Jy-
Suppose further that there exists
continuous on
u{y\t) + j:w'r{t)-j;Py' + j;
=
=
and
=
for
all
t
jump
£ (0,1]:
(19)
all
e>0,
(20)
and
J-{wiO),y(0-),0) = J'{w(0) - Ax'{0),yiO-)
Then J{w{0),y{0~),0) = J'{w{0),y{0~),Q) and x'
8.3
/?Ax'(0),0).
(21)
optimal consumption pattern.
Example
To demonstrate how
policy,
is the
+
to use the verification theorem in constructing an optimal consumption
we provide a closed form
solution for the optimal consumption problem in an infinite
20
Optimization
8
horizon economy with a constant interest rate
°°
This departure from our
calculations.
Our
The agent
seeks to maximize
1
-e-''y{trdt
I
given u;(0) and j/(0~) and where y{t)
r.
a
given by equation (4).
is
We
take
horizon setup in earlier sections
finite
topologies generalize in a standard
way to
is
a <
1
and
S
>
0.
for tractability of
this infinite horizon
measure, m, that
economy
by replacing the Lebesgue measure on [0,oo) with a
finite
continuous with respect to the Lebesgue measure.
For example, we can take Tn{A)
e~^^dt for
r.
all
Lebesgue measureable
sets A.
We
for this case,
which
is
simpler than in the case of finite horizon
separable in time and in {w,y).
With a
value function should have the form e~^^J{w,y) at time
differential inequality
oo) specified above satisfy
[0,
Simplicity follows from the observation that the
programs, needs a minor modification.
is
for
t
some function
Q
independent off. The second boundary condition
J(0,t/)
=
r
first
boundary condition
in (18)
and simply assert that
f
a
a{aii
needs modification as there
We
the agent does not have a preference for final wealth.
reader,
Then the
boundary condition
this
is
=
lime-*'j(«;(O,J/(r))
0,
(22)
in (18) simplifies to
=
e-'^l-iye-l'^'-'^r ds
Jo
The
J.
becomes
I.
is
abuse of notation, the
slight
maxi
:\^ + J^wT-JyPy-6J,l3Jy-jAJ =
which
=
we advocate.
the notion of continuity that
value function
absolutely
leave the details to the reader but one
can easily show that the agent's preferences for consumption on
The verification theorem
is
-\-
is
(23)
c)
no
date and, hence,
final
again leave the details to the
replaced by
0,
(24)
t\oo
Readers famiUar with
for all budget-feasible policies.
infinite horizon
programs would
rec-
ognize this as a "transversality condition".
We
wiU show that the key feature of the solution
to average past consumption
A:',
w
y. If
then the optimal behavior
increases and y declines
till
is
starts with
-'t^
the ratio
strictly greater
consumption, reducing
id
a
critical ratio
the agent starts at time zero with
to invest
at the rate which keeps the ratio
is
—
all
-
fc',
and increasing
reaches k'
.
From then
strictly less
w
than
If,
on, the agent consumes
on the other hand, the agent
then the optimal behavior
j/,
Jj'^J.
of wealth
the wealth in the riskless asset and wait while
equal to k' forever.
than
>
/:*
to bring
is
to take a "gulp" of
^ immediately to k'
gulp, the optimal consumption occurs at the rate that keeps the ratio
^
.
Following this
equal to k' forever.
To ensure existence of a solution, we make the following assumption. This assumption
guarantees, among other things, that the value function will be concave.
8
21
Optimization
Assumption
6
Now we
the problem satisfy
The parameters of
1
+ 00 >
> ar
6
0,
First
preceed as follows.
we
the policy of consumption that keeps
"Y^ >
it,
and
investigate
equal to
^
or after a period of no consumption
such a policy, denoted J''{w,y),
Proposition 10 J''{w,y)
is
A;
The
k.
an
initial
"gulp"
if
agent's life-time utility from
of the sufficient conditions for optimahty:
A
ky
^
if
if
{S
.
forever, after
'^^
A=
r
given by:
j''{w,y)=
where
-
6
the A:-ratio consumption policies:
all
>
rjo^ <
if
many
satisfies
- a)P >
(1
- a/3(^))~' -
(<^
+
and B =
a/3)~^
{S
^ <
^ >
—
y
Jt
k
- a0{^))~^- Furthermore,
j''
is
continuous, concave, has continuous first derivatives and satisfies
yl
+ rwJt-PyJ^-SJ'' =
o
Jt-PJ' =
together with the boundary condition (23)
However, there
This
is
fc-ratio policy is
and
if
(25)
the transversality condition (24)-
j'' satisfies
exactly one k so that
^<k,
^>k,
if
the diflferential inequality of (22).
then the optimal policy:
Proposition 11 Let
^ + (!-«)
— ar
and note
that k'
by Assumption
>
follows that the k' -ratio policy
The
is
1.
j'''
the optimal solution for the agent's problem.
reader can easily verify that k*
associated J^'
is
is
the unique ratio with the property that the
twice continuously differentiable over the whole
We summarize
of size
domain {w,y).
the solution in the following proposition.
Proposition 12 The optimal
initial gulp
satisfies the differential inequality (22). It then
solution
A = ^^^^^^ [{S -
is
to
consume
+
ar)«;(0)
at rate c(t)
-
-fi^wit), after an
(^ + (1 - a))j/(0-)]
,
if
^^
>
k'
,
or
1
after a period of
It is
interesting to
starts with the
who
no consumption of length
same
compare
initial
this
t'
=
log
3q<,
'
"^^^
,
if
"^o^ <
'^'*-
consumption policy to that adopted by a consumer who
wealth w{0) and the same past consumption experience j/(0~)
seeks to maximize:
Jo
a
9
22
References
where
c{t) is the
consumption rate
at
t
and we assume that
can be easily verified that his optimal consumption rate
> ar and
6
1
- a >
0.
It
a linear function of his wealth:
is
c{t)='-f^w{t).
The propensity
consume
to
additive preferences.
is
consume
c{t)
when
rw{t), and hence
wealth; see figure 3.
start
if
his
A
wealth
is
6
=
An
r.
his wealth at
agent with time-additive preferences
any time
will
be a constant equal to his
consumer with non-time additive preferences, however,
by consuming a "gulp" of
and then keep the
with time-additive and non-time-
for agents
However, the quantity of consumption and the trajectory of wealth
are different. Consider the case
=
same
the
level of his
size
*"*"'
;^0
—K
if his
wealth
is
will
initial
will either
too high relative to J/(0~),
wealth constant forever, or he will wait for
t'
=
^^r^ log
yZl
,
too low relative to j/(0~), and then start consuming the interest on his wealth
forever.
The parameter
mines the
which controls the rate of "decay" of previous consumption, deter-
(3,
size of the initial "gulp" or the length of the initial waiting period,
level of the
subsequent constant wealth. The higher the value of
the initial gulp,
if
/3,
and hence the
the smaller the size of
the solution calls for a "gulp", and the shorter the initial waiting period,
in case the solution entails a waiting period.
This
is
an intuitive result since a higher
/?
implies that past consumption experience has "smaller" effect on current, and hence total,
satisfaction.
Therefore,
cumulate wealth, the
effect of j/(0~)
period before the ratio
calls for
an
initial
when the optimal
^
reaches
"gulp", higher
and hence the "gulp" contributes
P choose
solution entails an initial waiting period to ac-
decays faster with higher
its critical value.
/?
means that the
(3
Similarly,
leading to shorter waiting
when the optimal
effect of the
less to future satisfaction.
solution
"gulp" will decay faster
Therefore, agents with higher
smaller initial "gulps".
References
9
Banach Lattices and Positive Operators, Springer- Verlag,
1.
Schaefer, H.,
2.
P. Billingsley (1968),
New
3.
Berlin, 1974.
Convergence of Probability Measures, John Wiley and Sons,
York.
A. Blaquiere (1985), "Impulsive Optimal Control with Finite or Infinite
Time
Hori-
zon", Journal of Optimization Theory and Applications 46.
4.
G. Constantinides (1990), "Habit Formation:
Puzzle", Journal of Political
5.
P.
Economy
A
Resolution of the Equity
Premium
98, 519-543.
Fishburn (1970), Utility Theory for Decision Making, John Wiley and Sons,
New
York.
6.
W. Fleming and
Springer- Verlag.
R. Rishel (1975), De(erm/n/s(/c and Stochastic Optimal Control,
9
23
References
7.
A. Hindy and C.
sumption:
A
Huang
(1990a),
On
Intertemporal Preferences for Uncertain Con-
Continuous Time Approach, Sloan School of Management, Massachusetts
Institute of Technology.
8.
A. Hindy and C.
Huang (1990b), Optimal Consumption and
Portfolio Rules with
Local Substitution, Graduate School of Business, Stanford University.
9.
C. Huang, Information Structure and Equilibrium Asset Prices, Journal of Economic
Theory 35:33-71, 1985.
10.
R. B. Holmes (1975), Geometric Functiona.1 Analysis and
its
Applications, Springer-
Verlag, Berlin.
11.
L. Jones (1981),
Commodity
Differentiation in
Economic Modelling, Unpublished
Ph.D. Dissertation, University of California, Berkeley.
12.
L.
Jones (1984), "A Competitive Model of Product Differentiation", Econometrica 52,
507-530.
13.
M. Magill (1981),
14.
A.
Mas
"Infinite
Colell (1975),
Horizon Programs", Econometrica 49, 679-711.
"A Model
of Equilibrium with Differentiated Commodities",
Journal of Mathematical Economics
15.
A.
Mas
Colell
(
1986),
2,
263-296.
"The Price Equilibrium Existence Problem
in
Topological Vector
Lattices", Econometrica 54, 1039-1053.
16.
Mas
in
Vector Lattices", Journal of Economic Theory
17. J.
18.
Colell
and
S.
Richard (1991), "A
to the Existence of Equilibria
53, 1-11.
Musielak (1983), Orlicz Spaces and Modular Spaces, Springer- Verlag, Berlin.
H. Ryder and G. Heal (1973), "Optimal Growth with Intertemporally Dependent
Preferences",
19.
New Approach
A.
Review of Economic Studies
40, 1-31.
Sundaresan (1989), "Intertemporally Dependent Preferences and the Volatility of
Consumption and Wealth", Review of Financial Economics 2, 73-89.
S.
A
24
Proofs
Appendix
A
Proofs
Fix a continuous, strictly increasing and convex function rj from [0,oo) to [0,oo) with
= and 77(00) = 00. Define /,,(x,a), for x £
and a £ (0,oo), by
X
77(0)
f\i\x{t)\/a)dt
Jv ix,a)=
+
r,{\xil)\/a).
Jo
Since functions x are of bounded variation, the integral
and
certainly well defined for every x
is
a.
We
record a
Lemma A A
lemma
first.
sequence {x„}
in
X
converges to zero in
•
\\
\\r,
if
and only
if,
for every scalar
=
0.
> 0, lim„ Ir,{xn,a)
Proof. See Musielak [1983, theorem
a
1.6].
I
1. We want to show that order intervals are weakly compact. In
sequential compactness in the norm topology itself. The proof follows
standard lines, and so we will provide only a sketch.
For X* > X, let / be the order interval [a;.,i']. We will assume that i. = 0, this is
without loss of generality because all of our topologies are linear. Let Q be any countable
dense set of points in [0, 1) that excludes all points of discontinuity of x*.
Proof of Proposition
fact,
we can show
Now X E I if and only if a: is nondecreasing and x' — x \s nondecreasing. Let {in} be any
sequence out of/. By the usual diagonalization procedure, we can produce a function y such
that, along a subsequence (which we will hereafter not bother to mention), lim„ Xn{t) = y{t)
at all arguments t £ Q. Since each x„ is nondecreasing, so is y. Define a function x by
x{t) = limsjtj/(<) for t £ [0,1), where the limit is taken over sequences converging to t
from (strictly) above along the set Q and x{l) = limni„(l). Since y is nondecreasing,
so is X. Moreover, y is right continuous (and has left limits) by construction. Clearly
lim„a:n(0 = a;(<) at all points t that are not points of discontinuity of i - that is, for
almost every t. And lim„i„(l) = i(l) by construction. Since all the x„ and x are bounded
above by x'(l), we can apply dominated convergence and Lemma A to conclude that i„
approaches x in 7^.
We are done once we show that x* > x > 0. We have already remarked that x is
nondecreasing (i.e., x > 0). So let us carefully sketch the other half: If i'(0 - x{t) <
—
then right continuity of both functions implies that we can find
values q >
from Q such that x'{q) — x{q) < x'{q') - x{q'). (The case of < = 1 is left
to the reader.) By increasing q' slightly to be some q" < q \i need be (if q' is a point of
x'{t')
x{t') for
t
>
t'
,
q'
discontinuity of x, this gives x*{q)
large n, x'{q)
-
Xn{q)
<
x'{<f')
Proof of Proposition
-
We
—
y{q)
Xniq").
< x'{q") - y{q") for q > 9". Hence
I
Which is a contradiction.
for sufficiently
note first that for any rj, %, is topologically equivalent to
Ta,, for any scalar a. Hence, without loss of generality, we can assume that 7^(1) = 1. Note
that this implies, by the convexity of 7/, that Ti(r) > r for all r > 1. Now it is evident that
Hence, if for fixed 77 (with q{l) = 1) we let 77i(r) = max{|r|,7/(r)}
111' fo'' l^'i
\\v ^
(which is convex as it is the maximum of two convex functions), we have that T is no
stronger than 7^, for any 77. The proof of the first part of Proposition 2 is thus shown if we
show that, for any 7/, 7^ is equivalent to T^,. Of course, Ti, can be no stronger, so we only
II
II
,
2.
A
25
Proofs
For this use Lemma A: If ar„ —
T^, then !„ —
need to show that if x„ —
7^,
7^,
—
0. Suppose that for some fixed a, /,„(in,a) does not go
then for every a > 0, /,,(a;„,a)
to zero - look along some subsequence where the limit exceeds 4/M for some integer M.
Now ljj[Xn,(i) can be written as the sum of three terms:
>
.
J
7?(l„(0/a)l{|x„(t)|/a>l}<^<
+
ViXn{t)/a)l{\^^(t)\/a<\}dt
J
+
77(i„(l)/a).
Pick n large enough so that the
is positive, each must go to zero in n.
term is less than 1/M. If the second is to go to zero, since tj is strictly positive except
at the argument zero, we can find n large enough so that the Lesbesgue measure of the
And for the third term to go to zero, it is
1 > Xn{t)/a > 1/A/} < 1/M.
set {t G [0,1]
necessary that, for n large enough, x„(l)/a < 1/M. Picking n large enough so that all this
happen at once, when we expand /^, (x„,a) in analogous fashion, we will find that (i) the
first term in the sum is less than 1/A/, since it is identical with the first term above, (ii)
the third term is less than 1/M, directly from our selection of n, and (iii) the second term
is less than 2/A/, when we overestimate T]i(Xnit)/a) in the second term with the function
that is 1/A/ if the argument is less than 1/A/ and 1 is the argument is between 1/A/ and 1.
Since this is so for all n above some level, we contradict the supposition.
Next we must show that, for any ry, T,, is no stronger than the total variatio n and
Skorohod topologies. Since total variation is a (linear) norm topology, it suffices to show
in total
that if i„ —»
in total variation, then i„ —
%,. Apply Lemma A - if Xn —
variation, then the same is true for Xn/a for any scalar a > 0. The total variation norm
And dominated
is larger than the sup norm, so the same must be true in the sup norm.
convergence gives the result. As for the Skorohod topology, Billingsley [1968, page 123,
problem 2] states the result for the L^ norm topology, which immediately gives the result
for T. For general 77, note that if Xn -^ x in the Skorohod topology, then for n sufficiently
large, sup, |x„(/)| < sup( \x{t)\ + 1/n. Fixing any a, then, and evaluating Ir,{xn - x,a), we
know that the arguments of t) will be bounded above by K/a = 2sup( |x(/)|/a + 2/(an).
Convergence of /,,(in - x,a) to zero, then, follows immediately if Ii{Xn — x,a) converges
to zero, where /] means the integral using the function r]*{t) = t in place of rj, since we
can overestimate r)(r) with rT)(K/a). Convergence of /i(x„ - x,a) to zero follows from the
I
problem in Billingsley, and we are done.
Since each term
first
:
Proof of Proposition 4. We need
T^. Fixing a scalar
Xn - Vn —
>
that an upper
bound on
/,,(a:„
-
j/„,a)
to
a
is
show that
>
0,
if
x„ and
?/„
are as in the (2.3), then
the techniques employed just above show
^7?((3a;„(l)
+
2)/a)
+
7?((x„(l)
-
j/„(l))/a).
The
second term vanishes for any a since a;„(l) - j/n(l) < pi^n^Vn) < 1/"- And the first term
vanishes for any a: By assumption, Xn(l)//i(n) -^ 0, and thus (3x„(l) + 2)/(an(n)) -^ 0.
Hence for any e there is A^, such that for all n > N,, (3x„(l) + 2)/a < €^l(n). Apply
T]
to both sides (t; is monotone), and use the shape of 77, for f < 1 to conclude that
I
r;((3x„(l) + 2)/a) < r]{(fi{n)) < (r](n{n)) - en, for all n > N,.
Proof of Proposition 5. In A'^., the weak topology is described by the Prohorov metric.
Accordingly, the topology is sequential, and it suffices to show that a sequence is convergent
in one of the two topologies if and only if it is convergent in the other. Suppose first that
sufficiently large so that
x„ — X in Tr,. Fix f > 0, and, applying Lemma A, pick
Ir,{xn - x,() < € ioT n > N. We claim that this implies that p(x„,i) < e. For if not, then
there exists some t such that x„(/ -\- () + ( < x(/) or such that x„(< - f ) - f > x{t). Suppose
that the former is true, and suppose further that t e[0,l- t]. See figure Al. In calculating
/^(xn - x,f), when we integrate 7/((x„(<) - x{t))() over [t,t + e], the integrand exceeds
N
7/(1)
=
1.
(Recall that
77(1)
=
1
is
wlog.) Hence the integral
is
at least
f,
a contradiction.
If
A
26
Proofs
(summed) term in /,,(i„ - x,c) will exceed 1,
again a contradiction. The case where, for some t, Xn(t — () — (> x{t) is handled similarly.
Conversely, suppose that i„ — x in the weak topology. Fix a constant a > 0. We will
show that /,,(x„ - x,a) -* 0, so that !„ — i T^ follows by Lemma A. If p(a:„,x) —* 0,
t
>
I
then
-
Iti(1)
>
and the
f,
We
zero in n.
m,
let
^
Now
in Ir,{xn,a)
converges to
we can bound
/q
t/ f
^"' '~^^
''
j
dt
above by
fmax{x{k/Tn),Xn{k/Tn)} — m\n{x{{k—l)/Tn),Xn{{k—\)/m)}
°
^
"^ibTi
x{t
term
need, therefore, only be concerned with the first (integral) term. For a given
N{m) be sufficiently large so that for n > N{m), p(xn,x) < 1/m. Recalfing
that both i„ and x are nondecreasing,
1
last
1(1)1 —^ 0, so that the final
immediate that |a;„(l)-
it is
integer
-
then x(l)
€,
since p(x„,ar)
— 1/m) - 1/m,
< 1/m, max{a;(<),i„(<)) < x(t + 1/m) + 1/m and min{x(<), !„(<)} >
summation given is bounded above by
so the
fx{{k+l)/m)-x{{k-2)/m) + 2/m\
Jli,
1
-SH
)
-a
i,From the shape of?; (convex, equal to zero at zero, and increasing), this in turn
is
bounded
above by
/y.
J_
argument of
For fixed a, the
assertion.
I
Proof OF Proposition
f,
y G 5f+ and
i
l)/m)-x((A:-2)/m) + 2/m \ ^
x((fc+
6
[0, 1]
6.
Suppose not. By joint continuity, there
such that w((r
+
r)/2,
+
2\
upper bound vanishes with m, proving our
fixed, so this
is
rj
/ 3x(l)
1
j/,
^
t)
exists
two scalars
r
and
+
u(f, y, t)/2. Without loss
u{r,y,t)/2. By joint continuity
u{r, y, t)/'2
of generality, assume that u({r + f)/2,y,t) > u(r, y, /)/2 +
again, there exists e >
and some interval / containing /, and some interval A containing
y such that u((r + f)/2,w,s) - e > u{r,w,s)/2 + u(f,w,s)/2 for all s G / and w £ A.
Consider a sequence of absolutely continuous consumption patterns constructed as follows:
Off of /, consume the pattern m{s), to be described shortly, in each a;„. On /, subdivide
/ into 2n equal sized intervals, and consume at rate r on the even subintervals and f on
the odd. This sequence of consumption patterns converges in the Prohorov metric to the
consumption pattern x that has x'{s) = m(s) off/ and x'{s) = {r + f)/2 on /. Choose Tn{s)
A for all 5 G [0,1]. Note that this is always feasible. By Proposition 5,
Xn -* X in T^. Proposition 7 shows that lim„_oc /o^ u{x',yn,t)dt = /q u(x',y,t)dt. But for
such that y{s) 6
any
j/„,
we
have:
u{x',yn,t)dt>
/
Jo
Taking limits of both sides as n
u(x',y,t)dt
=
J'
is
not continuous in T^.
u{x'^,yr„t)dt
—
oo,
/
u(x',yn,t)dt
lim
»
we
+
eX{I).
get:
+
f-^(/),
>
lim
/
u{x'yn,t)dt-\-(X{I).
"-*°° Jo
n-'oo Jo
Hence U{x) > lim„_oo t^(^n)
V
I
Jo
where A(/)
Finally, note that
if
is
the Lebesgue measure of /. Thus
= a{y,t)r + f3{y,t) for {z,y) 6
u{z,y,t)
-
A
27
Proofs
(0,oo) X (0,oo) and
if
u
is
a and
jointly continuous, then
/3
must be continuous functions.
I
1. We prove the necessity part of the assertion first. Suppose that
using the Hahn-Banach theorem
continuous linear functional on A'. We extend
to L'' to be a linear functional continuous in %,. Denote this extension still by 4>. Musielak
[1983, theorem 13.17] shows that 4> can be represented as
Proof of Theorem
(j!)
a
is
Ti,
(f>
f\it)y{t)dt
<i>(z)=
+
zil)yil)
(26)
Jo
for
some
y
Now
fix
L^
€.
.
X e
X
Since i
.
is
we can
a right-continuous bounded variation function,
use integration by parts to rewrite 4>(x) as follows. First, define f(t) = y(l)
=
[0, 1]. Note that / is absolutely continuous with f'{t) - -y{t) and /(I)
V? e
+
/(
j/(l).
y(s)ds
Then
integration by parts gives
[\(t)y(t)dt
+
x{l)y{l)
= - I
x(t)df{i)
+ x{\)f(\)= [
f(t)dx{t).
Jo-
Jo
Jo
We
have thus proved the only if part as x e A' is an arbitrary element of A'.
Next we turn to the sufficiency part. First reversing the above integration by parts to
show that
can be written as (26) with a y 6 L'^' Musielak [1983, corollary 13.14] then
shows that
is continuous in %j.
.
<t>
<f)
The
last assertion is obvious.
I
Proof of Proposition 7. Fix a topology T^. Let x„ be a sequence of consumption plans
that converge to x in T^. Assume, without loss of generality, that sup„ ||A'n||^ < oo. First
observe that
x„ -+
if
a;
in T^, then
lim
n— oo
Now
we
consider yjl)
-
[
i„
/' |xn(0
^x
-
in
T. Hence,
x{t)\ dt
+
|x„(l)
-
x{l)\]
=
(27)
0.
Jo
y{t)
=
fl^ki(!) ^(^'^)
('^^"C
"
^)
"
'^^^^
"
Integrating by parts,
^'^^-
get:
yn{t)-y{t)
= -(xn{klit))-x{k^it)))0{tJ-ki{t))-
(Xr,{t-S)-X(t-S)]e2{t,s)ds,
/
(28)
where ^2(^«) denotes the partial derivative of ^(<,s) with respect to its second argument.
But since both Q and its derivative are bounded, the left-hand side in (28) is bounded
by Mi|x„(^-i(<)) - a:(/:i(0)l + ^h[ll \in{i - s) - x(< - s)\dsl where M, and Ah are two
constants.
iFrom
,
we can
this,
easily conclude that
lim„_oo
[/o IVnit)
- y(OI
dt
is
less
than or equal
toMi\imn^oo[lo\xAt)-x{t)\dt]+M2\irn^-.oo[!o!o\''n(t-s)-x(t-s)\dsdt\,mthecase
when
x(t
-
ki{t)
is
s)\dsdt
y{t)\di
-
0.
a constant,
,
^lim„_oo[/o kn(0 -
in the case
when
<
ci
<
\k[{t)\
x(0|<^i]
<
Ci-
+ M2lim„_oo[/o
/o l^nit
-
s)
In both cases, lim„_oo[/o IVnCO
-
This follows from (27) and the uniform convergence theorem applied to the
inside intergral in the second term of the previous expression.
A
28
Proofs
Now
y.
It
consider the convergence of u(j/„). First, assume that u is Lipschitz continuous in
then follows that |u(j/n) — w(y)| < a|j/„ — y\, where a is independent of y. It then
- u{y)\dt
follows that lim„_,oo[/o |w(!/n)
< alim„^oo[/J
\yn{t)
-
y{t]\d.t
=
0.
It
then
U{xn) converge to U{x).
Next consider the case when u is merely continuously differentiable in y. For every
integer m, define Um{y) = u(^)X[o,i] + '^(y)^[^<y<m] + w('")X[m,oo)- For a fixed m, the
follows that
function u^ is Lipschitz continuous in y, and hence the arguments in the previous paragraph
show that Um{Xn) converge to Um{x), where ZVm(i) is defined exactly as U{x), except that
u is replaced by Um- Now fix x and the corresponding y. Let A = {t E. [Q,\]:u(y[t)J) >
0} and let A'^ = [0, 1] \ A. Note that either A or A'^ might be of measure zero. It is
— oo.
clear that u,n(t/(<),0 converge monotonically to u{y{t),t) on both A and A'^ as
Applying the monotone convergence theorem, we conclude that Um[^) = J^ "m(2/(0' ^^ +
— oo. It then follows that:
Jac Ujn{y{t),t)dt converge to ZV(ar) as
m
m
=
lim U{xn)
lim
Hm{Xn) = lim Hm{x) = H{x).
Proof of Proposition 8. Theorems 1 and 2 establish that the cone defining the order
dual corresponds to nonnegative functions /. Hence in the order dual, the join of / and
g (for f,g £ X* in the obvious embedding) would be given by max{/,5}, and the meet
by min{/,g}, these being taken pointwise. Now the absolute value of the derivative of
max{f,g] for absolutely continuous / and g is (almost everywhere) less than |/'| + \g'\.
Thus if f,g G X*, maoi{f,g} and ni'm{f,g} are also elements of X^. This completes the
proof.
I
Proof of Proposition 9. Assume that the agent adopts a budget feasible consumption
pattern x with associated wealth w and y. For convenience of notation, we at times denote
J'(w{t),yit-),t),J:;,iw{t),yit-),t),j;{w{t),y{t-),t),3indJ^{wit),y{t-),t)hyJ'it),j;^{t),
Jy{t),
and
Jt{t), respectively.
J^u{y{s),s)ds
+ E.
We
have
+ r{w{t),y{r),t) =
/,V' {J'js)rw{s)ds
-
J^uiy{s),s)ds
J;(5)(ii(5)
+
+ riw(0),yiO-),0) +
j;{s)0idx{s)
- y{s)ds) +
Y:,{j'ir,^)
-
J:{s)ds),
(29)
where the equality follows from fundamental theorem of calculus and we have set tq = 0.
Note that the boundary conditions for J* ensure that the left-hand expression is the correct
expression of utility even for consumption plans that exhaust all the wealth before time 1.
By the hypothesis that J* is concave in its first two arguments, we know that
r{rn -
J'ir,)
< J'jTiXwiT^) -
wir,))
Substituting this into (29), while noting that w{r^)
+
j;(r.)(y(r,)
-
- w{t) = x(t~) -
2/(7")).
x(t), gives
rl
u{y{s),s)ds
j:
<
+
J'{w{t),y{t-),t)
J'{w{0)MO-),0) + f\uiyis),s) + J'Js)wis)Tis) - j;{s)0y{s) + J:{s)]ds
Jo
+ j\l3J;{s)-J'Js)]dx{s)<J'iwiO),y(0-)^0),
J'{r,))
A
29
Proofs
where the second inequality follows from the hypothesis that the integrands are negative
I
and I is increasing.
Proof of Th EOREM
arguments
With the hypotheses added
3.
in addition to the
ones
Proposition
in
identical to those used to prove the proposition prove this theorem.
Proof of Proposition
9,
I
Suppose that - >
10.
to bring the ratio immediately to k
k. The size of the initial "gulp" required
= (w
ky)/{\ + /3k) and we define J^{w,y) =
A
is
J^{w - A,y +
/3A). Concavity of J^ in both it? and y follows from assumption 1, and the
rest of the proposition can be easily verified by direct computations. For the transversality
<
condition note that for any point {w{t),y{t~)), j''(w{t),y(t~))
7.
Noting that
any
for
7/3e*~*'''°"'''u>(0),
feasible policy, y{t~)
<
the transversality condition follows.
Proof of Proposition
^
- bJ^' <
/Sy
we only need
-
if
>
r
<
k'
to prove that
and
<0
-
if
.
y
inequality, consider any point x = {w,y) such that w
be the point on the intersection of the line w — k*y
first
to Figure 2, let a
<
y
PJ^'
-Jt'
y
To prove the
some constant
e~^* J^(w(t),y{t~))
I
11. In light of Proposition 10,
^a + j'^rW - Jf
7j/(<~)", for
and hence
/3iy(0)e''',
-«
through the point x with slope 7^ =
the boundary w = k'y, to which one would jump
line passing
In
>
k*y.
Referring
and the straight
words,
other
a is the point on
one starts at x. By construction,
=
=
J^'{a). Let f{w,y)
+ Jt'rw- J^' (3y, and note that f(a) - 6j'''{a) = 0.
J'^'ix)
Applying the fundamental theorem of calculus along the straight line connecting a and x,
we obtain f{x) - f{a) = j^(f^dw + fydy). Noting that along the line connecting a and
if
^
we have dy = —fidw, and that du; > in the direction from a to
is sufficient to conclude that
fw — 0fy <
X,
yl +
Q
Computing
for k'
=
f^,
- Pfy
in
j^\^.j^'py^sj''
<o
"
the region
w>
x_,
then follows that
it
->r.
if
y
k*y, the reader can easily verify that
/„,
-
fSfy
<
^^p^.
To prove the second
w < k'y. We can write
0J!:'
inequality, consider the function J^*
=
-
fiJy' in the region
where
where
y^-'gi^)
/3
r
where
A
is
assumption
1,
hence g{z)
dg
dz
]
00 &s z
derivative of 9,
A(S +
r
a/?)
+P
]
^
r
+
a/?
/3
Note that 5(1) =
given in proposition 10.
Computing the
+
/?
0,
and that
00.
we
get
_«±h5
/J_ _
U-
a(3 \
+ aP
k'(r+3))
k'{r
+ (3)J
6
d
-ar)
ziP + r)
0(6
1
+
^J
-^^ >
0,
by
A
^°
Proofs
Substituting k' in the expression for
for 2
>
1.
It
.
then follows that J^'
J5b
194
-
^, we conclude
I3J^'
>
that ^a
0, for all
=
for 2
=
1,
and that
points {w, y) such that
w<
^>
k'y.
I
Date Due
f -^^i.
t99A
F»-0®
Lib-26-67
Mil
3
TOflO
UBRARieS
00711512
3