:-
(UBfiARIES
.H414
—
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
ON INTERTEMPORAL PREFERENCES WITH A
CONTINUOUS TIME DIMENSION II:
THE CASE OF UNCERTAINTY
by
Ayman Hindy and Chi-fu Huang
WP# 2105-89
March 1989
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
^241989]
ON INTERTEriPORAL PREFERENCES WITH A
CONTINUOUS TIME DIMENSION II:
THE CASE OF UNCERTAINTY
by
Ayman Hindy and Chi-fu Huang
WP# 2105-89
March 1989
JUL 2 4
1989
ON INTERTEMPORAL PREFERENCES WITH A
CONTINUOUS TIME DIMENSION II:
THE CASE OF UNCERTAINTY
Ayman Hindy and
School of
Chi-fu Huang*
Management
Massachusetts Institute of Technology
Cambridge,
02139
MA
March 1989
Abstract
This paper proposes a family of topologies on the space of consumption patterns
time under uncertainty.
continuous
Preferences continuous in any of the proposed topologies treat con-
sumptions at nearly adjacent dates
The
in
cis
almost perfect substitutes except possibly at surprises.
topological duals of the family of proposed topologies essentially contain processes that are
the
sums
for
consumption come from the duals, consumptions
of processes of absolutely continuous paths and martingales.
prices except possibly at surprises. In particular,
Brownian motion, the duals are composed of
if
when
the information structure
Ito processes.
there are no arbitrage opportunities.
our topologies on standard models of choice
in
if
equilibrium prices
at nearly adjacent dates have almost equal
of long-lived assets in economies populated with agents
topologies
Thus
We
is
generated by a
discuss the properties of prices
whose preferences are continuous
We
also investigate
in
our
some implications
of
continuous time as well as on recent models of
non time-separable representations of preferences.
'The authors are thankful for the helpful conversations with David Kreps throughout the years this paper
was in preparation. Huang is also grateful for research support under the Batterymarch Fellowship Program.
Introduction and
1
summary
Mom,
I
can not eat any spinach now.
my
had
I just
dinner.
Anonymous
summary
Introduction and
1
No
"standard" economic agent could use the above convincing argument to delay his consump-
tion.
"Standard" economic models assume that very recent consumption has no
effect
on one's
current appetite; an cissumption that each of us repeatedly violates.
In addition,
we
frequently violate an important implication of the "standard models" con-
cerning the prices of
Eissets.
be much higher than
will
If
one knows for sure that the price of a certain asset tomorrow
today then one would be willing to delay his consumption
it is
day and use the proceeds to buy such an
else
who
is
asset. Alternatively,
willing to delay his consumption, pay
him a
the funds to realize the gain in the price of the asset.
to increase
and become very
Our behavior would
tomorrow.
If
new information might
and use
force today's price
no one were willing to
jump
delay consumption for one day for a small fee, then a situation of a big
periods in the absence of any
one
one could borrow from someone
relatively low rate of interest
close to the price of the asset
for
in prices over short
prevail in equilibrium.
In fact, this
is
a
prediction of "standard" models in which continuous changes in prices are obtained mainly by
We
exogenously specifying continuously varying endowments.
one develop a model
ask: could
of preferences for consumption over time under uncertainty that agrees with our economic intuition in
of prices
of the
which past consumption has an
is
is
on current
and
utility
in
which the continuity
a phenomenon implied by continuity of preferences and independent of the nature
endowments? This paper
This
effect
is
an attempt to address these
the second part of a series of two papers.
which we henceforth abbreviate as
H&K,
The
first
issues.
Huang and Kreps
part,
addresses the following questions:
How might
(1987),
one rep-
what are the appropriate
resent a consumption pattern in continuous time under certainty and
topologies on the space of consumption patterns that capture the idea that consumptions at
nearly adjacent dates are almost perfect substitutes? Moreover,
rium prices take when individuals
the
in
what form would the
economy have preferences continuous
in the
equilib-
appropriate
topologies?
"Standard" answers exist for the questions raised by U&iK:
time interval, say
is
[0,1], is
represented by a real-valued function
the consumption "rate" at time
functions in an 1/
A
norm topology
t.
for
Two
some
1
c
consumption pattern on a
:
[0,1]
consumption patterns are close
< p <
oo.
An
—
>
if
5R+,
where
c{t)
they are close as
agent's preferences are represented by
1
Introduction and
summary
2
j} u{c{t),t)dt, a time-additive functional of consumption rates, where, using the terminology
of
Arrow and Kurz
p
consumption pattern
c
y{t) is interpreted as the
and
a "felicity" function for consumption at time
is
come from L' with
equilibrium prices
at time zero of a
(1970), u{c,t)
+
^
=
!>
^^^ topological dual of I/; that
is,
can be represented as /q c{t)y{t)dt, where y
t.
The
the price
is
from L'
time zero price of one unit of consumption per unit time at time
t.
H&K
in
argued that these standard answers are unsatisfactory for the following reasons. First,
modeling consumption over time, one should allow consumption at rates as well as
observed consumption behavior such as having meals
the 1/ topology on the space of consumption rates
is
so strong along the time dimension
that consumptions at nearly adjacent dates are perfect nonsubstitutes!
the strong topology, the space of equilibrium prices
is
gulps -
Second and more important,
in gulps.
is
in
too rich.
As
a consequence of
includes, for example,
It
discontinuous functions of time and continuous functions of unbounded variation signifying
that equilibrium prices for consumptions at nearly adjacent dates are either very different or,
even though continuous, fluctuate
The most
general
way
in a
nowhere differentiable manner.
for representing a
by an increasing^ and positive function on
consumption pattern over time under certainty
[0,1].
Let x be such a function.
the accumulated consumption from time zero to time
of time,
The
its
t.
If
is
x{t) denotes
an absolutely continuous function
derivatives exist almost everywhere and can be interpreted as consumption rates.
discontinuities of x are the gulps of consumption.
space of these consumption patterns,
among
i
Then
H&K
Letting
introduce a family of
X
be the linear span of the
norm
topologies on
X
so that,
other things, consumptions at nearly adjacent dates are almost perfect substitutes.
example of
is
this family of
norm
topologies
is
the
L''
An
topology on accumulated consumption,
where the Z^-norm of a consumption pattern i
is
given by:
{C\x{t)\^dt
+
ix(i)r)p.
Jo
HiiK
also
show that preferences continuous
in
any of the norm topologies can be represented
by maximizing the time-additive functional of consumption rates
only
if
the felicity functions are linear!
The equilibrium
in
the "standard answers"
prices basically
come from absolutely
continuous functions - prices of consumptions at nearly adjacent dates are almost equal and
change over time
in
an almost differentiable manner.
'Note that one might expect equilibrium prices for consumption to fluctuate widely in an economy under
uncertainty due to temporal resolution of uncertainty, however; see Huang (1985a, 1985b).
^Throughout this paper we will use weak relations: increasing means nondecreasing, positive means nonnegative, etc.
When
the relations are strict, we will say, for example, strictly increasing.
1
Introduction and
summary
3
of this paper
The purpose
to develop topologies on the space of consumption patterns
is
under uncertainty that capture the notion that consumptions at adjacent dates are almost
A
perfect substitutes.
under uncertainty
is
crucial distinction
between an economy under certainty and an economy
that in the latter the passage of time also reveals partially the true state of
nature, or reveals information. In a world of uncertainty, the preferences of an individual over
consumption patterns may be affected by the way uncertainty
which any uncertainty
for
new
is
Thus
dates, which
it is
is
certainly different from a world which
A world
in
is
"sudden" and
"full of
unreasonable to require a priori that consumptions at nearly adjacent
may be random, be almost
perfect substitutes unless there are no "surprises" there.
we would not expect the equilibrium
Similarly,
resolved over time.
resolved gradually over time, with ample "warning" and "preparation"
bits of information
surprises".
is
prices to be continuous except at
(random) times
of no surprise.
We
introduce a family of
topologies on the linear span of the set of consumption
norm
patterns under uncertainty, which
is
composed of processes with positive and increasing paths.
This family of topologies are natural generalizations of the H<5iK topologies.
they reduce to the
H&K
topologies
when the
true state of nature
consumptions at nearly adjacent dates, where there
almost perfect substitutes.
(to
be defined)
in
We
is
In particular,
and
revealed at time
is
no discontinuity of information, are
then show that preferences continuous and uniformly proper
any one of the family of topologies exhibit intuitively appealing economic
properties.
In general equilibrium theory, equilibrium prices
Thus we
come from the
topological dual spaces.
also characterize the topological duals of the suggested family of
show that they are
essentially
functions.
topologies and
composed of processes which are sums of processes
This
continuous paths and martingales.
the case of certainty the
norm
shadow
is
prices for
a very natural result.
consumption are
Hence preferences that treat consumption
H&K
of absolutely
have shown that
in
essentially absolutely continuous
at nearly adjacent dates as almost perfect
substitutes will give rise to nearly equal prices for consumption at nearly adjacent dates and
these prices vary over time in an almost differentiable fashion; an intuitively attractive conclusion. In the case of uncertainty a
new element, the pattern
sample path properties of equilibrium
nent of the price process.
at surprises
It is
and can fluctuate
prices.
known that
in a
manner (due
effect is
captured
a martingale can
in
the martingale compo-
make discontinuous changes
only
nowhere differentiable fashion. Thus equilibrium prices
consumption are continuous except possibly
entiable
This
of information flow, affects the
at surprises
and can fluctuate
to temporal resolution of uncertainty).
in
a nowhere
for
differ-
This agrees with our economic
Introduction and
1
summary
4
intuition.
We
also investigate properties of arbitrage-free price processes in a
economy where an arbitrage opportunity
ket
from our family of topologies.
It is
shown
securities
between lumpvsum ex-dividend dates, price
that,
In particular,
the information struc-
if
generated by a Brownian motion and accumulated dividend process of a security
is
absolutely continuous process, the ex-dividend price process for this security
It is
mar-
defined using concepts of continuity derived
is
processes are continuous except possibly at surprises.
ture
dynamic
is
is
an
an Ito process.
worthwhile to point out that earlier models of equilibrium using the standard repre-
sentation of utility had to rely on exogenous factors in addition to preferences to characterize
Huang
the sample path properties of equilibrium prices; see Duffie (1986) and
example,
in the ceise of a
lutely continuous
Brownian motion
filtration,
accumulated dividend process
endowment process
is.
the price process for a security with abso-
will not
be an Ito process unless the aggregate
consumption
In our setup, however, since
For
(1987).
at nearly adjacent dates are
almost perfect substitutes at times of no surprise, price processes of securities with absolutely
continuous accumulated dividends will be Ito processes independently of the properties of the
aggregate endowment process.
As
for the existence of
an Arrow-Debreu equilibrium
whose preferences are continuous
in
in
one of our topologies, we have
opens up a question about the existence of an equilibrium
rest of this
paper
is
and an information structure
is
F =
[0, 1].
Taken
£
[0, l]}i
{Tt',t
time
7
,
Known
economies of our type.
eis
primitive
is
specifies
how
a probability space (n,
Q
where each u E
will consider,
P)
denotes a state of nature,
P
is
the probability beliefs
and F, an increasing family of
7
distinguishable events in
1,
,',
are revealed from time
to
1.
A
ing,
fields of
in
the collection of distinguishable events at time
about possible events held by the individuals we
sub-sigma
to report.
organized as follows. Section 2 formulates the stochastic environ-
ment under study with a time span
7, a sigma-field,
little
an equilibrium are not satisfied by our topologies. This
sufficient conditions for the existence of
The
an economy populated with agents
consumption pattern x
=
{x{t);t
€
[0,1]}
and right-continuous sample paths which
is
is
a stochastic process having positive, increas-
consistent with
variable x{t) denotes the accumulated consumption from time
of consumption patterns and
X so that
We
X be the linear span of X+.
preferences continuous with respect to
put forth a wishlist for a topology
wishlist in Section 3.
The agenda on our
T
it
Our
F
or adapted to F.
to time
task
is
t.
Let
X-(-
The random
be the space
to define a topology
is
on
exhibit economically desirable properties.
and introduce a family of topologies to
wishlist
T
two-fold: First, an
satisfy this
economy under
certainty
^
Introduction and
1
is
sf>ecial case of
a
summary
an economy under uncertainty where the true state of nature
In such event,
at time 0.
5
necessitates that the
we would
norms
of
like
H&K
our topology to agree with the
H<kK be used path by path
is
revealed
This
topology.
by state) on a consumption
(state
pattern. For the L^ example mentioned above, the path-wise distance between x
6
X+
and
is
where
i(a;, t) is
in (1)
p
is
the value of the
random
a function of the state of nature
trade-off of consumption across time
is
when the
variable x{t)
the
w
- there
same
is
state of nature
time
in
may
w. Note that
no a priori reason to expect that the
for all states.
One
Second, consider substitutability of consumption across states.
a time in a particular event
is
unit of consumption at
be a close substitute to one unit of consumption at the same
another event for an individual.
But there
is
no economic reason to expect that
all
individuals with continuous preferences consider consumptions in different events to be close
substitutes.
Thus the topology on
X should not a priori build in substitutability of consumption
On
across states in an arbitrary manner.
consumption
in
an event which
is
the other hand,
it is
quite intuitive that one unit of
very unlikely to occur should be close to not consuming at
all.
With
all
these considerations in mind, a natural
struction in the previous paragraph
is
way
of aggregating the path-wise con-
to just take expectation. Taking expectations embodies
the notion that consumptions at two disjoint events are perfect nonsubstitutes except
when
both events are negligible in probability and thus the two consumption patterns are almost
indistinguishable from zero. Taking expectations also embodies the notion that, at any time,
the differences of preferences for consumption in two events with the
decreasing function in the degree of overlap of the two events.
We
same probability
is
a
here remind the reader that
defining such a strong topology and considering continuous preferences certainly does not rule
out preferences that actually consider consumptions across states to be close substitutes. They
will
be special cases of preferences continuous
above, except
now
that p
is
in the "strong" topology. In
is
a standard U'
same U' example
nonrandom, the norm of a consumption pattern x G
{^^j\x[t)ydt +
which
the
\x[\)Y
X+
is
then
(2)
norm but on accumulated consumptions
rather than on consumption
rates.
^The reader may have noticed that
(2)
is
not
finite for
attention to a subset of X-|. depending on the topology.
every i
€
X.)..
Thus we
will
have to restrict our
1
summary
Introduction and
we develop some
In section 4
defined in Section
ties of the
6
3.
topological and uniform properties of the family of topologies
The reader
will see that the family of topologies inherit
Hi:K topologies when those
many
proper-
properties are properly "aggregated" across states by
taking expectations. Basically, the family of topologies exhibit the desired economic property:
consumptions at nearly adjacent dates where there are no discontinuities of information are
almost perfect substitutes.
Since uniformly proper preferences (see
Mas
Colell (1986)) have
been needed for the general equilibrium theory, we also investigate properties of preferences
that are continuous and uniformly proper.
Section 5 characterizes the topological duals of the family of topologies under consideration.
We show
there that the topological dual spaces corresponding to our topology - spaces which
contain the shadow price of consumption - consist of processes that can be decomposed into
the
sum
of two components: an absolutely continuous part and a martingale part.
We
show that a topological dual space corresponding to each one of our topologies may
a sublattice
in
known
the order dual; a property
further
fail
to be
to be sufficient, together with other things,
to guarantee existence of a Walrasian equilibrium in our model.
In particular, in the case of
Brownian motion information, the duals spaces are not sublattices of the order duals.
Section 6 examines
up.
how standard models with
time-additive utility functions fare in our set
Similar to the results of HA:K, preferences represented by time-additive utility functions
over consumption rates are continuous in any of our topologies
In addition,
we examine some
consumption on current
are not continuous in the sense that
In section 7
curities in
Bergman
utility,
we
they are linear.
(1985), Constantinides (1988), Heaton
we examine the implications
dynamic
asset markets.
Our
In particular,
most of them imply preferences that
of our topologies on the prices of long-lived se-
discussion
under conditions of "no arbitrage."
in section 5.
find that
we advocate.
we show that
is
We show
long lived securities inherit the properties of the
tures,
if
and Sundaresan (1988). Although such representations have elements that capture the
effect of pzist
prices
and only
of the prevalent representations in the literature of "non time-
additive" utility functions such as those in
(1988),
if
along two
lines.
First,
we examine
that in this case the prices over time of
shadow
in the case of
prices of
consumption that we discuss
Brownian motion information
and absent any arbitrage opportunities, the prices of long
lived securities
continuous accumulated dividend processes will be Ito processes. This result
by the continuity of preferences. Second,
continuous trading
in
we
asset
struc-
with absolutely
is
purely driven
discuss similar issues in the case of markets with
dynamic equilibrium, under the assumption that such an equilibrium
exists. Section 8 contains
concluding remarks.
Formulation
2
7
Formulation
2
Consider an economic agent
is
who
lives in
a world of uncertainty from time
a single consumption good available for consumption at any time
modeled by a probability space
w € n
J,P). Each
(fl,
a complete description of one possible realization of
time
to time
and
at time 1
We
This
is
/
sigma-field
There
Uncertainty
is
represents a state of nature which
is
t
[0,1].
exogenous sources of uncertainty from
represents the collection of events which are distinguishable
a probability meeisure on
modeled by a
is
We
P
The
1.
{CI,
7).
take as given the time evolution of our agent's knowledge about the states of nature.
fields of J;
t.
1.
all
G
to time
that
filtration
T,
'^
7t ii s
assume that T
=
7i,
F
filtration
is,
F=
{Tt;
<
Interpret
that
t.
t
E
which
\0,l]},
7t
an increasing family of sub sigma-
is
as the information that the agent has at time
known
1,
and the
suppose that the information structure
satisfies
the true state of nature will be
is,
at time
satisfies the following usual conditions:
1.
F
is
complete in that 7o contains
2.
F
is
right continuous in that /(
we
Besides these usual conditions,
=
all
P-null sets;
Tt+,
where
will further
=
7t+
^»-
At<«
a regularity condition, which almost always holds in applications.
Before we do that, some
definitions are in order.
T Q—
Definition 1 The function
:
[0,oo] is
an optional time with respect
{ujen:T{uj)<t}eTt
An
optional time can always be interpreted to be the
condition
be
{wen:
<
T{oj)
known whether a
E
t}
certain event
Definition 2 An optional time
times {Tn} such that T„
Tn
y T.
Tt in
< T
The sequence {Tn}
Intuition suggests that
if
T
a.s.
is
the
F
if
VtG[0,l].
first
time a certain event happens. The
the above definition then says that at any time
happened or
is
to
will
not.
said to be predictable
and on
t, it
the set
{T >
if there exists a
0}, almost surely,
sequence of optional
Tn < Tn+i <
T
and
said to announce T.
first
time an event happens
is
announced by the occurrence of
a sequence of other events, then the event will not take one by surprise. This intuition turns
out not to be correct for general information structures and
information structure:
is
valid for a quasi left-continuous
Formulation
2
An
Definition 3
and
its
8
F
information structure
quasi
is
left- continuous if
for a predictable time
T
announcing sequence {T„} we have
oo
^r
V
=
^T„
=
7t-
,
n=l
where 7t
is
a sigma-field of events prior to
Af]{T <
We
For example, the
or a combination of the two
is
filtration
7®
For each
consisting of all events
y
is
is
quasi left-continuous. This
Y{ui,-)
fi,
:
a mapping
[0,1] -> dt is
to be adapted to
constraint:
F
for each
if
E
t
[0,1],
V n
:
x
[0, 1]
—
to
F
that
9i
7 and
life
all
/^-measurable. This
is
t
consumption from time
the Borel sigma-field of
t
F (-,«): fi -» !R
The process Y is said
G
is
[0, l],
a natural information
the processes to appear will be measurable and adapted
is
represented by a process x whose sam-
and right-continuous with
to time
t
We
in state w.
x{uj,t)
denoting the accumulated
denote the set of such processes by
linear span of X-(., the space of processes having right-continuous
sample paths,
will
only on a subset of
preferences over
"continuous."
X+.
and bounded variation
be denoted by X.
For technical reasons, which will be
clear later,
this subset for
now by
we
£.^
will consider preferences defined
The agent
is
assumed to have
which are given by a complete and transitive binary relation > that
£,
Our
X+. Denote
made
task
is
to define a topology, say
T
,
is
on C such that preferences continuous
with respect to which exhibit desirable economic properties. This
now
measurable with
cannot depend on the information yet to be
time consumption pattern of an agent
ple paths are positive, increasing,
The
Y{t)
is
unless otherwise mentioned.
The
is
the subject to which
we
turn.
*It
has been shown by
Meyer (1963)
at predictable optional times (defined
is
>
& sample path and for each
the value of the process at time
revealed in the future. For brevity,
without much
completed.^
is
a random variable, which we will usually simply use Y{t) to denote.
is
is
generated by a diffusion process, a Poisson process,
S([0,l]), the product sigmarfield generated by
we
such that
for allte\0,l].
7t,
F
AE 7
quasi left-continuous once the filtration
measurable stochastic process
respect to
[0,1].
e
assume throughout our analysis that
loss of generality.
A
t}
T
that any information structure generated by a process that
with respect to
its
natural filtration) and has the strong
is
continuous
Markov property
quasi left-continuous.
^This subset
in fact
depends on the topology chosen. The reader can think of the the classical Banach space
all the random variables and is "topology-dependent."
example: L''(Q,7,P) certainly does not include
A
3
A
3
norm
family of
topologies
9
norm
family of
Before proceeding, we will
topologies
review the family of
first briefly
H<kK norm
This space, denoted by X,
span of the space of consumption patterns under certainty.
linear space
An
composed of functions on
[0, 1]
to time
and
A topology
t.
fi{oo)
inverse of
=
in the
oo that
/i, r;
:
i € X+,
H&K
x{t) denotes the
,
=
a
an increasing, positive, and right-
is
accumulated consumption from time
family corresponds to a function
SR+ with f7(0)
>
X+
strictly increasing, continuous,
is
—
!R+
If
[0, 1].
is
that are right-continuous and of bounded variation.
element of the positive orthant of X, denoted by
continuous function on
topologies on the linear
=
oo
r,{\x{t)\/a)dt
+
r,(|i(l)|/a)
—
St^.
:
and concave.
r?(oo)
and
/i
*
!R+
with
Defining
rj
=
/i(0)
=
n~^
the
,
strictly increaising, continuous,
is
and
Then
convex.
inf{a
>
/"
:
<
1}
Jo
defines a
norm
H&K
the above norm.
Two
(1)
x E X. Let > be a preference relation continuous
for
showed that the following properties are
X > y and y
> Xn
for all
(2) Sizable shifts in
n
—
»
is
€ X+
,
n implies
=
inf{f
^
y for
y
>
—
|in(i)
ij
x{t)\
=
then Xn
0,
>
:
i{<
+
e)
+
e
>
y{t)
>
x{t
-
e)
-
e, t
y for
>:
that
y,
and y
uniformly proper
is
in
>;
in for
all
amount
of time
is
all
n implies
insignificant:
propemets
will
[0, l]}.
n implies y
>:
If
p(xn,x)
—
>
as
x.
the direction X|m)> the indicator function of
regarded negligible. This
into the picture:
definition of uniform,
e
[O, l].)
the sense of Mais Colell (1986)^, shift of an increasingly large
The formal
>
x.
n implies x >
all
across a decreasingly small
comes
lim„_oo sup^giQ
the Prohorov metric on the space of increasing functions on
oo, then Xn
1], in
if
let
Moreover, for
[t,
by >.
consumption across small amounts of time are regarded as
p{x, y)
(This
satisfied
topology defined by
patterns of consumption that have almost equal accumulated consumption at every
point in time are close:
for I, y
in the
be given
later.
amount
is
of consumption
where the function
/i
3
A
(3)
Let {x„} and {t/n} be two sequences of consumption patterns with the properties that
norm
family of
topologies
{i„(l)} and {y„(l)} are both
Xn
+
Xt
>
Vn,
Litereilly,
where {in(l)}
means
(3)
if
aw;ross
which consumption
the
sequences in
(l)
(3)
and
p(in,yn) < 1/", then for
o{fi{n))
means lim„^oo
amount
of
=
0.
consumption shifted increases to
and
incresises to infinity
n
x„(l)//i(fi)
if
large n,
all sufficiently
infinity at a rate
the decreasingly small time interval
shifted goes to zero faster than
1/n goes to zero, then the
shift is
(2) are topological properties
while (3)
is
a uniform property since the
diverge rather than converge.
With these above
First, note that
H<kK
facts about the
tions that determine a family of
tainty.
(ii)
(i)
terms of preferences.
insignificant in
Note that
is
and
o(/i(rj)),
is
slower than the rate at which
10
norm
degenerates to an HitK topology
X
is
to two considera-
X.
a special case of an economy under uncer-
to a case of the latter where the true state of nature
at time 0. Therefore, a candidate topology
time zero, the topology on
we now turn
topologies on subspaces of
an economy under certainty
The former corresponds
||x||
topologies in mind,
when
Jo
is
revealed
under uncertainty should have the property that
=
7i.
That
is, if
the true state
u> E.
Q
it
were known at
should be generated by the "norm:"
=inf{a>0: /
+
<p{u,\x{u,t)\/a)dt
^{<^,\x{uj,l)\/a)
<
l),
Jo
for
some
creasing,
tp{u,-)
:
5R+
—
5R+ with <p[oj,0)
>
and convex. Note that n{u,
•)
=
=
and ^(a;,oo)
(p{uj,
•)~Ms a
/i
in
=
oo that
is
continuous, strictly in-
HA:K that measures how
fast shifts
of consumption across time increase to infinity in property (3) above. (Here, however,
depend upon the state of nature since every state of nature can be viewed
under certainty and there
time will be identical
that an
in
is
no a priori
we expect
>1 is
is
expect that the trade-off of consumption over
of an i
G X.
P
should be "close" to not consuming at
from
A
except
all.
On
no reason to expect that one unit of consumption at time
a close substitute for one unit of consumption at the same time but
disjoint
economy
that one unit of consumption at any time and in an event which
negligible in terms of probability
hand, a priori, there
may
each one of these different economies.) This consideration necessitates
H<kK norm be used on a path
Second,
reeison to
as a different
/i
when both A and
in
is
the other
t
in
event
another event A'
A' are negligible in terms of probability P, since in
which case, both consumption patterns are "close" to
These two considerations suggest that we use
H&K
zero.
path-by-path and "pa^te" together these
path-wise constructions by taking expectation according to P. This leads us to the notion of
A
3
norm
family of
topologies
11
(Musielak (1983)
a Generalized Orlicz space.
is
The
a good reference.)
following
is
a brief
introduction.
Consider the measure space (n x [0,1],0,P x A), where
the sigma-field generated by
P
the Lebesgue measure on [0,1], and
mapping on
fJ
X
[0, 1]
that
that an optional process
is
F
the processes adapted to
all
x A
is
the optional sigma-field -
is
having right-continuous paths, A
P
the product measure generated by
meeisurable with respect to
an optional process.
is
and
It is
A.
is
A
known
adapted to F; see Chung and Williams (1983). Denote the space of
is
optional processes with equivalence
P
x A almost everywhere by O. Note that
X
is
a subspace
of O.
$
Let
be the collection of functions
^ n
:
x !R+
—
5R+ meeisurable with respect to
»
J
B(!R)
with, almost surely,
<p{u),0)
=
and
=
v'(a;,oo)
oo,
that are continuous, strictly increasing, convex, and integrable in that
(p{ijj,
/
z)P{d(jj)
<
Vz >
0.
for a,
^ >
0,
x 5R+
—
oo
Jn
A
modular
is
(a) e(0)
=
(b) ^(i)
=
(c)
e(-i)
(d)
^{ax
For
<p
a function ^
:
O—
+
[0,oo]
such that
0;
implies i
-
+
=
0;
+
^(y) for x, y
e(x);
^y) < ^{x)
€ ^ and
for all i
€ O, ^(w,
G
O
|i(a;, <)|)
and
:
n
»
a
5R+
-h /?
is
=
1.
measurable with respect to
7 ® S (5R) and
^W
=
In /o
V>{<^,
\x{uj,t)\)dtP{duj)
+
/n ^(u;,
\x{uj,
l)\)P{du)
(3)
^E[l^cpi\x{t)\)dt
is
+
r{\x{l)\)]
a modular on O.
The modular
V
The
^ defines the so called generalized Orlicz space,
xeO-.E
/V(7|x(t)l)dt
+
vp(7l^(l)l)
U' where
,
as
7 -»
0-f-
(4)
set
^=\xeO:E f\{\xmdt + ^{\x{l)\)
Jo
< oo
(5)
,
3
A
family of
norm
topologies
12
will be called a generalized Orlicz class. It
O
the smallest vector subspace of
in
Lq
X
that Lq
The
containing Lq.
a convex subset of L'^ and
is
largest vector subspace of
O
L^
is
contained
is
E^ = <x e O: E
An
known
is
€ E^
Remark
<p['j\x{t)\) dt
1^
+
V7>ol.
<oo
<p[t\x{l)\)
(6)
said to be a finite element of L^.
is
1 If
x)
(p{u!,
=
<p{x) is
independent ofu, we say that
Orlicz class, respectively. Also, in the case of certainty, there
L"^
and Lq
are Orlicz space
no distinction between L^
is
,
and
Lg,
and Ef.
We
will henceforth restrict
our attention to E'^ on which we define a norm:
,
Definition 4 Given the function
=
\\x\\^
it is
E ^, we
mL>0:E
(For the fact that (7) defines a
Here
<p
norm on
E'^
As we mentioned
earlier,
X
is
,
\\
=
\z\'^
some
for
where
•
||
||
is
1
Proof. The
first
E'^.
(if''',!!
by
\\-\\^
oo, then
a proper subset of O, therefore £'^ contains processes whose
(£''',11
•
a
||^) is
A
P
•
||,^)
is a
normed space
Now
=
putting f '^
E'^
normed space with compact order
f]^, the
intervals,
with norm-compact order intervals.
proof of the second assertion
•
||
||^
peistes together the
- as we envisioned
earlier.
is
•
||
\\^
norm on
a
is
provided in Appendix A.
path-wise
H&K
E'^ and if^
is
a
I
construction by taking expec-
Henceforth, denote the topology on
if '^
generated
by T^.
Denote the positive orthant of f '^ by f^
then a consumption pattern. The cone
i,y e
< p <
equivalent to the standard 1/ norm.
assertion follows from the fact that
Note that the norm
tation under
1
as defined in (7) but restricted to £'^.
Proposition
subspace of
(7)
L''(n X [0,1],0,P X A),
sample paths are not of bounded variation or right-continuous.
following proposition shows that
on E'^ as follows: Vi G E'^
•
\\
Musielak (198S).)
see
if <p{ijJ, \z\)
L^ = L^ = E^ =
in Definition 4 is
norm
l\[^-^)dt + ^{^-^) <l|.
worthwhile to remark that
and the norm defined
define a
S'^.
We
say x
is
S^
h <f^f|X+,
which
defines an order on f
"greater" than y, denoted by i
>
y, if
i
-
is
'''
y
a cone.
in
Each i € f^
is
the following way: Let
G f^.
3
A
family of
norm
topologies
For the analysis to follow,
13
we
shall cissume that our agent's preferences
>
are defined on
if
^
and are continuous with respect to T^. Using the terminology of general equilibrium theory,
C^ and f^
A
are referred to as the commodity space and the consumption set, respectively.
natural question that arises from the above definition of commodity spaces
The answer
spaces and their corresponding topologies compare.
is
to such a question
how
is
these
given in
the following proposition.
Proposition 2 Let
<
<P2{<^, z)
where h
is
(p{uj,z)
—
all
is
some
It is
Theorem
the topologies {T^;
T
is
all y?
easily verified that, for all
<tp{uj,\z\)
random
—
a.e.
strictly positive constants.
<i''c
fcz
x X
Then
there
is
is
a weakest one - the one generated by
formally stated in the following corollary.
e $.
e ^,
h{u>)
PxA-a.e.
Vz >
The
variable h on {Q,T,P)-
assertion then follows from
I
2.
2 //
<p
+
The commodity space corresponding
Remark
ki,
P
I
8.5).
G $},
ip
weaker than T^ for
positive integrable
Proposition
Q and
and
stronger than T^j.
\z\
for
Vz >
h{u))
and denoted henceforth by T. This
\z\
Corollary 1
Proof.
+
ki(pi{(jj,k2z)
See Musielak (1983,
Among
satisfy:
a positive integrable function on
£fi c f'2, and T^,
Proof.
& ^
<pi,(p2
ipi
written as: There
and
is
<p2
ki,k2
are independent of
>
and
zq
(P2{z)
Now that we
>
<
=
to <p{uj,z)
ui,
\z\
will
be denoted by £.
then the condition in Proposition 2 can be
such that
ki<pi{k2z)
Vr >
Zq.
have defined a family of commodity spaces and their corresponding topologies,
the next item on our agenda
is
to see
whether continuous (and uniformly proper) preferences
give rise to economically desirable behavior.
4
and uniform properties of T^
Tof)ological
Topological and uniform properties of 7^
4
In this section
we
whether the topology T^ gives economically reasonable sense
will investigate
f^
of "closeness" for consumption patterns in
Tip
14
inherits
many
for all <p
€
^.
H<kK topology when those
properties of a
The reader
will find
out that
properties are appropriately ag-
gregated across states. In particular, consumptions at nearly adjacent dates where there
no
is
discontinuity of information are almost perfect substitutes - a natural generalization of HiiK.
We
first
record two facts about {S'^,T,p).
Proposition 3 Suppose
Then (f',7^)
that [Cl,T,P) is separable.
separable
is a
normed
topo-
logical vector space.
Proof.
Recall that [Q,
such that for every
A
where
7 P)
,
>
e
is
separable
if
whatever small and
A
T
E.
denotes the symmetric difference of two
dense set of (f '^, T^)
{Bn G
there exists a countable set
there exists
Then
sets.
it is
B„ such
7; n
=
1,2,
.
.
P{AABn) <
that
.}
e,
easy to see that a countable
the set of right-continuous bounded simple processes that change their
is
values at rational time points, take rational values, and vanish except at finite subsets of {fin}.
(A process
[0,1],
Y
is
simple
there exists a finite subdivision
if
and random variables a,ai,
Y{u;,t)
=
Remark
a{u)x{o}{t)
+ E7:I
.
.
.
=
C[0,
l],
J,P)
is
fact that this probability space
An immediate consequence
is
<
ti
<
•
•
<
tn
=
I
o{
I
separable. For example, in
the space of continuous function on
generated by open sets defined by the sup
to
7t_-measurable and bounded such that:
a,- is
Mu;)xit,.t.^,]{t).)
3 In most applications, (n,
cial markets, CI
,a„_i, where
=
norm on
C[0,l], and
[0, 1],
P
is
J
many models
is
the
of finan-
the Borel sigma-field
Wiener measure. The
separable can be found, for example, in Billingsley (1968).
of Proposition 3
is
that any continuous preference relation
> on
separable (f^, T^) has a numerical representation; see Debreu (1954).
Next we
will
examine
preference relation in
process,
we
First,
will also
we
in
HA^K
what
alternative forms the three properties satisfied by a continuous
as reviewed in the beginning of Section 3 continue to hold. In the
show that absolutely continuous consumption patterns are dense.
give an example to
show that two consumption patterns that have almost equal
accumulated consumption at every point
be close.
in
time and
in
almost every state of nature may not
4
Topological and uniform properties of 7^
Example
as
n —*
pE^.
1 Fix
and
oo,
the
let
Bn e T
Let
random
15
c B„ V
such that B„+i
be
¥„ >
variable
be
n,
such that
P(B„) >
and P[B„) ->
0,
= n/P[Bn)
(p[ijj,Yn{oj))
a.s.
Consider the sequence of consumption patterns
where XB„
That
indicator function of the set Bn-
** the
until time I, at
ift<
=
Xn['^,t)
which time
=
otherwise. Let x{t)
1;
i„ provides for no consumption
is,
provides Yn units of consumption in the event
it
for all
t.
It is
easy to see that
lim
sup
=
\xn{uj,t)\
Bn and nothing
0, a.s.
"-'~«elo,il
But
E
Note that
^
<p{xn{t))dt
-
||i„
—
x\\^
E
i/
>
f
+ ^{x„{l))^=
and only
-
<p{-i{xn{t)
<p{u;,Yn{uj))P{du;)
j^
>
for all 7
if
+
x{t)))dt
=
n
oo as n
oo.
0,
>p{i{xn{l)
-
->
x{l)))
as
n-^
00;
.Jo
Theorem
see Musielak (198S,
Xn
>i
y for
In the
n
all
1.6, p.
imply that x >
to
S).
Thus
||i„
—
as n —* 00
i||^ 7^
y.
above example, x„ converges uniformly to zero pointwise.
however, in(l) increeises to
x, supj^rg
vanishing probability.
The
ji
l^nC'^, t)
Along the
sets
{Bn},
such a way that (p[(jj,Xn{<^,i)) grows to infinity faster
infinity in
than the probability of B„ decreases to
Xn to converge to
and we do not expect
~
zero.
Thus
2:(w,t)|
a:„
does not converge to zero
cannot grow too
fast
in T^.
For
even on sets of gradually
following proposition gives an alternative form of property (1) of
H&K.
Proposition 4 Let x,y,Xn €
and
if there
exists a
supj^fQi] |i„(cti,t)
implies y
>
Proof. By
eis
n
—
»
-
if' for all n.
random
x{ijj,i)\
<
variable
K{(jj)
K
a.s.,
//
lim„_oo
sup^gfo,!] |in(w,t)
with the property that
then in
>:
y for
all
-
=
x{ijj,t)\
E[^{X)] <
a.s.
such that
00,
n implies x > y and y > Xn
x.
the continuity of
00. Recall
>:
from Example
E
1^
with respect to T^, we only have to show that
1
<p{l\Xn{t)
that
-
it
suffices to
X{t)\) dt
+
show
for
any 7 >
^(7|X„(1) - l(l)l)
0.
||in
-
x\\p —>
,
4
Topological and uniform properties of T^
The
16
assertion then follows from the Lebesgue convergence theorem and the fact that (p[uj,z)
continuous in z and
=
ip[(jj,0)
is
I
0.
Second, we want to show that sizable shift of consumption within the "information con-
enough
straint" over a small
here
is
One
absent in H<JiK.
advanced to time
t
—
in the
e
interval
regarded as insignificant. The information constraint
is
unit of consumption at time
same event
for f
>
t
in
an event
however small, since
A E
A may
may
7t
not be
not be a distin-
guishable event in Tt-e and doing so will violate the information constraint that consumption
On
patterns be eidapted to F.
the other hand, one unit of consumption at time
can always be delayed to any time
A.
A
>
t
event. Moreover, one unit of
consumption
Appendix
following proposition formalizes these discussions, whose proof is contained in
definition
is
Definition 5 Let
needed.
T
and S
be
two optional times. The stochastic interval {T,S]
{{u/,t)
Let
same
in the
advanced to an instant before by the nature of a surprise.
at a "surprise" cannot be
The
s
T< 5
an event
in
t
be two optional times.
:
It is
T(w) <
known
t
<
is
the set
S{uj)}.
E
that [T, S]
and thus the process
x{<^,t) =X[T.l]{<^,t),
denoting a consumption pattern that provides for one unit of consumption at time T,
and therefore adapted
Proposition 5 Let
an optional time for
P{T =
T
an optional time.
be
Fix k >
all n.
1}
=
Then
kx\T,i\
all
=
and X
kxiT+i.,1]
is
<^+
/"'' <ill
=
i'„
=
>:
as
x„ for
n
and only
if
T
all
n implies y
T+
predictable
—
and {Tn}
x\\^
is
—
>
as
n
—
»
>:
x.
On
Tn < T, and on the
kx[T„,i],
is
and
^
is
also
— CO.
a sequence of optional times with
||i'„
if
>
kx\T,i]-
Tn < T„+i < T.
Putting
k
Then
0.
n implies x > y and y
hand, suppose that {Tn}
^
and put
PX[r+i,il - *:Xir,i]L -*
Thus x„ y y for
optional
(to F).
Xn
Suppose that
is
oo
an announcing sequence.
set
the other
{T > 0}
4
Topological and uniform properties of 7^
Proof.
See Appendix A.
17
I
Proposition 5 states that delaying the consumption of k units, even for large
k, for
a small
period of time will be regarded insignificant by an agent with preference relation continuous
in T^.
However, advancing consumption over a small interval from an optional time to an
earlier optional
that
is,
time
is
regarded insignificant
and only
if
if
the optional time
is
predictable;
consumptions at a random time and at random times instants before are almost perfect
substitutes
if
and only
instants before.
It is
there
if
no
is
left
discontinuity of information at those
worthwhile perhaps to
any deterministic time
recall that
is
random times
predictable and
thus consumptions at nearly adjacent deterministic times are always almost perfect substitutes!
Discontinuity of information can only occur at
Poisson process jumps
Remark
is
an optional time except when
With the
T
T
T+ -
is
aid of Proposition 5,
caveat.
Under
T—
is,
-
we now show
is
Tn
=T—
time a
is in
general not
^.
in the following proposition that the set of
dense
in
certainty, one unit of
if '^.
This
is
the counterpart of a result
consumption at time
approximated arbitrarily closely by consuming at an average rate "around"
X[i/2,il
first
deterministic. Thus an announcing sequence of a predictable
is
in general cannot he constructed by putting
H<kK with one
T
an optional time whenever
absolutely continuous consumption patterns
in
For example, the
times.
a surprise that happens at a random time.
4 Note that although
optional time
random
t
(.
E
(0, 1)
can be
For example,
can be approximated arbitrarily closely by three sequences,
-nit)
= -(^-^ + ^)X|l-^,i.^,(0 + X(i,^„(0,
(8)
x4t)
= n((-^ +
(9)
-n{t)
=
^)x(i_i,ij{0
n(t-i)X|i,i + i](0
+
+
X(i,i](0.
X(i + i,i|(0-
(10)
=
The sequences above
shift
average rate. They
all
converge to consuming one unit at time 1/2 in any of the
From Proposition
5,
however, we know that consumption at nonpredictable optional times
consumption around, before, and
cannot be approximated by
earlier
consumption. Consumption at
approximated by delayed consumption except at the
time and creates no problem. This observation
proof
is
contained in Appendix A.
after
is
final
date
t
t
1/2, respectively, at an
H&K topologies.
all
times can nevertheless be
=
I,
which
is
not a
random
formalized in the following proposition, whose
Topological and uniform properties of T^
4
Proposition 6 The
18
set of absolutely continuous
consumption patterns
in i'^ is dense with
respect to the topology Tp.
Proposition 5 also suggests a more general measure of "shifts" of consumption in the spirit
of the Prohorov metric used by H<kK.
€ f^ and
Let x,y
= inf{e>
pA^'V)
The
e
define a Prohorov metric path-by-path:
0:i(t
neighborhood of a path
around
x{uj,
distance
•)
that
is
x{uj,
f.w)
+
e
>
y{t,u)
any other path
•) is
x{ijj,-)
is
within
The
of the total consumption under y at
f
idea here
sizable shifts of
less
than
pattern
e
in
is
that
p^{x,y)
if
is
t
left
say
K
,
Hence
in state uj, the
c
if
y(w,
•)
(
within
is
a
e
totd consumption under
some time no more than
e
away from
t.
across states of nature, then
5 implies that as
we can allow p^[x,y)
As
it
interval
G £^ with
—
earlier
to be large on a set of small probability.
a sequence of consumption patterns (i„)
7 Let x,Xn
f
>
0,
the
e
time since
turns out, Puj[^,y) being uniformly small
is
counter example can be constructed along the lines of Example
Proposition
[0,1]}.
and down and to the right
any y that advances consumption to
will not contain
a.s. for
>
€
width. Such shifts should not drastically change an agent's utility for the original
too strong, and
—
e,u) -e,Vf
that lives entirely within a "sleeve"
uniformly smaller than
that will violate the information constraint.
A
•)
At nonpredictable times, however, Proposition
x.
Pu{xn,x)
t/(w,
-
consumption at predictable optional times can only occur across a time
neighborhood of x
is
x{t
at every point {t,x{uj,t)).
of x(a;,-) in the Prohorov sense, then at every time
X
>
determined by moving up and to the
each direction) from
e in
+
too weak for
\\xn
(Note that
-
i||^
—
»
0.
1.)
the property that there exists an
T -measurable
function,
such that
^
Pu.(in,x)<—
and
P-as.
Y.\'p[K)\<oo.
n
Then
\\xn
—
x\\p
—
>
x.
implies that y
Proof.
>
as
n
—
>
See Appendix A.
oo.
Thus in ^ y for
all
n implies that x > y and y > Xn for
all
n
I
This proposition concludes our investigation into how continuous preferences behave with
respect to "shifts" of consumption.
shows the
fact that (if', T^)
Proposition 8 {S'^,Tp)
is
is
With the
aid of Proposition 7, the following proposition
not a topological vector
not a topological vector
lattice.
lattice.
Topological and uniform properties of T^
4
Proof. The sequence
Proposition
now
Third, we
consumption
of sure consumption patterns Xli/2,1]
But the
7.
19
converges to zero by
X\{n+i)/2n,i\
positive part (or the negative part) obviously does not.
consider
how our
topologies perform
shifted over a decreasingly smaller
is
~
amount being
long as the increjisingly larger
<p,
amount
amount
increasingly larger
of
of time. Intuition suggests that as
shifted goes to infinity slowly enough, the shift
should be regarded as negligible. The meeisure of
infinity is naturally related to
when an
I
how
slowly the
amount
of a shift increeises to
the state-by-state inverse of which meeisures the
same thing
corresponding economy under certainty.
in a
We
first
put
=
/I
<p~^, the inverse of ip state-by-state.
and with n[uj,0)
strictly increasing, continuous, concave,
proposition essentially shows that
if
the
amount
=
is eaisily
It
and ^(u;,oo)
seen that ^l{u),)
=
The
00.
is
following
shifted increases to infinity slower than
/i
grows
to infinity state by state except possibly on a set of small probability, then the shift will be
negligible.
=
Proposition 9 Let n
and {yn(w,
{x„{(jj, 1)}
(p~^
and
let
{x„} and
and
1)} are both o(/i(a;,n))
Pu{Xn,yn) <
where
n
—
K[ijj) is
an
7 -measurable
n
be
two sequences in
f^
Suppose that
.
satisfy
p-a.S.,
function such that E[^(ii')]
<
00.
Then
-
||x„
y„||i^
—*
as
00.
Proof.
See Appendix A.
I
The sequences {xn} and
hope
{«/„} in
when >
^
Proposition 9 are both divergent and thus
we could not
for the validity of statements such as
I*
I*
{t/n}
0,
is
example of a
—
and
strict
VnWip
i„
for n sufficiently large
>- j/„,
(11)
continuous and strictly monotonically increasing in the direction of i* G £^ with
where
W^n
+
>-
is
the strict preference relation derived from >.
linear preference relation that violates (11).
—
as n
»
—+
00, I*
monotonicity of
However, (11)
is
Definition 6
(Mas
>:
+ 1„ - t/n may
to conclude that
lie
outside of
In
The reason
Appendix B we
for this
f^ and we cannot
i'+i„-y„ y
for large
is
give an
that although
use the continuity
n and thus i'
+ i„ >
y„.
a property of uniformly proper preferences defined formally below:
Colell (1986)) Preferences
respect to T^ in the direction x'
G f^ with
the origin such that, for every x
E f^,
v
x'
eV
,
> on
(f^ are said to be
^
if there exists a
and
scalar a
>
0,
x
uniformly proper with
T^ open neighborhood
- ax' + av
"^
x.
V
of
5
Duality
20
The uniform properness
comes from the
that X
fact that x'
— ax' + av
>:
i
is
a form of strong monotonicity.
is
V
and
The uniform
must be chosen independently of
allowed to
fail
x.
part of properness
important to note
It is
by virtue of the fact that x — ax'
+
av
^ £^
.
Mas
(See
Colell (1986) for further details.)
Proposition 10 Let
the sequences {in}
uniformly proper in the direction x'
x'
Proof. Rather than mimicking
+ Xn
+ 1„ — y„ is strictly preferred
that x' + Xn > j/n for large n.
if >:
1«
Vn
is
we
use a different argument.
it
!„ -
t/„
+
is
5
Duality
lies
inside this
open
it is
a uniform property.
for
On an open
^ be
the price at time
be represented
in the
set con-
continuous preferences.
Since
continuous preferences on an
like
Arrow-Debreu
sort, equilibrium prices
linear functionals continuous in the topology with respect to
the topological dual spaced Let
i/'(i) is
and
follows
it
come
which agents'
preferences are assumed to be continuous. This space of continuous linear functionals
Then
set
satisfy (11).
In standard general equilibrium theory of the
from the space of
Richard
not a topological property of continuous preferences - there
becomes a topological property
f^, they
is
can be extended to an open
extended preferences by continuity. Then
for the
continuous and uniformly proper preferences on S+ behave
set containing
Suppose that >
OO.
uniformly proper,
exists continuous preferences that violate (11);
open
9.
I
In the positive orthant, (11)
taining S'^, however, (11)
" ~*
9, for large n, i*
to
I*
>-
the proof of H<kK,
By Proposition
f^.
{Vn} ^c as in Proposition
Then
.
and Zame (1986) have shown that
set containing
<"*<'
is
called
a continuous linear functional that gives equilibrium prices.
of the consumption claim i.
We
will find
out soon that
\p
can
form
^{x)=-E\r
f{t)dx{t)
iGf^,
Uowhere, roughly, /
is
a process that
is
decomposable into two parts: a process having absolutely
continuous sample path and a martingale.
one unit of consumption at time
^More
is its
formally, (f", T^)
topological dual.
ie
t
in state
We
tv.
interpret f{oj,t) to be the time
shadow
price of
Since a martingale can have discontinuities only
a topological linear space and the space of continuous linear functionals on
(£'*',
T^)
5
Duality
21
at nonpredictable optional times (see
consumption
at nearly adjacent
at surprises.
It is
known
that a continuous martingale
sumption can fluctuate widely
structure
time
is
is
VI. 14)),
we thus conclude
that
(random) dates have almost the same shadow prices except
Thus when information
variation.
Meyer (1966, Theorem
is
either a constant or of
revealed continuously throughout,
a nondifiFerentiable
in
is
shadow
In particular,
faishion.
if
unbounded
prices for con-
the information
generated by a Brownian motion, the shadow price process for consumption over
an Ito process.
We now
proceed to characterize the topological dual of (f *', T^), denoted by f '^
.
We
shall
make
use of the concept of complementary generalized Orlicz spaces for most of our analysis
here.
We
provide some definitions.
Definition 7 A function
•
lim^^o+
^^^ =
• limi_oo ^^''^
Remark
5
<p
By
=
^
E
N -function
€
n.
(p
an
Vcj
convexity, for
N -function,
we can write
v{uj,s) is the right-hand derivative of ip{uj,s) for a fixed
Definition 8 Let
v'{uj,a)
=
(p
E
^
an
be
<
sup{s: v{uj,s)
if the
following conditions hold:
e Q; and
V'^
oo
said to be an
is
ct},
N -function,
and
(p{u;,\x\)
=
Jq
v[(jj,s)ds,
where
uj.
let v{uj,s)
be as in the above
remark.
Let
then
<p*{oj,x)=
v'{ui,a)da
/
Jo
is
said to be complementary to
mentary
to L'^ ,
Example
1.
The generalized Orlicz space
L'^
is
then said to be comple-
recall the definition of a generalized Orlicz space in (4).
2 (1) If<p{u;,x)
(2) If'p{u;,x)
The
where we
<p.
=
\x\''^''\l
=e' -x-1,
<
p(a;)
then 'p{uj,y)
=
<
(1
oo, then <p'{uj,y)
+
y)/n(l
+
y)
-
=
|y|«("),
where
:^ +
^=
y.
following proposition describes T^-continuous linear functionals on the space f'' in
terms of elements of
L'^
.
Duality
5
22
Proposition 11 Let
ip
C^
:
—*
yt is
<p
G ^
N -function
an
be
and with
complementary function
a
and only
a T^-continuou8 linear functional if
f
f with absolutely continuous sample paths, and with
f 9{t)dx[t)
&
L'^
if
,
(p'
Then
.
there exists an adapted process
so that
Vi€(f'^,
(12)
Jo-
=
where g[t)
+
f{t)
rn(t),
and
m(0=E[-/'(l)-/(l)|7,]
is
a martingale, or equivalently, g[t)
may
continuous paths but
Proof.
An
=
/(()
—
—
/'(I)
VtG[0,l]
/(I), which
(13)
is
not be adapted.
See Appendix A.
I
important special case of Proposition 11
when
is
<p{uj,z)
=
\z\''
case
when p =
The
following proposition gives the duality result for this 1/ family,
1
is
a process with absolutely
not covered
in
Proposition 11, however, since
with
is
|2;|
1
< p <
not an
whose proof
oo.
The
A^ function.
is left
for the
reader.
Proposition 12 Let
Then
tp
:
6'^
—
(p{(xi,z)
=
\z\^,
for alluj, with
<p <
1
oo. Let q be
a T^-continuous linear functional if
5R ts
and only
such that l/p+l/q
if there exists
=
1.
an adapted
process f with absolutely continuous sample paths so that
rP{x)
=E
ViGf^,
y^' g{t)dx{t)
(14)
where
if'j'il))
g{t)
=
f{t)
+
e L'(n X [0,ll,O,P X
E[-/'(l)-/(l)|J,l
a martingale, or equivalently, g{t)
continuous paths but
Note that
that are
sums
in the
is
ip is
=
f{t)
-
/'(I)
-
V<€[0,1]
/(I); which
(15)
is
a process with absolutely
not necessarily adapted.
above proposition, when p
=
1,
the dual space
of processes with Lipschitz continuous paths and
Another important
when
L'(n,7,F),
m{t), and
m(0 =
is
A) X
ceise
of the function
sisymptotically linear, in that for
(p
which
some
a;
is
is
composed of processes
bounded martingales.
not covered in the above propositions
G n, we have limi_oo
~^ =
a >
0.
is
In the
Duality
5
23
HiiK have shown that
case of certainty,
Lipschitz continuous functions.
We
the space of shadow prices in this case
is
the space of
obtain a similar characterization for the case of uncertainty
under the assumption that the function
"integrably asymptotically linear." This result
ip is
is
recorded in the following proposition.
Proposition 13 Ltl
there exists a
the function
random
K,
variable
ip
with the property that E[v?(/if)]
(p{oj,x)
< ax +
In this case, we have the equivalence £'^
and
T
are the consum.ption space,
and
=
£
,
€
yx>
and T^
Proof.
In
See Appendix A.
H&K,
and
<
is
the
same
C^
is
T
as
where we
,
the
is
=
|i|
sum
recall that t
for all
bounded martingale.
I
is
represented by a process
of an absolutely continuous process and a martingale.
A
It is
known
nearly
random adjacent dates
F
is
7; that
is,
structure
AE
revealed
if
then follows that prices for consumptions at
when
P{A\Tt)
=
is
Thus when F
prices of
consumption
is
continuous
is
if
and only
if all
An
there are no "surprises."
E[x/i|/t]
is
information
a continuous process for
the posterior probability of any event evolves continuously.
(1985a).
a martingale
It
are almost equal
said to be continuous
information structure
if
is
that the
martingale can have a discontinuity only at nonpredictable optional times; see,
example, Meyer (1966, Theorem VI. 14)
for
E n.
of a process with
sample path properties of a martingale are determined by the way information
over time:
u)
a continuous linear functional can be represented essentially by an absolutely
sum
the
0,
(16)
continuous function. Here, however, a continuous linear functional
that
>
oo, such that
the topology constructed using <p{x)
a
e
K{(jj) a.s.
In particular, this shows that an element of the dual space to
Lipschitz continuous sample paths
any
be integrably asymptotically linear, in that for
It is
all
known that an
optional times are predictable; see
Huang
continuous, the dual of f*' consists only of continuous processes and
at nearly adjacent
random dates
continuous on a stocheistic interval,
variation there (Fisk (1965)).
are almost equal.
it is
It is
also
either a constant or of
Hence there are cases where prices
for
known
that
unbounded
consumption
at nearly
adjacent dates are almost equal but fluctuate widely in a nowhere diff"erentiable fashion. This
comes about because of uncertainty. When there
prices for
is
no uncertainty, the martingale part of the
consumption disappears and prices degenerate to absolutely continuous functions of
time.
An
important special case of continuous information structure
Brownian motion.
is
is
when F
is
generated by a
In this case, the dual space only contains Ito processes. This characterization
given in the following proposition:
5
Duality
24
Proposition 14 Let
F
be
generated by a Brownian motion and
the dual spaces characterized in Propositions 11, 12
g{t)=
where f
is
to time,
where
f*
Jo
and
is
a
IS, then g is
an ltd process:
Jo
Brownian motion and where
I
g be an element of any of
f'{s)dB+ f9{sfdw{8),
a process with absolutely continuous paths
w
let
\9{t)f dt
and
denotes
/'
9 is adapted to
< oo
F
derivative with respect
its
and
satisfies:
a.s.
Jo
Proof. From
Propositions 11, 12, and 13,
absolutely continuous process and rn{t)
fact that a martingale
is
we know
An example
The
a martingale.
adapted to a Brownian motion
an Ito integral (see Clark (1970, Theorems 3 and
process jumps. That
a discontinuity only at a "surprise" which occurs
we
Before
general
ples.
fail
The
we show below
leave this section,
when
is
Q—
{wi,W2}
o^rid
We
the filtration be
{n,0}
j
^'-\
Assume
ceise,
is
a martingale
shadow
obtained by
may
have a
price process can have
the Poisson process jumps.
demonstrate this by showing two exam-
example uses a very specialized information structure, while the second
3 Let
an
can always be represented as
when information
the
is,
context of a Brownian motion information structure, which
Example
is
that the dual spaces characterized above in
to be sub-lattices of their order duals.
first
m{t), where /
I
4)).
observing the realizations of a simple Poisson process. In this
when the Poisson
+
f{t)
assertion then follows from the
filtration
of a discontinuous information structure
discontinuity only
=
that g{t)
{{a;i},{a;2},n,0}
is
is
in the
the case of prevalent applications.
composed of
vte
VtG
[0,1/2),
[1/2,1].
that each of the two possible states has a strictly positive probability of occurrence.
Define two processes with absolutely continuous sample paths as follows:
Vwen
/i(<^,t)
4((-l/2)
-4{t-l/2)
1
-1
t/a;
te
[0,1/2)
= wiVtG
[i
Vw G
2t
5),
./a;=a;2Vte[|,|),
t/a; = wiVtG[|,l],
t/w =a;2Vt G [f,l].
f2{u,t)
^
[
I
l-4(t-l/2)
1
fi t G [0, 1/2)
i/a;=a;iVtG[i,l],
j/w = a;2Vt G [i,l].
Note that
-/(.,.)-/!«.,.)={-'
;^:::;;
-/,k.)-/;k.)={a
:^:=:;:
25
Duality
5
=
verified that gi{t)
It is easily
fi{t)
Propositions 11, 12, and IS for
i
=
— E[/,(l) +
/,'(l)|/tl '»«« "» the
dual spaces characterized in
Note that the order dual of C^
1,2.
processes g that are measurable with respect to
T®
is
composed of positive
B{[0,1]) so that the integral
•1
E / 9it)dx{t)
.Jois
well-defined for x
that
is,
dual of
<
gi
f^
E C^
<
g2 ifgi{'^,t)
as
The order generated by
.
From Proposition
g2{u),t) a.e.
11,
this
example, we can think of
g,{uj,t)
the pointwise
minimum
of gi
=
fi{u,t)
and
-
3,
as
-
fi{i^,l)
f'ii^A)-
g^ is
-1
t/w
4(t-l/2)-l
ifuj
-\+2t
ifu
t/w
A?2(w,
?l(w,
t/u;
On
[0,
the interval
must
1)
we can think of the topological
composed of processes not necessarily adapted but with absolutely continuous
sample paths. In
Then
the order dual is then the pointwise order;
[0, j),
gi
A
be a constant,
^2
'i'**
impossible to write gi
it is
absolutely continuous paths
different derivatives along
and
a martingale
A
and thus
52
it
= a;i tG [0 i),
= uxte\\,\),
= uite\\,\],
= W2 «G [0, i),
= a;2 te[|,l].
wi and
^^s
a
sum
ui^-
Since a martingale on
of an adapted process with
cannot be an element of the topological
dual of S'^.
The next example
Example 4
is
in the context of a
filtration.
Let the filtration be a Brownian motion filtration.
that the dual contains only Ito processes.
is a
Brownian motion
generalized ltd process in that
it
It is
can
be
well
known
that the
From Proposition I4 we know
minimum
of two Ito processes
decomposed into two parts: a continuous process
with bounded variation sample paths and an Ito integral (see Harrison (1985),^6).
taking the pointwise
may have
minimum
That
is,
by
of two Ito processes, one creates a process whose time trend part
a singular component.
Thus the space of
Ito processes is not a lattice.
We
note, in
contrast, that the space of generalized Ito processed is indeed a lattice.
A
generalized Ito process
is
a continuous process with
bounded variation sample
patlis plus
an Ito integral.
,
Comparison with standard models
6
26
Comparison with standard models
6
In this section
we
will see
how
the standard models fare in the context of our current model.
Using a standard model, one assumes that an agent maximizes his exp>ected
utility of the
following form
U{x)
for absolutely
tion.
-E\f\{x'{t),t)dt\,
continuous consumption pattern, where u{z,t)
(17)
is
a time-additive "felicity" func-
This model does not permit gulps of consumption. This creates no major problem. Recall
from Prop)osition 6 that the
We
=
set of absolutely continuous
can therefore follow HA:K
consumption patterns
is
dense in f ^.
the expected utility of a consumption pattern involv-
in defining
ing gulps to be the limit of the expected utilities of a sequence of approximating absolutely
continuous consumption patterns.
The above procedure
what
in
follows that
Our proof
linear.
and the
fact that a
;9
:
:
f^ —
[0, 1]
-»
5R is
!R
jointly continuous then
is
nonrandom consumption
u{z,t)
:
3f+ x [0,1]
continuous in T^ only
such that u{z,t)
Proof. See Appendix
The converse
HiiK
identical to that of
Proposition 15 Let
U
u{z,t)
if
is
work provided that U{x)
will
A.
=
a{t)z
in
9?
>
U{x)
ceise
is T;^
We
show, however,
continuous only
if
u[z,
t) is
of certainty by using Proposition 7
6e jointly continuous.
if there exists
+
7^-continuous.
certainly feasible in a world of uncertainty.
is
—
the
is
The
continuous functions a
utility
:
[0, l]
function
—
»
!R
and
l3{t).
I
of the above proposition
is
a direct application of the duality results in Sec-
tion 5.
Proposition 16 Let
where
oc'{t) is
is
sition 11.
a.
a{t)z
+
0{t).
Then
U
of (17)
is
continuous in T^
if a'
e
L'^'
that
1/(1):=
U
=
the derivative of a(t).
Proof. Observe
Thus
u{z,t)
e[/"
a{t)x'{t)dt
+
I
0[t)dt
linear functional plus a constant.
=Ef/"
Then the
a{t)dx{t)+ f ^{t)dt
.
assertion follows directly from Propo-
I
The above
analysis reveals that the standard representation of utility as the expectation of
the integral of felicity of current consumption
fails
(except in a very special case) to produce
6
Compeirison with standard models
27
The problem
preferences continuous in our topologies.
is
that the additive form of
how
current consumption implies that consumption at an earlier date, no matter
has no effect on current
This
satisfeiction.
recent
it is,
with our concept of preference
in direct conflict
is
felicity of
continuity in which consumptions at nearby dates are close substitutes
if
there
is
no discontinuity
of information.
The standard model
also fails to capture our
economic intuition
we deal with
the single perishable consumption good case that
of durable goods.
One could
purchaise of durable good
in situations different
here.
Take
from
example the case
for
interpret the consumption plan x{t) as the level of accumulated
up to time
only a few times over the horizon
t.
A durable
[0, 1].
good
is
most often acquired
in single units,
and
However, the owner of the good receives a continuous
flow of services from the earlier acquired durable goods. His level of satisfaction from
owning
the good at any time should reflect his enjoyment from the services provided by the good, in
spite of the fact that the
We
would
like to
the standard model,
topologies.
good was purchased
in the past.
propose an alternative representation of
is
mathematically tractable, and gives
To achieve
this objective,
standard model. However, the
but also on consumption
we
rise to preferences
will express utility as
felicity at
time
Our construction goes
in the "recent past."
u>,
which
is
differentiable in
bounded over u. An example of such a process
e{u;,t)
Now
using the process
6,
in
our
felicity as in
the
depends not only on consumption
t
t,
is
and whose
is
first
spirit of
continuous
an integral of
a consumption plan. Consider an adapted process 6{uj,t), which
the states of nature
which keeps the
utility,
at
(,
as follows; let x(w,t) be
uniformly bounded across
derivatives are uniformly
the Gauss-kernel given by:
^^e^.
=
time
(18)
construct from x another process i which gives a path-by-path
weighted average of recent past consumption. In other words, put
x{ui,t)=[ 0{oj,s)dx{tThe new process x
will
model. Let u{x,t) be a
X.
s).
(19)
be the element over which preferences are expressed using the standard
felicity
function which
is
state-independent, continuous and concave
in
Let
U{x)
= E[/
u{x,t)dt].
(20)
Jo
The
fact that
our topologies
under these conditions, the
is
utility function given in (20) is
recorded in the following proposition.
continuous
in
any of
7
and general equilibrium
Issues related to arbitrage
Proposition 17 Let
the averaging process 6
and
28
the felicity function
u satisfy the conditions
described above. Let U{x) be constructed as in (SO). Preferences represented by
in
U
are continuous
any T^.
Proof. See Appendix A
It is
I
important to point out that
many
authors have attempted to capture the effects of
the time complementarity of consumption using functional forms that appear to be similar to
It
(20).
turns out, however, that there
is
a crucial difference. In our proposed representation
of utilities, the felicity function u depends only on the "smoothed" consumption process x
and time.
In contrast,
most "non time-additive" formulations
in the literature posit a felicity
function, say v, which takes as arguments the current consumption "rate" x'{t) together with
the "smoothed" earlier consumption x{i).
V{x)
=
Thus one represents preferences
by:
-E[l\{x',x,t)dt\.
(21)
Jo
Notable examples of such representations include Bergman (1985), Constantinides (1988),
Heaton (1988), and Sundaresan (1988). Inclusion of the "smoothed" recent consumption
in
the
instantaneous utility function v captures the effect of past consumption on one's current satisfaction.
However, including current consumption
of preferences in the sense
We
cases.
defined in (21)
—
?R
is
and
7
One
x
v(c, 2, (): !R^
[0, 1] be
continuous in T^ only
^: 5R+
x
—
[0,1]
measure such that v{c,z,t)
Proof.
for in this
paper except possibly for uninteresting special
record this fact in the following proposition.
Proposition 18 Let
[0,1]
we argue
in the felicity function destroys the continuity
See Appendix A.
—
!R,
»
a{z, t)c
and
+
jointly continuous.
if there exists jointly
a subset
^{z,
t)
for all z
of
^+
G
A.
utility
function
V^:
£^ —
»
9?
continuous functions a: 9?+ x
with a strictly positive Lebesgue
I
Issues related to arbitrage
of the central topics in
A
The
modern
and general equilibrium
financial theories concerns properties of security prices
there are no arbitrage opportunities. Using a similar
when
commodity space and a Mackey topology
generated by the space of bounded and continuous processes and barring arbitrage opportunities,
Huang (1985b) has shown
optional times
when
there are no
that security prices over time are continuous at predictable
lump-sum dividends.
In particular, price processes will be
and general equilibrium
Issues related to arbitrage
7
29
lump-sum ex-dividend dates
generalized Ito processes between
if
the information structure
is
generated by a Brownian motion. However, requiring the dual space of the commodity space
to have only continuous processes
a bit too strong.
is
implies that consumption at nearly
It
adjacent dates are almost perfect substitutes even at times of surprise.
norm
economically more reaisonable
all
Huang (1985b)
the results of
N
in the following
which are available
securities
eiccumulated dividend process i„
G
time
is,
there
1 is
is
zero.
Assume that
trading strategy 6
is
we assume
in [0, 1]. Security
There
separable.
is
n
is
denoted by
that Xn(0)
=
n and Sn(l)
for all
=
and the ex-dividend price of a security
its
0;
at
an A'^-dimensional process that prescribes the portfolio
Without getting
strategy for the traded securities.
(if^,T^)
Let 5„ denote the ex-dividend price process for security
S'^.
no accumulated dividend at time
A
get essentially
manner.
any time
for trading
n. Since securities are traded ex-dividends,
that
we
topologies discussed in previous sections,
Take the commodity space to be (f',T^)are
Using the family of
into technical details,
we
shall only allow
agents to use simple trading strategies - strategies that are adapted, bounded, left-continuous,
marketed
if
non-random time
m{t)-m{O) = 0{OyS{O)+ [
9{sydS{s)+[
there
is
at a finite
by-path; alternatively,
consumption patterns.
is
points.^ shall say
E
f '^
is
we
x{t)
say that
It is
—
{xi{t),.
m
is
clear that
a simple free lunch
if
.
.
,Xf^{t))^
financed by
M
is
6.
a linear
^(0)^5(0) <
,
and the integrals are defined path-
M denote the space of marketed
subspace oi
An element m G M
Let
S'^.
m£
0,
^(s)^di(s),
(f^,
and
m^
0.
Barring simple free
M has a unique price at time m(0) + ^(0)^5(0), where
we can define a linear functional on M by 7r(m) = m(0) 6[0)^m{0).
lunches, each
m
a strategy $ so that
where ^ denotes "transpose,"
financed by ^
number
of
and change their values
mE
0,
9 finances
m, and
4-
Now we
shall
show that the
market does not admit
{(m„, Xn) €
MX
<f *';
n
linear functional
free lunches,
=
1,2,
.
.
.}
m„ -
tt
has a nice representation
a concept due to Kreps (1981).
and a bundle
i„ e S^
Vn,
/:
€
f + with k
||x„
-
<
0.
^
A free
if
lunch
the securities
is
a sequence
such that
k\\^ -> 0,
and
liminf„7r(m„)
To consider
more general than simple strategies, we will first need to discuss how general stochastic
which we do not want to do here. Interested readers should consult Huang (1985b) for a
strategies
intergals are defined,
discussion in a context similar to our setup here.
,
7
Issues related to arbitrage and general equilibrium
Suppose that there are no
there exists an extension of r to
and X
^
Let
tp,
,
that
and Huang (1986) shows that
of Duffie
strictly positive in that ip{x)
is
>
if
x G £^
0.
(p
some
oi S'^
all
satisfy conditions of Propositions 11, 12, or 13, then
=E
V-Cx)
for
Theorem A.l
free lunches.
30
strictly positive process g,
ViGf-^,
[1^' 9{t)dx{t)
which
the
is
sum
of a martingale
and a process of absolutely
continuous sample paths. Let Sm{t) be the ex-dividend price at time
from Proposition
5.1 of
Huang (1985b)
E
Note that the
first
S^
predictable optional times
m
if
m
is
€ M.
in
is
a martingale and thus the properties
suppose that
In particular,
be continuous at
between discontinuities of m, Sm
also continuous there and,
is
S^ must
For example,
a natural way.
F
is
generated by a
an absolutely continuous process. Since a martingale then can be
continuous process, the numerator of (22)
is
an
implies that
Sm
is
an Ito process.
is
is
The denominator
Ito process.
plus a process of absolutely continuous paths and thus
lemma
follows
^''^
represented by an Ito integral and the second term in the numerator of (22)
positive, Ito's
It
m
can have discontinuities only at surprises.
Brownian motion and
m
- !^9{s)dm{s)
[j^ 9{s)dm{s)\Tt]
term of the numerator of (22)
of g translate into those of
of
that
=
'-^'^
t
an Ito process. These are
all
is
an absolutely
a martingale
Since g
is
strictly
economically appealing
properties of price processes for securities over time.
As
for the existence of
an Arrow-Debreu equilibrium
space if^ equipped with the topology T^, however,
an empty
formly proper,
among
lattice,
then an equilibrium
proper, there exists an equilibrium; see
is
Mas
a topological vector lattice nor
former
recall
is
an economy with a commodity
to offer. Since
little
two
ensured
Proposition
an Arrow-Debreu equilibrium
in
8; for
its
Colell
its
dual
if
order dual and
is
if
(f '^, Tp)
agents' preferences are uniif,
if
among
other things, the
preferences are uniformly
and Richard (1987). Unfortunately, neither
is
the latter see
our economy
{£^,T^) has
possibilities. First,
other things; see Meis Colell (1986). Second,
topological dual of {S'^, T^) were a sublattice of
for the
we have
interior, existing general equilibrium theories offer
were a topological vector
(f'^jT^)
in
a sublattice of
Example
3 and
its
4.
order dual in general;
Whether there
exists
an open question.
However, provided that there exists an Arrow-Debreu equilibrium, one can implement the
Arrow-Debreu equilibrium by continuously trading a few long-lived
securities as in DufTie (1986)
8
Concluding remarks
DufiRe and
Huang
31
(1985),
and Huang (1987) and thus establish the existence of a dynamic equi-
librium with dynamically complete markets. In Duffie (1986) and
sp£ice is the
patterns
is
Huang
commodity
(1987), the
space of consumption rates and the sense of closeness between two consumption
defined by the standard Z/
norm on consumption
rates as functions of (w,t).
no surprise.
at nearly adjacent dates are perfect nonsubstitutes even at times of
consumption
As a consequence, the
shadow
properties of
prices for
consumption over time
Thus
in
a pure ex-
change equilibrium depend crucially on the properties of the aggregate endowment process.
For example, in the case of a Browni2in motion
filtration, the price
process for a security with
absolutely continuous accumulated dividend process will not be an Ito process unless the aggre-
gate
since
endowment process
is.
This does not conform with our intuition. In our setup, however,
consumption at nearly adjacent dates are almost perfect substitutes
prise, price processes of securities
no sur-
with absolutely continuous accumulated dividends will be Ito
processes independently of the properties of the aggregate
8
at times of
endowment
process.
Concluding remarks
we have advanced a family
In this paper
of topologies defining closeness
between consumption
patterns over time under uncertainty that capture the intuitive idea that consumptions at nearly
adjacent dates are almost perfect substitutes
we have
tried to formalize
is
if
there are no surprises there.
certainly not new.
Our contribution
and, more important, characterizing the topological dual spaces.
topologies
is
is
We
The
intuitive idea
topologizing this idea
feel
that our choice of
the natural one that conceptualizes the aforementioned idea for the following two
reaisons. First, in the
degenerate
ccise
where the true state of nature
is
revealed at time 0, we
reach the conclusion that the shadow prices of consumption are absolutely continuous functions.
Hence, preferences continuous in our topologies give
rise to
equilibrium prices in which the price
of consumption at adjacent dates are almost equal and in which prices change smoothly in a
differentiable
manner.
We
believe that this
is
an intuitively attractive
result.
Second, in the nondegenerate case, the topological duals are the natural generalizations of
those under certainty.
In this case, prices are processes that
can be decomposed as the
sum
of two components: a process of absolutely continuous sample path and a martingale. In the
case of uncertainty a
new element,
properties of equilibrium prices.
price process.
It is
Thus equilibrium
the pattern of information flow, affects the sample path
This
effect is
captured in the martingale component of the
known that a martingale can make discontinuous changes only
prices for
at surprises.
consumption are continuous except possibly at surprises. This
is
9
References
32
also an intuitively appealing result
which holds independently of the nature of the endowment
process.
The proposed family
properties
ical
known
of natural topologies, however, does not give rise to certain
to be sufficient for the existence of an
mathemat-
Arrow-Debreu equilibrium. The
resolution of this issue should be of high priority.
References
9
1.
K. Arrow and M. Kurz, Public Investment, the Rate of Return, and Optimal Fiscal Policy,
Johns Hopkins Press, Baltimore, 1970.
2.
Y. Bergman,
Time
Preferences and Capital Asset Pricing Model, Journal of Financial
Ekonomics, Volume 14, No.l, 1985.
3.
P. Billingsley,
Convergence of Probability Measures, John Wiley and Sons,
Inc.,
New
York, 1968.
4.
G. Constantinides, Habit Formation:
A
Resolution of the Equity
Premium
Puzzle, mimeo.
University of Chicago, 1988.
5.
K. Chung and R. Williams,
An
Introduction to Stochastic Integration, Birhkauser Boston
Inc., 1983.
6.
J.
Clark,
The Representation
Annals of Mathematical
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of Functionals of
Brownian Motion by Stochastic
IntegraJs,
Statistics, 41, 1970, pp. 1282-1295.
G. Debreu, Representation of a Preference Ordering by a Numerical Function, In Decision
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Coombs, and R. Davis. John Wiley
&
Sons,
New
York,
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C. Delleicherie and P. Meyer, Probabilities and Potential A: General Theory, North Hol-
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New
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Holland Publishing Company,
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York, 1978.
New
York, 1982.
D. Duffie, Stochastic Equilibria: Existence, Spanning Number, and the "No Expected
Gain from Trade" Hypothesis, Econometrica 54, 1986, pp. 1161-1184.
9
References
11.
33
D. Duffie and C. Huang, 1985. Implementing Arrow-Debreu Equilibria by Continuous
Trading of Few Long-lived Securities, EA:onometric& 53, 1985, pp. 1337-1356.
12.
D. Duffie and C. Huang, Multiperiod Security Markets with Differential Information:
Martingales and Resolution Times, Journiil of Matbem&tic&l Ek:ononiics 15, 1986, pp. 283303.
13.
D. Fisk, Quasi-martingales, Tr&ns. Amer. Math. Soc. 120, 1965, pp. 369-389.
14.
M.
Harrison. 1985, Brownian Motion and Stochastic Flow Systems, John Wiley
Inc.,
15. J.
Sons
York, 1985.
Heaton, The Interaction between Time-nonseparable Preferences and Time Aggrega-
tion,
16. C.
New
&c
mimeo. University of Chicago, 1988.
Huang, Information Structure and Equilibrium Asset
Prices, Journal of
Economic
Theory 35, 1985a, pp. 33-71.
17.
C. Huang, Information Structures and Viable Price Systems, Journal of Mathematical
Economics 14, 1985b, pp. 215-240.
18. C.
Huang, An Intertemporal General Equilibrium Asset Pricing Model: The Case of
Diffusion Information. Econometrica 55, 1987, pp. 117-142.
19. C.
I:
20. L.
Huang and D. Kreps, On Intertemporal
The Case
of Certainty,
MIT mimeo,
Preferences with a Continuous
Time Dimension
1987.
Kantorovich and G. Akilov, Functional Analysis, Pergamon Press, 1982.
21. D. Kreps, Arbitrage
and Equilibrium
Journal of Mathematical Economics
22. A. Mcis Colell,
The
in
Economies with
Infinitely
Many Commodities,
8, 1981, 15-35.
Price Equilibrium Existence Problem in Topological Vector Lattices,
Econometrica 54, 1986, pp. 1039-1053.
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Mas
Colell
Lattices,
and
S.
Richard,
A New
Approach
to the Existence of Equilibrium in Vector
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Meyer, Probabj/ity and Potentiais, Ginn
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Meyer, Decomposition of Supermartingales: The Uniqueness Theorem,
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(Blaisdell), Boston, 1966.
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26. J. Musielak, Orlicz
Sp&ces and Modular Spaces, Lecture Notes
in
Mathematics, Springer-
Verlag No. 1034, 1983.
27. S.
Richard and
W. Zame, Proper
Preferences and Quasiconcave Utility Functions, Journal
of Mathematical Economics 15, 1986, pp. 231-247.
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An
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Sundaresan, Portfolio Choice with Intertemporally Dependent Preferences.
tions for
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forthcoming
in the
Implica-
Review
Appendix
A
Proofs
Appendix
35
A
Proofs
PROOF OF Proposition
Proof.
without
Let i'
i:
>
i. be elements of £*' and let / denote the order interval [i,,i*]. Assume
= 0, since our topologies are linear. We will show that / is
and closed and therefore compact; see Schaefer (1980, p. 25).
loss of generality that i.
bounded
To show that / is totally bounded it suffices to show that
number of elements of /, {i„; n = 1,2,..., A^(e)} such that
totedly
for
any
e
>
there exists a finite
n=l
where O^
is
a T^-neighborhood of i, with
we know
•
diameter
||^
||
less
than
e.
Fix
>
e
From
0.
N[€) >
such that Pu{^,y) ^ 2;*(a;, 1)/A''(f) implies
there exists
Proposition 7,
~
Construct N[e) processes with right-continuous, nonnegative,
y||^ < f/2 for x,y G C^
ll^^
and nondecreasing sample paths in the following fashion. For each state uj, begin first by
building a Prohorov sleeve around i'(w,-) with Prohorov distance x'{ijj,\)/N{e). Let ii(a;,-)
be the function prescribed by the lower boundary of that sleeve. Formally, let
.
ii(a;,(
+
=
i*(a;, 1)/A^(e))
max{i*(a;,«)
where we have used the convention that x'(w,
=
t)
- i*(w, \)/N{e),Q}
if <
<
0.
Vt
G
5R,
Next we define !„ recursively
as
follows:
i„+i(w,t
+
=
i*(w,l)/7V(e))
max{i„(a;,0 - x'{uj,\)/N[t),0)
Vt
e
3t.
Note better that the processes i„ constructed may not be adapted. By construction, p^^{xn,in+i] <
i*(w,l)/iV(e) and thus ||i„ - i„+i||^ < e/2 V n = 1,2,. ..,N{t) - 1.
Suppose for the time being that the processes i„ are adapted. Let
Ol„
It is
= {zef^:||i-i„||^<e/2}.
then clear that
N(e)
n=l
and thus /
is
totally
bounded.
Now
if
i„ are not adapted, pick any element x„ £ 0\
and put
0',^^{xeEt:\\x-x^\\^<e),
we claim that 0%
C 0\
.
11^ ~"
To
see this, let i
^"llv
—
II'''
~
G Ol
^nllv
'
.
I!'''"
Then
~
if.
^nllip
'^
o
9
^
^'
We thus conclude that / is totally bounded.
Next we want to show that 7 is closed. Let x„ G I and ||in - x\\^ —
as n — oo
X G f^. Then there exists a subsequence of (i„) that converges P x A-a.e. to x. The
X G I follows from arguments similar to those in the proof of Proposition 1 of H(5iK.
by triangle inequality.
»
>
for
some
fact that
I
Appendix
A
Proofs
36
Proof of Proposition
Proof.
First,
we show
5:
that ^X[T,il
^
^ C^
for all optional
'P{')kX\T,i]{t))dt
time
T
and
all
Jt
>
0.
Note that
<Y.\^[lk)\ <c»,
where the
first inequality follows from the faict that ip is strictly increasing and the second
inequality follows from the hypothesis that (p is integrable.
T +
Second, the fact that
is
^
an optional time whenever
T
is
can be seen by directly
checking the definition;
[uen-.T + - <t) = {ujen-.T <t- -)e
n
Third, the hypothesis that
lim
n—•oo
E
/^
'p{iKx[T.i]{t)
n
f
is
7._x
c 7t>ite
[0,1].
r^
integrable and Lebesgue convergence theorem imply that
= E
- X[T+i,i](0))^
= E
J^
^(7*Jim(xiT.il(0-X|T+i,il(0))<^'
=0.
where the second and the third equalities follow from the fact that 'p{uj^ z) is continuous in z
and is zero at z = 0, respectively. Thus, by the continuity of >:, i„ >: y for all n implies x > y
and y > Xn for all n implies that y > x.
Fourth, suppose that {T„} is a sequence of optional times with Tn < T, and T^ < T and
This implies that
Tn < Tn+i on the set {T > 0} and that fcx[T„,i] ~* ^X|T,il m
lim
n—>oo
E
7 > 0. This implies that
subsequence {T„^} such that
for all
I
filHxiTAlit) -
kx[T„,l] ~* ^XfT.il in /*
nji^oo^l^-.'^l^'^''^
which implies that Tn^
The proof
— T
a.s.
Proof. The proof
~
and thus
for the last cissertion
Proof of Proposition
X[T„.l]{t)))dt
is
X[T,i](<^,0
T
is
oo
x A-measure, and hence there exists a
=
P
X A -
a.e.,
predictable.
just the reverse of the above paragraph.
I
6:
of Proposition 3 essentially implies that the set of right-continuous, and
bounded processes that change their values at a finite number of optional times is dense in
(f'^.T^). It then suffices to show that one unit of consumption at an optional time can be
approximated arbitrarily closely by a sequence of absolutely consumption patterns. We will
show here that consumption at a predictable time can be approximated by consuming at "rate"
instants before. The rest of the proof is left for the reader.
Appendix
Let
that T
T
>
A
Proofs
37
be predictable and {T„} be its announcing sequence. Assume without loss of generality
a.s. (On the set {T = 0}, we can shift consunnption to the right at rates.) Putting
5„^E[T|JrJ,
it
can be verified that 5„
know Sn > Tn
is
an optional time by checking the definition. Since
T>
T„
a.s.,
we
Define
a.s.
X{t)
=
X\T,l]{t)
^n{t)
=
- Tn
^X[T„,S„l(0+X(S„,l)(0'->n ~ -in
t
Note that !„
is
an absolutely continuous consumption pattern. One can verify that i„
Proof of Proposition
Proof. Note
again that
E
I
||i„
-
i||^
—
»
0, if
let
and only
+
<p['l\Xn{t)-x{t)\)dt
1.6).
Note that
P({p(i„,a:) >
For any integer m,
>
7:
Theorem
see Musielak (1983,
—
N{m)
for
if
any 7 >
<p['l\Xn{l)-x{l)\)
e[/)(i„, x)] -> 0, then
if
^0
-})
as
n
> ^
be such that P(lp{xn,x)
^ 00.
[)
< ^,
for all
n > N{m).
Next write:
E j
for
n
^{l\x4t) -
x{t)\) dt
+
v?(7|i„(l)
-
i(l)l)
+
'1
'2
'
> N{m), where
^"'"
=
L,
.
^\f\{tMt)-x{t)\)dt
+ p{'iMi)-x{i)\)
^
^
^
\
P{du;),
•'{p(»n,l)>^}L-'0
7^"
=
/
,
We
will
show that both
\
f\Uxn{t)
-
x{t)\) dt
integrals converge to zero, as
+ J-rMi) -
m
(and hence n)
x{l)\)
—
>
P{dco).
00.
First consider /J"'". Since
^
<p{l\xn{t)
-
x(f)i)
dt<
j
<p{i{x„{l)
+
x{t))) dt
< <p{l{x„{l) +
1(1)))
Vw e n,
i
in
A
Appendix
Proofs
38
we have
<p[i\x„{t)-x{t)\)dt
I
+
<p['j\x„{l)-x{l)\)
<
2^(7(i„(l)
<
2<p{i{K
+
—
or
i(l)))
2x{l)))
But since E[^(A')] < oo and x G f^, it then follows that /["'"
Lebesgue dominated convergence theorem.
Now consider /^'". Note first that for w G {Pui{xn,x) < ^}, we have
|i„(l)-x(l)|<-
+
eis
m,n
—
oo,
by
<p(i\x„{l)-x{\)\)<J^).
Thus
^(7|x„(l)-i(l)|)p(cfo.)<E[^(^)'
/
Next consider the
integral part.
For any
w G
— }, we
{p(in,i) <
can bound the inside
integral by:
mm
— ^<P\'^\^^'^{^{^-,—)^^n{^-,—)') - min{i(a;,
m
But since
rrj
\
^
for precisely those
sample functions p(a;„,i) <
max{i(a;,(),i„(u;,()}
<
),in(w,
m
— we
i(ai,t H
m
)
m
)})
'
.
j
have
H
m
and
min{i(a;,f),i„(u;,t)}
so the
summation above
is
is
a:(<^,f
m
)
m
m
)
{k-2)
- <^.
m )-!))
convex, equal to zero at zero and increasing, this
— <p\l2_^
,
bounded by:
m^^ \\
Finally, as y?
>
\x{uj,—-—)-x{uj,
{k-2)
2
^
m
m
j
in
turn
is
bounded
< ^v^(7(3x(w,l) +
Therefore,
[':p(iMt)-x{t)\)dtP{duj)<-E
f
C'" <
and /^'"
—
as
m,n —
»
-E
m
¥p(7(3i{l)
oo, proving our result.
+
2))
I
v^(7(3i(l)
+E ^(^)
+
2))
2)).
by:
Appendix
A
39
Proofs
Proof of Proposition
9:
Proof. For any ui E Q, and for any 7
Proposition 7, we get:
^(w,7|a:n(w,t)
-
-^<p{uj,i{Sxn{u;,
1)
/
>
using the
0,
y„(a;,t)|) dt
+
same arguments
v?(<^,7|x„(a;, l)
-
as in the proof of
ya(<^, 1)1)
Jo
IT
<
n
The second term on
+
2))
+
2<p{<jj,
7|x„(w,
-
1)
y„(a;, 1)|).
the right-side of the inequality vanishes for any 7, since
ir
|a:n('^, 1)
- yn(w,l)| <
—^•
<
Pu,{Xn,yn)
n
It then follows from Lebesgue convergence theorem, continuity of
the hypothesis that E[v?(if )] < 00 that
E
The
first
term vanishes
2y.(7|x„(l)-y„(l)|)
^0
Hence, for any 6 however small, there
WnyNs:
<p{u),
•)
—
^[^,0)
=
0,
and
00.
since,
i„(u;,l)/M(a;,n)
Applying
n
as
92(0;, 2) in z,
to both sides of the
thus
is A'^^,
7(3i„(w,
+
1)
2)//i{a;, n)
->
0.
such that:
i{ix„{uj,l)
+
2)
<
8^i{u,n).
above relation, and using the shape of
<p,
for all 5
1,
we
00,
we
<
conclude that:
K
-^¥'c.(7{3xn(w,l)
n
+
2)
K
K
n
n
< ^^'p^[8^i[uj,n)) < -^6<p^{^[oj,n)) < K^8
n > NsNoting that by Jensen's inequality, y?(E[ii']) < E[v7(^)] < 00 and since
I
conclude that ||i„ — yn|L ~* 0, by Lebesgue convergence theorem.
for all
Proof for Proposition
Proof.
E[v2(ii^)]
<
ii:
Integration by parts path-by-path gives
fP{x)
=E
'lo'-9{t)dx{t)]
=E
=E
'Jlf{t)dx{t)
=E
'-J^x{t)f'{t)dt-x{l)f '{!)],
+
Jlm{t)dx{t)]
'f{l)x{l)-j^x{t)f'it)dt
+
(23)
m{l)x{l]]
Appendix
where
A
40
Proofs
denotes the derivative of / and we have used the fact that
/'
E[ /' m{t)dx{t)]
=
E[
f
m{l)dx{t)]
=
E[m(l)i(l)]
(24)
Jo-
Jo-
and Meyer (1982, VI. 57)).
Musielak (1983, Corollary 13.14, p. 87) shows that (23) is a linear functional continuous
This is the sufficiency part.
T^ if f € L'^
in the third equality (see Dellacherie
in
.
Now
he a, T^o-continuous linear functional. By the
can be extended to be a continuous linear functional on the whole of
Hahn-Banach theorem,
E'^, of which we recall the definition from (6). Let * denote this extension. We first show that
consider the necessity part. Let
xp
£'*'
:
t/*
*
:
E'^ —* 5R can be represented in the form:
^{x)
for
some y E
{i„} G L^
is
L"^
= Ey\{t)y{t)dt +
For this, we
.
E
/^
<p{lMt) -
modular
x(t)l) dt
The sequence {in} € L^ converges
E
1^
<p{lMt) -
see Musielak (1983,
Theorem
(25)
A
consider the notion of modular convergence:
first
said to converge in
^xeE^
x{l)y{l)
in
+
to x,
X{t)\) dt
+
to x,
if
for
i(l)l)
and only
some 7 >
0.
if
^(7|ln(l) - 1(1)1)
Norm
1.6, p. 3).
-
y?(7|x„(l)
norm
sequence
if
for all
7 >
0;
convergence of x„ clearly implies modular con-
vergence.
Musielak (1983, Theorem 13.17,
on
(p
p. 88)
shows that (25)
is
true under the additional restriction
that
<p{u),
Vio >
there exists
>
c
such that
x)
>
c
for
i
>
iq
Vcli
G n.
Musielak (1983, Theorem 13.15, p. 87), however, shows that this restriction is required to guarantee that a norm continuous linear functional on L'^ is also modular continuous. In our
formulation, we do not need this restriction, since we do not deal with the generalized Orlicz
space L^. We only consider the subspace E'^, and on this subspace norm convergence and
modular convergence are equivalent; Musielak (1983, 5.2.B, p. 18). We therefore conclude that
restricting our attention to £"^ allows us to relax the above described condition. The interested
reader can eaisily verify this by consulting Musielak (1983, Theorems 13.15 and 13.17).
Now
define
f{u,t)
= - f
Jo
y{uj,s)ds
Vte{0,l]ujeQ,
which is clearly adapted and path-wise absolutely continuous. Denoting the derivative of f{u), t)
with respect to t by f'{uj,t), it is clear that f'{t) = —y{t). Reversing the arguments in deriving
(23),
we prove
the necessity part.
I
Appendix
A
Proofs
41
Proof of Proposition
Proof.
p
of £
is
,
Let
then it
13:
be integrably asymptotically linear. We will first
also an element of t'^. Note the following:
IkL = nj^
show that
¥'(ix(OI)X{|x(e)|<A-}dt
+
V5(|x(l)|)x{x(i)<K}]
+
E[|^ ¥'(|x(OI)X{|i(0|>A-}dt
+
¥'(|a:(l)|)X{x(i)>iC}]
<
2E
[<p{K)\
+ oE
f |i(t)|dt+|i(l)|
+2e <
if
i
is
an element
oo.
Jo
an element of f ^.
is weaker than T^, we only need to show that convergence in T implies convergence
in T,p. Consider a sequence of elements {in} that converges to i in T. This implies that {xn}
converges in the product measure generated by P and Lebesgue measure to x. Consider now
Hence i
Since
is
T
E[[j
'0
+ E[y
E[^
<
+ aE
<P{\Xn{t)
-
x{t)\)X{\x„{t)-z{t)\<K} dt
+
<p{\Xn{l)
-
l(l) |)X{x„(l)-x(l)<if }]
<p{\Xr,{t)
-
x{t)\)x{\z„{t)-x{t)\>K} dt
+
<P{\X„{1)
-
l(l)|)X{x„(l)-x(l)>/f}]
filxnit)
-
x{t)\)x{\x4t)-z[t)\<K} dt
+
<p{\xn{l)
-
a:(l)|)X{x„(i)-x(i)<A-}]
-
x{t)\dt
+
2e.
y
\xr,{t)
+
\xn{l)
-
x{l)\
I
The
first term on the right-hand side of the inequality converges to zero by Lebesgue convergence
theorem. The second term goes to zero as n —» oo by hypothesis. Since e is arbitrarily small,
\\xn
—
x\\^
—
»
as
n
—
OO.
Proof of Proposition
I
15:
Proof. Suppose
that u is not linear. Then there exists two scalars r,f and t € [O, l] such that
u(r,0/2
+ u(f,«)/2. Without loss of generality, assume that u((r + f)/2>0 >
i^
u(r, t)/2 + u(f,t)/2. By joint continuity, there exists e >
and some interval / containing ( such
that ui^r + r)/2, s) - e > u(r, s)/2 + u(f s)/2 for all s € /. Consider a sequence of nonrandom
absolutely continuous consumption patterns constructed cis follows: Off of /, consume at rate
1 in each !„.
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) = 1 off / and x\s) = (r + f)/2
on /. By Proposition 7, i„ —» i in T^. But U[x) > C/(i„) + eA(/), where A(/) is the Lebesgue
measure of /. Thus U is not continuous in T^.
I
u{{r
+
f)/2,t)
,
Appendix
A
Proofs
42
Proof of Proposition
Proof.
T^.
17:
Fix a topology T^. Let i„ be a sequence of consumption plans that converge to x in
loss of generality, that sup„ ||vV„||
< oo. We will first show that u{xn,i]
Assume, without
converge to ti(i,t) in the product measure i/ = P x A, where P is the probability measure on
(n,7), and A is the Lebesgue measure on the Borel sigma-field on [0,1].
First observe that if i„ — i in T^, then i„ — i in T. Hence,
»
»
jjm^Eiy |i„(0-i(OI'i« +
This implies that i„ converge to i
in the
=
0.
(26)
product measure v, and that /q linCO ~ ^(01 ^^ con-
verges to zero in P-measure. Therefore, for any
> k}) < i, and P[l^ |x„(0 -
|2:n(l)-i(l)|]
m, we can
find
A'^(tti)
such that I'fKw,
t):
|z„(w,t)-
> ^] < ^, for all n > N{m).
Now consider i„(w, t) - x{uj,t) = /q 0{uj,s) dxri{u>,t - s) - dx{u, t — s). Integrating by
and using the assumption that 9[ijo,k) vanishes, we get:
^KOI
Xn{oJ,
t)
- x{u, t) = -
(i„(o>;, t)
-
x{t)\dt
i(u;, t))
0{u,O)
-
where 6,[u),s) denotes the partial derivative of
and its derivative are uniformly bounded over
Mi\xn{uj,t)
From
-
this
x{uj,t)\
+
we can
,
Milfg
\x„{<jj,t)
—
(i„(a;,
j
6[uj,s)
uj,
x{<jj,t)\dt],
t
-
s)
-
x{uj,
t
with respect to
-
s.
s)) 9,{u, s) ds, (27)
But
the left-hand side in (27)
where
Mi
and
M2
parts,
is
since both 6
bounded by
are two constants.
easily conclude that
i^({{oj,ty.\xr.{uj,t)-x{uj,t)\>
>
+
-{Mi
m
Mi)})
'
<m
Vn>Ar(m).
(28)
in ^-measure to x. Now consider the convergence of u(xn). First, assume
uniformly continuous in x. Since the felicity function u is state-independent, and
uniformly continuous in i, it then follows that if |i„ — i| < ci then |u(in) ~ "(i)l < ck^i,
where a is independent of w, and x. We use this assumption of uniform continuity, together
with (28) to conclude that u{x„) converges in iz-measure to u{x). By Jensen's inequality, we
Hence f„ converge
that u
is
have sup„E[/q u{xn)dt] < sup„u(E[/q Xndt]) < 00. It then follows by Lebesgue dominated
convergence theorem that U{x„) converge to U{x).
Next consider the ceise when u is merely continuous in x. For every integer m, define
Um(^) = "(^)X(J-<i<ml- ^°^ a fixed m, the function u^ is uniformly continuous in x, and hence
the arguments in the previous paragraph
defined exactly as U{x), except that u
to u[x) as
m—
»
00.
show that [/„(!„) converge to Um{x), where Ujn{x) is
substituted by u^- But Um(i) converge monotonically
Applying the monotone convergence theorem, we conclude that
lim U(xn)
n —»oo
This shows that the
is
=
lim Um{xn)
m,n —oo
utility definition given
= Hm
tn—^oo
Um.{x)
=
U(x).
by (20) gives preferences continuous
in T^.
I
Appendix B
Example
of a 7^ continuous preference that violates (11)
Proof of proposition
Proof. Suppose
not.
By
43
18:
joint continuity, there exists
two scalars r,f and
+ r)/2, z, i) ^ t;(r, z, i)/2 + u{f, z, t)/2 for all values of z. Without
that v{{r + f)/2,z,i) > v[r,z,t)l2 + v(f,z,t)/2. By joint continuity
and some interval / containing t such that v((r + f)/2,z,s) - e >
t
G
[0, 1]
such that
loss of generality,
v((r
assume
again, there exists
e
>
v{r,z,s)/2 + v{f,z,s)/2
for all s E I. Consider a sequence of nonrandom absolutely continuous consumption patterns
constructed as follows: Off of /, consume at rate 1 in each in- 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) = 1 oflf / and x'{s) = (r + f)/2 on /. By Proposition 7, i„ —» i in Tp. Proposition 17
shows that lim„_oo
=
dt
/o v{x',Xn,t)
/q v{x',x,t) dt.
/ v{x',x„,t)dt> f
Jo
But
for
v{x'„,Xr„t)dt
any i„, we have:
+
eX{I).
Jo
Taking limits of both sides as n
/
—
>
v(x',x,t)dt= lim
oo,
Hence V(i) > lim„_oo y{xn)
I
continuous in To.
Appendix B
+
get:
v(x',Xn,t)dt> lim / v(x',Xn,t)dt
"^°oJo
/
"-'°°Jo
Jo
we
(^^{I)i
Example
where A(/)
is
+
the Lebesgue measure of
€X(I).
/.
Thus
V
is
not
of a 7^ continuous preference that
violates (11)
We first show that any linear preferences with the marginal utility being the optional projection
of a continuous and uniformly bounded process are continuous in all of T^ for all tp G ^.
f-.Q x [0,1] —
adapted, with the the properties that
Proposition 19 Let
f{u>,.)EC[0,l]
5R
6e
and
an
J x B{^) -measurable
where C[0, l] denotes the space of continuous functions on
by:
of f denoted
,
<
esssup^ max/(u;,t)
[0,1].
process, not necessarily
oo,
Define the optional projection
f
/•(a;,0
Note that by construction, the process
f
is
=
E[/(t)|7/.
adapted. Preference relations which are represented
by:
x>y
if
E
C r{t)dx{t) >E f r{t)dy[t)
Jo
Jo
are continuous in TL,
,
Vi.yef^^
Appendix B
Example
of a T^ continuous preference that violates (11)
Proof. Given Tn.i e f^, and
E
lim
n—»oo
we show
that
|i„(l)-i(l]
=
/* as in the hypothesis,
f
|i„(t)-i(0|(it
+
44
if:
0,
(29)
Jo
then
lim
n—'oo
C r{t)dxr,[t)~ Jof r{t)dx{t)
£
0.
(30)
Jo
is continuous in T and thus in 7^ for all v? € $ since, by Corollciry 1, T is weaker than
any T^. To prove this, we will use the fact that since /' is the optional projection of / then
Then >
E
f r{t)
Dellacherie and
Therefore, for any
pf
|x„(0 -
VI.57).
Now
x{t)\ dt
+
(29) implies that:
|x„(i)
-
m
>
0,
choose
N{m)
|xn(0 -
+
(3i:
(
;J;|)
such that for
x{t)\ dt
=
>
i(i)|]
I [f^
P{[ [[
Write (30)
Jo
Meyer (1982, Theorem
Jim
Vi„ G f ^i
f f[t)dx^[t)
dXr.it)
Jo
|x„(l)
n > N[m):
all
-
>
:r(l)|]
< 1.
1 1j
as:
r fit)
diXr.it)
-X{t))
+
1
Jo
'2
>
where
C"
=
/-"
=
/
[' fi<^,t)diXr.iu,,t)-xiu,t))Pidu)
r fiu,t)diXr.iuJ,t)-xiuj,t))Pidu),
[
Jn'r-"Jo
and where
n;"'"
n;
We now show
=
LG
n: \j' |i„(0
<w e
n:
I
- i(0|
+
|i„(l)
|x„(f)-x(t)|d<
+
li„(l)-i(l)|
that both /["'" and /^'" converge to zero
First consider /|"'".
Assume
-
df
to begin with that /(w,
bls
.)
i(l)
m
G
<
-\,n>Nim),
m
>M.n>^(H
l],
on
[0,l].
oo.
the space of continuously
This implies that fticjj,t) < K^ < oo
J-measurable function K, where we use /( to denote dfi(jj,t)/dt.
differentiable functions
—
(and hence n)
C^[0,
for
t
G
[0,1] for
some
Appendix B
For
Example
G D^'",
einy a;
of a T^ continuous preference that violates (11)
consider /q /(w,
t) <i(i„(a;, t)
-
45
x{uj,t)). Integration
by parts gives
/ f{uj,t)d{xn{'^,t)-x{u;,t))
Jo
=
(i„(a;, 1)
- x{u,
-
1)) /(a;, 1)
(i„(a;, 0)
-
-
x{oj, 0)) /(a;, 0)
[i„(a;, t)
^
-
x{u, t)]ft{oj,t) dt
Hence
/
Jo
f{uj,t)d{x„{uj,t)-x{uj,t))
< -/(a;,l) + (x„(0)-i(0))/(a;,0) +
m
By
and the
the right continuity of the sample functions of {in} and x
are constants, we have
lim
In(0)
—'u^j
m.n—'oo
K^
I
-
l{0)
fact that in(0)
and x[0)
0.
Tri,r(
Thus, as
m—
»
oo,
we
get
lim
r fit)d{xr.{t)-x{t)) = o
But
U
^^
Vo/efl
^
ni.n—*oo J r\
'
m
'
n^'".
n>N{m)
if
lim
r f{u;,t)d{xr,{u,t)-x{uj,t))^0
V/(a;,
.)
e C^O,
l]
Vo;
G f]
(J
Q^'",
m n>N(m)
"
jim r/(a;,0d(2:„(u;,0-^{'^,0) =
V/(a;,
.)
£ C[0,
l]
Va;
G
Q n>Af(ml
U "^"1
»"
see Billingsley (1968,
Theorem
In addition, since for all
7.1, p. 42).
wG
fij"'"
we
have:
f{u;,t)d{xr,{u,t)-x{u;,t))<A[j
j
d{{xr,{t)
-
x{t))]
<
A{Xr.{l)
-
where
A=
sup sup
\f{(^,t)\,
wentelo.i)
and since from
(29)
we have that
I^n(l)-X(l)|
it
as
n
—
>
oo,
then follows from Strook (1987, Exercise III. 3. 19) that 7^'" —
Now consider T^*'"- Since P(n^'") — from (31), and since
as
n —+
>
[' fit)d{x„{t)-x{t))<A{x4l)-x{l))
Jo
therefore /j
'
—
»
as n,
m—
>
oo by Lebesgue convergence theorem.
I
oo.
x{l)),
Appendix B
Example
Proposition 20 There
Proof.
for all
It suffices
I € £!^
46
of a T^ continuous preference that violates (11)
> on f^ continuous
exist preferences
to provide an example. Let
=
^
and
ip~^
in 6^, that do not satisfy (11).
i*
let
€ C^ be such
that x-\-x'
>-
x
Define the continuous function:
.
/(w,()
=
QyJl/tx{l/t)
+
a > a' = E^f f [t)
where
1
dx' (t)
.
Let
= E a^l//i(l/0+l
/•(a;,0
and define preferences on f^ by
x>y
E /'/•di >E f'f'dy
if
Jo
Jo
By construction and by Proposition 19,
sequences
=E
Cr.(w,0
It is
clear that
both sequences
nces
and
y/m(")]xo,
continuous
is
>:
in
=E
y„(a;, <)
T^.
Next consider the two
y'^(n) Xi/n-
satisfy the requirements of Proposition 9.
However,
and w:
e[I^
fit) d{x„[t)
+
x'{t))]
=
+ e[^/^'
a*
and
J
f{uj,t)dy„{u;,t)= [l
+ ayJl/n{uj,n))B[y/^)].
Hence
B[iy'{t)dy„]
=
e[\/^] +aE[y^]E[-yi/M(n)]
>
Q+
E[y^],
where we have used Jensen's Inequality and the
E
r f'{t)dx^{t)
=E
/
fact that
f{t)dx^{t)
Vi„ G f ^;
Jo
Dellacherie and
Meyer (1982, Theorem
VI. 57).
Hence
for
r f'{t)d{x4t)+x'{t)) <E
Jo
any
rr{t)dy4t)
Jo
and therefore:
lim In
n—•oo
which proves that
>:
violates (11).
h^k'I
U59
I
-
I*
lim
^ n—
»oo
n:
y^,
for every
n
Date Due
FBI.
8
1994
Lib-26-67
MIT
3
TOflD
LIBRARIES DUPL
1
DD5b7M30
1