3 9080 02239 2317
HI*
M
.M415
no. 00-30
DEWEY
i3i
15
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
INVESTMENT AND INFORMATION VALUE
FOR A RISK AVERSE FIRM
Susan Athey,
MIT
Working Paper 00-30
November 2000
Room E52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Social Science Research
Network Paper Collection at
http://papers.ssrn.com
ST1TUTE
NOV
8
LIBRARIES
Investment and Information Value for a Risk Averse Firm
Susan Athey,
MIT
November
2000
2,
Abstract
This paper analyzes the problem faced by a risk-averse firm considering
in a risky project.
The
firm receives a signal about the value of the project.
sary and sufficient conditions on the signal distribution such that
nondecreasing in the realization of the signal, and
to their ex ante information value. Finally,
(ii)
identify a class of utility functions for
(i)
different signals
we provide
We
which agents who are
to invest
derive neces-
the agent's investment
is
can be ranked according
conditions under which
compare the incentive to acquire information across agents with
we
how much
it is
possible to
different risk preferences,
less risk averse
and
purchase more
information.
JEL
Classification
Numbers: C44, C60, D81.
Keywords: Portfolio problem, value of information, stochastic orderings, comparative statics
under uncertainty, monotone likelihood ratio order, monotone probability ratio order.
*Some
of the comparative statics results in this paper originally appeared in
an
MIT working
paper entitled "Comparative Statics Under Uncertainty: Single Crossing Properties and LogSupermodularity," which has since been shortened to eliminate these results.
to Christian Gollier, Jonathan Levin, Paul Milgrom,
discussions,
I
am
John Roberts, and Ed Schlee
grateful
for helpful
and the National Science Foundation, graduate fellowship and Grants SBR-9631760
and SES-9983820,
for financial support; I
am
also grateful to the
Cowles Foundation at Yale
University and the Hoover Institution for their hospitality and support. Corresponding address:
MIT
Department of Economics, E52-252C, 50 Memorial Drive, Cambridge,
athey@mit.edu, http://web.mit.edu/athey/www/.
MA 02142.
Email:
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/investmentinformOOathe
Introduction
1
This paper analyzes the problem faced by a risk-averse firm
project with uncertain returns.
project.
The paper
The
who must choose an investment
in a
firm can potentially purchase a signal about the quality of the
consider three comparative statics questions. First, under
what conditions on
the information structure will higher realizations of the signal lead to higher investments on the
we ensure that one
part of the firm? Second, under what conditions can
source) will provide higher ex ante value than another for
all
signal
(i.e.,
one information
such risk averse firms? Third, what
conditions on preferences guarantee that one agent will be willing to pay
more than another
for
better information?
It
turns out that the answers to the three questions are closely related. Consider the
tion. Formally, the firm faces uncertainty
to
G
Q C
signal
is
M..
The agent has a
X, with
given by Fxiwi'l^)-
investment problem
is
The
and observes the realization of a
X
x G
ques-
about the state of the world, W, with typical realization
prior, H(-),
typical realization
first
C
The
K.
posterior F\y\x('\ x )
is
partially informative
on
signal distribution, conditional
computed according
to Bayes' rule.
W
The
=
w,
firm's
given as follows:
max
/
w x {Lo\x),
u(i\{a,uj))dF
\
where uisa
and
(typically concave) utility function,
on the investment, a G A. The marginal returns
state of the world
(cj).
The optimal
statics question concerns necessary
the agent's level of investment
is
policy
and
is
7r
represents a return function which depends
to investment, -^7r(a,u>), are nondecreasing in the
denoted a*{x). In this context, the
sufficient conditions
first
comparative
on the signal distribution such that
nondecreasing in x.
For two extreme cases of assumptions on risk preferences, the answer to this question has been
established in the existing literature. First,
if
condition for investment to be nondecreasing
the agent
risk neutral, the necessary
is
and
that x orders the poterior distribution
is
sufficient
Fw x (-\x)
\
according to First-Order Stochastic Dominance (FOSD).
1
Next suppose that
risk preferences are
unrestricted (not necessarily risk averse). In this case, a necessary and sufficient condition for the
optimal choice of a to be nondecreasing in x
Monotone Likelihood Ratio Order (MLR).
versus low realizations of
W,
2
is
that
x orders the posterior distribution by the
MLR requires
The
/vy|x( w ffl a; )//W'|.x'( w i.l a; )> i s nondecreasing in x.
pointed out the disadvantages of the
MLR order:
it is
very stringent, and
l
See Ormiston and Schlee (1992) or Athey (1999) for alternative proofs.
2
See Ormiston and Schlee (1993) or Athey (forthcoming).
restriction that
7r a
assumption that n
(a,cj) crosses 0, at
is
that the relative likelihood of high
most once, from below,
supermodular (Athey, forthcoming).
More
it is
Many
authors have
equivalent to requiring
generally, this result holds imposing only the
as a function of
a;,
rather than our stronger maintained
that the distribution be ordered by
FOSD
conditional on any two-point set of states. Thus, a small
change in probability mass affecting only two points in the distribution can upset the order.
Now
consider incorporating risk aversion. Several important special cases of this problem have
been considered
au
+
—
(1
For example, the standard portfolio problem has n(a,u>)
in the literature.
where
o)u)q,
u>q is
=
the risk-free rate of return. Eeckhoudt and Gollier (1995) showed in
the portfolio problem that a sufficient, but not necessary, condition on
increase her investment in response to a higher signal
is
F
for a risk-averse agent to
that the posterior distribution,
Fw x (-\x),
\
satisfies
the "Monotone Probability Ratio Order"
low versus high realizations of
this ordering is
FOSD
allow for
Many
more
Fw \x{ u h\ x
W,
restrictive
than
FOSD
some "averaging" over
)
(MPR), which
/ Fwixi^L^)
but
states
than the
fairly special.
=
3
MLR. Both
As expected,
the
MPR and
Sandmo's (1971) model of a firm facing demand
(P(a)+u>)a — c(a), Milgrom (1994) shows that any comparative statics
For example, we might
shocks, for example by allowing
first
main
result
is
a;
to
Sandmo model. This
like to generalize
make
the
demand curve
less elastic.
that the finding from the portfolio problem can be generalized:
MPR order,
nondecreasing whenever the agent
Furthermore, although the
is
risk averse.
condition for comparative statics in the portfolio problem,
allow for
Now consider
of information.
functional form
the model to allow for non-additive
higher signals x increase the distribution in the
when we
x.
and thus lead to more robust orderings.
result that holds in the portfolio problem, also holds in the
Our
nondecreasing in
other economic examples motivate a more general investment return function than that
uncertainty, where Tr(a,uj)
demand
is
,
less restrictive
of the portfolio problem. In a generalized version of
is still
entails that the probability ratio for
if
the optimal investment policy will be
we show
that
MPR is not a necessary
it is
more general (supermodular) investment return functions
a necessary condition
it.
our second question, which concerns the agent's preferences over different sources
For example, an investor might have access to "inside information" about asset
A
returns, or she might purchase investment advice.
firm might engage in market research, or
it
might learn from experience or experimentation. Consumers might have access to different sources
of information about product quality.
Formally, in the ex ante information gathering problem,
the firm chooses between information sources, indexed by a parameter
9 generates a signal
posterior
X
9
and
,
and chooses an
after the agent observes the realization of this signal, she
/
max
/
w x e{u>\x)dFx e{x)
u{ir{a,u;))dF
Gollier (1995)
shows that a necessary condition
is
forms a
—
c(9).
\
6>ee
posterior distribution
Each information source
action. Formally, the agent solves:
max
3
6.
for
comparative statics in the portfolio problem
ordered by "greater central riskiness."
We
will
is
that the
not pursue that order here, because
not apply directly to problems with a structure more general than the portfolio problem.
it
does
we
In this problem,
and
Fwx
states,
<>,
Our approach
(1988)
derive necessary and sufficient conditions
on the
joint distribution of signals
such that higher values of 9 increase the agent's ex ante expected
utility.
Lehmann
to analyzing the agent's information preferences builds directly on
AL
and Athey and Levin (2000) (henceforth AL). Lehmann (1988) and
when
of deriving informativeness orders,
payoff functions are restricted to
lie
in
study the problem
some
U. 4 These
set
authors further restrict attention to "monotone decision problems." These are decision problems
where
(for a
given 9 and U)
all
posteriors in the set
such that, for every decision-maker in the class
Of
signal lead to higher actions.
of the
course,
it is
U
{Fw X 9 ('\ x )i x £
i
%
can be totally ordered
}
under consideration, higher realizations of the
just this comparative statics result that
is
the subject
question addressed in this paper.
first
Lehmann
(1988) studies the particular case where
U
is
the set of
single crossing property; Persico (1996) applies this order to several
all
payoffs that satisfy a
economic problems, including
AL
generalize
to allow for different classes of payoff functions. In particular,
AL derive
portfolio problems, but without fully exploiting the consequences of risk aversion.
Lehmann's (1988) approach
an information ordering
problem when the firm
However,
AL
for
is
supermodular payoff functions, which would apply to the investment
risk neutral.
did not consider the special structure that arises in the case of a risk-averse firm.
This paper provides a
new information
MPR,
is
ranking for the risk-averse firm
is
the risk averse firm, the
Just as the order over posteriors for
order for this case.
weaker than the
MLR and stronger
than FOSD, the information
weaker than Lehmann's order and stronger than AL's information
ranking for supermodular payoff functions.
Our
third question concerns conditions
on agent preferences that imply
ranked according to the information order identified above, one agent
than another
for a
more informative
On
the other hand, an agent
On
the one hand, an agent
who
is
more
may
risk-averse
not find
it
may
choose a policy that
The approach
Lehmann
is
of
(1988) and
necessary and sufficient for
many
authors
(see, for
(1951) informativeness order
is
how
who
is
AL
all
more
signal.
responsive to
on an approach taken
contrasts with the approach taken by Blackwell (1951, 1953). Black-
decision-makers to have a higher ex ante expected value from the decision
example,
LeCam
(1964),
Lehmann
(1988) and
often too strong to be useful in applications.
exploits additional structure that arises in
not comparable using Blackwell's order.
is less
risk
as valuable to purchase better information.
to the comparative statics of information gathering builds
problem. As
AL
be willing to pay more
Our approach
well's order
signals are
pay to avoid the uncertainty that comes with an uninformative
the realizations of signals, and thus
4
if
signal. Notice that there is not a clear intuition for
aversion should affect willingness-to-pay for information.
risk-averse might be willing to
will
that,
economic problems, allowing
AL) have pointed
The approach
for the
of
ranking of
out, Blackwell's
Lehmann
many
(1988)
and
signals that are
by Persico (2000)
for the case of single crossing functions,
and generalized by
AL to
arbitrary sets of
payoff functions U. This paper identifies (rather stringent) sufficient conditions on risk preferences
and
so that one agent will purchase better information than another,
family of utility functions for which agents
who
in particular,
we
identify a
are less risk averse are willing to pay
more
for
information.
The Investment Problem
2
Stochastic Single Crossing and Comparative Statics
2.1
In this section,
we introduce and
the paper. Let
X
and
on Q. Let TZ be the
we
define the set
if
R
be compact subsets of R, and
set of
UR
UR =
For example,
Q
for all
function /
x >
xq.
:
{u
|
u
X
:
X— R
>
we
P,Q
/
Q
—> R, and
V xh >
x that
weak
is
is
have:
are
1
//,
for
set of probability distributions
to R. For a given set
Q,
%l, u(xh, )
single crossing at xq
R (ZlZ,
weak
UR
is
-
u(xl,
€ R}.
•)
the set of supermodular functions
single crossing
if
f(x)
if
<
there exists
for all
x < xo and f{x) >
some x$ such that f(x)
is
single
5
E.V- Then
r{u))dP{oj)
Q
>
P
dominates
implies
Shannon (1995)) shows that
comparative statics
be the
bounded and measurable).
/
in the stochastic single crossing order for R, if
r(u))dQ(uj)
are stochastic single crossing orderings useful?
application of
Lemma
x
the set of nondecreasing functions,
is
crossing at xo- Then,
Why
V
as the set of functions with ^-incremental returns. Formally,
The function
Definition 1 Let
let
bounded, measurable functions mapping
(with incremental returns to
A
characterize several stochastic orders that will be used throughout
The
>
for
all
following
r
6 R.
Lemma
(which follows as an
stochastic single crossing orders can be used to derive
results.
all
order for R, then for
xu >
all
xi, Fw\x('\ x h) dominates Fw\x('\ x l) in the stochastic single crossing
g €
UR
,
there exists a nondecreasing selection a*(x)
from
argmax a f g(a,u))dFw x (u\x).
\
5
Given the assumptions we use
crossing order in terms of
property.
We
in this paper, all of
our results would also hold
Milgrom and Shannon's (1994)
if
we denned the
single crossing property rather
use the weak single crossing property for simplicity.
stochastic single
than the weak single crossing
r
Now
us define several important examples of stochastic single crossing orders, each cor-
let
We
responding to a different set of payoff functions R.
we
subsequently,
will identify the sets
P
Suppose that
First, consider
Second,
p,
with
FOSD.
let \x
Q
and
Q y FO SD P
Formally,
be a measure on
Q
denoted
P-, if
:
r is
-.
q(uio)
q.
and
i-i
single crossing m
•
weak
•
lo
at+ loq
a..e.-/j,.
P{u)q)
Q
>~lr-u>
Fw x {-\x)
x orders
€ Q.
Q are absolutely continuous with respect to
P dominates Q according to single crossing of likelihood
In other words, the likelihood of a state
requiring that
P
for all lo
6
gH
—
—gM
—
and suppose that supp(P)=supp(<2)=fi.
QM < P(u>)
if
such that
£1
p and
^-lr-ujo
(smaller) with q than p. If
which they are stochastic single crossing orders.
for
are probability distributions,
strictly positive densities
ratios at ujq,
R
begin by defining the orders directly;
by
>
lo
P f° r
MLR
(<)loq relative to the likelihood of loq
/^-almost
is
Q
then
all loq,
is
greater
>~lr P- In this languange,
equivalent to requiring that, for
all
xh >
#l>
\
Pw\x{'\ xh) >~lr Pw\x('\ x l)- Equivalently, the density fw\x must be log-supermodular, so that
fw\x{u xH )l fw\x( u} xL )
1S
nondecreasing in
to.
\
\
Third,
we can
similarly define single crossing of probability ratios, replacing the density with
Q ypR-uj P if grj) — puJ) s weak single crossing in lo at loq. If Q )~pr-u P
for all ljo, then Q >~pr P. When Q ypR P, then for every u>q, the likelihood of state loq relative
to the likelihood of all states lo < u>q is higher for Q than P. Requiring that x orders Fw x (-\x) by
the distribution:
i
\
MPR
is
equivalent to requiring that, for
all
xh > x l,
Pw\x('\ x h) >~pr Pw\x('\ x l)- Equivalently,
F\v\x must be log-supermodular.
It
MPR is weaker than MLR (since log-supermodularity is preserved
than FOSD (since the MPR requires that Fw\x{u xH I Pw\x(CJ xL
can be verified directly that
by integration) and stronger
)
\
is
nondecreasing in
to,
and the upper bound of the
2 Suppose that supp(P)=supp(Q)=Cl.
r nondecreasing.
if
and only
Q y PR
The
Q
Q{-
\
>~lr P-
(i)
The
1).
Q yFOSD P
following
if
an d on h
o)
1
too)
if J r
jj,-
1
1
provides
the relationships between them.
W G {wi,^}) >~FOSD P(- W S {^1,^2}) for
for
Q{- W < w
^FOSD P(- W <
(Hi)
lemma
dQ > f rdP for all
almost
all loq
G
all
{^i,^} G
Q
if
and only
Q,
if
p.
idea of this
two-point
6
if
(ii)
is
MLR, MPR, and FOSD, and
alternative characterizations of the
Lemma
ratio
)
\
We say
set,
Lemma
while the
that a function g
can be stated in words:
MPR is
:
CI
equivalent to
— R satisfies single
pair (using the measure induced by
p,
on
CI
2
),
g
MLR is equivalent to FOSD conditional on any
FOSD
conditional on any lower interval of the form
crossing almost everywhere
satisfies single crossing
(a.e.-|i)
on {wi,w«}.
if,
for
almost every
(ui l ,uj
h
)
{ui
:
u <
u>o}.
Parts
(i)
and
As a consequence
the definitions.
Lemma
of this
(ii)
of
(i)
and
are well-known, while part
x orders
(iii),
Fw x {-\x)
follows
(iii)
by the
MPR
if
by checking
and only
if
\
E^fr^^x, W < wo]
is
nondecreasing in x for
MLR order must
Note that because the
uo 6
all
Q and
r nondecreasing.
all
be checked for every pair of
states,
independent of
it is
H and a signal distribution Fx \wi ^ x orders the posteriors
Fw \x{-\x) by MLR, the same conclusion holds H replaced with any other prior distribution.
In contrast, the MPR order might hold for the posteriors generated by a given prior H and a signal
More
the prior.
precisely, given a prior
if
distribution
Fx w
,
while
it
Fx w
7
might
fail for
is
the posteriors generated by another prior
G
and the same
\
signal distribution
.
\
Our next step
is
to define several classes of payoff functions of interest.
which are weak single crossing, while rSC^o)
RND
wo!
the se ^
js
5C
r nondecreasing; _R
f
that, for
The
is
the set of
all r(-)
a \\ r (.) which are weak single crossing at
'
is
the set of
all
(
3( a;
°)
is
the set of
and quasi-concave, where the function changes direction
are in iJ
R sc
sc<3( a'o)
some
for
some
An
ujq-
important subset of
r nondecreasing, g{uj)
following
Lemma
=
l{ aJ < Wo }(w)
•
at ljq, while
n sc Q( u} o)
r(u>).
all r(-)
js
Clearly,
which are single crossing
R SC Q
is
the set of
the set of r which
all
functions g such
RND C RSCQ C Rsc
.
characterizes the stochastic single crossing orders for these payoff func-
tions.
Lemma 3 Suppose that supp(P)=supp(Q)=Q.
ND if and only if Q >~fosd P(i) Q dominates P in the stochastic single crossing order for R
(ii) Suppose that either fl is finite, or else P and Q have continuous, strictly positive densities with
respect to a measure
(a) For loq £ J7, Q dominates P in the stochastic single crossing order for
ftSC{uj
p. (b) Q dominates P in the stochastic single crossing order for
if an(j on iy ^ q y LR _
R sc if and only if Q y LR P.
sc ^ u
(iii) (a) Foruio > inf fi, Q dominates P in the stochastic single crossing order for R
if and
sc
®, if and
only if Q >~pr-cj P- (b) Q dominates P in the stochastic single crossing order for R
only if Q >-p R P.
,
/j,.
)^
UJo
,
°>
The proof
7
follows from
However, under these conditions the
MPR
would hold
for
3
and
4,
and Athey (1999). 8
any prior distribution formed from
H
(using Bayes'
W
< a.
by conditioning on the event
Athey (forthcoming) focuses on problems where
rule)
8
Athey (forthcoming), Theorems
,
(1994) single crossing property, rather than the
weak
R
is
the set of functions that satisfy Milgrom and Shannon's
single crossing property used here (where r
may become
positive
Athey (forthcoming), we may allow for supp(Q) to be greater than supp(P) (in the strong set
sc it can be shown that
order). When we consider the larger set R
Q dominates P in the stochastic single crossing
sc
order for
only if supp(P)=supp(Q)=fi. We maintain the assumption supp(P)=supp(Q)=f2 as a prerequisite
and then return to
0). In
,
R
rather than a conclusion in order to
make
it
easier to state the definitions of the relevant ratio orders.
.
To connect
part
of
(iii)
Lemma 2
the function defined by g(w)
=
and part
l[ UJ<u}0 y(oj)
of Lemma
(iii)
t{uj) is in
observe that
3,
H sc Q(u o) Further,
if r(-) is
Lemma
nondecreasing,
2 implies that
if
Q ^PR P,
Jr(u>)-l {uJ<u o} (u;)dQ(uj)
,
Q(wo)
l {ul<UJo }{u)dP{u>)
J r(to)
>
_
P(wo)
or, rearranging,
l {w<W0} ( W )dQ(w)
fr{ui)
>
|^| y r(«)
•
•
l {w<a o} HdP(u;).
,
This in turn implies
l {ltJ<llJo} (u,)dP(u,)
fr(u>)
In words,
form
Q
P
dominates
l{o,< Wo }(w)
•
=>
is
}(
u; )
r ( w )-
'
The
•
l {w<a o} (u;)dQ( W )
,
nondecreasing, a subset of
the counter-examples required for necessity in part
l{w<w
fr(w)
>
0.
in the stochastic single crossing order for the set of functions of the
where r
r(u>),
>
(iii)
of
Rsc<2.
Lemma
Further,
we can construct
3 using functions of the
form
following characterization lemma, proved in the Appendix, builds on this
logic.
Lemma
single
supp(P)=supp(Q)=Q. For to q > inffi, Q dominates
crossing order for R sc Qv^o) ( & q U ivalently, Q >~pk_ wo P) if and only if
4 Suppose
that
fr(u)dQ(u) >
2.2
^p\
I'r(u)dP-(u>) for all r
G
P
in the stochastic
RSG Q^)
(1)
Comparative Statics in the Investment Problem
Consider the investment problem of a risk-averse firm, as described in the Introduction, where the
firm has access to the signal
X and solves max ae
^i
J u(Tr(a,Lu))dFyy\x(Lo\x)
.
In our analysis,
we
will
refer to the following assumptions.
Al
For each x G X, supp{F-\y^x('\ x ))— ^-
A2
Either
fi is finite,
or else, for each
x G X,
Fw \x('\ x
)
has a strictly positive and continuous
density.
A3
u and
it
are
C2
,
the derivatives are absolutely continuous, and A, Q, and
X
are compact,
convex subsets of K.
A4
For
all 8,
x, the
J u{ix{a,w))dFw \x{u\x)
optimal choice of a
is
is
quasi-concave in a and
interior.
C3
.
Further, assume that for each
A5
7r(a, ui) is
nondecreasing in u, and further, the expected value of the project,
J ir(a, co)dFw x ( w x )
l
\
nondecreasing in
is
a.
Assumptions Al and A2 simplify the analysis by allowing us to work with the
without treating separately the possibility of moving support.
that
Q
if
is
not
finite,
Further,
A2
MLR and MPR
allows us to assume
the posterior distribution has a well-defined, continuous density, and that
The
each x, the likelihood ratio f(u>'\x)/f(u)"\x) exists for almost every (u',io") pair in Q.
for
differentiability
assumptions (A3) simplify the notation and analysis. Assumption
A4
allows us to
ignore the possibility of multiple optima, and to assume that the first-order conditions characterize
the optimal choice of investment (we clearly cannot impose the assumption
neutral, however).
when
the firm
AL
(A3) and (A4) can be relaxed; see Athey (forthcoming) and
Assumption A5 says that higher
states are better,
and that on average, the returns
are non-negative. These assumptions are satisfied in the portfolio problem,
is risk-
for details.
to investment
and they highlight the
elements of the portfolio problem that are critical for comparative statics results. Finally, note that
paper focuses on the case where w
this
is
u
represents
lead to
monotone
supermodular, incorporating the idea that
the marginal returns to the investment.
Now consider the role of risk preferences in deriving orders over posteriors that
decision problems.
The
following Proposition summarizes the comparative statics results for two
extreme cases: firms whose
risk preferences are unrestricted,
Proposition 5 Assume A1-A2, and suppose
«*(•) is nondecreasing for all
i\
supermodular,
if
u nondecreasing
only
if
Fw \x{'\ x
and only
(that
)
*s
all
is,
Fw x (-\x)
\
*s
risk neutral firms.
there is a unique optimal action for each x.
u nondecreasing and
if
and
linear (that
ordered by
FOSD.
(ii)
all
a*(-) is nondecreasing for all
firms with arbitrary risk preferences) and
ordered by
neutral firms), and
all risk
is,
(i)
all
n supermodular,
if
and
MLR.
Proof. Under the assumptions of the proposition, weak single crossing of the incremental returns to
a
is
necessary and sufficient for the comparative statics prediction,
= u o it,
g(u>,a)
Lemma
3.
functions.
Now
(ii)
where u
is
Taking the
Apply
Lemma
nondecreasing and
class of linear,
7ris
supermodular, unless u
However,
nondecreasing and concave, ^g{uj,a)
all
.
Apply
3.
monotonicity restrictions.
not
UR
nondecreasing u returns the class of supermodular payoff
preserved by nondecreasing, concave transformations.
ir is
Taking the class of functions
supermodular, generates the set
consider the problem faced by a risk-averse firm.
supermodular when
(i)
for
is
Thus, g(u>,a)
linear, or else
any g{u,a)
satisfies
Notice that supermodularity
= u
o
7r
u
is
where
= u
o
tt
convex and
7ris
not
not necessarily
tv
satisfies
some
supermodular and u
the single crossing property.
But,
functions with the single crossing property can be generated in this way.
8
is
is
if
u
is
is
concave,
Since the relevant set of payoff functions for risk averse firms
ular payoff functions, but smaller than the set of payoffs
larger than the set of
is
whose marginal returns
supermod-
satisfy the single
crossing property, the comparative statics result for risk averse firms requires an intermediate con-
on the
dition
distribution. It turns out this condition
order for the set
of functions R sc ® defined
given that u' (Ti(a,uj))n a (a,Lj)
is
all ix
MPR,
is
if
and only
nondecreasing in x for
is
x orders
if
MLR and stronger than FOSD.
Eeckhoudt and Gollier (1995), who establish the
is
somewhat
Fw x (-\x)
<
all
The
u concave and nonde-
MPR.
by
for
comparative
find x,
u concave, and
approximate
concave and
7T
it is
strictly larger
is
we construct
in
we can
x, u'(Tr(x,u>))-J^Tr(x,u>),
In other words, the set of functions of the form g
supermodular, contains (and
of
r(w); but, for any such function
supermodular, so that the marginal returns to
this function.
The proof
result for the portfolio problem.
l^ <a y(uj)
3 use payoff functions of the form
statics,
sufficiency part of this result generalizes
necessity builds on our discussion from the last subsection: the counter-examples
Lemma
surprising,
R sc ®.
Appendix. As expected, the condition
result is in the
weaker than
This
in the last subsection.
Then &*()
supermodular,
The proof of this
equivalent to the stochastic single crossing
not necessarily in the set
Proposition 6 Assume A1-A5.
creasing and
is
=
u
o
it,
where u
is
UR
than)
3
The Value
Now
consider the problem of comparing the value of different signals. In particular, suppose the
of Information
{X e }g £ Q, where
indexed by
8.
Formally, the firm has a fixed prior, H(-), about the state of the world. Then, the firm chooses
8,
firm can choose from a family of signals
which determines the
set of possible posteriors
the quality of the signal
{Fw x o (-\x) x €
,
X
e
}
is
as well as the likelihood of
\
each. After observing the realization of the signal
X — x,
e
the firm forms the posterior
FW X
9 ('\
\
and solves max a£ _4 f u(Tr(a,to))dFw
When
x e(u)\x).
evaluating the problem from an ex ante perspective, for each
signal distribution,
noted
\
Fwx e(-,
•).
Fx w
ei
The
,
probability
x),
joint distribution
and
8,
the prior
H
and the
determine a joint distribution over states and signal realizations, de-
marginal distribution of the signal
Pr(W < u\X e <
X)
is
denoted
Fx »{x\W <
mass functions or
Fwx e(-,
•)
is
referred to as the information structure.
Fx e(-).
The notation i<V(u;|X
w) represents Pr(X e
< x\W <
co).
<
The
x) represents
The corresponding
densities are denoted by /.
For a given payoff function
:
AxQ—
>
M.
(such as g
=
u
o
77),
the ex ante value of information
given by
is
V*{9;g)=
max
/
g(a,u)dFwlx e(uj\x) dFxe {x).
/
J x e laeA J n
In the next subsection,
we review and extend the
-2-1
conditions such that -§sV*{9\
uo
90
77)
>
results
AL
from
that give necessary and sufficient
for all payoff functions in a given class.
Monotone Information Orders
3.1
AL
develop a theory which can be used to order information structures for payoff functions of the
form g
AxX—
»
:
UR
functions
As
,
A
higher value of 9 provides a "more informative" signal for a set of payoff
£§V*(9;g)
if
Lehmann
in
R.
(1988),
>
g
for all
UR
<E
.
AL's approach considers only "Monotone Decision Problems." That
is,
the set of admissable information structures must be "i2-ordered," so that the stochastic single
crossing order for R, denoted >-r,
all
xh >
UR
g G
is
is
a complete order on {Fw{-\x)} X £X-
xl, Fw{-\xh) >~r F\v(-\xl).
If
Fwx
13 implies that for
argmax ae ^ f g(a, Lj)dFw X B{u>\x) such
there exists a selection a*(x;9) from
,
Lemma
.R-ordered,
is
e
In other words, for
all
that a*(x;9)
\
nondecreasing in x; thus, throughout we assume that the agent chooses a nondecreasing policy.
For each of the different sets of payoff functions
restrictions implied for
The
restriction to
we have
considered,
Lemma
3 describes the
Fwx e.
monotone decision problems allows us
when the
decision-maker's preferences. In particular,
inherits properties
R
from
g(a,cj).
For example,
if
g
make use
to
decision policy
is
is
of our knowledge of the
nondecreasing, g(a*{x\
supermodular and
Fw x b(-\x)
is
9), to)
ordered by
\
FOSD, then
An
a*(x; 9)
nondecreasing in x and g(a*(x; 9),oj)
realized
X
e
supermodular in
has no intrinsic meaning to a decision-maker:
a transformation
Y = T(X
Why
9
is
9);
;
T X
:
x
so long as the ordering
like,
6 —
>
Y
d
is
Our approach
is
is
that the value attached to a
all
that matters after the signal
preserved. For example,
and
strictly increasing in x,
[0, 1]
we
for
each
9,
It
consider
consider the signal
as
X8
.
plays a central role in our characterization of the
to ranking information structures can be outlined as follows.
XH
and
XL
L To do so,
it is
sufficient to
9^, with corresponding signals
X
"more informative" than
.
.
We
will provide conditions
show that
for
that expected payoffs are higher than
when
under
each g G
there exists a policy (not necessarily optimal) that a decision-maker can use with signal
X
are free to
we can
must give the decision-maker the same payoffs
such a normalization important?
9h >
XH
e
having access to
information orders.
Consider
(x, u>).
the posterior distribution over the state of the world. For this reason,
is
normalize the signal as we
which
is
important thing to note about the setup of the problem
realization of
is
is
XH
,
UR
,
such
the decision-maker uses the optimal policy with signal
L
.
10
Consider how this might be accomplished.
o*(-;8l)
9h)
£*(;
'
'
L —*
A. This yields expected payoffs V*(#l; p).
X
XH —
^4,
>
For example,
function a(x; 9h)
=
L
A=
if
a*(x; 8i)
Instead, our approach
[0, 1],
—x
(3*(-,8 L )
[0,1]
:
A
X =
X
mined.
in the
and X H =
we have not
policy,
then to construct a policy,
is
XL
rather than
[-100, 100],
if
)
yields ex ante
how
a*(x; 8 L )
T X9 x Q —
Y H = T{X H ;8 H
L
and
)
>
:
).
a*(x;8 L ), so that using
to construct
=
x, the the
[0, 1],
and
rescale
Next, we define
L ) with signal
/?*(•;
YL
is
Then, our approach to ranking information structures
.
access to signal
YH
can do better than the agent with
YL
#l)- Since the realizations of both
who has
YH
access to
yet said anything about
how
main
following proposition includes the
results
YH
and
in [0,1],
lie
.
this transformation, T, should
turns out that the answer depends on which class of payoff functions
It
The
tion.
/?*(•;
well-defined for an agent
course,
=
who has
L
access to Y using the same policy,
Of
XH
not immediately obvious
it is
Y L = T(X L ;8 L
by fi*(T(x;8 L );8 L )
reduces to showing that an agent
is
goal
to choose a particular transformation,
is
equivalent to using «*(•; Ql) with signal
this policy
[0, 1]
and take the optimal
,
not well-defined.
is
each signal, defining two new signals
-»
Our
that (when the decision-maker has access to
expected payoffs greater than V*{&L;g). Notice that
this policy.
UR
Consider g G
is
be deter-
under considera-
about information orderings;
proved
it is
Appendix.
Proposition 7 Let
H
be a prior probability distribution over Q, let
eterized signal distribution,
and
let
Fwx
Fx w
be a
e<
smoothly param-
be the corresponding family of information structures.
e
Assume A1-A2.
(i)
Suppose that for each
8,
Fw x b(-\x)
ordered by
is
\
and
all
g supermodular,
Fw
XB
1
by >~lr-uo
i
T(x;u>,8)
For
(Hi)
all u>o
8,
T(x;u,9)
For
is
if,
letting
T{x\ 8)
ordered by
Fw x e{-\x)
is
\
n x for eac h ^o)- Then
™ 8 for all y
> F OSD
ordered by the
-§qV*{8\ g)
>
Then -§gV*(8;g) >
x.
for
MLR
all
in
8 and
x
all
G
is,
R
g G U
all
6
(MIO-spm)
[0, 1].
(that
for
Fw x e(-\x)
\
,
if
is
and only
ordered
if if
we
= Fx w (x\u)),
e\
G
fi
and y G
[0, 1],
Suppose that for each
by ypn-ui
and only
> T~\y; 8))
Suppose that for each
(ii)
let
{-
if
FOSD in
= Fx e(x),
in
x for each
8,
Fw
{-
1
X9
Fw x e(-\x)
loq).
i
is
-1
> T (y;u;o,0))
ordered by the
is
ordered by )^lr-u>
MPR
in
x
(that
is,
in 8.
Fw x e(-\x)
(MIO-sc)
is
ordered
\
Then j^V*(8\g) >
for
all
g G
UR
,
if
and
only, if
we
let
= Fx e(x\W <u),
all loq
>
mtft,
and y G
[0, 1],
Fw{-
\
X9
> T~
l
{y\LOQ,d)) is ordered by
>pr- Uo
in 8.
(MIO-scq)
11
Part
of this proposition
(ii)
originally
is
due to Lehmann (1988);
AL. Part
of single crossing orders over average posteriors by
new
was
first
stated in terms
due to AL, while part
is
(i)
it
(iii)
is
(MIO-sc) has been applied in economics in the area of principal-agent problems (Kim,
here.
and auctions
1995; Jewitt, 1997)
AL
(Persico, 2000).
apply (MIO-spm) to a number of economic
problems, including adverse selection problems and oligopoly games.
To
interpret the three conditions, consider 9 H
Notice that
all
>
6 L and the corresponding signals,
X H and X L
.
three conditions are comparisons across information structures of the average over
"high" posteriors (that
is,
posteriors corresponding to high signal realizations) where posteriors are
,
ordered by the relevant stochastic single crossing order. Each monotone information order requires
XH
that the "high" posteriors for the good signal,
the bad signal,
X
L
when
are "higher" than the "high" posteriors for
,
monotone
correspond to higher investments. For example, consider U R
Recall that
.
the agent uses a
agent whose return to investment changes direction at
chooses a policy where, for some y, she invests
implies that,
cjo),
if
the agent instead observes signal
when
XH
and
,
let
If this
ujq.
X
policy, higher signal realizations
L
A=
{0,1}, and consider an
agent observes signal
> F~\{y\W <
invests exactly
she
,
Then, (MIO-scq)
u>o).
when
XL
XH
> F~},(y\W <
the average over the agent's posteriors conditional on investment are higher (according to
Since the average of the posteriors
is
the prior, the conditions of Proposition 7 likewise require
that the "low" posteriors for the good signal are "lower" than the "low" posteriors for the bad
signal.
Thus, another way to interpret the conditions
is
that they require the posteriors of an agent
given a good signal to be more "spread out" in the appropriate stochastic order.
The
three
they also
monotone information orders
differ in
not only in the relevant order over posteriors;
that different "normalizations," T, of the signals are used.
are tailored to the class of payoff functions,
Observe that in part
u>q
and they are derived
(and similarly, in part
(ii)
(MIO-sc) by requiring that
why, observe that
differ
Fw
{\
X
9
>T
in the proof of the proposition.
we cannot
(iii)),
_1
ordered by the
(y; lj, 9)) is
appears twice in the statement of (MIO-sc):
normalization T, and second,
we check
The normalizations
first,
simplify the statement of
MLR in
we
9 for
condition on
that the posteriors are ordered by >-lr-uj
that the crossing point specified for the stochastic order
is
the
same
-
^
To
see
ujq in
the
all uj.
important
is
as the crossing point used in
the normalization T.
Finally,
jjR
rpj^
"o
^
to
we note
that
we could
also
orcj ers over posteriors
AL establish that (MIO-spm)
Lemma 2 (ii)). To verify that
the definition of (MIO-scq).
It
modify parts
(ii)
and
(iii)
to consider the sets
would simply be checked only
holds for
all priors, if
(MIO-scq)
is
and only
if
UR
"°
for the relevant ojq.
(MIO-sc) holds
stronger than (MIO-spm), simply
analogous
(this is
let ujq
=
supfi in
can further be shown that, given a prior H, (MIO-scq) holds,
12
and
if
and
only
if
(MIO-spm) holds
= H(u\W <
form G(u>)
for all priors of the
Lemma
a) (analogous to
2
(hi)).
Monotone Information Orders
Characterizations of
3.1.1
This section provides two alternative characterizations of (MIO-sc) and (MIO-scq). The charac-
may be
terizations
independently useful, because
it is
stated in terms of signal distributions rather
than posterior distributions.
The
Lemma
For
(c)
characterization
first
8
For
(b)
G
)
x Q, and
[0, 1]
(y,w
all
G
)
9h >
9; (d) for all
(ii)
The following four conditions are
(i)
all (y,u)
all
x Q, and
[0, 1]
Ol, F~]j
>
v
ui
v
all
(Fx l(x\W —
,
<
Fx
cj
loq)
(F~l(y\W
e
Fx
,
For
(y,w
all
For
G
)
all (y,LO
in 6; (d) for
all
Condition
G
)
6H
x Q, and
[0, 1]
>
[0, 1]
6L
is
Lemma is
we seek
MLR) we
signal
when
X
L
,
if
Fx l{X \W =
V-
We
of the world
v
uiq)
say that
is
v
>
XH
Fx w
is
e
<
most
Now
y.
e
,
is,
)
|
W<w
)
is
(ii) is
new
MLR,
size,
to
that
more informative than
thereafter.
r,
is
in
all
x G
XL
.
nonincreasing in
nondecreasing
is
for
u>
Lehmann
all
x G
(1988),
XL
and
.
AL
here. First, consider the interpretation of
W>
loq
against the alternative
with probability no greater than
implement a
is,
XL
test of size y
signal,
we
if
given the size of test y,
is
smaller with
XH
,
With
and consider
Fx h (X h \W =
if
the true state
X H than X L
up when the true
ojq.
reject the null
XH
a test that rejects the null
if,
W=
when
.
state
Part
is
v
(i)(c)
<
ujq.
provides a uniformly more powerful
L
.
consider the interpretation of part
investment,
for
a test of size y. Observe that (given posteriors that
In the terminology of classical statistics, for a fixed size y,
Now
loq
nondecreasing in
requires that the probability of rejecting the null hypothesis goes
X
is
)
likely to reject the null incorrectly
same
nonincreasing in
v)
\
wo, the probability of rejecting the null
hypothesis test than
W=
suppose we gain access to another
test of the
is
|
ordered by (MIO-scq);
to test the null hypothesis that
the posteriors are ordered by
l
implementing a hypothesis
w o) <
are
)
the statement of (MIO-sc) given by
true, that
is
ui
nondecreasing in
is
test that rejects the null hypothesis
y when the null hypothesis
are ordered by the
=
e
,
equivalent to (i)(d); part
and we want a
u>o,
all
e
Fx (F~}(y\W < loq) W < v) is
< w Fx (F~j(y\W < uj \W <v)
u>
F~ H (Fx l(x\W < u
,
part (i)(b). Suppose that
<
x Q, and
>
(a)
l
(i)(d) of this
show that (MIO-sc)
v
all
[F~] (y\W
e
W = wo)
\
Fx w is ordered by (MIO-sc); (b)
= loq) \W = v) is nonincreasing in 9;
equivalent: (a)
The following four conditions are equivalent:
9; (c)
W
given as follows.
is
# sc,(2( w o)_
Suppose that
A=
j
n wor d S) r
{0, 1}.
Then,
(ii)(c).
is
for
Suppose that the agent's marginal return to
nondecreasing on
each
13
9,
u>
<
ujq,
and remains positive
consider the policy where, for
some
y,
the
agent chooses to invest
that
it
the true state of the world
if
Xe
greater than
is
in the set {lo
is
<
u>
:
F~]{y\W <
v}, where v
<
loq,
that the agent chooses to invest. In contrast, by part (h)(b),
less likely
makes
signal
the realization of
if
more
it
Part
(ii)(c)
requires
a "better' signal makes
when
«>woa
"better"
chooses to invest. Given the agent's preferences, these
likely that the agent
changes increase ex ante expected
a>n).
utility.
Notice that, unlike (MIO-sc), (MIO-scq) places requirements on the "average" signal distribu-
That
tions.
W.
is,
in order to
compute
Fx l{-
|
W<
we need
u>q),
prior distribution over
Thus, while (MIO-sc) can be interpreted in terms of "classical" hypothesis testing in
giving conditions for signals to be ranked for
make use
we must be
As AL
more
discuss in
of payoff functions R SC Q
of the additional structure on the set
detail, in order
MPR
order and (MIO-scq) are unchanged for a given signal family
allow for an alternative prior of the form H{lo)
The next lemma
= H(lu\W <
Fx e(-\u;)
of
x that implements
only
if
9
(i)
than
The
y.
is
=
F~],
supermodular in
—
Define x(y;9,uo)
supermodular in
=
w (y
F~]{y
(6,ljq), that
\
o>
W < uq).
9 (
interpret the result of
X
'Lemma
and
W, where
the "cutoff" level
Then
u>o).
Fx w
e
is
ordered by (MIO-sc)
if
and
is,
is
nondecreasing in
loq
for
all x.
(2)
Then
if
Fx w
e
is
ordered by (MIO-scq), x(y;9,u)o)
is
)
,
,
is
nondecreasing in
u>o
for
all x.
(3)
)
Lemma
9
positive
dependence
(ii),
-^j-x{y\ 9,loq)
is
a measure of the positive dependence
defined to increase expected payoffs for the
is
To see this, we
H = ~§gF o\ (x\LJo) > 0. Then,
)
(2)
x W
9 can also be used to establish directly the relationship between MIO-sc and MIO-scq.
can define h{uj ;v) for
is
how
9
is,
fx e{x\W <cj
s
W=
places conditions on
signal,
)
4 FX__x\W <u)—
between
Suppose that
u>o)
—-22
To
\
(6,loq), that
7)qFX 9(x\W =
—
—
—
=
fx e(x\W
(ii)
Lemma
following
changes in 8 and wo-
this policy
Define x(y; 9,ojq)
x(y;8,iVo)
we
than the inverse distributions. The
an investor follows the policy of investing when the realization of the normalized
;9,u)o), is greater
Lemma
if
builds on the characterizations from parts (i)(b) and (ii)(b), stating the condi-
characterizations can be understood in terms of the transformation function, T.
T(X
),
v).
tions directly in terms of (average) signal distributions rather
e
R sc
(rather than
willing to specify a restricted class of prior distributions. Notice, however, that given
a prior H, the
for all 9,
statistics,
possible priors, (MIO-scq) requires a Bayesian
all
perspective, in particular, a pre-specified prior distribution.
to
know the
to
nondecreasing in
ljq
L
i)
e {v ,v"}
as follows:
when
h{uio;if) is
exactly
log-supermodular (Karlin, 1968),
(2)
h(u
;T)
L
)
=
e
f
(x\ujo), h(uj ;r}
log-supermodular. Since integrals of log-supermodular functions are
nondecreasing in
u>o
implies (3)
14
is
nondecreasing
in u>
-
relevant class of decision-makers (in this case, the
MPR)
the positive dependence (in terms of the
Finally, consider briefly
r
nondecreasing
is
if
Condition
.
X
between
9
and W.
(3) requires
that 9 increases
10
an analogous characterization of (MIO-spm). In particular, notice that
and only
The requirement reduces
MPR)
to
if it is
J|
in _R
~Pi(X
e
?M. Now,
5C,<
< F~}{y)
(MIO-spm). Equivalently, -§gFWX 9(v,F~l(y))
\
>
consider
W
<
v)
for all
Lemma 8,
>
part
for all v,y.
(ii)(b),
This
is
when
cjq
= u.
equivalent to
n
v,y.
Information Orders for a Risk- Averse Investor
3.2
For the problem of a risk-averse investor, payoffs satisfy single crossing of incremental returns, and
so
we can apply (MIO-sc). However, (MIO-sc)
must hold conditional on any pair of
is
potentially a very strong condition. Its restrictions
states. In contrast,
(MIO-scq) allows some "averaging" over
states.
Thus, we would
like to relax
(MIO-sc)
.
The next
result establishes that
(MIO-scq)
is
the relevant
information order for risk-averse investors.
Proposition 10 Assume A1-A5. Assume
Then
x.
if
for
result
is
all
Fw x e(-\x)
i
u nondecreasing and concave and
all tt
is
ordered by the
supermodular,
if
MPR
in
and only
(MIO-scq) holds.
The proof of this
it
>
-ggV*(9;u,Tr)
that for each 9,
in the
Appendix.
It builds
uses the characterization of (MIO-scq) derived in
The Information
4
The information
Lemma
= argmax
^"
is
9.
/
/
Jx
Juj
wx e(oj,x) - c{9),
u(-n(a*(x\9,p),Lo);p)dF
p,
which describes the agent's
risk preferences. Consider
the following question about the comparative statics on information gathering:
To
When
of information acquired by the firm increase in p?
begin,
we
present a result that applies to the class of payoffs
UR
10
Persico (2000) provides an analogous interpretation of MIO-sc. See also Jewitt (1997).
11
See
AL
and
stated as follows:
where we have introduced a new parameter,
amount
7,
Acquisition Problem
acquisition problem
9*{p)
on the proofs of Propositions 5 and
for further characterizations of
(MIO-spm)
in
terms of "marginal-preserving spreads."
15
does the
X
Proposition 11 Let
H
be a prior probability distribution
and
eterized signal distribution,
Assume A1-A2. Suppose
that 6
w(x,u)
If for allx H
When
Fw be
orders Fwx
let
let
f2,
FX W
9\
be a
smoothly param-
the corresponding family of information structures.
9
by (MIO-scq). Define
e
=
over
- g(a*{x;9,p L ),uj;p L ).
g(a*(x;8,p H ),w;p H )
> xL w(x H ,co) -w{x L ,u) 6 RSCQ
,
then -§gV(9,u)
,
>
-§gV{0,v).
the relevant functions are differentiable, the requirement of the proposition
is
that
^0^g(a*(x;6,p),uj;p) £ R sc ® In other words, p makes the incremental returns to investment
"more R SC Q." For example, this condition would be satisfied if g(a*(x;9,p),u>;p) = x r(u>), for
.
p
r
G R sc ®. This
result
is
analogous to results established for
R sc
in Persico (2000)
and
R ND
in
AL.
Now
lio
consider applying this to a risk-averse investor.
problem, where ir(a,u)
= aw +
—
(1
a)wo
(this
is
Let us restrict attention to the portfo-
not necessary, but
it
simplifies
an already
complicated analysis). Further, we add the following assumption:
A6
There
exists a c
>
such that for each
Further, the investor
The
satiation condition
is
is
=
a*(x;9,p)
eventually satiated: ui(c
clearly undesirable, but
Consider small changes in
$(x,uj;8,p)
9,
p,
it
Co;
p)
>
=
c for all realizations of the signal.
0.
greatly simplifies the argument.
and define
—[a*(x;0,p)uu(ir(a*(x;9,p),u);p)]
dp
u 112 (iT(a*(x;9,p),Lo);p) a*(x;8,p)
-a*p (x;9,p)
too
uui(^(a*(x;9,p),iv);p)
Uu(Tv(a*(x;9,p),u});p)
+a*(x;9,p)
+
Then, we have the following comparative
Proposition 12 Assume A1-A5,
let
(without loss of generality) that each
for
all 9,
Fw x e{-\x)
\
ing to (MIO-scq).
is
statics result for information acquisition.
A=
X8
ordered in the
u in (iT{a*(x;9,p),Lj);p) Tr(a*(x;9,p),Lo)
is
[0,1],
and suppose
normalized so that
MPR
in x,
Then, whenever ty(x,w;9,p)
nondecreasing.
16
<
= aw + (1 — a)u>o. Assume
-§gFx e(x) = 0. Suppose further that
ir(a,u})
and 9 orders
and a*
is
the signal family
Fwx
e
accord-
supermodular in (x,p), 9*{p)
is
To
make
if
interpret these conditions, observe that
the agent
more
a*
if
is
make the
in (x, p)
sensitive to signal realizations. Typically, this effect
* <
Then, $ <
p reduces risk aversion. The requirement that
to zero, only the
supermodular
term
first
utility function
important.
is
more concave, which
is
Kin <
0.
higher values of p
would be
more complicated.
if
,
If
a*
likely to
is
close
hold
enough
In words, higher values of p
typically corresponds to greater risk aversion. This
captures the idea that for a given decision policy, greater risk aversion makes information more
valuable.
The
If
a*
other terms in capture the effects that arise because p changes the optimal decision policy.
>
(i.e.,
higher values of p
make
the agent invest more), then p can be interpreted most
naturally as a decrease in risk aversion.
second term
is
Under the same
negative.
relative prudence,
—
U
^(^.) W,
is less
is
>
1.
>
and um(w,p)
conditions, the third term
than
a future (multiplicatively separable) risk
in addition, ojq
If,
negative
is
then the
0,
if
the agent's
Since an agent's incentive to save in anticipation of
increasing in relative prudence, this requirement can be
understood as a restriction on the extent to which an agent
is
willing to
pay to avoid exposure to
a multiplicative risk.
What more can we
Consider
first
an agent with
CARA
increase in p (a reduction in risk aversion)
a*
is
and a*
mean x and
assumptions are required to obtain this
\I>
supermodular
in (x,p)?
makes the agent more responsive
<
result.
some 7 >
the
13
an
to signal realizations,
for posterior beliefs, additional
Now, consider our
are that, for
If
variance independent of x,
supermodular in (x,p). For more general functional forms
verified that sufficient conditions for
is
12
preferences, with coefficient of risk aversion —p.
posterior beliefs have a normal distribution with
i.e.,
^ <
say about preferences for which
on ^.
restriction
0, 1
> —p >
7,
It
can be
and wo and
inf
Q
are both greater than 2/7.
Now
consider another specific functional form, a quadratic utility function, where
to provide a definitive prediction. Let ujq
p
>
Co.
Then
define
u
:
Q—
>
M.
=
0,
G
(0, 1) for all x.
is
Note that
Q=
[w,o>]
= pm — —m
The
with
Co
>
>
lo,
and consider
.
not necessarily supermodular. Suppose that
Eiy^lX 6 =
x]/Ew[Vl/2 |^' e
=
x]
first-order conditions for optimality entail that
<**{x,p)
12
[0, 1],
possible
by
u(m)
Notice that u(au)
A=
it is
=
E W [W\X B = x]
P
E W [W 2 \X 9 = x]
this utility function does not satisfy our satiation requirement, so to
apply our proposition directly we
would need to extend the arguments of Proposition 12.
13
Persico (1996) also analyzes this case, using Lehmann's (1988) information order.
17
If
Fw \x
9 ('\
x)
is
ordered by the
that the optimal policy
is
MPR,
the optimal policy
supermodular in
(x, p)
nondecreasing in x. This in turn implies
is
Observe further that
.
E w [W\X e = x]
Ew [W 2 \X 9 = x]'
/
*/
n
^
^
x
a
a
a *,(x;6,p)u
n (7r(a (x;0,p),u>),p) = -p/
Since the optimal policy
0).
non-negative, the expression
is
nonincreasing in p (so that ty(x,Lu;
is
6, p)
<
Notice that
~u
—
—{m;p) —
"
\
(
«'
l
p-m
a
and
d (~ u
—
—— fm;p)^ =
dp\u>
-i
"
(
,y
y
'J
*.
(p-m) 2
So, risk aversion decreases with p. In this case, the result says that as the agent
averse, the policy
To apply
of signals,
The
becomes more
sensitive to information,
{X
less risk
and information becomes more valuable.
these results, suppose that an investor can choose
9
becomes
among
the
members
of a family
8 € 0}, where the posteriors from each signal are totally according to
:
might correspond to
different signals
different opportunities for information gathering,
as annual reports of a firm, a full auditor's report, or being a
manager of the
willingness to pay for signals in this family will be increasing in 6,
if
each
ir^Q-ordered, and 9 orders the family by (MIO-scq). Then, the results in
to provide conditions under
which
MPR.
less risk averse
firm.
member
The
such
investor's
of the family
this section
is
can be used
agents purchase signals that more informative
according to (MIO-scq).
Finally, observe that
(MIO-sc) in place of
Persico (1996) or
5
if
we
are willing to impose the
MPR and
AL
more stringent requirements of
MLR
and
(MIO-scq), the requirements on risk preferences can be relaxed. See
for details.
Conclusions
This paper derives
sufficient conditions for
comparative statics on investment and preference order-
ings over information structures for the problem faced by a risk averse firm,
make an investment
where the firm must
decision on the basis of a noisy signal about the investment returns.
of payoff functions generated
by
class of decision
functions, but smaller than the set of payoffs
problems
is
The
set
larger than the set of supermodular
whose marginal returns are
single crossing. Since the
orders over posteriors and information structures induced by the set of single crossing payoffs are
fairly restrictive,
the weaker orders derived in this paper are potentially more useful in applications.
Using these orderings, we analyze how the value of information changes with
We
show that
information.
for a fixed policy,
making a
However, a decrease in
utility function
risk aversion often
18
risk preferences.
more concave increases the demand
makes an agent more responsive
for
to an
informative signal, and for some utility functions (such as quadratic), this effect dominates. Then,
who
agents
Appendix
6
Proof of
Lemma
necessity,
observe that R sc Q(uo)
P
Q
>-r
of
Rsc Q(
and only
if
UJ o)^
The
4:
fact that (1)
\iQy R P
j
g
is
sufficient follows
& c j ose(j convex cone.
there exists a p
if
>
R = RSCQ{u
for
To prove the
G RSCQ("o)
of RSCQ(wo)_
and only
if
That
t
)/P{u
But, for any closed convex cone R,
l{ UJ<u>0 }(u))
Thus,
).
holds for
if it
by
directly, observe that
£SCQ(u>o)
is,
= Q(u
and
6 R. u In the case
= — l{ ul<UJQ }(u))
t(lj)
holds for
(1)
for all r
r e rSCQ{u>
all
)
and {r
)
for
:
the un j on
ig
o
Q >pr-u
to
>
wo;
f
all
linearity of the integral, (1) holds for all
r
6 E SCQ ^°\ where
E SCQ{-^
is
a set of extreme points
R SC Q("°)
a set of functions such that any r e
is
the following sets: {-1{ W < W0 }( W )}>
K^O —
^-{uokuko}^)}- Then,
can be formed
£,5C(?( o) The se t
r(u) = l{ a <u,<Lj }{u)},
aJ
of
(iii)
Lemma
Consider
6:
and a
a;
<
>
loo
It
u' (TT(a,u}))-i^it(a,cj) is
so that 7r(a,w)
and
u' ra
for u>
Lemma
positive
,
>
13 Consider n H
<
oj
,
straightforward to verify that (1)
first necessity.
The counter-examples
equivalent
is
required for necessity
a
•
we can
also find a
r(u>) for all
is
u, and then given this
7r,
this,
choose u so that v!
choose n
«
when
1
concave.
Before beginning,
sufficiency.
Now,
r(o>).
u concave, a n supermodular, and an
j
This u
u>o.
o»o,
approximately equal to l{ a; < U }(w) •r(w). To see
>
g(u>,T]
r;
L
,
we need
H) >
and suppose that for either
g(u,r) L ) for all
and continuous, and suppose
crossing at
L
a
for
a lemma.
For variants on this
Athey (forthcoming).
Suppose further that
rj
=
remains to establish
result, see
:
3 can be constructed using functions of the form l{ w < Wo }(w)
observe that given r nondecreasing and
a such that
it is
ir
-
P-
Proof of Proposition
in part
implies
and only
if
^
using convex combinations, positive scalar multiples, and limits of functions in
E SCQ{u
For
definitions.
^ desired.
)
more
result
p
if
=
r{u>)
from checking the
Jn rdQ > p JQ rdP
such that
checking the latter inequality for
that the inequality holds only
r
more information.
are less risk averse purchase
ujq.
letl(u>o)
=
,T]
H )/k(uJo,r] L ).
/ g(ut,
f]
satisfies
=
n L or
E Q. Suppose
that k{-\r] H )/k{u)Q\r] H )
Then f g(u,T])k(u},r))du)
k(u
lj
r\
weak
—
77
= H
r}
,
that k(-;rj L )
g(-,n)
and
k{-;rj L ) / k{u)Q\r] L )
single crossing in
r\.
is
e
R sc ^o)
k(-;r}
H
weak
)
are
single
Further, for any n H
>
Then,
H )k{u), H )du) >
rj
l(u
)
/
g(u,
r/
L )A:(w,
L )dw.
T]
'See Jewitt (1986), Gollier and Kimball (1995), and Athey (1999) for alternative proofs.
19
(4)
9(^,1^^, Vh)^ >
/
The
first
R sc (^o)^
in
because
^g
<
1
)
(>) t t°'^| for
$»4 >
is
and
it,
u;
where g(u,T} L )
is
<
The second
positive.
(>)0. If instead,
inequality
it is g(-,r]
H
)
that
sec0 nd inequality hold replacing g(u>,r] L ) with g(u,r) H ); then,
0,
g(u>,T]
nondecreasing in
Fix u and
^g
e q Ua ij^y an( j
H"o,riH )
since g
g{u,fi L )k{uj,T) H )dw
J
inequality follows by our assumptions on g, since k
follows because yufl!
is
13: Suppose that g(-,r)L ) G ft scV°). Then,
Lemma
Proof of
H
H )k{u,ri L )dw >
/ g(uj,r) L )k{u;,r] L )duj
77.
represent the marginal returns to a given x, which are set equal to
let cf){a,x)
zero for an interior optimum:
<p(a,x)=
w x {u\x).
(FOC)
u'(-n(a,u)))-n a {a,u))dF
I
\
Integrating by parts,
=
<f>(a,x)
we have
w x {ui\x)
I ir a {a,uj)dF
u'(ir(a,u>))
-
u"(n(a,uj))
ir a
w x (t\x)duj
(a,t)dF
i
\
=
+ ip(a,x),
£(a,x)
where we define
£{a,x) =u'{-n(a,£j))
n a (,a,u)dFw x (u>\x)
/
(5)
\
ip(a,x)
= - f u"(n(a,u))E w [ira(a,W)\x,W <u>]Fw x (u>\x)du.
(6)
\
Fix x H
(a)
by the
> xL
,
fix
Since u
an
is
MPR, Lemma
in x,
Now
by the
let
Ku o)=
nondecreasing,
implies that £(a,x#)
(b)
and
a,
7r a
(a,u;) is
weak
>
single crossing at
lies
u>
,
F0SD
-
'M <
1.
Fw x (-\x)
and x orders
\
13 implies that £{a,x H )
>
£(a,x L )- Since £(a,x L )
>
and
l(u>
)
<
1,
this
l(uJcj)£(a,XL).
consider the second term,
MPR
^S\£y Since the MPR im P
and since
TT a
(6).
(a,uj) is
By Lemma
2,
nondecreasing in
20
Ew[n a (a, W)\x,W <
u>.
Further,
7r a
(a, o>)
u]
is
nondecreasing
nondecreasing
in u>
Ew
implies that
[ir a
W) \x, W <
(a,
to] is
single crossing in u. Since u"
W<
imply that -u"(7r(a,cj))Eiy[7r a (a, W)\x,
<
by assumption, these
and
satisfies single crossing in u>
u>]
facts
nondecreasing
is
in x.
> xL
Fix x H
4>(a,x H )
>
)(j)(a,x L ).
l(uj
Lemma
Then, by
.
and the comparative
(i)
and
H >
)
Fw
Combined with
<f>(a,x) satisfies
weak
that implies
(a),
single crossing in x,
statics conclusion follows.
7:
Our approach
to the proof of
the prior, an equivalent condition to (MIO-scq)
Fx e(X \W
)ip(a,x L ).
(iii)
follows the proof provided in
AL
Before beginning, we recall that because the expected value of the posteriors
(ii).
e
l(Lo
Thus, we have established that
Proof of Proposition
for
13, tp(a,x
is
that for
>
all u>
inf
we
ft, if
T(x;iO ,9)
let
is
=
<u>o),
{1
X e < T-\y;
lu
,
0)) is
ordered by
For the moment, we do not specify T;
it
will
>-
P h-^
-
in
9 for
y e
all
(MIO-scq')
[0, 1].
be determined in the course of our analysis. Define
V{9;g,P,T)= f f g(P(y),u;)dFW:X e(u;,T- 1 (y;9))
L(U r ,(3,0h,@l)
Let
be the minimized value of the following program:
inf
g£U R
subject
V(6 H ;g,P,T)-V(e L ;g,P,T)
€ argmax
to: for all y, (3(y)
/
wX
g{a,u))dF
9
(oj\X
e
— T~
1
(y;&i)).
\
If
we consider small changes
In contrast,
6h
if
is
in 8, the envelope
larger than 6i
theorem implies that -§gV*{6;g)
by a discrete increment,
su.ppL(U R ,j3, 0h,6l), since the decision-maker might
under
/?*(•;
6jj,
while the program that determines
Ql)- Therefore, a finding that
L(U R
,
j3,
UR
in the class
;
for differential
changes in
To proceed, we begin by showing
comparisons
for small
changes in
8,
6,
XH
is
is
g£U R\V*{6 h] g)
non-negative for
the condition
is
is
—
.
utility functions take the
In such a problem, a
a
=
1 if
X
9
> T~
l
{y\ 9),
form g(a,u>)
=
nondecreasing provides
all (3
XL
for decision-makers
necessary and sufficient.
necessary and sufficient for informativeness
restrict attention to
U R We then
a subset of agents in
allows only a binary choice, a
a r(u), where
G
r(-) is restricted to lie in
otherwise. Fix a particular y e
21
and
{0, 1},
monotone decision policy takes the following form:
and
>
to use the policy
.
argue that the informativeness rankings are sufficient to compare informativeness for
U R The special class of decision problems
V*(6L',g)}
choose a different (and better) policy
more informative than
that (MIO-scq)
when we
-§-V(rj;g, /?*(•; 9),T)\r,-
L constrains the agent with signal 6h
9h, 6l)
a sufficient condition for the conclusion that
like to
in.i
=
[0, 1],
for
all
all
admissable
R.
some y G
and denote
agents in
[0, 1], let
this policy
by (P.
Substituting in
=
/?
/3*,
and noting that the optimal policy
will specify that
a
=
1
as soon as the
expected returns are positive, the minimization problem can be rewritten as follows:
Fw {lu\X h > T- (y;^ ,^))Pr(X ff > T- {y;u, Q ,0 H ))
-Fw {u\XL >T-\y-,u Q ,e L )¥z{X L >T-\y-,u ,eL ))
1
inf
r(cu)d
/
rcRJn
subject
to:
l
w (u\T(X L ;u)Q,0 L = y) = 0.
/ r(u)dF
)
Jo,
The
constraint
equivalent to the requirement that, given y,
is
where y corresponds to the optimal policy
L(U R ,py,9
H ,9 L
will
)
be non-negative
f r(cv)du
Jet
when
for this r,
we
the agent has access to
there exists a multiplier A(y) such that, for
if
XL
all
15
.
r
Then,
e R, u
[FWiX L{u,T- 1 (y-,u Q ,eL ))-Fw xH {u,T- 1 {y-,L}Q ,eH )\
(7)
,
L
-\{y) I r{uj)dFw {io\T{X
Jn
where we used Bayes' rule (together with the
oj
]
,e L )
fact that the
R
are only concerned with r £
= y) >
0,
expected value of the posteriors
is
the
prior) to simplify the expression in brackets.
R—
Focus now on the case where
l{w<w
}(
u; )
Fw xL {u
,
and when
= -l{ w<wo }(w)
Fw x „{u
,T-\y-uj ,8 L )) -
1
In words,
when
loq),
the policy function
j
n
particular.
First,
,T- l {y-LO
,8 H ))
=
X(y)
Fw (cu
Bayes' rule implies that X(y)
is
checking
(7)
when
|
=
XL
=
T-\y-io Q ,9 L )).
yFw{^o) — yFw(^o)
minimizing the difference between expected payoffs at signals 6h and
\{y)
=
0,
by Bayes'
rule, (7)
holds
if
/ r{u))dFw (u\X
'
?£'<£fer»'»
w
Pr(x^<r
1
(y;
o,fc'L))
/
Jn
and only
L
16
if:
<T-\yw,0 L )) >
^
'« rfF
We
The approach we
when
6>£.
i
x"
^
;
(8)
^*"».»-»-
L
have ignored the possibility that fn r(u>)dFw{w\T(X 9 L ) = y) might be discontinuous
this possibility by making the constraint an inequality does not affect the analysis.
15
=
written as a function of the signal using the "correct"
normalization, then optimality of the policy function does not pose a binding constraint
When
=
r{to)
requires that
,
= Fx e{x\W <
Letting T(x;loo 9)
0.
r{uS)
RSCQi^o)
in y.
Allowing
for
take here (looking for a multiplier, A, that makes (7) positive) can alternatively be phrased in
terms of Gollier and Kimball's (1995) "Diffidence Theorem"; see also Jewitt (1986) or Athey (1999)
of this approach using tools
from functional
analysis.
22
for
a discussion
F
9
Then, using Bayes'
we have
rule,
Pt(X h <T-\y;cu
Pi(X L <T-i(y;<v
H )) _ Fw xH {uj ,T-
,9
l
Fw^L^T-\y- u>
'
,9 L ))
w {uo\X H <T- {y;uj
w Q \X^ < T-^(y;u
l
{y-u ,6 H ))/F
.
L ))/
,
,6 H ))
(9)
'
{u}
,9 L ))
Using Bayes' rule again and substituting in the definition of T, we have
Fw X H{cv
x
""
V^o.^H^o^l))
Substituting (10) into
and then
(9),
J
Fw (lo
F
Fw (u
Lemma
Finally,
\
)
\
W<
W<
uj
)
1.
oj
(10)
)
f
/
))
„
4 implies that (11) holds for
all
established that (MIO-scq')
H
for all y, which in turn implies that X
is
rWdFwlu,
y and
all
r
(11)
X H < F-l(y\W
< Wo
x
|
,/n
))
>
))
|
u>
|
we have
holds. Thus,
w {u X L < F-\{y\W < u
r(io)dF
L
f
)
in turn substituting (9) into (8) yields
X L < F~l(y\W < u
w,<
XyflH ;< FFx- H (y\W
I
FxH (F- H (y\W < co
FxL (F-l(y\W < u
,T-\y;u;Q,e H ))
^
£ rSCQ(u,
and only
)^ if
equivalent to the result that
is
more informative than
X L for
))
l(U
if (MIO-scq')
rSCQ( " o)
,
/^,
H
agents in this restricted
all
r G i? 5C
'
class of decision-makers (with utility functions given
by
ar(cj), for
Consider a more direct approach to proving that (MIO-scq)
Lemma
to applying
appropriate T, (11)
utility
XH
with
r G rSCQ{uj
)
y-^
^
4) for the informativeness
is
satisfied if
ancj
^
the agent observes
X
,
and only
if
when she
=
K=
r(u)
J
so that investing
-
1
construct a
is
new
r{u)dFw {u>
when
XL
|
X L = F-l(y\W <
> F~ L (y\W <
X
lvq)
L
.
u>
Oh = 0l + d9
(that
is,
))
(that
T
Suppose that (11)
.
K, adding
K
was chosen
F
/ w
is,
some
fails for
(3^ is
is
no
optimal when
l{ w<Wo }(u;) to r does not
for just this reason.
XH
{lo
\
X L = F~l(y\W < w
using the policy
Observe that
an agent with payoffs af are lower with
(11) fails. If
from the result that given the
Then,
where
the agent with payoffs af, with signal
utility for
start
payoff function, for which
.
l{ w<tJo }(w),
a necessary condition (as opposed
not necessarily optimal for the agent, there
L Observe that, given any constant
K
3( aJ o)).
an agent with payoff a r has greater ex ante expected
whether (11) holds; indeed, the transformation
define f(u)
We
comparison.
uses the policy {$
Since the policy
We now
immediate contradiction.
affect
X
than with
L
is
<
/?*) is
)),
optimal policy
for
SC Q^ UJ °) Then, ex ante expected
f G R
.
than with
we consider a small change
X
L using the
policy
in 9), the
ffi,
because
envelope theorem implies
that increasing 9 decreases ex ante expected payoffs for the agent with payoff function af.
Thus
where
far,
we have
established informativeness orderings for a special class of decision problems,
utility functions take the
form g(a,uj)
=
a
-r(u>). It is
23
straightforward (see
AL)
to
show that
,
6L )
>
the monotone information orders are further sufficient for larger classes of decision problems.
most
this
easily,
observe that
when (A1-A4)
/
J\Q.
'[o,i]
we may
hold,
w
/ -^9{(3{y)Mdu>F
Jn °y
integrate by parts to yield V(8; g,
To
/?,
see
T)
{LO,T-\y-6))dy.
=
(12)
(To see an approach without using integration by parts, see the proof of Proposition 12 below.)
Then, the inner integral of
y
r
(12)
nonincreasing in 9 whenever
is
= y, and T(x;lo ,6) = Fx e(x\W < w (so that
G RSC QM, (MIO-scq) implies that (7) holds.
Proof of Lemma 8: Consider part (ii) (part
between
(b)
>
u>o,
and v
and
(d).
Fx „
Suppose that
Fx l
v) to
(FX L(xy \W <
as desired for (d). Finally, apply
both
u>
and
sides,
)\W <
Fx h
is
{-\W
io
< lvq)
^-#(/?(y),w),
But we have shown that
0).
for this T, if
analogous). First, consider the relationship
{F-\(y\W <
letting
xy
< F~ Hl (FxL
)
=
equivalent to requiring that, for
<v)< FxL
(F~ Hl (y\W < u )\W
But then, apply F'j, (-\W <
(i) is
This
(b) holds.
=
X(y)
)
holds for r(u>)
(7)
)\W <
v)
= F~\(y\W < w
),
and
>
6H
9L
.
we have
<v)\W<v),
(x y \W
to both sides,
u>
all
= Fx l
let y'
(x y \W
<
v)
.
This
reduces to
FxL (F~l
<
we can exchange
Since
ujq
if (c)
holds. Thus, (b)
Now
v,
(y'\W
and
(c)
the roles of
< Fx h (F"i
)
and
u>q
v, so
(y'\W
FxS (-\W < w
),
7.
v)
\W<u>
(13)
).
9h >
if
and only
(a) implies (b),
building
all
0l,
are equivalent.
>
In particular, fix v
(MIO-scq') implies that
= 1^ <v \(lo) .Finally,
<
that (13) holds for
consider the relationship between (a) and (b). First
on the proof of Proposition
r v (uj)
<v)\W< w
and
loq,
holds for
(8)
all
we show that
recall that, using the
r
£ rSCQ(uj
use Bayes' rule and the fact that Pr(w
<
v)
is
)
_
transformation
Theil) check
^
for
constant in 6 to write the
condition in terms of conditional probabilities, as in (b).
Now, we show that
(b) implies (a). Fix v
and only
nonincreasing in
is
equivalent to (11). In turn, (11) for rv
if
(8)
Then, we have just argued that
ujq.
holds for rv
is
6, if
>
is
.
The proof of Proposition 7
Fx
e
(F~l(y\W < o>o)|W <
establishes that (8)
equivalent to
Fw {v\X L <F-\{y\W<^)) _ Fw {v\X H <Fx H {y\W<^))
>
"
Fw (w X H < F~ H {y\W < w ))
Fw (lo X L < Fx l(y\W < w ))
x
l
'
Q
'
'
|
I
Finally, consider the case
condition
(c)
requires that (8)
that implies that (14)
where v
<
fails for
rv
wo, and recall that (b) implies
.
This
is
fails.
24
(c).
Using Bayes'
equivalent to requiring that (11)
fails.
rule,
Finally,
v)
Then, we have established that
>~pr-uj
in Q,
Proof of
^ desired for
Lemma
(MIO-scq) from
9:
Lemma
8,
if (b)
holds, then i*V(-
(MIO-scq'), which
We
prove part
part
(ii)
(ii);
=
Thus,
^ T>i{X
e
(this
part
X >(*y\W<v)-
10:
)
W<
|
v)
M
(MIO-scq) holds
If
if
(i) is
if
analogous. Recall the characterization of
we
=
xy
let
<
xy \W<v) m
x(y; 0,ujq)
= F~](y\W <
uq),
for all v
>u
UR
)
"
W<U}Q)
\
and only
with an appropriate u and n.
us assume without loss of generality that
and only
if
,
<o;
if
(3)
holds for
all x.
U
Necessity follows from Proposition 7 because, as argued above,
payoff g G
we use the
Fx°(xy\W
Mxy
does not affect the information content of
(MIO-scq) hold).
ordered by
'de
Proof of Proposition
let
is
Fxe (xy \W <v) + fx e(x y \W < v)—F-l(y\W < u>„)
< F~l(y\W < w
Before beginning,
uo))
F~l(y\W <u; )\W <v)
p
we can approximate any
< F~}(y\W <
equivalent to (a).
Notice that,
(b).
^?T(X e <
de
is
X6
|
fact that
Pr(W <
X
,
so
X
e
is
Now
consider sufficiency.
normalized so that -§gFx e(x)
=
MPR
or
also does not affect
it
with
v) does not vary
8,
whether
Lemma
9 implies that
if
Fwx e(v,x)
oe
.
is
nondecreasing in v for
all x.
(15)
W\X' (v\x)
Now, observe that by the envelope theorem,
^V*(6;uon) =
Integrating by parts (which
is
valid
it
follows that
d_
J
Ju(7t(a*(x;8), >))d
by (A3)) and
w,x» (uj,x)
letting subscripts denote partial derivatives,
we
have:
- fa*x (x;9)
d
fu'(iT(a*(x;e),Lj))ir a (a*{x;e),u;)dul
Taking the inner integral and integrating
it
FW,a-»(w,x)
dx.
by parts again (grouping together the
of the integrand, analogous to the proof of Proposition 6),
we
oe
we have the
last
(16)
two terms
following condition which
will refer to as (II):
-u'(7r(a*(x;0),w)) fir a (a*{x e),oj)d u
;
-qqFw<x b{u),x)
(II-a)
U "(7r (a*(x;0),a;))7r a; (a*(x;0),a;)
-I
g?
J?*, *»(<**(*;
tr,.v
0),*H:i F
m
25
w,x»^< x )
m FWiX »{u),x)
du:
(Il-b)
To ensure that
the
first
-§qV*(8\
u
o n)
>
to establish that this expression
0, it suffices
positive
is
whenever
order conditions (FOC) hold. Those, too, can be integrated by parts (as in the proof of
the comparative statics proposition). Multiplying by -1, the marginal returns to a are given by:
f n a (a*{x; 8),uj)d^Fw{X 9(u>\x)
-u'(Tr(a*{x;0),u>))
+
Notice that the
L\
t
]
term of
first
^"
w\x
>
e)^)d
\r.^a(a*(x
£oo*.("-(^).WlS^
(FOC-a)
this expression
is
(FOC-a)
Fwlx o(uj\x)duj
(FOC-b)
'
u
non-positive, since
is
nondecreasing and
the expected marginal returns to a are positive by assumption. Thus, a necessary condition for the
sum
of the two terms to equal zero
Our approach
holds only
by showing
implies (II-b)>
(a)
Consider
we have
established that
implies that
(II)
>
0.
(FOC)
Let us proceed
always under our assumptions, and second showing that (FOC-b) >
(Il-a).
Integrating by parts again yields
(a*(x-e),u>)
TT a
— Fx
where the sign follows since -§qFx b{x)
because A.
-§qF
wx b{uj-,x)
>
=
e(x)
-
/
n auJ (a*(x;e),ui)—FWX 9(u,x)dw >0,
by our normalization of
x,
by supermodularity of
(as discussed above, the latter inequality
under our normalization of
Now
non-negative.
is
0.
-u'(ir(a%x;e),u))
(b)
as follows. Since
is
show that (FOC-b) >
to
0, it suffices
that (II-a)>
first
that the second term (FOC-b)
to the remainder of the proof
(FOC-b) >
if
is
x,
§§Fx e(x) =
consider the second term.
(17)
7T,
and
equivalent to
(MIO-spm)
= Fw x e(iv\x),
and define
is
0).
=
Kl{u)
Define
K{u\kl)
\
Kh(u) = K(cj\kh) =
(15)).
lp{lo\ki) is
weak
13.Let 0(u;,k)
in
—u
and
Then, by
K
H
(
\
Notice that
-§gF
= P^
Let <p{u\n)
follows because
WX 9(LL>,x).
-7r a (a*(x; 8), t)dt ffij^
> K
L
\ \
for
t
<
lo
g(cj, k) is
Lemma
is
non-negative and nonincreasing (by
First, observe that ip(u>\K
— 7r a (a, t)
is
is
nonincreasing in
nonincreasing in
13, since
Proof of Proposition
be the joint distribution of
p,
H >
)
t).
^"rl
is
Next we apply
t.
is
nondecreasing in w, $^ g(u, K,)K{u\K)du}
who
11:
W
(II)
Fix
and
>
w
Ye
.
p(uj\k l ) (this
Second, note that
weak
nondecreasing in K by our previous arguments, and since u"
Thus, we have established that
parameter
/'l
-u"(n(a*(x;8),u))-Tr LJ (a*(x;9),ui)-ip(uj\K), where g(u>,KL)
crossing in k. In other words, (FOC-b) >
utility
-
K
and — iT a (a,t)
single crossing in —a;, since
=
H
Lemma
single crossing
<
and
satisfies
tt^
weak
>
0.
single
implies (II-b)> 0, as desired.
0,
which implies that -§gV*{6\ u
G
fi,
and define
o n)
>
0.
Y = Fx o{X \W <
e
e
oj
).
Let
G WY
e
Let (3*(-;8,p) be the optimal policy for an agent with
observes the signal
Ye
.
Further, define w(y,u)
26
=
g((3*(y;8,p H ),u>;p H )
-
g((3*(y;9,p L ),Lu;p L ).
By
the Envelope Theorem,
-^V*(9;g(.;pH ))
Assume
A
that w(y,u>)
for all
i
Then
is finite.
=
= w(yu u)
-
^V
there
on
is
some
[yi,yi+1 ). Let r^uo)
= w{y
[ri(W)
Y e+de > yi
|
Pr (Y e+de >
~\
-
yi )
i=2
<
...
and note that
<
n
j/w+i, such
£ rSCQ{u
)
if
£>,y
[n(W)
|
Ye >
yi ]
Pr (y 9 >
>
y>)
0,
j=2
we showed
in the proof of Proposition 7, holds for all r,
(MIO-scq) holds (simply use the
to rerrange (10), (9),
is
Jj),
<
yi
n
J2®w
if
<
with y\
,
Lo)-tu{y i _ u
il
- &V(0,g(-,p L )) >
n
which,
.
:
series of cut-points {yi}™=1
Then, ^V(0,g(-,p H ))
2,...,N.
=ff "Md^ {G w^(u y)}
*(9,g(.;p L ))
high).
The
and
case where
Proof of Proposition
is
ftSCQ{ujo) an(j a jj
fact that the expectation of the posteriors
(11) so that in each case
A
£
we condition on
y^
jf
an(j on iy
must equal the
prior
the probability that the signal
convex follows via a limiting argument.
12: Use subscripts to denote partial derivatives. For each p, the
first-
order conditions for optimality define a*{x;9,p). Building on our integration by parts ((FOC-a)
and (FOC-b) from above), and using our assumptions about the upper
the first-order conditions can be written as follows for each
j
u u (n(a*(x;0,p),uj);p) a*(x;9,p)
first
V(x,uj;9,p)
we can
and
simplify (Il-a)
increasing information condition
I *(x,u>;e,p)
(t\x)
increasing in p
dO P W,X<>
9
ao
~dS^'w,x
Fw x e(uj\x)dw =
0.
\
Fw x o(u>\x)du) =
(oj\x)
and use the
utility,
differentiate with respect to p, yielding
Fw{x e(t\x
^ (t-LJ )dff
J—oo
Juj
(II-b),
(II) is
we can
W\X
J—oo
Jul
Likewise,
(t-wo)d,
/
:
'Fw\x°Hx)
order conditions hold as an identity,
on the marginal
p:
W\X
)d
J—oo
Jul
Since the
-u
(t
/
limit
0.
(18)
\
fact that
^gFx g(x) =
to
0,
show that the
if
(t,x)
(
(V i
d_
39
x)
FWX 9(ut,x)duj >
(19)
:
of Proposition 10 (starting with part (b)) showed that
The proof
(18) holds,
then an expression analogous to (19) holds. In particular,
portfolio problem,
ir
u (a*(x;0,p),uj)
will
when an
is
(FOC-b) and
(II-b) are identical to (18)
replaced with $>{x;9,p). Thus,
have (18) implies
(19).
27
if
and
we simply
(19)
expression analogous to
for the special case of the
when
u"(ir(a*(x;9,p),u}))
establish that ^(x,u;;9,p)
<
0,
we
However, there
is
one more
effect to consider.
(II)
just the inner integral of (16),
is
represents the change in ex ante expected payoffs from an increase in
provided conditions under which
and
if
the posteriors are ordered by
and a% p
7
>
nondecreasing in
(II) is
MPR,
a*
>
p.
So (16)
0.
When
9.
Our arguments thus
(MIO-scq) holds,
nondecreasing in p
is
which
if
far
(II) is positive,
*(x, w;
9, p)
<
0.
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(1999): "Characterizing Properties of Stochastic Objective Functions,"
MIT Work-
ing Paper 96- 1R.
ATHEY, SUSAN (forthcoming):
"Monotone Comparative
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Quarterly
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Athey, Susan and Jonathan Levin
Problems,"
BLACKWELL, David
MIT Working
(1951): "Comparisons of Experiments,"
(1953):
Statistics, 24,
in
Monotone Decision
Paper 98-24.
Symposium on Mathematical
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"The Value of Information
(1998):
Proceedings of the Second Berkeley
Statistics,
"Equivalent Comparisons of Experiments," Annals of Mathematical
265-272.
BOYER, M. AND R. E. KlHLSTROM, Ed.
Bayesian Models in Economic Theory.
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"Demand
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Risky Assets and the Mono-
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I:
(2):
JEWITT, Ian
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KlHLSTROM, RICHARD
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