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DEWEY
B31
M415
working paper
department
of
economics
Comparative Statics Under Uncertainty:
Single Crossing Properties
and Log Supermodularity
Susan Athey
No. 96-22 qW-P^K
Revised Version
July, 1996
July, 1998
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
Comparative Statics Under Uncertainty:
Single Crossing Properties
and Log Supermodularity
Susan Athey
No. 96-22 IW'P^K
Revised Version
July, 1996
July, 1998
MASSACHUSETTS
INSTITUTE OF
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
wggagUf
SEP l 6 1998
LIBRARIES
COMPARATIVE STATICS UNDER UNCERTAINTY:
SINGLE CROSSING PROPERTIES AND
LOG-SUPERMODULARITY
Susan Athey*
MITandNBER
1994
Last Revision: March, 1998
First Draft: June,
ABSTRACT:
This paper develops necessary and sufficient conditions for
monotone comparative
optimization problems.
statics
The
predictions
in
several
of
classes
stochastic
results are formulated so as to highlight the tradeoffs
between assumptions about payoff functions and assumptions about probability
distributions; they characterize "rninimal sufficient conditions" on a pair of
functions (for example, a utility function and a probability distribution) so that the
expected utility satisfies necessary and sufficient conditions for comparative statics
predictions. The paper considers two main classes of assumptions on primitives:
single crossing properties and log-supermodularity.
Single crossing properties
arise naturally in portfolio investment problems and auction games.
Logsupermodularity is closely related to several commonly studied economic
properties,
including decreasing absolute risk aversion,
affiliation
of random
The results are used to
extend the existing literature on investment problems and games of incomplete
information, including auction games and pricing games.
variables,
and the monotone likelihood
KEYWORDS:
monotone
ratio property.
Monotone comparative
statics,
single
crossing,
affiliation,
likelihood ratio, log-supermodularity, investment under uncertainty, risk
aversion.
*I
am indebted to Paul Milgrom and John
Roberts for their advice and encouragement.
I
would
further like to thank
Kyle Bagwell, Peter Diamond, Joshua Gans, Christian Gollier, Bengt Holmstrom, Ian Jewitt, Miles Kimball,
Jonathan Levin, Eric Maskin, Preston McAfee, Ed Schlee, Chris Shannon, Scott Stein, Nancy Stokey, and three
anonymous
comments. Comments from seminar participants at the Australian National
University, Harvard/MIT Theory Workshop, Pennsylvania State University, University of Montreal, Yale
University, the 1997 summer meetings of the econometric society in Pasadena, the 1997 summer meeting of the
NBER Asset Pricing group, are gratefully acknowledged. Generous financial support was provided by the National
Science Foundation (graduate fellowship and Grant SBR9631760) and the State Farm Companies Foundation.
Correspondence: MTT Department of Economics, E52-251B, Cambridge,
02139; email: athey@mit.edu; url:
referees for useful
MA
http://mit.edu/athey/www.
Introduction
1.
Since
Samuelson,
comparative
statics predictions.
attention (Topkis (1978),
(1994)), and
have
economists
studied
and applied systematic tools for deriving
Recently, the theory of comparative statics has received renewed
Milgrom and Roberts (1990a, 1990b, 1994), Milgrom and Shannon
two main themes have emerged.
First, the
new
literature stresses the utility
general and widely applicable theorems about comparative
emphasizes the role of robustness of conclusions to changes
Second,
statics.
The
comparative
results
statics
supermodularity,
1
In this paper,
that
arise
literature
economic theory
in
shows
a
that
comparative
many of
on three main
rely
literature
specification of models,
in the
searching for the weakest possible conditions which guarantee that
conclusion holds across a family of models.
this
of having
statics
the robust
properties:
log-supermodularity, and single crossing properties.
we
study comparative statics in stochastic optimization problems, focusing on
characterizations of single crossing properties and log-supermodularity (Athey (1995) studies
supermodularity in stochastic problems).
this paper.
The
Two
main classes of applications motivate the
results in
concerns investment under uncertainty, where an agent makes a risky
first
investment, and her choice varies with her risk preferences or the distribution over the uncertain
returns.
first
Second, the paper considers games of incomplete information, such as pricing games and
where the goal
price auctions,
is
to
which a player's action
find conditions under
nondecreasing in her (privately observed) type.
One
reason such a finding
is
useful
is
that
is
Athey
(1997) shows that whenever such a comparative statics result holds, a pure strategy Nash
equilibrium
(PSNE)
The theorems
will exist.
in this
paper are designed to highlight the issues involved in selecting between
different assumptions about a given problem, focusing
on the
tradeoffs
between assumptions about
payoff functions and probability distributions. The general class of problems under consideration
= f u(x,s)f(s,6)dju(s), where each bold variable is a vector
For
vector, and 6 represents an exogenous parameter.
can be written U(x,0)
of real numbers, x
represents a choice
several classes of
restrictions
on the
primitives (u and/)
sufficient conditions for
are nondecreasing with
The
analysis builds
which
a comparative
arise in the literature, this
statics conclusion, that is, the
on the work of Milgrom and Shannon (1994), who show
single crossing property in (x;0). In this paper,
1
A
positive function
conclusion that the choices x
6.
condition for the optimal choice of x, x*(0), to be nondecreasing in
not just for a particular
paper derives necessary and
utility
we show
if
we
is
that a necessary
that U(x,0) satisfies the
desire that comparative statics hold
function, but for all utility functions in a given class,
on a product
set is
supermodular
if
it
will often
be
increasing any one variable increases the returns to
increasing each of the other variables; for differentiable functions this corresponds to non-negative cross-partial
derivatives between each pair of variables.
A
supermodular. See Section 2 for formal definitions.
function is log-supermodular if the log of that function
is
necessary that U(x,&) satisfies a stronger condition than single crossing.
of U(x,0)
is identified, it
call
the desired property
remains to find the "best" pair of sufficient conditions on u
Most theorems
generate the comparative statics predictions.
what we
Once
"minimal pairs of
sufficient conditions": they
for the comparative statics conclusion,
and
in the
paper are stated
and/
terms of
in
provide sufficient conditions on u
further, neither
to
and/
of the conditions can be weakened
without strengthening the condition on the other primitive.
We begin by studying log-supermodular (henceforth abbreviated log-spm) problems.
primitives arise naturally in
spm
if
many
contexts: for example, an agent's marginal utility u'(w+s)
in (w,s) if the utility function satisfies decreasing absolute risk aversion,
demand becomes
A density is log-spm if
less elastic as e increases.
from auction theory or
if it satisfies
Log-supermodularity
is
Log-spm
the
monotone likelihood
it
and D{p;z)
is
log-
log-spm
is
has the property affiliation
ratio property.
a very convenient property for working with integrals, because
products of log-spm functions are log-spm, and
integrating with respect to a
(ii)
(i)
log-spm density
preserves both log-spm and single crossing properties. However, a critical feature for our analysis
of log-supermodularity and the single crossing property
in stochastic
problems
is
that (unlike
supermodularity), these properties are not preserved by arbitrary positive combinations.
The
first
main comparative
statics
theorem of the paper
by showing that log-supermodularity of U(x,0)
is
increase in response to exogenous changes in
6 for
establish necessary
and
sufficient conditions
is
established in
two
steps.
We
begin
necessary to ensure that the optimal choices of x
all payoffs
We
u which are log-spm.
on the density/ to guarantee
this conclusion:
then
/must be
log-spm as welL The results can be applied to establish relationships between several commonly
used orders over distributions
absolute risk aversion
We
is
further consider a
in investment theory; they
preserved
when independent
more novel
application,
can also be used to show that decreasing
or affiliated background risks are introduced.
game between
a pricing
multiple
(possibly
asymmetric) firms with private information about their marginal costs. The analysis characterizes
conditions under which each firm's price increases in
results
its
marginal cost. Finally,
we show
that the
can be used to extend and unify some existing results (Milgrom and Weber, 1982; Whitt
(1982)) about monotonicity of conditional expectations; these results are applied to derive
comparative
statics predictions
Our second
variable,
set
when payoffs
of comparative
statics
are supermodular.
theorems concern problems with a single random
where one of the primitive functions
log-spm.
properties,
We
satisfies
a single crossing property and the other
is
study necessary and sufficient conditions for the preservation of single crossing
and thus comparative
statics results,
under uncertainty.
The
results are
extended
in
order to exploit the additional structure imposed by portfolio and investment problems. Further, in
a
first
price auction with heterogeneous players and
under which a player's optimal bid
is
common
nondecreasing in her
values, the results imply conditions
signal.
Finally, in
problems of the form
]v(x,y,s)dF(s,0),
we
applied to signaling
The
characterize single crossing of the x-y indifference curves; the results are
games and consumption-savings problems.
results in this
paper build directly on a
set
of results from the
concerns "totally positive kernels," where for the case of
positivity
of order 2 (TP-2)
supermodularity
is
equivalent to log-spm.
preserved by integration,
is
strictly positive bivariate function, total
Lehmann (1955) proved
while
which
statistics literature,
and Rubin
Karlin
that bivariate log-
(1956)
studied
the
preservation of single crossing properties under integration with respect to log-spm densities.
number of papers
monograph presents
Following
multivariate functions,
integration.
number of
total positivity
though comparative
of order 2 (MTP-2)
MTP-2
Karlin and Rinott (1980) present the theory of
statics results are
statics.
Only a few papers
in
not
densities in their study of affiliated
on the preservation of
random
economics have exploited the
variables in auctions.
single crossing properties
over risk aversion and associate comparative
risk aversion
power
results
of
can be recast as statements about log-spm of a marginal
This paper extends the existing literature in several ways.
this
features of log-spm
in his analysis
he makes use of the
it is
in the
Jewitt (1987) exploits the
and bivariate log-spm
statics;
preserved
among them. Thus,
Milgrom and Weber (1982) independently discovered many of the
literature.
is
to
functions together with a
surprising that results developed primarily for other applications have such
study of comparative
sign changes
Ahlswede and Daykin (1979) extended the theory
showing that multivariate
variety of useful applications,
somewhat
this,
and Karlin' s (1968)
this relationship,
the general theory of the preservation of an arbitrary
under integration.
by
have exploited
in the statistics literature
A
work
of orderings
fact that orderings
over
utility. 2
The
results
from the
statistics
of basic mathematical properties of stochastic problems. This paper
literature identify a set
customizes the results for application to the analysis of comparative statics in economic problems,
especially
in
relation
to
the recent lattice-theoretic approach,
and develops the remaining
mathematical results required to provide a set of theorems about "minimal pairs" of sufficient
conditions for comparative statics results.
several of
results:
which themselves represent new
we
Each theorem
is
illustrated
results. Further,
we
with economic applications,
identify the limits of the necessity
formally identify the smallest class of functions that must be admissible to derive
necessary conditions.
The
structure available in
economic problems allows the stated conditions to be weakened, and they
applications are chosen to highlight the extent to
which the additional
motivate several extensions to the basic theorems that exploit additional structure.
In the economics literature,
many
authors have studied comparative statics in stochastic
optimization problems in the context of classic investment under uncertainty problems, 3 asking
2 See also Athey et al
(1994),
3
Some
who apply the sufficiency conditions
notable contributions include
Diamond and
to an organizational design problem.
Stiglitz (1974),
Eeckhoudt and Gollier (1995), Gollier (1995),
and Meilijson (1990), Meyer and Ormiston
Jewitt (1986, 1987, 1988b, 1989), Kimball (1990, 1993), Landsberger
questions about
how
an agent's investment decisions change with the nature of the risk or
uncertainty in the economic environment, or with the characteristics of the utility function.
paper shows
how many
of the results of the existing
framework with fewer simplifying assumptions, and
work
This
is
stochastic problems. 4 Vives (1990)
shows
suggests
it
also related to Vives (1990) and
literature
can be incorporated
some new
Athey (1995), who study supermodularity
by integration (which
that supermodularity is preserved
all
payoff functions.
Athey (1995)
theorems about monotonicity of ]u(s)f(s;Q)ds
analogous results about other properties "P" of ju(s)f(s;Q)ds
is
in
also
which are preserved by convex combinations (including monotonicity and
concavity), showing that
property which
in a unified
extensions.
follows since arbitrary sums of supermodular functions are supermodular).
studies properties
This
in G, in the case
preserved by convex combinations, and further u
However, the necessary and
conclusions emphasized in
paper
this
is
sufficient
correspond to
in
is in
where "P"
a closed convex
a
of
set
condition for comparative statics
a single crossing property, which
is
not preserved by
thus, different approaches are required in this paper. 5
convex combinations;
is
Independently,
Gollier and Kimball (1995a, b) have also exploited convex analysis, developing several unifying
theorems for the study of the comparative
The paper proceeds
as follows.
statics associated
with
Section 2 introduces definitions.
problems, while Section 4 focuses on single crossing properties
Section 5 concludes and presents Table
variable.
about comparative
statics
proved
risk.
in this paper.
1,
in
Section 3 explores log-spm
problems with a
which summarizes the
six
single
random
main theorems
Proofs not stated in the text are in the Appendix.
Definitions
2.
The following two
and
links
them
sections introduce the main classes of properties to be studied in this paper,
to comparative statics theorems.
single crossing properties, supermodularity
primitives,
turns out that
some of
and log-supermodularity,
arise
the
same
properties,
both as properties of
and as the properties of stochastic objective functions which are equivalent
comparative
2.1.
It
to
statics predictions.
Single Crossing Properties
We will be concerned with a variety of different single crossing properties,
Definition 1 Lef g:9?-»$R.
WSCl(t
),
ifg(t)<0for
(i)
all t<t
g(t) satisfies
,
detailed below.
weak single crossing about a point to, denoted
t>t while g(t) satisfies weak single crossing
and g(t)>Ofor all
,
(1983, 1985, 1989), Ormiston (1992), and Ormiston and Schlee (1992, 1993); see Scarsini (1994) for a survey of
the main results involving risk and risk aversion.
4 See also Hadar and Russell (1978) and Ormiston (1992),
used to derive comparative
5
The
statics results
who show
that the theory of stochastic
dominance can be
based on supermodularity.
single crossing property is only preserved under convex combinations if the "crossing point"
Section 5 and Athey (1998) for
more discussion of this
point.
is fixed;
see
(WSC1)
(ii)
if there exists
a
to
such that g
for all
and g(t)>Ofor
to'<t<t ",
= h(x H ;t) - h(xL ;t)
(iv) h(x,y,t) satisfies
The
if there exist t '<to"
two variables (SC2) in
SCI. WSC2(t
satisfies
to return to
(x;t)
after
1).
it
Weak SCI
becomes
is
variables,
SC2, requires
used to derive comparative
differentiable
SClforall
git)
PL
positive. Single
cross zero at most once, from below, as a function
it
all xh>Xl,
allows the
returns to
t;
for
g(t) crosses zero, at
two
of
if,
and WSC2 are defined analogously.
)
crossing in
jc
such that g(t)<Ofor all t<to, g(t)=0
single crossing in three variables (SC3) if h(x,b(x),t) satisfies
most once, from below (Figure
g
t
of SCI simply says that
definition
function
in
all t>t ".
(Hi) h(x,t) satisfies single crossing in
g(t)
satisfies WSCl(to).
(SCI)
g(t) satisfies single crossing
that the incremental
non-
statics results in
Figure
problems or problems which are not quasi-
Milgrom and Shannon (1994) show
argmax h(x,t) is monotone nondecreasing in t for
concave.
1:
g(t) satisfies
SCI
in
t.
that the set
B
if
and only
be applied repeatedly throughout the paper. 7
Finally,
all
if
h
satisfies
SC2
in (x;t); 6 this
xefl
result will
show
that for suitably well-behaved functions,
crossing property, that
2.2.
is,
hi(x,y,t)/\h2(x,y,t)\
SC3
Milgrom and Shannon (1994)
equivalent to the Spence-Mirrlees single
is
nondecreasing in
t.
Log-Supermodularity and Related Properties
For bivariate functions, supermodularity and log-supermodularity are stronger than SC2.
Supermodularity requires that the incremental returns to increasing
must be nondecreasing (rather than SCI)
that the relative returns,
h(x H ;t)/h(x L ;t)
in
,
t;
g(t)
x,
- h(x H ;t) - h(x L \t)
,
log-supermodularity of a positive function requires
are nondecreasing in
t.
Thus, as properties of objective
The
functions, both are sufficient for comparative statics predictions.
multivariate version of
supermodularity simply requires that the relationship just described holds for each pair of variables.
Topkis (1978) proves that
for
all i
The
a
&j. If h
is
order
h
twice differentiable, h
is
positive, then
definitions
partial
if
h
is
log-spm if and only
can be stated more generally using
>,
the
xvy = inf{z|z>x,z>y}
operations
and
is
"meet"
(v)
supermodular
if
log(h(x))
is
if
and only
supermodular.
lattice theoretic notation.
and "join"
XAy = sup{zjz<x, z<y}
(for
(a)
-^- h(x) >
if
are
9T with
Given a
defined
as
a requirement about every pair of choices of x.
the quantification over constraint sets.
class of models,
7
where
Shannon (1995)
Many of the comparative
statics
X and
follows:
the usual order, these
6 Notice that the comparative statics conclusion involves a quantification over constraint sets. This
is
set
theorems of
this
is
because
SC2
paper eliminate
Instead, the theorems require the comparative statics result to hold across a
that class is sufficiently rich so that a global condition is required for a robust conclusion.
establishes a related result for
of a nondecreasing optimizer.
WSC2,
namely,
WSC2 is necessary and sufficient
for the existence
.
maximum and component-wise minimum,
represent the component-wise
X together with a partial order,
a set
Definition 2 Let (X,>) be a
.
such that the set
is
A function h:X-^>SR
lattice.
A lattice is
closed under meet and join.
is
supermodular
v y) + h{x a y) > h(x) + h(y) h is log-supermodular (log-spm)^
for all x,y<=X, h(x v y) h(x a y) > h(x) h(y)
h(x
respectively).
.
for all x,ysX,
if,
it is
if
non-negative, 9 and,
•
Log-supermodularity arises
spm are both
in
many
comparative
sufficient conditions for
Shannon (1994)).
contexts in economics.
Further, observe that
A
log-spm.
elasticity,
demand
sums of supermodular functions are supermodular, and
DP (P,t)/D(P,t)
e(P) = P-
log-spm
in (w,s)
asset) if
and only
(where
if
-
,
consider
D(P,t)
is
F(s; 0))
hazard rate which
nonincreasing in
is
0.
is
and only
if
the joint density, /(s),
monotone likelihood
constant in
0,
F
and
f{s;0 H )j f(s;0 L )
is
ratio
has
a
density /,
nondecreasing in s for
price
the
if
U'(w+s)
is
if
that a vector of
A parameterized
(DARA).
1-F(s;0)
log-spm, that
is
is, if
Milgrom and Weber (1982)
random
variables vector s
is
log-spm (almost everywhere).
is
order (MLR).
primitives are
A marginal utility function
nondecreasing in
show
only
in
wealth and a represents the return to a risky
Finally, log-supermodularity is closely related to a
as
t.
and
if
In their study of auctions,
define a related property, affiliation, and
affiliated if
log-spm
is
initial
a natural property to study
some examples where economic
nondecreasing in
often represents
is
the utility satisfies decreasing absolute risk aversion
distribution F{s;8) has a
f(s; 0)/(l
w
Now
function
(Topkis (1978); Milgrom and
statics predictions
products of log-spm functions are log-spm; thus log-spm
multiplicatively separable problems.
supermodularity and log-
First,
When
H
>
known
the support of F(s;0), denoted supp[F], 10
MLR
then the
all
property of probability distributions
requires
is,/is
L , that
that
log-spm
likelihood
the
is
ratio
However, we wish
11
to define this property for a larger class of distributions, perhaps distributions with varying support
and without densities (with respect to Lebesgue measure) everywhere. 12
Definition 3 Let C{s;0 H ,0L ) =-j{F(s;0L )
F(s;0) according to the
dF{s\0)l dC(s;0H ,0L )
+ F(s;0h )). The parameter 0indexes
Monotone Likelihood Ratio Order (MLR)
is
log-spm for C-almost
all Sh>sl
mlR
if,
for all
the distribution
H >
L,
2
.
8 Karlin and Rinott (1980) referred to this property as multivariate total positivity of order 2.
9
We
include non-negativity in the definition to avoid restating the qualification throughout the
text.
Although
supermodularity can be checked pairwise (see Topkis, 1978), Lorentz (1953) and Perlman and Olkin (1980)
establish that the pairwise characterization of log-spm requires additional assumptions, such as strict positivity (at
least
throughout "order intervals").
10 Formally, supp[F]
=
$ F(s + e) - F(s - e) >
Ve > 0}
Note that the MLR implies First Order Stochastic Dominance (FOSD) (but not the reverse): the MLR requires
that for any two-point set K, the distribution conditional on K satisfies FOSD (this is further discussed in Athey
11
(1995)).
12 In particular,
absolute continuity of F(-;dH) with respect to F{-;6d
consequence of the definition, not prerequisite for comparability.
on the intersection of
their supports is a
Note
that
we have
not restricted
F to
be a probability
distribution.
when studying
This paper further makes use of an ordering over sets
function and
how
sets
of optimizers of a
they change with exogenous parameters, as well as in applications where
parameters or choice variables change the domain of integration for an expectation. The order
known
as Veinott's strong set order, defined as follows:
Definition
a
in
is
4 A
set
A
B
greater than a set
is
A and any binB, avb gA and
a /\b
the strong set order (SSO) iffor any th
>
in the strong set
gB.A
vl,
order (SSO), written A>B,
set-valued function A(t)
A
A(th)> A(tl).
set
A
is
if,
for any
nondecreasing in
is
a sublattice
if and
only
if
A>A.
If
of
a set A(r)
this set are
is
nondecreasing. 13
(SSO)
set is increasing
statics (see
nondecreasing in r in the strong set order, then the lowest and highest elements
in a,
This
and
Topkis 1978) and
Comparative
3.
in
[a^bj x [a^fcj
set
our analysis of Section
that
In this section,
assumed
to
we
\sfdfj. is
will focus
both
and further such a
,
study of comparative
in the
3.
of
functions
form
the
U(x,O)=\u(x,s)f(s,0)dfi(s).
u and /are bounded, measurable functions, bold variables
are real vectors of finite dimension, and
allow for the possibility that
arise
2
Log-Supermodular Payoffs and Densities
objective
Throughout the paper, we assume
a sublattice of SR
is
These properties
i=l,2.
bi,
Statics with
considers
section
Any
ju is
a non-negative
cr-finite
product measure.
a probability measure, but do not require
on problems where one or both of the
We
thus
it.
u and
primitives,
/,
are
be non-negative and log-spm. Applications to investment problems and pricing games
are provided in Sections 3.1 and 3.2, while problems with conditional expectations are considered
in
The question we seek
Section 3.3.
comparative
statics: that is,
when does
to ask in this context is a question about
the following condition hold?
x(6)= argmaxU(x,0)=\ u(x,s)f(s,9)d^(s)
By Milgrom and Shannon
(1994),
(MCSa) holds and
is
smaller than any other set in the strong set order,
optimum
(following
wish to allow for some generality
(MCSa) hold
class
for
of log-spm
all
u.
u
in
The
some
in the specification
of functions?
following result
is
the
first
is
(MCSa)
0.
nondecreasing in B,
Since an empty set
we do
Milgrom and Shannon
class
nondecreasing in
further x*
U is quasi-supermodular in x 14 and satisfies SC2 in (x;0).
existence of an
is
if
and only
if
always larger and
not state an assumption about the
In our context, however,
(1994)).
of the primitives. Thus,
We begin by
we
ask,
we
when does
considering the question for the
step in the analysis of this question.
Lemma 1
(i)
monotone
Suppose u and fare nonnegative, and uf>Ofor son a set ofpositive fi-measure. Then
(MCSa) holds for all u(x,s) log-spm, if and only if(ii) U(x,0) is log-spm for all w(x,s) log-spm
13 For
14 If
more discussion of the strong
set order, see
Milgrom and Shannon (1994).
X is a product set, U is quasi-supermodular if and only if it satisfies SC2 in
each pair (x&j).
Proof: If U(x,&)
(1994) show
must be quasi-supermodular, which Milgrom and Shannon
implies the comparative statics conclusion. Now suppose that U(x,0) fails to be
log-spm, then
is
Our assumptions imply U(x,0)>O. Consider first xe9? 2 Then,
an xw > xu, xw > x2l and#/ > 6l such that U(xihJC2h,0h)IU{xujcw,8h) <
log-supermodular for some
there exists
it
u.
.
U(xwJC2l,6l)IU{x\lJC2l,9l). Lety=U(xiLjC2L,0L)/U(x lHj:2L,&L)>O. Then,
and b(xi H) =
y.
Since log-supermodularity
is
let b{x{)
=
ifx&XiH,
1
preserved by multiplication, v{x,-)su{x,-)-b{x{)
is
= yU(xwJC2H,6H)IU(xi L jC2H,dH) < 1 =
= V(x lHr>C2L,0L)/V(XlLr>C2L,0L). ThuS, V{X\H^2L,6L) = V{XlL^2L,6L)
log-supermodular. But then, V{xwJC2h,6h)IV{xilJC2h,6h)
YU(XihJ2L,6h)/U(XilJC2L,0h)
while V(x\ HJC2h,0h)
statics conclusion.
< V(xil*K2h,0h), violating quasi-supermodularity and thus the comparative
The argument can be easily extended to multi-dimensional x.
From Milgrom and Shannon's
SC2
that U(x,0) is
much
result,
Lemma
follows that
it
log-spm.
in (x;0) for all w(x,s)
This
(ii) is
1
somewhat
is
equivalent to the statement
surprising since log-spm
is
a
stronger property than SC2. However, the result then motivates the main technical question
for this section:
Observe
when
U(x, 6) log-spm, given that one of the primitives
is
that characterizing
Thus,
immediate
is
not
ln(j"w(x,s)/(s,#)d/i(s)).
To
function.
literature,
which
it
will
log-spm of
U is
we
address this problem
be one of the main tools
log-spm?
non- trivial, since the function
properties
that
is
that
hold for ln(w)
< h 2 (s v
/^(s')
will
s') -fc 4 (s
h
(
statics.
While
we will use this result in a
(and immediate) consequence of
preserved by integration.
k(.s)
Then (L2.1)
=
To
g(y,s), h2(s)
see
=
spm, which makes
log-spm
this result
arbitrary sums, only to
variety of
(L2.1)
further explore a variety
statics is that
log-supermodularity
is
this, set
g(y',s), fh(s)
in y.
= g(y v y' ,s) and
fk(s)
,
log-spm
= g(yAy',s).
in (y,s), while (L2.2)
reduces to the conclusion
is
Recall that arbitrary sums of log-spm functions are not log-
somewhat
The preservation of log-spm under
surprising.
1
)
But notice
+ #(y>s
integration
2
is
),
that
Lemma
2 does not apply to
when g is log-spm in
all
arguments.
especially useful for analyzing expected
Since arbitrary products of log-spm functions are log-spm, a sufficient
condition for Jw(x,G,s)/(s,x,0)d//(s) to be log-spm
Karlin and Rinott give
They
(L2.2)
ways throughout the paper, the most important
2 for comparative
sums of the form gty.s
values of payoff functions.
statistics
though they do not consider the problem of comparative
Lemma
states exactly that #(y,s) is
that jg(y,s)d{i(s) is
for
(i=l,...,4),
as') forju-almost all s,s'e9T
Karlin and Rinott (1980) provide a simple proof of this lemma.
statistics,
hold
in this paper.
J/i l (s)rf/i(s)-J/22 (s^(s)^J^(s>/Ms)-J^(s)rfMs).
of interesting applications in
not a linear
introduce a result from the
Lemma 2 (Ahlswede and Daykin, 1979) Consider four nonnegative functions,
where h^.W —> 5R. Then condition (L2.1) implies (L2.2):
h\(s)
ln(-) is
many examples of densities
is that
k(x,0,s) and/(s,x,0) are
that are log-spm,
log-spm
and thus preserve log-spm of
a payoff function. 15
Despite the fact that log-spm of primitives arises naturally in
still
somewhat
hold.
we
Thus,
restrictive.
many economic
problems,
it is
consider the issue of necessary conditions for log-spm to
Before stating the necessity theorem
we
introduce a definition that will allow us to state
concisely theorems about stochastic objective functions throughout the paper. All of the results in
this
paper concern problems of the form ju(x,s)f(s,Q)d^(s)
find pairs of hypotheses about
thus look for what
Definition
and
,
in
such problems,
u and / which guarantee that a property of
we call a minimal pair of sufficient conditions,
we
wish to
£/(x,0) will hold.
We
defined as follows:
5 Two hypotheses H-A and H-B are a minimal pair of sufficient conditions (MPSC)
C if: (i) Given H-B, H-A is equivalent to C. (ii) Given H-A, H-B is equivalent
for the conclusion
toC.
This definition introduces a phrase to describe the idea that
contexts,
hypothesis
H-A
H-B
H-B
and
Definition 5 will hold
will
we
will search for the
is
is
no logical requirement
does, the main results of this paper satisfy
(i)
used to
weakest
all
that part
Remark Theorem
is sufficiently
«:9l' x
9T ->
and f
<R +
(x,0).
1 requires that
x has
is
:
SR" x
9T
-^
9? + ,
which
is
proved formally
Then
at least two components (1=2) ifn>2, so that the class of
even in cases where n>2 and 1=1, log-supermodularity off in
conclusion (Tl-C) to hold whenever (Tl-A) does.
1,
of
three parts of the
where
make log-supermodularity offin s a necessary
rich to
(ii)
state the following theorem.
log-spm in
ls u(x,s)f(s,Q)dfi(s)
Theorem
and
n>2 implies l,m >2.
is log-spm a.e.-^i; and (B)f(s,Q) is log-spm a.e.-ju; are a MPSC for (C)
Theorem 1 Suppose
(A) w(x,s)
be reversed. Though there
whenever
This definition
definition.
u),
(such as an assumption on f) which preserves the conclusion; in other problems,
the roles of H-A
u's
be given (such as an assumption on
will
are looking for a pair of
cannot be weakened without placing further structure on the problem. In
sufficient conditions that
some
we
(si,
0) is
condition. However,
necessary for the
in the appendix, states that, not only
do (A) and (B) imply
the conclusion, but neither restriction can be relaxed without placing additional restrictions on the
other primitive.
While
subsequent theorems
different conclusions.
in
we
Theorem
1,
comparative
develop the proof of Theorem
The proof of Theorem
15
on u and/ are the same (log-spm),
will pair different conditions,
Together with
sufficient conditions for
the conditions
1
1,
and
Lemma
in
our
such as single crossing and log-spm for
1, this result
statics conclusions.
can be used to provide necessary and
Before proceeding to that
result,
we
identify its limitations.
can be understood with reference to the stochastic dominance
For example, symmetric, positively correlated normal or absolute normal random vectors have a logsupermodular density (but arbitrary positively correlated normal random vectors do not; Karlin and Rinott (1980)
give restrictions on the covariance matrix which suffice); and multivariate logistic, F, and gamma distributions have
log-supermodular densities.
}
literature,
and more generally Athey (1995), which exploits a "convex cone" approach
Athey (1995) shows
characterizing properties of stochastic objective functions.
problems,
if
one wishes to establishes that a property
P
often necessary and sufficient to check that
it is
The extreme
P holds for
holds for
U(x,0) for
u
all
in the set
points can be thought of as test functions; for example,
nondecreasing functions, the set of
are nondecreasing in
test functions is the set
Athey (1995) shows
s.
that this
all
u
to
that in stochastic
in
a given class
71,
of extreme points of 7i.
%
when
is
the set of
of indicator functions for sets Li(s) that
approach
is
when P
the right one
a
is
property preserved by convex combinations, unlike log-supermodularity.
Despite the fact that the "convex cone" approach does not apply directly here, an analogous
intuition
can
still
be developed.
supermodular functions
to be a
is
Consider the hypothesis that the
necessary condition for Tl-C:
test functions satisfies
Lemma 3
only
if
/(s,0)
Be (x)
is
But,
(x;0).
it
remains to establish that
this set
1a(i)(s) is
is
log-spm in
log-spm in
(s,r) if and
s if and only if
A
.(svs')-l,,,
.(sas')^I,.,
^s)-1,,
,(s')-
only ifA(z)
is
a
of
nondecreasing
sublattice.
1a<i)(x) is
The
is
log-spm
in (s,z) if
definitions imply that if the
right-hand side equals one, the left-hand side must equal one as well. Likewise, 1a(s)
spm in s if and
defined
log-supermodular almost everywhere-n will
is
Proof: Simply verify the inequalities and check the definitions.
1..
where
Tl-A. The following lemma can be used:
(strong set order). Further, 1a(s)
if:
,
This set of test functions will certainly yield Tl-B as a
x.
be log-supermodular in
The indicator function
and only
of "test functions" for log-
the set of indicator functions of the form l Bt00
cube of length e around the point
Jslfli (x)(s)/(s,8)rf//(s)
set
only
if:
v s') -1 a (sas')>1^ (s)
1 A (s
•
1 A (s')
.
is
log-
Again, this corresponds exactly to
the definition.
It is
straightforward to verify that sets of the form x,[a,,&] are sublattices, as desired.
such sets are nondecreasing (strong set order)
role for the strong set order
Lemma 3
when
will also
be useful
and sublattices
in
many
each endpoint.
in
in the analysis
applications, for
Lemma
Further,
3 leads to an important
of stochastic optimization problems.
example
(as the pricing
game of Section
3.2)
the choice variables or parameters affect the domain of integration.
The analogy
to the "test functions" approach,
and thus the proof of Theorem
1,
can be made
complete with the following lemma.
Lemma 4
(Test functions for log-spm problems) Define the set
V(P)= {h:3x and 0<e</3 such that w(x,s) = 1 B<r (x) (s)
Then U(x,Q)=ju(x,s)f(s,Q)dju(s)
only if for some fi>0, U(x,Q)
Lemma 4
is
is
log-spm in (x,0) whenever u(x,s)
Theorem
monotonic, and thus Theorem
we
1
by
1
of u will change the conclusion of Theorem
In contrast,
is
log-spm
a.e.-/x., if
and
log-spm whenever ue7(J3).
establishes the limits of
nondecreasing.
.
1.
identifying
which additional regularity properties
For example, the elements of
7(/?) are clearly
not
does not hold under the additional assumption that u
can clearly approximate the elements of
10
1(J5)
is
with smooth
.
functions, so smoothness restrictions will not alter the conclusion of Theorem
Together,
comparative
comparative
Lemma
1,
Theorem
theorem of
statics
Lemma
and
1,
1.
3 can be used to establish the
main
first
the paper, which states necessary and sufficient conditions for
problems with log-supermodular payoff functions.
statics in
Corollary 1.1 (Comparative Statics with Log-Supermodular Primitives) Consider functions
m
m
w:M xM xW^W+f:M\Vi ->M +,Aj:V{-+2*\j=l,..,J. Let A(x) = r^. =1 y A'(r,). Letx(Q,x,B)
l
= argmaxxeB W(x,0,T)=JseA(T) w(x,Q,s)f(s;6)dju(s)
(1) (Necessary
Then
(2)
x* (9,1,1?) is nondecreasing in
6 for all
(General sufficiency) x\Q,x,B)
and, for each j,
Qe%
and sufficient conditions) Suppose A(x) is a sublattice,
A J (Tj)
is
if
and only iff is log-spm
nondecreasing in (Q,x,B)
if
wand fare
a.e.-/u
on A(t).
log-supermodular,
nondecreasing (strong set order) in x
r
Proof: (1) Necessity: by
supermodularity of
is
w log-spm,
and l>2 ifn>2.
Win
Lemma
(x,0).
1,
the comparative statics conclusion
Then we can apply Theorem
equivalent to log-
is
Sufficiency follows by: (2)
1.
Since products of log-spm functions are log-spm, the function
w(x,Q,s)f(s;Q)-l AAlr(r,)is)-LA ; ,
,<s) is
\Tj)
Note
log-spm
not divided by the probability that se A(x). Further, observe that (1)
of a minimal pair of
if
choice variables. In general, log-spm
While Corollary
The reason
sufficient conditions.
conditions for comparative statics
we
quantify over
is sufficient,
1.1 is straightforward
is
all
that
Lemma
1
is
not stated
in
in
terms
only yields necessary
payoffs «(x,s), where u contains the
but not necessary, for
MCSa.
given the building blocks outlined above,
problems a structure commonly encountered
it
shows
that
economic theory can be analyzed by checking a
few simple conditions. The following applications
illustrate the
theorem.
Applications to Investment Under Uncertainty
3.1.
An
immediate implication of Corollary
commonly used
in the literature
integrating a function
also
Lemma 3.
by
that the objective function analyzed in Corollary 1.1 is not a conditional expectation, since
we have
(that
in (s,x,G,x)
is,
the density
be
log-spm, 16
is
makes
it
More
is
on investment under
more
likely to
that
a ratio orderings over distributions,
uncertainty, can
be log-spm.
easily
compared: loosely,
If a distribution satisfies the
which can be shown to be stronger than
Dominance
generally,
be
MLR order
log-spm), then the corresponding cumulative distribution function F(s;0) will
This will in turn imply that
Stochastic
1.1
if
we
J^ F(s; G)ds
First
log-spm, which
is
is
Order Stochastic Dominance.
stronger than Second Order
#does not change the mean of s.
can consider any
nondecreasing in s for g,h>0. The easiest
16 See Eeckhoudt and Gollier (1995)
who term
way
ratio
ordering
to
this into
this the
fit
which
our framework
monotone probability
ratio
MPR shift is sufficient for a risk-averse investor to increase his portfolio allocation.
11
specifies
(MPR)
is
g(s)/h(s)
is
to define u(x,s)
on
that
order,
and show
that an
{0,1 }x<R,
and
let u(x,s)
=
g(s)
if jc=1,
and
let u(x,s)
g(s)/h(s) is nondecreasing in s, E[g{s)\6]IE[h{s)\6\
1
shows
that
=
h(s)
Then, ]ff(s;6)
if jc=0.
must be nondecreasing
log-spm of/ is the weakest condition which preserves every
An example from the economics
of uncertainty
Further,
in 9.
the Arrow-Pratt coefficient of risk aversion,
is
log-spm
if
and only
response to an
if
R(w;u)
risk.
w and
functions of Lemma 4, so necessity cannot be obtained using
are preserved with respect to
it
then follows from
background
Theorem
prudence condition, 17 where prudence
and related properties
If R(w;u) is decreasing,
be decreasing as well, implying that R(w;U)
is
defined by -u"\w)lu'\w). li
More
consider orderings of utilities over risk aversion (not just those induced by a
parameterize the agent's risk aversion directly, and
R(w;u(-;0))
is
w
nonincreasing in
nonincreasing in
statistically
test
is
Similar analyses apply to other ratio orderings, such as Kimball's (1990) decreasing
decreasing.
Finally,
considering a
for risk averse agents.
1
= \u(w+s)f{s)ds.
Let U(w)
risks.
that U'(w+t) will
1
Theorem
that orderings over risk aversion
is
when
u additively precludes the
s enter
is
then implies that in
1
the agent will be less risk averse
But, observe that the fact that
A further consequence of Theorem 1
(DARA) and
lu(w+s)f(s;&)ds, and observe that u'(w+s)
w (DARA). Theorem
nonincreasing in
is
MLR shift in the risky asset s,
new, independent
=
Let U(w;0)
Theorem
ratio ordering.
R(w;u)= -u"(w)/u'(w). Suppose that an agent has decreasing absolute risk aversion
faces a risk ^ with distribution F(s;0).
log-spm and
is
6 and
Theorem
1
w.
Other
and
U
0,
ratio orderings
allows us to consider
let
U(w,&)
=
Then
so that R(w;U(-;0))
if
is
can be similarly analyzed.
more general background
dependent on the risk of primary
can
Let 6
shift in wealth).
ju(w+s;0)f(s)ds.
these properties,
inherits
we
generally,
risks,
those which might be
Let z be the "primary"
interest.
risk,
and
let s
be a
vector of assets in which the agent also has a position, represented by the vector of positive
portfolio weights a.
The
agent's
utility is
given by U(z, 6)
=
\u(z+a-s;0)f(s\z)ds.
risks are affiliated, that is,/(s,z) is log-spm. But, applying the multivariate
risk aversion with respect to z (that
and
decreases risk aversion.
An
conditional density of y=a-s given
is,
Suppose
Theorem
R(r,U,0)) will be nonincreasing in
6, if
u
1,
that the
the agent's
satisfies
DARA
alternative, but distinct sufficient condition requires that the
z, g(y\z), is
log-spm.
This analysis thus provides several extensions to the existing literature. Jewitt (1987) showed
that the
"more
risk averse" ordering is preserved
by a
MLR
shift in
the distribution, while Pratt
(1988) established that risk aversion orderings are preserved by expectations. Eeckhoudt, Gollier,
and Schlesinger (1996) provide additional conditions on distributions under which a
a background risk decreases risk aversion for a
17
Kimball (1990) shows that agents
savings"
18
It is
results
when
who
are
DARA
more prudent
investor.
will
engage
in
The above
FOSD
shift in
analysis generalizes
a greater amount of "precautionary
anticipating future uncertainty.
interesting to observe that since a
smooth function g(x+y)
can also be proved using the fact that log-convexity
is
is
supermodular
1979); but, the correspondence between convexity and supermodularity breaks
12
if
and only
if it is
convex, these
preserved under expectations (Marshall and Olkin,
down
with
more than two
variables.
.
these results by showing that
and further
all
.
of the results extend to other
ratio orderings,
such as prudence,
DARA
are preserved by affiliated background risks for
risk aversion orderings
investors.
Log-Supermodular Games of Incomplete Information
3.2.
This section establishes sufficient conditions for a class of games of incomplete information to
have a
PSNE
in
Consider a game of incomplete information between
nondecreasing strategies.
many players, each of whom has private information about her own
XiU).
Vives (1990) observed that the theory of supermodular games (Topkis, 1979) can be
applied to
games of incomplete information.
supermodular
in
is,
true that the
if
we
each player's payoff
is
i>,(x)
has strategic complementarities in
PSNE exists.
consider the same problem but assume that
game has
V;(x) is
log-spm,
strategic complementarities in stategies in the
different approach:
games of incomplete information where an
xfoFy^Xfot)), whenever
Formally, suppose that h(t)
v,(x,t) (so that
game
the
This in turn implies that a
However, we can instead take a
(i.e.
that if
a pointwise increase in an opponent's strategy leads to a pointwise increase
in a player's best response).
In contrast,
He showed
the realizations of actions x,
strategies %,{tj) (that
type
and chooses a strategy
type, U,
is
is
of her opponents use nondecreasing
all
that a
PSNE
nondecreasing
in
exists in
her own
strategies.
Let player
the joint density over types.
no longer
sense just described.
Athey (1997) shows
individual player's strategy
it is
/'s utility
be given by
opponents' types might influence payoffs both directly and indirectly, through the
opponents' choices of actions). Thus, a player's payoff given a realization of types can be written
Ui(Xi£)=v,{Xi,%-i(t-i),t).
for expected payoffs:
Taking the expectation over opponent types yields the following expression
£/,(*,•,?,)
= Js_
>
« (Xi,t)^(t-;|f -)<A-/.
f
l
The following
result gives sufficient
conditions for a player's best response to nondecreasing strategies to be nondecreasing.
Proposition! Let /z:9T—>9?+, where h(t)
convex support. For
i=l,..,n, let hi(Li\ti)
payoffs as above. Then (i)for all
i,
is
a probability
density.
Assume
that h has
be the conditional density o/t_, given
#(?,)
= argmax^ U(x„t )
t
is
u,
a fixed,
and define
nondecreasing (strong set order)
and log-spm, if and only if(ii) h(t) is log-spm almost
everywhere-Lebesgue on the support oft
Proof: Sufficiency follows from Lemma 2 and Milgrom and Shannon (1994). Following the
proof of Theorem 1, it is possible to show that Uifati) is log-supermodular for all u (x ,t) log-
in tifor all %,{%) nondecreasing for j*i,
v,-
t
supermodular, only
if,
for
all t^.
>
t*,
and
all t,"
> t,
L
,
hi (X"i \t?)hi {t':i \th >
t
k (t^t" Mt
H
L
i\t,
)
But, since for a positive function, log-spm can be checked pointwise, this condition holds for
all i if
and only
if ft is
log-spm. Apply
Spulber (1995) recently analyzed
alters the results
profits.
how
1
asymmetric information about a firms' cost parameters
of a Bertrand pricing model, showing that
prices are increasing in costs,
expected
Lemma
Spulber' s
there exists an equilibrium where
and further firms price above marginal cost and have positive
model assumes
that costs are independently
13
and
identically distributed,
and
that values are private; Proposition
and
signals,
to imperfect substitutes.
easily generalizes his result to asymmetric, affiliated
1
Let
v,(x,t)
the vector of marginal costs, and D,(x) gives
By Theorem
1,
demand
the expected
Demand
set
is
demand
that the elasticity of
function
is
where x
(x,— r,)A(x),
demand
opponent uses a nondecreasing strategy, and
condition
=
to firm
is
Z),(x) is
when
i
log-spm
if
is
the signals are affiliated, each
interpretation of the latter
a non-increasing function of the other firms' prices.
CES, transcendental
functions which satisfy these criteria include logit,
logarithmic, and a
of linear demand functions (see Milgrom and Roberts (1990b) and Topkis (1979)).
when
the
goods are perfect
expected demand
substitutes,
when each opponent uses a nondecreasing
AU)j .,l,
t
and the
set {ty.
Then, by
%j(tj)>Xi}
Lemma 3
Thus, a
PSNE
are affiliated and
is
2( , 2)> ,
i
also log-spm.
is
demand
strategy, expected
a2 )-l
;rn( , n)>Xi
2.1,
is
log-spm and continuous, or
To
see
this,
Further,
note that
given by
1
when
expected demand must be log-spm
Xi is nondecreasing.
when
the density
is.
whenever marginal cost parameters
exists in nondecreasing pricing strategies
demand
yt_
(r 2 )/z (t_ 1 |r 1
is
nondecreasing (strong set order) in X\
and Corollary
t is
prices are x.
The
log-spm.
the vector of prices,
in the
case of perfect substitutes.
Conditional Stochastic Monotonicity and Comparative Statics
3.3.
Corollary 2. 1 applies to stochastic problems where the domain of integration
is
restricted, but
not to conditional expectations. Conditional expectations of multivariate payoff functions arise in
a number of economic applications, such as the "mineral rights auction" (Milgrom and Weber,
1982). This section studies comparative statics
when the
agent's objective takes the form
08
U(x,aA(T))=hu(x,s)f(s\0,A(T))dM (s)=jA(T) u(x,s)-
The comparative
statics condition
of interest
x*(6,T,B)=argmaxxeB U(x,(M(t))
Before proceeding,
we need
H
assumption
interactions
neither
is
clearly
,s) is
Our focus on supermodularity of
for this class of payoff functions
1:
Of
is
(x;s), if
nondecreasing differences in
course, U(x,9A{r))
is
(MCSb)
say that u(x,s) satisfies nondecreasing
all
H L
x >x
.
In terms of the vector
in s, since
satisfies additional
s,
no assumptions on the
s are imposed; however, nondecreasing differences
monotonicity
is
restrictions.
U(x,6\A(t)) in the study of comparative statics in problems
motivated by the following result (Athey, 1995), which
for a given t, U(x,0A{t)) satisfies
nondecreasing differences in
satisfy
is
weaker than log-supermodularity of u
weaker nor stronger than log-spm, unless u
Lemma
we
dfi{8).
(0,r).
nondecreasing in s for
between the components of
analogous to
problem
nondecreasing in
another definition:
L
differences in (x;s) if u(x ,s)-u(x
this
is
in this
'?
•f
and only
if
U(x,QA(t))
SC2
in (x,&)
in (x;0) is
for
all
u which
supermodular for
all
is
satisfy
u which
(x;s).
supermodular in (x,6)
14
if
and only
if
h
U(x ,9A(t)) -£/(/, <9IA(r))
is
.
nondecreasing
G(0\A(t))
in
H
for
L
x >x
all
= jA g(s)f(s,0\A(T))dju(s)
and
,
is
thus
problem
the
nondecreasing
reduces
and r for
in
all
checking
that
g nondecreasing.
The
to
following theorem applies to this problem:
Theorem 2 Consider f-M n+l -»
is
log-spm
a.e.-ju;
are a
9? +
Then (A) g:S -»
.
MPS C for (C) G{0\A{r))
is
9? is
nondecreasing
a.e.-^i;
nondecreasing in r and
and (B)
/(s; 0)
for all A(t)
nondecreasing in the strong set order.
Theorem 2
a generalization of a result which has received attention in both
is
economics (Sarkar (1968), Whitt (1982), Milgrom and Weber (1982)). 19
sufficiency using
Lemma 2,
consider the case where g
To
statistics
see the proof of
nonnegative. Consider rH > rL and
is
and
H
>
0l,
and define the following functions:
h
L
(s)
= g(s)-lS z A(ZL) (s)-f(s;0 L )
h3 (s) =
It is
,
/k(s)
=
and
fuis)
g(.s)-l SGA(lH) (s)-f(s\0n),
lse/Hr „,(s)- f(s;0 H )
= l^r^s)- f(s;0 L )
straightforward to verify under our assumptions that
Lemma 3
and the
fact that
log-spm
is
,
ft1
(s)-/j2 (s')
< ^(svs')-ft,(sAS
preserved by multiplication); thus,
Lemma 2 gives
r
)
(using
us
Rearranging the inequaHties gives the desired ranking, G(0l\Al)<G(0h\Ah).
The
stochastic monotonicity results can then be exploited to derive necessary and sufficient
conditions for comparative statics conclusion in problems of the U{x,0,z).
In particular,
we have
the following corollary:
Corollary 2.1 (Comparative Statics with Conditional Expectations) Consider any A(r), and
suppose F(s,0)
a probability
is
nondecreasing differences in
Finally,
we examine
Then
distribution.
(s;0), if and
only
MCSb holds for all u
if f(s,0)
is
the limits of the necessity part of
Theorem
construct require us to condition on an arbitrary sublattice.
additional structure
arises very
and
let
on the
commonly
A(t)=1 {j<T
comparative
}
.
set
A, log-spm
is
4.
Then
assume there
MCSb holds for all u(x,s)
Comparative
economic problem places
is
is
a single
to
be log-spm.
and restrict attention to
supermodular, if and only ifF(s;0) is log-spm.
and Densities
This section studies single crossing properties in problems where there
for all sublattices
variable,
se%
Statics with Single-Crossing Payoffs
19 Sarkar
(1968) establishes sufficiency,
random
not necessary for monotone
we require the distribution
Corollary 2.2 Consider a probability distribution F(s\0),
Mi)=1{s<t\.
If the
In this case, log-supermodularity of the density
but instead,
The counter-examples we
2.
no longer necessary. Consider a particular case which
in applications (such as auctions):
statics predictions,
which satisfy
log-supermodular a.e.-ju.
and Whitt (1982)
studies conditions under
15
a single random
which G(QA)
A; Milgrom and Weber (1982) study conditions under which G(6L4)
a limited class of shifts in A.
is
is
is
increasing in
increasing in
6 and
6
also in
Multivariate generalizations of the results are sufficiently restrictive that they are not
variable.
However, many problems
considered here.
auctions, and signaling
games can be
fruitfully
the theory of investment under uncertainty,
in
analyzed with a single random variable.
Problems of the form U(x,0)=ju(x,s)f(s;0)dju(s), where
variables are real numbers, are a
all
special case of those considered in Section 3. This section seeks to relax the assumption that both
log-spm In
primitives are
in
Problems which
(x;s).
we
particular,
fit
into
consider the weaker assumption that u(x,s)
framework include mineral
this
investment under uncertainty problems. The problem of interest
x*(6,B)=aigmaxxeB U(x,0)=ju(x,s)f(s;0)d{i(s)
Our
results
from Section 3 give some
necessary condition for
To
see
satisfy
this,
(MCSc)
observe that since
is
f{s,6) is
log-spm
do not establish
However, since SC2
a.e.-jj..
u SC2 and /log-spm are
that
is
SC2
U(x,0) and u(x,s) satisfy
satisfy
SCI. Thus,
we
if
and only
from
easily analyzed
sufficient for
additional applications.
crossing property (SC3).
SCI
(MCSc) holds
By
for
which
Corollary 1.1, this implies that
results
(MCSc). To analyze
first
all u(x,s)
from Section 3
this question,
it
difference with respect to x, since
xh>Xl, U(xh,0)-U(xl,0) and u(xh,s)-u(xl,s)
satisfies
SCI
in 0.
SCI of G{0) and
applications which are
that perspective, such as the portfolio problem.
This section develops
Section 4.2 considers
(MSCc)
directly,
and provides
Section 4.3 considers comparative statics using the Spence-Mirlees single
The main results
in Stochastic
The main
all
Section 4.1 analyzes
the main technical ideas of Section 4.
4.1.
iha.tf(s,0) is log-spm. a.e.-|i.
is
weaker than log-spm our
for
(MCSc)
0.
study conditions under which G(0)=j g(s)k(s,0)d/i(s)
Section 4 proceeds as follows.
most
if,
if
log-spm.
be useful to transform the problem by taking a the
will
SC2
u which are
all u(x,s)
is
into this problem: they imply that a
weaker than log-spm,
SC2, then (MCSc) must hold for
SC2
games and
rights auction
nondecreasing in
initial insight
to hold for all
SC2
is
satisfies
are summarized in Table 1 in the conclusion.
Problems
result of this section finds the
minimal
our single crossing
sufficient conditions for
conclusion.
Theorem 3 Let K(s;0) be a
satisfies
MLR;
are a
distribution.
(A) g(s) satisfies
MPSCfor (C) G{0)=\ g(s)dK(s\0)
Extensions to Theorem 3: 21 Theorem 3 also holds
if: (i)
SCI
in (s) a.e.-jJ-°;
satisfies
SCI
there exists
in
and (B)
K(s;0)
0.
k and ju such that
s
K(s;0)= j_oo k(t,0)d^(t) for all
(ii)
20
g depends on
A
in
which case (B)
under the additional
We will say that g(s) satisfies SCI
hold for almost
21
directly,
9,
is
equivalent to k
restrictions that
almost everywhere-//
(a.e.-//) if
the product measure on SR induced by
is
16
a.e.-/i;
piece wise continuous in
conditions (a)
and
(b)
of the definition of SCI
//)
(sH,sd pairs in 9* such that sh>sl-
version of this theorem which gives minimal sufficient conditions for strict
Athey (1996).
log-spm
2
2
all (w.r.t.
g
is
single crossing
is
provided in
and g
nondecreasing in 6 (see also Theorem 5 below for another sufficient condition);
is
WSC1, provided
only
(Hi) g(s) satisfies
The theory of
supp[K(s;0)]
constant in
is
0.
the preservation of single crossing properties under uncertainty has a long
Karlin and Rubin (1956) establish sufficiency, and Karlin (1968)
history in the statistics literature.
233-237) analyzes necessary conditions, 22 under the absolute continuity assumption of
(pp.
Extension
and some additional regularity conditions (including the assumption that
(i)
least three points). 23
may be
assumption
When
a probability distribution, the ex ante absolute continuity
is
A!"
has at
undesirable; however,
k represents a
if
utility
function,
it
is
the right
assumption.
We
will outline the
show
applications will
proof
that
functions which
is
Lemma 4
did in
to
surprisingly simple,
is
and our
produce extensions of Theorem
and Section
3.3),
we
In
3.
identify the smallest class of
required to generate the counter-examples used to prove necessity; this will be
useful for analyzing the tradeoffs
we
sufficiency proof
can be easily modified
it
we
addition, in this section (as
The
in the text.
illustrate in applications
between
how
alternative assumptions
on
commonly encountered
additional
primitives.
In Section 4.1.1,
restrictions
on g or k can be
used to relax (T3-A) and (T3-B).
Consider
which
first sufficiency.
guarantees
(4.1-B)
Suppose
for simplicity that k(s,0)>O. Define l(s)= k(s;0 H )/k(s;0L ),
Notice
nondecreasing.
is
\g(s)k(s;6 H )dn(s)= J g(s)(k(s;0 H )/k(s;0 L ))k(s;0 L )d^(s)
first
.
that,
for
a
given
H
>9L
,
Let s be a point where g crosses
zero. Then:
J
g(s)k(s;
=
6 H )dn(s) =
J
g(s)l(s)k(s-
-JIk(*)|'(*)fc(s; e L w(s)
e L )d/t(s)
+ \l g(s)i(s)k(s- e L )dfi{s)
(4. i)
s
^ -l{sa )\^\g{s)\k{s-e L )dfi{s) + l( So )Jlg(s)k(s;0 L )d{i(s) =
The second
equality holds because
g
l(s o )lg(s)k(s-0 L )dju(s)
SCI, while the inequality follows since monotonicity
satisfies
of / implies the following condition, which is necessary (4.1) to hold:
l(s)
<
l(so)
for s<so, while l(s)
>
l(s
)
for s>s
(4.2) requires that the likelihood ratio l(s) satisfies
show how
fixing the crossing point
addition, /(so)<L then
this
nondecreasing,
The
22
23
1
when
case arises
WSCl(so). (In Section 4.1.1 below,
can allow us to exploit (4.2) as a
to
jfcfa;^)
is
sufficient condition).
In that case,
a probability distribution (not a density).
must always be
less
than
1,
an anonymous referee
which
is its
who directed me to this
necessity proof provided in Karlin (1968)
preservation of an arbitrary
we
will
If,
in
g(s)k(s;0H )d/*(s)^Q implies ]g(s)k(s;0 L )dju(s)<jg(s)k(s;0 H )dju(s);
necessity parts of this theorem can be proved
am grateful
The
it
j
(4.2)
-
number of
is
sign changes,
complicated construction.
17
value at the highest
s.
by constructing counterexamples, which
theorem.
designed to solve a
and thus
if l(s) is
it
much more
general problem (about the
requires additional regularity assumptions and a
}
.
place
all
of the weight on the
failures.
Thus, the proof of Theorem 3 makes use of the following:
Lemma 5 (Test functions for single crossing problems)
0iP)-
andsL <sH
{§' 3a,b>0, fi>s,8>0,
g(s)=bfor sg(sh-s,Sh+s),
"%(fl)-
ft>£>0 and s
{k- 3<2>0,
Then Theorem 3 holds
s.t.
sets:
g(s)=-afor sg(sl-s,sl +s) and
\g\<<5 elsewhere,
and g
k(s)=afor s&(s
for any P>0,
if,
s.t.
Define the following
SCI
and k-Q elsewhere}.
satisfies
-e,So+e),
(4.1 -A) is replaced with (4.1
A
g(s)e#(fi);
')
Theorem 3 also
holds if (4.1 -B) is replaced with (4. IB') k(s,6)&1i:(p).
As
Theorem
in
Theorem
assumed
is
(for example, if
necessity parts of the theorem break
Now
consider
how
so that k(s;6H) places
low points
k
is
all
on
sets
down, as shown
in Section 4.2.1.
g
fails
(T3-A), then there
But then,
//.
,
G(9)
fails
SCI
since
g does.
of positive measure, SH and SL such that increasing
two
sets
relative to
SH
Then we can
construct a g(s) that
is
negative
on
all
If
A;
places
,
Sl, positive
sH
such
be defined
of the weight on
fails
(T3-B), then
Sh,
and close to
else.
not place ex ante restrictions on
how
moves with
supp[AXs;#)]
(T3-C) holds whenever (T3-A) does are summarized
If j g(s)dK(s; 6) satisfies
SCI
in
The
6.
in the following
6 whenever g
satisfies
Theorem
3 does
restrictions implied
when
Lemma.
SCI
in
s,
(i)for all
6h >9l,
absolutely continuous with respect to K(s;0L) on (infs supp|X(.s;ft/)], sup s supp[^T(5';ft.)]),
K(s;Gh)
is
and
supp[K(s;0)]
(ii)
<
more weight on SL
on
However, the proof of necessity of (T3-B) involves some additional work.
Lemma 6
sL
is
k(s;0) can
of the weight on high points sH while k(s;$L) places
there exist
zero everywhere
monotonicity of one of the
If
will.
SL and SH of positive measure
s L . This function is log-spm, but
-
change the conclusion of
will not
a probability distribution, not a density), then the
the counter-examples are used. If
that g(sz.)>0, but g(sH)<0,
the
on g
placing smoothness assumptions
but monotonicity or curvature assumptions
3,
primitives
1,
The following
is
nondecreasing in the strong set order.
applications
show how
additional structure present in particular problems can be
exploited; they also generalize the result to allow k to cross zero.
4.1.1.
Extensions and Applications to Investment Problems
This section uses
Theorem 3
and the decision problem of a
to analyze
risk averse
two
firm
classic problems, the portfolio investment
We develop
extensions to
problem
Theorem 3 which allow
us to generalize several comparative statics results previously established only for special
functional forms.
Consider
a risky asset
J
first
s,
the standard portfolio problem, where an agent with
and receives payoffs u((w- x)r + sx)
u'dw- x)r + sx)(s - r)f(s,6)ds
Notice that s-r
satisfies single crossing,
and
.
The
first
initial
wealth
w
invests
x
in
order conditions are given by
(4.3)
further, the crossing point is fixed at sv=r for all utility
18
functions. This section identifies
how (T3-B)
can be weakened using
this additional structure.
We
begin with an additional definition.
Definition
5 We
WSCR(so)
if(i) l(s
will
)
say that k(s;0) satisfies weak single crossing of ratios about so, denoted
= lin\_„ k(s,0 H )/k(s,0 L ) exists, (ii) supp[K(s,0)) is nondecreasing in the
o
strong set order, (Hi) either
k>0 or k satisfies WSCl(so), and
-
(iv) k(s, 0H)lk{s, 6>l)
l(s
)
satisfies
WSCl(so)fors where k(s;eL)k(s\0H)>O.
may appear
While the WSCRfao) condition
particular,
To
next subsection.
until the
WSCR(so)
k(s, 6l)
allows that the function k could
it
for
all So if
and only
cross exactly once, at
with G,
known
it is
start,
if
If
Sq.
is
A:
A:
is
we
unfamiliar,
itself
is
it
potentially quite useful.
cross zero, though
observe that
if
log-spm. Further,
k
is
we
In
will not exploit that fact
nonnegative, then k(s,0) satisfies
WSCR(so) can be
a probability distribution and the
satisfied if k(s,0H)
mean of s does
that such a single crossing property of the distribution implies that
and
not change
9 indexes k
according to second order stochastic dominance.
To
see
how
the
weak
of ratios can be used to establish single crossing
single crossing property
conclusions, recall that the inequality in (4.1) holds whenever (4.2) holds.
and for k>0, (4.2)
will hold
is
whenever
Theorem
equivalent to WSCRfao). Thus,
k(s,0) satisfies
WSCl(so) in
(s) a.e.-ju,
and (B)
is
any
point,
how
k(s;0) satisfies
this result relates to
we must
Theorem
3.
If
we
will
we
is
we
restricted to
be log-spm.
can apply Theorem 4 to
is
nondecreasing
desire a comparative statics result for
satisfies
all r,
WSC1
around
Consider an
(4.3), letting
in
g=s-r
6 whenever f(s,0)
log-supermodularity off
be required. In many problems the relevant range for the risk-free rate may be smaller than the
support of the risk-free asset, and thus
Theorem 4 has
The portfolio problem has been widely
for
WSCl(s<>), (4.1)
relax (T3-A) to allow for
and k=u'f. Then the optimal portfolio allocation x*( 6,B, r)
if
satisfies
A'o
&.
strengthen (T3-B) so that k
WSCR(r). But,
g
6 and k is nonnegative. Then (A) g(s)
WSCR(s ) a.e.-//, are a MPSCfor (C)
application of this result. In the portfolio problem,
satisfies
that
constant in
G(0)=jg(s)k(s,0)dfi(s) satisfies SCI in
Consider
we know
fixed
WSCR(so).
Suppose that supp(K)
4:
if
However, for a
studied.
more general investment problems, where
n(x,s),
or
make
and the
state
;r
is
far
fewer results have been obtained
potentially risk-averse firms invest in a risky projects
Suppose
as follows: max^z, ju(7r(x,s),0)f(s,O)d^i(s), and the solution set
represents a general return function which depends
of the world,
some exogenous parameter
risk aversion.
However,
pricing or quantity decisions under uncertainty about demand.
agent's objective
x*{6JB). Thus,
content.
The
s.
0,
Notice that in
agent's
utility
initial
19
denoted
on the investment amount,
x,
wealth or some other factor relating to
problem, the crossing point of %
as in the portfolio problem.
an
depends on the returns to the project as well as
which might represent
this
is
that
is
not automatically determined
When this
denoted j u
l
objective function
is
differentiable,
(7!:(x,s),0)7r z (x,s)f(s,0)dju(s)
,
satisfy
SCI. However, we might
for the possibility that the agent faces discrete choices
is
between investments, or
n cannot
endogenously determined (so that regularity properties of
this
problem,
we
and
Theorem
introduce another extension to
Theorem 5 Suppose
0;
check that the marginal returns to
suffices to
it
that
log-spm
(Hi) k(s, 0) is
supp(K)
(i)
is
constant in
also wish to allow
that the function
To
be assumed).
n
analyze
3, as follows:
0, (ii) g(s,
0) satisfies
Then G(0)=j g(s,0)k(s, O)dju(s)
a.e.-ju.
jc,
WSCR(so)
satisfies
SCI
a.e.-fifor all
in 0.
Proof: Consider the case where g crosses
(otherwise, the expectation is always nonnegative) and £=0 only at s=s and k>0; the other cases can be handled in a manner similar to
the proof of Theorem 3. Then, we extend (4.1) as follows:
,
jg(s;0 H )k(s;0H )dM (s)
= lim
"*<>
^^^
g{s\0 L )k(s;0 L )
g(s;0 L )k(s;0 L )dM (s)
J
J
The inequality follows, as in Theorem 3, because g
WSCRfao) since g and k do.
Proposition 2 Consider the problem
increasing
(A)
and differentiable24
7i(x,s) satisfies
MPSCfor the
SC2
in
max
y and
conclusion (C) x*(0,B)
n(x,s) is
is
Proof: (A) and (B) imply (C): If %
nondecreasing in
is
differentiable in
whether fu^n^x, s),0)x x (x, s)f(s,0)d{i(s)
apply
Theorem 3.
Then,
let g(s,0)
nondecreasing in
(B) ut (y,0)f(s,0)
satisfies
is
.
s.
Assume
log-spm. in (s,y,0)
a-e.-^i, 25
are a
0.
x and B
SCI.
= u{n{xH,s),0)-u{n{xL,s),0), and let k(s;0)
if
that u(y,0) is
Then:
is
a convex
u
is
set,
We can let g=K
x,
Now consider the investor's choice between two
preserved under monotone transformations, so that
gk satisfies
WSClfao), and
\u(7r(x,s),0)f(s,0)d/x(s)
and
in (x;s) a.e.-/i,
satisfies
we can analyze
and
values of x,
k=u
let
xH > xL
=f(s;0). First, observe that
nondecreasing in
x
its first
and
-f,
.
SC2
is
argument,
SC2 in (x;s), and g{s,0) satisfies SCI in s. Now, consider
satisfies WSCR(5 ). Let s be the crossing point of nx First
then by (A), u(n(x,s),0) must satisfy
conditions under which g(s,0)
restrict attention to s>so,
h
is
where
log-spm in (a,b,0) for
all
g{s,9)=h(n(xL,s),7i(xH,s),9) is
g(s;0H )/g(s;0L )
is
.
M.Xh,s) >7i(xl,s). Define h(a,b,0)=j
log-spm
in (s,0)
24 This
25 If u
is
is
where %
is differentiable;
not essential but
on s>So
it
since
n
is
,
nondecreasing in
On the other hand,
Then, g{s\0H )j g{s;0L )
log-spm, and WSCR(.So) holds. Then apply
the case
Ui(y,0)dy
and note that
a<b, by Corollary 2.1 and (B). This in turn implies that
nondecreasing in s on s>So.
g(s,0)=-h(7i(xH,s),7i(xL,s),0).
=a
is
if
s,
and thus
s<So, n(XH,s)<n(xL,s),
and
nondecreasing in s on s<so since h
Theorem 5. Necessity follows by Theorem
more general case.
is
3 for
the proof is omitted for the
simplifies the proof.
not everywhere differentiable in
its first
argument, the corresponding hypothesis can be stated as follows:
[u(y H ,0)-u(y L ,8)]-f(s;0) is log-supermodular in (yL ,yH,6,s) for all
20
yL < yH
.
Hypothesis (B)
MLR
generates an
6 decreases
is satisfied if (i)
the investor's absolute risk aversion, and
This result provides a general statement of two basic results
shift in F.
theory of investment under uncertainty, illustrating that log-supermodularity
is
(ii)
6
in the
behind the
seemingly unrelated conditions on the distribution and the investor's risk aversion. The proof of
Proposition 2 further highlights an application of the tools from Section
The
of
analysis
section establishes several generalizations of the existing investment
this
This literature typically considers the problem where the objective
literature.
Landsberger and Meihjson (1990) shows that the
quasi-concave.
strictly
comparative
statics in the portfolio
comparative
statics results are
between
risk aversion
and the
Sandmo's 1971
h(x)s, as in
also hold for
preserved under the
MLR.
classic
A
MLR,
differentiable
MLR
and
is sufficient
for
and suggest a behavioral relationship
few papers consider comparative
model of a firm facing demand
Sandmo's model. Proposition 2 extends the
statics
when
7i(x,s)
=
The focus of
uncertainty.
statics results
derived for the portfolio
analysis further, highlighting the
supermodularity of k, but not multiplicative separability,
fact that
is
problem, and Ormiston and Schlee (1993) show that arbitrary
Milgrom's (1994) analysis was to show that comparative
problem
3.
is critical
for the comparative
conclusions of Proposition 2. 26 Thus, the comparative statics results from portfolio theory
statics
and Sandmo's model can be extended to more general models of firm objectives.
SC2 and Comparative
4.2.
The main comparative
statics
Statics
Theorem of Section 4
follows immediately from
Theorem
3:
SC2
in
Corollary 3.1(Comparative Statics with Single Crossing Payoffs) (A) u(x,s) satisfies
(x;s) a.e.-.p.
and (B) K{s;8)
satisfies
MLR;
are a
MPSCfor (C) MSCc
Corollary 3.1 has an interesting interpretation. Recall that
the necessary
to
and
sufficient condition for the choice
be nondecreasing
in s.
SC2 of u(x,s)
MLR
will
hold
shifts are
when
s
is
(condition C3.1-A)
is
of x which maximizes u(x,s) (under certainty)
Thus, Corollary 3.1 gives necessary and sufficient conditions for the
preservation of comparative statics results under uncertainty. 28
known,
holds for all sets B. 27
unknown
Any
result that holds
but the distribution of s experiences an
MLR
when \y
Further,
shift.
the weakest distributional shifts that guarantee that conclusion.
is
The
result
is
with two examples.
illustrated
26 Building from a paper by Eeckhoudt and Gollier (1995) for portfolio problems, Athey (1998) uses the techniques
of this paper to extend this result to incorporate the restriction that investors are also risk averse; under that
condition,
it
follows that the restriction that/ is log-spm can be weakened to allow that only the distribution, F,
is
log-spm.
27
The quantification over constraint
sets
B can be dropped for the conclusion
that (B) is necessary.
28 Jewitt (1987) and Ormiston and Schlee (1993) also give this interpretation in their analyses.
Schlee (1993) explicitly analyze the preservation of comparative statics with respect to
show, under additional regularity assumptions, that single crossing of u
hold for
all
MLR shifts.
21
is
MLR
Ormiston and
shifts,
and further
a necessary condition for the result to
The Choice of Distribution and Changes in Risk Preferences
4.2.1.
This section considers the consequences for Corollary 3.1 of placing additional assumptions on
The context
risk preferences.
is
a problem where the choice variable
is
a parameter of a
probability distribution (for example, the agent chooses effort in a principal-agent problem, or
makes an investment
3.1 can then
be applied systematically to provide succinct proofs of some existing
further to suggest
Formally,
and the exogenous parameter describes risk preferences. Corollary
decision),
let
some new
ones.
u{s,6) be an agent's utility function,
probability distribution F(s;x)
This
an example where
is
it
and
results,
is
=
$ x f(t;x)dju(t).
and
The agent
useful to have results that
be a density with an associated
let f(syc)
max xeB ]u(s,$)f(s;x)dju(s).
solves
do not
rely
on concavity of the
objective:
concavity of the objective in x requires additional assumptions (see Jewitt (1988) and Athey
(1995)), which
may or may
We will now
not be reasonable in a given application.
SC2
study several sets of sufficient conditions for the agent's objective to satisfy
in (x;0),
each corresponding to different classes of applications. For
where u
satisfies
enough
regularity conditions so that the integration
arg max xsB ju(s, 0)f(s; x)dji(s)
=
arg max xeB
simplicity, consider the case
by parts
is valid.
Then: 29
- jus (s, 0)F(s; x)ds
= arg max ^b us (s, 6)\sf (s; x)ds + juss (s, 0)\*=s F(t; x)dt ds
The following
result follows directly
from these equations and Corollary
3.1.
Of course,
it
can
be further extended to allow for restrictions on higher order derivatives.
Propositions Consider the following conclusion: (C)x*(0,B)=argmaxx € BJ u(s,O)f(s;x)d^(s)
s
nondecreasing 6 for all B. In each of the following, (A) and (B) are a
Additional Assumptions
and {s\u(s,&)*0}
(i)
u(s,d)>0,
W)
us (s,0)>O, and {s\us (s,0)*O}
is
constant in
is
constant in
MPSCfor (C):
A
B
u(s,6) is log-spm. a.e.-{i.
f(s;x) is
u£s, 6)
is
F(s;x) is
log-spm. in
WSC2
in (x;s) a.e.-fi
WSC2
in (x;s) a.e.-Lb.
(s,-6)a.e.-n.
e.
(Hi)
6.
is
\sf(s\ x)ds
is
constant in
x,
w„<0, and
1
u ss (s,0)\
is
log-spm. in
\l^F(t\x)dt
isWSC2in
(s,-6) a.e.-fu.
{s\uss (s,0)*Q} is constant in 6.
In each of
relies crucially
(i)-(iii),
the fact that (B)
(x;s) a.e.-Lb.
is
necessary for the conclusion to hold whenever (A) holds
on the nonmono tonicity of the relevant function
in (A).
Thus, while the sufficient
conditions and the necessity of (A) in each case are quite general, one should be
drawing conclusions about the necessity of (B).
However, there are applications
appropriate for each of these results. Consider each of
29
The
notation
s.
(i)-(iii)
and s indicates the bounds of the support of F.
22
more
in turn.
careful in
that are
To
project
is
is
with Corollary 3.1: the single crossing condition on the payoff to
(i)
replaced by a single crossing condition on the density. Case
restricted to offer a stochastic
that will
the support of s
crossing of ratios
Now,
consider Part
WSC2
assumption,
it
is
distribution; that
In case
(iii),
an agent, and the uncertainty
of a probability density
stronger than
is
log-spm
That
is,
in (s,-0)).
F(s^H)
is
if
a principal
about the allocation
is
relative return to
equivalent to
weak
single
FOSD but weaker than the MLR.
is
Further, (B) requires that the distribution F(s;x)
F(s%)
crosses
at
most once, from below.
x decreases the mean as well as the
Under
this
riskiness of the
might incorporate a mean-risk tradeoff.
We see
the agents are restricted to be risk averse.
agent's prudence (as defined in Section 3. 1)
Of these
might apply
log-spm when higher types have a larger
possible that increasing
is, it
(i)
the'
Hypothesis (A) requires that the agent's Arrow-Pratt risk aversion
(ii).
in (x;s).
is
to
WSC2
fixed,
is
(WSCR), which is
nondecreasing in #(i.e., us
satisfies
mechanism
be received. 30 The payoff u
When
s.
compare
begin,
results, only
(ii)
and only
if
if
that
j^ F(v, x)dv
x always
satisfies
many authors have
further studied the result (such as Jewitt (1987, 1989)).
WSC2 in (x;s).
Diamond and
has received attention in the literature.
established the sufficiency side of the relationship, and
increases with an
The example
Stiglitz
(1974)
since exploited and
further illustrates
Jewitt's (1987) analysis of risk aversion relates to this paper: Jewitt (1987)
shows
how
that (A)
is
necessary and sufficient for (C) to hold whenever (B) does.
Mineral Rights Auction with Asymmetries and Risk Aversion
4.2.2.
PSNE
This section studies existence of a
in
nondecreasing strategies in Milgrom and Weber's
(1982) model of a mineral rights auction, generalized to allow for risk averse, asymmetric bidders
whose
functions are not necessarily differentiable, reserve prices
utility
when bidding
The
units
may be
consider n asymmetric bidders.
where each agent's
Ui(bj,su s2 ) is
The
signals
u (b
i
i
,s l ,s2 )=j Ui(y
where there are two bidders;
Suppose
utility
nondecreasing in
have a joint density
density of s-i given
if
,
that bidders
when we
difficulties will arise
one and two observe
and
signals s
{
(-&,, s l ,s 2 )
h{si,si)
- bt)g(y\s u s2 )dv
30 Such uncertainty might arise
by bidders, and
s2
,
(written {4(£t,£i,£2 )) satisfies
written f-,{s-i\si).
si is
differ
discrete. 31
analysis begins with the case
respectively,
may
and supermodular
in (bi,sj),j=\,2.
(4.4)
with respect to Lebesgue measure, and the conditional
To
see an example where assumption (4.4)
where v
is affiliated
with
si
and
52,
and
«. is
the principal cannot observe the agent's choice perfectly, or
if
is satisfied, let
nondecreasing
the principal must
design an error-prone bureaucratic system to carry out the regulation.
31
As
in the pricing
game
studied in Section 2, this
bidding functions, so Vives (1990)
is
may not be applied
23
not a
game
with strategic complementarities between players
to establish existence of
a PSNE.
and concave. 32
When player two
given her signal (s
t
uses the bidding function J3(s 2 ), then the set of best reply bids for player one
can be written (assuming
)
= argmaxj^ u
b' (s\)
{
When
l
to
bH are
lost
+jl
l
Ul (b
l
2
,s l ,s 2 )l bl=mSi) (s 2
)f2 (s2 \s
negative for low values of s2
strictly
with
on the region where
to
and the
be,
WSC2 in
,
(t\;
s2 )
when
The
.
returns to increasing the bid from
effect is zero for s2 so high that
even bH does not win. For each
(ii)
to
Theorem
Proposition 3 Consider the 2-bidder mineral rights model, where the
above, the joint density h
variables
Then
nondecreasing (strong set order) in
biiSi) is
that in auction
affiliated,
pennies) or else
w, is
and
and
all
signal of all of the
either the players
utility function satisfies
4.4
and the support of the
fhisi)
which
is
nondecreasing in
Si.
as the
first
price auction described above, a
choose from a
finite set
of bids
(i.e.,
bidding in
model
symmetric
opponents use the same symmetric bidding function, then only the
maximum
opponents
will
extend
the opponents are using the
same
necessarily have the highest bid.
were only two bidders.
asymmetric auctions with
n bidders?
are affiliated
strategies,
when
the distribution
is
symmetric.
Further,
whichever opponent has the highest signal
to this
if
will
problem exactly as
if
Unfortunately, the results do not in general extend to n-bidder,
value elements.
affiliation
is affiliated
to
Then we can apply Proposition 3
common
use asymmetric strategies),
this
be relevant to bidder one. Define sm = max(s2,..,sn ). Milgrom
(si,sm )
the highest bidder
3.1):
If the bidders face a
try to
and Weber (1982) show that
there
and Corollary
continuous.
What happens when we
distribution,
games such
3,
bid, the
of Proposition 3 holds, and further, the type distributions are
exists if the conclusion
atomless and
(affiliated),
Suppose that bidder 2 uses a strategy
is fixed.
Athey (1997) shows
PSNE
log-spm almost everywhere
is
where she would
raising the bid causes the player to win,
.
S2.
)ds 2
the opponent bids less than be, the returns
returns are increasing in s 2 This yields (using Extension
random
l
bidder two plays a nondecreasing strategy, (4.4) implies that bidder one's payoff
are increasing in
have
broken randomly):
(b u s l ,s 2 )l bl>/KS2) (s 2 )fz (s 2 \s )ds 2
function given a realization of s 2 satisfies
bL
ties are
of the signals
Under asymmetric
is
distributions (or if players
not sufficient to guarantee that the signal of
with a given player's signal.
The conclusions we can draw about
the equilibria of auction
games based on Proposition 3
are
stronger than those available in the existing literature, which has typically considered only
continuous bidding units, and either symmetry or stronger distributional assumptions, such as
private values or independent signals. 33
32
To
see why, note that
in (b h v).
Sj
By Athey (1995),
33 See Lebrun (1996),
who
and
Sj
each induce a
first
order stochastic dominance shift on F, and
supermodularity of the expectation in
(bj,si)
and (bh sj)
«. is
supermodular
follows.
uses a differential equations approach to derive existence in an asymmetric independent
private values auction.
24
s
SC3 and The Spence-Mirrlees
4.3.
Single Crossing Property (SC3)
This section extends the results about single crossing to consider single crossing of indifference
curves, providing general comparative statics theorems and applications to an education signaling
game and
The
a consumption-savings problem.
many other mechanism
screening games, as well as
SM
The
problems.
condition
3
h
arbitrary differentiable function
satisfies -§;h(x,y,t)*
single crossing property is central to the
and
analysis of monotonicity in standard signaling
design
SM
-» 9?
9?
:
an
for
that
given as follows:
is
&h(x,y,t)/\§h(x,y,t)\
is
nondecreasing in
(SM)
t.
When
defined,
the (x,y) indifference curves are well
SM
is
equivalent to the requirement that
the indifference curves cross at most once as a function of
We
(Figure 2).
t
will
make use
following assumption which guarantees that the (jc,y)-indifference curves are well-behaved
also possible to generalize
SM to
the case
where h
is
we
not differentiable, but
maintain
the
(it is
(WB)
for
simplicity):
h
is
differentiable in (x,y); -^h(x,y,t)
Milgrom and Shannon (1994) show
V (x, y, 0) = j
s
v(x, y, s)f(s;0)dp(s)
Theorem 6 Assume
and.(B)f(s;&)
is
that v
log-spm
3
:
5R
—»
a.e.-ji,
that
SM
This section characterizes the
;
under (WB), (SM)
is
(WB). Then (A)
MPSC for (C)
]s
v(x,y,s) satisfies
1).
SC3
form
3).
a.e.-p,
and the Spence-Mirlees SCP) Theorem 6 also
is
holds
nondecreasing
in
if 4.7-C
6for all B
&:<R-*9?.
Theorem 6
more
(Definition
v(x,y,s)f(s;9)dp(s) satisfies SC3.
replaced with (C) x*(Q,B)= arg max, e B Jv(*, b(x), s)f(s; 0)dp(s)
and all
SC3
equivalent to
single crossing property for objective functions of the
9? satisfies
are a
(WB)
the (;c,y)-indifference curves are closed curves.
with the following theorem (analogous to Theorem
,
Corollary 6.1 (Comparative Statics
is
*
is
restrictive.
Sufficiency
hx(s)=\v y (x,y,s)\f(s;6 L )
and note
that
Theorem
closely related to
,
in
although the class of payoff functions considered
3,
Theorem 6 can
also
be shown using
h 2 (s)=vx (x,y,s)f(sr,eH ) h 3 (s)= Vjc (x,y,s)f(s;0 H )
,
Lemma
2.
Let:
h4 (s)=\vy (x,y,sj{f(s;0L ),
A and B imply:
vx (x,y,s L ) f(sH ;0L )
i
i
y,s L )\ f(s L ;0 L )
Lemma 2 that
\v (x,y,
y
H )\ f(s L ;0H )
£ V(x,y,0L )/\§V(x,y,0L
-\
< J: V(x,y,0H )/\§ V(x,y,
This theorem can be applied to an education signaling model, where
choice of education, y
is
(4-5)
i
\v (x,
y
This in turn implies by
vx (x,y,s H ) f(sH ;6H )
<
i
monetary income, and
25
is
x
is
H \.
represents a worker's
a noisy signal of the worker's
ability, s (for
example, the workers' experience in previous schooling).
satisfy
will
If the
worker's preferences u(x,y,s)
SM and higher signals increase the likelihood of high ability,
the worker's education choice
be nondecreasing
in the signal
#for any wage function w(x).
on
In another example, consider a variation
denote an agent's
initial
wealth, and
The agent
value function of savings.
where the value of
1
5 is
unknown
let
the standard consumption-savings problem.
x denote
Let z
savings and b(x) the (endogenously determined)
further has a non-tradable
endowment
at the time that the savings decision is
distribution over s at the time the savings decision is
made
is
(0) of a risky asset (s),
made.
The
probability
The parameter
given by F(s;6).
might represent the quality or quantity of an exogenously (or previously) determined endowment.
The
agent's utility given a realization of s
nondecreasing. Then, our agent solves
two conditions are a minimal pair of
max je[0
is
,
z]
u(z + s-x,b(x))
,
which
is
assumed
to
\u(z + s-x,b{x))dF{s;6). Then, the following
sufficient conditions for the
comparative
statics
conclusion
that savings increases in &. (A) the marginal rate of substitution of current for future utility,
is
nondecreasing in
s,
and (B)
F
MLR.
satisfies
be
FOSD
Thus, a
improvement
uju 2
,
in the distribution
over the consumption good does not necessarily increase savings.
Conclusions
5.
The goal of
this
paper has been to study systematic conditions for comparative
show how
predictions in stochastic problems, and to
applications.
As
the results can be used in
statics
economic
such, this paper derives a few main theorems (summarized in Table 1) which can
be used to characterize properties of stochastic objective functions that
context of comparative statics problems in economics.
The
arise frequently in the
properties considered in this paper, the
various single crossing properties and log-supermodularity, are each necessary and sufficient for
comparative
an appropriately defined class of problems.
statics in
results are also analyzed,
each of which
motivated by the requirements of different economic
is
problems. Because the properties studied in
predictions,
in
do not
rely
on
differentiability
Several variations of these
this
paper, and the corresponding comparative statics
or concavity, the results from
paper can be applied
this
a wider variety of economic contexts than similar results from the existing literature.
It
turns out that a
conditions for
statistics
all
few
results
from the
of the properties studied
literature did not
statistics literature are at
in this paper.
emphasize comparative
This
statics.
is
work behind
the sufficient
especially surprising because the
However, the common
structure
underlying the results in this paper provide a useful perspective on the relationships between
seemingly unrelated problems. Building on the results from
statistics, this
paper characterizes the
conditions which economists can use directly for comparative statics predictions.
Finally, this
paper focuses on results about necessity as well as their
defining the tradeoffs that
limitations,
sharply
must occur between weakening and strengthening assumptions about
various components of economic models.
These tradeoffs are highlighted
26
in
Table
1,
which
further provides guidance about
problems.
and other
which properties
will
be most appropriate
in different classes
of
Thus, together with Athey (1995)'s analysis of stochastic supermodularity, concavity,
differential properties, the results in this
paper can be used to identify exactly which
assumptions are the right ones to guarantee robust monotone comparative
wide variety of stochastic problems.
27
statics predictions in a
Table
Thm#
A: Hypothesis
on u (a.e.-//)
Thml;
Cor
m(x,s)>0
log-
is
B: Hypothesis
1:
on/
Summary
\
<2:
y(s,9)
is
Equivalent Comparative
Statics Conclusion
log-spm.
arg
!u(x,s)fls,Q)d/j(s) is log-
max Ju(x,s)/(s,0)d/Xs)
1.1
Thm2;
w(x,s)
spm.
is
spm.
Cor 2.1
in (x,sd
V
j{s,Q)
is
log-spm.
Thm3;
in (x,6).
w(x,.y) satisfies
tin(9,B).
arg max Ja"(x,s) -.
d//(s)
Ja«(x,s) t
d/Xs)
lf(s,0)ds
i.
is
spm.
T
in (x,8,A).
in (9 Aff)-
arg
j{s,6) is log-spm.
max lu{x,s)f{s, 9)
d/j(s)
3.1
SC2
Thm6,
Cor
MCS:
Conclusion
(a.e.-//)
spm.
Cor
of Results
satisfies
in (x;s).
u(x,y,s)
f(,s,0) is
log-spm.
SC2
in (*;#).
tintfforallfl.
arg max lu(x,b(x),s)fis, 0)d/4.s)
Iu(x,y,s)fis,0)dfj(s)
6.1
satisfies
SC3.
satisfies
SC3.
Extensions: assume, in addition, that supp[F(s;0)]
Thm4
u{x,s) satisfies
WSC2(^
Thm5
u(x,s) satisfies
WSCR(.y
In each row:
further,
).
C
is
A
j{s,0)>O satisfies
is
tin#forall&:<R->.9L
constant in
arg max ju(x,s]f{s, 0) d/x(s)
Su(x,s)f{s,d)d{j(s)
WSCR(so).
satisfies
j{s,0) is log-spm.
SC2
mB
in (x;0).
).
and B are a minimal pair of
SC2
t
in
6 for
all
B.
aigmaxSu(x,s)f{s,0) dp(s)
iu(.x,s)f{s,0)d^s)
satisfies
6.
xeB
in (x;d).
tin
for all 5.
sufficient conditions (Definition 5) for the conclusion
C;
equivalent to the comparative statics result in column 4.
Notation and Definitions: Bold variables are vectors
negative; spm. indicates supermodular,
and
in 91"; italicized variables are real
increasing in the strong set order (Definition 4);
SC2
indicates single crossing of incremental returns to x,
and SC3 indicates single crossing of x-y indifference curves (Definition
with a fixed crossing point, s (Definition
crossing point s (Definition 5).
Arrows
1);
numbers; /is non-
log-spm. indicates log-supermodular (Definition 2); sets are
WSCR(^
indicate
weak
28
)
indicates
weak
monotonicity.
1);
WSC2(s
)
indicates
weak SC2
single crossing of ratios at a fixed
Appendix
6.
Proof of Lemma 4 and Theorem 1: Sufficiency in Theorem 1 follows from Lemma 2;
sufficiency in Lemma 4 will follow from Lemma 2 since the test functions are log-spm by Lemma
3. Necessity in Lemma 4 will be established below in (1) and (2), noticing that our counterexamples come from the relevant set. Necessity in Lemma 4 implies necessity in Theorem 1. We
will treat necessity for/, the argument for u is of course analogous:
(1)
«=1: Let v{A\9)=\A f(s;0)dju(s)
=
and SH{s)r^SL(s)=0. Let u(xl,s)
Pick any two intervals of length
.
1 SlU) (s)
,
and
let u(xh,s)
=
e,
1 Sh ( C) (s)
.
such that Sh{s) > Sl(s)
Then
ju(x L ,s)f(s,0)d/t(s) = v(S L \0) and ju(x H ,s)f(s,0)d/x(s) = v(SH \O). Since u
is
log-spm then
ju(x,s)f(s,0)dju(s) must be log-spm by (C),i.e., v{Sh,0h)v{Sl,0l)>v{Su0h)v{Sh,0l). Standard
limiting
arguments
Martingale convergence theorem as applied in the appendix of Milgrom
(i.e.
and Weber (1982)) imply \hatf(s,0) must be log-spm //-almost everywhere.
n>2: Let v{A\0)=jA f(s;Q)dju(s)
(2)
[h
- jr
,
(h
+ 1) -57]
x
• •
~ ir
[in
>
(in
+ 1) -jr] and
let Q'(s)
X=
yv
>
and take any a,b £91". Further,
Define u(x,s)
w(y
a
z,s)
=
onX as follows:
le
(aAb) (s)
,
It is
.
We partition 9?" into n-cubes of the form
.
let
=
w(y,s)
{y, z,
le
.
w (s)
,
z,
be the unique cube containing
ya
u(z,s)
where each element of X
z},
=
s.
l ff(b) (s)
straightforward to verify that u
is
«(y
,
v z,s) =
lG
.
Consider a
t,
is distinct.
(avb) (s)
,
and
log-spm. If J«(x,s)/(s,0)a?//(s)
is
v b)|9 v 9') KG' (a a b)|0 v 9') > v{Q' (a)|9) KG' (b)|9') for aU
9,9'. Since this must hold for all t and for all a,b, we can use the Martingale convergence theorem
to conclude that /(a v b)|9 v 9') /(a a b)|9 a 9') > /(a|9) /(b|9') for //-almost all a,b (recalling
log-spm,
follows that v(Q' (a
it
•
•
•
•
that // is a product measure).
Proof of Theorem
bounded below,
2:
let
(A) and (B) imply (C): Sufficiency
M
inf{s(s)}
otherwise,
;
same arguments with g(s)+M, taking the
in the text for
limit
00 if
H
(A) and (C) imply (B): Consider a,b such that v (Q' (b)) v {Q' (a))
•
nothing to check) and such that each element of {a,b,avb,aAb}
in the
proof of Theorem
AL={G'(b),G'(aAb)}. Choose g(s)
further so that lc(s)
is
Li* g(s)f(s;0 L )d/j(s)
reduces
=
1,
and
let
and
,
l c (s) so that Q'(avb),Q'(b)
let
/(s;
H )dju(s)
(otherwise there
is
Let Q'(s) be the unique
Ah={ Q'(a),Q'(avb) and
}
c C and Q'(aAb),Q'(a) <£ C, and
,
< Jse/1 „
>
is
is distinct.
nondecreasing. Then, G(-\A H 0h)>G(-\Al, 0h)
jseAH
is
M be an arbitrary positive constant, and apply
as M approaches
g unbounded below.
L
cube containing s defined
g>0. If g
let
sea
the
shown
is
if
and only
if
^(s)/(s; 0„)dfi(s) seAL /(s;
j
L )d/i(s)
,
which
to:
v -(G'(b))-[v w (G'(avb)) + v H (G'(a))]<v H (G'(avb))-[v L (G'(b)) + v t (G'(aAb))].
z
Simplifying gives v (Q'(b))-v H (Q'(a))< v H (Q'(avb))- v
L
all t
and
if
all t if
Jsee
,
t
(Q (a/\b)). Since
this
must hold for
we conclude \haXfl$,0) must be log-spm for //-almost all a,b and all 0.
imply (A): Consider a < b and let A L=Q'(a) and A^=Q'(b). Then G(-\A H)>G(-\A L)
all a,
(B) and (C)
only
L
(a)
b,
s(s)/(s|0,G'(a))rf//(s)
and only
if g(a)<g(b)
for
//
^
L
G1(b)
-almost
all
s(s)/(s|0,G'(b))rf//(s)
,
which
is
true for
all
(ii)
implies
(i):
a<b.
Since [l/F(a;0)]j^l
29
t
< a (t)dF(t,0) is
and
a<b and
s
Proof of Corollary 2.2:
if
a probability
,
a and
distribution for all
and a and
all
and
Rewriting the
for
all b.
all
b<a, which
spm of F and
a for
all
is
is
apply a result from Athey (1995): G{0\a)
and only
we have
(ii):
in
6 for
txa. This
for
in
nondecreasing
is
all a, all
is
nondecreasing
in
g(s)= l sib (s)
[1/F(a; 0)]j°dF(s; 8) nondecreasing in
and a
b<a. Log-
in <9for all
for
and only
if
1
if
G(0\a)
b<a, which implies that 1- F(b;0)/F(a;0)
all
nondecreasing
is
- F(b; 0)/F(a; 6)
is
F is a distribution.
true since
(i)
g nondecreasing
all
From above, we check that
b<a.
all
is
Simply apply
implies
nondecreasing
is
1- F(b;0)/F(a;0)
nondecreasing in a for
(i)
G{ 0\a)
if
F is a distribution yield the result.
g(s)= ls >b(s) and
Proof of Lemma
if
true if and only if
nondecreasing in a, which
all
we can
latter condition,
the fact that
Likewise, G(-|a)
in
0,
g nondecreasing,
is
nondecreasing
equivalent to log-spm of F.
is
6: Pick
6H
>9 L
Define measures v/.and vh as follows: VL(A)=j dK(s;0 L )
A
.
and vh(A) = \A dK{s;0 H ). Define a =
inf [s\s
esupp[K(-;0H )]} and b
= supl^s esupp[ £"(;#/.)]}.
Define h(s;0)=dK(s;0H )/dC(s;0H ,0L). Let D=supp[K(-;0 L )]<Jsu-pp[K(;0 H )]. Since the behavior
of h(s;0) outside of
D will not matter, we will restrict attention to D.
The proof proceeds
in
several steps.
Part
(a): If
Part
>
{b):
where a <
For any S =
and Vh(S)
g
g(s)=.9
b.
(s L ,s H ] c: [a,b]
= 0. Note that
vL(S) >
,
< K(sL ;0H )
implies that Vh(S)
> 0.
Proof: Suppose that vL (S)
since a<SL- If supTp[K(s L ;0 H )]^SL, then define
for 5 g(-co,s l ), while g(s)=K(s L ;0 L )/v L ([s L ,<x>)) for s e[s L ,co).
follows. g(s)=-l
define
we
a>b, then the conclusions hold automatically. Throughout the rest of the proof,
treat the case
as follows: g(s)=-l for s £(-<»,$*.),
K{s l \9h)I^h{[sh,'x:>))
g(s)=K(sL ;0 L )/v L (S) for
for 5 e[s//,oo)
Now,
.
it is
.yeS,
g
as
Otherwise,
and
straightforward to verify that
SCI
is
violated for this g.
Part
>
(c):
For any S =
and vL(S)
= 0.
If
(s L ,s H ] <- [a, b]
K(s L ;0 L )=Q, then
g(s)=K(s H ;0H )/(2 V//([%,oo)) for
^)=-Vl([^,°o))/^(^;^)
se[jH,<»).
Now,
it is
>
Vh(S)
,
implies that
define
s e[,S//,oo)
g
vL(S) >
0. Proof:
Suppose
as follows: g(i')=-l for s e(-co,s H )
,
that vh(S)
while
Otherwise, define g as follows:
.
g(s)=-v„([s H ,<x>))/v H (S) for seS, and g(s)=l for
for s e(-oo,s L ),
straightforward to verify that
SCI
is
violated for this g.
Part
(d): supp[AT(-;^ w )]>supp[A!'(-;^ L )] in the
strong set order. Proof: Parts (c) and (d) imply that
vh
absolutely continuous with respect to Vl
on
is
that
if s'
esupp[ £(•;#/,)] and s"eswpp[K(-\0 L )],
s',s"e supp[AT(-;
to s"<b.
L )]r\smpp[K(-;
Likewise
H )]
.
Since
s<£
we may restrict attention
[a,b],
on
[a,b],
supp[#(-;
Lemma Al G{0)=\ g(s;0)k(s;0)d^(s)
conditions: (i)(a)
nondecreasing in
For each
0.
0,
else supp[AT(-;#)] is constant in
we are
satisfies
g(s, 0) satisfies
(i)(b) k(s;0) is
L )] for all
to s'>a. But, if (as
and
log-spm
on
30
now
suffices to
show
s>b,
we may restrict attention
we argued in part
[a,b],
(b))
vh
is
then
done.
SCI
WSC1
in
under the following
sufficient
in s a.e.-/i; for ju-almost all
a.e-fi. (i)(c)
0.
It
and s">s', then
absolutely continuous with respect to vl and vice versa
supp[/sr(-;0 w )]=supp[#(-;0 L )]
and vice versa.
Either g(s, 0) satisfies
s,
SCI
g(s, 0) is
in s a.e-fi,
or
.
Proof of Lemma Al: Pick
g(s,0 L )>O for
that
definition
almost
ju-
L )>O,
H ) > g(s,
G(0h)>O by
and
J"
>
we
and g(s,
s
L)
g(s;0 L )k(s;0L)dju(s)>(>)O. Choose
<
for
almost
//-
since supp[K(;0H)]
This in turn implies
(i)(b)).
.
First,
nonnegative would imply that g(s,0i.)=O
(
exists
s
by the
if
in the
strong set
we can conclude
is
that
degenerate, since this
But then our hypothesis
on supp[K(;0L)]. Since swpp[K(-;0)]
turn implies G(0h)=O by (i)(c).
this in
so
s
supp[K(;0H)],
G(0l)>O, then
set order that supp[K(-;0L)]<s o as well.
s
note that
nondecreasing
is
that, for all s in
if
<
all s
Second, observe that the case where s\xpp[K(;0H)]<s o
(i)(c).
nondecreasing in the strong set order,
So,
that
which implies G(0h)>O. Further,
would imply by the strong
is
s
Suppose
.
supp[K(-;0H)]> s
Lemma 6
order (applying
G(0l)
all
L
of WSC1). Let s = min{5 > s and s €supp[K(-;0 L )]}
So £supp[AT(- ;#«)], then
g(s,
H >
a.e.-//
that
is
consider the third case where So e supp[AT(- ;#/)], but there exist s',s"e supp[AT(;ft/)] such
that s'<s <s".
surrounding s
Notice
•
It
by the strong
that,
set order, s\xpp[K(-;0H)]=s\xpp[K(-;0L)]
further implies that g(s,0L)<O for
everything in the set
all
s
interval
e supp[K(-;0L)]\s\xpp[K(-;0H)] (because
below suppfA^- ;#/)], which contains
lies
on an
s
By
).
same reasoning,
the
g(s,0O>O on supp[tf(-;6k)]\supp[#(-;&)].
Now,
define a modified likelihood ratio
/ (s),
as follows:
/
(s)=0 for s£supp[K(-;0L)],
1
(s)=k(s;0 H )/k(s;0 L ) for sesupp[K(-;0t)] and k(s;0L)>O, and then extend the function so that
/
(s)
ratio
= maxiliml (s'),liml (s')j
for SGSupp[K(-;0L)] and k(s;0L)=Q (recalling that the likelihood
can be assumed to be nondecreasing
Thus,
(i)(b)).
on an open
we know
/
in s
on
supp|X(-;ft.)] without loss of generality
by
(f )>0, since s e supp[#(-;#z.)] and since supp|X(-;#/)]=supp[#(;#z.)]
interval surrounding s
.
We use this to establish:
\g{s;0 H )k{s;0H )d^s)>] g(s;0 L )k(s;0H )dM(s)>S g(s;0j(s)k(s;0 L )dM (s)
2
"^(*o)
The
first
£
L )\k{s;0 L
\g(s;
W(s) + /(£„)£ g(s\
L )k{s;
L )dfi{s)
inequahty follows by the fact that g(s,0 H ) > g(s,0 L )
the definition of
and since g(s,0L)>O on
/ (s)
WSC1
of g. The third inequahty
since k
is
log-spm.
The
is
=
/(£„)/ gfr
The second
.
supp[AT(-;ft/)]\supp[AT(-;ft.)].
true because
/ (s) is
last equality is definitional.
nondecreasing
Thus, since
/
L )dju(s)
by
inequality follows
The
a.e.-/i
(s)>0,
L )k(s;
equality follows
by
on supp|X(-;&.)]
G( #l)>(>)0 imphes
G(6W)>(>)0.
Proof of Theorem 3 and Lemma 5: Sufficiency follows from Lemma Al Necessity will follow
by constructing counter-examples from the relevant sets. Necessity of (T3-A) follows by
.
observing that for any sl<sh, if we let SL (e)= (s L - e,s L ] and SH (s) = (s H - e,s H ] the function
H
f
fc(5,6 ')=l5 L(£) , k(s,0 )=l Slli£) is log-spm. Thus, (T3-C) will imply that g is SCI a.e.-|a. Necessity of
,
Define a, b,
(T3-B):
First, notice that if a
h(s;0H)
=
h,
>
and h(s\0L)
This implies that h(s;0)
and
b,
D as in the proof of Lemma 6.
then h(s,0)
= 2 on
is
is
log-spm
supp[/sT(-;
L )]
,
a.e.-//.
To
while h(s;0L)
It suffices
see
=
this,
to consider
log-spm of h.
observe that a > b implies that
and h(s;0H)
= 2 on
supp[#(-;
H )]
log-spm.
Now, suppose a<b. Let B = supp[K(-;0 L )]r\supp[K(;0 H )] which we have shown is equivalent
to Dn[a,b]- Pick any sl,sh&B, and define SL (s)= (s L - e,s L ] and SH (e) = (s H -s,s H ], such that
,
31
6
-e>
sH
x
.
we will supress the e in our notation. Suppose further that sl,sh ^B,
v H (SH )-v L (S L )<v H (S L )-v L (SH ). By definition, if s L > a then v H ([a,s L -e])>0; then, by
but
sL
.
For the moment,
,
- e ]) >
absolute continuity, v L ([a,s L
and thus K(s L -
s;
L)
>
.
Let g(s;8) be defined as
,co))-v l (Sh )
s Vl([s l
=
follows: g(s;8)
8
K{s L -s-e L )
-1
<
with
But
this g.
this implies that for
5>0 such
any s positive and
Vh(Sh (£))• Vl(Sl(s)) ^ V/zOSl (£•))• v L (SH (e))
But
.
e[s L ,<x)\S H
sgS„
szSL
straightforward to verify that there exists a
It is
s
[v l (Sh)
that the single crossing property fails
in the relevant range,
log-spm a.e.-C on B.
this implies that h{s;6) is
Since supp[K(-;9 H )]>supp[K(-;8 L )] in the strong set order, this implies that h(s;6)
a.e.-C
is
log-spm
on D.
Proof of Theorem
6:
Under the assumptions of the theorem, v(x,y,s) has SC3
(i):
u(x,s;b)
=
v(x,b(x),s) has
and only
if
U(x,0;b) = V(x,b(x),0) has
v(;t,y,.s)
Theorem
SC2
in (x;s) for all functions b.
SC2 in
if
and only
Furthermore, V(x,y,6) has
SC3
we know
that
So,
(x;0) for all functions b.
if
if
if
SC2 in (x;s) for all functions b. If k is log-spm &.e.-ju, then
has SC2 in (x;s) for all functions b. But this in turn implies that
has SC3, then u(x,s;b) has
3.1 implies that U(x,&,b)
V(x,y,0) hasSC3.
Consider any f(s;0). Let F(s;8)
(ii):
does not
fails
SCI
(MLR), then
satisfy
(this is
function g.
=
s
\_aa f{t;6)din{t)
there exists a continuous
The working paper shows
.
g which
We know that,
since
g
is
which are the boundaries of the region where
d=
cs
s\xp s(&{s\g(s)
= 0}
.
Now, we can
find a
S>
g(s)
= 0:
so that \g(s)dF{s;9)
Lemma 5).
continuous and crosses zero only once,
Now,
nondecreasing in some neighborhood of the crossing point/region.
points,
SCI
satisfies
a continuous approximation to the test functions from
that if F(s;0)
it
Consider
this
must be monotone
define the following
c = infSG ^{s\g(s)
- 0},
and two corresponding points,
= sup J<f {s\g(s) = -8} and ds = mfs>d {s\g(s) = 8} such that
,
(c,d)cz(cs,ds)
and g
is
new function, a(s), as follows: a(s) = 8 for s &{cs,dg),
Now, pick any x H > x L and let v(x, y,s) = x- g(s)/(x H - L ) + a(s) y
nondecreasing on (c&ds). Let us define a
a(s)
=
Igfa)!
elsewhere.
,
Since a(s) and g(s) are continuous and a(s)
%l%
=
g(s)/((x H - x L )
oc(s)) is
> 0,
theorem as well as SC3. Now, lv(x,y,s)dF(s;6)
satisfies
SC2 in
(x;0) for all b.
construction \g(s)dF(s;&)
Thus, Sv(x,y,s)dF(s;0)
fails
(iii):
If v(x,y,s) fails the
then,
Theorem 3
fails
v satisfies
nondecreasing in
Let&(x)
= 0.
SCI, which
s.
satisfies
SC3
Finally,
satisfies the
if
and only
But, v(x H ,Q,s)-v(x L ,0,s)
assumptions of the
if iv(x,b(x),s)dF(s;0)
= g(s), and by
in turn implies that jv(x,O,s)dF(s;0) fails
SC2
in (x;6).
SC3.
SC3, then there
exists a b(x) such that v(x,b(x)>s) fails
implies that there exists anf(s;0) which
jv(x,b(x),s)f(s;0)d/i(s) fails
(WB).
Thus, v
SC2 in
(x;0).
is
log-spm
a.e.-//
SC2 in
such
(x;s).
that
We conclude that lv(x,y,s)f{s;0)d^s) fails SC3.
32
But
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and S. Stern,
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Date Due
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