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working paper
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CHARACTERIZING PROPERTIES OF
STOCHASTIC OBJECTIVE FUNCTIONS
Susan Athey
No. 96-1R
(No. 96-1
August 1998
October 1996)
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
CHARACTERIZING PROPERTIES OF
STOCHASTIC OBJECTIVE FUNCTIONS
Susan Athey
No. 96-1R
(No. 96-1
August 1998
October 1996)
MASSACHUSETTS
INSTITUTE OF
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
CHARACTERIZING PROPERTIES OF STOCHASTIC
OBJECTIVE FUNCTIONS
Susan Athey*
First Draft: March 1994
Last Revised: April 1998
Massachusetts Institute of Technology
Department of Economics
E52-251B
Cambridge,
MA
02142
EMAIL: athey@mit.edu
ABSTRACT:
objective
This paper develops tools for analyzing properties of stochastic
functions
which take the form V(x,0) = ju(x,s)dF(s;Q).
The paper
analyzes the relationship between properties of the primitive functions, such as utility
functions u and probability distributions F, and properties of the stochastic objective.
The methods
where the utility function is restricted
to lie in a set of functions which is a "closed convex cone" (examples of such sets
include increasing functions, concave functions, or supermodular functions). It is
shown that approaches previously applied to characterize monotonicity of V (that is,
stochastic dominance theorems) can be used to establish other properties of V as well.
The first part of the paper establishes necessary and sufficient conditions for V to
satisfy "closed convex cone properties" such as monotonicity, supermodularity, and
concavity, in the parameter 0. Then, we consider necessary and sufficient conditions
for monotone comparative statics predictions, building on the results of Milgrom and
Shannon (1994). A new property of payoff functions is introduced, called lare designed to address problems
supermodularity, which
is
shown to be necessary and
supermodular in x (a property which
predictions).
The results
KEYWORDS:
single
crossing
sufficient for V(x,Q) to
be quasi-
in turn, necessary for comparative statics
are illustrated with applications.
Stochastic
properties,
monotone comparative
is,
dominance, supermodularity, quasi-supermodularity,
concavity, economics of uncertainty, investment,
statics.
*This paper represents a substantial revision of my August 1995 working paper of the same title, which was based on a
my Stanford University Ph.D. dissertation. I am very grateful to my dissertation advisors Paul Milgrom and
John Roberts, as well as Meg Meyer and Preston McAfee, for their numerous insights and suggestions. I would also like
chapter of
to thank Kyle Bagwell,
Scarsini,
Don Brown,
Armin Schmutzler,
Darrell Duffle, Joshua Gans, Ian Jewitt,
Scott Stern,
Don
Dave Kreps, Jonathan Levin, Marco
Topkis, two anonymous referees, and seminar participants at Australian
National University, Brown, Berkeley, Boston University, California Institute of Technology, Chicago, Harvard, LSE,
MIT, Northwestern, Pennsylvania, Princeton, Rochester, Stanford, Texas, Toulouse, U.L. de Bruxelles, University of
New South Wales,
Yale, and conference participants at the Stanford Institute of Theoretical Economics and the World
Congress of the Econometric Society in Tokyo for helpful discussions. All errors are of course my own. Financial
support from the National Science Foundation, the State Farm Companies Foundation, and a Stanford University
Lieberman Fellowship is gratefully acknowledged.
INTRODUCTION
1
This paper studies optimization problems where the objective function can be written in the form
V(\,Q)=\u(x,s)dF(s;Q), where u
is
a payoff function,
F
a probability distribution, and 8 and s
is
For example, the payoff function u might represent an agent's
are real vectors.
profits, the vector s
might represent features of the current
and 6 might represent an agent's investments,
state
utility
or a firm's
of the world, and the elements of x
effort decisions, other agent's choices, or the nature of
the exogenous uncertainty in the agent's environment.
Problems which take
this
form
many
arise in
contexts in economics. For example, the theory of
the firm often considers firm choices about investments
interaction
between firms often takes place
firm has private information about
pricing games.
how
is
a large literature concerning the comparative statics of
and decision-making under uncertainty, where the central questions
distributions over asset returns,
this
paper
statics predictions in
is to
and background
add to the
such problems.
To
set
initial
wealth, characteristics of the
risks. 1
of methods which can be used to derive comparative
that end, the
properties of the objective function, V(x,0), based
paper develops theorems to characterize
on properties of the payoff function,
problems often place some structure on the payoff function; for example, a
assumed
be nondecreasing and concave, while a multivariate
to
restrictions
on
cross-partial derivatives.
payoff functions.
strategic
Examples include signaling games and
investment responds to changes in risk preferences,
The goal of
And
returns.
an environment of incomplete information, where each
costs or inventory.
In another example, there
portfolio investment decisions
concern
its
in
which have uncertain
utility
Economic
function might be
profit function
These assumptions then determine a
u.
set
might have sign
U
of admissible
This paper derives theorems which, for a given set U, establish conditions on
probability distributions
which are equivalent
The paper focuses on
to properties of V(x,9).
sets
payoff functions, U, which are restricted to satisfy "closed convex cone" properties, that
properties
which
are preserved under affine transformations
and
limits;
of
is,
examples include the
properties nondecreasing or concave.
We proceed to
under which
sets
V satisfies
V in
two
steps.
The
first
"convex cone" properties. For example,
step
we
is to
characterize conditions
characterize
how
restrictions
on
of payoff functions, U, correspond to necessary and sufficient conditions on probability
distributions
1
analyze properties of
such that
V
is
For examples, see Rothschild and
Gollier (1995), Jewitt (1986,
supermodular in
Stiglitz
(1970, 1971),
G,
so that investments
Diamond and
1987, 1989), Kimball (1990,
Stiglitz (1974),
0-,
and
Oj
are mutually
Eeckhoudt and Gollier (1995),
1993), Landsberger and Meilijson (1990),
Ormiston (1983, 1985, 1989), Ormiston (1992), Ormiston and Schlee (1992, 1993), Ross (1981).
Meyer and
complementary for i&j.
Building on these results, the second part of the paper then focuses on
characterizing the conditions (single crossing properties and quasi-supermodularity 2 ) which have
been shown
be necessary and sufficient for comparative
to
(Milgrom and
Although single crossing properties and quasi-supermodularity are not closed
Shannon, 1994).
convex cone
predictions
statics
we show
properties,
that the techniques
developed for closed convex cones can be
generalized to analyze comparative statics properties.
Consider
first
the analysis of closed
convex cone properties of
Since such properties are
V.
preserved by arbitrary sums, closed convex cone properties of w(x,s) in x are inherited by V.
somewhat more
subtle to analyze properties of
admissible payoff functions, U.
on the existing theory of
sufficient conditions
some
set U.
If the
property of
stochastic dominance.
on F(s;0) such
V
in 0, while exploiting restrictions
V which
is
desired
monotonicity,
is
on the
It is
we can
build
A stochastic dominance theorem gives necessary and
that V(x,Q) is nondecreasing in 0, for all payoff functions
Thus, each set of payoffs
of
set
U induces
u in
a partial order over probability distributions. This
paper begins by introducing some abstract definitions which formalize an approach to stochastic
dominance which has been used
An initial
implicitly,
and
result in this paper is that stochastic
in the following way.
Such a theorem
set
is
useful if
T is
of "test functions."
is
T might be
First
nondecreasing in
for all
we
We characterize exactly how
small
T must be equal
T can
can use a set of
test functions
and the constant functions 1 and
to stochastic
m in some other
when
T which
— 1. Applying
is
can be thought
be a valid
all
indicator functions
approach yields the familiar
nonincreasing in
on a case by case
basis. 3
formulates and proves general theorems about the approach, and establishes that
used to check other properties of
2
Formal definitions are given
V
in 0.
set
convex cone of U. In
for all
dominance based on convex cones has been applied
classes of payoff functions
We
set T.
s.
in previous
studies (for example, Brumelle and Vickson, 1975) to characterize monotonicity of V(x,Q) in
few commonly encountered
it is
U is the set of univariate,
contains
this
T
be: in order to
to the closed
Order Stochastic Dominance Order, which requires that F(s;Q)
The approach
for all u in U,
smaller and easier to check than U, so that the set
a set of "extreme points" for U. For example,
nondecreasing functions,
for upper intervals,
is
nondecreasing in
of test functions for U, the closed convex cone of
particular,
dominance orders can be completely characterized
Instead of checking that V(x,0)
necessary and sufficient to check that V(x,Q)
of as a
less often explicitly, in the literature.
In particular, building directly
it
for a
This paper
can be also be
from our
results
about
in Section 3.1.
3 Independently,
Gollier
and Kimball (1995a, 1995b) have advocated a more abstract approach to stochastic dominance
theorems analogous to the one in this paper. In contrast to Gollier and Kimball, this paper focuses on characterizing a
wide variety of properties of V in G. Further, we provide necessary conditions for a set of functions T to be a valid
test functions for U, an exercise which requires us to identify the "right" topology of closure for this purpose.
set
of
stochastic dominance,
we show
convex cone property, the
we
that if the property
"test functions"
desire for
V (call
approach described above
we show
a subset of these closed convex cone properties,
is
it
property P) in
is
also applicable.
a closed
Further, for
that the
approach cannot be improved
new
classes of theorems, such as
upon.
Thus, the paper establishes several potentially interesting
"stochastic
theorems"
supermodularity
The
and "stochastic concavity theorem."
stochastic
supermodularity theorems can be used to prove comparative statics predictions in decision problems
and games. For example, they can be used
in
a firm's profit function, or
to
show when two
complementary
risky investments are
when investments by two
firms
are
or
substitutes
strategic
complements.
In Section 3,
we
build on this approach to characterize comparative statics properties of V.
We
begin with the case where x and 6 are scalars, and study conditions under which the optimal choice
of x
nondecreasing in
is
supermodular; but
of
V in x,
A
0.
sufficient condition for comparative statics predictions is that
supermodularity
if
is
the desired property,
we
can simply take the
first
u(x ,s)-u(xL ,s). However,
it
will also
difference
be useful to characterize the necessary and sufficient condition
for comparative statics, the single crossing property
cross zero, at
(which requires that the incremental returns to x
most once and from below, as a function of
We
6).
establish that the conditions
required for single crossing are in general weaker than those required for supermodularity.
surprising result
conditions on
F can
other words, if
U
where m(a^,s)-m(^,s)=1 and
v,
Our
is
is
large
much
all
enough
ueU,
easier to
large
enough
,s)^-1, then
to include at least
no weakening of the
if
and only
if
V(x,0)
is
check than single crossing
applied),
knowing
supermodular for
all
wet/.
x
is
Since
in this context (in particular, existing
that
it is
necessary for
some
classes of
useful.
final goal is to characterize
Milgrom and Shannon (1994)
supermodular in (x }yK2). However,
we have uncovered
statics properties
of the function
V
exogenous change
this property is not
is
in
some parameter
6:
V
0.
x and x2
to
must be quasi-
preserved by convex combinations, and thus
not sufficient to establish that
V is
quasi-supermodular. This
the general principle underlying the results of Ormiston and Schlee (1992),
few
x or
in
x
quasi-supermodularity of w(x,s) in x
essentially this result for a
comparative
identify a necessary condition for the optimal choices of
jointly increase in response to an
4 Thus,
v(jc ,s)-v(jc
l
in the sense just described, then the optimal choice of
dominance theorems may be
is
w
But, a
be obtained by considering single crossing as opposed to supermodularity. 4 In
nondecreasing in 6 for
supermodularity
U is
our set of admissible payoff functions
is that if
two functions u and
problems
is
and apply the theory of stochastic dominance described above, based on the properties of
H
stochastic
V
specific classes of payoff functions.
who
establish
paper identifies a
new
property,
V
sufficient to guarantee that
supermodularity,
supermodularity,
yet
it
The paper proceeds
by
necessary and
call /-supermodularity, that is in fact
quasi-supermodular.
preserved
is
may be
is
which we
convex
This property
combinations.
is
closely related to quasi-
Unfortunately,
quasi-
like
difficult to check.
Section 2 considers properties of
as follows.
admissible payoff functions U.
V
in
based on the
Section 3 considers comparative statics properties of V.
set
of
Section 4
concludes.
PROPERTIES OF
2
V(x,0)
IN 9
This section begins by studying monotonicity of
V in
mathematical constructs and lemmas required for what
In this context,
0.
we
call the
we develop
"closed convex cone" or "test
functions" approach to characterizing properties of stochastic objective functions.
to apply the techniques to characterize other "closed
identify a set of properties for
2.1
A
Unified
which
this
main
the
We
convex cone" properties of
V
then proceed
in 0,
and
we
approach cannot be improved upon.
Framework for Stochastic Dominance
This section introduces the framework which
as an abstract class of theorems,
we
will use to discuss stochastic
and to draw precise
parallels
dominance theorems
between stochastic dominance
theorems and other types of theorems.
We
allow for multidimensional payoff functions and probability distributions,
using the
following notation: the set of probability distributions on
9T
F:9T —> [0,1]
will use the notation A"Q to represent the
set
all
Further, for a given parameter space 0,
.
we
is
denoted A" , with typical element
of parameterized probability distributions F:9?" X0-»[O,1] such that such that
0g 0. With
this notation,
we formally
F(;0)e A"
define a stochastic dominance theorem.
Definition 1 Consider a pair of sets ofpayofffunctions (U, T), with typical elements m:9?"
pair (U,T)
FeA"e
is
for
a stochastic dominance pair
iffor all parameter
spaces
—> SR
.
The
with a partial order and all
,
ju(s)dF(s;6)
J
t(s)dF(s;d)
Definition
pairs of sets
1
is
is
nondecreasing for all ueU,
if and
only
if
nondecreasing for all teT.
clarifies the structure
of stochastic dominance theorems.
,
These theorems
of payoff functions which have the following property: given a parameterized
probability distribution F, checking that all of the functions in the set {ju(s)dF(s;Q)\u
nondecreasing
identify
is
equivalent to checking that
4
all
e u\
are
of the functions in the set <At(s)dF(s;6)\t e T\ are
Stochastic dominance theorems are useful because, in general, the set
nondecreasing.
T
is
smaller
than the set U.
Definition
1
differs
from the existing
existing literature generally
G, viewing
to 9t.
u(s)dF(s) and
J
In contrast,
bilinear functional
line,
we
Brumelle and Vickson, 1975),
literature (i.e.,
compares the expected value of two probability
in that the
distributions, say
F
and
u(s)dG(s) as two different linear functional mapping payoff functions
J
by parameterizing the probability distribution and viewing
mapping payoff functions and (parameterized) probability
are able to create an analogy
J
u(s)dF(s;6) as a
distributions to the real
between stochastic dominance theorems and stochastic
supermodularity theorems, an analogy which would not be obvious using the standard constructions.
The
dominance and stochastic
stochastic
The
first
goal of this section
be a stochastic dominance
is
become more
of this definition will
utility
a closed convex cone
is
P theorems,
ccc(G)
is
formalize the relationship between
P
(such as supermodularity).
and sufficient conditions for the pair (U,T) to
We will build on the concept of a closed convex cone, where a set G
g e G implies a g + fig e G for a and P positive, and further G is
pair.
2
if
g\
2
l
Thus, an example
is
the set of nondecreasing functions.
two constant functions, {u(s) =
l}
vj{u(s)
= -i).
Then:
U,T sets of bounded, measurable functions, and consider
a stochastic dominance pair if and only if
1 Consider
Then (U,T)
is
the
weak
topology. 5
ccc(Ukj {1,-1}) = ccc(Tu {1,-1})
Remark:
detail. 6
(2.1)
If we eliminate the requirement that
condition (2.1) can be replace
We
begin by interpreting Theorem
First,
F is a probability distribution
observe that unless
ue U. For example,
U might
crossing functions, that
The weak topology
is
is,
in Definition 1,
by ccc{U) = ccc(T).
T is
1;
the following subsection establishes the proof in
a subset of U, there
is
theorems provide conditions which are easier to check than
5
The
defined to be the smallest closed convex cone which contains G. Let {1,-1} denote the
set containing the
Theorem
when we
for other properties
to identify necessary
closed in a the appropriate topology.
set
clear
be a
set
which
is
no guarantee
J
that stochastic
some
dominance
u(s)dF(s;6) nondecreasing in
for all
not a closed convex cone (such as the set of single
functions which cross zero once from below).
formally defined below in Section 2.2;
it is
Its
closed convex cone
the weakest topology which
makes
jB
u(s)dF(s)
continuous
in both u and F.
6 In the context
of specific sets of payoff functions U, the existing literature identifies similar conditions for the
corresponding stochastic dominance theorems, though a different topology is used. For example, Brumelle and Vickson
(1975) argue that (2.1) is sufficient for several examples, using the topology of monotone convergence. In this paper,
using the abstract definition we have developed for a "stochastic dominance pair," we are able to formally prove that the
sufficient conditions hypothesized
by Brumelle and Vickson (1975) are
theorem, not just particular examples.
dominance
pair
is
new
here.
Further, the result that (2.1)
in fact sufficient for
is
any
stochastic
dominance
also necessary for (U,T) to be a stochastic
1
,
might be much larger (for the case of single crossing functions, the closed convex cone
all
functions).
which
In that case,
check
easier to
is
stochastic
it
that
might by
expected payoffs are nondecreasing
dominance theorems are most
When U contains
difficult to find a subset, T, of that closed
likely to
the constant functions and
be useful when
is
in 9.
is
the set of
convex cone for
Thus, (2.1) indicates that
U is itself a closed convex cone.
a closed convex cone, (2.1) becomes:
U=ccc(T\j{1-1})
In principle, the
(2.2)
most useful
T is
the smallest set
general, there will not be a unique smallest set.
no convex combination of u and v
u,v in E{U),
sets
9?,
whose closed convex cone
A set E{U) is
is in
E(U)
the set of indicator functions
is
Finally, because (2.2) is necessary
when U=ccc(Tu {1,-1} we know
,
that
M
{
and
we
U is
if
U.
However,
U
a set of extreme points of
E{U). For simplicity,
of extreme points which are closed. For example,
is
we
if,
for
in
all
will also consider only
the set of nondecreasing functions
on
a e 9? } ?
,
be a stochastic dominance pair
sufficient for (U,T) to
cannot do any better than letting
The
a set of extreme
points of U.
Table
summarizes the main stochastic dominance theorems which have appeared
I
economics or
statistics literature to date.
dominance theorems as well
restrictions
on the
(for
the
in
There are potentially many other univariate stochastic
example, theorems where the
set
of payoff functions imposes
third derivative of the payoff function); our analysis will indicate
how
these can be
generated.
TABLE I
STOCHASTIC DOMINANCE THEOREMS*
Sets of Payoff Functions,
:
U
Sets of Test Functions,
T
Condition on Distribution
for Stochastic
0)
(ii)
U F0 m {u|w:9t ->
U
so
m
{u\u:<R ->
91,
T F°={t\t(s) = l la „
nondecr.}
% concave}
T
s°
=
{t\t(s)
u{r|f (s)
(iii)
U S0M =\u
u: 91
—> 5R,
nondecr. ,
concave
7
Notice that there are
many
1
(s),
ae*}
-F(a;0)Tin0Va.
)
= -s}
=
Tsou = j,|,(5) _
a
-\_F{s,6)ds
min(a, s), a e 9?
^^
Dominance
5)>
j
Q g <RJ
\'_F{s,e)ds^
in
-j"_F(s,6)dt
Tin0Va.
6\fa.
!
possible sets of extreme points.
of sets bounded by rational numbers.
And we could
We could consider,
for example, only indicator functions
multiply the indicator functions by a positive constant.
h:9?
(iv)
—>
2
'(^2)=
nondecreasing,]
91,
(• supermodular
|
t
a,,a 2
1 |^.-)(*l>-
:l
F(a p a 2 ;0)
t-,^.)(*2).l
e9?
-F(a,;0)-F(a2 ;0) Tin
J
-F(a,;0), -F(a 2 ;6)
0;
T
[u\u: 9t
(v)
2
—> 9?,
'(^2) = V.-A)' 1 !*,.-)^)']
supermodular}
in
V(a,,a 2 )-
F(a p a 2 ;0) Tin
0;
>
t
a ,a2 e
t\t( Sl
,s 2 )=
"(V s
)=
f(s,,s2 )
=
2
[u\u: 9J
(vi)
2
—> 9?,
nondecreasing}
t and 1 A
The next
F(a,;0), F(a 2 ;0) const
91
x
(5,,
-l la
^
Sl ),
a,e9l}
"W)^' «
1 A (5,,^2 ),
V (ai,a
in
e^}
2
whereAc9t 2 ]
52 ) nondecreasing
<*F(s,0)
f
two lemmas which
2.2 Mathematical Underpinnings
will
Tin0VAs.t.
l A (5,,s 2 ) is nondecr.
j
section introduces the details of the mathematical results underlying
further proves
2 ).
Theorem
results are variations
and a Proof of Theorem
and
1
on theorems from the theories of
judm.
linear functional analysis, topological
vector spaces, and linear algebra; the main contribution of this section
function spaces and topology and restate the problem in such a
to solve the stochastic
,
be used throughout the paper.
We begin by proving some general mathematical results about linear functional of the form
Our
1
way
that
define the appropriate
is to
we can
dominance problem. One feature highlighted by
adapt these theorems
this analysis is there is
no
"art" to finding the right topology or notion of closure in the context of stochastic problems: the right
topology
is
simply the weakest one which makes the functional J u
work of Bourbaki (1987)
the
in linear functional analysis,
we
dm
continuous.
By
appealing to
highlight the simplicity and generality
of the closed convex cone approach.
We will work with a class of objects known as finite signed measures on 9T denoted ftp. Any
finite signed measure m has a "Jordan decomposition" (see Royden (1968), pp. 235-236), so that
m = m + -mr, where each component is a positive, finite measure. We will be especially interested
,
in finite signed
set
measures which have the property that
of all non-zero
finite
because elements of
F
and
F
2
1
and
F1
so that
signed measures which have this property by ^*;
this set
can always be written as
= j[F — F
l
fi
are probability distributions; likewise, for any
2
the measure F - F e ^*.
1
,
jdm = 0,
2
],
jdm + =jdm~
we
.
Denote the
are interested in this set
where
two probability
it
is
a positive scalar,
distributions
F
1
and
we
In this section,
me^.
will prove results
inequalities of the
we
linear,
is
Jw(s)dF'(s)
> Jw(s)dF
2
can
without
(s) for the
be easier to work with
in
loss
of
generality
there
we? *
a
is
5
/5(u,m)
= \udm. Then
P theorems
/?(?>",%*) is
9T
be a subset of measurable payoff functions on
PiA n ,B")
a separated duality. For consistency
is
We now define the appropriate topology,
the topology with the fewest
dual of P*)
Bourbaki
N(u; £, (m,
is
(1987,
,
. .
.
,
m
k ))
set
open
exactly the set
p.
n.43),
UeP*,
refer to
j
)|
< e >, where
is
any
me%*
our results hold
fl,
* fl 2
,
such that
if
we
A"
let
like to find the coarsest topology (that
a
as
there
is
the dual cone of the set U, denoted if,
M*=lf*
there
for
B" be any subset of %*, so long as
let
is,
continuous linear functionals on P*
all
this is the
uses
in
weak topology
basis
cr(P*,%*)
neighborhoods
on
of
By
P*.
the
form
a neighborhood corresponding to each
and each e>0.
u e U}. Likewise, for a set
M=lf, we
topology
be useful
use the pair (P*,^1) in our formal analysis.
where would
m)|m e 3£'};
{/?(-,
this
we will
,
arbitrary: all of
such that the set of
sets)
= 1 u max] fi(u - u, m
finite set (m,,...,m t )c3£*
Given a
a
9T and we
notation
Define the bilinear functional
.
^u 2
somewhat
(P*,2#*) is
as
"nondecreasing."
is,
x
inequality
will
it
a separated duality: that
I
}
all
u dm>0, where
The former
P other than
for properties
such that /3(«,m )^/3(w,m2 ), and for any u
P(u ,m) ^ P(u 2 ,m). s Our choice of
(the
J
this
terms of proving our main results, and further
Let P* be the set of bounded, measurable payoff functions on
x ^f -> SR by
interpret
appropriate pair of probability distributions.
proving results in Section 4 about stochastic
p:£>"
form
Since positive scalar multiples will not affect this inequality, and because the integral
operator
will
which involve
is
defined by
M of measures, the dual M'={u:
is
as the
U={m:
J
u(s)dm(s)>0 for
ju(s)dm(s)>0 for
second dual of U. Bourbaki (1987) shows
all
that dual
meM}.
If
cones can be
characterized in the following way:
Lemma 1
Consider (P*,^) with the weak topology. Suppose
convex cone. Further,
Lemma
Lemma 2
1
U
8
1
and McTW. Then
U
is
a closed
*=ccc(U) and M**=ccc(M).
can be related to our condition (2.1) in the following way.
ccc(T)=ccc(U),
if and
only
if
Proof: Suppose ccc(T)= ccc(U).
T
UcP
are closed
U'=f'.
By Lemma
1,
T"=ccc(T)= ccc(U). Also by
convex cones, and thus U'"=U and t"'=T
The boundedness assumption guarantees
.
Lemma
1,
if and
Taking the dual of
that the integral of the payoff function exists. It is possible to place other
on the payoff functions and the space of finite signed measures so that the pair is a separated duality, in
which case the arguments below would be unchanged; for example, it is possible to restrict the payoff functions and the
signed measures using a "bounding function." For more discussions of separated dualities, see Bourbaki (1987, p. 11.41).
restrictions
.
.
f'=ccc(U)=U**
by
Lemma
Lemma
1
Now
suppose f=U*. Then
f'=U*\ which
follows that
it
implies ccc(T)= ccc(U).
2 refers to a condition which
functions are not included.
attention to measures
stochastic
f"=U***.
yields
.
is
The following Lemma
me^*, and
specializes
states the result in a
dominance formulation. The proof of
theorem, rather than relying on
Lemma 2,
same
close, but not exactly the
way which
Lemma
this
Lemma
as, (2.1): the
2 to the case where
will
map most
constant
we
restrict
closely into our
based directly on the Hahn-Banach
is
of the result and the role of
in order to clarify the structure
the constant functions in the analysis.
Lemma 3
Consider a pair of sets of payofffunctions (U, 7), where
the following two conditions are equivalent:
U and T are subsets ofP*.
(i)
{me$'\ludm>0 Vwe U] = {me$'\judm>0 V«g7"}.
(ii)
ccc(U U{1 -1}) = ccc(T U{1,-1})
Proof of Lemma
3: First consider
(ii)
\me$"\\udm>Q Vwe£/} =
Since U<zccc(U
LHS
in the
set
u{ 1,-1})
of
(2.3).
It
,
it
implies
(i).
We begin by establishing that:
{mef \judm>0
suffices to
show
that
Then
|X is
Vmeccc(£/u{1,-1})}.
RHS
in the
of (2.3) whenever
set
follows for the constant functions because
(2.3)
J
dm = -j dm = 0.
it is
This
U u{l,-l} because Jw, dm > and
dm + a ju dm>0 when a,,a >0. It
follows for positive combinations of functions in
J«2
dm>0
implies J[a,w,
+ a 2 u2 ]dm = a ju
l
follows for the closure of the set because
J
u
dm >
Under
continuous in
the hypothesis that
T to
with
is
u,
oiP*^)
from above
ccc(U
T
is
(ii).
topology
is
,
apply (2.3) replacing
Define
U = ccc{U u {1,-1})
that the set
e U such
U
ef
and
we can take c =
is
f = ccc(T u {1,-1})
that
iiif
of continuous linear functionals on V* is exactly the
facts,
.
We know that
T
convex,
set
a corollary to the Hahn-Banach theorem implies that
closed and convex, theie exists a constant c and a m,
Since {1,-1}
and
generated from a family of open, convex neighborhoods. Recall
hyperplane)- so that /?(«, nu)>c for
argue
such that
c /(A)
u{ 1,-1}) = ccc(!T u {1,-1}) we can
loss of generality) that there exists a u
m)\m e %"}. Using these
since
we have chosen the weakest topology
get the result.
Suppose (without
{/?(-,
2
2
2
and for a continuous function, f{A)
Now we prove that (i) implies
the
l
all
ue T and
Oef
,
j3(«,
eW (a separating
nu)<c.
as well. Thus, /3(0,m,)
= 0>c. Now we will
without loss of generality. Suppose not. Then there exists a
uef
.
such that c < P(u,nu) = c < 0. Choose any positive scalar
c
p
>—>
such that p
(which
1
c
implies that pc
<
c
).
Since
hypothesis that /J(w,m.)
Because {1,-1} e
and thus
judrtu
f
,
>
f
a cone, pw e
is
c for
all
for all
ue
,
< 0, which
violates condition
fails,
closure, is critical for determining the
j udrtu
<
there exists a
for
measure m.e£* such
some u € ccc(U
u {1,-1})
This m.
.
The key consequence of our construction
F
1
2
1
such that
am.=F -F
for
that
is
F(-;6,)=F
.
decreasing for
Theorem
particular,
1
is that
some a>0. Thus,
m.e$?, which implies that there exists
this separating
letting
2.3
0={9„,0J, and
nondecreasing in 9
continuous. Thus,
if
VmgT,
but
the
two
\u(s)dF(s;Q)
some
sets are equal
weak topology
Other Closed Convex Cone Properties ofV(x,B)
weak
and
is
1
less familiar than
ccc(U u{ 1,-1}) = ccc(Tu {1,-1})
/J(-,m) continuous, (2.1) will follow for the
1
F{-\Q^=F* and
letting
others; in
dominance theorems have been proved by showing
monotone convergence of the convex cones of
F
hyperplane can be used to generate a
makes use of the weak topology, which might be
is
appropriately
,
This counter-example establishes the proof of Theorem
such that (5(-,m)
By
Lemma 3 has produced the following result:
judnu>0 for all u e ccc(T u {1,-1}) but
ugU.
u(s)dF(s;Q)
existing stochastic
,
which determines the
form of the hyperplane.
we have
\
g f but
for all u
thus the topology,
Vickson, 1975; Topkis, 1968). However, by definition, the
which makes
w dm, >
Then,
some
closure under
J
= -/3(l,m.) = 0,
the separating hyperplane. 10
counterexample to a stochastic dominance theorem, by
2
that
p\l,m»)
(i).
choosing the topology and the two spaces of functions,
if (ii)
c, contradicting the
0.
,
dm, = 0. 9 So, m, e £" and we have shown
J
= pc <
f we conclude that
The proof uses a separating hyperplane argument, and
meaning of
But, (5(pu,nu)
.
we let c =
u e U. So,
and p\w,m.) >
f
is
that the
(Brumelle and
the coarsest topology
for
some
other topology
topology.
in 6.
This subsection derives necessary and sufficient conditions for the objective function, ju(s)dF(s;d),
to satisfy properties
P
other than "nondecreasing in 9," for example, the properties supermodular or
We ask two questions:
concave
in 9.
9 See also
McAfee and Reny (1992)
(i)
For what properties
for a related application of the
P is
the closed
convex cone method of
Hahn-Banach theorem, where
the separating
hyperplane also takes the form of an element of £*.
10
Notice that
establishes the
if
F
is
not restricted to be a probability distribution,
remark following Theorem
1.
10
it is
not important that
m&%*, and
thus
Lemma
2
P
proving stochastic
theorems valid?
convex cone approach
To
we
begin,
(ii)
For what properties
we
introduce a construct which
satisfies property
P on
denote the set of
P
convex
set,,
which are defined so
f
" is well-defined
We
that,
we
establish that the closed
while for supermodularity,
it
theorems, which are precisely
P
are interested in properties
h:Q p —>9?
given a function
,
together with
the statement "h(Q)
and takes on the following values: "true" or
such parameter spaces
all
P
will call stochastic
analogous to stochastic dominance theorems.
QP
can
exactly the right one, as in the case of stochastic dominance?
is
parameter spaces
P
Q p When P represents
.
must be a
lattice. Further, for
Qp
concavity,
"false." Let
can be any
we
a given property P,
will
define the set of admissible parameter spaces together with probability distributions parameterized
on those spaces:
n
VP s {(F,e p )\Q P
e
QP
n
F e A Qp }
and
.
Definition 2 Consider a pair of sets of payofffunctions (U,T), with typical elements «:9t"
pair (U,T)
is
a stochastic
J
u(s)dF(s; 6)
J
u{s)dF(s; 8) satisfies property
With
this definition in place,
method of proving
is
P pair iffor all
Vp
satisfies property P on Q p for all ue U,
(F,Q P ) e
stochastic
it
Definition 3
and
The
:
if and
only
if
be straightforward to show that the closed convex cone
dominance theorems can be extended
to all stochastic
P
theorems,
if
P
"CCC property," defined as follows:
A property P is a CCC property if the set offunctions g:Q P —> 9?
a closed convex cone, where closure
.
P on O p for all ueT.
will
a closed convex cone property, or a
—> 9?
is
which
satisfy
P forms
taken with respect to the topology of pointwise convergence,
if constant functions satisfy P.
Note
while
that
we
functions,
The
we
are using closure under the topology of pointwise convergence for properties P,
are using closure under the
weak topology
property "constant."
and supermodular are
is
CCC.
which places a sign
Further, any property
Finally, since the intersection of
all
the property "nondecreasing
The following
result
and convex"
shows
a stochastic
stochastic
P pair.
a
is
a
CCC property.
11
on a mixed
two closed convex cones
is itself
CCC property.
convex cone approach applies
Equivalently, if ccc(U
P pair.
properties, as is the
partial
a closed
For example,
CCC property.
that the closed
Theorem 2 Suppose property P
is
is
CCC
restriction
convex cone, any of these properties can be combined to yield another
(U,T)
payoff
U and T.
properties nondecreasing, concave,
derivative
(as defined in Section 2.2) for sets of
If (U,T)
is
to
CCC properties.
a stochastic dominance pair, then
then (U,T) is a
u {1,-1}) = ccc(T u {1,-1})
,
6
We establish a series of implications, referring to the following condition:
First, (i) (2.4)
holds
Vu e T
(ii)
implies
functional and
Third,
(iii)
(2.4)
implies
and constant functions satisfy
Second,
P
u(s)dF(s;Q) satisfies
J
(2.4) holds \/u
Vw e ccc{ T u {1,-1} ). To
P is a CCC property, and
J
u(s)dF(s; 6)
J
This theorem generates
.
many new
2.3.1
=
l,
is
a linear
see this, recall that a subset of a
,
J
all
the convergent nets of
continuous for
wa (s) dF(s;
)
—> J
all \i,
for
any net ua
u(s) dF(s;
)
.
Since
the pointwise limit of a net of functions which
P as well.
the hypothesis of the theorem and since
P
theorems, where the (U,T) pairs which
on stochastic dominance
literature
nondecreasing
respectively, is a stochastic
P(-;p.) is
classes of stochastic
theorem generalizes a
to
,
Precisely analogous arguments establish the symmetric implication.
have been identified in the large
corresponding
is
Vm e U, by
Finally, (iv) implies (2.4) holds
special case, this
1,-1 } because \ dF(s;8)
s
contains the limits of
then given 6
u(s)dF(s; 6) satisfies
u {1,-1})
it
Since the linear functional
set.
T u {1,-1} such that «„-»«,
Uczccc(U
{
P is a CCC property.
elements in that
satisfy P, then
TU
Vm e cc( T u {1,-1} ), because the integral
(2.4) holds
topological space is closed if and only if
in
e
P by definition.
implies (iv) (2.4) holds
(iii)
(ii)
result
functions
are also stochastic
by Topkis (1968), who proves
and
indicator
functions
of
P
pairs.
As a
that the (t/,7) pair
nondecreasing
sets,
P pair when P is a CCC property.
Some Simple Examples
Using Theorem
2,
we
can apply the results of Table
I to
problems of stochastic supermodularity.
Consider the following optimization problem:
max
z
In our
to
improve
costs
example, the optimizer
first
(s).
its
is
a firm making investments in research and development
production process, and in particular
The
returns to research
nonincreasing in
its
searches for
ways
is
parameterized by
t.
statics
nondecreasing in
t
Suppose
distribution
that the firm's payoffs are
c(z).
Then, Theorem
theorem of Milgrom and Shannon (1994)) implies
for all c
12
and
all
(z)
to reduce its unit production
production costs, and that the firm's investment has a cost,
2 (together with a comparative
(z) is
it
and development are uncertain, and the probability
over the firm's future production costs
investment
\u{s)dF(s;z,t)-c(z)
u nonincreasing,
if
and only
that
if F(s;z,t) is
supermodular
Intuitively, the goal
in (z,t).
weight towards lower realizations of
its
of the firm's investment in research
When F
unit cost.
is
supermodular, the parameter
the "sensitivity" of the probability distribution to investments in research:
correspond to probability distributions where research
In the second example,
we
more
is
choice of effort,
let c(z)
be the cost of
all
and
let t
Then expected
assignment or production technology.
nondecreasing in z for
effort,
we
though
u nondecreasing and concave,
(z),
as in the classic
Let s represent the agent's wealth
will not
model
that here), let z
be the
be a parameter which describes the job
utility
if
t
effective at lowering unit costs.
formulation of the principal-agent problem (Holmstrom, 1979).
rule,
indexes
t
higher values of
consider a risk-averse worker's choice of effort
(which might be determined by a sharing
to shift probability
is
(ignoring c(z) for the
and only
if z shifts
moment)
F according
to
is
second
pa
order monotonic stochastic dominance, that
F(s;z,t)ds nomncreasing in z for
is,
Js=—«•
Next, observe that the optimal choice of effort
F(s;z,t)ds
-J
concavity"
The
is
result:
supermodular
is
expected
concave
latter result is particularly useful, since
and only
in z if
it
all c(z),
and only
if
if
Finally, consider the analogous "stochastic
in (z,t) for all a.
utility is
nondecreasing for
is
all a.
if
—
pa
F(s;z,t)ds
I
is
concave for
all a.
can be used to establish that the First-Order Approach
valid in the analysis of principal-agent problems.
In fact, Jewitt (1988) establishes just this result,
applying the more standard approach of integration by parts.
Necessary and Sufficient Conditions for "Stochastic P Theorems"
2.3.2
This section considers the question of whether the "test functions" approach can be improved
upon
precise,
pairs
P
for "stochastic
it
will
when P is a property other than nondecreasing. To be more
introduce some additional notation. We let S JD7 be the set of (U,T)
theorems,"
be helpful
to
-
which are stochastic dominance
some property P. Then, observe
pairs,
we
that
and
endeavor in
theorem
To
set
U
only
property "constant in 6."
functions, and the set
clearly
is
is
the set of
1L SPT
all
all
stochastic
= Z 5/57
P
for arbitrary closed
one could check a stochastic
of the extreme points of U, E(U).
which properties
P
pairs for
Our
P
final
are such that (U,T) satisfies a stochastic
P
if (2.1) holds.
see an example where (2.1)
lu(s)dF(s;0)
Z spr be
for the possibility that
without necessarily checking
this section is to identify
if and
let
have not established that
convex cone properties P. This opens the door
theorem for a
we
is
too strong, consider a simpler example of a
Take the case of
f = -U
F0
constant in 9
.
if
The
pair
and only
(U
if
not a stochastic dominance pair.
13
U F0
F0
,
the set of
,f)
satisfies
-lu(s)dF(s;6)
all
CCC
property, the
univariate, nondecreasing payoff
a "stochastic constant theorem," since
is
constant in 6.
However,
U FO ,f)
X 5/ r = JL SDT
In general, the result that
(U,T)
a stochastic supermodularity pair
is
dominance. For example,
and
is
,
U
this set
U
dominance
it
may be
U are
T have
theorem
This allows us to bypass the step of checking
immediate.
is
same closed convex cone
the
does not
literature
make such
P pair if and only
are a stochastic
jdm?=l)
ccc(U
if
such
directly (further,
G(;G
follows:
for
distributions. But, observe that
= {G
1
vG 2 ,*)
1
1
,*)
J
Thus,
J
if
we
The
is
some
it
can also be expressed as follows:
G(;Q
supermodular, since the
1
)
= G(;Q 2 ) = -^—.
Then,
\udnu>0
if
dnC
)
latter entails
v G 2 ) - J u(s)dG(sid ) + J u(s)dG(s& a G 2 ) - J u(s)dG(s$ 2 ) >
1
can conclude that for
u dG(;Q)
for
^,
and
dnC
1
1
(normalized so that
\udnu<0
but
,
we
3,
Define a parameterized distribution, G(;G), as
)
Jw dG(;Q)
u(s)dG(sid
m*e^*
a
Lemma
aG 2 ,6 2 }.
J
and only
exists
u e ccc(T kj {1,-1})
all
vG 2 = G(;G aG 2 ) = 7
1
there
Recall that, in
.
= \[ng + n£ —nu -nu].
Q SPM
let
u {1,-1}) ^ccc(Tu {1,-1}),
supermodular, for which (U,T)
This nu, our separating hyperplane, can be expressed as the difference
between two probability
nu
most of the stochastic
a statement explicitly).
if ccc(U u {1,-1}) = ccc(T u {1,-1})
$udnu>0
that
ueccc(Uv {1,-1}).
Now,
on stochastic
has been analyzed in the stochastic dominance literature, then the corresponding
and
that
literature
determined by an economic problem,
We begin this section by giving an example of a second property,
showed
easier to verify whether or not
by drawing from the existing
the characteristics of a set
if
stochastic supermodularity
whether
useful because
fails to
this
G, \udG(;Q)
be supermodular, and
critical feature
is
supermodular for
all
.
u e ccc(T
u {1,-1})
,
yet
we contradict the definition of a "stochastic P theorem."
of supermodularity in
example
this
is that
the separating hyperplane, an
element of ^*, can be used to generate the needed counterexample to the stochastic supermodularity
theorem.
More
generally,
we
will define a set of properties called Linear Difference Properties
(LDPs), properties which also have
The
first
important attribute of
this feature.
LDPs
is
that they
can be represented in terms of sign
on inequalities involving linear combinations of the function evaluated
For example, g(G)
is
nondecreasing on
if
and only
if
H
g(Q )-g(Q
This statement specifies a set of inequalities, where each inequality
function evaluated at a high parameter value and a
g(6)
is
supermodular on
if
and only
if
g(Q*
14
at different
L
)>0
is
parameter values.
for all
G"
>G L
in 0.
a difference between the
low parameter value. To take another example,
vQ )- g(Q*) + g(Q aQ 2 )- g(Q 2 )>0
2
restrictions
}
for all G',6
2
in
,
sum
Again, this statement specifies a set of inequalities, this time involving the
0.
of two
differences.
we
In both cases,
can represent every inequality by the parameter values and the coefficients
which are placed on the corresponding function
For example,
values.
the
for
property
nondecreasing, each inequality involves the coefficient vector (1,-1) and a parameter vector of the
H
L
form (G^G* ), where 1 is the coefficient on g(Q ) and -1 is the coefficient on g(Q ) Thus, g(G)
w
L
w
if and only if for all vector pairs (a, <]>) = ( (1,-1) (G
G ) ) such that G > 6^
is nondecreasing on
-
.
,
,
^T._ a, g(fy)
= g(Q H )-g(Q L )>0
vector pairs of the form
Xt,«i g(^ ) = g(Q
i
l
=
(cc,<|>)
on
supermodular
Likewise, in the case of supermodularity,
.
1
((1,-1,1,-1), (G
and
if
only
ve 2 ,e',e' aG 2 ,G 2 )).
for
if,
1
2
1
2
are interested in
In this case,
such
all
vQ )-g(Q ) + g(Q AG )-g(G )>0. More
2
we
g(Q)
vector
is
pairs,
generally, consider the following
definition.
A property P is a linear difference property
Definition 4:
there exists a positive integer k
@,Q e
c
I
and a
a G SR*, ]£._, a, = 0,
(a,(|))
(f>
if,
for any parameter space
collection of pairs of vectors,
0™
e
\,
so that conditions
(i)
@q
,
Q P e@ p
,
where
and (ii) are equivalent for any
g:Q P ^3i:
The
P on Q p
(i)
g(Q) satisfies
(H)
lUargi^ZOforall
definition of
&Q
a
(a,«)))
P
on
P
."
e
eQf
on the above discussion; we
builds directly
inequality representation of
the coefficients
.
Note
first that in
will refer to
the construction of
In the
,
as the "linear
we have
be useful when
to vary with the vectors of parameters; this will
multivariate concavity has a linear inequality representation.
CQ
&Q
allowed
we show
that
above examples, nondecreasing
and supermodular both have linear inequality representations. In the case of nondecreasing, given a
parameter space
<?e ND
=
Q ND
,
= (1-D;
{(oc,40|a
4>
= (6",G L );
6"^ e G ND
For the property supermodular, given a parameter space
^e sw =
Observe
zero.
It
is
{(«,(t))|a
that in the
= (l-U-l);
<t>
= (G' vG 2 ,G
1
,G
1
,
6" > Q L }.
Q SPM
,
aG 2 ,G 2 );
the appropriate set is
G',G
2
€0 5PA,}.
examples of nondecreasing and supermodular, the components of
easy to show that
if
combination of differences between
the definition requires X*=i
X,*=i
jc,.
a,
's.
=
,
then
£*=1 a
.
(
jc,
a sum
to
can be represented as a linear
This motivates the word "difference" in the name, since
ot.=0.
15
Lemma 4
P
If property
is
QP
Proof: Consider
(a,§) e
&e
•
e<d p
If
.
,
P on
+ a 2 ft
[«ift Ofc )
0,,
(<t>i
)]
.
Then
The
which
Definition 5
require
3.
<3 P
(<fc )
'
:
any (a,§) e
ft
,
,
&e
and any a,,a,>0,
+ «2 S=i «,
ft (<t>,
)
.
which
nonnegative by
is
This argument can be extended to sequences or nets of
may be more
difficult to check.
A property P satisfies the single inequality condition
VP
n
wiSR"—>SR, there exists (F,0) e
The
P on
for
on 0,, then X*=i a £(<t>,) =
for all
Now
suppose
that g (9)
by Definition 3.
constant in
satisfy P.
we
condition
last
is
= «i Xti «,
assumption and by Definition
functions
g(9)
and thus g(9) satisfies
and g 2 (9) satisfy
Zli «/
a linear difference property, then Pe CCC.
such that
\udm >
if and
only
if,
if
for any
me^ and any
ju(s)dF'(s;9)
single inequality condition is satisfied if for each vector pair
satisfies
P on
0.
there exists an
(a,<}>),
appropriate parameter space and probability distribution so that the inequality corresponding to
(a,<})) is critical in
is
determining whether \u(s)dG(s;Q)
a strong condition:
this subsection.
it is
what we verified
P for arbitrary payoff functions
satisfies
in the discussion of supermodularity at the
While the single inequality condition may seem close
u.
It
beginning of
to the desired conclusion,
we
have been unable to find alternative conditions which are easy to verify (though see the working
One
paper (Athey, 1995) for assumptions which are "more primitive.")
not always well-defined on the restricted parameter space which
single vector
issue
is that
P is
defined by the components of a
is
0.
Consider several examples which serve to
illustrate the single inequality condition.
already established that nondecreasing and supermodular satisfy the condition.
any discrete generalization of a sign restriction on a mixed
manner analogous
we
the property
partial derivative
We
have
Next, observe that
can be represented in a
to our representation of supermodularity at the start of this subsection.
Finally,
consider concavity.
Example: "Concave"
Proof: Let
satisfies the single inequality condition.
CV
=
[9,1]
.
Define G(;G)
= G(;l) = 7^-7 G( ; 9) = (1 - 29)
,
jdirC
for0<9<l/2, and
G(-; 9)
= (29 -1)7^— + (2-29)-^\dmt
fails to
be concave
if
J
definition, this holds if
u(s)dG(s;j)
J
16
"'
for 1/2<9<1. Then, fw(s)JG(s;9)
Js
\dmt
< j j u(s)dG(s;0) + \ j u(s)dG(s;l)
u(s)dmt < \ J u(s)dnu
+ 29
7^
jdnC
\dmt
+jj u(s)dtru
,
or
J
.
But, with the above
u(s)dnu
<
.
.
interesting to note that in general, the intersection of
It is
two properties which
However, since the
inequality condition does not necessarily satisfy the single inequality condition.
intersection of
two
CCC
properties
is
approach for the intersection of two
this,
it is
CCC
LDP
property,
inequality condition. This
an arbitrary
that lattice
and
lattice
-^^-f(x)>0
that
(ii)
is
we
can always apply the closed convex cone
properties that satisfy the single failure condition.
interesting to note that supermodularity,
be defined as requiring
can be
a
which
Given
for a suitably differentiable function /(x) can
for all i±j,
is
in fact an
LDP
and
which
true because supermodularity is a property
when
satisfy the single
satisfies the single
can be defined on
(i)
the lattice has four or fewer points, supermodularity of a function
on
can be expressed in terms of a single inequality, so that the single inequality condition
satisfied.
We now state the final result of this section,
inequality condition, then the
stochastic
which
same mathematical
is
that if
P
is
an
LDP
and
satisfies the single
structure underlies the stochastic
P
theorem as a
dominance theorem.
Theorem 3 IfP
an
is
LDP and the single inequality condition holds,
then conditions
(i)-(iii)
are
equivalent:
P pair.
(i)
(U,T)
is
a stochastic
(ii)
(U,T)
is
a stochastic dominance pair.
(ii)
ccc(C/u{l,-l})
Proof:
Lemma 4 implies that P is
equivalence of
(i)
= ccc(ru{l,-l}).
implies
(iii).
(ii)
and
(iii),
Suppose
and Theorem 2 gives
(iii) fails,
and there
short-hand for the closed convex cones, as in
there exists m,
e £* such that J u dm,
>
J
u(s)dF(s;Q) satisfies
stochastic
P
Z JFT = H SDT
.
implies
So
(i).
5 e U such
it
establishes the
remains to show that
ug
that
1
T
(the sets are
Lemma 3). Then by the proof of Lemma 3,
for all
uef
,
but J udm.
(F,Q) e
n
"D P
<
By
.
such that
J
the single
u(s)dF(s;Q)
stated as follows: if
P
is
an
LDP
and
sets
P
satisfies the single
This result establishes that the closed convex approach to
theorems exactly the right one for this class of properties. Note that Theorem 3 holds
without any assumptions about continuity, differentiability, or other such properties.
on
fails
P for all we f
The consequence of Theorem 3 can be
inequality condition, then
(iii)
exists a
inequality condition, this implies that there exists
on 0, but
Theorem
a closed convex cone property.
Assumptions
of payoff functions might be relevant for a particular stochastic dominance pair (U,T); these
assumptions would then be inherited by the corresponding stochastic supermodularity theorem.
Linearity of the functional l%u,m) in m, however,
17
is critical.
we
In the next section,
discuss applications of
theorems which can be derived using Theorem
some of
the
new
stochastic supermodularity
3.
Applications: Stochastic Supermodularity Theorems with Supermodular Payoffs
2.4
This section explores applications of the
from Theorem
3.
By Theorem
3,
new
stochastic supermodularity theorems
which follow
each of the (existing) stochastic dominance results corresponds to a
(new) stochastic supermodularity
result,
which may then be applied
to solve comparative statics
problems.
Consider Table
The
payoff functions.
interval of the
which gives the stochastic dominance theorem
I (v),
random
T
set
in
variable
s,
Table
,
I
for bivariate,
supermodular
(v) contains both the indicator functions for
each upper
and the negative of the indicator functions for each upper
interval.
Thus, the corresponding stochastic dominance theorem requires that the marginal distribution of
each random variable
is
unchanged by the parameter
This condition
6.
is
required because a
supermodular function does not necessarily satisfy any monotonicity properties.
In particular, a
supermodular function might be additively separable, and monotone decreasing or monotone
increasing.
Thus, an
FOSD
shift in either
marginal distribution will raise expected profits for some
supermodular payoff functions, and lower expected payoffs for others.
Now,
5,=
and
a,
s2
=a
2
.
Then
must be nondecreasing
When
(j ,,5 2 )gSR
consider a partition of the space
in
,
2
into four quadrants, delineated
the requirement that the function
6
6
specifies that
the marginal distributions are constant in 6, that
nondecreasing in
random
6.
is
We will refer to this type of shift as
variables; such a shift is beneficial
when
l [aix) (s )-l [a2 „ (s 2 )-dF(s ,s 2 ;6)
)
1
JJ
shifts probability
by the axes
mass
l
into the northeast quadrant.
equivalent to requiring that F(a u a 2 ;6)
is
an increase in the "interdependence" of the
the
random
variables are
complementary
in
increasing the payoff function.
The
intuition for the stochastic supermodularity
builds directly
from the stochastic dominance
theorem for the
intuition.
set
For two parameters to be complementary in
increasing the expected value of a supermodular payoff function, they
marginal distribution functions,
of supermodular payoffs
must not
and further they must be complementary
in
interact in the
increasing
the
interdependence of the random variables.
To
see a special case of
how Table
I (v)
might be used, consider the following example.
A firm
engages in product design for a product with two components, and the random variables, s and s 2
i
represent the qualities of the
increasing the quality of one
two components.
The components
component increases the
18
fit
together in such a
way
returns to quality in the other component.
,
that
The
two product design teams work
two teams
stochastic, but the outputs of the
Each team's output
develop the two different components.
to
are correlated (perhaps due to realizations of
is
random
events which affect the whole firm, or due to communication between the two teams).
Suppose the firm wishes
to set target qualities (or incentive contracts) for
each group, where an
increase in the target quality increases the expected quality the group will produce.
The above
gives necessary and sufficient conditions for supermodularity of the firm's objective.
would be
interpreted as follows: increasing the target quality for
increasing the target quality of the second group.
the cost of producing quality for
result
This result
one group increases the returns
to
Further, in response to an exogenous decrease in
team one, then the firm would find
it
optimal to raise the targets for
both teams.
The
conditions for "stochastic supermodularity" can be verified in a specific functional form
example.
Suppose
random
that the
covariance ((s ,s 2 ) ~ 5V7V(/x
x
Table
I
(v) establishes that if
1
G 12 >0).
,// 2 ,(7; ,0' 2 ,ct 12 );
u
G 12 >
Then checking
.
is,
u(s1 ,s2 )dF(s1 ,s2 ;fi l ,n 2 ,a n )
J
Intuitively,
when
the payoff function
one random variable increases the returns to the other, then
mean of one random
the
random
the conditions given in
supermodular, the expected profits are supermodular in the means
is
of each marginal distribution. That
for all u supermodular, if
variables have a bivariate normal distribution with a positive
!
supermodular
is
is
in a stochastic
variable increases the returns to raising the
variables have a positive covariance, increasing the
mean of
interact in the marginal distribution functions is satisfied in this
design example,
we
conclude that
if
),
environment, raising the
the other.
mean of one does
low values together. Of course, the requirement
(jj,,,|i 2
such that under certainty,
effectiveness of the other in terms of shifting probability weight into regions
variables realize high or
in
Further, since
not decrease the
where the random
that the parameters
example.
do not
Thus, in the product
the outputs of the product design teams are joint normal with
positive correlation, the target qualities of the
two teams
will
be complements in the firm's
profit
function.
Now we
turn briefly to multivariate, supermodular payoff functions.
a particularly complicated problem because changes
functions pose
distribution can potentially affect the
the high-order
mixed
interpretations
on such
application to
income
are
assumed
to
dominance
derivatives.
between
results.
One
all
However,
it
is
approach, followed by
often difficult to place
Meyer (1990)
Meyer (1990) considered supermodular
between three or more variables are ruled
out.
19
thus,
of the arguments of the payoff function are
inequality, is to consider objective functions
be zero.
the joint probability
co-movements of many random variables simultaneously;
partial derivatives
relevant for stochastic
in
Multivariate payoff
economic
in the context of an
where higher-order derivatives
objectives
Supermodular payoffs of
this
where
interactions
form can be used
to
represent a social welfare function that
is
Our
averse to inequality.
context as well; a stochastic supermodularity theorem could be used to check
complementary
when two
in that
policies are
expected social welfare.
in increasing
The following
be used
results could
result takes
an alternative approach, studying the case where the random variables
are independent.
Theorem 4 Let s be a vector of independent random
parameters such that for
variables. Further, suppose that
marginal distribution ofsi
i=l,...,n, the
is
a vector of
given by F^s^d.). Then the
is
following two conditions are equivalent:
(i)
For all supermodular payofffunctions
(ii)
Either (a) or (b)
is
i
Proof: For
i
all (ij) pairs,
J
(ii) (a)
or
{
i
is
Theorem 4
k
Va,e<K.
:
^ii;Oy).
($,-,$,)
since supermodularity
that the supermodularity condition
is
on the
preserved by
distributions
(b).
FOSD
shifts in
independent random variables are
expected value of supermodular payoff functions. 11
FOSD)
is
Thus,
if
a
complementary with increasing the others stochastically as well.
Returning to the joint normal distribution,
when 6
V0,">0f,
supermodular in a group of random variables, increasing one variable in the
stochastic sense (of
satisfied
in 6.
V6» >0f, Va.eSR.
;ej-)
supermodular in
and check
in increasing the
payoff function
i
;0*)<JF;(a.;0f)
This result says that parameters which induce
complementary
supermodular
S r\d
(v)
I
is
rewrite the expectation as
that the inner integral is
sums. Apply Table
—>S{, ju(s)dF(s;0)
J^;»,.*JWi f (i.x f ;e. w )-^;«,)
S
*I'
reduces to
i
= U,n, ^(a
J
Note
n
true:
(a)Fori = l,..,n,F (a ;6?)>F (a
(b)For
u:3i
represents the
if the
random
variables are independent, the conditions are
mean of random variable sh
result has potential applications in the study
of coordination problems in firms
the product design example from above) as well as in general investment problems.
Schmutzler (1995) apply
this result to
(recall
Athey and
analyze a firm's choices over investments in product design
and process innovation.
11
Since a supermodular function
(ii)(a)
or
(ii)(b) in
Theorem
is
also supermodular in the negative of all of
4.6.
20
its
arguments,
we
allow for either case
.
3.
COMPARATIVE STATICS PROPERTIES OF V(x,Q) IN (x,0)
This section builds on the analysis of Section 2 to characterize interactions between components
of x and components of
are necessary
and
in the function V.
sufficient for
Particular emphasis will be placed
monotone comparative
on properties which
statics predictions.
3.1 Definitions
Since this section will focus on properties relevant to comparative statics predictions,
Given a
introducing several relevant definitions.
set
X and
we
begin by
a partial order >, the operations "meet"
x v y = inf {zjz > x, z > y} and x a y = sup{z|z < x, z < y}
(v) and "join" (a) are defined as follows:
A lattice is a set X together with a partial order, such that the set is closed under meet and join.
6 Let (X>) be a
Definition
lattice,
(i)
A function h:X->Sl is quasi-supermodular
if,
for
all
x,yeX,
h(x a y) - h(y) > (>)0 implies h(x v y) - h(x) > (>)0 (ii) hissupermodularifforallx,yeX,
h(x v y) + h(x a y) > h(x) + h(y) (Hi) Ifh is nonnegative, it is log-supermodular (log-spm) 12
.
.
if,
allx,yeX, h(xvy)-h(xAy)>h(x)-h(y).
h(x",x_ k ) — h(x k ,x_ k )
is
nondecreasing
We will be particularly
In that case, if
1978); h
is
h
is
(iv)
in
h has increasing differences
x_ k for all x k
interested in the special case
>x k
for
(x k ;x_ k ) if
in
.
where our function of
interest is bivariate.
smooth, supermodularity corresponds to the restriction that -^-h(x)>0 (Topkis,
log-supermodular
h(x",x2 ) — h(x(',x2 )
is
if log(/j) is
supermodular. Equivalently, supermodularity requires that
nondecreasing in x2 for
same of h(x",x2 )/h(x(-,x2 ).
all
x"
l
>jc,
,
and log-supermodularity requires the
Quasi-supermodularity (Milgrom and Shannon,
1994) requires
something weaker: the incremental returns, h(x" ,x2 )-h(x[,x2 ), must satisfy a single crossing
property.
The following
definition lays out several variants of single crossing properties
which
will
be useful in our analysis.
Definition 7 Let
X and
be partially ordered sets,
(i)
g(6) satisfies single crossing (SCI) in
6 if,
for all 6H>9L g(6L)>(>)0 implies g(d„)>(>)0. (ii) h(x,6) satisfies single crossing of incremental
returns (SC2) in (x;6) if, for all xH >x[/ g(0) = h(x H ;6)-h(x L ;9) satisfies SCI.
,
Topkis
(1978,
supermodularity.
introduced
1979)
Building
x*(0,i?) = argmax/i(x,0)
is
on
his
some
basic
work,
nondecreasing in
comparative
Milgrom
(0,2?) in the
and
statics
Shannon
"strong set
theorems
(1994)
order" 13 if
on
based
show
and only
if
that
h
is
xeB
quasi-supermodular in x and satisfies
SC2 in
(x;0).
If
xe 9?,
the result simply requires that
12
Karlin and Rinott (1980) called log-supermodularity multivariate total positivity of order 2.
13
A>B in the strong set order if,
for
any
aeA
and beB,
21
avb&A and ciAbeB.
h
satisfies
.
.
SC2
Thus,
in (x;0).
to satisfy
SC2
we
satisfy
when Primitives Form a Closed Convex Cone
subsection motivates the study of necessary and sufficient conditions for V(\,G) to
last
SC2
0,
numbers. In
when
Since this property does not concern interactions between components of x or
in (x;0).
components of
real
sufficient conditions for V(x,6)
in (x;0).
3.2 Stochastic Single Crossing
The
on necessary and
will focus special attention
we change
this context,
V(x H ,d)-V(x L
However,
in
notation to consider properties of V(x, 6), where the italic variables are
,d)
it is
= l[u(x
clearly equivalent to study
H ,s)-u(x L
,s)yiF(s;d) satisfies
we might wish
other contexts,
supermodularity of V(x,6). Thus,
when
we
to
study
V(x, 0) satisfies
SCI
SC2
SC2, and
to study
in 6.
of V(x,0) in (6;x), or quasi-
present a theorem which allows the roles of the utility function
and the probability distribution to be interchanged:
we
analyze J u(s)dK(s;6)
,
but do not restrict
K to
be a probability distribution.
Theorem 5 Suppose
Then the following two
that ccc(T)=G. Let K(;ff) be a signed measure.
conditions are equivalent:
(i)
V(6)= J g(s)dK(s; 6)
(ii)
For all
satisfies
6,^.QL , there exists a
SCI
in
6for all ge G.
X>0 such
jt(s)dK(s;6 H )>k jt(s)dK(s;d L ) for
that
all
teT(that
Proof: Observe that (a) V{ej>Q implies V(6H)>0 for
geK(s,6,)' implies V(6H)>0,
(The
latter equality
definitions,
and
if
and only
if (c)
follows because (^(s.flj
all
is,
K(s,6H)
ge G,
if
K(s,9H) e (K(s,0,)'
-
X-K(s,e,)
and only
if (b)
e T).
ge G and
n G)' = ccc(K(s,0,) u G').
u G*)* = ^(s,^* n G", simply applying
G is assumed to be a closed convex cone, so that G=G"). In turn, (c) holds if
=
^(s,^ + a m(s) for some m(s)e G*, and some a,A^0. But then,
g(s)dK(s;6 H ) =X j g(s)dK(s;6 L ) + a J g(s)dm(s). Since f=G' by Lemma 2, and
I
and only
J
if (d)
K(s,8H)
g(s)dm(s)>0 by definition of G\
Now suppose that A^O,
e
A,
f such that
J
and suppose
follows that J g(s)dK(s;0 H )
that
ViO^O for some ge G.
>X
J
g(s)dK(s;d L )
Then,
we can
g(s)dK(s;d H )=0. But fc=Q implies that K(s,0H)-X K(s,6H)e
we have V(0J>O and V(6H)=0
the definition of
it
for
some ^(s,^) where
tf(s,0w )-A K(s,0H)e
find a K(s,6H )
f as well.
T, contradicting
SC 1
Closely related results have been applied in the economics literature by a few authors;
interpret the theorem,
Then,
and then return
to discuss the literature in the next subsection.
22
we
first
Theorem 5
Sufficiency in
jt(s)dK(s;6 L )>(>)0,
it
(that
is, (ii)
implies
will follow that
(i)) is
Now
To
consider necessity.
where g
g,
subject to the constraint that j g(s)dK(s;6 L )>0.
infimum must be nonnegative. This implies
geG
it
will follow that there exists a
more
must
course, this
G (a closed convex cone),
the single crossing property holds, the
if
Lagrange multiplier X such
exist a
Xj g(s)dK(s;6 L ) >
0.
all
geG.
equivalent to checking that K(s;6 H )-XK(s;6 L )sG'.
is
that
X such that
jg(s)dK(s;d H )-Xjg(s)dK(s;0 L )>O for
Of
Taking convex
familiar form, consider taking the
restricted to lie in
Then,
that there
inf jg(s)dK(s;6 H )-
Then
is
then whenever
yields the implication required.
r's
recast the result in a
with respect to
j g(s)dK(s;6 H )
If (ii) holds,
j t(s)dK(s;6 H )>XJ t(s)dK(s;6 L )>(>)0.
combinations and limits of sequences or nets of these
infimum of
straightforward.
we
Finally,
can apply
Lemma 2, which establishes that G* =T* when ccc{T)=G.
Theorem
Clearly,
Theorem
that
1
F
is
5 and
Theorem
A such
that
are very closely related.
a probability distribution and apply
monotonicity requires that K(s,6H)
positive
1
K(s,6H)
- K(s,6,) e
In contrast,
T'.
- X-KfaGJ e T\ The
latter
we
If
relax the restriction in
to the
problem above, stochastic
Theorem
5 requires that there exists a
it
condition
is less stringent,
and yields a
weaker conclusion (single crossing as opposed to monotonicity).
However,
Theorem
1
G contains the constant functions and K is a probability distribution,
if
and Theorem 5 coincide. In
Corollary 5.1 Suppose that
constant in
6.
{
1,-1 }e
V(6)=j g(s)dK(s; 0)
(ii)
J
t(s)dK(s;d)
Proof: Clearly,
.
we have the following
G and G=ccc(T\j{ 1,-1}).
If
(ii)
is
SCI
satisfies
nondecreasing
implies
(i),
since
8 for all
in
(ii)
This result
if and
j
dK(s; 6)
is
is
then
it
must follow
that
if
implies that condition (B) of Theorem 5 holds
{ 1,-1 e G, then condition (B) of Theorem 5 requires that
6,
only
te T.
}
constant in
all
corollary:
Suppose further that
6 for all ge G,
in
and jdK(s;0 H )<X jdK(s;8 L ). This holds only
holds for
turns out that
Then
(i)
fc= 1
particular,
it
if
j dK(s;d H )=X
J
when
dK(s; 6 H )>lj dK(s; 9 L )
jdK(s;d L ). 1ffdK(s;0)
is
X=l.
potentially quite powerful, since
payoff functions in the desired class
if
it
implies that in
and only
if
many problems,
the probability distribution
according to stochastic dominance. Thus, Corollary 5.1 reduces a potentially
23
single crossing
more
is
difficult
ordered
problem
F is a probability distribution,
g(s)dF(s;d) satisfies SCI for all g nondecreasing, if and only if F is ordered according to FOSD.
J
Likewise, J g(s)dF(s;6) satisfies SCI for all g concave, if and only if F is ordered according to
to
a well-studied problem.
SOSD. Below, we
For example,
if
will apply this result to comparative statics problems.
Relation to the Literature
3.2.1
The approach used
in our formal
proof of Theorem 5 builds on the approach taken in a note by
"more
Jewitt (1986) to characterize the notion of
(1995 a, b) also prove
"diffidence theorem."
satisfies
SCI
in
this result for the
The
Gollier and Kimball
case where seSR or seSR and s independent. They call the
Gollier and Kimball's diffidence theorem can be stated in our language
for all densities/, if
[\h{s)k{s,6l)ds
risk averse" (Ross (1981)).
2
as follows (taking h, a density, as given,
>X
implies that
it
- k(s,6,)]
a.e.
and assuming
and only
that seSR): ]k(s,6)h{s)ds
if there exists
a
X>0 such
-
\k{s,6)f{s)ds
that $h(s)k(s,6H)ds
— k(s,dH)
We can place their result in the above framework as follows.
If
we
some probability density h(s) and consider G={g:g=h-f for some probability density/}, then we
can let T(Gh ) = {g:g(s)=h(s)-l (a _ ea+e) (s) for £>0, aeSR}. When we restrict attention to a single
fix
random
variable, their result is equivalent to
Theorem
6.
Their method of proof
is
different
from the
one used here.
Gollier and Kimball
show
wide variety of problems
in the- theory of investment
under
uncertainty can be usefully studied with the diffidence theorem, and their papers provide
many
that a
examples and applications. To see the simplest example where the diffidence theorem applies,
index preferences.
Then SCI of
jh(s)k(s,G)ds
—
y{s)k(s,6H)ds
is
let
6
interpreted as follows: if an agent
with preferences 6L likes the gamble corresponding to density h better than gamble corresponding to
/, all
agents with preferences 6^>6L will also prefer gamble h to
The
contribution of Theorem 5 thus
is to
gamble/
provide a statement and formal proof of the theorem at
the appropriate level of generality for our purposes, using the basic theorems of topological vector
spaces, and then derive theorems
Thus,
we
which
establish
its
consequences for comparative
statics analysis.
extend the domain of applicability of the tools, and show that these tools can be usefully
connected to Milgrom and Shannon (1994)'s methods for comparative
interest is Corollary 5.1,
which does not appear
statics analysis.
in the previous literature; this corollary,
Of particular
which gives
conditions under which stochastic dominance and stochastic single crossing are equivalent, will
imply that in many economic situations, comparative
stochastic objective cannot
be improved upon. This
24
statics
is
theorems based on supermodularity of a
taken up in the next section.
Single Choice Problems
3.3
and Comparative
Statics
SC2
This section examines conditions under which V(x,6f)=j u(x,s)dK(s;0) satisfies
which
necessary and sufficient for comparative statics in univariate choice problems.
is
allow for
Since
we
K to be a signed measure (not necessarily a probability distribution), our analysis applies to
problems where the choice variable affects either a probability distribution, or a
Let
Corollary 5.2
9t*
in {x;6),
K be a finite signed measure,
—>SK such that ccc(Ui)=ccc(T),
and let
utility function.
Uu T be sets of payofffunctions mapping
and let U ={u:Xx3i —»SR: X a partially ordered set,
k
and
2
u(xH ,s)-u(xL ,s)e Ut for all xH >xL ).
(A) x'(6)=aigmaxA u(x,s)dK(s;6)
A>0 such
is
nondecreasing
in
8 for all ue
U
2,
if and
only
a
if there exists
that j t(s)dK(s;0 „)>X jt(s)dK(s;0 L ) for all te T.
(B) Suppose that
F is a probability distribution, and that
x\ff)=argmaxI u(x,s)dF(s;6)
\
nondecreasing
in
6 for all
is
nondecreasing
in
{
1,— 1 }e
6 for all us
U
U
2,
x
Then
.
if and
only
if
jt(s)dF(s;6)
is
te T.
Proof: Follows from Theorem 5 and Milgrom and Shannon (1994), except for the following
caveat:
Milgrom and Shannon (1994) show
x'(d,B)
is
because
that
SC2
is
necessary for the conclusion that
nondecreasing in (x,B). The quantification over
we
B can be dropped
are asking that the comparative statics result hold for
all
ue
our case
in
and
U^,
thus,
effectively, for all constraint sets B.
To
see
how
this result
could be applied, consider
choose between two distributions.
first
part (A), and suppose an agent
We wish to characterize how the choice of distributions
must
changes
with a parameter of the agent's preferences. Let/be a density for the probability distribution F(s;x),
and suppose
that
that
x indexes
de{dudH }. Then
and only
if
Corollary 5.2 (A)
some A>0 such
there exists
state the condition
F according
that g(s,6H)
is
nondecreasing function: that
distributions
J
result is
due
to
is
consider
nondecreasing
us that
J
F^^^fajxJ.
s.
Suppose for the moment
g(s;d)f(s;x)ds satisfies
is
SCI
that g(s,0H)— Xg{s,6,) is nondecreasing in
is
"more nondecreasing" than
to
SCI
That
g(s,0,).
If
for all such F, if
Another way
s.
g(s,6,)
to
and some other
we
consider instead
Second Order Stochastic Dominance, the corresponding
for all
is,
geG,
g(s,6H)
is
if
and only
if
there exists
some A>0 such
"more concave" than gis^J.
Ross (1981), and Jewitt (1986) showed how the method used
used to establish Ross'
Now
g(s,6H)
concave in
so that
must be a convex combination of
g(s;6)f(s;x)ds satisfies
gis^-Xgis^J
FOSD,
tells
which are indexed according
result is that
that
is,
to
in
The
latter
Theorem 5 can be
result.
part
(B)
of the
Corollary.
Checking
that
][u(x H ,s)-u(x L ,s)]dK(s;0)
equivalent to solving the stochastic dominance problem analyzed in
25
Lemma
is
1.
.
.
Examples of payoff functions
u(x,s)
such that the set Ui={g: g(s)=u(xH ,s)-u(xL ,s) for some x„>xL }
the conditions of Corollary 5.2 (B) include supermodular payoffs and payoffs
satisfy
u(xH ,s)-u(xL ,s)
nondecreasing and concave.
is
techniques, that the stochastic
where
Ormiston and Schlee (1992) show, using different
dominance orderings are necessary
for comparative statics in a
few
specific classes of payoff functions; thus, Corollary 5.2 (B) identifies the general principle behind the
results.
To
how
see
this result
Suppose
(1996).
might be applied, consider a noisy signaling game, as studied by Maggi
two
that there are
The
the quantity to produce.
first
firms, a first
mover has
y,
noisy signal
of the incumbent's choice.
(z)
private information about his
mover chooses a quantity
After observing
the first
mover and a second mover, and each firm chooses
(as is usual in
such models) that
is
7I2
when
marginal cost,
y.
but the second mover observes only a
The second mover forms a
that 712(^1,^2) is the expected profit to the entrant
suppose
(#,),
own
posterior F{q
the quantities chosen are
{
\z).
Suppose
q and q2 and
,
x
submodular. Then the second mover's best policy
is
given by
q2 (z)=aig max f n 2 {q q)dF(q
x ,
I
l
z)
1
Then, Corollary 5.2 (B) establishes that q 2 {z)
ordered according to
if F(qi\z) is
mover's
profits
mover's policy
be given by
is
is
nonincreasing in z for
FOSD. Now
consider the
and assume
Ttiiqi.q^f),
first
all t^
submodular,
mover's problem.
that this function is submodular.
if
and only
Let the
first
Then, the
first
given by
?i(7)=arg max J 7T, (q, q 2 (z), y)dG{z\ q)
1
Then, by Corollary 5.2(B), #,($
and only
if
function
is
and G{z\q)
is
is
nonincreasing for
ordered by
all
FOSD. By Athey
nondecreasing, a pure strategy
7C,
submodular and for
all
q2 nonincreasing,
if
(1997), whenever each player's best response
Nash equilibrium
exists in this
game
(also see
Vives
(1990) for an existence result for supermodular games with incomplete information).
While
it
might seem that most economic problems
fall
under the domain of Corollary 5.2 (B), the
following subsection discusses a class of payoffs which does not satisfy the relevant assumptions.
3.3.1
It
Single Crossing at a Point and the Portfolio Problem
might
at first
seem
that
most
sets
of payoff functions which arise in economics contain the
constant functions. However, portfolio theory motivates a class of payoff functions which does not:
the class of functions
which cross zero
at
a fixed point.
optimization problem:
26
In particular, consider the following
.
max
xe[OM
This
Jf
u(xs + (1 - x)r)dF(s; 6)
the standard portfolio problem.
is
In this problem, the marginal returns to investment for a
When
given realization of s are given by (s-r)u(xs+(l—x)r).
crosses zero from below, and
satisfies
weak
always crosses
at the
same
is
is
nondecreasing, this function
Let us say that a function g
point: s=r.
Notice
single crossing at s if s>(<)s implies g(s)>(<)0.
crossing (SCI) functions
crossing at s
it
utility is
that,
while the set of single
not a closed convex cone, the set of functions which satisfy
weak
single
a closed convex cone. Further, observe that a set of "test functions" for this set of
functions can be described as follows (for
some P>0):
T(s ;$)={t: BasSi, £,5<|3 such that t(s)=(l-2l{s<s })(l{min(a-e,s )<s<max(a+£,s )}+6)}
Notice that the constant functions are not in
this set
of test functions, and thus Corollary 5.1 and
Corollary 5.2(B) do not apply.
Athey (1998) shows
can be analyzed using a
crossing about s
approach
is
respect to
problems where payoffs (or incremental returns) are weak single
that
different,
and seemingly unrelated,
set
of tools.
based on the fact that single crossing properties are preserved when integrating with
While the
log-supermodular densities.
statistics
introduction to Karlin and Rinott (1980)) has typically
(for
literature
drawn sharp
distinctions
example, see the
between methods
based on convex cones, and methods based on "majorization" (Marshall and Olkin, 1979),
now briefly
The
establish a connection
Consider the following
result, also established in
(i)
(ii)
J
for all
is
Athey (1998) using a different method of proof:
nonnegative,
and suppose further
that k is continuous
and
Then the following are equivalent:
6.
g(s)k(s; 6)ds satisfies
For all 6H> 6U
will
between the two approaches.
Corollary 5.3 Fix s and suppose k(s;6)
strictly positive at s
we
SCI for all g which satisfy weak single crossing about So.
k(s; 6„)/k(s; 6L)-k(s
;
6H)/k(s ; 6L)
satisfies
weak single crossing about s
almost everywhere.
Proof:
By Theorem 5 and using the
SCI
j g(s)k(s;6)ds satisfies
if there exists
almost
all
s<s
A>0 such
.
that
for all
set
of test functions rCs ;P) for
g which
satisfy
fc(.s;0w )>AJfc(.s;&)
This implies that
£(•?<>;
weak
(3
arbitrarily small,
single crossing about s
for almost all s>s
,
and
,
if
and only
Jfc(.s;0w)<A.ifc(.s;ft,)
for
fl^^Cs,,;^. Substituting and applying the definition
of single crossing gives the desired conclusion.
To
interpret condition
(ii)
of Corollary 5.3, observe that k(s;8H)/k(s',dL)
the realization s of the risky asset. Condition
point, the likelihood ratio is greater than
below the crossing
(ii)
it is
is
a "likelihood ratio" for
requires that for realizations of s above the crossing
at the crossing point.
Likewise, for realizations of s
point, the likelihood ratio is lower than at the crossing point.
27
.
To connect
the
about log-supermodular densities, notice that
this result to results
crossing
of
point
g
advance,
in
k(s;dh )/k(s;6L)-k(s ;dh )/k(So,6L) satisfies
point, s
.
But
weak
then
we
is
forced
we do
not specify
check
to
that
for every possible crossing
single crossing about s
checking that k(s;0h )/k(s;6L )
that is equivalent to
be
will
if
nondecreasing in
s,
or that
it
is
log-
supermodular. Equivalently, k satisfies a monotone likelihood ratio property. Thus, despite the fact
that neither single crossing
nor log-supermodularity are closed convex cone properties, Theorem 5
can be used to establish a result relating the two properties.
To
see
will invest
how
this result
more
We conclude that an agent
can be applied, return to the portfolio problem.
in a risky asset in response to an increase in 6, if
asset returns according to condition
(ii)
in Corollary 5.3,
when
and only
s =r.
if
6
shifts the density
of
Athey (1998) provides further
applications.
Multivariate Choice Problems
3.4
and l-Supermodularity
This section derives comparative statics properties of stochastic objective functions
decision-maker's choice set
question of
when U(x,&)
we need to
introduce two
is
a lattice (for example, a vector of real numbers).
is
quasi-supermodular in
new
x.
Theorem
In order to apply
when
Consider
first
the
the
5 to this problem,
properties.
8 For a given Ae 9?„ consider a function h:X—>S{ (X a lattice). Then h is l-supermodular
at (x,y) with parameter X if h(xvy)-h(x) > A[/i(y)-7i(XAy)]
Ifh:XxQ->% where is a partially
ordered set, then h{x,9) satisfies l-increasing differences (l-ID) on YxT with parameter X if, for all
x„>xte Y and 0„>6>te V, h(xH ,dH)-h(xL ,dH) > XlKx^-hix^O,)]
Definition
.
the
It is
clear that /-supermodularity implies quasi-supermodularity,
two
definitions. Further, if h is
some A>0 such
that
h
is
which follows immediately from
bounded and quasi-supermodular, then
/-supermodular at (x,y) with parameter A.
supermodularity, as opposed to quasi-supermodularity,
/-supermodular with parameter
A
at (x,y), the
for each (x,y), there exists
But, a critical property of
for a fixed A,
is that if,
two functions
convex combination of the two functions
are both
will inherit
the /-supermodularity property. In other words, the definition of /-supermodularity gives us a
make two quasi-supermodular
functions comparable, so that
we may
/-
way
to
take convex combinations
without upsetting quasi-supermodularity.
We will
say that the function
some A>0 such
is
/-supermodular on a sublattice
that the function is /-supermodular at (x,y) with
Y if,
for each x,y
e
Y, there exists
parameter A, and likewise for /-ID.
Clearly, supermodularity is stronger than /-supermodularity, since the definition is satisfied for A=l.
However, /-supermodularity allows for some
points.
flexibility in the scaling
of a function
at different
For example, consider the sublattice y={(0,0),(0,l),(l,0),(l,l)} and a function h(x) which
28
satisfies increasing differences in (x \x2 )
t
(the
on {0,1}x{0,1}.
same) affine transformation of both h(0,0) and h(l,0), increasing differences
preserved, while /-I.D. in
(jc,;x2 )
When
attention
is
restricted to a four-point sublattice
this constant
not necessarily
h(-,Q)
does
x,.
{x,y,xvy,XAy}), log-supermodularity
(i.e.
To
also stronger than Z-supermodularity for positive functions h.
However, since
is
an affine transformation of
Intuitively, taking
will be.
not affect the sign of the incremental returns to
X
transformed by taking
If the function is
depends on the choice of x and
y,
we
see
this,
let
is
X=h(x)/h(xAy).
cannot necessarily find a single
for all (x,y) pairs in a given sublattice.
The following
result characterizes quasi-supermodularity
stochastic problems (where, for a distribution K,
with respect to
ji
and k
is
Theorem 6 Consider a
jx is
and the single crossing property in
a measure such that
K is
absolutely continuous
the density).
Let k:Xx9T—>9t. Suppose ccc(T)=G.
lattice X.
(A) The following conditions are equivalent:
(i)
For every y,zeX,
supermodular at
(ii)
(B) Let
there exists a
(y,z) with
A>0 such
parameter X
V(x)=j g(s)k(x,s)dn(s)
all teT, j t(s)k(x,s)diLl(s) is
for
that,
l-
.
quasi- supermodular in x.
is
Qbe a partially ordered set, and let fc:Xx0x9T—>%.
The following two conditions are
equivalent:
For every subset Y={xH ,x L }cX such
(Hi)
that
6H>6L there
differences on
(iv)
exists
,
YxT
a X>0 such
that,
that
for
xH>x L and every subset V= { 6H ,dL )
,
all teT,
J
C0 such
t(s)k(x,8,s)dfl(s) satisfies l-increasing
with parameter X.
V(x,8)=j g(s)k(x,e,s)dfi(s) satisfies
Proof: Consider part (A). V(x)
is
SC2
in (x;6).
quasi-supermodular in x
if
and only
if,
for every sublattice
F={y,z,yAZ,yvz}cX, Jg(s)[£(y vz,s)-%s)]^(s)>(>)0 implies
I
g(s)[k(z,s)-k(y Az,s)]d^i(s)>(>)0. Applying Theorem
there exists a
for all te T.
X>0 such
But
that
J
f(s)[fc(y
The comparative
statics
To
all g<=
G,
if
and only
if (i)
interpret this result, consider first the case
suppose
n
is
supermodular
Lebesgue.
at (y,z)
Then
(i)
requirement that
is
and
where
that
(iii)
J
and only
t(s)k(x,s)d^i(s) is
if
is
are satisfied.
G is the
X for almost all s. The
29
l-
x\ff)^aigmax x€B j g(s)k(x,6,s)dfi(s)
set
of
all
requires that for y,zeX, there exists a
with parameter
if
analogous.
at (y,z) for all te T. Part (B) is
consequence of Theorem 6
nondecreasing in (6,B) for
holds
v z, s) - k(y, s)]djiz(s) >Xj t(s)[k(z, s) - k(y a z, s)]d/i(s)
this in turn is equivalent to the
supermodular with parameter X
5, this in turn
probability densities, and
X>0 such
order of the quantifiers
that k(x,s)
is critical:
is l-
there
must be a single X for almost
supermodular in
all s.
As we discussed above, one
Since sums of supermodular functions are supermodular, V(x) will clearly be
x.
supermodular in x as well, a stronger result than
To
see another example, suppose that k
the set of nondecreasing utility functions,
is
and
required.
is
we can
dH
,
^>0 such
we wish
that F(s;xL ,6H )-F(s;xH ,6H)
H
Hh(s)-Hl (s), where
is
to
Then
is
check whether our choice of distribution
is
on the cumulative
to a
change
in
x
at
ordered by
6L
.
all
for all
xH>xL and 6H>6L
Alternatively,
s.
distribution of increasing
x
at
equal to a convex combination of F(s;xL ,6L)-F(s;xH ,dL) and a difference
FOSD
(Hh(s)<Hl(s)).
requiring that the shift in the distribution due to a change in
due
requires that, for
(iii)
> X[F(s;xL ,dL)-F(s;xH ,6L )]
interpret the condition as follows: the effect
F{s'^c L ,6H )-F{s'^c„,0H ), is
G
a density for the probability distribution F(s;x,0),
nondecreasing in a parameter of the distribution.
there exists a
sufficient condition is that k is
It is
Thus,
x
we
at Q„ is
interesting to return to the result of
can interpret the result as
"more
FOSD"
Theorem
1
than the shift
as applied to the
property supermodularity and the set of nondecreasing functions: a necessary condition for V(x,ff) to
be supermodular for
all
g nondecreasing
is that l-F(s-jc,0) is
supermodular
in (x,6) for all
s.
Thus,
the necessary condition for comparative statics is a bit weaker. This result could be applied to study
how
4.
a worker's choice of effort (x) varies with different task assignments
(0).
CONCLUSIONS
This paper develops systematic tools for analyzing properties of stochastic objective functions, in
particular properties
which are relevant
to deriving comparative statics predictions.
introduces abstract definitions which allow parallels to be
as stochastic
drawn between
The paper
classes of theorems, such
dominance theorems and theorems which characterize other properties of stochastic
objective functions, in particular properties
which
arise in the study
of comparative
statics analysis
such as supermodularity, single crossing properties, and quasi-supermodularity. The main results are
then stated as mappings between two sets of functions (such as restricted classes payoff functions
and probability
distributions),
where the mappings provide characterizations of when a stochastic
objective function satisfies desired properties.
The
results
can be applied to
many economic
problems, including portfolio theory, investment games, and games of incomplete information.
30
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DD
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Date Due
Lib-26-67