28
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
On Intertemporal Preferences in Continuous Time
The Case of Certainty
by
Chi-fu Huang and David Kreps
WP //2037-88
July 1988
MASSACHUSETTS
INSTITUTE- OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE. MASSACHUSETTS 02139
On Intertemporal Preferences in Continuous Time
The Case of Certainty
by
Chi-fu Huang and David Kreps
WP //2037-88
July 1988
On
Intertemporal Preferences in Continuous Time:
The Case
Chi-fu
of Certainty
*
Huang! and David Krepsf
Revised, June 1988
on the space of
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. Strictly concave time-additive utihty
Different topologies
certain
functions are not continuous in this topology, since they yield preferences
with the property that consumption at nearly adjacent dates are perfect
The topological duals of our suggested topologies are
spaces of absolutely continuous functions whose derivatives satisfy certain "boundedness" restrictions. A utility function that is continuous
in the suggested topologies and that captures an intuitively appealing
notion of time complementarity of consumption is provided.
non-substitutes.
1.
INTRODUCTION AND SUMMARY
Consider an economic agent
pose there
is
zero and one.
a single
We
who
lives
from time
ask the following questions:
First,
to time
is
at
if
we
=
t
1
.
Sup-
any time between
how might we
his lifetime? Second,
space of possible consumption patterns, what
is,
=
consumption commodity consumable
agenf s consumption pattern over
that
<
let
represent the
X^
denote the
an appropriate topology on
X^
;
one that gives an economically reasonable sense of "closeness", possess-
ing the appropriate intertemporal flavor and that, at the
same
time,
is
relatively
well-behaved mathematically? Third, what form will equilibrium prices take
if
they are given by a continuous and increasing linear functional?
Economic and
questions.
financial theorists
Consumption patterns
have long "known" the answers
are represented
to these
by nonnegative real-valued
The authors would like to thank Peter Diamond, Darrell Duffie, Oliver Hart,
Larry Jones, Andreu Mas-Colell, Scott Richard, and two anonymous referees
for comments. Itzhak Katznelson told us about Orlicz Spaces. Tong-sheng Sun
•
pointed out an error in an earlier version.
f School of Management, Massachusetts Institute of Technology
J Graduate School of Business, Stanford University
1
.
.
functions of time, c{t), where c{t) represents the rate of consumption at time
t
Two
.
such consumption patterns
functior\s; say, in the
sup-norm topxjiogy or
represented by maximization of U{c)
Arrow and Kurz
of
come from L«
We
,
and
c(t)
=
c{i) are "close" if
in
V
some
J^ u{c{t),t)dt
,
they are close as
topology. Preferences are
where, in the terminology
(1970), u is the felicity function of the individual.
And
prices
for the appropriate value of q
are dissatisfied with these standard answers.
The causes of our
we
dis-
satisfaction will
be recorded in greater detail below, but basically
consumption
one time should be something of a substitute for consumption
at
That
near-by times.
at other,
an agent for
is,
a while, until, say,
a large
meal
time .5001
(not at finite rate) should be possible.
.
The
at time, say,
.5,
feel that
should sustain
Consumption taken
in large gulps
gained from frequent equally
utility
spaced and equal sized gulps should approximate that gained from continuous
ingestion at the corresponding average rate, as should consumption at quickly
varying
rates.
And,
in
consequence, prices for consumption at dose-by times
should be approximately the same.
of the previous paragraph that will
We
set
out to find answers to the questions
conform
to these intuitions,
while at the same
time retaining mathematical properties important for economic analysis.
We
begin in Section 2 with a formulation of the consumption set and con-
sumption spaces. For the
set of feasible
consumption patterns,
of nonnegative, nondecreasing, right continuous with
on
[0,1],
denoted
X^.
For given x G A'+ and
cumulated consumption from time zero to time
(That
is,
in
terms of standard treatments, x{t)
model both continuous ingestion and gulping.
the space of
consumption bundles
the space of functions of
The space
A"
bounded
is
the linear
variation
comes equipped with
a
on
t
e
left limits
we
use the space
(RCLL) functions
[0,1], x{t)
denotes the
ac-
under consumption pattern x
t
=
c{s)ds)
J^
.
speak of net trades,
In order to
span of
[0, 1]
number
X+
This permits us to
.
Denoted
A'
,
this is
that are right continuous.
of topologies, any of which
could conceivably serve for our purposes. In order to investigate their
we
construct in Section 3 a "wish
Our wish
keeping
list
in
comes
mind
in
two
we
end,
that, in the
of properties for a suitable topology.
list"
parts. First
we
paper in terms of economics;
when
different
Once we have our wish
This
fare.
done
is
tially.
is
We
is
and Jones
||x||
we
=
will use
J^ \x{t)\dt
denoted hereafter by
topologies are
all
weak-+ topology with
fail
rather substan-
respect to continuous
we
[1984].)
endowed with
we
But, as
desire,
on economies where the commodity
this topology.
will see, the
find appealing
topolgies have
all
5, to a
family of
norm
one representative member of
+
T
\x{l)\
.)
We
for x
will
e
X
show
;
(See
weak topology
Mas-
lacks a
topologies. (For this inthis family,
the topology induced
in section 5 that,
topologically equivalent to the
different uniformities,
nomic
well the standard topologies
quickly look at the topologies induced by the
a substantial literature
turn instead, in Section
troduction,
we
how
uniform property that we find appealing.
certain
is
the
the heart of this
be (uniformly) "close" together.
Skorohod topology; these two
(i.e.,
is
are giving our intuition concerning
can see
the space of positive measures,
Colell [1975]
norm
We
to the story,
and more importantly,
analysis. Then,
we
will
we
add uncertainty
under the sup norm) has many of the properties
[0, 1]
and, indeed, there
space
the
The weak topology
functions on
list,
in Section 4.
norm and
total variation
essentially,
consumption patterns
to
The discussion here
desirable economic properties.
list
give useful mathematical properties,
want
will
do optimization, and conduct equilibrium
we
suitability,
weak
on
A'^.
given by the
this
norm
these
norm
by
,
topology. But they induce
and our norm topologies give the uniform property which
and which the weak topology
the properties
we
find desirable
fails to
provide. These
norm
from the point of view of eco-
intuition and, in addition, they are mathematically very well behaved; for
example preferences continuous
representations.
in these topologies
admit (continuous) numerical
we
In Section 6,
nomic models
(the integral of felicity of
T
are continuous in
of consumption
felicity of
ask whether the usual representation of preferences in eco-
is,
at
In Section 7,
^
if
and only
if
rates) gives preferences that
the marginal felicity of the rate
each point in time, constant. (That
is,
rate function for time t, u{c{t),t)
=
consumption
functions a and
does so
It
.
consumption
u{c{t),t)
if
a{t)c{t)
+
is
the
I3{t)
for
.)
we
turn to characterization of the topological dual spaces for
our topologies. In particular, the dual
(X,T) (denoted,
to
the space of Lipschitz continuous functions. In this section,
hereafter, as
we
X*)\s
also confirm
some
desirable mathematical properties of the dual spaces, notably that the duals are
vector lattices.
Section 8 concludes the paper wdth a brief discussion of an alternative for-
mulation, where the consumption space
X+
the
but rather
all
of
X
.
Then
is
the uniform condition that distinguishes
weak topology can be posed
is
T
from
as a local (topological) property.
There are two companion papers to
journal
taken to be not the positive orthant
this.
Following immediately in
Mas-Colell and Richard [1987], which does the hard
work
this
of establish-
ing existence of equilibrium for economics based on our (and other) topologies.
Appearing subsequently, Hindy and Huang [1988] generalizes and extends our
analysis to
certainty,
models
models where there
is
uncertainty and temporal resolution of that un-
and they consider the impact of our topology on some of the standard
in the theory of finance.
2.
THE CONSUMPTION SET AND CONSUMPTION SPACE
Consider an economic agent
who lives from
time zero to time one. This agent
can consume a consumption commodity at any time during his
natural to require that the agent
sumption commodity,
'
But see section
8.
^
and
life
consume only nonnegative amounts
that the
accumulated consumption
span.
It is
of the con-
is finite.
Thus
a
life-time
consumption pattern can be represented by a nonnegative nondecreas-
ing real-valued function x
consumption from time zero
consumption
time
to
at "rates" (if x{t)
x{t) is discontinuous),
(if
-+ $R, with
[0,1]
:
t
.
denoting the accumulated
x{t)
Note
that this representation permits
absolutely continuous) and in discrete lumps
is
and more besides. As x
is
nondecreasing,
it
has the
following properties:
For each
(i)
(ii)
t
G [0,1], both unilateral limits x{t-)
\im.3ii x{s)
exist
For each
e
t
and are
[0,1],
x
is
The only possible kind
and x{t+)
limj^^ x(5)
=
finite.
continuous
at
if
t
and only
x{t)
=
x{t+).
of discontinuity
is
a
x{t-)
(iii)
=
To avoid complications, we
=
shall further
\\mx(s)
=
jump.
assume
Vie
x(t),
if
that x
right continuous:
is
[0,1).
ait
That
is,
the agent
if
commodity during
time
at
f
.
consumes
the time interval {t-,t+), he in fact
To sum up, we
^
amount
a nontrivial stock
will represent the
of the consumption
consumes x{t+)-x{t-)
agenf s
consumption
life-time
pattern by a positive nondecreasing right continuous real-valued function x on
[0, 1]
.
Tl-ie
set of
Throughout
[r, 1]
will
That
is,
is
denoted by A'+
the paper, Xr for r e
[0, 1]
be denoted by
<
.
We
of any x e
defined.
a, 6,
denote the indicator function of
for
.
Also, scalars
...
will continue to
X
will
.
Xr represents a unit gulp of consumption at time r
Here we have used the convention
^
V<
.
such functions
use
this
that x{t)
=
x(l)
Vf
>
1
and
x{t)
=
convention throughout, so that the value
less than zero is always well
arguments greater than one or
X
Let
denote the linear span of Xj^
which
,
the space of right continu-
is
ous real-valued functions of .>ounded variation defined on
> and > on
induces orders
and
and
>
ii
if
-
^\
X^\
^2 ^
{0}
We
.
>
ii
X
will refer to
12
The cone
.
ii
if
-
12 £
X^
^+/
as the commodity space
to Xj, as the agent's consumption set.
Note
x"*"
12
way:
A" in the usual
[0, 1]
X
is
=
x(0).
.
=
Then we can define sup(x,y)
X, we
For x €
a lattice, in the usual fashion.
denote the positive (increasing) part of x
to
x+(0)
that
Note
that
- y^ +
(x
>
x(0)
if
will use
,
then
with inf(x,y)
J//
defined similarly.
Our agent
will
be assumed
to
have preferences over Xj^
mathematically by a complete and transitive binary relation
assume throughout
X+
Vx e
The reader may be put
always
will
would
suf-
.
xi-|-X2^Xi;
that they are strictly increasing in the direction of xo
assume
We
on A'+
that these preferences are increasing:
Vxi,X2GX+,
fice to
>:
that are given
and
Va >
0,
x
+
'
axo
>- x-
played by xo here.
off
by the special
role
that there is
some x e X+
in the direction of
strictly increasing,
although condition
ness assumption about to be discussed
(3)
It
which preference
is
below and our uniform proper-
would need
to
be modified accordingly.
For the analysis of Mas-Colell and Richard [1987], one would wish to use the
social
endowment
Moreover,
with
T+
we
in place of xo
will
assume
denoting the topology
•
that there
T
.
given a linear topology
relativized to
then be assumed to be continuous in
T
is
.
then preferences will
T+ and uniformly proper with
Continuity of preferences in a fixed topology
justification.
X+ And
is fairly
Uniform properness, on the other hand,
is
T on X;
respect to
standard and needs no
not so standard. This
condition, introduced
by Mas-Colell
[1986],
lated to the existence of equilibria for
commodity space and, indeed,
is
shown
there to be intimately re-
economies over an
infinite
optima
to the existence of individual
sumption budgeting problem when there are
many
infinitely
dimensional
in the con-
commodities. The
reader should consult Mas-Colell [1986] for further commentary; here,
merely
with respect
Preferences
scalar a
>
we
—
x
,
V
ax*
+ av
x
'^
3.
cally.
in
properties will
two
Ideally,
parts.
T
=
form properness and we
we
will
wish
LIST
for the
want
will
a locally
T
V
v e
and
,
proper with respect
to
to
T
FOR T
T ? The
topology
T
to
list
of desiderata
be well behaved mathemati-
convex, metric topology, and A' should be
this metric.
will (eventually)
for
,
•
A WISH
we
should be
X+
T-
a
^
xo
we want
First,
X^ and
.
complete and separable under
agents,
there exists a vector x* e
if
that preferences are uniformly
for the particular choice of i*
What
are said to be uniformly proper
of the origin such that, for every x e
assume
will
X+
on
>:
T
topology
to a linear
open neighborhood
comes
will
recall his definition:
Definition (Mas-Colell).
In fact,
we
Also, since
wish
to
we wish
to
invoke uni-
undertake optimization
be a linear topology. (That
is,
for
our
the vector space
operations are continuous.) In order to apply the equilibrium existence results
of Mas-Colell [1986], one
with respect
it
to
would wish
T, or (even)
will turn out, will be too
that
much
that the lattice operations are continuous
(A',
to
considerations. Happily, Mas-Colell
X+,T) forms
hope
for,
Banach
lattice.
But
this,
when we come upon economic
and Richard
To repeat: Our use of xo in this special
Any X* e A'^. could be used instead.
^
a
[1987] allow us to
role
is
weaken
this
purely to ease exposition.
desideratum substantially:
and the topological dual
to
The second and more
of the situation.
to
be
Our
and we
close,
We will want the order intervals to be weakly compact
order dual.
lattice in the
be a vector
substantial part of our wish
X
on
concerns the economics
consumption streams ought
intuition suggests that certain
will look for a topology
list
and
X+
that
accomodates
this
Consider the following requirements, phrased in terms of properties
intuition.
^.
of
Two
(1)
patterns of consumption that have almost equal accumulated
tion at every point in time
should be
sup
lim
close. Formally,
\xn{t)
—
=
x{t)\
consump-
if
0,
""*°°t6[0,l]
then x„
In
(1),
>:
y for
all
t
.
fixed size across small
we
p(i,y)
The idea of
this
tion path X
is
is
We
wish
,
=
metric
>:
x„ for
all
n implies y
amounts of time
this
are also regarded as insignificant.
known
metric on A^+
,
To do
the Prohorov metric.
let
inf{e
is
>
:
x(i
+
e)
+
depicted in figure
e
1;
>
y{t)
the
e
>
x{t
-
e)
-
e}.
neighborhood of a consump-
y
that lives entirely within a "sleeve"
determined by moving
e
up and
The
x .^
so that shifts in consumption of a
any other path
(e in each direction)
>i
consumption between x^ and x must be
weaken
to
begin by introducing a well
For X and y in A'+
that
,
differences in accumulated
small at every time
so,
n implies x >zy and y
from x
at
to the left
and down and
every point {t,x{t)). Hence
if
y
is
around x
to the right
within
e
of
model where
consumption rates, unless felicity is linear. We
will demonstrate this explicitly in section 6. The astute reader will also note that
(1) could be said more compactly as:
T'^ should be no stronger than the sup
"*
astute reader v^ll notice that this rules out the standard
utility is the integral of felicity of
norm topology
relativized to
A%
.
8
>
.
X
,
then at every time
consumption under
the total consumption under x
t ,
some time no more than
y at
is
within
away from
e
t
of the total
e
With
.
we
this
can pose our second requirement:
^0
p(i„,x)
If
(2)
y>zxn
Our
for
final
all
—
n
as
>
CO, then x„
n implies y
>:
x
>:
y for
amount
shift of
(2)
somewhat. The idea
of consumption across a small
It
may
As n goes
a positive scalar.
grows
We
large.
wish
a^^d
to infinity, the shift in time
in particular, p(anXi/2ianX(n+i)/2n) —^0), but
shifted
in (2)
amount
Suppose
is
of time
we
that
help to give a concrete example:
Consider comparing the consumption patterns c„xi/2
is
and
an increasingly larger amount of consumption over
an decreasingly smaller amount of time.
On
,
^
should not be of great consequence in terms of preference.
imagine instead the
y
>z
.
economic consideration sharpens
that the shift of a given
n implies x
all
if
a„X(n+i)/2n/ where
grows small (and
so,
a„ grows with n, the amount
to assert that, a priori,
it
should be possible
for
a„ to go to infinity slowly enough so that the difference in terms of preference
becomes
negligible uniformly.
strict
monotonicity of
>-
wish
to assert that the
"added
shift of a„
from 1/2
To be more
in the direction
to (n
utility"
+ l)/2;i,if
precise,
that
xo
we know, by
xo
in
slow enough. Or we might require that a^
On
is
a„
>-
OnXi/2
•
We
for the
a^ goes to infinity sufficiently slowly. For
we might
if
°nXi/2
from xo eventually compensates
example,
insist that,
+
presumed
the
o{n)
is
(i.e.,
anjn -^0), then the growth
is
o{n^l'^), or
o(lnn), in
order to come to the desired conclusion that the shift in consumption through
time
small relative to the
is
"sufficiently
time
is
slow" rate of growth
^
of consumption shifted. But there
in the
amount
shifted such that,
if
is
some
the shift in
small, the change in utility is small.
To measure what
that
amount
we
Or:
is
meant by
a "sufficiently
slow" rate of growth,
we suppose
are given a concave, continuous, strictly increasing function ^
T^ should be no
stronger than the
9
weak topology on
Xj^
:
[0,
oo)
—
such that
[0,oo),
=
fi{z)
z^Ip
numbers a„
since
(3)
/i
and n{n)
—
=
+
or ^{z)
,
;z
meant
is
as above,
/z
/i
that the condition
grows, so
all
,
x^
"slow enough."
We
We
are
{j/n}
+
^
xo
we do
two sequences of claims with
and
,
(ii)
=
z
.
.
fj.
We
is
the appropriate function
x'
e
X+
-worst element of A^(x)
the worst
the lower-right
we
It
are imagining that ^
not wish to prespecify what rate
/i
is
not very natural -
for a
is
/x
is
given consumption path x,
First, for a
that are within
is
x "delayed
consumption pattern
boundary of
the sleeve.
6, if
some impatience
is
possess this path.
What
and lowered"
if
by,
in
is
the
e
is:
and lowered by
in the sleeve
If
For each x G
e ".
what path x minimizes
this
we
is,
x
in
X+
We
the
would be
to this structural property
we
suggest at the
to
have
=
k
this
.
we
can
Suppose you
path "delayed
give you an additional xo
all
,
terms of
to that representation), then
amount, over
10
That
drawn around
one admits
added
Nt{x)
let
of x in the Prohorov metric.
most you are willing
compensation
how would one
given population of consumers?
ask of a given consumer: Take any path x € A'+ with i(l)
Now
then
becomes stronger the
observe, will hold for the sort of representation
end of section
< 1/n,
p(y„,a:„)
enters as part of the data of the
it
suggest that a natural property for consumer preferences
(which,
that
are willing to contemplate,
fixed
/z(z)
has been said to us that this condition
1,
Note
only suppose that some "slow enough" rate can be specified
suggest the following.
figure
o(/x(n)).
economic application.
denote the set of
>:
sequence of
Vn
agents in the economy;
consideration. But
discover what
we
=
a
pose the following condition.
when
economy under
It
we
depends on the
strongest
it is
given uniformly for
in a particular
a„/fi(n) -^ 0, or a„
{x„} and
if
For examples, think of
.
By "slow growth" of
.
growth" that
In general,
for all sufficiently large n
faster
1)
and {y„(l)} are both o{^{n))
{a:„(l)}
Note
oo as n -> oo
>
ln(2
the fastest "slow
is
must be concave.
For fixed
(i)
1
relative to
growth
linear
p >
for
=
//(O)
paths with x(l)
^P
front?
= kl
(For
preferences with representations as at the end of section
problem
answer
is tractible.) Finally,
how
does
this
amount change
do
these
in terms of
optimization
as k increases?
The
THREE CANDIDATE TOPOLOGIES
Several topologies on the space of functions of
by mathematicians. In
extensively analyzed
this
us fi.^
to this question, of course, gives
4.
6,
our vdsh
bounded
variation have been
we
this section,
ask
how
each of
list.
For the space of functions of bounded variation, a mathematically natural
norm
is total variation:
For x e
X
,
the total variation of x
,
denoted by TV(a-),
is
TV{x) =
|x(0)|
+
/
\dx{t)\.
Jo
Note well
that
positive cone,
variation
if
'
norm
to
have
X
be
will
need
to
have
we hope
then
we
it
lacks separability),
it
wish
In the first place, (1)
is
x„ given by Xn(t)
According
to
(i),
=
k/n for
t
the underlying time space
Although
at least as
violated: Consider the
€ {(k
-
strong as the
total
is
consumption patterns
l)/n,k/n] and x given by x{t)
Xn converging to x, but (of course)
norm. In the same way
have the sort of economic
to
topology
performs miserably in terms of our economic
we should have
not, in the total variation
when
a
with A'+ the
topology. But while this topology performs well mathematically
(although
list.
a topological vector lattice
a
(2) is violated.
continuum, measures time
qualities
we
=
t.
we do
Total variation,
much
too finely
desire.
getting a bit ahead of the game, let us observe that there
will be a point to these questions. The "faster" the function ^ rises, the "more
continuous" we expect equilibrium prices to be. This is not quite precise logically
- our characterization of equilibrium prices as coming from the topological order
^
it
is
dual will depend on uniform properness in norm topologies that are motivated
by but not implied by (3). But, in spirit, the idea is that the "smoother" are
consumer preferences, the more tightly we can characterize prices.
^
which we would want if we were to rely on the existence result of Mas Colell
[1986]
11
.
Consider next the Skorohod metric originally introduced on the space of
functions which are right
metric
atinuous and have
Let
defined as follows:*
is
continuous mappings of
For i,y G
which
c
X,
onto
[0, 1]
A denote
A
=
=
and A(l)
infimum of those positive
define S(i,y) to be the
there exists in
the class of strictly increasing,
A G A, then A(0)
itself. If
(RCLL). This
finite left limits
scalars
1
for
e
such that
a A
sup \\(t)~i\ <
e
<6[0,1]
and
sup
-
\x{\{t))
y{t)\
<
e.
te[o,i]
It
can be
shown
that
S
:
XxX
-^ R.^
is
a metric,
and
that {X, S)
a separable
is
metric space.
Moreover,
this
metric does possess
two consumption patterns
desire. Since
under the Skorohod metric,
note, in this regard, that
2n{t
-
1/2) for
p{^n, X1/2) -^
t
€
if
[1/2, (n
(so (2)
some
+
define x^
l)/2n],
by
comes
^
t
,
it is
for
for
1
t
>
not satisfied.
<
f
(n
We
1/2, x„(i)
+
=
l)/2n, then
should apply), but !„ does not approach xi/2 in the
is
also not very well
to exploiting the linear structure of the
In particular,
tn
=
(2) is
=
Xnit)
and x„{t)
Skorohod topology. The Skorohod topology
it
Unhappily,
we
sup norm are close
that are close in the
(1) is satisfied.
we
of the economic qualities that
behaved when
space of consumption claims.
not a linear topology. Whereas X(„ -^ Xt in this topology
their difference
x<„
-
Xi
does not approach zero. Hence
we
if
continue
our search.
One
other topology frequently used
vergence denoted by T^
C[0,
1]
®
A
,
on
X
is
the topology of
with => denoting convergence in
be the space of continuous functions on
good reference
is
Billingsley (1968).
12
[0, 1]
.
weak
this topology.
Then Xq
=^ x
if
conLet
and only
.
/ f{t)dxcit)-^ I
on the closed
(Integrals here are defined
as a first temn in the integral.)
separable
It is
(cf.
Holmes
interval
[0, 1]
,
so that /(O)x(O) enters
^
weak topology
Recall that the
V/€C[0,1].
f{t)dx{t)
Jq
Jo
a linear, Hausdorff, locally
is
[1975, exercise 22c]).
convex topology.
metrizable,
It is
by precisely the
Prohorov metric p (hence has a countable neighborhood base and
,
on A'+
,
but
From
not metrizable on
it is
the point of
on
Altogether,
all
of A'
).
It
it is
.
weak topology on
in the
.Y+ will obey
quite an attractive candidate (except that
has the further virtue
analyzed in the economics
(in
is
terms of
in
(3),
—
2^/TJJi(lJt)
linear preferences
by x ^
y
+
if
1.
for
ji ,
weak topology
and consider the continuous
/ fdx > J fdy
.
That
=
oo
is
weak
is
assumed.) Define
preferences are given by
is,
evaluating consumption plans at the "prices" given hy f
because /
and
any fixed function ^, however.
(Recall that //(oo)
tinuous and uniformly proper in the
(1)
Mas-Colell [1975] and Jones [1984].
some
so even for linear preferences: Fix
function f{t)
well
consequence) of having been thoroughly
literature, e.g., in
need not be well behaved
is
not metrizable
it is
Preferences that are continuous and uniformly proper in the
This
sequential)
view of economic desiderata, the weak topology
behaved: Preferences continuous
(2).
X
is
topology, and
.
it
A fortiori, y
is strictly
is
con-
increasing
strictly positive.
Consider x„
qualify in terms of
= \/M^Xo
(3).
=
and y„
\/A'(")X'i/n
In particular, x„(l)//i(n)
=
•
Note
1/ y/yjn)
—
that
»
.
a:„
and
rjn
But, for every
n,
/
<
fd[Xn
j fdyn
+
Xo)
= Va<^
=
•
1
(1
+ a/M^
+ 2VlM"))-
^
If we norm C[0, 1] with the sup norm, then the weak topology
topology on A' relative to C[0, 1]
13
is
the
weak- *
.
Roughly put, "shadow prices"
consumption
for
time zero rise
at times just after
very quickly - so that shifting an increasing amount of consumption from time
zero to time 1/n has quite an impact,
Condition
so that the
holds that there
(3)
is
a
if
should become negligible, but the
shift
to rise
increase
is still
we have
too
much.
^°
A FAMILY OF NORM TOPOLOGIES
total variation is
topology does better, but
convergence
in the
family of
is
violates (2)
this,
we
and
not
is
linear.
violate
(3).
In this section,
topologies which will be free of
the reader
is
must be introduced
an accessible reference.)
all
The topology
.
The Skorohod
The topology of weak
quite suitable, except that preferences continuous
(Musielak [1983]
contained,
it
X
too strong along the time dimension.
weak topology may
norm
To do
slow-
picked shadow prices (given
have discussed three possible topological structures on
generated by
proper
p>oint of (3) is that the
around zero quickly enough so that the prespecified slow-enough-
5.
We
increases quickly enough.
slow enough increase in the amount shifted
enough-increase must be prespecified, and
by / )
amount
that
we
and uniformly
will introduce a
these defects.
to the notion of
To make
this
an Orlicz space.
paper close
to self-
give a very quick introduction here, adequate for (and told in
terms oO our particular application.
To begin, consider the case where the function ^[z)
Imagine
that
we
defined the following
norm on
X
=
z^^p
,
for
some p >
1
:
i/p
l^llp
=
''dt
+
\x{l)\P
.Jo
^°
For any continuous function /, and for preferences defined from / as in
we can find a rate function /i that grows slowly enough so that (3)
would be satisfied for these preferences and for
Thus the order of quantifiers
in (3) is all important - ;i must be picked first, and then, what we have shown,
there are "nice" (i.e., linear) preferences continuous and uniformly proper in the
weak topology that violate (3) for this fi.
the example,
f.i
14
.
Since our functions x are
tegrals exist.
of
bounded
variation,
From standard Banach space
normed
separable
all
theory,
it is
clear that all these in-
clear that (X,
it is
•
||
||p)
is
a
topological linear space:
Nonnegativity of the norm, the triangle inequality, and the standard conditions in conjunction with the linear space operations
all
X
standard theory. Right continuity of the functions in
=
then 1
follow immediately from
ensure that
if
=
||a:||p
0,
0; note in this regard the special treatment of x(l). For separability,
note that simple right continuous functions that change values at (finitely many)
arguments and take on rational values are dense
rational
with
this
norm.
norm: Consider,
ation functions
2),l/(2u
-
3))
worth noting, perhaps,
It is
for
...
U
(
A+
,
however,
This has the right "flavor" for
•
where
if
lip
II
(3)
is
to
the difference in
amounts
as well, for the case n{z)
are scaled
by being raised
two consumption paths
What, then, do we do
if
y.{z)
obvious candidate analogy
is
to
,
ft
T]{oo)
Tj
to
^^
T]
=
continuous,
is
oo
.
in this
is
is
0(l/n)
begin by defining
[l/(2n
-
obvious
tj
to the
See proposition 4 for an exact statement.
.
in
Roughly
amounts,
power p Hence
.
dimension and
small.
^^
z^/'p ?
For a general
is
—
fi~^
.
From
is,
p. ,
an
the properties
7](0)
=
value repeatedly, extend
real line; that
15
U
z^^p
and
convex, and satisfies
to write absolute
be an even functions on the entire
1))
to the
=
in the time
not of the form
strictiy increasing,
To avoid having
-
complete.)
o(n}IP) in the space dimension, then the difference
of
X
in
has unbounded variation (and
trades off differences in consumption paths in time
differences in
hence
complete
These are Cauchy and converge
[1/2,1].
not right-continuous).
put,
,
example, the sequence of right continuous, bounded vari-
measurable function, but what they converge
is
L^
that A' will not be
that are indicator functior\s of [l/2n,l/(2n
a:„
U
in
for negative
and
jj.
and
arguments
X
,
=
let \i{x)
and
^l{-x)
let 7/(1)
=
lklU:=A^(^
Unhappily,
this
=
A
r]{x{t))dt
Hence
norm,
r]{x{l))y
is
a
homogenous
function, ^
the triangle inequality will not follow, nor will
more ingenious procedure
define the unit
+
for the
try,
for scalars a.
|a|||x||^
slightly
Then
.
won't work, because while z^^f
will not be, in general.
||ax||^
r?(-x)
X
circle in
,
C, := {x
denoted
eX:
f
C^,
is
as follows. For ^
and
as above,
77
as
,
rj{x{t))dt
+ r^{x{l)) =
1}.
Jo
The convexity of
"gauge"
to
for
ensures that this unit
77
measuring
all
other vectors x
be the unique positive scalar a that
because of the continuity of
this definition is
77,
||x||^
is
circle is
:
appropriately shaped to be a
Define, for x other than x
satisfies
x/a ^
It
Cr,
=
0,
||x||^
takes a proof, but
well defined. Moreover, for
=
fi{z)
z^^^
With
equivalent to the standard definition of the norm.
,
this
definition:
Proposition
(A',
1.
•
|1
||^)
is
a separable
normed
topological linear space.
the order operations given above, order intervals are
weakly compact
With
each of
in
these topologies.
We
provide the proof in the appendix for completeness, although
application of the theory of Orlicz spaces, of which this
Musielak
T;,
,
simple
an example.
(See
[1983].)
Fixing
by
is
this is a
/i
and the corresponding
and the
the case ^(z)
=
=
z
the topology induced
topology relativized to
T,,
case where fi{z)
77 ,
z'^Ip
,
we
A'-i-
is
denoted by
use notation such as Jp
,
by
T^
results are established in the appendix.
16
.
is
denoted
For the special
and we use Too
.
The following basic
||-||^
/
etc., for
Proposition 2. For
turn
The
is
any ^ and corresponding
weaker than both the Skorohod and
first
part of this proposition
Ti
r/,
is
weaker than T,
which
,
in
total variation topologies.
Since
is fairly intuitive.
t]
is
convex,
tends to
it
accentuate (more than linearly) large differences in the dimension of space. Note
"weaker" in
that
also that, in our
this proposition
norm
should be read as "no stronger than." Note
-
continuous. For example, xi/2
>:
that
is
looking for preferences on
that
>:
is
to T^
,
increasing as before.)
Proposition 3. Preferences
(2).
that are given
A'^.
some prespedfied
for
The following
for the "right" thing, insofar as
and
do not converge.
T+
complete, transitive, continuous in the
proper with respect
Moreover, .Y+
is
t]
binary relation
a
topology, and uniformly
(We
.
by
will continue to
results establish that
our desiderata were
that are continuous
>:
definitely not
X{n+i)/2n -^ 0/ but the corresponding positive
parts (or negative parts, or absolute values)
We will be
most
topologies, the lattice operations are
T^
separable under
we
are looking
T^
satisfy (1)
correct.
with respect
,
to
and hence every continuous
y on X+ admits
(complete and transitive) preference relation
assume
a continuous
numerical representation.
Satisfaction of (2) follows as a simple corollary to proposition 5 (which
proved
(1).
The
in the appendix),
separability of
which
X^
immediately implies the satisfaction of
follows standard lines and
of a numerical representation
for
in turn
is
is
is
omitted.
The
existence
standard given continuity and separability - see,
example, Fishburn [1970].
Proposition 4. Fix
/i
and the corresponding
?/
.
Suppose
that
y
is
T^
continu-
ous, increasing as above, and T^ uniformly proper /or the specific choice i*
Then y
satisfies (3).
17
=
xo
•
we prove one
Abbreviated proof. In the appendix,
as in
- x„
then y„
(3),
-4
yn
— Xn
is
eventually inside
Xo
+ ^n,
to
conclude
X
in the relation
T,,
.
.
Thus
V Apply
to xo
+
a;„
How
This leaves open one issue:
weak topology on X+
? It is clear
to the
Proposition 5.
weak
For any
topology on Xj^
and y„
are
of the origin,
y„
must not stand
it.
do
the topologies
T^ compare
with the
from the example of section 4 and proposition
4 that uniform properness with respect to
with respect
V
any neighborhood
xo+^n- Xo + J/n - 2:„ =
That does
.
If j:„
the definition of uniform properness to
.
that, evenhaally,
for
technical fact:
T^,
is
stronger than uniform properness
topology. But:
and
/i
T^
t] ,
is
topologically equivalent to the
weak
•
The proof of proposition 5
is
essentially technical,
and so
it
too
is
consigned
to
the appendix.
Hence we
Uniform properness
reaffirm (the obvious):
is
not a topological
property
6.
ON THE STANDARD MODEL AND THE
The standard model
T^
TOPOLOGIES
of preferences for consumption through time assumes
that preferences are defined over
consumption
in rates
and are given by
a nu-
merical representation of the form
U{x)=
u{x'{t),t)dt,
/
Jo
where
x'
denotes the
one specializes
first
further, to u{x'{t)^t)
ing function, although for
continuous
derivative of accumulated consumption x
in its
=
.
Typically,
e""^ f{x'{t)) with /(•) a concave, increas-
what follows
it
two arguments.
IS
will suffice to
assume
that u
is
jointly
,
Preferences so given are not defined for consumption in gulps (where x
we might
discontinuous), but
preferences to general x
approximate x
:
consider the following procedure for extending
Find a sequence of absolutely continuous x„
=
and then define U{x)
,
is
lim„_oo U{,^n)
>
hoping
that
that the limit
well defined and independent of the approximating sequence x„
.
We
will
is
want
preferences to be T^+ continuous, so the sense of approximation had better be no
weaker than
to
convergence - and
T^
approximate
continuous x^
we have
arbitrarily closely (in
.
Hence
already remarked that
terms of T^+
any x G
)
program does have some hope
this
A'-j.
it
is
possible
by absolutely
of working.
It
will
only work, though, in very special cases:
Proposition
(a)
6.
Suppose
that preferences
on
by
are represented
A'^.
U{x)=
:)= /
/
the utility function
u{x'{t),t)dt
u{x'it),
(*)
Jo
for
some
(jointly)
continuous function u
sumption paths X.
to absolutely
Conversely,
(for
some
if,
77 ),
for all absolutely continuous con-
these preferences are continuous in
continuous consumption paths
the form u{r^i)
(b)
If
,
—
OL{i)r
+
(for
any
where a and ^ are
/?(<),
for a preference relation
>:
on A'+
t] ),
T^
,
relativized
then u must have
(clearly) continuous.
that
is
continuous in
7^
preferences for absolutely continuous paths can be represented
by
[7(x)= /
a{t)x'{t)
+ /3{t)dt
Jo
for continuous functions
a and
U{x)
/3
then
,
= f
a{t)dx{t)
Jo
represents preferences on
all
of
A'^.
19
.
And
for
M
any continuous function a
preferences on
Proof.
Suppose
as in
(+),
r
for
and
some
concave in
is
not
r', {u{r,t)+u{r',t))/2
its first
This
.
continuous for
the
is
Then
affine.
for
12
all
on absolutely continuous paths
u(-,i)
t,
<u{(r + r')/2,t)
u{r',t))/2
T^
given hy i*-k) are
that preferences
and
arguments
X+
are represented
some (nonnegative)
u{(r+r')/2,t). Suppose that {u{r,t)+
^
more natural
argument. But the other case
case, thinking of u as being
handled by a virtually identical
is
argument.
Then by
the continuity of
around
positive length
—
r')/2,t')
t
such that for
Consider
e.^^
u, there exists
a
all
t'
>
e
el
and some
{u{r,t')
,
+
construct x„ as follows: subdivide
,
consume
at rate
r
at rate r
on
I
=
1
the even subintervals
for a bit of time, then at
for
<
^ J and
+
U{x) > U{xn)
x'{t)
<
u{r',t'))/2
u{{r
+
r'
,
=
and
is
each
on the odd. (That
r'
and then
(r+r')/2 for
eA(J), where A(J)
in
1
a-„
.
into 2n equal sized subintervals,
consumption paths converges weakly (hence
x'{t)
of
sequence of (absolutely continuous) consumption
paths constructed as follows: Off of I, consume at rate
I
J
interval
in
t
at
.
)
and
consume
This sequence of
etc.)
,
T^
any
eJ
r
is,
On
to the
path x that has
Hence, by the representation,
the length of the interval J.
It is
easy
to
turn this into a contradiction of the supposed continuity of preferences.
To
is
finish part (a), note that if u(r,f)
jointly continuous, then
For part
(b),
lutely continuous are
of this
all
dense
is that,
of
(in
A'^.
(^Hk-)
,
a{t)r
^{t) for r £ (0,oo)
if
u
functions.
T^
and the consumption paths which are abso-
any of these topologies)
except
and
gives a weakly continuous (hence
when
"felicity of
constant marginal felicity at every point in time
'^
+
must be continuous
fS
note simply that
continuous) function on
The point
a and
=
(a
.
consumption rates" exhibits
case not of great interest in the
This does not assert that preferences so defined are
See the discussion at the end of section 7.
'"^
The cases t =
and t = 1 present no difficulty
20
in A'+
T^
uniformly proper.
standard model
literature), the
in the
example given
norm
tof)ology, so
be continuous in our topologies. Indeed,
converge to x in the sup
in the proof of part (a), the a:„
any treatment of preferences
that asked for property (1)
would not admit
The reason why should be
ing steadily
fails to
the standard
Unless marginal
clear:
consumption through time
for
model
in the literature.
felicity is constant,
consum-
not well approximated in the standard model by consumption in
is
"quickly" and properly spaced gulps or even by consumption at quickly varying
in the
sort of
model might then be used
Following Jones (1984, p.525),
we make
problem with the standard model
at time
t
of felicity for
some time
some length
5
that
continuous 6
[0, 1
the
+
:
One
[—k,0]
k]
—
>
standard model?
to replace the
The
the following tentative proposal:
consumption
from
at time
generates
t
consumption taken
in at time
way
of time thereafter - put the other
at
time
t
felicity
A more
gone immediately.
it is
depends on consumption not only
in the recent past.
G
is
only; the "pleasure" derived
model would be one where
^
close
sup norm.
What
for
made
cumulative consumption paths of such patterns can be
rates. Yet the
produces
around,
a
apt
flow
felicity at
but on consumption
s
can imagine a model of this running as follows: For
[0,oo)
and consumption pattern x 6
by Fx{t) = /°^ eis)dx{t +
5)
^^
That
.
is,
A'^.
Fx{t)
define Fx{t)
,
is
a
"weighted
average" of recent past consumption, where 6 gives the weighting function.
Then apply
the usual construction of utility
from
felicity
by defining
/•i+fc
U{x)=
for
some
u{Fx{t))dt,
/
Jo
well behaved "instantaneous felicity function" u
the reader, but one can generate conditions
gives
^'^
T
uniformly proper preferences.
Since this
is
only an example,
Note that ^(0) =
will
6{0) =
to savour a jolt of consumption.
.
on u and
We
leave details to
6 so that
U
so defined
(Indeed, one can generalize this quite
make matters
come to mean
21
.
easy by assuming that 0{-k) —
that it takes some time to begin
weighting function 6 can depend on calender time,
substantially, so that the
Since
etc.)
we have no particular reason
preferences (except perhaps for
this
model
further.
its
to
advocate
mathematical
this specific representation of
tractibility),
we
wall not
pursue
But the form given does indicate the sort of "intertemporal
substitution of consumption at near-by times" that motivated this study.
Indeed,
need
it
has been said to us that this form
That
for the analysis here.
is (to
is
so natural that
it
obviates the
quote from a referee report), "one can think
of the actual consumption process as the process of 'burning' calories,
of transaction costs associated with eating."
for
that
how
We agree with
actually motivates our study. That
it
food
is
The discreteness of the eating process occurs presumably because
quite smooth.
beUeve
which
is
is, if
this
one has
statement, but
this picture in
we
mind
translated into calories or satisfaction, then the sorts of topologies
we have advanced
(or the
weak topology) become more
natural than the standard
model. The payoff then comes from the resulting characterization of equilibrium
prices;
if
preferences are "smoother" (continuous in a weaker topology), then
prices (as continuous linear functional) are taken
So
we
turn
now
from
a
more
to the issue of characterizing (positive and)
functionals for our
norm
continuous linear
topologies.
DUALITY
7.
For the duration of
restricted class.
this section,
we assume
that
some
pair ^
and
7/
as in
Section 5 has been fixed.
The companion paper by Mas-Colell and Richard
be possible
to establish existence of
are increasing
[1987] tells us that
it
will
equilibrium in economies where preferences
and uniformly proper
in
our topology
if
we
can establish that the
topological dual forms a vector lattice in the order dual. Moreover, equilibrium
prices for these equilibria will be
our aim
drawn from
the topological order dual.
in this section is to characterize the topological
which we denote by X*
,
and
to verify that A'*
22
Hence
dual space of (X,T;,),
does form a vector
lattice.
We
will not give
(e.g.,
complete answers here, as the theory
in Musielak, 1983, section 13). But
we
well developed elsewhere
is
will give
some
reader can see the basic flow of the argument. In particular,
dual to (A',T), denoted A'*,
(essentially) the
is
of details, so that the
we
will
show
that the
space of Lipschitz continuous
functions.
As
a first step,
we show
that every continuous linear function can
sented as integration against some continuous function on
is
no
larger than
And
C[0,1].)
[0, 1]
be repre-
(That
.
is.
A'*
the order dual will correspond to nonnegative
continuous functions.
Proposition
7.
l(
(f)
e A'*
,
then there
some / e
is
C[0,
1]
such that
= f mdx{t).
<f>{x)
(0)
Jo
Moreover, the positive cone in A'* will correspond to continuous functions /
that are nonnegative.
(Recall that
Proof.
(^
when we
write J^
,
we mean
The proof follows standard
€ A'*
.
For
t
Xt„ —* Xt in T^
that that,
by
[0,1], define
E
,
we know
linearity,
function x € A'+
And
by
(0) on A'+
.
proposition
is trivial.
<f)
f{t)
that
/
=
is
and so
(l>{xt)
we
only sketch
Since t„ -^
t
a continuous function
in
[0,1]
on
[0, 1]
a sequence of step functions, to obtain that
then extend to
If
/ gives
a
all
some
implies that
.
Next show
,
of
X
by
linearity.
The
final
=
>
f{t)
f fdx >
0. Conversely,
if
/
statement of the
in section 4
23
(<>),
is
and /
is
nonnegative,
.
course, not every continuous function / gives rise to a
example given
has the form
<f)
continuous linear functional via
J fdxt
then for any nondecreasing x
linear functional - the
First fix
it.
has the form (0) on step functions. Approximate any
in the positive orthant, then
Of
lines,
the integral over the closed interval.)
shows
this.
T^,
continuous
This example also
.
shows
the fundamental thing that can
because, while continuous,
continuous linear function,
a
relative to
/i
go wrong - the function / may
it
varies "too quickly"
r
e
[0, 1]
,
define
n
n
= sup{^
-
\f{t,)
f{s,)\
<
:
51
<
<i
<
...
<
<
5„
<
<„
is,
Vf{r)
(i)
is
maximum
the
is
(ii)
<
r).
absolute variation of / over a disjoint finite
whose lengths sum
to r or less. Since
/
is
continuous
defined on a compact interval, the following facts are easily established:
The function / has unbounded variation over
if
s,)
i=l
collection of subintervals
and
-
Y^{t,
1,
1=1
That
give
over short time intervals.
For a given continuous function / and scalar
Vf{r)
fail to
=
Vf{r)
oo for
The function /
r
all
>
the interval
if
[0, 1]
and only
.
and only
is
absolutely continuous on [0,1]
is
the definition of absolute continuity
if
if
limr-^o ^/(r)
=
0.
The second of these
The
is
first
does have a
=
that Vf{l)
oo
The point of
condition on /
Proposition
according
A
8.
.
bit of content, since the definition of
But
/
is to
unbounded
variation
not hard to get the stronger statement.
with
we
it,
can give a simple necessary
give a T^ continuous function.
continuous function / gives a T^ continuous linear functional
(0) only
to
it is
this definition is that,
if
- no proof is necessary.
if
there
is
K
some constant
such that ^{\/r)vj{r) <
K
for all r
Proof.
Suppose
^(l/r)v/(r)
Si)
<
T
=
N
.
Then we can
and /i(l/r)^,
or on the
xt^{t)
>
to the contrary that for
r?
|/(i!,)
-
find ^i
/(-^i)!
-
f{t,))
X fi{l/r)
<
>
ii
A'-
A'^
<
.
Note why
24
there exists
...
<
5n
<
t„
some
r
such that
such that ^.(^ -
Construct an element x;v within
1^(0 =
unit circle as follows:
sign(/(5,)
every
for
<
it is
G [<„,5n); for
t
6 [5„,^n),
that x/v lives within the
r/
s
unit circle:
which
=
a:A'(l)
—
r]{x^'{t))
lem.)
on
.
This
1.
only
Corollary
the linear functional given
unit circle.
T]
Corollary
(<>)
J^ f{t)dxf^{t).
is,
if
Then / €
measure
r
or less, on
It
ti
=
|/(i,)-
does not present a prob-
s,+i
by / and (0) takes on unbounded values
cannot, therefore, be T, continuous.
A
function / G C[0,1] gives a T^ continuous linear functional via
/
is
absolutely continuous.
Suppose
2.
set of
by construction, precisely /^(l/r)X^
(The reader should check that the case
Thus
the
nonzero on a
is
l/r.
Now evaluate
f{si)\
0, and x^it)
that
is
rj
asymptotically linear; rj{r)/r —y k for
some
C[0,1] gives a T^ continuous linear functional via (0) only
^'
if
>
.
/
is
Lipschitz continuous.
Corollary
1
follows immediately the second fact given above and the proposition
(and the fact that /^(1/r) -+00 as
if
77
we
is
asymptotically linear, then so
of the order 0{l/{ti^
Corollary
/'
^
is
For corollary
0).
>
.
And
if
/
is
2,
use the fact that
not Lipschitzian, then
can find a sequence of intervals [5^,<;v] of vanishing small length such that
\f{tN)- f{sN)\/{tN-SN) >
is
—
r
1
-
N
sim))
,
.
But then
and our
f^{l / {t
criterion will
provides a further harvest.
differentiate almost everywhere and
as the derivative of /,
/
we
is
is
f{t)dx{t)
=
Proposition
9.
If
a
/
is
be violated.
absolutely continuous, then
the integral of
/(l)x(l)
its
it
derivative. Writing
- /
x{t)f'{t)dt.
(00)
Jo
included in the
first integral,
from other applications of integration by
which includes
If
will be (asymptotically)
can integrate by parts in (0) to obtain
^0
(Note that /(O)x(O)
n - n))
converse to corollary
/ € C[0,
1]
is
parts.)
so this
may look a bit different
Hence we obtain
the following,
2:
Lipschitz continuous, then / G A^* for every
26
tj
.
/
Proof. If
K
constant
Lipschitz continuous, then
is
Hence
.
for
any x on the
rj{xil))+
K
<
|/'|
some
unit circle, since
rj
Tj{x{t))dt
f
(almost everywhere) for
=
1.
Jo
we
can use the convexity of
and thus
t]
to
conclude that
1
and
that
<
T]{x{l))
x(t)dA <
r](
1.
I
Using (00),
us an immediate upper bound on
this gives
A")^(l) for X from the
is
bounded, and thus
unit circle.
tj
Hence
X*
where
is
well
that
=
ri{r)
r
worked
.
r]*{s)
=
sup{r5 —
r]*{s)
=
5',
function,
the
norm
of
r -+
as
T]{r)
r
:
it is
•
||
||^.
;
i.e.,
for
for
If,
much more
complete char-
a
1
.
Then
/
for
K
/'
,
Assume
T]{r)
—
What
if
r^
to give a
for the
for
1
moment
ri*{f'/K)
T]{r)/r 7^
> p >
rj*
00
,
by
then
T^ continuous linear
live in the Orlicz
development can be found
general setting).
The theory
the "conjugate" function
rj
example,
some constant
(in a
space given by
must be
integrable and
in Musielak, chapter 13
as r —>
0?
If
77
is
not
which case we know the answer from proposition 9 above), then we
can create an
past
.
necessary and sufficient that
integrate to one. This
77
0}
where 1/p + I/5 =
must
linear (in
Define from
0.
>
we have
that are not asymptotically linear?
77
and runs roughly as follows:
out,
—^0
ri{r)/r
2,
asympototically linear, including the special case
is
77
What
by (0)
continuous.
it is
if
f{t)dx{t) of (sup|/(<)|+
the linear functional defined
Putting together proposition 9 and corollary
acterization of
J^
some
r]'
which does meet
level r
.
The proof
depends only on the behavior of
this
requirement and which
of proposition 2 in the
77
for large
26
is
identical with
appendix shows
that T^
arguments, so the dual with respect
,
.
to
T]
will
be the same as with respect
We
Musielak.
to
and the
r/' ,
leave this to the reader, only to note the special case of
to state that the dual space
a lattice in the order dual,
is
by
latter is characterized
and
to
T](r)
make two
=
r^
further
remarks.
Proposition 10.
If
=
ri{r)
rP
for
some
1
for all absolutely continuous functions
(<0>)
for q satisfying l/p
+
=
l/q
any
Proposition 11. For
the space A'*
rj ,
nonnegative functions /
(for
the
X*
f,g e
oo, then
given by the form
/'
lie in
L',
Hence
.
is
a lattice in the order dual.
cone defining the order dual corresponds
in the
order dual, the join of / and
g
obvious embedding) would be given by max{/, y}, and
in the
of the derivative of max{/, ^}
less
is
/ whose derivatives
meet by min{/, ^}, these being taken pointudse.
everywhere)
X*
1
Proof. Proposition 7 establishes that the
to
<p<
than
|/'|
+
for absolutely
\g'\.
The
Now
the absolute value
continuous / and g
rest is clear
is
(almost
from the characterization of
A'* given above.
Remark
we
1.
Lipschitz continuity of /
2.
=
Since
U
77
.
proper
it
is
In proposition 6,
a, U{x)
/o
is
a{t)dx{t)
that,
framework of
are "within" the usual
Remark
show
Propositions 9 and 10
the
we
same
once
we
the classic
as /' G L°^
are able to integrate
Banach spaces;
by
parts,
in particular,
.
asserted that for any continuous function of time
would give preferences
that are
T^
continuous for
all
(manifestly) linear, for preferences of this form to be T^ uniformly
would be necessary
for
q e X*
;
we now know
the conditions
needed
for that to be true.
8.
To
this point,
PREFERENCES DEFINED
we have
ON
A'
taken an approach where preferences are defined only
over nondecreasing consumption paths x G
27
A+
.
In
some treatments
of infinite
,
dimensional commodity spaces
(e.g.,
be an entire linear space. In
to
defined on
(although
on
[1981]), the
this case, this
consumption
we
could no rely so heavily on the
of
X ,\n proofs).
all
X
on some open subset of
(or, at least,
set is taken
would mean having preferences
weak
that contains
somewhat
taken this approach, our exposition would be
Had we
able
X
of
all
Kreps
topology, which
In place of requirement
(3),
we
X+
.)
less tortured
is
not metriz-
could pose something
like:
For any sequence
(3')
3;+(l)
=
>
then X
mean
o{^{n))
y
it
(or
ing in
and
,
and
for
y
if
>:
we want
to
+
2;„
that
(i)
x for
n, then y
all
if
Xn-\-
>:
^
in the sense of (3')
,
we
are
discussion at
Clearly,
(3')
all;
on
all
is
is (strictly)
There
of
^^
A
;
A
).
weak
no need
topology.
Property
set d(x,y)
(3')
=
to
(ii)
for all y
we
fails in
to
the
weak
our norm
invoke uniform propemess in the
And we know
all (strictly)
norm
less requir-
^^
weak
smaller than the dual of the
(Of course, the
and
(By x~ here,
we move
of A' there are sequences convergent in our
whether our T^ topologies are
all
is
a local property.
are not convergent in the
x>zy
more or
This
to zero.
topology (as the example given before indicates), and so
this.
•
\/n
.)
such sequences {x„} converge
topologies to ensure
=
p(x+,x~)
choose a topology such that preferences continuous
more) are well behaved
(3')
such that
.Y
every x and y from A',
the negative part of x^
Essentially, since
in
,
from
{x„}
topology.
T^,
topologies that
that the dual space A'*
One wonders,
therefore,
weaker than the weak topology (on
topologies are not the weakest topologies that
may suggest to the reader another metric topology on all of
— y)'^ (x — y)~ This does not give a norm, clearly, and we
p{{x
)
,
are not certain that it gives a metric: the triangle inequality is not at all obvious.
But if it is a metric, it gives a topology that seems too weak to us. In particular,
would imply properties like (3) without any restrictions at all on the rate of
growth of the Xn and y„ And it would give very uninteresting equilibrium
prices: The only continuous linear functional on A' that is continuous in the
it
.
sense that ^(x„) ->
if
J(x„,0) ->
is
the trivial
28
(?i>(x)
=
0.
give A'* for the dual.)
Our guess
counterexample nor proof.
is
that this
not true, but
is
we
can offer neither
^^
REFERENCES
Kenneth
J.
Arrow and Mordechai Kurz
and Optimal
Fiscal Policy,
(1970), Public Investment, the Rate of Return,
Johns Hopkins Press, Baltimore.
Patrick Billingsley (1968), Convergence of Probability Measures, John Wiley
New
and Sons,
York.
Peter Fishburn (1970), Utility Theory for Decision Making, John Wiley and Sons,
New
York.
Ayman Hindy and
Chi-fu
Huang
Continuous Time Dimension,
Richard
B.
Holmes
II:
(1988),
"On Intertemporal
Preferences with a
Models with Uncertainty", mimeo.
(1975), Geometric Functional Analysis
and
its
Applications, Springer-
Verlag, Berlin.
Larry Jones (1984),
"A Competitive Model
of Product Differentiation", Economet-
rica 52, 507-530.
David Kreps (1981), "Arbitrage and Equilibrium in Economies with
Many Commodities", Journal of Mathematical Economics 8, 15-35.
Andreu Mas
Colell (1975),
"A Model
of Equilibrium with Differentiated
modities", Journal of Mathematical Economics
Andreu Mas
ical
Colell (1986),
Infinitely
2,
Com-
263-296.
"The Price Equilibrium Existence Problem
in
Topolog-
Vector Lattices", Econometrica 54, 1039-1053.
Andrew Mas
and Scott Richard (1987), "A
Vector Lattices", mimeo.
Colell
of Equilibria in
New
Approach
to the Existence
Julian Musielak (1983), Orlicz Spaces and Modular Spaces, Springer- Verlag, Berlin.
^^
A
referee suggests the following counterexample, for the case
where con-
sumption takes place not on the time
interval [0,1] but instead on [0,oo). For
be the bounded variation function that rises
each positive integer TV, let i^
to 1/A^ at times l,3,...,2iV - 1 and that falls back to zero at ttmes 2,...,2A^.
Then x^ converges to zero in the weak topology, but it does not do so in any
of our norms. We are not certain how to adapt this example to our setting with
a bounded time interval.
29
Scott Richard
and William Zame
Utility Functions",
H.
L.
(1986), "Proper Preferences
and Quasiconcave
forthcom'ng in the Journal of Mathematical Economics.
Royden (1968), Real Analysis, Macmillan, London.
30
:
.
APPENDIX
Fix a continuous, strictly increasing
with
7/(0)
=
and
=
7;(oo)
and convex function
and
for every x
Lemma
Al. For x
G,with
in
We
a
^
+
lima-.oo
'^t;!^:,^)
and
that
is
||x||^
appendix,
we
=
linria^o
part
1,
and) only
(if
that scalar
=
-'^r/l^jCi)
=
cxd
.
such that
.
identically zero.
triangle inequality,
is
is
=
||6x||
•
||
.)
||
Of
course,
y{t) \
(
+
x{\)
/
J^^
||y||
for scalar
+
Hlkll +
+
||0||
,
is
so
that
evident. For the
h is
77
/ ||x||(x(0/||x||)+||y||(y(0/||y||) \
l|xii
we have
apparent that for every x that
|6|||x||
Hlkll + lly|ir
H
just defined,
I^
(For the remainder of the
1.
and write
77
it is
use the convexity of
\( x{t) +
r^
right
a
The equation
we
is
unique scalar a >
such that I^{x,a) = 1
well defined, and it is zero only at the function x
not identically zero, there
our candidate norm
=
Ijj{x,a)
From Lemma Al,
Because x
0.
In terms of the function
1.
a
x
if
suppress the dependence on
defined to be
Vo
decreasing and continuous
is strictly
leave the details to the reader with one observation:
Proof of Proposition
is
=
certainly well defined
is
shown.
easily
is
by
T]{x{l)/a).
variation, the integral
0, the function I^(x,a)
continuous, I,j{x,a)
is
bounded
Moreover, the following
.
[0,oo) to [0,oo)
00. Define /^(x,a), for x E A' and a G (0, 00),
T](x{t)/a)dt
Since functions x are of
from
77
y{l)
\_
I|y|i;~
||x||(x(l)/||x||)
^i^
||,||
+
+
||y||(y(l)/||y|
||y||
<_iL
+
|2:||
||y|
Nl
Ikli
Hence
Next,
||x
+
y||
<
||x||
we must show
a useful
lemma (now
+
+
XI
+
,11^1
„
lkll
l|yll
+
,
„
X
1
^
1.
liy|
||y||
We
separability.
that
we know
pause
that
space).
31
for a
we have
a
moment
in order to state
normed
topological linear
Lemma A2. A sequence {x„}
every scalar
a>0,
X
in
lim„ /;,(in,a)
=
converges to zero in
if
•
||
1|
and only
for
if,
0.
Suppose that, for every scalar a > 0, lim„ Iji{xn,a) = 0. Then for every
scalar a >
and N{a) sufficiently large so that I^{xn,a) < 1 for all n > N{a),
we know that ||x„|| < a for all n > N{a). Thus ||x„|| -> 0, and i„ -> 0.
(i.e.,
||x„|| -> 0). Fix a > 0, suppose that n is
Conversely, suppose x„ ^
sufficiently large so that ||x„|| < a, and write
Proof.
+
Ir,{x„,a)^ I T}{xnit)/a)dt
r){x(l)/a)
Jo
a
where
\\x
m\,,^^f^il)
Hikn
<feJx
a
a
\
\\Xn\\ /
Jo
the inequality follows
from the convexity of
\X,
xl,
ry
.
The
right
hand side
clearly
has limit zero.
Proof of proposition
1,
change values
at
that
most
Q
denoted
continuous functions
set of all right
finitely times, at rational
rational values only. This set,
dense
Take the
completed.
is
countable.
As
is
well
known, Q
under the L^ norm. Moreover,
in the set of all integrable funcrions,
any fixed function of bounded variation x
arguments, and that take on
,
we
is
for
can find a sequence of functions
Q which approximate the given function in the L^ norm, which differ
from X in the sup norm by at most the total variation of x plus one, and that
take on values at t = 1 that have as limit x(l) From this, it is easy to see that
for a given function x £ X and a given integer n we can pick some t/n € Q
such that Ij,{x — yn, 1/n) < l/n Apply lemma A2 to conclude that y„ — x
from
.
,
>
.
.
To complete the proof of proposition 1, we need to show that order intervals
are weakly compact. In fact, we can show sequential compactness in the norm
topology
itself.
The proof follows standard
lines,
and so we
will provide only a
sketch.
For X*
this is
>
X,
let
/ be the order interval
without loss of generality because
any countable dense
set of points in
[0,
[x.,x'].
all
32
will
assume
that x,
of our topologies are linear. Let
1) that
of X*.
We
excludes
all
=
Q
0,
be
points of discontinuity
.
Now
X e I
and only
ii
x
if
is
.
—
nondecreasing and x*
x
is
nondecreasing. Let
{xn} be any sequence out of 7. By the usual diagonalLzation procedure, we can
produce a function y such that, along a subsequence (which we will hereafter
=
not bother to mention), limnXn(i)
Xn
nondecreasing, so
is
at all
arguments
t
e Q- Since each
—
Define a function x by x{t)
y.
is
y{t)
for
limsity{t)
G [0, 1), where the limit is taken over sequences converging to t from (strictly)
above along the set Q and x(l) = limna^n(l)- Since y is nondecreasing, so is
<
Moreover, y is right continuous (and has left limits) by construction. Clearly
lim„ Xn(<) = x(^) at all points t that are not points of discontinuity of x - that
X
.
for almost every
is,
t
And
.
= x(l) by construction. Since all the x^
we can apply dominated convergence and
lim„ x„(l)
and X are bounded above by x*(l),
lemma A2
We
are
to
conclude that x„ approaches x in T^
done once we show
>
that x*
>
i
We
.
have already remarked
that x
> 0). So let us carefully sketch the other half: If x*{t) —
x{t) < x*(t') — x{t') for t > t' then right continuity of both functions implies that
we can find values q> q' from Q such that x*(g) - x{q) < x*{q') - x{q') (The
case oi t — 1 is left to the reader.) By increasing q' slightly to be some q" < q if
need be (if 5' is a point of discontinuity of x this gives x*{q) — y{q) < x*{q") —
is
nondecreasing
(i.e.,
x
,
.
,
>
y{q") for q
Which
q"
Hence
.
We note first that for any 77, T^ is topologically equivalent
any scalar a Hence, without loss of generality, we can assume that
Note that this implies, by the convexity of rj that ri{r) > r for all
to Tajj for
r
T]
>
=
1
,
Now
it
is
77(1)
=
1
(with
maximum
any
t]
.
of
any
only need
we
)
Xn
to
—
let
two convex
T^
rj ,
If
evident that
The proof of
for
A2:
2.
.
1.
.
is
Tr,
that
,
the limit exceeds
J
sum
||^
first
if
for
that
part of proposirion 2
x„
for
||^'
||
we have
Tr,^
—
>
.
Of
course,
>
rj
>
T
is
is
integer
>
.
Now
if
convex as
is
shown
T^,
—
M
Hence,
for fixed
if
the
it is
T^^
we show
,
.
for
that,
can be no stronger, so
T^,
,
.
no stronger than
thus
I^(xn,a)
look along
go to zero
some
t]'
(which
T^, then x„ -*
then for every a
4/M
<
.= max{|r|,77(r)}
functions),
equivalent to
show
>
the
•
||
T]i{r)
fixed a, /^j(x„,a) does not
the
x*{q")-Xri{q")
a contradiction.
is
Proof of proposition
T/(l)
<
n, x'{q)-Xn{q)
for sufficiently large
we
lemma
for some
For this use
Suppose that
some subsequence where
.
/^(x„,a) can be written as
of three terms:
V{^n{t)/a)l{i^„lt)\/a>i]dt
Since each term
so that the
first
is
+
positive, each
term
is less
than
T](xr,{t)/a)l[i^^(t)\/a<i]dt
must go
1/M
.
33
If
to zero in
the second
n
.
+
7y(x„(l)/a).
Pick n large enough
is to
go
to zero, since
t/
.
positive except at the
is strictly
that the
And
argument
Lesbesgue measure of the
set
e
{t
we
zero,
[0, 1]
can find n large enough so
>
1
:
> 1/M} < 1/M
x„(i)/a
go to zero, it is necessary that, for n large enough,
in(l)/a < 1/M. Picking n large enough so that all this happen at once, when
we expand /^j(x„,a) in analogous fashion, we will find that (i) the first term in
the sum is less than 1/M, since it is identical with the first term above, (ii) the
for the third
term
to
1/M directly from our selection of n and (iii) the second
term is less than 2/M, when we overestimate r]i{xn{t)/a) in the second term
with the function that is 1/M if the argument is less than 1/A/ and 1 is the
argument is between 1/M and 1 Since this is so for all n above some level,
we contradict the supposition.
term
third
than
less
is
,
,
.
Next we must show that,
and Skorohod topologies.
for
show
—+
suffices to
if
Xn -+
The
that
a:„
if
any
norm
is
no stronger than the
is
Since total variation
in total variation,
same
in total variation, then the
total variation
T^
tj ,
is
norm
a (linear)
is
then !„
total variation
—
T^
>
any
true for a:„/a for
topology,
it
Apply A2 -
.
scalar a
>
.
sup norm, so the same must be true in
convergence gives the result. As for the Skorohod
page 123, problem 2] states the result for the L^ norm
larger than the
And dominated
the sup norm.
topology, Billingsley [1968,
T
topology, which immediately gives the result for
For general
.
note that
77 ,
if
—
X in the Skorohod topology, then for n sufficiently large, sup^ kn(OI ^
supj \x{t)\ + 1/n Fixing any a , then, and evaluating /,,(xn — i, a) , we know that
x„
>
.
the arguments of
Convergence of
will
77
J,,(x„
-
Ji(xn
—
77 ,
since
means
/i
5.
can overestimate
In A'+
,
the
Accordingly, the topology
77(r)
sequence
is
sequential,
is
^
if
2/(an).
Ii(xn
—
'q*{t)
x,a)
=
t
.
described by the Prohorov
and
convergent in one of the two topologies
+
Convergence of
and we are done.
with rr]{K/a)
weak topology
is
if
the integral using the function
x,a) to zero follows from the problem in Billingsley,
Proof of Proposition
metric.
we
\x{t)\/a
x,a) to zero, then, follows immediately
converges to zero, where
in place of
K/a = 2supj
be bounded above by
it
suffices to
and only
show
that a
convergent
if it is
> 0, and, applying lemma
A2, pick N sufficiently large so that /,,(x„
x,e) < e for n > N
We claim
that this implies that p(x„,x) < e For if not, then there exists some t such that
^nit + e) + e < x{t) or such that x„(< - e) - e > x{t). Suppose that the former
is true, and suppose further that t e [0,1 - e].
See figure Al. In calculating
Ir,{xn - 2:, e) when we integrate T]{{xn{t) - ^it))^) over [t, t + e], the integrand
Suppose
in the other.
first
that x^
x in T^
.
Fix
e
-
.
.
,
exceeds
77(1)
=
1. (Recall that
e, a contradiction.
If
t
>
I
-
77(1)
e,
=
1
is
then x(l)
34
Hence the
- x„(l) > e, and
wlog.)
integral
the last
is at
least
(summed)
,
term in
some
i
/,,(xn
Xn{t
,
- x, e) will exceed 1 again a contradiction. The
- e) - e > x{t) is handled similarly.
case where, for
,
^ x in the weak
/^(x„ -x,a) ^ 0, so that x„
>
lemma A2.
Conversely, suppose that Xn
topology. Fix a constant a
We
-> x
If
will
show
p(x„,x)
that
—
0, then
^
it
immediate
is
term in 7;,(x„,a) converges
with the
first (integral)
large so that for n
nondecreasing,
we
that
to zero in
n
.
-
|xn(l)
We
,
X(l)| -> 0,
m,
so that the
.
final
N{m) be
sufficiently
that both x^
and x are
let
< 1/m. RecaUing
p(xn,x)
follows by
need, therefore, only be concerned
term. For a given integer
> N{m)
T^
can bound
above by
1
m
^
a
\
^
Jt=i
Now since
x{t
/max{x{k/m),Xn{k/m)] -mm{x({k -l)/'m),Xn{{k - l)/m)}
-^
p(xn,x)
— 1/m) — 1/m,
< 1/m, max{x(<),x„(i)} < x(i+l/m)+l/m and
so the summation given
_^^
(
k=\
V^
m
^
From the shape of
is bounded above by
+ l)/m) -
x{{k
is
x{{k
min{x(i),Xn(i)}
bounded above by
- 2)/m) + 2/m
a
(convex, equal to zero at zero, and increasing), this in turn
77
x{{k
+
l)/m)
-
xi{k
- 2)/m) + 2/m
\k=\
^
1
m
For fixed a
,
the
argument of
r/
is
/3x(l)
+
2
a
\
fixed, so this
upper bound vanishes with
m
proving our assertion.
Proof of the technical assertion in the proof of proposition
x„ and
T/n
are as in the
(2.3),
then x„
techniques employed just above
1
/3x„(l)
n
\
a
show
— 1/„
—
>
that an
+ 2\
T^
4.
.
We
Fixing a
upper bound on
/^x„(l)-y„(l)
J
35
show
scalar a >
need
to
/,,(xn
—
that
if
0, the
yn,a)
is
>
,
The second term vanishes
for
any a since
a;„(l)
—
y„(l)
< p{xn,yn) < \/n And
.
By assumption, a:n(l)//i(n) — 0, and thus
(3x„(l) + 2)/(a/x(n)) — 0. Hence for any e there is A''^ such that for all n > N(
to both sides (
(3x„(l)+2)/a < efi(n) Apply
is monotone), and use the shape
the
first
term vanishes
for r-^y
a:
>
>
.
of
r?,
for all
for
e
<
1
to
7/
r;
conclude that r7((3i„(l)
n > N(.
36
+
2)/a)
<
T]{efi{n))
<
er]{fi{n))
=
en,
A neighborhood of the function x in the Prohorov
Another function y is within epsilon of the function x
y is connpleteiy within the "sleeve" around x that is drawn.
Note well the hatched part of the sleeve near the origin, which
exists because we implicitly think of x as being extended for
negative arguments having the value zero.
Figure
1
:
metric.
if
t+e
t
Figure A1
each
.
The dashed box
of length epsilon.
is
a square whose sides are
7131
U53
Date Due
OEC
8
A99A'
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Lib-26-67
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