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THESIS

This is to certify that the

dissertation entitled

INDEPENDENT INCREASES IN RISK
AND THEIR COMPARATIVE STATICS

presented by

HELEI QU

has been accepted towards fulfillment
of the requirements for

Ph.D. . Economics
degree in

 

 

Date March 25, 1994

 

 

 

 

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MSU II An Affirmative Action/Equal Opportunlty lnotIIqun
, Walla-9.1

MEPENDENT INCREASES IN RISK AND
THEIR COMPARATIVE STATICS

BY

Helei Qu

AN ABSTRACT OF A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

1994

AUBSTJIAITT

INDEPENDENT INCREASES IN RISK AND
THEIR COMPARATIVE STATICS

BY
Helei Qu

This paper introduces a special type of Rothschild and
Stiglitz increases in risk. This is prompted by the fact
that Rothschild and Stiglitz increases in risk are too
broadly defined to generate determinate comparative statics
in most decision models. Typically very severe restrictions
on the utility functions are needed in order for the
comparative static effect to be determinate. An alternative
is to further restrict the increases in risk. Many cases of
special Rothschild and Stiglitz increases in risk have thus
been proposed and examined. We introduce our own increase
in risk in this paper.

An independent increase in risk further restricts a
Rothschild and Stiglitz increase in risk by requiring E to
be independent of E. Random variable y is an independent
increase in risk from random variable § if

Si=4 E + E,
where E is independent of E and E(E) = 0. An independent

increase in risk is a special Rothschild and Stiglitz

increase in risk as E(E) = E(E|x) = 0.

Under an independent increase in risk, the distribution
of random variable E is the same no matter what realized
value of random variable E is. The distribution of E is
independent of x. This uniform property makes the
comparative static effect of an independent increase in risk
determinate in many instances.

Like a strong increase in risk, an independent increase
in risk is also a generalization of an introduction of risk.
An independent increase in risk can, however, generate any
Rothschild and Stiglitz increases in risk, when the initial
random variable E is degenerate at a point.

An independent increase in risk imposes no restrictions
on the two distribution functions in the center of the
supports. The two CDF’s may cross many times. The support
of F(x) is, however, contained inside the support of C(y).

The conditions for generating comparative statics are
acceptable for an independent increase in risk. The
independent increases in risk have a wide range of
applications. Background risk, savings and uncertainty,
asset proportion, portfolio problem and the output level of

a competitive firm are few examples.

Dedicated to My Parents, Yuee Li and Jinlian Qu

iv

ACKNOWLEDGENIENT

A dissertation cannot be accomplished without the
support of many people. This is especially true for this
dissertation. There would not be this dissertation without
the supports from my committee members and many friends.

During this long process, my adviser, Professor Jack
Meyer suffered as much as I did if not more than I did. Dr.
Meyer read many drafts of this dissertation and made lots of
comments in addition to his insightful guidance on this
subject. His efforts have always been appreciated.

I am fortunate to have Dr. Ching-Fan Chung on my
committee. His kindness, sincere efforts made this
dissertation possible. His courage helped me through some
of the most difficult times. His supports are beyond the
expression of English language. Without him this
dissertation can not be finished.

The same can be said to Professor Carl Davidson,
another member of my committee. Numerous faculties and
students have helped me through this ordeal, Professor
Lawrence W. Martin, Professor Robert H. Rashe, Dr. Gyemyung
Choi and John "Bud" Schulz, to name a few. It would be too
tedious to list every one here.

Mr. Hailong Qian is worth mentioning specifically. It

was Mr. Qian who came to the author's rescue and read part

V

of this dissertation when the author was at difficulty. He
was always there extending a helpful hand when the author
needed it. And he shared the excitements and frustrations
with the author as this dissertation progressed. Without
him this dissertation would have been still in working. His
friendship is always appreciated.

During my prolonged education, I lost the most precious
person in my life, My Dear Mother, Yuee Li. I was not at
her side when she passed away 22 December 1991, which is and
will be the most regrettable moment in my life. Dear Mom,
you were, are and will be with me every step along the long
life journey.

My dear father, Jinlian Qu, has been giving me his best
support through my education. My beautiful sister, Lanlan
Qu, has been helping the family through so many turbulent
years, while I have been far away. My brother-in-law,
Rongjun Liu, is a great help to the family, and this is
always appreciated. Without the support of my family, I
would not have finished this.

This dissertation is dedicated to my beloved parents.

vi

TABLE OF CONTENTS

LIST OF FIGURES OOOOOOOOOOOOOOOOOOOOOOOOOOO0.0.0.0.... Viii
CHAPTER 1; INTRODUCTION ........ ....... ..... ...... ..... 1

CHAPTER 2: LITERATURE REVIEW oooooooooooo .............. 7
2.1 CDF Approach to Risk Changes ------------------ 10
2.1.1 An Impossibility Theorem ............... 11
2.1.2 Strong Increases in Risk ~-------------- 13

2.1.3 Relatively Strong Increases in Risk
2.1.4 Relatively Weak Increases in Risk ...... 16

2.1.5 Monotone Likelihood Ratio Change

in Randomness ........................... 18

O
H
h

2.1.6 Summary .......................... ..... . 20
2.2 Deterministic Transformation Ap roach ......... 22
2.2.1 Deterministic Transforma ions -~ ------- - 23

2.2.2 Simple Increases in Risk and
Comparative Statics .............. ....... 25
CHAPTER 3: INDEPENDENT INCREASES IN RISK ----°--- ------- 29

3.1 Independent Increases in Risk ----------------- 29
3.2 Recovering the Independent Random Variable ---- 43
Appenclix ooooooooooooooooooooooooooooooooooooooocoo 49

CHAPTER 4: COMPARATIVE STATIC EFFECT ...... ----------- .. 51
4.1 Comparative Statics 0.0000000000000000. ........ 51

4.2 Examples of Application -------- ----- ~--------- 61
4.2.1 Savings and Uncertainty --- ----- -- ----- - 61

4.2.2 Asset Proportion ....................... 69

4.2.3 More Applications ....... ............... 73

REFERENCE ...... . ...... ........ ........ . ................ 73

vii

LIST OF FIGURES

Figure 2.1 ............................................. 21
Figure 3.1 ............................................. 33
Figure 3,2 ...... ......... .............................. 39
Figure 3,3 ..... ................... ..................... 42
Figure 3.4 ...................... ..... .................. 44

Figure 3.5 ....................................... . ..... 46

viii

CHAPTER 1
INTRODUCTION

In economics there are many unknowns. For instance,
the determinants of supply and demand and therefore the
market price have significant stochastic components. Like
any other science, economics takes these into consideration.
Including randomness enriches the content of economics.
Explaining the impact of uncertainty is an important aspect
of economic theory. Risk and uncertainty theory is the
branch of economics that deals with these issues.

Risk, uncertainty or lack of information is represented
by including random variables as parameters in a decision
model. A random variable has a probability distribution
function (PDF) and a cumulative distribution function (CDF).
These describe all the possible outcomes and the likelihood
that each of the outcomes occurs. We will use the PDF and
CDF to describe the risk and uncertainty.

Von Neumann and Morgenstern (1944) place conditions on
preferences over the random outcomes. These conditions,
known as expected utility theory, imply that the ranking of
random alternatives is given by the expectation of the
utility of the possible outcomes.

An economic agent’s preferences can be classified into

risk averse, risk loving or risk neutral according to their

2
attitude towards risk. Some economic agents buy lottery
tickets as well as insure their properties. That is, there
are agents who do not belong to any of the above groups,
they are both risk averse and risk loving.

Different economic agents have different attitudes
toward risk, some are more risk averse than others. Risk
neutral economic agents are those who are indifferent
between taking a random outcome and taking the certain
expected value of the random outcome. Risk averters are
those who prefer the expected value of a random outcome to
the random outcome itself. Risk lovers are those who prefer
a random outcome to the certain expected value of the random
outcome. These attitudes are reflected in the shape of the
utility function. Risk averters have a concave utility
function, risk lovers have a convex utility function, and
risk neutral economic agents have a linear utility function.
Concave utility functions are usually best suited to the
maximization of expected utility.

An economic agent's attitude toward risk can be
measured. Arrow (1965) and Pratt (1964) use absolute and
relative risk aversion to measure the curvature of the
utility functions. Absolute risk aversion is defined as
A(z) = -u"(z)/u’(z), and relative risk aversion is defined
as R(z) = -z-u"(z)/u’(z) = z-A(z), where u is the utility
function, z is the outcome parameter. A(z) 2 0 is for the

risk averters, A(z) s 0 for the risk lovers and A(z) = 0 for

3
the risk neutral economic agents.

Risk aversion depends on the level of the outcome
parameter. When A(z) decreases as 2 increases, this is
referred to as decreasing absolute risk aversion (DARA).
Increasing absolute risk aversion (IARA) occurs when A(z)
increases as 2 increases. Finally, constant absolute risk
aversion (CARA) prevails when A(z) does not change as 2
changes. Arrow (1965) argues that absolute risk aversion
A(z) is a decreasing function (DARA) of z where z is wealth.

Similar definitions apply to relative risk aversion.
Arrow (1965) argues that the relative risk aversion R(z) is
an increasing function (IRRA) of wealth. IRRA means that if
both the wealth and the size of the random variable are
increased in the same proportion, the willingness to accept
the risk should decrease.

When one group of economic agents prefer one random
variable to another, this generates the definition of
dominance among the random variables, or stochastic
dominance. Hanoch and Levy (1969) define dominance for all
agents with non-decreasing utility functions and also for
those with non-decreasing and concave utility functions.
Radar and Russell (1969) also give these definitions and
call them first order stochastic dominance (FSD) and second
order stochastic dominance (SSD), respectively. Hanoch and
Levy, Radar and Russell prove that a distribution F(x) FSD a

distribution G(x) if and only if all economic agents whose

4
utility function is non-decreasing in x prefer F(x) to G(x).
Similarly F(x) SSD G(x) if and only if all economic agents
whose utility function is non—decreasing and concave in x
prefer F(x) to G(x).

Stochastic dominance is a unanimous preference concept.
FSD is for the group of non-decreasing utility functions,
including risk averters, risk lovers and risk neutral
economic agents. SSD applies to non-decreasing and concave
utility functions, that is only risk averters are in this
group.

A related concept defined by Rothschild and Stiglitz
(R—S) (1970) is a Mean Preserving Spread (MPS) increase in
risk. Rothschild and Stiglitz give three definitions of a
risk change. They consider unanimous preference for all
economic agents with concave utility functions.
Distribution G(x) is a Rothschild and Stiglitz increase in
risk from F(x) if the risk averters prefer F(x) to G(x) and
F(x) and G(x) have the same means. Note that u(x) is not
required to be increasing or decreasing.

The most commonly studied decision model contains only
one random variable. Often this is presented in a one
random variable, one Choice parameter and one outcome
parameter (1-1-1) format, Choi (1992). In this model, the
final outcome is a function of the random variable, the
choice parameter and possibly other exogenous parameters.

This is the model used in much of the research in risk and

5
uncertainty theory. The model is employed in chapters 2 and
3 and the notation will be introduced at that time.

Both the stochastic dominance and the Rothschild and
Stiglitz increase in risk definitions allow two particular
comparative static questions to be posed in this decision
mode. When the random variable undergoes an FSD, SSD or
Rothschild and Stiglitz risk change, how does expected
utility change, and how does the decision made by the agent
change? These comparative static questions are an important
part of risk and uncertainty analysis over the past twenty
years.

This dissertation is organized as follows. In chapter
2, we will review the literature on different approaches to
changes in randomness and their comparative statics and the
findings which have been published so far. Changes in
randomness are usually represented as CDF changes. The
initial and the final CDF are specified and their difference
is restricted. The comparative static question this
dissertation focuses on concerns the change in the optimal
choice parameter as the CDF changes. This change will
always have an already known effect on expected utility.

The general MPS, FSD or SSD changes are often too broad to
generate determinate comparative static results. Much
research therefore has focused on how to generate special
types of changes in randomness to obtain determinate

comparative static results. These special types of changes

6
in randomness and their comparative static results are also
reviewed in chapter 2.

Chapter 3 introduces an independent increase in risk,
which is the main focus of this dissertation. An
independent increase in risk changes a random variable by
adding an independent random variable to it. This generates
a special type of Rothschild and Stiglitz increase in risk
for which determinate comparative static results can be
demonstrated.

Associated with independent increases in risk are
Independent Mean Preserving Spreads (IMPS). An IMPS is a
set of MP8 that differ from one another by linear shifts.

In chapter 3 we connect independent increases in risk and
IMPS.

The last chapter is devoted to comparative statics. To
get determinate comparative statics after all is the purpose
of introducing the independent increases in risk. An
independent increase in risk indeed generates determinate
comparative statics under quite general conditions. We will

see this and examples of applications in chapter 4.

CHAPTER 2
LITERATURE REVIEW

In this chapter we will review the literature on
changes in randomness and their comparative statics.
Stochastic dominance and Mean Preserving Spread (MPS)
increases in risk are perhaps the two most important
concepts in the risk and uncertainty literature, we
therefore start with these concepts.

In this paper the notations and assumptions are as
follows, E and E represent the random variables with CDF’s
F(x) and G(y) respectively, F(x) has bounded support [b,B]
and G(y) [a,A], a s b s c s C s B s A, c and C are two
points inside the supports. A and E represent the exogenous
non-random parameters, a represents the choice parameter, u
is the utility function. Thus a 1-1-1 model is Eu[z(x,a)],
where z is the outcome. Economic agents choose a to
maximize expected utility. To guarantee an interior
solution, erx,a) = 0 for a finite a V x e [a,A] is assumed.
zwxx,a) < 0 is also assumed to guarantee the second order
condition for maximization is satisfied.

Hanoch and Levy (1969) and Hadar and Russell (1969)
define first order stochastic dominance (FSD) and second

order stochastic dominance (SSD) as follows.

8
Definition 2.1 (Hanoch and Levy, Hadar and Russell):
(1). CDF F(x) is said to be at least as large as CDF G(x)
in the sense of FSD if and only if G(x) 2 F(x) V x e [a,A];
(2). CDF F(x) is said to be at least as large as CDF G(x)
in the sense of SSD if and only if S§[G(x) - F(x)]dx 2 0 V y

e [a,A].

These definitions of stochastic dominance are for CDF’s not
for random variables. Hanoch and Levy and Hadar and Russell
also show that the dominating distribution is unanimously
preferred to the other distribution by a certain group of
economic agents. F(x) FSD G(x) if and only if all economic
agents with an increasing utility function prefer F(x) to
G(x). F(x) SSD G(x) if and only if all economic agents with
an increasing and concave utility function prefer F(x) to
G(x). Thus the impact of a CDF change on expected utility
is known, the impact on the choice variable is of concern
here.

Rothschild and Stiglitz (1970) give three definitions
of an increase in risk and prove that these definitions are

equivalent. We state this result as a definition:

Definition 2.2 (Rothschild and Stiglitz): The following
three definitions of an increase in risk are equivalent.
(1). Every risk averter prefers random variable E to random

variable E, that is” Sfiu(x)dF(x) 2:Sfiu(x)dG(x),

where u" s 0;

(2). Random variable E is equal in distribution to random
variable E plus some noise E. That is, E =4 E + E, where E
satisfies E(E|x) = 0 and "=fi’" represents "is equal in
distribution to";

(3)- (a) SZ[G(x) ‘ 1‘71!)de Z 0, V Y 6 [8111];

(b) Sfi[G(x) - F(x)]dx = 0, where [a,A] contains the supports

of E and E.

These definitions define an MP8 increase in risk and
definition (3) is often referred to as the integral
conditions of an MP8 increase in risk. An MP8 increase in
risk moves probability mass from the center of a
distribution to its two tails while preserving the mean.
This change produces a new distribution which has the same
mean as the original distribution and is defined to be a
risk increase. Definition (1) shows the impact on expected
utility. Rothschild and Stiglitz also provide the framework
for determining the impact of a risk change on the decision
made by an expected utility maximizing agent. The method of
defining a risk change and determining its comparative
static effect focuses on the integral conditions in
definition (3). Rothschild and Stiglitz (1971) use this CDF
approach and most have followed ever since.

An alternative method of representing a change in the

riskiness of a random variable involves transforming the

10
random variable and has been used by Sandmo (1969, 1970,
1971) and others, and it is formalized under the name
deterministic transformation by Meyer and Ormiston (1989).
A deterministic transformation changes the random variable
and hence the CDF by deterministic function t(x), which maps
every realization value of random variable E into a new
point. Appropriate restrictions on t(x) will generate MPS,
FSD or SSD changes in randomness. This method has proven to
be a simple and effective way to represent a random variable
change, we will review this method in section 2.

Over the years economists have found that a general
Rothschild and Stiglitz increase in risk or an FSD or SSD
change in randomness is too broad to yield determinate
comparative static results. Typically very severe
restrictions on utility functions are needed in order for
the comparative static results to be determinate. An
alternative is to further restrict the change in randomness.
Many subsets of these FSD, SSD and Rothschild and Stiglitz
changes in randomness have thus been proposed and examined,
these changes and their comparative static results are

reviewed next.

2.1 CDF Approach to Risk Change and its Comparative Statics

CDF approach to comparative static analysis specifies
the initial and the final CDF's and then compares the

optimal values for the choice parameter under the two CDF's.

11
The change in the optimal values is the comparative static
effect of the CDF change. In this section, we shall
investigate the comparative static effect of CDF changes
referred to as strong increases in risk, relatively strong
increases in risk, relatively weak increases in risk and
monotonic likelihood ratio risk changes and stochastic

dominance.

5 2.1.1 An Impossibility Theorem

For an expected utility maximizer Eu[z(x,a)], his
optimal choice parameter a satisfies first order condition
(FOC) Eu’[z(x,a)]-z;cx,a) = 0 in the 1-1-1 model. By the
first definition of an MP8 increase in risk, if
u’[z(x,a)]-z;Lx,a) is concave in E, Rothschild and Stiglitz
then conclude that an MP8 increase in risk will decrease a.
Rothschild and Stiglitz continue to explore what this
condition implies in specific economic models. Concavity of
u'[z(x,d)]-z;(x,a) is a general requirement, it is very
restrictive when translated into conditions on utility
function u(z) and payoff function z(x,a).

Meyer and Ormiston (1983) ask when an arbitrary
Rothschild and Stiglitz increase in risk causes all risk
averse economic agents to adjust their optimal choice
parameters in the same direction in a 1-1—1 model.
Unfortunately the answer to this question is negative, they

have the following theorem regarding this problem.

12
Theorem 2.1 (Meyer and Ormiston): Rothschild and Stiglitz
increases in risk cause all risk averse economic agents to
decrease optimal choice parameter a if and only if there

exists a do such that za(x, a0) = 0 V x e [a,A].

Result concerning the same problem with regarding to FSD is

in the following corollary.

Corollary 2.2 (Meyer and Ormiston): All changes in a CDF
such that the final CDF is dominated in the sense of FSD by
the initial CDF cause all risk averse economic agents to
decrease the choice parameter if and only if there exists a

(10 such that za(x,a0) = 0 V x e [a,A].

An SSD shift is a combination of an FSD and an MP8 shifts,
Hadar and Sec (1990). A similar result can thus be derived
for an SSD change in randomness.

The implication of condition za(x,a0) = 0 for all x is
that optimal choice parameter¢%,does not depend on random
variable E. Obviously this restriction eliminates all
interesting models. A general Rothschild and Stiglitz
increase in risk does not yield determinate comparative
static results for all risk averse economic agents in any
meaningful 1-1-1 model. This leaves us two choices. One is
to further restrict the utility function such as requiring

it to be DARA. The other choice is to restrict the change

13
in randomness. Most literature on this subject defines
special types of changes in randomness. These special types
of changes in randomness and their comparative statics are

reviewed next.

5 2.1.2 Strong Increases in Risk

Meyer and Ormiston (1985) introduce a strong increase
in risk. A strong increase in risk transfers probability
mass from the original interval [b,B], which contains the

support of initial CDF F(x), to intervals [a,b] and [B,A].

Definition 2.3 (Meyer and Ormiston): CDF G(x) is a strong
increase in risk from CDF F(x) if their difference G(x) -
F(x) satisfies the following conditions.

(a)- 52mm - F(x)de 2 o, v y e [a,AJ;

(b). mam - F(xudx =0 ;

(C). G(x) - F(x) is non-increasing in interval (b,B).

A strong increase in risk is a generalization of an
introduction of risk, which is an increase in risk from an
initial non-random situation. Note that the two CDF’s only
cross once in the case of a strong increase in risk. The
opposite is however not true. Not all CDF pairs which cross
once are strong increases in risk.

Meyer and Ormiston (1985) have the following theorem

regarding a strong increase in risk.

14
Theorem 2.2 (Meyer and Ormiston): Assume that decision
makers choose a to maximize Eu[z(x,a)], where u’ 2 0, u" s
0, then all risk averse economic agents, when faced with a
strong increase in risk, will decrease the optimal value of
a if
(a). z,‘ 2 0 and Zen 5 0 Vx e [a,A];

(b). z“, 2 0, 3m s 0 Vx e [a,A].

Condition z,2:0 together with u’ 2 0 guarantees that the
higher values of random variable are preferred to the lower
values. The case where %:S 0 can be treated the same way
with minor modifications. The conditions 2” 2 0 and.zgu:s 0
are restrictions needed to generate the comparative static

results, they are conditions on the payoff function.

5 2.1.3 Relatively strong Increases in Risk

A relatively strong increase in risk is proposed by
Black and Bulkley (1989) to be less restrictive than a
strong increase in risk. A relatively strong increase in
risk allows some of the probability mass that is transferred
to intervals [a,b] and [B,A] in the case of a strong
increase in risk to stay inside interval [b,B] and yet
preserves the comparative static result associated with a

strong increase in risk.

15
Definition 2.4 (Black and Bulkley): CDF G(x) is a relatively
strong increase in risk from CDF F(x) if:
(a). mean - F(x2de = o;
(b). For all points in interval [c,C], G(x) - F(x) is non-
increasing; For all points outside this interval, G(x) -
F(x) is non-decreasing, where c and C are two points inside
[b,B] and c s C;
(c). f(x)/g(x) is non-decreasing in interval [b,c);

(d). f(x)/g(x) is non-increasing in interval (C,B].

Conditions (c) and (d) relax a strong increase in risk. A
relatively strong increase in risk allows the density
function of the riskier random variable to be bigger than
that of the less riskier random variable in intervals [b,c)
and (C,B].

Assume a, and a6 are the optimal choice parameters
under distributions F(x) and G(x) respectively. Black and
Bulkley (1989) have the following theorem concerning a

relatively strong increase in risk.

Theorem 2.3 (Black and Bulkley): The sufficient conditions
for aai<¢y,for all risk averse economic agents are:

(a). G(x) represents a relatively strong increase in risk
from F(x);

(b). 2x20, zm20,z SOandzm<0forallxandcu

w

16
This theorem requires no more conditions on utility function
than those required by a strong increase in risk.

The difference between a strong increase in risk and a
relatively strong increase in risk is that density function
f(x) is bigger than density function g(x) on entire interval
[b,B] under a strong increase in risk, but under a
relatively strong increase in risk density function f(x) may

be smaller than g(x) in intervals [b,c] and (C,B].

5 2.1.4 Relatively Weak Increases in Risk

Dionne, Eeckhoudt and Gollier (1993) propose a
relatively weak increase in risk when studying a special
type of payoff function. They study models with a payoff
function that is linear in both random variable and choice
parameter. This group of payoff functions satisfies the
restrictions on payoff functions required by a relatively
strong increase in risk.

The model Dionne et al. use is a general linear model
z(x,a) = a(x-A) + 5, where A and £ are the exogenous
parameters. There are two cases a 2 0 and a s 0. In an
application, the sign of a is usually known. We only
consider a 2 0, for a s 0 can be handled with minor
modifications. The conditions on the payoff function are
> 0, zm = z,“ = zw = 0. These stronger conditions on

z(x,a) allow one to relax some restrictions on the type of

changes in randomness. We start with the definition of a

17
relatively weak increase in risk, assume two density

functions cross at two points c and C.

Definition 2.5 (Dionne, Eeckhoudt and Collier): CDF G(x)
represents a relatively weak increase in risk from CDF F(x)
if for parameter 7
(i). When 7 e [c,C],
(a) ”(G(x) ' F(X)]dx = 0;
(b) For all points in interval [c,C], G(x) - F(x)
is non-increasing; For all points outside the
interval, G(x) - F(x) is non-decreasing;
(ii). When 7 e [b,c), then besides conditions (a) and (b)
one needs the following conditions;
(6) f(X)/g(X) S f(7)/9(7). b s X s 7; f(X)/9(X) 2
f(7)/9(7), 7 S x < c;
(iii). When 7 e (C,B], then besides conditions (a) and (b)
one needs the following conditions;
(d) f(x)/g(x) 2 f(7)/9(7)z C < X S 7; f(X)/9(X) S

f(7)/g(v). 7 s x s B.

7 plays a critical role in this definition, different
conditions are needed when 7 belongs to different intervals.
On intervals [b,c) and (C,BJ a relatively strong increase in
risk imposes condition on f(x)/g(x) on the entire intervals,
while a relatively weak increase in risk imposes condition

on f(x)/g(x) relative to point 7, or f(7)/g(7).

18
The comparative static result for a relatively weak

increase in risk is in the following theorem.

Theorem 2.4 (Dionne, Eeckhoudt and Gollier): Suppose that
cy.and Co are the optimal choice parameters under CDF’s F(x)
and G(x) respectively and.a¢ is an interior solution. Then
the sufficient conditions for a6:5¢y,for all strictly risk
averse economic agents are:

(a). G(x) represents a relatively weak increase in risk
from F(x);

(b). Payoff function z(x,a) is a linear function of both

random variable E and choice parameter a, zu = 2m,= 0.

This theorem depends on parameter 7, if 7 is in different

intervals, the required conditions are different.

5 2.1.5 Monotone Likelihood Ratio Changes in Randomness
Ormiston and Schlee (1991) use Monotone Likelihood
Ratio (MLR) stochastic dominance which is a subset of FSD to
study the tradeoff between restricting the utility function

and restricting the types of changes in randomness.

Ormiston and Schlee specify a set of economic agents whose
behavior is known under certainty, then they investigate the
behavior of these economic agents under uncertainty. For a
class of economic agents whose preferences under certainty

are such that the optimal choice parameter increases with

19
the increase of non-random variable x, what type of CDF
change for E when E is random causes all economic agents in
this class to increase optimal choice parameter a under
uncertainty?

CDF F(x) has a support of [b,A] and CDF G(x) has a
support of [a,B]. m(x) is a non-negative and non-decreasing
function. H(x) is the difference between the two CDF’s H(x)
= G(x) - F(x). Ormiston and Schlee have the following MLR

definition.

Definition 2.6 (Ormiston and Schlee): CDF G(x) is MLR
dominated by CDF F(x) if there exists a non-negative and
non-decreasing function m(x) defined in interval [b,B] and
the following conditions are satisfied.

(a). H(x) = G(x) in [a,b);

(b). dH(x) = [1-m(x)]dG(x) in [b,B], where dH(x) is the
derivative of H(x), dG(x) is the derivative of G(x);

(c). H(x) = -F(x) in (B,A].

Ormiston and Schlee consider model Eu(x,a), where E is
the random variable, a is the Choice parameter. They deal
with random variable and choice parameter directly without
payoff function z. If a, and a6 maximize the expected
utility function Eu(x,a) under CDF's F(x) and G(x)

respectively, then

20
Theorem 2.5 (Ormiston and Schlee): The following conditions

are equivalent.

(a). CF 2 aa whenever CDF G(x) is MLR dominated by CDF

F(x);
(b). um(x,a) 2 0 whenever uafix,a) = 0.
Condition um(x,a) 2 0 whenever uaLx,a) = 0 is a condition

under certainty. This theorem says that whenever economic
agents increase their optimal choice parameters under
certainty, they will increase their optimal choice

parameters under uncertainty if facing an MLR risk shift.

5 2.1.6 Summary: Relationship Among Different Type of
Changes in Randomness

It is important to see the relationship among these
special types of changes in randomness. In terms of
density function, a strong increase in risk requires that
the density function g(x) of the riskier random variable,
Figure 2.1, to be smaller than that of the less risky random
variable in interval [b,B], the riskier random variable also
distributes in intervals [a,b] and [B,A]. A relatively
strong increase in risk allows the density function of the
riskier random variable be bigger than that of the less
riskier random variable in two intervals [b,c) and (C,B].
But in interval [b,c), f(x)/g(x) is non-decreasing; And in

interval (C,B], f(x)/g(x) is non-increasing. The MLR

21

 

 

1'igure 2.1. f(x) is the density function of the initial
random variable, g(x) is the density function
of the final random variable.

22
requires that the support of the dominated random variable
is somewhere lower than that of the dominating random
variable, and d[G(x)-F(x)] = [1-m(x)]dG(x) is satisfied in
interval [b,B], where m(x) is a non-negative and non-
decreasing function.

We can see that a strong increase in risk is the most
restrictive type of increase in risk. It implies a
relatively strong increase in risk which in turn implies a
relatively weak increase in risk. A relatively strong
increase in risk and a relatively weak increase in risk have

different restrictions in intervals [b,c) and (C,B].

2.2 Deterministic Transformation Approach

The deterministic transformation approach changes the
initial random variable and hence the CDF by mapping each
possible outcome of the random variable into a new value.
Meyer and Ormiston (1989) define deterministic function t(x)
as deterministic transformation function which transforms
random variable E into a new random variable. Deterministic
function t(x) is a non-decreasing, continuous and piecewise
differentiable function. The non-decreasing assumption
combined with the monotonic preferences for outcomes ensures
that the transformation does not reverse the preference
ordering over the various possible outcomes of the original
random variable. A special group of deterministic

transformations are simple transformations. We will review

23

the comparative static result for simple transformations.

5 2.2.1 Deterministic Transformations

To study the marginal risk change effect of price,
Sandmo (1971) transforms random variable E into a new random
variable (7E + 0), where 7 is the multiplicative shift
parameter and 0 is the additive shift parameter. A change
in 0 will only change the mean of the random variable,
Sandmo gives the following result regarding the changes in

parameter 0. The model is Eu[ax+k(a)+§].

Theorem 2.6 (Sandmo): The decreasing absolute risk aversion
(DARA) is a sufficient condition for optimal choice

parameter a to increase with an increase in parameter 0.

This result is a local one, it has to be evaluated at 7 = 1
and 0 = 0.

A change in 7 (from point 7 = 1 and 0 = 0) will change
the mean of random variable E. We however only need a mean
preserving increase in risk, Sandmo therefore reduces 0
simultaneously. To restore the mean we need d0/d7 = -u,
where u is the mean of random variable E. Ishii (1977)
proves a comparative static result for a change in

parameter 7.

24
Theorem 2.7 (Ishii): Decreasing absolute risk aversion
(DARA) is a sufficient condition for optimal choice variable

a to decrease with an increase in parameter 7.

The transformation Sandmo uses is a prototype of a
deterministic transformation. Meyer and Ormiston (1989)
prove that a general deterministic transformation is a
fourth characterization of a Rothschild and Stiglitz MPS
increase in risk. They have the following theorem

concerning transformation function t(x).

Theorem 2.8 (Meyer and Ormiston): Deterministic
transformation t(x) represents a Rothschild and Stiglitz
increase in risk for the random variable given by CDF F(x)
if function k(x) = t(x) - x satisfies the following
conditions:

(a)- I: k(x2dF(x)

0, where [a,A] is the support of E;

(b). I: k(X)dF(X)

IA

0, V y e [a,A].

The advantage of deterministic transformation is that
the changes in randomness can be restricted in different
ways from that under the CDF approach. Restricting function
t(x) beyond those in theorem 2.8 will generate special
changes in randomness. Meyer (1989) uses deterministic

transformation to define an FSD change.

25
Theorem 2.9 (Meyer): Transformed random variable t(x)
dominates initial random variable E in the sense of an FSD

if and only if [t(x)-x]-f(x) 2 0 V x e [a,A].

The FSD dominating CDF lies below the initial CDF, it
generates a greater expected utility for all economic agents
with non-decreasing utility functions.

For SSD risk changes, Meyer has:

Theorem 2.10 (Meyer): Transformed random variable t(x)
dominates initial random variable E in the sense of an SSD

if and only if S§[t(x)-x]f(x)dx 2 0, V y e [a,A].

Recall that SSD changes in randomness yield a higher
expected utility for all economic agents with a non-

decreasing and concave utility function.

5 2.2.2 Simple Increases in Risk and Comparative Statics
One way to further restrict t(x) so that determinate
comparative static results can be obtained is to restrict
the difference between the two random variables, k(x) = t(x)
- x. Simple transformations require k(x) to be a monotonic
function. A simple transformation produces a special
Rothschild and Stiglitz increase in risk. A simple increase
in risk is the case where k’(x) 2 0 if k(x) differentiable,

it implies that there exists a value.f'6'[a,A] such that

26

the values of E to the right or to the left of XEIare moved
away from x’ as the risk increases. The original random
variable is stretched out around particular value xd'to get
the new random variable.

The economic agents maximize expected utility function
Eu[z(x,a)]. The comparative static result from the simple

transformation is in the following theorem.

Theorem 2.11 (Meyer and Ormiston): The economic agents
choosing a to maximize Eu[z(x,a)] will decrease optimal
choice parameter a when the random variable undergoes a
simple increase in risk, if

(a). Utility function u(z) displays decreasing absolute
risk aversion (DARA);

(b). 2x20, znso, zm20andzmso.

This is a generalization of Sandmo and Ishii’s results for
the competitive firm. Condition DARA is widely used and
accepted.

The simple transformation, like the general
deterministic transformation, can generate FSD and SSD
stochastic dominate changes in randomness. Comparative
statics for these FSD and SSD changes are also possible.
Ormiston (1990) defines a simple FSD transformation as

following.

27
Definition 2.7 (Ormiston): The random variable given by
simple transformation t(x) first degree stochastically
dominates (FSD) random variable E if and only if function

k(x) = t(x) - x satisfies k(x) 2 0 V x e [a,A].

The comparative static result for this FSD simple

transformation is in the following theorem.

Theorem 2.12 (Ormiston): The optimal value of the choice
parameter increases for any simple FSD transformation if
(a). u’ > 0, u“ s 0 and A’ s 0;
(b). zx>0, znsOandzw20;

(c). k(x) > 0 and k’(x) s 0.

A simple SSD change in randomness is an SSD shift

generated from a simple transformation which is defined as

the following.

Definition 2.8 (Ormiston): The random variable given by
simple transformation t(x) second degree stochastically
dominates (SSD) random variable E if function k(x) = t(x) -

x satisfies.5§.k(x)dF(x) 2 0 V y e [a,A].

The comparative static result for a simple SSD

transformation is in the following theorem.

28
Theorem 2.13 (Ormiston): The optimal value of the choice
parameter increases for any simple SSD transformation if
(a). u’ > 0, u" s 0 and A’ s 0;
(b). z>0,z so, zm20andzwso;

(c). k’(x) s o and I; k(x)dF(x) 2 o v y e [a,A]_.

A simple SSD transformation requires not only that 2; be
monotonically increasing in random variable E, but also it
requires that 2;.be concave in the random variable. A
relaxation of the restriction on the transformation requires
an extra condition on the payoff function, that is 2;“ s 0.
This highlights the tradeoff between the restrictions on the
payoff function and the restrictions on the types of changes
in randomness.

We have reviewed strong increases in risk, relatively
strong increases in risk and relatively weak increases in
risk. All these are subset of MP5. We also reviewed the
alternative approach to comparative statics, the

transformation approach. In chapter three we will introduce

a new increase in risk, an independent increase in risk.

CHAPTER 3
INDEPENDENT INCREASES IN RISK

In this Chapter we will introduce a special type of
increase in risk, an independent increase in risk. A random
variable E is said to be an independent increase in risk
from the random variable E if E =4 E + E, where E is
independent of E and E(E) = 0. Here, " =4 " represents "is
equal in distribution to". Independent increases in risk
can produce determinate comparative statics in the 1-1-1
models which will be discussed in chapter 4. In this
chapter, section 1 discusses stochastic transformations and
defines independent increases in risk. In section 2, we
will provide a method to recover the independent random

variable given the distribution of E and E.

3.1 Independent Increases In Risk

The second of Rothschild and Stiglitz's three
definitions of an increase in risk is that random variable E

is riskier than random variable E if
d

9=§+z,

where

29

30

E(Elx) = 0.

An independent increase in risk further restricts the
Rothschild and Stiglitz's definition by requiring E to be

independent of E.

Definition 3.1: A random variable E is an independent

increase in risk from the random variable E if

3i=¥ E + E,

where E is independent of E and E(E) = 0.

To better understand what an independent increase in
risk is, we shall examine the conditions placed on the CDF's
for E and E for the special case of discrete random
variables with a finite number of mass points. The supports
of all the random variables are assumed to be contained in
compact intervals on the real line. Assume that random
variable E has probability function f(x) and CDF F(x) with
support [b,B]. The probability function f(x) has mass p,==
f(x,) at x, < :r2 < - - - < x". Let E be a random variable with
the probability function h(e) and CDF H(e) on [gush]. We
assume E is independent of E and E(E) = 0. The probability
function 11(6) has mass g, = h(ej) at e, < 52 < < em so
that E(E) =J§1qj.ej = 0. Note that e, = 0 for some j is

possible. Suppose g(x) and G(x), respectively, are the

31

probability function and CDF of the random variable E which

is defined by

E="E+E.

Then the probability function g(x) has mass pi]. = p,-qj = g(xy.)

at x9. , where

3*:
Ill

x,+ e}. fori =1, ”-11, j =1, H, m.

We may denote the support of the random variable E by [a,A].
Let s(x) be the difference between the two probability

functions g and f:

S(1‘!) = g(x) " f(X)-

Also, let S(x) be the difference between the two

corresponding CDF’s:
S(x) = G(x) - F(x).

According to the Rothschild and Stiglitz, since E is an R-S
increase in risk from E, s(x) can be decomposed into the sum
of a number of MPS’s.

In the following analysis we will examine the
properties of s(x) in details. Specifically, we first
concentrate on the behavior of s(x) over a subinterval of
its support and then extend the analysis to the entire
support. Initially to simplify the analysis, we make an

important assumption about the "noise" random variable E: we

32

assume the support [6,, 6",] of E is relatively narrower in
comparison with the support [b,B] of E. In particular, if

we define

><
III

E, + 6, and x; 5x, + Em, for 1' = 1, on, n,

then the subintervals [x,',X;], i = 1, on, n, do not overlap.
That is, the length of the support [6,, GM] of E is shorter

than the distance between any two adjacent points x, and EN.
(The analysis without such a restriction is in Appendix A.)

Since E(E) = 0, we have 6, < 0, 6 > 0, and x,' < x, < x,7, for

all 1'. Moreover, we have x, a x, + 6, 6 [x,’,x;], for all i and
'j, which implies the support of E, as well as that of s(x),
are included in 1Q1[x,',x,'].

Given the non-overlapping subintervals [x,',x;],

i = 1, no, n, let us define s,(x) to be the restriction of

s(x) on [x,',x;]. That is,

_ {3(x), for x 6 [x,/,x,”];

0, otherwise.
A careful inspection of the density function g(x) and the
definition of s(x) reveals that the mass of discrete
function s,(x) are all positive except at point x, and the
sum of all the mass is always zero. So each s,(x) is an MP8
function. Moreover the shapes of all s,(x), 1' = 1, no, n,
are all proportional to each other with p, being the

n
proportionality factors. Finally, we note s(x) = £15,.(x) .
1:

33
To further explore the property that the shapes of
54x) are proportional, we introduce the following

definition of linear shift.

Definition 3.2: The discrete MPS sJX) is a linear shift

from s,(x) if

Si(x) = >‘i°SI(X+Ei)I for X 5 [XI/Xi] and X + 5i 5 [Xi/Xi]:
where
A,- = pi/pll
£1“ = (X, - X,),

are referred to as the linear shift parameters.

Since s,(y) = s,(y-£,) />\, with y = x + 5,, s,(x) is also a
linear shift of s,(x) if s,(x) is a linear shift of s,(x) .
Let us examine the relationships among a set of MP8

which differ by linear shifts.

Definition 3.3: A discrete Independent Mean Preserving
Spread (IMPS) for the discrete random variable E is a set of
discrete MPS's {sdxj,i = 1, ---, n} in which 5417 are

linear shifts of each other.

The IMPS is the key concept in determining the integral

conditions for an independent increase in risk.

34
Define cumulative function s,(x) from s,(x) as follows.

SI“) 3 fax-5'1“)“ = Esrixy)s

x,“

where x, a x. + 6.. Note that s,(x) are step functions.

Since

St“) = 231“,” = E 11510;; + 5:) = E APIN-W) = A75105 + 51):
’u“ ‘0“ 311‘“!

s,(x) is also a linear shift of s,(x) . In fact all s,(x),

i = 1, --- n, are linear shifts of each other. Since

I
n

s(x) 32:12.00. sac) = I: smdt and 5.0:) = I: s.(t)dt, we
:2
have S(x) =iE=1S‘(x) . S(x) is also a step function.

Now, let us define T(x) as

w) = fa’S(t)dt = [jam - F(t) 1dr.

Note the Rothschild and Stiglitz’s conditions for G(x) to be

an MP8 increase in risk from F(x) are

5* [G(t)-F(t)]dt _>_ o, for x e [a,A];

(1) T(X)
(2) TM)

I: [G(t)-F(t)]dt 0.

where [61,11] is the support of G(x) - F(x) . Define T,(x) by

Tm = fjsxndx. for x e Ix.’.x.”1.

n
so T(x) =1221T,(x) . Note that T(x) and T,(x) are all
continuous functions. Since s,(x) is an MP8, we have T,(x) 2

0, for x 6 [X;,X,7], and T,(x;) = 0 by the Rothschild and

35

Stiglitz's conditions. Moreover, since

Tux) = (fund:

= fjA.S.(t+£.)d(r+e.)
=fx+€lA‘S1(Z)dzl z = t +51
mi,
= *1 T1(x + £1):
!n(x), i = 1, ---, n, are also linear shifts of each other
if s,(x), i = 1, ---, n, are.

For an IMPS increase in risk, T(x) can be decomposed
into T(x) =l§1T,(x), where T,(x) are linear shifts of each
other with the shift parameters determined by the
probability distribution of E. Note that iffn(x) are
linear shifts of each other, then 34x) are also linear
shifts of each other, and so are sdx7.

The relationship between the independent increases in

risk and the independent mean preserving spreads is

presented in the following theorem:

Theorem 3.1: Given two discrete random variables E and E,

the following two statements are equivalent.

Statement 1:
T(x) ==Sj [G(t)-F(t)]dt satisfies the following three
conditions:

11. T(x) 2 0. for x 6 [6,11];

36
12. T(A) = 0,-
I3. T(x) can be decomposed into
B
Tm = 1.211300.
where

Ti(x) = >‘I'°TI(X+EI')'

for x 6 [x,',x,7], for x + E, 6 [x,',x,'], A, = p,/p,, and

E, = (x, - X,). Also T,(x) 2 0 and T,(x,7) = 0.

Statement II:
E =4 E + E, where E is discrete and independent of E

and E(E) = 0.

Proof: I implies II. Assume that T(x) = S: [G(t)-F(t)]dt
satisfies the three integral conditions. Then S(x) =
11:15,“) and s,(x) = A,-S,(x+£,); that is, s,(x) differ from each
other by linear shifts. Note that G(x) - F(x) = S(x) =
11:15,”) .

Define F, (x) = F(x) + s,(x) , then F, (x) is a discrete
MPS increase in risk from F(x). Therefore, there exits a
random variable E, such that E(E,|x) = 0, E has a CDF F(x) ,
and E + E, has a CDF F, (x) according to the Rothschild and
Stiglitz's conditions. s,(x) contains all the information

needed to determine the random variable E,. Since S, (x) is

non-zero only in the interval [X1316], F,(x) and F(x) differ

37

on [x[,X,'], as shown in Figure 3.1.

 

s,(x), x1’sx<x1;
F1(x)=‘f(x1)+ 81m. x. s x < x1”:
_F(x), x1’lsx 3A.

The joint CDF of E and E, is F,(x) = F(x) ~H,(x) , where H,(x)

is the conditional CDF of E, given E = X,.

I /
f(x1)H1(x) . x1 5 x < x1];
F105) = //
FUL x1sst.
Hence,
Em, x, s x < x1,
f(x1)
”1“) =‘
S1(x) //
1 + —, x, s x s x, .
f(x1)

 

Define F,(x) = F(x) + s,(x) , then F2(x) is an MP5
increase in risk from F(x). Again, there exists a random
variable E2 such that E(E,|x,) = 0, E has a CDF F(x) , and E +
E2 has a CDF F2(x) according to the Rothschild and Stiglitz’s
conditions. s,(x) = S, (x+£2) -f(x,) /f(x,) contains all the
information needed to determine the random variable E2.
Since s,(x) at 0 only when x 6 [x,',x;], F,(x) and F(x) differ

on [xz'fié], as shown in Figure 3.2.

38

 

F(x)

 

 

 

 

 

«Figure 3.1. F,(x) and F(x) differ on [x,',x,'].

39

 

 

 

 

 

 

F(x)
I
J d
F,(x)
_I _-
F
x1 x2, x2 x2" x3

F1Quite 3.2. F,(x) and F(x) differ on [x,',x§].

40

 

r1111), x1 sx<x1:;
f(t1)+32(x). xésx<x2i
172(1) 3* ,,
f(x1) +f(x2) +Sa(x). xasx<x2;
,F(x), xélsst.

The joint CDF of E and E, is F,(x) = F(x) ~H,(x) , where H,(x)

is the conditional CDF of E, given X,.

 

ff(x1)s x15x<lei

F2(x) = Ma) +f(x2)H2(x). x; s x < xé’;

,F(x), xzflsst.

Hence,
g(x)-.ftx.) for.) ' ”2““2’
2 _

S2(3) St (x + 52) //
1+—=1+—, .
_ f(x2) f(x1) "2 ”“2

 

Thus we have H,(x) = H, (x+£,) , for x 6 [x,',x,] and for
x + f, 6 [x,,x,']; that is, the two conditional CDF's H, and H,
are the same. These random variables therefore have the
same conditional distributions E,|x, =" E,|x,.

By repeating the same process, we then have H,(x) =
H, (x+£,) , for all i, which implies that Ei and E, have the
same conditional distributions E,|x, =4 E,|x,, for all 1'. Let

Ebe

41

then E is independent of E and E(E) = 0.

Define G(x) = F(x) +i§15,(x), then G(x) differs from
F(x) by a sequence of MPS's which differ by linear shifts.
From SJE), i = 1, ---, n, we could determine the random

4 E + E, where E is independent of E

variable E such that E =
and E(E) = 0.

II implies I. As to the proof of the inverse part, we
note the previous argument that leads to the IMPS conditions

in the first part of this section is actually a proof of

this. Q.E.D.

Example 3.1: Consider a random variable E

Then E + E is
1 1
_6: "9. -4: _; .2:
i 8 8

which is an IMPS increase in risk from E. Note that the
assumption x,’ < x,” is not imposed in this example. In this
example, G(x) - F(x) changes signs from positive to negative

twice and there are four MPS's as shown in Figure 3.3. The

42

 

 

 

 

 

 

 

 

 

P — p I I I
F(x) I +

- I
I I I I _ fl
. 600

_+ I I I
A»
-6 -4 -Q 0 2 4 6
FigurI 3.3. G(x) differs from F(x) by four MPS’s and G(x)

is an IMPS increase in risk from F(x).

43
four MPS's are over the subintervals {-6,-2], [~4,0], [0,4]
and [2,6], respectively. All four MPS's are linear shifts
of each other. Actually the four MPS’s are identical in

this example since Xi==1 for all i.

3.2 Recovering the Independent Random Variable

Theorem 3.1 establishes the equivalence between an IMPS
and an independent increase in risk for a discrete random
variable. It does not, however, show how to find the
independent random variable E given the two random variables
E and E. Constructing such an independent random variable
without imposing the condition 1:} < Xi,“ is the subject of
this section.

Suppose that we have two random variables E and E with
the CDF's G(y) and F(x), respectively, as shown in Figure
3.4. At each xi, F(x) increases by pn- = f(x,), for i = 1,
~--, n, where xiare ordered as xi<th4. G(y) has steps at
Y]. with height p”. = g(yj), for j = 1, ---, m, where Y} are
ordered as yj < y)“.

Suppose E is an IMPS increase in risk from E. We now
demonstrate how to recover the independent random variable E
from the two density functions f(x) and g(y). It is easy to
see that the support of E is [Yr-x1, ym-xn] and the mass of
the random variable E at y, - x, is necessarily g(y,) /f(x,) by
the independence assumption. Therefore, the first mass

point of the random variable E is {yI-XI, g(y,) /f(x,)}.

44

P09

 

 

 

 

 

F J
yl 3’2 x1 y3 y4 x2 y5
Figure 3.4. How to find the CDF of independent random

variable E from two CDF’s.

45

By subtracting f(x,.)g(y,) /f(x,) from g(y) at
(x, + y, - XI) for every 1 to get rid of y,, we are left with
a new function which can be denoted as g”. This new
function g' has at most (111 - 1) mass points at, say,
y; < < y;-

We will repeat the same procedure to guy?) as we did to
g(y) to get the second mass point of E. Since gflj”) at y;
is derived from f(x) at.xy, the second mass point of E is
therefore {YE-x1, g‘ (YE) /r (x1) } -

By subtracting f(x,.) g°(y;) /f(x,) from g'(y°) at

(x, + y; - x,), we then have another new function which will
be denoted as g". It has at least one point fewer than 9'
does. Again we can get the third point of E which is
{y;‘-x,. g“ (y?) /f (x1) }.

Continuing this process until all the mass points of
g(y) are deleted, at which we then recover all the mass
points of the random variable E. Note that if any of the
above steps fails, we can infer that E is not an IMPS

increase in risk from E.

Example 3.2: Assume a random variable E has the CDF F(x)

and the density function f(x) as in Figure 3.5. So E is

I 2
_; 1’ _ I
{0’ 3 3}

If a random variable E has a CDF G(y) and a density function

46

 

 

 

 

 

F(x)
....................... ,_._.|
r — '— _ G(y)
.F —
| .
-2 4 0 1 2 3 4
Figure 3.5. G(y) is an IMPS increase in risk from F(x),

G(y) can be broken into two parts in this
example, the dotted line.

47
g(y), and is defined by

1 3 2
_2: —; ‘1: -1. —
{ 9 9 “

:il;££.
9.}

9

Then it can be verified that G(y) is an MP8 increase in risk
from F(x). By assuming G(y) is an IMPS increase in risk
from F(x), we should be able to find a random variable E
such that E is independent of E, E(E) = 0, and E =" E + E.
Since the mass of E at -2 is derived from E at 0, the
probability mass of the independent random variable E is
(1/9)/(1/3) = 1/3 and the first point of E is {-2, 1/3}. By
subtracting f(xJ/3 at x,-2 from g(y), that is, subtracting

1/9 at -2 and 2/9 at -1 from g(y), we get 9H3?)
'1: l; 0) 2; 3: l; 4: "' 0
9 9 9 9

9H3”) at -1 is derived from f(x) at 0, So the second point
of E is {-1, 1/3}. By subtracting f(xJ/3 at x,-1 from
9W3”), that is, subtracting 1/9 at -1 and 2/9 at 0 from

9711'). we have 9"(y")

391:4,Z}'
{ 9 9

g”(y") at 3 is derived from f(x) at 0 and the third point of
E is {3, 1/3}. By subtracting f(xJ/3 at x,+-3 from g”(y”),
that is, subtracting 1/9 at 3 and 2/9 at 4 from g"(y”), we

‘then have g”'=={0}. Consequently we have recovered the

48

random variable E

'which is mean preserving, and E(E) = 0. We have shown that

(3(y) is indeed an independent increase in risk from F(x).

APPENDIX

49

Appendix

In this appendix we relax the non-overlapping condition

x; < xi}, and show that the proof of Theorem 3.1 still holds

with some minor modifications.

Suppose s, (x) is a discrete function at points

{x,+e,, ..., xii-em} and S”, (x) a discrete function at points
{Em-+6" no, x,+,+e,,,}. Let us assume x, + a} = x,“ + 6,, or
Xawu for some positive integers j and k, then s(x) at

xg=
Now, let us assume

this point is equal to s(xy) + sum”) .

s (Xmuk) 0 when we define s,(x) , and s(xg-) = 0 when we

With such redefinitions, both s,(x) and

define s,“ (x) .
Therefore,

s,+,(x) still have the same properties as before.
S, (x) and T,(x) also have the same properties as before.

With the non-overlapping restriction, if x e [x,;,x,:] for
Some positive integer k, then

n
sat) =i§1s.m = smu-

That is, if x is in subinterval [xg,x,;] of [a,A], then s,(x)

= O fori¢k.
Without the non-overlapping restriction, for some

p°Sitive integers k and j, x may be in j > 1 subintervals,

When x E [x,,',x,,], on, x e [x,2+j,x,;+j], then

n
50‘) = E 530‘) = SM") +

That is, these j subintervals overlap.

50
S(x) and T(x) also have the same interpretations as
s(x) does without the non-overlapping condition. That is,
for some positive integers k and j, when x e [xpdfi], --- x

I

e [x,;+j,x;+j], because s,(xm) = 0, we have
n
5(X) =i§13;(x) = 5k“) + + Sm“)-

For T(x), we have, for some positive integers k and j, when
x e [x,;,x;], no, x e [x;+j,x,:+j], because Ti(xim) = O.

n
Tm =i§1T.-(X) = Tax) + + T..,-(x)-

n
Theorem 3.1 remains true except that T(x) = 231T,(x) has
1:

a different interpretation.

CHAPTER 4
COMPARATIVE STATIC RESULTS

In chapter 3, we introduced independent increases in
risk and IMPS, now we will study the comparative statics of
the independent increases in risk. An independent
stochastic transformation can also be used to generate first
order stochastic dominance (FSD) and second order stochastic
dominance (SSD) changes in randomness, we will consider the
comparative statics for these changes in randomness as well.

We first provide a comparative static result for an
independent increase in risk in section 1, and then proceed
to consider its applications in section 2. The comparative
statics is presented in a one random variable, one choice
parameter and one argument (1-1-1) model. The 1-1-1 model
can be written as Eu[z(x,a)], where E is the random
variable, a is the choice parameter and z is the argument.
An economic agent chooses a to maximize the expected utility

Eu[z(x,a)].

4. 1 Comparative Statics

This section concentrates on the comparative statics of
1-1-1 models. The increases in risk (changes in randomness)

aare accomplished by the independent transformation. A

51

52

random variable E is transformed into (E + E), where E is

independent of E and E(E) = 0. When E has a non-positive

support, E FSD E + E, and E SSD E + E when E(E) s 0.
Before our main theorem, we prove the following

corollary.

Corollary 4.1: Assume:

(a). Random variable E has a support [a,A] and a density
function h(e) and CDF H(e);

(b). W(€) and v(e) are two continuous and differentiable
functions of e, w’(e) s 0 and v’(e) 2 0;

Then Ew(e) -v(e) s EW(€) -EV(€) .

Proof: Let W(e) = w(e) - Ew(e) and V(e) = v(e) - Ev(e), we
then need to prove E W(e)-V(e) s 0. Let dK(e) = V(€)°dH(6),
we have K(e) = S; V(t) ~dH(t) + K(a) and

5: We) 'V(6) ~dH(e)

5: WE) -dK(e)

W(e)-K(e) I: - S: K(a)-Wm ~de

W(6)°[S.‘. V(t)-dH(t)+K(a)J I:
- S: W'm ~[Sz V(t) -dH(t)+K(a)J-de

- I: two-(S; V(t)-dHrtn-de s 0-
'The last inequality is because W'(e) = w'(e) s 0, and since
5;: V(t)-dH(t) = o and v'(t) = v’(t) 2 0 then 5; V(t)-dH(t) s

(D. This completes the proof. Q.E.D.

53
We now prove our main theorem of the chapter concerning
the comparative static effect of an independent increase in

risk.

Theorem 4.1: Assume:
(a). Utility function u satisfies u’ 2 0 and u" s 0, A’(u)
s 0 and P’(u) s 0, where A(u) = -u"/u’, P(u) = -u”’/u";

(b). 2 20, z

x

m S 0, zour 2 0, 2w 5 0 and A’(z) s 0, where
11(2) = 43/2,;
(c). E is replaced by E + E, where E is independent of E
and E(E) = 0;

An economic agent maximizing Eu[z(E,a)] will decrease

optimal choice parameter a.

Proof: The FCC after the independent increase in risk is
Edi u’[z(x+e,a)]'z;(x+e,a).

Since u’[z(x+e,a)] is a decreasing function of E and
z;Lx+e,a) is an increasing function of E, the following
inequality follows
E,u’[z(x+e,a)]°za(x+e,a)
S E,u’[z(x+e,a)J-Ecza(x+e,a)
s E,u’[2(x+e,a)]'za(x,a) .
The last inequality is because za(x+e,a) is concave in E.
Let v(x+e) = -u’[z(x+e,a)], where a is the constant

1:hat maximizes the expected utility before the independent

>46!

54

increase in risk. Since

v’(x+e) = -u"[z(x+6,a)]'zx(x+e,a) 2 0,

v"(x+e) = -u”’[z(x+e,a)]°zf(x+e,a)

- u"[z(x+e,a)]'zn(x+e,a) 5 0,

therefore v(x+e) is an increasing and concave function of E.
Define risk premium ¢(x,e) for risk E under function V(X+6)
as E‘v(x+e) = v[x-¢(x, 6)], (ME, 6) is then positive.

Let A(v) = -v"(x+e) /v’(x+e), then

A(v) = -u”’[z(x+e,a)]'zx(x+e,a)/u”[z(x+e,a)]

- zn(x+e,a) /zx(x+e,a)

P(u)-zx(x+e,a) + A(z) 2, 0,
A’(V) ov’(x+e) = P’(u) -u'[z(x+e,a)]-zf(x+e,a)
+ P(u) 'zn(x+e,a) + A’(z) 'zx(x+e,a) S 0.
Hence we conclude A’(V) s 0 and therefore ¢x(x, e) s 0 by
Pratt (1964).
Now the FCC after the independent increase in risk can
be written as
E115, u’[z(x+e,a)]°za(x+e,a)
s E,.E, u’[z(x+e,a)]°za(x,a)
= -E,,.EI V(x+e) °za(x,a)
= -E. var-w -z.,(x.a)
= E. U’[2(X-¢.a)J-Z.(x,a)
= Ex {u’[z(x-¢,a)]/u’[z(x,a)]}ou’[Z(x,a)]°za(x,a)
S 0,
Lf D = u’[z(x-¢,a)]/u’[z(x,a)] is positive and decreasing in

’5': Exu’[z(x,a)]°za(x,a) = 0 is the FCC before the

‘d

55

independent increase in risk.

Next we prove that D is positive and decreasing in E.
D is positive obviously. The derivative of D with respect
to E has the same sign as the numerator of the derivative,
which is

u"[z(x-¢,a)] 'z,(x-¢,a) ' (1"l’.) 'U’[Z(Xza)]

' U'[Z(X'¢za)]'u"[z(xra)I'zdxla)

‘1"[Z(X'¢ra)]'zx(xta)”17200001

IA

- u’[Z(x-¢.a)]°U"[Z(X.a)J'Z.(X.a)
= u’[z(x‘¢:a)]°zx(xza)'u’[Z(Xra)]'
{u"[2(x-¢,a)]/u’[z(x-¢,a)] ‘ u"[z(x,a)]/u’[z(x,a)]}
= u’[Z(X'¢,a)J'zx(xla)'U’[Z(X,a)]°{A(u[z(xpa)])
' A(u[z(x-¢,a)])}
s o, if A’(u) s o.
¢,(x,e) s 0 if A’(v) S 0. P’(u) s 0 and A’(z) s 0 are
sufficient conditions for A’(v) s 0.
Thus the FCC after the independent increase in risk is
.negative. In order to maximize the expected utility, a has

‘to'be decreased. This completes the proof. Q.E.D.

For a Rothschild and Stiglitz increase in risk, the
(itistribution of random variable E may be different for
different values of random variable E. The distribution of

‘2? (iepends on x. As the random variable E moves across its

Support, it may be a different 2' that is added to 3?. Under

an independent increase in risk, the distribution of random

'I‘

56

variable E is the same no matter what realized value random
variable E takes. The distribution of E is independent of
x. The increase in risk is the same for different value of
random variable E. This uniform property makes the
comparative statics for an independent increase in risk
attractive.

Like a strong increase in risk, an independent increase
in risk is also a generalization of an introduction of risk.
An independent increase in risk, however, generates a
general Rothschild and Stiglitz increase in risk, when the
initial random variable is degenerate at a point.

An independent increase in risk imposes no restrictions
on the two distribution functions in the center of the
supports. The two CDF's may cross many times. The support
of F(x) is however contained in the support of G(y) for an
independent increase in risk. A strong increase in risk, a
:relatively strong increase in risk and a relatively weak
:increase in risk all require that F(x) - G(x) is non-

decreasing in the center of the supports, and is non-
.iJucreasing at the two ends of the supports.
P(u) = -u ’"/u" in our proof is termed absolute prudence
lDd?’ Kimball (1990) who first introduces it in a precautionary
Eial‘ling problem. Kimball defines precautionary premium

¢ (W, x) as the quantity satisfying

u ' [w-¢ (wt) J = E.u ' (w+x2 .

57

where w is the final wealth and E is the random variable.
It can be showed that ¢(w,x) is approximately equal to
-(1/2)-0241”7u", where qzis.the variance of random variable
E and P(u) = -u ”’/u" is the absolute prudence.

There are some basic assumptions about the absolute
prudence. Kimball argues that the absolute prudence is the
propensity to prepare oneself in the face of uncertainty.
The absolute prudence is assumed to be a decreasing function
of the initial wealth and it is greater than the absolute
risk aversion measure if the utility function is decreasing
absolute risk averse (DARA). A positive absolute prudence
is a necessary condition for DARA, A’(u) = A(u)-[A(u) -
P(u)]. Note that in our proof ¢(x,e) is the risk premium
under function V(x+e) and is the precautionary premium under
utility function u[z(x+e,a)].

Eeckhoudt, Gollier and Schlesinger (1992) use absolute
prudence in their background risk study. An economic agent
has a utility function u(w), where w = E + a-E is the final
wealth. Random variable E is an exogenous and unavoidable
background risk whose CDF is initially GHQU. There is a
second source of uncertainty due to the existence of an
independent and endogenous risk E with CDF F(x). 0 is the
choice parameter. Eeckhoudt, Gollier and Schlesinger
consider the impact on optimal choice parameter a when an

agent faces a change in the distribution of unavoidable risk

58

E from CDF G, (y) to G2(y) .

The method Eeckhoudt, Gollier and Schlesinger use is
independent transformation. When initial random variable E
first order stochastically dominates (FSD) E + E, where E
has CDF G,(y) and E + E has CDF Gz(y) and E is the
independent noise, the optimal choice parameter decreases if
the utility function is DARA. When random variable E second
order stochastically dominates (SSD) E + E, the optimal
choice parameter decreases if the utility function is
standard risk aversion, Kimball (1993). A utility function
is standard risk aversion if it is DARA (A’(u) s 0) and

decreasing absolute prudence (DAP) (P’(u) s 0).

If independent random variable E has a non-positive
support, random variable E FSD random variable (E + E). We
have the following theorem regarding this FSD change in

randomness.

Theorem 4.2: Assume:

(a). Utility function u satisfies u’ z 0, u" s 0 and A’(u)
S 0;
(b). 2,20, ZnSOandszO;

(c). E is replaced by E + E, where E is independent of E
and E has a non-positive support;
Then an economic agent maximizing Eu[z(x,a)] will

decrease optimal choice parameter a.

59

Proof: The FCC after an FSD change in randomness is
ErE, u'[z (x+e,a) ] -za(x+e,a) .

Since E has a non-positive support and zm(x,a) 2 0, we have
za(x+e,a) s za(x,a), therefore

E,u’[z(x+e,a)]-za(x+e,a)

S Etu’[z(x+e,a)]-za(x,a).
The FCC can be written as

15.3,)?!¢ u’[z(x+e,a)]-za(x+e,a)

s E,.E. u'[2(x+e.a)J-z.(x.a)

E,{E,u’[z(x+e,a)]/u’[z(x,a)J}-u’[z(x,a)j-za(x,a).
The expression Egu’[z(x+e,a)]/u'[z(x,a)] is positive and
decreasing in E. It is obviously positive. Its first order
derivative with respect to E has the same sign as its
numerator which is

u’[z(x,a)J-E,u"[z(x+e,a)]-zx(x+e,a)

- u"[z(x,a)]-zx(x,a) ~E,u'[z(x+e,a)]

IA

1173090)]'E.U"[Z(X+E,a)]'zx(xra)

' u"[2(x,a)]-zx(x,a)~Etu’[z(x+e,a)]
= u’[z(x,a)]'zx(xra)'E.U’[Z(X+€,a)]'

{u"[z(x+e,a)]/u’[z(x+e,a)] ' u"[z(x,a)]/u’[z(x,a)]}

= u’[z(x,a)]-z,(x,a)~E,u’[z(x+e,a)]-{A(u[z(x,a)])

" A(u[Z(X+€,a)J)}
u’[z(xta)]'zx(xra)'E.U’[Z(X+€,a)]°{A(UIZ(X,0)J)

‘ E,A(u[z(x+e,a)])}
so, if A’(u) s 0.

IA

60
The last inequality is because u’[z(x+e,a)] is decreasing in
E and A(u[z(x,a)]) - A(u[z(x+e,a)]) is increasing in E. Now

we have
E.{E.U’[Z(X+€.a)J/U’[Z(X,a)]}°U’[Z(X,a)J-Z.(X,a) s 0.

That is, the FCC is negative. In order to maximize the

expected utility, a has to be decreased. Q.E.D.

An FSD change in randomness is stronger than an MP8
increase in risk, it therefore needs less restrictive
conditions on the utility functions than an MP8 increase in
risk does. A similar comparative static result is also
available for an SSD change in randomness. An SSD shift is
a combination of an FSD and an MP8 shifts, Hadar and Sec
(1990). The comparative static result is then easily
obtained. The comparative static result is in the following

theorem.

Theorem 4.3: Assume:

(a). Utility function u satisfies u’ 2 0 and u" s 0, A’(u)
S 0 and P’(u) S 0, where A(u) = -u"/u’, P(u) = -u”’/u";
(b). 2 2 0, z < 0, z 2 0, z s 0 and A’(z) _<_ 0, where

x XX _ at at:

11(2) = -zn/zx;
(c). E is replaced by E + E, where E is independent of E
and E(E) s 0;

An economic agent maximizing Eu[z(x,a)] will decrease

61

optimal choice parameter a.

Proof: Combining the proofs of Theorems 4.1 and 4.2, we can

prove this theorem. Q.E.D.

An SSD change in randomness is the least restricted
change in randomness of the three and it thus needs the most
restrictive conditions on the utility functions. This shows
the tradeoff between the restrictions on the changes in

randomness and the restrictions on the utility functions.

4.2 Examples of Applications

In this section, we will examine some applications of
the independent increases in risk theorem. Rothschild and
Stiglitz (1971) have discussed the savings and uncertainty,
portfolio, a firm's production, the output level of a
competitive firm and multi-stage planning models, we will

consider these models here.

5 4.2.1 Savings and Uncertainty

Rothschild and Stiglitz (1971) consider an economic
agent who has a given wealth W0 which he wishes to allocate
between consumption today and consumption tomorrow. The
savings today yields a random return e tomorrow, the
expected utility is Eu(C“C§), where C,is the consumption in

period 1, i = 1,2. This is a two period consumption model.

 

62
Rothschild and Stiglitz assume that the utility
function is strictly increasing, strictly concave and

separable,
E'WCuCz) = 11(0)) + (1'6)°EU(02).

where C, = (1-s)-W0 and C2 = s-Wo-e, s is the savings rate, 6
is the time discount rate. The first order condition (FOC)

is
u’[(l-s) .w0] = Eu’ (s-Wo-e) . (1-6) -e.

Rothschild and Stiglitz then conclude that an increase in
risk in the sense of MP8 will increase or decrease the
optimal savings 5 depends on the concavity of u’(s-W0-e) . (1-
6)-e.

Sandmo (1970) studies a two period consumption model
without assuming the separable utility function. The first
period budget constraint facing the economic agent is I,==Cy
+ 3,, where I, is the income in the first period, S, is the
savings and C, is the consumption. The second period
consumption is C2 = I2 + 5,-5, where E is the rate of return,
1} is the income in the second period. The economic agent
maximizes the utility function u[C,,I,+(I,-C,)£], where C, is
the choice parameter.

Sandmo assumes that the second period income is a
random variable. Replacing I2 with E, he rewrites the model

as Eu[a,x+(x-a)£], where a -cy is the choice parameter, A

63

and E are the exogenous parameters and E = I, is the random
variable. To study the comparative static effect of the
random variable, Sandmo replaces the random variable with 7
+ O-E, where 7 is the additive shift parameter, 0 > 0 is the
multiplicative shift parameter. The comparative static
result concerning an increase in parameter 7 is in the

following theorem.

Theorem 4.4 (Sandmo): Assume:

(a). Utility function u satisfies u’ 2 0, u" s 0 and

an ' £4122 > 01'

(b). 70 is replaced by 7,, and 'y, 2 70;

(c). Eu[a,, (y,+0x)+(}\-a,)£] is maximized at a,, 1' = 0,1;
Then a, > do if the utility function is decreasing

absolute risk aversion (DARA).

This result is a local one, because it has to be evaluated
at point 0 = 1, 7 = 0.

Dardanoni (1988) discusses the same problem. Instead
of taking the consumption in the first period as the choice
parameter, Dardanoni takes the savings in the first period
as the choice parameter. The model becomes Eu[k-a,x+a£].
Dardanoni considers a Rothschild and Stiglitz increase in
risk for the additive shift random variable. The
comparative static result is stated in the following

theorem.

64

Theorem 4.5 (Dardanoni): Assume:
(a). Utility function u satisfies u’ 2 0, u" s 0 and
11,2 - E-u22 > 0;
(b). E0 is replaced by E’, where E1 is a Rothschild and
Stiglitz increase in risk from E”;
(c) . Eu[)\-a,,x"+a,£] is maximized at a,, i = 0,1;

Then a, 2 do if the absolute risk aversion, A2 =

-un/u2, is non-increasing in a.

The assumption that A,is non-increasing in a means
that: (1) . A, decreases when the random component of the
utility function increases; (2)..A,increases when the
certain component of the utility function increases. The
random component is the second argument and the certain
component is the first argument. This assumption was used
by Sandmo (1969).

Dardanoni also considers a multiplicative shift
problem. The model is Eu[A-a,£+ax]. He limits the choice
variable a 2 0. The comparative static result is in the

following theorem.

Theorem 4.6 (Dardanoni): Assume:

(a). Utility function u satisfies u’ 2 0 and u" s 0 and
u” - E-u,2 > 0;

(b). a is an increasing function of x under certainty;

(c) . E0 is replaced by E’, where E’ is a Rothschild and

65

Stiglitz increase in risk from E”;

(d). Eu[)\-a,,x‘a,+£] is maximized at 02,, i = 0,1;

Then a, 2 do, if E = 0 and the relative risk aversion is

non-decreasing in a.

The relative risk aversion is defined as
R(uz) = -ax-u22()\-a, ax) /u,()\-a, ax) .

For E ¢ 0, the comparative static result will still hold,
after replacing the relative risk aversion R(uz) with the
proportional risk aversion P(ufl. The proportional risk

aversion is defined as
P(uz) = -ax-u22()\-a,£+ax) /u2()\-a,£+ax) .

The savings and uncertainty model is a two argument model,
and it is actually presented in a one random variable, one
choice parameter and two argument format (1-1-2), Choi
(1992).

Now we will study a general savings and uncertainty
model, Eu[X-a,z(x,a)], where z is the second argument, a is
the choice parameter. When random variable E undergoes an
independent increase in risk, the comparative static result

is presented in the following theorem.

66
Theorem 4.7: Assume:
(a). Utility function u is increasing and concave in both
arguments and 11,2 2 0, 11,22 5 0, 1.1222 2 0;

(b).

"N

2 o, 2

xx

so,zzo,z

a s 0, z 2 0;

ca w

~

(c). x is replaced by E + E, where E is independent of E
and E(E) = 0;
Then an economic agent maximizing Eu[A-a,z(x,a)] will

increase the optimal choice parameter a.

Proof: The first order condition (FOC) before the
independent increase in risk is
-Eu,[)\-a,z(x,a)] + Eu2[>.-a,z(x,a)]°za(x,a) = 0.
The second order condition (SOC) is
E11,, - z-Eun-za + Eun'zi + EUZ'ZM s 0.
The FCC after the independent transformation is
-E,E,u,[>\-a,z(x+e,a)] + ErE,u2[)\-a,z(x+e,a)]-za(x+e,a) .

The first term in the above expression, C(e) ==-1y[x-

a,z(x+e,a)], is convex in 6, because

C’ = -un°a,s 0,
_ _ . 2 _ .
C" ’ 11122 Z: ”12 Zn 3 0°
Therefore

Eruzn-a. z (X+e, 02)] 2 WM k-a. z (x, a) J -

 

67

The second term, D(e) = u2[)\-a,z(x+e,a)]oza(x+e,a) , is
also convex in 6, because
D’ = uzfzx-za + u2°z s 0,

(I!

— e 2e e e . . e e

(XII-

So that
E,u2[X-a,z(x+e,a)]-2a(x+e,a) 2 u2[)\-a,z(x,a)]-za(x,a) .
Thus we have

-E,E,u,[)\-a, z (x+e, a) ] + E,E,u,[k-a, z (x+e, a) ] -za (x+e, a)

Z -E,U,[)\-a,z(x,a)] + Exu,[)\-a,z(x,a)]oza(x,a) = 0°

The FCC after the independent transformation is
positive under the old optimal choice parameter, to maximize
the expected utility, the choice parameter thus has to be

increased. Q.E.D.

The comparative static result for an FSD independent
transformation is also possible. We present it in the

following theorem.

Theorem 4.8: Assume:
(a). Utility function u is increasing and concave in both
arguments and u” 2 0,19” s 0;

(b). 2

1

20,2 20,2,“50;

a

(c). E is replaced by E + E, where E is independent of E

68

and E has a non-positive support;
Then an economic agent maximizing Eu[X-a,z(x,a)] will

increase the optimal choice parameter a.

Proof: The first order condition (FCC) and the second order
condition (SOC) are the same as those in theorem 4.7. The

FCC after the transformation is
-E,E,u,[)\-a,z(x+6,a)] + ErE,u2[)\-a,z(x+6,a)]-za(x+6,a) .

The first term, C(6) = flh[A-G,Z(X+6,a)], is decreasing

in E, because C' = -un'a,s 0. Thus
E.-U,[>\-a. z (we. a) J .>. win-a. z (X. a) J .

The second term, D(6) ==Lb[k-a,2(x+6,a)]-zg(x+6,a), is

decreasing in E, because 0' = ufl°agza-+z§°z s 0. So that

E¢u2[x-alz(x+6la)J°za(X+Ela) Z u2[x-alz(xla)]oza(xla)’
Therefore we have

-E,E,u,[>‘-a,z(x+6,a)] + E,E,u2[k-a,z(x+6,a)]~za(x+6,a)

2 -E,u,[)\-a,z(x,a)] + Exu2[>s-a,z(x,a)]-za(x,a) = 0.

This proves that the FCC is positive, to maximize the
expected utility, the choice parameter thus has to be

increased. Q.E.D.

69

5 4.2.2 Asset Proportion

Rothschild and Stiglitz (1971) consider a one safe and
one risky asset portfolio model. An economic agent
allocates his asset in money, which yields zero return, and
a risky asset, which yields a random return of e. The final
wealth is W(a):=lfiy(a-e+1), where a is the proportion held
in the risky asset. The expected utility is Eu[W(a)] and

the FCC for maximization is
WO-Eu’[W(a)]-e = 0.

And the comparative statics of an increase in risk once
again depends on the concavity of u’[W(a)]-e.

Hadar and Sec (1988) discuss an asset proportion
problem in a two risky asset portfolio model. There is a
correlation term in two random variable models. When one
random variable changes, the correlation term often changes,
too, which introduces a new dimension into the models. This
makes the two random variable models very difficult to
study. To avoid this problem, many models assume
independence between the two random variables.

When a risk averse agent diversifies between two risky
assets, the proportion held in the FSD, SSD or MP8
dominating asset is not necessarily greater than half. For
the case of FSD, Hadar and Sec (1988) offer the following
intuitive explanation. When random variable E FSD random

variable E, not only risk averse agents prefer E to E, but

70

also the risk loving agents prefer E to E. That is, E has
some characteristics which make it attractive to both risk
loving agents and risk averse agents. Thus risk averse
agents may invest more in E.

The economic agent is assumed to maximize the expected
utility function Eu(z), where z = aE + (1-a)E is the final
wealth, E and E are the two random variables representing
the returns of the two risky assets, a is the proportion
held in asset E. The two random variables are assumed to be
independent and have CDF's F(x) and G(y) respectively, the
joint CDF then is H(x,y) = F(x)-G(y).

Hadar and Sec has the following regarding FSD and Mean

Preserving Contraction (MPC).

Theorem 4.9 (Hadar and Seo): Assume:
(a). Utility function u satisfies u’ > 0, u" s 0 and u’” 2
0 (for MPC only);
(b). E and E are stochastically independent;
(c). Eu[ax+(1-a)y] is maximized at am;

Then (10 2 1/2 for E FSD E if and only if u’(z) -z is
non-decreasing in z 2 0.

Then an 2 1/2 for E MPC E if and only if u'(z) -z is

concave in z 2 0.

It is worth to mention that the payoff function z is linear

in both random variables.

71
Assume that in a two random variable model,
Eu[z(x,y,a)], the independent random variable is added to
random variable E, and it is independent of both E and E.
First we have the following property regarding the

independent transformation.

Theorem 4.10: An independent random variable E transforms
random variable E in a two random variable model
Eu[z(x,y,a)] and it is independent of both E and E, then the
independent transformation does not alter the covariance

between random variables E and E.

Proof: The covariance after the independent transformation
is COV(E+E,E), the covariance before the transformation is
COV(E,E). COV(E+E,E) = cov(E,E) + cov(E,E) = cov(E,E).
Q.E.D.

Now we are ready to consider the asset proportion
problem using the independent transformation. The expected
utility function is EU[ax+(1-a)y], where a is the choice
parameter. Assume short term buying and selling is not
allowed, 0 s a s 1, and E and E are stochastically
dependent, the joint cumulative density function is H(x,y).
If one random variable is an independent transformation of
the other one, then the proportion held in the less risky
asset is greater than 1/2. We state this result in the

following theorem.

72

Theorem 4.11: Assume:

(a). Utility function u satisfies u' > 0 and u" s 0;
(b). Random variable E is obtained from E by adding an
independent random variable E where E(E) = 0;

(c). .Eu[ax+(1-a)y] is maximized at aa;

Then 00 2 1/2.

Proof: It is sufficient to show that the first order
condition for a 2 1/2 is greater than zero. The FCC is
Equ’[ax+(1-a)y]-(x-y)
= Euu’[ax+(1-a)(x+6)]-(-6)
= Efizu’[(x+(1-a)6)]-(-6).
Let a 2 1/2, then
Eeu'[x+(1-a)6]o (-6)
2 1.‘.',u'[Jir+(1-oz)6]-£.'¢ (-6) = 0,
since u’[x+(1-a)6] is decreasing in 6 and (-6) is decreasing
in 6. So that E;u’[x+(1-a)6]-(-6) 2 0 for a 2 1/2.

Q.E.D.

Note that the above proof is independent of a, FOC 2 0
if a = 1. Thus we can say that the maximization occurs at
a=1.

Similarly, we have the results for FSD changes in

randomness, which is presented the following theorem.

73
Theorem 4.12: Assume:
(a). Utility function u satisfies u’ > 0 and u" s 0;
(b). Random variable E is obtained from E by adding an
independent random variable E, where E has a non-positive
support;
(c). Eu[ax+(1-a)y] is maximized at do;

Then do = 1.

Proof: It is sufficient to show the FCC is greater than
zero for a = 1. The FOC is
Equ’[ax+(1-a)y]-(x-y)

= Euu’[ax+(1-a)(x+6)]-(-6)

ExE,u’[(x+(1-a) 6)]-(-6) 2 0.
The FOC is greater than 0 for a = 1. To maximize the

expected utility, a has to be set to one. Q.E.D.

5 4.2.3 More Applications

In this part, we will consider more applications: a
firm’s production problem, the choice of output level for a
competitive firm and a multi-stage planning problem, all of
which have been studied by Rothschild and Stiglitz (1971).

A firm’s production problem is this: the output level Q
for next period is uncertain and the firm wishes to minimize
the expected cost of producing Q. The cost consists of

labor (L) and capital (K), the cost function, C(L,K), is

74

c = r-K + w-L(K,Q) ,

where r is the cost of capital and w that of labor. Capital
can not vary in the short run. L(K,Q) is the labor required
to produce Q given capital K, which is convex in Q if the
production function is concave. Hence, an increase in risk
always leads to an increase in the expected cost.

What happens to the optimum level of K when the output
level Q changes to random? This is a standard expected
utility function model, except that we are to minimize the
expected cost instead of maximization. The FOC is

.L.=E§!E&§L

w 6K
So that, a sufficient condition for decreasing K when faced
with an increase in risk is the concavity of dL/dk.

Writing this model in our notations, we have
z(x,a) = a-r + w-L(x,a),

where E is the output level, a is the capital. Given the
convexity of L(x,a), we have the convexity of z(x,a). By
the comparative statics theorem in section 1, an independent
increase in risk in the output will increase the capital
needed.

A related problem is the competitive firm's output
problem. Rothschild and Stiglitz (1971) assume that a firm

chooses the output level for tomorrow, although the price p

75
of output Q is uncertain. The firm is assumed to maximize

the expected utility of profit, Eu(n), where the profit is
n = p'Q - C(Q)I

where C(Q), as before, is the cost function and is assumed

to be convex, p is the random price. The FOC is
Eu’(")°[P’C’(Q)J = 0-

Since the concavity of utility function, there is always
less output under uncertainty than under certainty.

Sandmo (1971) and Ishii (1977) study the comparative
statics of the same problem using linear transformation.
They replace the price p with (7-p + 0), where 7 > 0 is the
multiplicative shift parameter and 0 is the additive one.
Changes in parameter 7 generate special Rothschild and
Stiglitz increases in risk and changes in parameter 0
generate special FSD changes in randomness. Decreasing
absolute risk aversion is a necessary and sufficient
condition for increasing of output when there is an increase
in parameter 0. DARA is a sufficient condition for
decreasing of output when there is an increase in parameter
7.

This model, in our notations, is z(x,a) = a-x - C(a).
Given the convexity of C(a), z(x,a) is concave. Therefore
by Theorem 4.1, an independent increase in risk will

decrease the choice parameter a, that is, the output level

76
will decrease when the firm is faced with an increase in
risk.
We consider our last application next, the multi-stage
planning problem of Rothschild and Stiglitz’s (1971). In a
simple economy, the final consumption good is produced by

labor and an intermediate commodity y:

Q = P(LZIY) I

while y is produced by labor alone:

Y = M(Ll) + e,

where e is the random variable associated with the
production of y. The constraint on labor is L = L,-+.Lr
The social planner's problem is to allocate the labor

between the two sectors efficiently. The FOC is

E(Pl -P2°M’) = 00

If e becomes riskier in the sense of Rothschild and Stiglitz
increases in risk, what happens to the allocation of labor

between the two sectors depends on the sign of

P122 ‘ MI°P222°

Rewriting this model in our notations, we have

Eu[>.-a,z(x,a)] = EP[L-L,,M(L,)+e].

This is a two argument model as in the savings and

77
uncertainty problem. Thus the results (theorem 4.7) we got
from the savings and uncertainty problem are also applicable
here.

We have considered the savings and uncertainty, asset
proportion, a firm's production, output level of a
competitive firm and multi-stage planing problems. It can
be seen that the independent transformation is a useful tool
in a wide range of applications. It is nice in generating

determinate comparative statics.

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lREflHERflflNCflflS

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