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CHANGING DISTRIBUTIONS IN THE STRONG SENSE AND
THEIR COMPARATIVE STATICS: A RATIO APPROACH
presented by

Soojong Kim

has been accepted towards fulﬁllment
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Ph . D . degree in Economics

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1/98 GICIRCJDmDmNS-au

CHANGING DISTRIBUTIONS IN THE STRONG SENSE AND
THEIR COMPARATIVE STATICS: A RATIO APPROACH

By

Soojong Kim

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Economics

1998

ABSTRACT

CHANGING DISTRIBUTIONS IN THE STRONG SENSE AND
THEIR COMPARATIVE STATICS: A RATIO APPROACH

By

Soojong Kim

This dissertation analyzes the comparative static effects on the choice behavior of
an economic agent who faces an uncertain situation when the given random situation
changes into another. The decision maker’s problem is to choose b to maximize
E [u(z(x, b))] where x is an exogenous random variable, and z is the payoff function
which, as the argument of utility, depends on both the choice variable and the random
variable. Given this general type of decision model, the existing comparative static
results are developed by imposing restrictions on the following three components: (i) the
set of changes in randomness; (ii) the set of decision makers; and (iii) the structure of the
concerned decision model.

The purpose of this study is to develop general comparative static statements
regarding various types of shiﬁs in probability distribution function (pdt) or cumulative
distribution function (CDF) of the given random variable. This dissertation concerns six
special types of ﬁrst-degree stochastic dominance shiﬁs in chapters 3 and 4, four types of
Rothschild-Stiglitz sense of increases in risk in chapters 5 and 6, and two types of
second-degree stochastic dominance shifts in chapter 7. For each given type of change in

the randomness, we develop conditions on the structure of the decision model and on the

risk preferences of the decision makers that are sufﬁcient for making a general
comparative static statement.

There are three special features in this dissertation. First, deﬁning the subsets of
changes in the randomness, the ratio between an initial and a ﬁnal pdt’ s or C DF’s is often
restricted to be monotonic increasing or decreasing. Thus, we call this analysis a ‘ratio
approach’ to comparative static problem. Second, the general comparative static
statement developed for each of these subsets contains a relatively large class of decision
makers, such as ‘all individuals with non-decreasing utility functions,’ ‘all risk averse
agents,’ or ‘all risk averse agents with non-negative third derivative of utility functions.’
In this sense, all the changes in randomness examined in this study are referred to as
‘CDF changes in the strong sense.’

Lastly, a special technique is used to generalize existing comparative static
statements. It is often shown that: a CDF change that satisﬁes one type of CDF change
can always be decomposed into a series of CDF changes that satisfy another type of CDF
change. Then the comparative static statement made for the latter type of CDF change
can also be applied for the former type of CDF change, without requiring any additional
assumption. Using this technique, this study improves the robustness of the existing

comparative static statements within the context of the ratio approach.

Copyright by
Soojong Kim
1 998

Dedicated to My Parents and Brother

ACKNOWLEDGMENTS

I would like to say ‘thanks’ to Professor Jack Meyer, my dissertation chairperson,
for sharing his professional knowledge and valuable time with me and every page of my
dissertation. His insights and recommendations improved the quality of my dissertation
signiﬁcantly. I also thank other committee members, Professor Carl Davidson and
Professor Lawrence Martin for their helpful comments.

During my graduate education at this department, I have enjoyed and learned so
much from discussions with my fellow graduate students. Thanks to Dongwook Han,
Soomyung Jang, Suyeol Ryu, Sanghyop Lee, Taejung Kim, Younghoon Seo and the rest
of the graduate students I worked with.

I am grateful to my family for their support throughout my graduate studies. My
ﬁrst thanks must go to my parents, parents-in-law, brother and sisters. I would especially
like to give heartﬁil thanks to my wife, Younghee, and my two sons, Jeongheon and
Jeongin, for their love and encouragement, and for always believing that I could succeed

at my goals.

vi

TABLE OF CONTENTS

LIST OF TABLES ...................................................................................................... ix

LIST OF FIGURES .................................................................................................... x
CHAPTER 1

INTRODUCTION ...................................................................................................... 1
CHAPTER 2

LITERATURE REVIEW ........................................................................................... 15

2.1 General Stochastic Dominance Selection Rules .................................... 15

2.1.1 Basic Deﬁnitions of Stochastic Dominance Orders ................... 16

2.1.2 Stochastic Dominance Selection Rules ...................................... 18

2.2 Comparative Static Analysis .................................................................. l9 .

2.2.1 An Overview .............................................................................. 20

2.2.2 Deterministic Transformation Approach ................................... 22

2.2.3 CDF Approach ........................................................................... 26

. 2.2.4 Ratio Approach .......................................................................... 29

2.3 Concluding Remarks .............................................................................. 31
CHAPTER 3

RELATIONSHIPS AMONG SUBSETS OF FSD SHIFTS ....................................... 32

3.1 Important Subsets of FSD Shiﬁs ........................................................... 33

3.1.1 Basic Deﬁnitions of Subsets of FSD Shifts ............................... 33

3.1.2 Graphical Examples ................................................................... 41

3.2 Some Properties among Subsets of FSD Shifts ..................................... 50
CHAPTER 4

COMPARATIVE STATIC RESULTS WITH SUBSETS OF FSD SHIFTS ............ 60

4.1 Literature Review and Preliminary Discussions .................................... 61

4.1.1 Monotone Likelihood Ratio F SD Shifts .................................... 61

4.1.2 Monotone Probability Ratio FSD Shifts .................................... 64

4.2 Comparative Static Analysis: Generalized Results ................................ 69
CHAPTER 5

RELATIONSHIPS AMONG SUBSETS OF R-S INCREASES IN RISK ................ 81

5.1 Important Subsets of R-S Increases in Risk ........................................... 82

5.1.1 Basic Deﬁnitions of Subsets of R-S Increases in Risk .............. 82

5.1.2 Graphical Examples ................................................................... 88

vii

5.2 Some Properties among Subsets of R-S Increases in Risk .................... 97

CHAPTER 6
COMPARATIVE STATIC RESULTS WITH SUBSETS OF R-S INCREASES IN
RISK ......................................................................................................................... 109
6.1 Literature Review and Preliminary Discussions .................................... 110
6.2 Properties of the Function u’(z)z,, ......................................................... 116
6.3 Comparative Static Analysis .................................................................. 120
CHAPTER 7
SUMMARY AND SOME REMARKS FOR COMPARATIVE STATICS WITH
SUBSETS OF SSD SHIFTS ....................................................................................... 127
7.1 Summary ................................................................................................ 127
7.1.1 Relationships among Subsets of CDF Changes ......................... 128
7.1.2 Comparative Static Analysis ...................................................... 133
7.2 Remarks for Comparative Statics Regarding Subsets of SSD Shifts 137
7.2.1 Subsets of SSD Shifts ............................................................... 138
7.2.2 Comparative Statics for Subsets of SSD Shiﬁs ......................... 140
REFERENCES .......................................................................................................... 146

viii

LIST OF TABLES

Table 6.1 Different solutions when b 2 O ................................................................. 1 14
Table 7.1 Comparative static results regarding subsets of F SD shifts ...................... 134
Table 7.2 Comparative static results regarding subsets of R-S increases in risk ..... 136

ix

Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6
Figure 3.7
Figure 3.8
Figure 3.9

Figure 3.10
Figure 3.11
Figure 3.12
Figure 3.13

Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5

Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5

LIST OF FIGURES

Example 3.1: F MLR G ......................................................................... 42
Example 3.2: F L-MLR G ..................................................................... 42
Example 3.3: F L-MLR G (a discrete case) ........................................... 43
Example 3.4: F L-MLR G ..................................................................... 43
Example 3.5: F R-MLR G ..................................................................... 44
Example 3.6: F MPR G ......................................................................... 44
CDF representation of example 3.6 ....................................................... 45
Example 3.7: F MPR G (a discrete case) ............................................... 45
CDF representation of example 3.7 ....................................................... 46
Example 3.8: F k-F SD G ....................................................................... 46
Example 3.9: F k-MPR G ...................................................................... 47
F MPR G that is the sum of ﬁve L-MLR shifts ..................................... 57
F k-MPR G that is the sum of a k-FSD and an MPR ............................. 59
A special case of c-F SD shift ................................................................. 68
F L-MLR G with k e [c, x3] .................................................................. 74
F R-MLR G with k e[x2, c] ................................................................. 74
A c-FSD shift ........................................................................................ 79
A c-MPR shift ....................................................................................... 79
Example 5.1: Mean-preserving truncation from G to F ......................... 89
Example 5.2: G SIR F ............................................................................ 89
Example 5.3: G RSIR F ......................................................................... 90
Example 5.4: G L-RSIR F ..................................................................... 90
Example 5.5: G L-RSIR F (a discrete case) ........................................... 91

Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9
Figure 5.10
Figure 5.11
Figure 5.12

Figure 7.1
Figure 7.2
Figure 7.3
Figure 7.4
Figure 7.5
Figure 7.6
Figure 7.7

Example 5.6: G L-RSIR F ..................................................................... 91
Example 5.7: G ESIR F ......................................................................... 92
Example 5.8: G ESIR F (a discrete case) ............................................... 92
CDF representation of example 5.8 ....................................................... 93
Example 5.9: G L-ESIR F ..................................................................... 93
G ESIR Gl ESIR F but not G ESIR F ................................................... 102
G ESIR F that is the sum of four RSIR shifts ........................................ 108
Relationships among subsets of FSD shifts ........................................... 129
Relationships among subsets of R-S increases in risk ........................... 132
An ESSSD shiﬁ that is the sum ofan MPR and an ESIR ..................... 141
An L-ESSSD shift that is the sum of an MPR and an L-ESIR .............. 141
A shift showing that F ESSSD GI ESSSD G but not F ESSSD G ....... 143
A shift that is the sum of an ESSSD and a c-FSD ................................. 144
A shift that is the sum of an L-ESSSD and a c-FSD ............................. 144 7

xi

Chapter 1
INTRODUCTION

Uncertainty is an element of our environment and accordingly it affects every
aspect of our life. Economists have recognized the pervasive inﬂuence of uncertainty on
the choice behavior of economic agents. However, only after the pioneering work of von
Neumann and Morgenstern (1947), who formalized the expected utility theory, has the
analysis of risk and uncertainty been one of the main themes in the economics literature.
During the last ﬁve decades, there has been much progress in the economic theory of
uncertainty. Today the issue of uncertainty is often included in many ﬁelds of study such
as ﬁrms’ production, stock market, insurance, futures markets, public policy,
international trade and ﬁnance.

In the study of choice under uncertainty, there are two main topics. One is to ﬁnd
selection rules g6verning the choice behavior of an individual who faces an uncertain
situation. Every choice available to the individual gives a random outcome represented
by a probability density function (pdt) or a cumulative distribution function (CDF). Then
one is concerned with the question: which is the preferred probability distribution among
those that are feasible? The stochastic dominance (SD) approach and the more traditional
mean-variance analysis are two such selection rules. Each gives necessary and/or
sufﬁcient conditions on a pair of risky1 prospects for one to be preferred to the other by

all decision makers in some appropriate set.

 

' Throughout the paper, the terms ‘uncertain,’ ‘random,’ and ‘risky’ are used interchangingly.

The second main topic is to ﬁnd predictable changes in choice behavior when a
given random situation changes into another. In other words, one is often interested in
ﬁnding necessary and/or sufﬁcient conditions for determining the effect of a particular
type of change in randomness on the optimal choice made by a decision maker. These
general comparative static statements usually contain restrictions on the changes in pdf or
CDF of the concerned random variable, the set of decision makers, and the structure of
the given decision model. General SD orders play an important role in this comparative
static analysis. Most research papers are concerned with particular types of CDF changes
which are subsets of ‘ﬁrst-degree stochastic dominance’2 (FSD) shifts, ‘second-degree
stochastic dominance’3 (SSD) shifts, or ‘mean-preserving spreads’4 (MPS) which are
‘increases in risk in the Rothschild-Stiglitz (R-S) sense.’

Many researchers have provided general comparative static statements dealing
with the effects of various types of changes in the distribution of the random variable.
Recent papers by Black and Bulkley (1989) and Landsberger and Meilij son (1990) have
shown the importance of the likelihood ratio5 between a pair of pdf’ 5. Imposing
monotonicity restrictions on the likelihood ratio, they deﬁne particular types of shifts
from one pdf to another pdf. Black and Bulkley introduce the concept of a ‘relatively
strong increase in risk’ (RSIR) which deﬁnes a subset of R-S increases in risk.
Landsberger and Meilij son’s analysis concerns a ‘monotone likelihood ratio’ (MLR)
order which is widely used in the statistical literature. MLR shifts are a subset of general
FSD shifts. In another recent paper, Eeckhoudt and Gollier (1995) give a deﬁnition of a
‘monotone probability ratio’ (MPR) order, imposing a monotonicity restriction on the

probability ratio6 between a pair of CDF’s. The MPR order also speciﬁes a subset of

 

2 The concept of FSD is introduced by Quirk and Saposnik ( 1962) and Schneeweiss (1969).

3 The concept of SSD is introduced by Hadar and Russell (1969, 1971) and Hanoch and Levy (1969).

4 In order to give a formal deﬁnition of an increase in risk, Rothschild and Stiglitz (1970) introduce the
concept of MP8.

5 Given a pair of pdt’sﬁx) and g(x), the likelihood ratio means the ratio of one pdf to the other.

6 Given a "pair of CDF’s F (x) and G(x), the probability ratio means the ratio of one CDF to the other.

F SD changes and is more general than the MLR order. The comparative static statements
developed in these papers contain relatively weak restrictions on the set of decision
makers. All individuals with non-decreasing utility functions are allowed in Landsberger
and Meilij son’s analysis, and all risk averse decision makers are considered in the other
two cited papers. The likelihood ratio for pdf’ 3 and the probability ratio for C DF ’3 play
an important role in the analysis. In order to differentiate this particular work from the
rest, we call their analysis a ‘ratio approach’ to the comparative statics problem.

In this dissertation, the analysis of comparative statics under uncertainty is
continued. Our main purpose is to improve the robustness of the existing comparative
static results within the context of the ratio approach. The results contained in the ratio
approach literature are generalized by weakening'various assumptions and considering
more general subsets of CDF changes. Two such subsets of FSD shifts are speciﬁed by
the concepts of a ‘left-side monotone likelihood ratio’ (L-MLR) order and a ‘right-side
monotone likelihood ratio’ (R-MLR) order which are deﬁned by imposing monotonicity
restrictions on the likelihood ratio between a pair of pdf’ 8. These are closely related to
the subsets given by the MLR and the MPR orders, and the basic relationships among
these subsets are as follows: The set of all CDF changes that satisfy the MLR order is a
subset of the set of all CDF changes that satisfy the L—MLR order, which, in turn, is a
subset of the set of all CDF changes that satisfy the MPR order. Also, the set that
satisﬁes the MLR order is a subset of the set satisfying the R-MLR order.

Two more subsets of FSD shifts are speciﬁed. One is deﬁned by the concept of
an ‘FSD improvement with respect to a point’ (point-F SD). This subset is not related in a
known way to the above four subsets of FSD shifts. The other is speciﬁed by the concept
of an ‘MPR with respect to a point’ (point-MPR). Deﬁned by weakening the conditions
required for an MPR shift, the point-MPR order speciﬁes a more general type of CDF
change than an MPR shift. The sets of CDF changes deﬁned by these two orders are

determined by a given point. Let’s denote, for a given point k, the orders as k-FSD and k-

MPR. Then these two orders are related to each other in such a way that a k-MPR shift
can always be understood as the sum of two F SD shifts, one MPR and the other k-F SD.
All these subsets of F SD shifts are used to improve the robustness of the existing
comparative static results regarding the subsets of F SD shifts.

Regarding the case of R-S increases in risk, three subsets are considered when
doing comparative static analysis. These sets are deﬁned by introducing three special
concepts of R-S increases in risk. A ‘left-side relatively strong increase in risk’ (L-RSIR)
is deﬁned by imposing a weaker condition than the one used by Black and Bulkley in
deﬁning an RSIR. An L-RSIR requires the monotonicity of the likelihood ratio only on
the left-side of the given pair of pdf’ 5. Monotonicity restrictions on the probability ratio
between a pair of CDF’s are used for obtaining the concepts of an ‘extended strong
increase in risk’ (ESIR) and a ‘left-side extended strong increase in risk’ (L-ESIR). Each
of the three concepts speciﬁes a more general class of R-S increases in risk than does the
RSIR order.7 Basic relations among these subsets of R-S increases in risk are: The set of
all CDF changes that satisfy the RSIR order is a subset of the set of all CDF changes that
satisfy the ESIR or the L-RSIR order, which, in turn, are subsets of the set of all CDF
changes that satisfy the L-ESIR order. For each of these subsets, we develop a general
comparative static statement, and the case of ESIR shifts improves the robustness of the
results in Black and Bulkley’s analysis.

All the comparative static results obtained in this dissertation are associated with
relatively large sets of decision makers; either the set of all individuals with non-
decreasing utility functions, the set of all risk averse agents, or the set of risk averse
agents with non-negative third derivative of their utility functions. These sets include
utility functions representing quite plausible preferences, such as the ones exhibiting

decreasing absolute risk aversion (DARA) generally accepted as a reasonable attitude

 

7 We often interchangingly use the following terms: an ‘order’ between a pair of CDF’s, a ‘type’ of CDF
change, and a ‘concept’ of CDF change. Each means a deﬁnition of partial rankings (orderings) over
probability distributions.

toward risk. These weak restrictions on the set of decision makers require a relatively
strong restrictions on the changes in CDF for making general comparative static
statements. In this sense, the changes in distributions examined in this study are all
referred to as ‘CDF changes in the strong sense.’ Next the decision model used in the
comparative static analysis is discussed.

An economic decision model involving randomness typically includes the
following components: (i) random exogenous parameters; (ii) choice variables; (iii) an
objective function; and (iv) a set of decision makers. In some decision models, more than
one random and one choice variable are included, or utility functions depend on more8
than one argument. For many of these decision models, it is generally difﬁcult to make
determinate comparative static statements. If the decision model includes multiple
sources of randomness, every random variable affects the choice variables, and a change
in the distribution of one random variable is generally accompanied by a simultaneous
change in the distributions of the other random variables. Thus, even in the case of one
choice variable, the presence of stochastic dependence among random variables makes it
difﬁcult to analyze the comparative statics for changing distribution of any one random
variable. To avoid this problem, research papers often assume that the random variables
are independent of one another, allowing one to change while the others are unchanged.
This assumption allows the comparative static analysis for a change in any one random
variable, using the standard SD orders as the restrictions on the change in the random
variable. An example of this is Hadar and Seo’s (1990) analysis of the portfolio model
with more than one risky asset.

When there are several choice variables, more than one ﬁrst-order condition
deﬁnes the optimal solution, and this makes the comparative static analysis more difﬁcult

than for the case of one choice variable. Most papers studying the comparative statics

 

8 That is, the objective function maps into R" space where n > 1, rather than R'.

assume that the ﬁrst and the second-order conditions are satisﬁed to guarantee an interior
unique solution. For the models with only one choice variable, the comparative static
effect of a change in randomness is determined by the direction of the change in marginal
expected utility of the choice variable (calculated at the initial optimal choice level). This
is not true for the cases with more than one choice variable because of the interactions
among the choice variables.

Comparative static analysis for models with more than one choice variable is
usually possible by imposing additional restrictions on the concerned decision model.
Examples of two choice variable decision models include Batra and Ullah (1974) who
examine a competitive ﬁrm’s input decisions with two inputs under output price
uncertainty. Feder, Just and Schmitz (1977) deal with an international trade model, and
Katz, Paroush and Kahana (1982) examine a price discriminating ﬁrm’s optimal sales in
two markets under price uncertainty in one of the two markets. Regarding a general form
model with one random and two decision variables, Choi (1992) examines a special case
of a corner solution.

When a utility function depends on two or more outcome values, that is, the
outcome ﬁmction is multidimensional, the standard risk aversion measures introduced by
Pratt (1964) and Arrow (1971) are not able to be used when describing the risk preference
of decision makers. In order to obtain general comparative static statements for these
models, researchers need other deﬁnitions of risk aversion measures. Sandmo (1970)
analyzes the saving behavior in the two-period income-capital model and uses his own
deﬁnition of the ‘temporal risk aversion’ measure. Dardanoni (1988) treats a somewhat
general form of two-outcome decision model and the obtained results depend on the
‘proportional risk aversion’ measure introduced by Menezes and Hanson (1970). There
are other examples of imposing restrictions on the multidimensional utility function, such
as Rothschild and Stiglitz (1971) and Mirrnan (1971) who analyze a two-period

consumption-saving model, using an additively separable utility function.

The decision model used in this study includes only one random and one choice
variable and utility depends only on one argument, that is, the objective function is single
dimensional. Relatively standard notation for the general form of this one-argument

decision model is:

mlax E[u(z(x, b))] (1.1)

where b is a scalar choice variable and x is an exogenous scalar random variable which
interact to yield, 2, the outcome which is the argument of utility. Many authorsg have
used this general decision model in their comparative static analysis. In this framework,
problems involving multidimensionality are avoided because utility depends only on the
value of a single dimensional outcome given by the intermediate function 2.

Though the model (1.1) is a simple form, a variety of economic decision problems
are of this type. Included are the often studied examples of a competitive ﬁrm facing
random output price analyzed in Sandmo (1971), and the portfolio choice model with one
risky and one riskless asset in Rothschild and Stiglitz (1971) and in F ishbum and Porter
(1976), and the coinsurance problem in Dionne, Eeckhoudt and Gollier (1993). Sandmo
considers a perfectly competitive ﬁrm choosing output level q to maximize expected
utility of proﬁts when the output price p is uncertain. In this model, the intermediate
function 2 is understood as proﬁts 7r given by II = pq — c(q) — F where c(q) is the
variable cost function and F is the ﬁxed cost. In the portfolio model, an individual is
assumed to have initial wealth W0 which is to be allocated between one riskless asset
paying a zero rate of return and the risky asset paying a random rate of return e. In this
case, the function 2 is understood as terminal wealth W given'by W = WO[1+ be] where b

is the proportion of WO invested in the risky asset.

 

9 To name just a few, they are Feder (1977), Kraus (1979), Katz (1981), Meyer and Orrniston (1983, 1985.
1989), Black and Bulkley (1989).

Coinsurance is a contract that reimburses the policy holder a certain speciﬁed
percentage for a loss occurred. In the standard coinsurance problem, the payoff function
is given by the ﬁnal wealth W = W0 - x + bx — bky where W0 is the initial value of a
given asset, x the random loss with mean it , b the coinsurance rate, big the insurance
premium, and [Lu the marginal cost of insurance rate. An individual chooses b to
maximize expected utility from the ﬁnal wealth. Typically the coinsurance rate b belongs
to the interval [0, l], the marginal cost of insurance rate [Lu is less than the maximum
loss and larger than the minimum loss, and the. maximum loss is usually less than initial
wealth W5. Other examples of models of this simple form include the problem of hiring
workers in F eder (1977), the cooperative ﬁrm model in Paroush and Kahana (1980). By
using the general form (1.1), each result developed in this dissertation is applicable to
these speciﬁc models as well as others.

Regarding the decision model, there are several assumptions needed for
conducting comparative static analysis. All decision makers are assumed to maximize
expected utility. In other words, a decision maker always chooses that alternative which
yields the highest mathematical expectation of a von Neumann-Morgenstem utility
function u(-) which is deﬁned over the set of all outcomes. The utility function u and the
payoff function 2 are assumed to be continuous and thrice differentiable with respect to
their arguments. In some results, it is enough for the functions to be differentiable only
once or twice. Restrictions on these functions are often needed to guarantee an interior
unique solution as well as to generate deﬁnite comparative static statements. To simplify
the analysis, we follow the literature and focus only on the case where z,(x, b) _>. 0. With
the assumption u’(z) .>_ 0, this implies that higher values of the random variable are
preferred to lower values. The case where zx(x, b) .<_ O can be handled with an

appropriate redeﬁnition of the random variable. '0

 

'0 The coinsurance problem cited above is of this case, and thus, if we redeﬁne the random loss x as x '7— y

where "7" means “has the same distribution as,” the outcome function for y satisﬁes 21,,(y, b) 2 O.

The random variable x is assumed to have its initial and ﬁnal distributionsll
characterized by CDF’s G(x) and F (x), with their corresponding pdf’ 5 g(x) and ﬁx),
respectively. Our study includes all the cases where the random variable x is continuous.
discrete or mixed. While the results are usually proved for the continuous case (using
integrals for summation), they can be adapted for the case where x is discrete or mixed.
Let b6 and b. denote the optimal choice'2 of the problem (1.1), under the CDF’s G and F,
respectively. We analyze the comparative static effects on the sign of b,,. — b“. for
changes in CDF from G to F. The supports of x under both the distributions are assumed
to be ﬁnite intervals. The notations for these ﬁnite supports are introduced as needed.

This dissertation is organized as follows. In the next chapter we will give a short
review of the literature concerning the SD approach and the previous analysis concerning
the comparative statics under uncertainty. The review of the SD selection rules is for the
univariate case and the review of the comparative static analysis is limited to the decision
models of the form (1.1). General comparative static statements are obtained by
imposing restrictions on the following components: (i) the structure of the decision
model; (ii) the set of decision makers; and (iii) the set of changes in distribution of the
random variable. In an analysis, relatively strong restrictions on one component are
usually associated with relatively weak restrictions on the other two components.

Previous work on this topic can be classiﬁed according to the restrictions placed
on these components. Structural restrictions are sometimes imposed on payoff functions.
Most papers concerning the decision model (1.1) adopt some restrictions such as 2, Z O,
z,”r 2 0 and 2,,” S 0. Since many economic problems already satisfy the restrictions, they
are not very harmful for conducting comparative static analysis. As somewhat stronger

restrictions, authors sometimes restrict the concerned payoffs to be linear in the choice

 

” To keep the notation for F to be stochastically dominant over G, the CDF’s F and G for the case of an R-
S increase in risk are regarded as an initial and a ﬁnal distribution of x, respectively.

‘2 Some parts of our study follows the usual assumption that the ﬁrst and the second-order conditions are
satisﬁed for deﬁning an optimal solution of the problem (1.1), but the results concerning subsets of FSD
shifts are established without requiring the second-order condition.

10

variable and/or the random variable. This restriction on the structure of the decision
model limits the analysis to more specialized decision problems.

Given a decision model structure, some research imposes relatively strong
restrictions on the changes in the pdf or CDF and relatively weak restrictions on the risk
preference of decision makers, and other research treats the reverse case. The studies
using relatively strong restrictions on the changes in distributions usually consider proper
subsets of the set of FSD changes or R-S increases in risk.

According to the type of restrictions used for deﬁning sets of changes in pdf or
CDF, there are three approaches to the comparative static analysis. The deterministic
transformation approach obtains a change in distribution by transforming an initial
random variable into another random variable through a given detenninistic function.
The CDF approach directly imposes a restriction on the relation between the initial and
the ﬁnal CDF’s G and F or pdfs g and f, often restricting G — F. Recently, interesting
sets of changes are deﬁned by imposing restrictions on the ratio of pdf’s or the ratio of
CDF’s. In order to differentiate these changes from the others, the studies employing
them are said to use the ‘ratio approach.’ The focus in this dissertation is on the ratio
approach to comparative static analysis.

In chapters 3 and 4, we examine subsets of FSD shifts. For the ratio approach,
they include the MLR shifts used in Landsberger and Meilij son (1990) and the MPR
shifts in Eeckhoudt and Gollier (1995). We deﬁne four additional subsets of the set of
general FSD shifts. They are speciﬁed by the L-MLR, the R-MLR, the point-FSD, and
the point-MPR orders. Chapter 3 discusses the relationships among these subsets of FSD
shifts. The restrictions used in deﬁning these subsets are discussed and graphical
examples are given to illuminate the basic relations among the subsets. For the basic
relations, it is shown that the MLR order speciﬁes, a special case of the L-MLR or the R-
MLR order, and that the L-MLR order deﬁnes a special case of the MPR order which, in

turn, gives a special case of the point-MPR order.

The property of transitivity is also discussed with regard to these orders.
Transitivity holds for both the MLR and the MPR orders, but not for the L-MLR or the
R-MLR. Given a point k, both the k-F SD and the k-MPR orders also display transitivity.
Whether a CDF order is transitive or not has important implications in doing comparative
static analysis. If a given CDF order is sufficient for a general comparative static
statement, then any shift that can be decomposed into a series of shifts satisfying the CDF
order also has a predictable outcome. Thus, if the given order is not transitive, the
admissible set of CDF changes for the comparative static result include more shifts than
are included by the CDF order itself. The L-MLR order is generally non-transitive, but
has an important relationship with the MPR order resulting from transitivity. We show
that an MPR shift can always be decomposed into a series of L-MLR shifts. Another
important relationship exists with regard to the point-MPR order. Given a point k, a k-
MPR shift can always be decomposed into two shifts, one k-FSD and the other MPR. In
chapter 4, these properties are used to prove some generalized comparative static
statements.

Chapter 4 provides general comparative static statements for the subsets of FSD
shifts presented in chapter 3. Each developed result gives a generalization of an existing
comparative static result. The results in Landsberger and Meilijson (1990), and
Eeckhoudt and Gollier (1995) are for decision models with linear payoffs. This chapter
shows that the same results also hold for general non-linear decision models, without
imposing additional assumptions. The extension of Eeckhoudt and Gollier’s result is
demonstrated using the L-MLR order and its relationship with the MPR order presented
in chapter 3. The proposed comparative static statement for the set of MLR shifts applies
to the set of all decision makers with non-decreasing utility functions, and the one for the
set of MPR shifts applies to the set of all risk averse decision makers.

In addition, for each of the two comparative static results, we explore the trade-off

between the structural restrictions on the decision model and the restrictions on the

change in the distribution. When the concerned payoff (outcome) function is restricted to
be linear in the choice variable b (2b,, = O ), even more general classes of changes in CDF
yield determinate comparative static statements. Given a decision model satisfying the
linearity restriction, the value of x satisfying 2,, = 0 does not depend on the choice
variable b. Let c be such the point of x. Then two generalized comparative static results
are derived. One is that, in addition to the set of MLR shifts, L-MLR shifts satisfying the
monotonicity restriction on the left-side of the point c, and R-MLR shifts satisfying the
monotonicity restriction on the right-side of the point c are also sufﬁcient for Landsberger
and Meilijson’s result. The other is that the set of c-FSD shifts and the set of c-MPR
shifts are also sufﬁcient for Eeckhoudt and Gollier’s result. Each generalization implies
that, for a ﬁxed set of decision makers, the ﬁnding is extended to a more general class of
CDF changes with the cost of adding the linearity restriction on the decision model.

Chapters 5 and 6 treat the cases of R-S increases in risk. In addition to the
existing subsets of R-S increases in risk, we deﬁne three additional subsets of R-S
increases in risk speciﬁed by the CDF orders: ESIR, L-RSIR and L-ESIR. These are
more general than the RSIR order deﬁned by Black and Bulkley (1989). The restrictions
on the pdf ratio used in deﬁning the RSIR and the L-RSIR orders are replaced by the
restrictions on the CDF ratio when we deﬁne the ESIR and L-ESIR orders. In chapter 5,
the relationships among these orders are examined and the basic relations are illustrated
with some graphical examples. It is shown that the monotonicity restrictions on the CDF
ratio are less severe than the ones on the pdf ratio. Thus, we have the basic relationships
among the orders as: the RSIR order speciﬁes a special case of the L-RSIR or the ESIR
order, which, in turn, give a special case of the L-ESIR order.

In general, transitivity does not hold for these CDF orders. This implies that.
given one of these orders, a CDF change that can be decomposed into two or more
distinct shifts that are generated by the order may not satisfy the order. This guarantees

that, if a general comparative static statement is made for the CDF order, the order is only

sufﬁcient but not necessary for the result. Though they are all non-transitive, there exist
two important relationships between the RSIR and the ESIR orders, and between the L—
RSIR and the L-ESIR orders. They are: (i) an ESIR shift can always be decomposed into
a series of RSIR shifts; and (ii) an L-ESIR shift can always be decomposed into a series
of L-RSIR shifts. Using these properties, chapter 6 provides two useful generalizations
of the comparative static results given in the same chapter.

Chapter 6 gives general comparative static statements regarding the subsets of R-
S increases in risk, presented in chapter 5. The comparative static result in Black and
Bulkley (1989) is generalized to the set of ESIR shifts which is larger than the set of
RSIR shifts. This generalization is, without any cost of additional assumption. The
proposed comparative static statement is for the set of all risk averse decision makers
(same as in Black and Bulkley) and the set of ESIR shifts.

When the concerned payoff is restricted to be linear in the random variable,
another extension of Black and Bulkley’s result is given by using a trade-off between the
restrictions on the set of decision makers and the restrictions on the set of R-S increases
in risk. With the given structural restriction, the proposed result contains, compared with
Black and Bulkley’s result, a larger set of changes in CDF and a smaller set of decision
makers. That is, the set of risk averse agents with non-negative third derivative of utility
functions and the set of L-RSIR shifts which includes all RSIR shifts are contained in the
result. A further generalized result is provided in order to include a more general class of
CDF changes. The set of L-ESIR shifts replaces the set of L-RSIR shifts in the result,
and this generalization is directly comes from the relationship between the L-RSIR and
the L-ESIR orders given in chapter 5.

Finally, chapter 7 gives a summary for the results developed in the previous
chapters. Also some remarks regarding the comparative static results for subsets of
second-degree stochastic dominance (SSD) shifts are given in section 7.2. We show that

the results obtained in previous chapters can be applied for cases of SSD shifts. In

general, the SSD order deﬁnes a more general type of changes in distribution of a random
variable than does the F SD order or the concept of an R-S increase in risk. According to
Hadar and Seo (1990), any SSD change can be decomposed into an FSD shift and an R-S
decrease in risk (mean-preserving contract (MPC)). This can be applied for deﬁning
subsets of the set of SSD shifts which are combinations of FSD shifts and R-S decreases
in risk, which satisfy the CDF orders deﬁned in chapters 3 and 5.

Two subsets of SSD shifts are deﬁned: ‘extended strong SSD’ (ESSSD) shifts and
‘left-side extended strong SSD’ (L-ESSSD) shifts. An ESSSD shift can always be
decomposed into an MPR and an ESIR, and an L-ESSSD shift can always be understood
as the sum of an MPR and an L-ESIR. These properties allow one to make general
comparative static statements for these subsets, by combining the results obtained in

chapters 4 and 6.

Chapter 2

LITERATURE REVIEW

There are two main topics in the analysis of choice under uncertainty. One is to
ﬁnd selection rules governing the choice behavior under uncertainty, and the other is to
analyze the comparative static effects on the choice made by an economic agent when a
given random situation changes into another. This chapter reviews a selected portion of
the literature on these two topics. Section 2.1 presents the well-known standard
stochastic dominance (SD) selection rules. Section 2.2 gives a brief review of the
literature concerning the comparative static analysis. Finally, section 2.3 contains some
remarks speciﬁc to this study. For expositional ease, let x be a random variable
characterized by an initial CDF G(x) and a ﬁnal CDF F (x), and if not stated otherwise, the

supports of these CDF’s are assumed to be in the ﬁnite interval [0, l].

2.1 General Stochastic Dominance Selection Rules

When a decision maker is confronted with alternative uncertain prospects, he
wants to choose to maximize expected utility. SD selection rules provide a partial
ranking of these alternatives. Each selection rule gives a necessary and/or sufﬁcient
condition on a pair of risky prospects (a pair of distribution functions) for one to be

preferred to the other by all individuals with non-decreasing and/or concave utility

15

functions. Restricting ourselves to the case of univariate distributions, we give the

deﬁnitions of the standard SD orders and then explain several SD selection rules.

2.1.1 Basic Deﬁnitions of Stochastic Dominance Orders

We review three13 types of SD orders for pairs of C DF’s. The ﬁrst-degree
stochastic dominance (FSD) introduced by Quirk and Saposnik (1962) and Schneeweiss
(1969), and the second-degree stochastic dominance (SSD) introduced by Hadar and
Russell (1969) and Hanoch and Levy (1969) are deﬁned as follows:

Deﬁnition 2.1. F (x) is said to stochastically dominate G(x) in the ﬁrst-degree
(denoted by F FSD G) if and only if
G(x)— F(x) 2 O, for all x e[0, 1].

Deﬁnition 2.2. F (x) is said to stochastically dominate G(x) in the second-degree

(denoted by F SSD G) if and only if

E[G(x) — F(x)]dx _>. 0, for all s e [0, 1].

Each deﬁnition gives a partial ordering on a set of probability distributions. It is well
known and obvious from these deﬁnitions that if F FSD G then F SSD G. This implies
that the set of CDF’s that can be ordered by the concept of FSD is included in the set that
can be ordered by the concept of SSD. These stochastic dominance orderings have the
property of transitivity. That is, considering CDF’s E, F2 and P}. if F; FSD (SSD) F2
and F2 FSD (SSD) F,, then Fl FSD (SSD) F3.

 

'3 There are more stochastic dominance orderings, which are not concerned in this paper, such as the
traditional mean—variance ordering by Markowitz (1952, 1959) and Tobin (1965), the third and higher
degree stochastic dominance by Whitrnore (1970, 1989), and SSD with respect to a function by Meyer
(1977a,1977b).

Finally, Rothschild and Stiglitz (R-S) (1970) provide a general deﬁnition of an
‘increase in risk.’ According to R-S, the following three conditions are equivalent and

are used to deﬁne ‘G is riskier than F.’ The conditions are:

I. A random variable y can be obtained from another random variable x by an
addition of random noise a, where y and x are the random variables deﬁned by
CDF’s G and F, respectively, that is,

y 7x + 6,
where E (six) = 0 and “ f ” means “has the same distribution as.”
2. All risk averse individuals prefer F to G, that is, for every concave utility

function u(-),

£u(x)dF(x) Z £u(x)dG(x).

3. G can be obtained from F by a sequence of mean-preserving spreads (MPS). '4

The condition 3 can be stated with the well-known ‘integral conditions,’ being the
restrictions imposed on the difference G — F. Because the three conditions are

equivalent, an increase in risk is formally deﬁned by the integral conditions as:

Deﬁnition 2.3. G(x) is said to be riskier than F (x) in the Rothschild-Stiglitz sense
(denoted by G MPS F) if and only if

(a) ﬂaw — F(x)]dx = o

(b) £[G(x) - F(x)]dx 2 o, for all s e [0. 1].

 

" Using an alternative deﬁnition of a mean-preserving spread, Machina and Pratt (1997) give a more
explicit result that, for an arbitrary pair of distributions F and G satisfying the integral conditions given in
Deﬁnition 2.3, there always exists a sequence of MPS’s by which F converge directly to G.

The condition (a) implies that CDF’s F and G have the same mean, and the
condition (b) is the one used in deﬁning F SSD G. Thus an R-S increase in risk is a
special case of SSD, requiring in addition, equal means. It too gives a transitive partial
ordering on a set of probability distributions. That is, considering CDF’s Fl , F2 and F3 . if
F} MPS F2 and F2 MPS F3, then H MPS F3. For almost thirty years now, this
deﬁnition has replaced variance or standard deviation, as the basic concept for studying

comparative risk.

2.1.2 Stochastic Dominance Selection Rules
Three selections rules which are used in this dissertation are presented here. To
generate these rules, each of the above stochastic dominance relations is associated with a

particular set of utility functions. These sets are deﬁned as:

Definition 2.4. Assuming that a utility function u(-) is continuous. bounded and
twice differentiable for the interval [0, 1], deﬁne the following three sets as:
(a) U. = M)! u’ 2 0}
(b) U2 = {u(-)| u' 2 0 and u" S 0}
(c) U. = {u(-)l u" s 0}.

Denote, for utility function u, E U ,, and E U“. as the expected utility computed with
respect to CDF’s F and G, respectively. Then selection rules for the cases of F SD and
SSD are expressed as:

Theorem 2.1. EUF 2 EUG for every u in UI ifand only ifF FSD G.

Theorem 2.2. EU). 2 EU“. for every u in U2 if and only ifF SSD G.

19

The proof of FSD rule is given by Quirk and Saposnik (1962), Schneeweiss (1969‘),
Hadar and Russell (1969) and Hanoch and Levy (1969), and the proof of SSD rule is
given by the last two papers.

The last selection rule given in this sub-section is the one provided by Rothschild
and Stiglitz (1970). From the three equivalent conditions shown in sub-section 2.2.1. the

selection rule is expressed as: f

 

Theorem 2.3. EU,, 2 EUG for every u in U3 if and only ifG MPS F.

Concavity is the only restriction used in deﬁning the set U3. This general property plays
an important role in their companion paper, Rothschild and Stiglitz (1971), studying the

comparative static effects of an R-S increase in risk.

2.2 Comparative Static Analysis

The question of interest for comparative statics under uncertainty is to ﬁnd
necessary and/or sufﬁcient conditions for determining the direction of change in the
choice variable when a change in the random variable occurs. Many economic models
are used in addressing this question. Some are of a speciﬁc form and others are of a
general form. There are models including two ormore random variables, choice
variables and/or outcome values.l5 However, since our concern is restricted to the
general one-argument decision model (1.1) introduced in chapter 1, the literature review

is restricted to studies concerning decision models of this form.

 

'5 See the dissertation by Choi (1992), for the comparative static analysis concerning decision models with
multiple outcomes.

2.2.1 An Overview

Regarding the decision models of the form (1.1), the standard monotonicity or
concavity properties of utility functions and the standard SD shifts are not sufﬁcient for
making-general comparative static statements. That is, for an arbitrary F SD, MPS or SSD
shift in distribution, one can not say that every decision maker with a non-decreasing
and/or concave utility function changes his choice in the same direction. This is more
formally stated by Meyer and Orrniston (1983) who show that any FSD or any SSD shift
in distribution causes every risk-averse individual to adjust his choice level to the same
direction if and only if the optimal choice under certainty is independent of the value of
the random exogenous variable. Obviously, this is not an interesting case. Following
this line, all the existing comparative static results have used some restrictions on the
following three components:

(i) the structure of a given decision model
(ii) the set of decision makers concerned
(iii) the set of changes in distribution of the concerned random variable.

Imposing ‘relatively weak restrictions’ on one of the three components means that more
cases or a relatively larger class of the component are included in an analysis, and vice
versa. Generally speaking, when relatively strong restrictions are imposed on one
component, the derived comparative static results are usually associated with relatively
weak restrictions on the other components.

As the structural restrictions imposed on the general decision model (1.1), 2:‘ 2 0.
z,” 2 0 and 2,,”r S 0 are often assumed by many authors. Because many economic
problems already satisfy the assumptions, they are not very harmful for the comparative
static analysis. Some authors add the assumptions 2,, = 0 and/or 2,, = 0, that is, the
concerned payoffs are restricted to be linear in the choice variable and/or the random

variable. Two of the most often studied examples are the standard portfolio problem in

which the ﬁnal wealth is linear in both the choice and the random variable, and a
competitive ﬁrm model under price uncertainty in which the proﬁt function is linear in
the random variable. Other examples are found in Dionne, Eeckhoudt and Gollier (1993).
and Eeckhoudt and Gollier (1995). As an extreme case, Meyer and Onniston (1983) use
such a payoff that the optimal choice under certainty does not depend on the value of the
random variable. Thus the restriction on the ﬁrst component limits the analysis to special
economic problems.

Given a decision model, the comparative static question is to ﬁnd a set of changes
in CDF or pdf which is sufﬁcient for signing the effect on the choice variable selected by
an arbitrary decision maker in a speciﬁed set. In a general comparative static statement. a
relatively large (small) class of shifts in CDF is usually followed by relatively strong
(weak) restrictions on the risk preference of decision makers. The trade-offs between the
restrictions on these two components are often examined.

Examples using relatively strong restrictions on the risk preferences of agents
include Rothschild and Stiglitz (1971), Hadar and Russell (1978) and Hadar and Seo
(1990). They examine the general classes of FSD, MPS and/or SSD changes.
Concerning the standard portfolio choice problem, they found that, for an F SD shift in the
distribution of the return on a risky asset, relative risk aversion to be less than one is
sufﬁcient to cause investors to demand more of the risky asset. For an MPS shift in the
random return on a risky asset, the conditions that absolute risk aversion is decreasing
and relative risk aversion is increasing and smaller than one are sufﬁcient for investors to
decrease their demand for the risky asset. In these studies, large classes of changes in
distribution are considered but relatively strong restrictions are imposed on the attitude
toward risk of the concerned decision makers.

Many authors have used relatively strong restrictions on the third component, the
changes in randomness. They have found sufﬁcient conditions on the changes in

distribution which cause every individual in a fairly large set to adjust his choice in the

IJ
IQ

same direction. Subsets of F SD shifts are examined in Sandmo (1971), Landsberger and
Meilijson (I990), Orrniston (1992), Orrniston and Schlee (1993), and Eeckhoudt and
Gollier (1995), among others. Various subsets of MPS shifts or R-S increases in risk are
analyzed in Sandmo (1971), Ishii (1977), Kraus (1979), Eeckhoudt and Hansen (1980),
Katz (1981), Meyer and Orrniston (1985, 1989), Black and Bulkley (1989), Dionne,
Eeckhoudt and Gollier (1993), and more.l6

According to the type of restrictions used in deﬁning various sets of changes in
CDF or pdf, these studies can be divided into three approaches. Deterministic
transformation approach deﬁnes a set of changes in distribution by mapping every
possible initial outcome value into a new value through a given deterministic function.
CDF approach uses restrictions on the initial and the ﬁnal distribution functions G and F,
often imposing restrictions on the difference G — F. Other interesting sets are deﬁned by
imposing restrictions on the likelihood ratio between the two pdf’s or the probability ratio
between the two CDF’s. These studies are referred to as the ‘ratio approach.’ The

following sub-sections present each approach in more detail.

2.2.2 Deterministic Transformation Approach

I According to Meyer and Orrniston (1989), a single-valued function t(-), what they
call a ‘deterministic transformation function,’ can be used for deﬁning a type of change in
a random variable. That is, given an initial random variable x, a transformed random
variable y is deﬁned by y = t(x). The transformation function is assumed to be non-
decreasing, continuous and piecewise differentiable. The non-decreasing assumption is
related to the following two properties. First, combined with the monotonicity property
of utility functions, it ensures that the transformation does not reverse the preference
ordering over the set of all possible outcomes of the original random variable. Second.

letting the random variables x and y be characterized by CDF’s G and F, respectively, the

 

'6 More examples are found in Coes (1977), Feder ( 1977), Paroush and Kahana (1980), Gollier (1995).

 

non-decreasing assumption implies that G(x) = F (t(x)) for all possible outcomes ofx.
This non-decreasing assumption is necessary to make general statements about the effects
of some deterministic transformations of a random variable on the expected utility.

Using the transformation method, Meyer and Ormiston provide a fourth

characterization of an R-S increase in risk as follow:

Theorem 2.4. The random variable given by a transformation t(x) represents an
R-S increase in risk from an initial random variable x described by a C DF G(x) if and

only if the function k(x) s ((x) — x satisﬁes

(a) £k(x)dG(x) = 0

(b) £k(x)dG(x) s 0, for all s E[O, 1].

By the condition (a), the mean of the random variable is preserved. The condition (b)
guarantees that the initial random variable dominate the transformed random variable in
the second-degree. This theorem is proved by showing that the given transformation
reduces the expected utility for all risk averse decision makers.

Similarly, FSD and SSD shifts can also be characterized in terms of

transformations. According to Meyer (1989), we have:

Theorem 2.5. The random variable given by a transformation t(x) dominates an
initial random variable x described by a CDF G(x) in the ﬁrst-degree if and only if
k(x) E t(x) — x 2 O, for all x e [0, 1].

The given condition implies that the transformed CDF lies to the right side of the initial
CDF G. Thus, as we saw in section 2.1, the transformation gives a greater expected

utility for all individuals with non-decreasing utility functions.

Theorem 2.6. The random variable given by a transformation ((x) dominates an

initial random variable x described by a CDF G(x) in the second-degree if and only if the

function k(x)a t(x)—x satisﬁes Ek(x)dG(x)2 0, for all s E[O, I].

Only the condition (b) in Theorem 2.4 is used and the transformation gives a higher
expected utility for all risk averse economic agents.

Imposing further restrictions on the transformation function produces interesting
sets of changes in distribution within the context of comparative static analysis. A well-
known example is a linear transformation analyzed in Sandmo (1971) and Ishii (1977).

In their papers, the transformation function used is a linear form as:

y=y(x—f)+6+5c'
where 7 is a multiplicative shift parameter, I9 is an additive shift parameter and f is the
mean of the random variable x. An increase in 6(from y = l and I9 = 0) increases the
mean of the random variable without changing its shape or dispersion of the initial
distribution. Since it gives a parallel shift of the original CDF G(x) to the right, it deﬁnes
a type of FSD shift.

Sandmo, studying the behavior of a competitive ﬁrm under price uncertainty,
provides a general comparative static statement that, facing an increase in 9 in the random
price, the ﬁrm exhibiting decreasing absolute risk aversion increases its level of output.
Sandmo also examines an increase in the parameter 7(from 7 = 1 and I9 = 0). Preserving
the mean, this change causes a ‘stretching out’ of the initial distribution about its mean.
This transformation deﬁnes a type of R-S increases of risk. Treating the same problem in
Sandmo, Ishii gives a proof for the comparative static result that, facing an increase in y
in the random price, the ﬁrm exhibiting decreasing absolute risk aversion decreases its

level of output.

The comparative static results concerning the linear transformations are
generalized by introducing ‘simple deterministic transformations.’ Meyer and Ormiston
(1989) introduce a ‘simple increase in risk’ as a type of R-S increases in risk. and
Orrniston (1992) deﬁnes a ‘simple F SD transformation’ as a class of FSD shifts. These

are given as:

Deﬁnition 2.5. The random variable given by a transformation ((x) represents a
simple R-S decrease in risk from an initial random variable x described by a CDF G(x) if

the function k(x) 2' t(x) - x satisﬁes

(a) (k(chm = 0

(b) £k(x)dG(x)20, for allse[0, 1]

(c) k'(x) S 0.

Deﬁnition 2.6. The random variable given by a transformation ((x) represents a
simple F SD shift from an initial random variable x described by a C DF G(x) if
k(x) :- t(x) — x 2 O and k'(x) S O, for all x e [0, I].

These simple transformations generalize the above linear transformations by not requiring
the shifting or the stretching to be of uniform intensity. Using the general decision model
(1.1), each of the simple transformations allows one to make general comparative static
statement. Assume that u’(z) 2 0, u"(z) S 0 and z,,,, < 0. Meyer and Orrniston provide

the following theorem:

Theorem 2.7. Facing a simple R-S increase in risk in the random variable x, a
decision maker will decrease the optimal value of b. if

(a) u(z) displays decreasing absolute risk aversion

(b) 2, 2 0, 2‘u S 0, z," 2 0 and 2,,“ S 0.

Similarly, Onnistonl7 gives a general comparative static statement as:

Theorem 2.8. Facing a simple FSD shift in the random variable x, a decision
maker will increase the optimal value of b, if
(a) u(z) displays decreasing absolute risk aversion

(b) zr 2. 0. Z... S 0 and z,“ 2 0.

With these simple transformations, the comparative static results by Sandmo and
Ishii are generalized in two senses. First the same results are obtained for more general
classes of transformations, with no additional restrictions required. Second, the decision
model concerned is replaced by the general decision model which includes the

competitive ﬁrm model as a special case.

2.2.3 CDF Approach

CDF approach to comparative static analysis uses restrictions on the initial and
the ﬁnal CDF’s G and F to deﬁne sets of changes in distribution of the random variable.
Sandmo (1971) considers a special type of R-S increases in risk, namely, an ‘introduction
of risk’ or a ‘global increase in risk’ which implies an R-S increase in risk from an initial
non-random situation. He found that the optimal output level of production of a risk-
averse competitive ﬁrm is smaller when the price is uncertain, compared to the
production level when the price is known with certainty.

Sandmo’s negative effect of an introduction of risk is generalized by Kraus (1979)

and Katz(l981). Kraus presents the same result using the general one-argument decision

 

'7 Onniston (1992) also deﬁnes a simple SSD transformation and provided its comparative static theorem.
However, since an SSD shift can be considered as combinations of an FSD and an R-S decrease in risk or
mean-preserving contract (MPC), the result is not presented here.

model (1.1), that is, he derives a sufﬁcient condition for signing the effect ofa global
change in risk on the optimal level of b selected by an arbitrary risk averse decision

maker. The condition includes restrictions on the payoff function z(x, b) as follows:

Theorem 2.9. Assume that u' > 0, u" < 0, z, > 0 and z,,,, < 0, then all risk averse

decision makers, facing a global increase in risk in the random variable x, will decrease

the optimal value of b if z," 2 0 and 2,," S 0.

The condition in Theorem 2.9 is somewhat more generalized by Katz who provides a
shorter proof of the result.

Though the above results apply to a fairly large set of decision makers (all risk
averse agents), the restriction that risk change is a global change in risk is quite severe. A
global increase in risk is only a small subset of the set of all R-S increases in risk. This
limits signiﬁcantly the situations to which the result can be applied. Eeckhoudt and
Hansen (1981) analyze the comparative static effect for a less restrictive set of R-S
increases in risk. They introduce the concept of a ‘min-max truncation’ or a ‘mean-
preserving truncation.’ Let the supports of CDF’s G and F be the ﬁnite intervals. [xI . x ,]
and [x2, x3], respectively, where xl S x2 S x3 S x,. This is a slight modiﬁcation of the
original notation, using the points x2 and x3 as the minimum and the maximum points

where the truncation occurs. A mean-preserving truncation is deﬁned as:

Deﬁnition 2.7. F (x) represents a mean-preserving truncation from G(x) if their

difference G(x) - F (x) satisﬁes
(a) [‘ [G(x) — F(x)}dx = 0

(b) F(x): 0 for allx e[x,, x2), F(x) =1 for allx e(x3, x4], and G(x) = F(x) for all

x E[xz, x3].

Since the conditions (a) and (b) imply that f [G(x) — F (x)]dx 2 0 for all s e [.t., x , ], a

mean-preserving truncation is an R-S decrease in risk and it includes a global decrease in
risk as a special case. It speciﬁes a transfer of probability mass such that both the
truncated left-tail and right-tail probability mass are moved into the minimum and the
maximum end points, respectively. Eeckhoudt and Hansen show that, using Sandmo’s
competitive ﬁrm model, the optimal level of production of a risk averse ﬁrm increases
when a mean-preserving truncation occurs in the random price. Thus they improve the
robustness of Sandmo’s result because the same comparative static result is obtained for a
more general class of R-S increases in risk without additional assumptions required.
Meyer and Ormiston (1985) provide a ﬁirther generalization of the results in
Eeckhoudt and Hansen. They introduce a ‘strong increase in risk’ (SIR) as a concept of
an R-S increase in risk. The condition (b) in Deﬁnition 2.7 is replaced by a much less

restrictive form as:

Deﬁnition 2.8. G(x) represents a strong increase in risk from F (x) (denoted by F

SIR G) if their difference G(x) - F (x) satisﬁes'8
(a) [‘ [G(x) — F(x)]dx = 0

(b) G(x) — F (x) is non-increasing in x E[xz, x3].

An SIR speciﬁes a transfer of probability mass such that some probability mass from the

interval [x2, x3] is moved into both the tail intervals [x,, x2] and [x3, x4]. Thus it

deﬁnes a more general class of changes in distribution than the one by a mean-preserving

 

'3 The condition, used in the original paper, that [[G(x) - F (x)]dx .>. 0 for all s e [x,, x,] is omitted here

because it is implied by the conditions (a) and (b) in Deﬁnition 2.8.

truncation. Using the general decision model (1.1), Meyer and Ormiston’s comparative
static result improves the robustness of the previous results. Without any cost of .
additional assumptions, the results in Kraus and Katz are obtained for a more general
class of R-S increases in risk, and the result in Eeckhoudt and Hansen is extended to the
general decision model and to a more general class of changes in distribution.

All the subsets, presented in this sub-section, are deﬁned by imposing restrictions
on the difference between the initial and the ﬁnal CDF. They are all subsets of R-S
increases in risk. There are more studies giving further extensions of the above results.
In the next sub-section, we review the ratio approach which treats more sets of changes in

distribution, including subsets of FSD shifts as well as subsets of R-S increases in risk.

2.2.4 Ratio Approach

Recent papers such as Black and Bulkley (1989), Landsberger and Meilijson
(1990), Dionne, Eeckhoudt and Gollier (1993) and Eeckhoudt and Gollier (1995) have
shown the importance of the likelihood ratio between a pair of pdfs and the probability
ratio between a pair of CDF’s in deﬁning sets of changes in pdf or CDF. As mentioned
earlier, our comparative static analysis follows this ratio approach and thus these previous
papers are crucial to our comparative static analysis. In this subsection, we give a brief
review of the ratio approach and more detailed review is given in later chapters.

Landsberger and Meilij son introduce a monotone likelihood ratio (MLR) shift
which is deﬁned by imposing a monotonicity restriction on the likelihood ratio between a
pair of pdf 5. The set of MLR shifts is a subset of general F SD shifts. Using the standard
portfolio choice problem, Landsberger and Meilij son provide a general comparative static
statement that, when an MLR shift in the random return on a risky asset occurs, all
investors with non-decreasing utility functions increase their demand for the risky asset.

Another recent paper by Eeckhoudt and Gollier gives a deﬁnition of a monotone

probability ratio (MPR) shift by imposing a monotonicity restriction on the probability

ratio between a pair of CDF ’5. Being also a subset of general F SD shifts, the set of MPR
shifts is larger than the set of MLR shifts. Assuming that the concerned payoff is linear
in both the choice and the random variables, they show that an MPR shift in the random
variable causes all risk averse individuals to increase their optimal choice level.
Compared to Landsberger and Meilijson’s analysis, this implies a trade-off between the
restrictions used. That is, Eeckhoudt and Gollier provide a larger set of admissible
changes in distribution for the comparative static result with the cost of a smaller set of
decision makers. Since these two papers play an important role in our comparative static
analysis for the case of FSD shifts, more detailed discussions are given in chapter 4.

Black and Bulkley introduce a relatively strong increase in risk (RSIR) which is
deﬁned by imposing a monotonicity restriction on the likelihood ratio between a pair of
pdf’ 5. As a subset of R-S increases in risk, the set of RSIR shifts is larger than the set of
SIR shifts given in Deﬁnition 2.8. Black and Bulkley show that, based on the general
decision model (1.1), an RSIR shift reduces the level of optimal choice selected by an
arbitrary risk averse decision maker. Accordingly, this is a generalization of the results in
Meyer and Orrniston (1985). Without the cost of additional assumptions, the same result
is obtained for a more general class of changes in distribution. However the comparative
static results in these two papers require a common restriction that payoffs should be
strictly concave in the choice variable, that is, z,,,, < 0. This restriction excludes the
decision models in which payoffs are linear in the choice variable (2“, = 0) such as the
standard portfolio problem.

In general, when 2,, = 0, the second-order condition for the decision problem
(1.1) is not guaranteed and thus the cases of an unbounded and a comer solutions should
be examined for making general comparative static statements. Dionne, Eeckhoudt and
Gollier raise this issue and show that, when the concerned payoff is linear in the choice
variable b, the results in Black and Bulkley are also obtainable for a ‘relatively weak

increase in risk’ which is a more general concept than an RSIR. They examined the cases

of an unbounded and a comer solutions in the proof of their comparative static result.
More detailed discussions about these papers are given in chapter 6 which considers the

comparative static analysis for the cases of R-S increases in risk.

2.3 Concluding Remarks

Reviewing the comparative static analysis gives a general notion that. when
relatively strong restrictions are imposed on one of the component of the decision model,
the derived comparative static results are usually associated with relatively weak
restrictions on the other components. The comparative static statements developed in the
context of the ratio approach contain relatively weak restrictions on the set of decision
makers such as all risk averse agents or all individuals with non-decreasing utility
functions. Our study follows the ratio approach in order to generate interesting
comparative static statements. That is, most sets of changes in randomness examined in
this study are deﬁned by imposing monotonicity restrictions on the likelihood ratio
between a pair of pdf 5 or the probability ratio between a pair of CDF’s. The purpose of
this dissertation is to generalize and extend the existing comparative static results
presented in sub-section 2.2.3. The generalization includes the cases where the same
result is derived for a more general class of changes in distribution or for more general
decision problems. Some trade-offs between restrictions used in a general comparative
static statement are explored. In this study, we also examine the changes in the
admissible set of shifts in distribution when the linearity (with respect to the choice

and/or the random variable) restriction is added to the general decision model (1.1).

Chapter 3
RELATIONSHIPS AMONG SUBSETS OF F SD SHIFTS

This chapter gives the deﬁnitions for six special types of F SD shifts. Five
deﬁnitions result from imposing restrictions on the likelihood or the probability ratio
between an initial and a ﬁnal random variables. That is, either the ratio of the pdf’s, g
and f, or the corresponding CDF’s, G and F, is restricted. The other one is given by
imposing restrictions on the pdf difference g — f and the CDF difference G — F. Each
deﬁnition characterizes a class of CDF changes for which a general comparative static
statement is made in chapter 4. Before doing the comparative static analysis, this chapter
is used to present some important relationships among those six types of FSD changes.

This and the next chapter consider only subsets of FSD shifts, that is, the ﬁnal
distribution F always dominates the initial distribution G in the sense of ﬁrst degree
(denoted by F F SD G). Both the CDF’s G and F are assumed to have their points of
increase in bounded intervals. For notational convenience, we assume that the support of
G(x) is a ﬁnite interval [xv x3] and the support of F (x) is another ﬁnite interval [x2 , x 4]
where x, S x2 and x3 S x,. Deﬁning the supports in this way, we also assume that: for
the continuous case, G(x) > 0 for all x e(x,, co) and F(x) > 0 for all x 6(x2, co); and

for the discrete case, G(x)> 0 for all x E[xl, co) and F(x) > 0 for all x E[xz, oo). '9

 

'9 Deﬁning sets of CDF changes, the ratio of f to g or the ratio of F to G is often restricted to be non-
decreasing in the overlapped interval [x2, x3]. Then this assumption allows both the ratios to be strictly

positive for the interval [x2, x,] in a discrete case, and for the interval (x2,- x,] in a continuous case.
which are often used to simplify the proofs of the results developed in chapters 3 and 4.

32

3.1 Important Subsets of F SD Shifts

In this section, we give the deﬁnitions for six special types of FSD changes. They
include two previously introduced ones: a monotone likelihood ratio (MLR) order from
Landsberger and Meilijson (1990) and a monotone probability ratio (MPR) order from
Eeckhoudt and Gollier (1995); and four newly deﬁned ones: a left-sided monotone
likelihood ratio (L-MLR) order, a right-side monotone likelihood ratio (R-MLR) order, a
point-FSD shift, and a point-MPR shift. In sub-section 3.1.1, basic deﬁnitions of these
concepts are presented. Graphical examples in sub-section 3.1.2 are given to help the

reader understand the basic relations among the six orders.

3.1.1 Basic Deﬁnitions of Subsets of F SD Shifts .

In order to obtain a general comparative static statement in the portfolio problem,
Landsberger and Meilij son (1990) use the concept of an MLR which was previously
deﬁned in the statistical literature. The MLR order speciﬁes a type of FSD shift in a

distribution function. satisfying a monotonicity restriction on the ratio between a pair of

pdf 3 f and g.20

Deﬁnition 3.1. F (x) represents a monotone likelihood ratio FSD shift from G(x)
(denoted by F MLR G) if there exists a non-decreasing function h: [x2 , x3] —> [0, 00) such
thatf(x) = h(x)g(x) for all x E[x2 , x3].

According to the condition in Deﬁnition 3.1, g(x) 2 f (x) when h S 1 and g(x) S f (x)

when h 2 l, and the function h non-decreasing implies that the pdf’ 5 f and g cross only

 

‘0 Following our notation about the supports of CDF’s F and G, Deﬁnition 3.1 is slightly modiﬁed from the
original one in Landsberger and Meilijson’s paper.

34

once. Thus it is easy to conﬁrm that an MLR order is an F SD shift.21 Let k be the point
at which the crossing occurs, and we know that k 6 [x2, x3]. Then an MLR shift speciﬁes
a probability transformation such that some probability mass on the points to the left of
the point k is transferred to the points to right of the point k, requiring the ratio of the ﬁnal
pdf f to the initial pdf g to be non-decreasing.

The MLR order can be applied for cases of discrete or mixed random variables as
well. In general, the condition that the function h is non-decreasing in x 6 [x2, x3]
requires that the ratio of f to g is non—increasing for the interval. As we noted in footnote
19, if the ratio h is non-decreasing then h is positive for all x 6 (x2, x3] regardless of
discrete or continuous case. Thus the MLR condition also requires that, for the interval
(x2, x3), 22 if g(x) = 0 then f(x) = 0, and if f(x) = 0 then g(x) = O. This implies a
restriction that, for the interval (x2, x,), the outcome values which were possible under
the initial distribution must be also possible under the ﬁnal distribution, and the outcomes
which were previously impossible must have zero probability of occurring under the ﬁnal
distribution.

According to Landsberger and Meilijson (1990), and Orrniston and Schlee (1993)
who also use the MLR order in their comparative static analysis, there are special
examples of shifts that already possess the MLR order. First, we do not exclude the case
where x3 S x2, that is, all possible outcome values under the ﬁnal CDF F is larger than
those under the initial CDF G. For any shift of this type, it satisﬁes that F MLR G.
Second, consider a random variable which has only two outcomes. A transfer of
probability from the lower outcome to the higher one is an MLR shift, while an
increasing in the magnitude of either one or both outcomes (still the previously higher

outcome being larger than the new lower outcome) is not an MLR change. Third, if the

 

2' The MLR condition implies that G(x) 2 F(x) for all x e[x,, x4], and this is the condition for an FSD
shift (see Deﬁnition 2.1).

22 Both the end points are dropped from the interval [x2, x3 ], because it is possible that h(x2 ) = 0 or
h(x3) = so for continuous case.

random variable is continuous, then the MLR order gives complete orderings over the set
of all the uniform densities with the same left (right) endpoint. In this instance, a pdf
with a higher right (left) endpoint is MLR dominant over a pdf with a lower right (left)
endpoint.

Lastly, if an initial pdf is log-concave, adding a positive constant to the random
variable leads to an MLR change. To show this, assume that the ﬁnal random variable y
is obtained from the initial random variable x as y-jx + c, where c is a positive constant.
With the given pdf ofx, g(.), the pdfofy,ﬂ-) is given by f(t) = g(t - c) for all t e R.

Then the condition that g(t — c) / g(t) is non-decreasing in t is equivalent to

an -c> > g'm
g(t-c) - g(t)

 

and this is guaranteed by the condition that g’(x)/ g(x) = d In g(x)/ ab: is non-increasing
in x or equivalently that In g(x) is weakly concave. Examples possessing this property
include normal, uniform, and exponential distributions.23

A second specialized FSD change is deﬁned in the analysis of Eeckhoudt and
Gollier (1995). They impose a monotonicity restriction on the ratio between the two
CDF’s F and G, and refer to the change as a monotone probability ratio (MPR) change.

Its deﬁnition follows:24

Deﬁnition 3.2. F (x) represents a monotone probability ratio FSD shift from G(x)
(denoted by F MPR G) if there exists a non-decreasing function H: [x2 , x3] —5 [0, 1] such
that F(x) = H(x)G(x) for all x E[x2 , x3].

 

23 See Karlin and Rubin (1956) for more examples.
2’ With the same reason as in footnote 20, Deﬁnition 3.2 is a slight modiﬁcation of Deﬁnition 2 in
Eeckhoudt and Gollier.

36

The fact that H EU), 1] guarantees G(x)2 F(x) for all x e[x,, x4], and thus an MPR
shift is an F SD shift. Compared with an MLR shift, an MPR shift imposes less stringent
restrictions on the change in the random variable. The feature accruing to the MPR order
is that it does not restrict the number of times of crossing between the pdf’s f and g. This
comes from replacing the monotonicity restriction on the likelihood ratio h with the
monotonicity restriction on the probability ratio H. Thus, the assertion that the MLR
order implies the MPR order can be made by showing that the condition that the
likelihood ratio h is non-decreasing implies the condition that the probability ratio H is
non-decreasing. While the proof for this statement is formally given in the next section,
here we provide some cases showing that an MPR shift is not an MLR one.

One example satisfying the MPR but not the MLR condition is found in a shift in
a two-outcome random variable. Contrary to the MLR case, both the cases of a transfer
of probability from the lower outcome to the higher one and an increase in the magnitude
of lower outcome are MPR changes. Another notable thing is that, for the interval
(x3, x3), while F MLR G requires that if g(x) = 0 then f(x) = o, and if f(x) = 0 then
g(x) = 0. the condition F MPR G does not require that f (x) = 0 when g(x) = 0. Thus it
allows that f (x0) > 0 when g(x") = O, for some x° e(x2, x,). This is a useful extension
for discrete cases where the ﬁnal pdf f often has some points with positive probability
which were impossible under the initial pdf g. An example of this case is given in Figure
3.9 in sub-section 3.1.2. I

As Eeckhoudt and Gollier note, there is a class of CDF’s in which adding a
positive constant to the random variable leads to an MPR shift. Similar to the MLR case.
assume that the ﬁnal random variable y is obtained from the initial random variable x as

y j x + c where c is a positive constant, and that the initial CDF G is log-concave.

Then the ﬁnal CDF F, given by F(t) = G(t —- c) for all t e R, is an MPR shift from G.

This is because the condition that G(t - c) / G(t) is non-decreasing in t is equivalent to

G'(l-c) > G'(t)
G(t—c) _ G(t)

 

and this is guaranteed by the condition that G'(x)/ G(x) = d In G(x)/ air is non-
increasing in x or equivalently that In G(x) is weakly concave. Bagnoli and Bergstrom
(1991) show that the log-concavity of the CDF is less demanding than the log-concavity
of the pdf. Thus, examples possessing this property include the normal, uniform, and
exponential distributions which satisfy the log-concavity of the pdf. They also show that.
if the pdf is non-increasing, adding a positive constant to the random variable is an MPR
shift but does not lead to an MLR shift.

In addition to the MLR and the MPR orders, there are other interesting types of
changes in distribution. An MLR shift from g to f restricts the two pdfs to cross only
once, and requires the ratio of f to g to be non-decreasing for both side of the crossing
point. Relaxing some of these restrictions, we introduce two more special types of F SD
shifts. One, called a left-side monotone likelihood ratio (L-MLR) order, is obtained from
relaxing the monotonicity requirement for points to the right-side of the crossing point.
The other, called a right-side monotone likelihood ratio (R-MLR) order, drops the
monotonicity requirement to the left-side of the crossing point. Each of these relaxed
restrictions speciﬁes a class of FSD shifts including all MLR shifts. Thus all the
examples of shifts given in the discussion of the MLR case are also examples of the L-

MLR or the R-MLR shifts. Formally, these changes are deﬁned as:

Deﬁnition 3.3. F (x) represents a left-side monotone likelihood ratio F SD shift
from G(x) (denoted by F L-MLR G) if there exist a point k e [x2, x3] and a non-
decreasing function h: [x2, k) -) [0, 1] such that f (x) = h(x)g(x) for all x E[xz, k) and
g(x) S f(x) for all x e[k, x3] .25

 

2’ Only one crossing at a point k E[xz, x3] implies that f(x)S g(x) for all x E[xz, k) and f(x)2 g(x) for
all x e(k, x, ]. At the point It, it is usual that f (k) = g(k), but if the pdf g or f is discontinuous at k. both the
cases of f (k)< g(k) and f(k)> g(k) are possible, and thus Deﬁnition 3.3 includes the case where

38

Deﬁnition 3.4. F (x) represents a right-side monotone likelihood ratio F SD shift
from G(x) (denoted by F R-MLR G) if there exist a point k 6 [x2 , x3] and a non-
decreasing function h:(k, x3] —> [1, 00) such that g(x) 2 f (x) for all x E[xz. k]

f(x) = h(x)g(x) for all x e(k, x3].

Both the L-MLR and the R-MLR conditions require that the pdf’s f and‘g cross only once
at the point k and that g(x) 2 f (x) for all points to the left-side of k and g(x) S f (x) for
all points to the right-side of k. This implies that each of the two orders deﬁnes an F SD
shift. Since an L-MLR requires the condition of monotone likelihood ratio only for the
left-side of the point k, and an R-MLR requires the monotonicity only for the right-side of
the point k, both the orders are more general than the MLR order. An L-MLR shift
speciﬁes a probability transformation such that a decreasing proportion of probability
mass of the left-side of the point k is transferred to the right-side of the point k. An R-
MLR shift gives a transformation that some probability mass taken from the left-side of k
is transferred to the right-side of k, keeping the ratio of the ﬁnal pdf to the initial pdf non-
decreasing only for the right-side of k.

In order to give special examples of an L-MLR and an R-MLR shift, consider an
initial random variable which has only two outcomes. If the lower (higher) outcome
becomes larger than before, then it is the case of an L-MLR shift where k = x2 (an R-
MLR shift where k = x3 ). Generally, given an initial pdfg(x) with its support [x,, x3],
let’s consider a conditional pdf g’ , given by g’ (t) = g(t)/G(T) with its support [x,, T]
where T S x,. Then an MLR shift in g’ gives an L-MLR shift in the initial g(x). If we
deﬁne a conditional pdf g’(t) = g(t)/ [1—G(T)] with its support [T, x,] where xl S T, an
MLR shift in g’ gives an R-MLR shift in the initial g(x).

 

f(x)S g(x) for all x E[xz, k] and f(x)2 g(x) for all x e(k, x,]. This discussion is applied for all the
other deﬁnitions given in this section.

With the restrictions used, it is obvious that an MLR shift is also an L-MLR or an
R-MLR shift, but not vice versa. Similarly, the restrictions for the MLR condition that,
for the interval (x2, x3), if g(x) = 0 then f(x) = 0, and if f(x) = 0 then g(x) = 0, is
somewhat relaxed. The condition F L-MLR G allows that f (x0) > 0 when g(x") = 0,
for some x0 e [k, x,), and the condition F R-MLR G allows that g(xo) > 0 when
f (x0) = 0, for some x0 6 (x2 , k]. Another basic relationship exists between the L-MLR
and the MPR orders, although there is no direct relationship26 between the R-MLR and
the MPR orders. In section 3.2, we show that the monotonicity restriction on the
likelihood ratio between a pair of pdf 3 for the MLR and the L-MLR shift implies the
monotone probability ratio between the corresponding CDF ’5. Thus the MLR order
implies the L-MLR order, which in turn implies the MPR order.

In what follows, two more types of F SD shifts are deﬁned. As a special feature,
these two CDF orders are speciﬁed by a given point and thus each set of CDF changes
satisfying the orders is determined by the given point. One is called an ‘FSD
improvement with respect to a point’ (point-F SD). This is deﬁned by imposing
restrictions on the CDF difference G - F and the pdf difference g - f, and the
restrictions depend on a given point. A point-FSD shift does not use any monotonicity
restriction on the likelihood or the probability ratio between a pair of distributions. This

is deﬁned as:

Deﬁnition 3.5. Given a point k 6 [x2, x3], F (x) represents an F SD shift with
V respect to the point k from G(x) (denoted by F k-FSD G) if G(x) 2 F (x) for all
x e[x.. x4], and 'g(x)S f(x) for all x E[k, x3].

 

2" While F R-MLR G allows that g(xo) > 0 when f (x0) = 0, for some x° 6 (x2 , k], which is not allowed

in F MPR G, an MPR shift allows multiple crossings between the two pdfs, which is not allowed in an R-
MLR shift.

40

Since the condition includes the general FSD condition, a k-FSD shift is also an FSD
shift. In addition to the condition F FSD G, a k-FSD change requires only the restriction
that, given the point k, G(x) — F (x) is non-increasing or g(x) S f (x) for the interval
[k, x,]. For the left-side of the point k, the only restriction required is F FSD G. Thus,
as the given point k becomes large, the restriction on the point-FSD becomes less severe.
That is, for the given points kl and k2 where kI S k2, all kl -FSD shifts are also k2 -FSD
shifts, but not vice versa. For any FSD change from g to f, there exists a point
k E[xz, x3] satisfying the restriction that g(x) S f (x) for all x E[k, x,]. This implies
that any FSD change can be classiﬁed as a k-F SD shift, for an appropriately chosen k.
Compared with the above four deﬁnitions of F SD shifts, a k-F SD does not restrict
the number of times of crossing between the pdf 5 f and g, nor is there any monotonicity
restriction on the likelihood or the probability ratio between the two distributions. While
the set of CDF changes deﬁned by the point-F SD depends on a given point k, it generally
includes shifts that do not satisfy any of the four types of F SD shifts deﬁned above.
Finally, another CDF order which depends on a given point is called an ‘MPR
with respect to a point’ (point-MPR). This is obtained from weakening the conditions
required for an MPR shift, and the degree of the weakened restriction depends on a given
point. A point-MPR requires the monotonicity restriction on the CDF ratio only for the

points to the right-side of the given point, and the change is deﬁned as:

Deﬁnition 3.6. Given a point k 6 [x2, x3], F (x) represents a monotone

probability ratio FSD shift with respect to the point k from G(x) (denoted by F k-MPR G)
' if, with the ratio function Hgiven by H:[x2, x3] —> [0, 1] such that F(x) = H(x)G(x),
the function H is non-decreasing in x E[k, x3] and H(x) S H(k) for all x E[xz, k].

The MPR condition is relaxed for the interval [x2, k] in which a k-MPR does not

require the CDF ratio function H to be non-decreasing. Similar to the k-FSD case, as the

4i

given point k becomes large, the restriction on the point-F SD becomes less severe. That
is, for the given points kl and k2 where kl S k2, all kI -MPR shifts are included in the set
of k2 -MPR shifts. As a special case, the set of MPR shifts is equal to the set of k-MPR
shifts where k = x2. A point-MPR is related to a point-F SD shift in such a way that,
given a point k, a k-MPR shift can always be decomposed into two shifts, one MPR and

the other k-FSD. This is formally given in section 3.2.

3.1.2 Graphical Examples

This sub-section provides graphical examples of the six concepts of FSD shifts
deﬁned above, and shows the basic relationships among them. In order to contrast the
differences among those concepts, we consider the following nine examples of FSD
shifts, seven continuous cases and two discrete cases. Examining each example, we can

visualize the basic relationships discussed in section 3.1.1.

(1) Example 3.1: FMLR G in Figure 3.1. As Figure 3.1 shows, the pdfs g andf
cross only once at a point k. The likelihood ratio function h given in Deﬁnition 3.1 can
be deﬁned by h = f / g for the interval [x2, x3]. Since the function h is non-decreasing
in the interval, the shift from G to F satisﬁes the MLR condition. To show that the ratio
f / g is non-decreasing is equivalent to show that f ' - g 2 f - g'. When g at 0, this is

equivalent to the following condition:

f’/g'2f/g,wheng’>0
f’20,wheng'=0
f’lg'Sf/g,wheng'<0.

Graphically, this implies that, when g 2 f and g' > 0 ( g’ < 0), the tangent lines

1g and I, in Figure 3.1 do not meet to the left (must meet to the right) direction in the

 

42

 

f(x).g(X)
1g ./
l 3 . [s
,, m
g(x)
x, x2 c2, ‘ k c2 x3 x,

Figure 3.1. Example 3.1: FMLR G.

f(x).g(X)

 

 

 

x2 k c x3 x4

Figure 3.2. Example 3.2: F L-MLR G.

43

 

 

 

xl x2=k x3=x4

Figure 3.3. Example 3.3: F L-MLR G (a discrete case).

M). g(X)

h——-—-—------_—--- ------

 

29.
L

---—————-——-‘

 

 

-’--"“-‘--‘--

 

 

31

Figure 3.4. Example 3.4: F L-MLR G.

x2 k T x3 -

l
R
b

44

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ﬁx).g(X)
f(x)
g(x
)2, x2 c1 52 k x, x,
Figure 3.5. Example 3.5: FR-MLR G.
f(x)t8(x)
f(x)
g(x)
xl I x2 , c, c2 c3 x3 x4

Figure 3.6. Example 3.6: F MPR G.

45'

 

 

 

Figure 3.7. CDF representation of example 3.6.

g(x)

 

 

f(x)

IIIIIIIIIIIIIIIII

III-"'l'll

'I‘llllln'l

 

 

Figure 3.8. Example 3.7: F MPR G (a discrete case).

46

 

 

 

 

 

 

 

 

 

 

 

r __________ l
I
I
I
r____;F(x)
I
I
I
I
G(x)
r """""" ' '
I
__________ I
I
r--_-J
I
l -
x1 x2 CI c2 x, x,

Figure 3.9. CDF representation of example 3.7.

 

 

 

Figure 3.10. Example 3.8: Fk-FSD G.

47

 

 

 

Figure 3.11. Example 3.9: F k-MPR G.

space above x axis, and when g S f and g’ < 0 ( g' > 0), the tangent lines IX and I, do

not meet to the right (must meet to the left) direction in the space above x axis.

(2) Example 3.2: F L-MLR G in Figure 3.2. This is also an example of single
crossing between the pdf’s g and f Let’s deﬁne the likelihood ratio function h as
h = f / g for the interval [x2, k). Then the ratio is non-decreasing in the interval and
thus the shift from G to F satisﬁes the L-MLR condition. The ﬁgure shows that the two
tangent lines at point c (g(c) S f (c) and g'(c) < 0) meet in the right direction, which
implies that, at the point c, the ratio f / g is decreasing. Thus the shift cannot be an

MLR change.

(3) Example 3.3: F L-MLR G in Figure 3.3. Example 3.3 considers a shift in a
two-outcome random variable. The initial random variable has only two outcomes,
x1 and x3, with equal probability of occurring. Increasing the lower value xI to x2 leads
to an L-MLR shift. Let x2 = k, then the domain of the non-decreasing ratio function h is

empty and f (x) 2 g(x) for all x e [k, x3], which satisfy the L-MLR condition. The shift

48

cannot be an MLR shift because the ratio of f to g is + so at x2, and one at x,, that is, the
likelihood ratio function h should be decreasing at least one point in the interval [x2, x3 ].
In this case, if the higher outcome x3 increases with the lower outcome unchanged then it

is the case of an R-iMLR shift.

(4) Example 3.4: F L-MLR G in Figure 3.4. This example shows that, given an
initial pdf g(x) with its support [x1, x3], an MLR type of shift occurred only on the left-
side subinterval such as [xl , T] where T S x3 rives an L-MLR shift in the initial pdf g.
Let’s deﬁne a conditional pdf g’ (t) = g(t)/ G(T) with its support [x,, T], then the shift
shown in the ﬁgure can be understood as an MLR shift of the conditional pdf g’ . Since
the ratio f to g is constant and less than one for the interval [x2, k), and f 2 g for all

x e [k, x3], it is obvious that F L-MLR G.

(5) Example 3.5: F R-MLR G in Figure 3.5. There is only one crossing between
the pdf's g and f Since the shift in this example satisﬁes that f S g for all x 6 [x2, k),
and that the ratio f to g is increasing and larger than one for the interval [k, x,], it is an
R-MLR shift. The shift cannot be an MLR. As an evidence, the ﬁgure shows that,
f = 0 and g > 0 for the interval [CI , c2], which is not allowed for an MLR shift.

(6) Example 3.6: F MPR G in Figure 3.6 and 3.7. As Figure 3.6 shows, the pdfs
g and f cross three times at the points cl , c2 and c3, and thus the example cannot be an
MLR, an L-MLR or an R-MLR shift. In Figure 3.7, we can see the CDF representation
of example 3.6. The probability ratio function H given in Deﬁnition 3.2 can be deﬁned
by H = F / G for the interval [x2, x3]. Since the function H is non-decreasing in the
interval, the shift satisﬁes the MPR condition. In order to see this graphically, we can use
the same logic used in the MLR example 3.1. G is always non-decreasing and greater

than F in an F SD change. Thus, if the tangent lines [G(z G(x) in the given example) and

49

1,. for any point in the interval [x2, x3] do not meet to the left direction in the space

above x axis, the CDF ratio F / G is non-decreasing.

(7) Example 3.7: F MPR G in Figure 3.8 and 3.9. This is an example of a discrete
case. To see that the shift is an MPR shift, consider the CDF representation of the shift
given in Figure 3.9. Since the CDF ratio F to G is non-decreasing for the interval
[x2, x,], it is obvious that F MPR G. Figure 3.8 shows that the difference g - f changes
sign at three points x2, c1 and c2. This implies that the pdfs g and f cross three times,
and thus the shift cannot be an MLR, an L-MLR or an R—MLR change. Note that, at
point c2 which is in the interval (x2, x3), f (c2) > O and g(cz) = O, which is not allowed
for an MLR shift. Thus, this is an example showing that the MPR order is a useful
extension for discrete cases where the ﬁnal pdf f often has some points with positive

probability which were impossible under the initial pdf g.

(8) Example 3.8: F k-F SD G in Figure 3.10. Given the point k, the fact that the
difference G - F is decreasing for the interval [k, x3] guarantees the shift to be a k-F SD
shift. Since the ratio of F to G is decreasing for the interval [c,, c2], this example cannot
be an MPR shift. We noted that, as the given point k becomes large, the restriction on the
point-F SD shift becomes less severe. This implies that, if the selected point is

sufficiently large, all the above examples are also the examples of point-FSD shifts.

(9) Example 3.9: F k—MPR G in Figure 3.11. As the last example concerned in
this sub-section, we present a point-MPR shift. Given the point k, the ﬁgure shows that
the ratio F / G is non-decreasing for the interval [k, x3], and F (x)/ G(x) S F (k)/ G(k)
for all x 6 [x2, k]. Thus example 3.9 satisﬁes the k-MPR conditions, but it cannot be an
MPR shift because the CDF ratio is decreasing for the interval [CI , c2 ]. Note that the set

of MPR shifts is a subset of the set of k-MPR shifts, for any given k 6 [x2 , x3]. The shift

50

also does not satisfy the k-F SD condition because the difference G — F is increasing for

some points in the interval [k, x3].

3.2 Some Properties among Subsets of F SD Shifts

This section points out several relationships among the concepts of FSD shifts,
deﬁned in section 3.1. They include important properties which play a key role for
developing general comparative static statements in chapter 4. First, we discuss the basic

relationships among the orders generated by these concepts.
Property 3.1. FMLR G :> FL-MLR G :> FMPR G :> Fk—MPR G.

Proof. The claim that “if F MLR G then F L-MLR G” is obvious. It is because

the restriction that the pdf ratio function h(x) is non-decreasing in the interval [x2, x3] ,

which is required for an MLR shift, is relaxed for an L-MLR shift in such a way that, for

the interval of x satisfying h(x)21, h need not be non-decreasing. The relationship that
“if F MPR G then F k-MPR G” is also easily veriﬁed by the fact that the condition that
the CDF ratio function H(x) is non-decreasing for the interval [x2, x3] implies
H(x) S H(k) for all x E[xz, k].

Consider the second relationship “if F L-MLR G then F MPR G.” Since G > 0

for all x e (x,, 00), we can deﬁne a function H for the interval [x2, x3] as,27

H(x) = F(x)/G(x), for all x E[xz, x3].

 

27 If x, = x2 and x is continuous, then G(xz) = 0 and the proof is also true for this case with the function H
deﬁned as, ‘
0, when x = x2 .
: {F(x)/ G(x), when x e(x2, x3].

5l

To prove the relationship, since H 2 0, it is sufﬁcient to show that the condition F L-

MLR G implies that H is non-decreasing and less than or equal to one for the interval. It

is clear that H S l for all x 6 [x2, x3] because an L-MLR shift in which the pdfs g and f
have only a single crossing is an FSD shift. For the interval, the condition F / G non-

decreasing is equivalent to:
f(x)G(x)—g(x)F(x)20, for allx E[xz, x3]. (3.1)

With the condition F L-MLR G, divide the interval [x2, x3] into two sub-intervals, one
such that g 2 f and the other such that g S f. When g S f, the condition (3.1) is
satisﬁed because G 2 F. When g 2 f, the L-MLR condition implies that

(i) if g(x) = 0, then f(x) = 0

(ii) if g(x)¢ 0, then h(x) = f(x)/ g(x). .
Thus, for all the values of x such that g = O, the condition (3.1) is satisﬁed. For case (ii),

the condition (3.1) can be written as,
h(x)G(x) — F(x) 2. 0, for all x 6 [x2, x,] such that g(x) ¢ 0. (3.2)

Since h(x2 )G(x2)— F (x2) 2 0 and the condition h non-decreasing in x implies that the LHS

of (3.2) is non-decreasing, i.e.,
é’[h(x)G(x)— F(x)]/ 6x = h'G + hg - f = h'G Z O,

the condition (3.2) is satisﬁed. Hence the condition (3.1) is satisﬁed for all x such that

g .>. f. This completes the proof. Q.E.D.

Now we have formal relationships among the subsets of FSD changes. Let’s
deﬁne the following ﬁve subsets of FSD shifts, each representing a set of pairs of CDF’s
(F, G) such as:

52

0 = {(F. G): FMLR 6}
r2,,,,, = {(F, G)| FL-MLR G}
0..-... = {(F, G)| FR— MLR G}
{2.0, = {(F, 0)] FMPR G}

9am = {(F, G)! Fk-MPR G}.

Then Property 3.1 implies.
QM, c: {2,41, c [2,”, c Qk_w,
and, since an R-MLR shift is less demanding than an MLR shift,
9m. C 0R- M1.-

Next we show that the four CDF orders, MLR, MPR, k-FSD, and k—MPR have the

property of transitivity.
Property 3.2. If F, MLR F, and F, MLR R, then F3 MLR F].

Proof. Let the support of F; (and its corresponding pdf ﬂ) be a ﬁnite interval
[x,’, x,’,], i=1, 2, 3, where x} S x,2 S x,3 and x}, S x}, S x2. Then the given two MLR
shifts imply that there exist non-decreasing functions h,:[x,2, x,',] —) [0, co) and
h,:[x,3, x:]-§[0, 00) such that f, =1th for x e[x,2, xi] and f3 = h,f, for x e[x,’, xi],
respectively. This implies that, for the interval {x}, x,',], f, = h,fl and f3 = h, f,, and thus
f3 = h,h,fl . Since hl - h, is also non-decreasing and non-negative for the interval

{x}, x,',], and by Deﬁnition 3.1, F, MLR F]. Q.E.D.

Property 3.3. If F, MPR F, and F, MPR F,, then F, MPR Fl.

53

Proof. With the same supports deﬁned in the proof of Property 3.2, the given two
MPR shifts imply that there exist non-decreasing functions H,:[x,2, x,',] a [0, l] and
H,:[x,3, xj]->[0, 1] such that F, =H,F, for x E[xf, x,',] and F, = H,F, for x e[x,3, xi],
respectively. This implies that, for the interval {x}, x,',], F, = H,F, and F, = H,F,, and
thus F, = H, H,F; . Since H, . H, is also non-decreasing and between zero and one for all

x E[xf, xL], and by Deﬁnition 3.2, F, MPR F,. Q.E.D.
Property 3.4. Given a point k, if F, k-FSD F, and F, k-FSD F, then F, k-FSD F, .

Proof. With the same supports given in the proof of Property 3.2, the given two
k-F SD shifts imply that F, (x)- F,(x) is non-negative for all x e R and non-increasing for
all x E[k, x,',], and that F,(x)— F,(x) is non-negative for all x e R and non-increasing for
all x E[k, xi]. This implies that F,(x)— F,(x) which is the sum of F,(x)-—F,(x) and
F,(x)— F,(x) is non-negative for all x e R and non-increasing for all x E[k, x,',]. Thus.

by Deﬁnition 3.5, F, k-FSD 15,. Q.E.D.
Property 3.5. Given a point k, if F, k-MPR F, and F, k-MPR F, then F, k-MPR F.

Proof. With the same notations used in the proof of Property 3.3, the given two
k-MPR shifts imply that the ratio function H,:[x,2, x,',] —) [0, l] is non-decreasing for all
x E[k, x,',], and H,(x)SH,(k) for all x E[xf, k], and that the ftmction H,:[x,3, xﬁ]—>[O, 1] is
non-decreasing for all x E[k, xi] and H,(x) S H,(k) for all x e[x,‘, k]. With
F, = H, H,F for the interval [x,’, x,',], this implies that H, - H, is also non-decreasing
and between zero and one for all x e [k, xi], and H,(x)H, (x) S H,(k)H,(k) for all
x e[x,3, k]. Thus, by Deﬁnition 3.6, F, k-MPR E. Q.E.D.

54

From Property 3.2, 3.3, 3.4 and 3.5, we have a general statement that a series of
MLR (MPR, k-F SD, k-MPR) shifts results in an MLR (MPR, k-FSD, k-MPR) shift. That
is, a series of n +1 (n .>. 1) number of MLR shifts such that F MLR G" MLR Gm, MLR
G, MLR G, implies that F MLR G, and the same is applied for the other three CDF
orders. This can be stated in another way that any shift that can be decomposed into a
series of MLR (MPR, k-FSD, k-MPR) shifts is also an MLR (MPR, k-F SD, k—MPR) shift.
Thus, any shift that does not satisfy the MLR (MPR, k-FSD, k-MPR) order cannot be
decomposed into a series of MLR (MPR, k-FSD, k-MPR) shifts.

Transitivity has important implications with regard to comparative static analysis
under uncertainty. As we noted in chapter 2, one strategy to make a general comparative
static statement is to ﬁnd an admissible set of changes in CDF. Consider a CDF order
which deﬁnes a particular type (set) of changes in CDF, and make assumptions so that the
comparative static effect of any CDF change generated by the order is determinable for a
given decision model considering all decision makers in a given set. Then the admissible
set of changes in CDF for the comparative static result can be extended to include every
shift that can be decomposed into a series of shifts which are all satisfying the given CDF
order. If the given order has the property of transitivity, then the extended set is just the
same as the set speciﬁed by the initial order and thus there is no gain in the comparative
static result. The above four CDF orders, MLR, MPR, k-FSD, and k-MPR are of this
type because they are transitive.

If the order is not transitive, the extended set of changes in CDF includes more
changes that do not satisfy the given order. This implies that the order is only a sufﬁcient
but not necessary condition for the comparative static result and implies a possibility of
extension giving an efﬁciency gain in the result. As a special case, consider that an
arbitrary change in CDF generated by a CDF order can be decomposed into a series of

shifts which are all satisfying another CDF order. If the latter order is sufﬁcient for a

55

general comparative static result, then all the shifts satisfying the former one are also
admissible for the result.

Transitivity does not hold for the L-MLR order. It is easy to ﬁnd an example that
the sum of two single crossing L-MLR changes, from f, to f, and from f, to f,, can
involve multiple crossings between 1, and f,. In general, the relationships F, L-MLR
F, and F, L-MLR F, do not necessarily mean F, L-MLR F}. In this case, since an L-
MLR shift is also an MPR shift, Property 3.3 implies F, MPR F, . In other words, a
CDF change that can be decomposed into a series of L-MLR shifts is not always an L-
MLR shift but is an MPR shift. Considering the reverse case, there exists an important
relationship between the L-MLR order and the MPR order. The next property shows that
an MPR shift can always be decomposed into a series of L-MLR shifts.

Property 3.6. Any MPR shift can be decomposed into a series of L-MLR shifts,
that is, if F MPR G, then there exists a series of CDF’s G, , - - - , G” such that F L-MLR G,
L-MLR G L-MLR L-MLR G, L-MLR G.

"—1

Proof. Given an arbitrary pair of pdf’ 3 f and g satisfying the condition F MPR G,

let’s deﬁne a function h: R —> [0, oo) as

rf(x)/ g(x), when g(x) at 0
h(x) = < the deﬁned last value off(x)/ g(x), when g(x) = f(x) = 0

 

Koo, when g(x) = 0 andf(x) at 0.

Under the MPR condition, the function h is generally not monotone and thus there are
points at which h changes from non-increasing to non-decreasing. Let’s denote all such
points as 0,, i=1, 2,~-, n where n .>. 028 and c,,<c,,_,< ---<c,, and for each c, deﬁne a

CDF G, and its corresponding pdf g, as

 

2‘ When n = o. it just implies that the MPR shift is an MLR shift.

56

xl,G(x), when x < c,
G,(x) =

(3.3)
F(x), when x 2 c,
and
x1,g(x), when x < c,
g.(x) = (3-4)
f(x), when x 2 c,

where 2,. = F (c,)/ G(c,), respectively. If we set G=G0 and F =G,,,, with 2,, =1, 2,”, = 0,
c0 = x, and c“, = x,, then from (3.4) we see that, for every i=1, 2,---, n +1, g, is

generally related to g,_, as

(A, / ll,_,)g,_,(x), whenx < c,
g(x) = J (h / 1,_,)g,_,(x)(= f(x)), when c, s x < c,_, (3.5)
(gt-1(x)(= f(x)), Whenx Z cI-I'

 

With the given condition F MPR G, if, / AH <1. From (3.3), the fact that G,(c,) = F (q)
and G,(x) S F (x) for all x e[x,, 6,.) implies that, at every point c,, g, meets f from the
below. This implies xi, S h(c,). For the interval [c,, c,_, ), it is known that the function h
is initially non-decreasing and changes to non-increasing through the end of the interval.
Accordingly, (3.5) implies that there exist a point k, e [c,, c,_, ) 29 and a non-decreasing
function h,:[x,, k,) -—> [0, 1] such that g(x) = h,(x)g,_l (x) for all x e[x,, k,) and

g,(x)2 g,_,(x) for all x e[k,. x4], Where h,(x) is given as,

A,//i,_,, forx e[x,, 6,)
[1,-(x =
h(x)/ 4,49 forx e[c,, k,).

Hence, by Deﬁnition 3.3, G, L-MLR G,_, for all i=1, 2,--, n+1. Q.E.D.

 

2" Ifg,(c,)2 g,_,(c,), then c, is the point k, and from (3.3),

8107) = (’I'i l’l'i-i)gi-i(x)s forx E[xl’ Ci)

gi(x) 2 gi-|(x)s forx 6[C,, x4]
which imply G, L-MLR G,_,. This case, as will be shown in Figure 3.12, leads to also GM L-MLR G,_, ,
and more generally, if g,,,(c,,,)2 g,._,(c,+ 1) for every j =0, I, m, m, then G L-MLR G,_,.

fund

57

To illustrate the result in Property 3.6, consider an example of an F SD shift given
in Figure 3.12. As the ﬁgure shows, since the pdfs g and f cross seven times, the shift is
neither an MLR nor an L-MLR shift. Following the method given in the proof of
Property 3.6, four intermediate pdfs g,'s, for i=1, 2, 3, 4, are deﬁned. The shift from
g to f is divided into ﬁve shifts, from g to g,, m, and from g, to f Applying the
conditions in Deﬁnition 3.3, it is easy to know that each shift is an L-MLR shift. Thus
the example shows that a shift from g to f, satisfying the MPR3o condition, can be

decomposed into ﬁve L-MLR shifts such that:
F L-MLR G4 L-MLR G, L-MLR G, L-MLR G, L-MLR G.

As we noted, since g,(c,) 2 g,(c,) and g,(c,) .>. g,(c4 ), the shift can be decomposed
into only three L-MLR shifts such that: F L-MLR G, L-MLR G, L-MLR G.

 

 

 

 

 

 

 

 

 

 

 

 

I l /

l \\ g1

\ 3 E

X 32 g(x)
I \ \ 2 2
l \ =
g,
x1 "‘2 9;]‘4 63:}: Liz P2 x3=cl=kl £4

Figure 3.12. F MPR G that is the sum of ﬁve L-MLR shifts.

 

3° Another implication from the example in Figure 3.1 I is that the shift from g to f is guaranteed to be an
MPR shift, without examining its CDF representation. It is because the ﬁve L-MLR shifts in the ﬁgure are
also understood as the ﬁve MPR shifts (by Property 3.1) which are transitive (by Property 3.3).

58

Property 3.6 shows the case that a shift satisfying one CDF order can be
decomposed into shifts satisfying another CDF order. In the next, as the last property
examined in this section, we show another case that a shift satisfying one CDF order can
be decomposed into shifts satisfying two other CDF orders. This interesting relationship
exists among the orders, MPR, k-FSD and k-MPR. While they all have the property of
transitivity, the following property shows that a k-MPR shift can always be understood as
the sum of two shifts, one k-FSD and the other MPR.

Property 3.7. Any k-MPR shift can be decomposed into two F SD shifts one k-
F SD and the other MPR, that is, if F k-MPR G, then there exists an intermediate CDF G,
such that F k-FSD G, MPR G.

Proof. With a shift from G to F satisfying the condition F k-MPR G where

k e[x,, x,], deﬁne an intermediate CDF G, as

Ii,G(x), whenx < k
G,(x) = (3-6)

F(x), whenx 2 k

where A, = F (k)/ G(k), respectively. Then we know that G, MPR G because the ratio of
G to G, is non-decreasing for all x e[x,, x,], and that F k-FSD G, because the k-MPR
condition implies that F (x)/ G(x) S A, for all x e [x,, k] which, in turn, implies

G,(x) 2 F(x) for all x e[x,, k]. Q.E.D.

A graphical example is given to illustrate the result in Property 3.7. The shift
from G to F shown in Figure 3.13 satisﬁes the condition F k-MPR G, but does not satisfy
the MPR or k-FSD condition given in Deﬁnition 3.2 and 3.5, respectively. The ﬁgure
shows that the intermediate CDF G, , given by using (3.6), satisﬁes the relationship that
F’k-FSD G, MPR G.

59

 

 

 

 

k x, x,

Figure 3.13. F k-MPR G that is the sum of a k-F SD and an MPR.

The last two properties can be considered as the examples showing that a CDF
order can be decomposed into shifts that can be generated by other one or more CDF
orders. These properties have important consequences with regard to providing general
comparative static statements. By Property 3.6, the comparative static statement for the
set of MPR shifts can be obtained from the one made for the L-MLR order. By Property
3.7, the comparative static statement for the set of k-MPR shifts can be obtained from the

ones made for the MPR and the k-FSD orders. These are shown in chapter 4.

Chapter 4
COMPARATIVE STATIC RESULTS WITH SUBSETS OF FSD SHIFTS

This chapter provides general comparative static statements regarding the classes
of F SD shifts discussed in chapter 3. Both the concepts of an MLR shift and an MPR
shift are considered in Landsberger and Meilijson (1990) and Eeckhoudt and Gollier
(1995), respectively. However their comparative static results are restricted to the special
case of the general framework (1.1) such that the concerned payoff function is linear in
both the choice and the random variables. Their results are generalized in this chapter by
showing that the same comparative static results hold for the general decision model
without any additional assumption required. The concept of an L-MLR shift and the
discussion given in chapter 3 are useful for the generalization of Eeckhoudt and Gollier’s
result. We also discuss trade-offs between the structural restrictions on the given decision
model'and the restrictions on the set of changes in the pdf or CDF. In particular, when
the concerned payoff function is restricted to be linear in the choice variable, it is possible
to make an even more general comparative static statement. Four CDF orders, L—MLR,
R-MLR, k-F SD, and k-MPR are used in these trade-offs.

Section 4.1 presents preliminary discussion about these two key papers and some
related work, and the main results in this chapter are provided in section 4.2. In this
chapter, let b6 and bF be the optimal choices under the CDF’s G and F, respectively, then
we examine the sign of bF — b6 when a shift from an initial CDF G to a ﬁnal CDF F

occurs.

60

61

4.1 Literature Review and Preliminary Discussions

Meyer and Orrniston (1983) show that an arbitrary FSD improvement causes
every risk averse agent to increase his level of choice variable if and only if its optimal
level under uncertainty is independent of the value of the random exogenous variable.
However, it is not an interesting problem because the restriction used on the structure of
the concerned decision model is too severe. Using the simple case of a one-risky and
one-safe asset portfolio problem, Fishburn and Porter (1976) also show that an FSD
improvement in the return of the risky asset does not necessarily induce a risk averse
investor to increase his demand for the risky asset. These papers generally imply that,
only for subsets of the set of general FSD shifts, “interesting comparative static statements
regarding the choice made by an arbitrary risk averse decision maker can be made. Two
recent papers by Landsberger and Meilij son (1990) and Eeckhoudt and Gollier (1995)
introduce two special types of FSD shiﬁs, an MLR shift and an MPR shift, respectively.
These were described in chapter 3. This section provides discussion concerning
comparative static results for these types of shifts, which are about to be generalized in

the next section.

4.1.1 Monotone Likelihood Ratio F SD Shifts

Landsberger and Meilij son made a general comparative static statement for the set
of MLR shifts. The decision model used in their analysis is the standard portfolio choice
problem. In that problem, an individual is assumed to have one unit (normalized) of
wealth which is to be allocated between one safe asset paying a zero rate of return and the
other risky asset paying a random rate of return e. Then the individual wants to maximize

his expected utility of ﬁnal wealth W given by:

W=1+be (4.1)

62

where b is the fraction of the unit wealth allocated to the risky asset. Landsberger and
Meilij son’s ﬁnding for this problem is that an MLR shift in the distribution of the random
return on the risky asset induces all investors with non-decreasing utility functions to
increase their demand for the risky asset. This result is very strong in the sense that the
comparative static statement is for a very large and general class of decision makers.

A point concerning their analysis is worthy of mention. Most other comparative
static analysis assumes that the ﬁrst and the second-order conditions for the maximum
problem (1.1) are satisﬁed to guarantee an interior bounded solution. Along with the
continuity and the differentiability assumptions, these conditions are then used to prove
comparative static results under uncertainty. That is, to prove that b, — b6 2 0 for a
particular type of change from G to F, or correspondingly from g to ﬁ it is sufﬁcient to

show that,

Qw.) = l: u'izix. b..>iz.<x. be)[g(x)- f(x)]dx s 0. (4.2)

As Dionne, Eeckhoudt and Gollier (1993) note, this approach can be quite restrictive and
may exclude some interesting cases such as a comer or an unbounded solution. In
particular, when decision makers are risk-neutral and the payoff function z(x, b) is linear
in the choice variable, the sufﬁcient second-order condition is not met and the expression
(4.2) is not as useful for the comparative static analysis. Landsberger and Meilijson’s
analysis relies on a very weak optimality condition of the decision problem. Their
analysis requires only the condition that the optimal solution b0 is determinable for an
initial CDF G, including an unbounded or a corner solution.31

The method used in the proof of their result is interesting because it includes the
cases of a corner and an unbounded solution along with the interior solution case. Let the

expected utility in the above portfolio problem be expressed as a function of b, the

 

3' Furthermore it does not require the continuity and the differentiability assumptions on the utility
functions and the payoff function W in (4.1).

63

fraction invested in the risky asset as: for an initial and a ﬁnal CDF’s G and F of the

random return e,

EU,,(b) = E,,[u(1+ be)]

and

EU,.(b) = E,.[u(l + be)]

respectively. To prove that b,,. S b,.. , it is sufficient to show that for any pair b, and b,,
ifb, S b, and EUG(b, ) S EUG(b,), then EUF(b,)S EU,..(b,). (4.3)

This is because (4.3) implies that E U ,.(b) S E U F(bG) for every b S b,,. which in turn
implies that the maximum of E U F(b) exists at a value of b larger than b,,. and thus

b,,. S bF. Landsberger and Meilijson show that an MLR shift from G to F satisﬁes the
condition (4.3) for the portfolio problem.

However Landsberger and Meilij son’s comparative static analysis is restricted to a
speciﬁc problem, i.e., the standard portfolio choice model. This is a special case of the
general framework (1.1) as the ﬁnal wealth W in (4.1) is linear in both the choice and the
random variables. Subsequently, Onniston and Schlee (1993) provide a comparative

static result based on the general decision model of the form:
mfx E[u(x, b)l - (44)

where a utility function, u, depends on both the choice variable b and the random
exogenous variable x. Since in this general framework the utility function may depend on
two or more outcome values, the general framework (1.1) is a special case of (4.4).
Treating the same subset of FSD changes, they examine a trade-off between assumptions

on risk preferences and ordinal preferences under certainty. They show that an MLR shift

64

induces all the individuals whose choices under certainty are non-decreasing in the
exogenous variable to increase their choice level.

The restriction imposed on ordinal preferences under certainty implies, in the
problems of the form (1.1), the set of decision makers with non-decreasing utility
functions. In this sense Orrniston and Schlee’s comparative static result is more general
than Landsberger and Meilij son’s result. However, the assumptions required for deﬁning
a unique solution of the problem (4.4) is quite restrictive. They also assume that the ﬁrst
and the second-order conditions are satisﬁed, and thus only the case of an interior
solution is examined. When the decision models are of the form (1.1) and have linear
payoffs such as the portfolio problem, the cases where a corner or an unbounded solution
prevails and the risk neutral decision maker are excluded from the analysis.

So far, we have discussed the comparative static analysis for the MLR order. In
section 4.2, we generalize Landsberger and Meilij son’s result by showing that the same
result is obtainable for the general decision model (1.1) for interior or non-interior
solution cases — an efﬁciency gain compared to the analysis of Orrniston and Schlee.
Furthermore, we provide another important generalization. Since the MLR order is
sufﬁcient (but not necessary) for Landsberger and Meilij son’s result, we show that the

same result is true for a more general class of FSD shifts than the MLR order.

4.1.2 Monotone Probability Ratio F SD Shifts

Eeckhoudt and Gollier’s paper is an extension of the Landsberger and Meilij son
contribution. They examine a trade-off between the restrictions on the set of decision
makers and the set of changes in distribution. While the MLR order concerns the ratio
between density functions, the MPR order is deﬁned using the ratio between cumulative
distribution functions. As we have seen in section 3.2, the MLR condition is a stronger
restriction than the MPR condition. Eeckhoudt and Gollier show that an MPR shift in the

distribution of the concerned random variable increases the choice level selected by an

65

arbitrary risk-averse decision maker. Thus, they consider a more general class of changes
in CDF, but their comparative static result is for a smaller set of decision makers.

The decision model used in their paper is restricted to the special case of (1.1),
requiring the payoff to be linear in both the choice and the random variables. Their

maximization is of the form:

mpx E[u(z0 + bx)] (4.5)

where 2,, is some given scalar. Examples of such linear payoffs are the standard portfolio
problem, a competitive ﬁrm with constant marginal costs facing uncertain output price,
and the coinsurance problem. Assuming the program (4.5) is concave in the choice
variable b, Eeckhoudt and Gollier prove their result by showing that the condition F MPR
G implies (4.2). Thus, the ﬁrst and the second-order conditions are assumed to be
satisﬁed for the optimal solution of the problem (4.5). They consider only the interior
solution case. For the proof, they used an important property (Lemma 3 in their paper)

that:
ifF MPR G, then E,,.[x: x S t] 2 E,,.[x: x S t] for all t e [x,, x,] (4.6)

where. E F [xl x S t] and E0 [xl x S t] denote the conditional (x S t) expectation of the
distributions F and G, respectively. Gollier (1995) shows that a modiﬁcation of the latter
condition in (4.6) has an important implication in the analysis of comparative statics for
decision models with linear payoffs. He provide a necessary and sufficient condition on a
shift in distribution, for signing its effect on the optimal choice made by an arbitrary risk

averse decision maker. The condition is as follows:

There exists a scalar 7 e R such that, for all t E[x, , x, ],

. . (4.7)
y ] xf(x)dx = 7F(t)EF(x; x s t) 2 G(tiE. (x; x s r) = j xg<x>dx.

66

It means that, regarding the decision model (4.5), b,,. - b,,. 2 0 if and only if a shift from G
to F satisﬁes the condition (4.7). The latter condition in (4.6) is a special case where

7 = l which is named as a linear stochastic dominance of factor I (LSD, ), and (4.6)
means that if F MPR G then F LSD, G.

According to Eeckhoudt and Gollier, there are advantages of using the MPR order
over the MLR order. Consider a shift from a random variable x (deﬁned by a CDF G) to
another random variable y (deﬁned by a CDF F) such that y = x — s where s is a positive
constant. In the portfolio problem this can beviewed as a per-tuiit tax 5 on the returns of
risky assets. With the presence of an income effect, the effect of such a shift on the
optimal investment is ambiguous. However, as Onniston and Schlee (1993) note, if the
original density function g is log-concave, then the shift satisﬁes that G MLR F and thus,
by Landsberger and Meilij son’s result, imposing a per-unit tax reduces investment in the
risky asset for all investors with non-decreasing utility functions. Similarly, Eeckhoudt
and Gollier show that if the original distribution function G is log-concave, then the shift
satisﬁes that G MPR F and thus imposing the per-unit tax reduces investment in the risky
asset for all risk averse investors. It is shown, by Bagnoli and Bergstrom (1991), that the
log-concavity restriction on a CDF is less stringent than the log-concavity of the
corresponding pdf and all non-decreasing density functions are included in the cases of
the log-concave CDF.

Another advantage of using the MPR order is related to the study by Eeckhoudt
and Hansen (1980) who analyze the effect of a minimum and a maximum price
regulations on the output level of a competitive ﬁrm. A minimum price regulation gives

a change from an initial price distribution G to another F such that

F(x) = 0 for all x < xmm and G(x) = F(x) for all x 2 xmm (4.8)

and a maximum price regulation gives a change such that

67

F(x) =1 for all x 2 x,,,” and G(x) = F(x) for all x < x (4.9)

max ’

where x,,,“ and xmam denote the minimum and the maximum price given each price
regulation, respectively. Eeckhoudt and Hansen found that, while the minimum price
regulation increases the output level of a risk-averse competitive ﬁrm, the effect of the
maximum price regulation on the output level of the ﬁrm is ambiguous. This lack of
symmetry between the two regulations can be explained by Eeckhoudt and Gollier’s
result. The minimum price regulation (4.8) implies F MPR G but the maximum price
regulation (4.9) does not imply G MPR F. However this explanation is restricted to the
ﬁrms with constant marginal costs because Eeckhoudt and Gollier examine only the
linear payoffs as shown in (4.5).

. Lastly Eeckhoudt and Gollier suggest a possible extension. Regarding the
decision model (4.5), the MPR order is only sufﬁcient (but not necessary) to obtain their
comparative static result. They provide a special type of changes in distribution which is
not an MPR shift but gives the same comparative static result. Let’s rewrite the payoff

function shown in the decision model (4.5) as,
z(x, b) = 2,, + b(x - c) (4.10)

where 20 and c are exogenous parameters. This linear form is used in Dionne, Eeckhoudt
and Gollier (1993). The point c is the value of x satisfying z,,(x, b) = 0, and it is
understood as the rate of return on a safe asset in the portfolio problem, the constant
marginal cost in the competitive ﬁrm model under price uncertainty, or the marginal cost
of insurance rate in the coinsurance problem. Eeckhoudt and Gollier gives a type of

change in CDF satisfying, '
G(x) 2 F(x) for all x e[x, , c] and G(x) = F(x) for all x E[C, x4 ], (4.11)

and they informally say that the type of shift satisﬁes the result b,,. 2 b,,.

68

 

 

 

x, x, c x, = x,

Figure 4.1. A special case of c-FSD shift.

Figure 4.1 gives an example of a shift satisfying the condition (4.1 1). It is easy to
see that the example is neither an MPR shift nor an MLR shift. According to Deﬁnition
3.5, given a point c, the shift in the ﬁgure is a special case of c-FSD shift. A c-FSD shift
requires the restriction that G(x) — F (x) is non-increasing in the interval [c, x, ], which
is weaker than the restriction, used in the condition (4.1 1), that G(x) = F (x) for all
x E[C, x4 }. Eeckhoudt and Gollier’s notion is formalized in the next section, using the
more general types of changes in CDF, c-FSD shifts and c-MPR shifts.

In section 4.2, the comparative static result in Eeckhoudt and Gollier’s analysis is
also generalized in two ways. One is obtained from generalizing their result to the
general one-argument decision model (1 .1). Proving this generalization, we follow the
same technique used in Landsberger and Meilijson’s analysis. Thus imposing a weak
restriction on the optimality of the problem, our result improves the efﬁciency of
Eeckhoudt and Gollier’s result. The other generalization is given when the payoff is
restricted to be linear in the choice variable b. This shows that the set of admissible
changes in CDF is extended from the set of MPR shifts to the set including both the

point-F SD shifts and the point-MPR shifts, given an appropriately chosen point.

69

4.2 Comparative Static Analysis: Generalized Results

Using the general one-argument decision model (1.1), this section provides
general comparative static statements concerning various subsets of FSD shifts. First we

examine the MLR shifts.

Theorem 4.1. For all decision makers with non-decreasing utility functions,
b,,. .>. b,, if
(a) F MLR G
(b) 2,,, 2 0.

Proof. With the CDF’s F and G given, each expected utility can be expressed as

a function of the choice variable b,

EU,.(b) = Eu[z(x,b)]f(x)dx
and

EUU(b) = £1u[z(x,b)]g(x)dx, '

respectively. To prove the theorem it suffices32 to show that, for a pair of points b, and

b,,
ifb, s b, and EU,,.(b,)S EU,,.(b,), then EU,,.(b,) s EU,,(b,). (4.12)

This is because (4.12) implies that A, = E U ,,(bG ) — E U ,,.(b) 2 O for every b S b,,. which
in turn implies b,, 2 b,,.33 Thus, assuming that 4,. = E U,;(b,)— EU,,.(b,) 2 0 where

b, < b,, we show that the following is non-negative,

 

32 We follow the technique used in Landsberger and Meilijson ( I990).

70

A. = £‘A(x)f(x)dx (4.13)

where A(x) = u[z(x, b,)]-u{z(x, b,)]. The difference z(x, b,)— z(x, b) is non-decreasing

in x, since 2,,, 2 0 by assumption. The assumption A6 = [3A(x)g(x)dx 2 0 excludes the

case where z(x, b,)—z(x, b,) S 0 for all x e [x,, x,] because it contradicts the
assumption, that is, with the assumption u’ 2 0, the case implies A S 0 for all x e [x, , x,]
and thus AG S 0. If z(x, b,)-z(x, b,)z 0 for all x e[x,, x,], then the assumption u' 2 0 '
implies that A 2 O for all x e [x,, x,,] and thus AF 2 0. Since these cases are true for any
FSD shift from G with its support [x,, x,] to F with its support [x,, x,] where x, S x,
and x, S x,, 34 they will not be considered any more in later analysis.

Now consider the case that, With b, , b, and the payoff function 2 given, there
exists a point x'(b,, b,, 2) e [x,, x,] such that the difference z(x, b,)—z(x, b,) is non-
positive for all x S x’ and non-negative for all x 2 x'. This implies that A S 0 for all
x S x’ and A 2 0 for all x 2 x’.35 According to Deﬁnition 3.1, the condition
h:[x,, x,] —> [0, oo) non-decreasing implies that there exists a point k e [x,, x,] such that

0ShSl forall xe[x,, k) and 1Sh<oo forall xe[k,x3I,0r OShSl forall

 

3" If there is no such pair of points b, and b, satisfying b, S b, and EU,;(b, ) S EU,;(b, ), this is the case of
comer solution and the optimal choice is the lowest among the feasible set. In this case, b, 2 b,, for any
shift from G to F.

3’ We do not exclude the case x, S x, where all the possible outcome values of x under F are higher than
under G. For any shift of this type, it is easy to see that A, 2 0.

35 This proof will be useful for the later comparative static result. However a shorter proof for the case
x'(b, , b,, 2) e[x,, x,] is suggested by Jack Meyer: According to Deﬁnition 3.1, if F MLR G then there

. exists a non-decreasing function Ir[x,, x,] —> [0, co) such that f (x) = h(x)g(x) for all x e[x,, x,].
Without loss of generality, let h(x) = 0 for the interval x e[x,, x, ). Since f (x) = 0 for all x e [x,, x,)
and A 2 0 for all x 2 x,, (4.13) can be rewritten as,

4F = [k(x)/mm [h(x)/(xxx = [homemade
Since A(x) changes its sign from negative to positive at x = x’ and h(x) is non-decreasing,
[h(xvttxigtxwx 2 h(x‘ ) [h(x)g(xxtx = h(x‘)4. 2 0.

Thus, with the assumption AG .>. O , we have A, 2 0. Q.E.D.

71

x e[x,, k] and 1S h < 00 for all x e(k, x,].36 At the point k, both cases h(k)2l and
h(k) S1 are possible. Since the proof for each case follows the same procedure, only the
case h(k) 21 is considered in the below. Without loss of generality, the non-decreasing

function h can be written as,

{1—6(x), forx e[x,, k)
h(x) =
1+ 77(x), forx e[k, x,]

where 6:1 forx e[x,, x,), 0S§Sl forx e[x,, k) and 7720.

Case (i): when x' s k. Since f(x) = 0 forx e [x,, x, ), (4.13) can be rewritten as,
A. = J:‘ A(x)f(x)dx
= J:A(x)i1 - 6(x)1g(x)dx + J," A(xtl + ntx)1g(x)dx + J: A(x)/meta (4.14)
Rearranging (4.14),
A. = 4.. + f4tx)i—6(x)ig(x)dx + f‘ A(x)n(x)g(x>dx + J: 1‘ A(x)f(x)dx.
Since A(x) change its sign from negative to positive at x = x' and g(x) is non-increasing,
a. 2 A. — 5(x‘) [A(x)g(X)dr + [‘Atx1rztx1gtx)dx + J: A(x)/ma.

Since 4, 2 0 by assumption, 0 _<_ 6(x')Sl and 4,. 2 [A(x)g(x)dx, we have 21,2 0.

Case (ii): when k S x'. For the ﬁrst term in the RHS of (4. 14), since A(x) S 0
and l—6(x)S1 for all x e[x,, k),

 

’6 Note that, as two special cases, if k = x, or k = x,, then the interval of x such that 0 S h S 1 or I S h < so
is null, respectively, and we can follow the same procedure for the proof of the cases.

72

[4(le - 6(x)1g(x)dx 211+ r7(x')]ff1(x)g(x)dr

and for the second term, since A(x) change its sign from negative to positive at x = x’

and r7(x) is non-decreasing,
[‘Atx1i1+ 77(x)]g(x)dx 20+ 406)] J: A(x)g(x)dx.
Thus from (4.14), we have
A. 2 [1+ um] I: A<x>g(x)dx + J: A(x)f(x)dx = 11+ «£114. + J: A(x)/coda:

and hence the assumption AG 2 0 implies that A, 2 0. Q.E.D.

Theorem 4.1 is a direct extension of Landsberger and Meilij son’s study, showing
that the same comparative static result holds for decision models with non-linear payoff
functions. Proving our result, we follow the same technique used in their study. Thus it
improves the robustness of their result without any additional cost of assumptions.

In addition if we closely examine the proof of Theorem 4.1, we see some notable
things. For the result in Theorem 4.1, if x’ S k then only the condition L-MLR which is
weaker than an MLR is sufficient, and if x’ 2 k then only the condition R-MLR which is
also weaker than an MLR is sufﬁcient. These ﬁndings allow us to make another
comparative static statement. In particular, if the concerned payoff is restricted to be

linear in the choice variable, a ftu'ther generalized result is possible.

Theorem 4.2. For all decision makers with non-decreasing utility functions,
b,,. 2 b,,. if

(a) z(x, b) is linear in b

(b) F L-MLR G with k E[C, x,] or F R-MLR G with k e[x,, c], where c is the value of x
satisfying z,,(x) = 0 and k is the point of crossing between the pdf’ 3 g and f

(c) 3,, 2 0.

Proof. Since the payoff function 2 is restricted to be linear in the choice variable
b, the point x’(b,, b,, 2) deﬁned in the proof of Theorem 4.1 is independent of the choice
values b, and b,. This implies that x' is the value of x satisfying z,,(x) = 0 and thus
x' = c. From the proof of Theorem 4.1, it is known that: (i) if k 2 x' (the point of single
crossing between the pdfs f and g is larger than the value of c), then only the condition F
L-MLR G is sufﬁcient for the result b,,. 2 b,,. , and (ii) if k S x' (the point of single
crossing is smaller than the value of c), then only the condition F R-MLR G is sufﬁcient

for the result b,.. 2 b,,. Q.E.D.

The trade-off used between Theorem 4.1 and 4.2 is that a larger class of changes
in CDF is allowable at the cost of the linearity assumption on the payoff function. If the
payoff function is linear in the choice variable, then, according as the point of crossing
between the pair of pdf’ 5 is smaller or larger than the point c, the restriction of monotone
likelihood ratio on either one of the left- or the right-side (of the point of crossing) is not
necessary for the result. Consider the linear payoff function given in (4.10), in which
x' = c. The point c can be, as we noted in sub-section 4.1.2, understood as the rate of
return on a safe asset in a portfolio problem, the constant marginal cost in a ﬁrm theory.
or‘the marginal cost of insurance rate in an insurance model. Two examples of FSD
shifts are shown in Figure 4.2 and 4.3, and both are satisfying the condition (b) in
Theorem 4.2. The result in Theorem 4.2 implies that, for both the shifts in the ﬁgures,
b,,. 2 b,,. for all the individuals with non-decreasing utility functions. Therefore the MLR

condition for the comparative static result in Landsberger and Meilijson's study is unduly

74

 

 

f(x). g(X)
ﬁr)
g(x
:, .t, c k x, x,
Figure 4.2. F L-MLR G with k E[C, x,].
fI-r). g(x)
f(x)
g(x)
x, x, k c x, x4

 

 

Figure 4.3. FR-MLR G with k e[x,, c].

75

restrictive for the case of linear payoff and Theorem 4.2 improves the robustness of their
result without additional cost of assumptions.

From this point forward, we replace the set of non-decreasing utility functions
with the set of non-decreasing and concave utility functions; that is, the concerned
decision makers are assumed to be risk-averse. In what follows, we show that adding this
restriction on the risk preferences of decision maker allows the expansion of the
addmissible changes in CDF for desirable comparative static statements. The next
theorem concerns the set of L-MLR shifts which is larger than the set of MLR shifts and

smaller than the set of MPR shifts.

Theorem 4.3. For all risk averse decision makers, b,,. 2 b,,. if
(a) F L-MLR G
(b) 2,, 2 0 and 2,,, 2 0.

Proof. We follow the technique and notation used in the proof of Theorem 4.1,
and as we noted, we consider only the case where x, S x' S x,. According to Deﬁnition
3.3, the condition F L-MLR G implies that there exists a point k e [x, , x,] such that
f(x) S g(x) for all x e[x,, k) and f (x) 2 g(x) for all x E[k, x,]. Since the proof for
the case that f(k) S g(k) and f(x) S g(x) for all x E[x, , k] and f(x) 2 g(x) for all
x e (k, x,] follows the same procedure, we omit the case. Hence, with the condition F

L-MLR G, f (x) can be expressed as,

{[1 —6(x)]g(x), forx e[x,', k)
f(X)= t [1+ ”(31806), forx Elk» x3)

f(x), forx e[x,, x,]

 

76

where 6 is non-increasing in x, 5 :1 for x e[x,, x,) and O S 5 S1 for x e[x,, k), and
27 is a non-negative function. Then, for the case where x’ S k, the proof is exactly same

as in the proof of Theorem 4.1. Consider the case where k S x'. Let’s rewrite (4.13) as.
21,, = J" A(x) f(x)dx + f_‘A(x)f(x)dx + J" ‘ A(x) f(x)dx. (4.15)
Integrating the ﬁrst term in RHS of (4. l 5) by parts, and by adding and subtracting,

21,, = A(x)F(x)

 

:, _ [F(X)F(X)dx+ [3A(x)f(x)dx+ J: A(x)f(x)dx

= A(x)[F(x)-G(X)II:; + A(x)G(x)

 

'_ £B(x)[F(x)—G(x)ldx — 1??me

+ [,‘A(x)if<x> — g(x)]dx + [.‘A(x)g(x>dx + I: 1‘ A<x>f <x>dx

where B = LL4/ dx = u’(z(x, b, ))z,(x, b, ) -u'(z(x, b, ))2, (x, b, ). By rearranging the above,

a, = 4, + A(x)[F(x)- G(x)]: — [gimme - G(x)]dx

+ ['.‘A(x)if<x)—g(x)idx+ J: A(x)f(x)dx. (4.16)

Note that z(x, b,) S z(x, b,) when x S x' and z, (x, b,) 2 z,(x, b,) when x 2 x, by the
assumption 2,,, 2 O. The assumptions u' 2 0, u" S 0 and z, 2 0 imply that B(x) 2 0 for
x e[x,, x’]. Since A S 0 for all x S x' and A 2 0 for all x 2 x', and the L-MLR
condition implies that F (x) S G(x) for all x e[x,, x,,] and f (x) 2 g(x) for all x E[k, x,],

the assumption AG 2 0 implies that A,,. 2 0. Q.E.D.

Compared with Theorem 4.1, the comparative static result in Theorem 4.3
includes a larger set of FSD changes and a smaller class of decision makers. Since we
follow the same technique used in the proof of Theorem 4.1, the result in Theorem 4.3 is

also established on the weak restriction on the optimality of the decision problem (1.1).

77

While the result itself is meaningful as a general comparative static statement, a further
generalization is given in the next theorem. From Property 3.6 developed in chapter 3. it
is known that an MPR shift can be decomposed into a series of L-MLR shifts. This
allows a direct generalization of the comparative static result in Theorem 4.3, without any

additional cost of assumptions.

Theorem 4.4. For all risk averse decision makers, b,,. 2 b,,. if

(a) F MPR G
(b) 2, 2 0 and 2,,, 2 0.

Proof. From Property 3.6 we know that an MPR shift can always be decomposed
into a series of L-MLR shifts. That is, if F MPR G then there exists a series of CDF’s
G,,---,G,, such that F L-MLR G" L-MLR G L-MLR L-MLR G, L-MLR G.

"-1

With the result in Theorem 4.3, it completes the proof. Q.E.D.

Theorem 4.4 is an easy and important generalization of Theorem 4.3, and it is also
an important generalization of the result in Eeckhoudt and Gollier's analysis. That is, the
MPR condition is sufﬁcient for the general decision model (1.1). Another efﬁciency gain
is that our result is established on the weak restriction on the optimal solution of the
decision problem. That is, the cases of a corner or an unbounded solution and the risk
neutral decision makers are not excluded from our result. This completes the Eeckhoudt
and Gollier's discussion about the lack of symmetry in the effects of the minimum and the
maximum price regulations in Eeckhoudt and Hansen (1980).

Finally, we formalize the intuition given by Eeckhoudt and Gollier. As we
discussed in sub-section 4.1.2, given a linear (in the choice variable b) payoff function,
they informally say that it is possible to determine the effect of the type of changes in

CDF deﬁned in (4.11) on the optimal choice made by an arbitrary risk averse decision

78

maker. It is also discussed that the shift is neither an MPR Shift nor an MLR shift, but a
special case of c-FSD shift. Hence it is meaningful to make a comparative static
statement for the set of c-FSD shifts.

Adding a structural restriction that the concerned payoff is linear in the choice
variable b, the extension of the admissible changes in CDF for the result in Theorem 4.4
is obtained from the point-MPR as well as the point-FSD shifts. Next theorem shows
that, given a point c, the set of c-F SD shifts is ﬁrst shown to be sufﬁcient for a desirable
comparative static statement and then the extension to the set of c-MPR shifts comes

directly from Property 3.7.

Theorem 4.5. For all risk averse decision makers, b,, 2 b,,. if
(a) z(x, b) is linear in b and z,,(c, b): 0
(b) F c-FSD G or Fc-MPR G

(c) z, 2 0 and 2,,, 2 0.

Proof. First consider the set of c-F SD shifts. We know that the linearity
assumption implies that the point x'( b, , b,, 2), deﬁned in the proof of Theorem 4.1, is
constant for all the pair of choice values b, and b,. Now x’ is the value of x satisfying
2,,(x) = 0 and thus x' = c. The condition Fc-FSD G implies that G(x) - F(x) 2 0 for
the interval x e[x,, x'] and g(x) - f (x) S 0 for the interval x E[x', x,]. This is the
case that k S x' in the proof of Theorem 4.3, and from (4.16) we know that the condition
is sufﬁcient for AF 2 0.

Now, given the proof for the set of c-F SD shifts, the proof for the set of c-MPR
shift can be completed by Property 3.7 and Theorem 3.4. By Property 3.7, it is known
that a c-MPR shift can always be decomposed into two FSD shifts, one c-FSD and the

other MPR. Q.E.D.

79

 

 

 

Figure 4.4. A c-FSD Shift.

 

 

 

Figure 4.5. A c-MPR shift.

80

Adding the linearity restriction, the comparative static result given in Theorem 4.5
includes a more general set of changes in CDF than the set of MPR shifts contained in the
result in Theorem 4.4. This is an example of a trade-off between the restrictions on the
set of changes in C DF and the structure of the concerned decision model. This result is
also a generalization of the comparative static result in Eeckhoudt and Gollier’s analysis.
Consider the two examples of FSD shifts shown in Figure 4.4 and 4.5. It is easy to see
that each of the examples is neither an MLR shift nor an MPR shift. Given the point c.
the shifts satisfy the conditions, F c-F SD G and F c-MPR G, respectively.

Now, regarding the decision models of the form (4.10), we know the result that. for both
the shifts, b, 2 b0 for all risk-averseldecision makers.

In this chapter, comparative static theorems are developed with regard to subsets
of the set of general FSD shifts. All the results developed in this chapter are associated
with relatively weak restrictions on the risk preference of decision makers, i.e., the set all
risk averse agents or the set of all individuals with non-decreasing utility functions. In
this sense, we refer all the subsets of CDF changes concerned in this chapter to “FSD

shifts in the strong sense.”

Chapter 5
RELATIONSHIPS AMONG SUBSETS OF R-S INCREASES IN RISK

This chapter gives the deﬁnitions for four special classes of R-S increases in risk.
Each class is speciﬁed by adding additional restrictions to the ‘integral conditions’ which
deﬁne the general concept of an R—S increase in risk (Deﬁnition 2.3). As in the case of
deﬁning subsets of FSD changes, monotonicity restrictions on the likelihood ratio
(between a pair of pdf’ 5) or on the probability ratio (between a fair of CDF’s) are also
important in deﬁning subsets of R-S increases in risk. For each subset, a general
comparative static statement is developed in chapter 6. Before doing this, this chapter
examines several important relationships among these four types of R-S increases in risk.

In chapters 5 and 6, Since only subsets of R-S increases in risk are considered, a
ﬁnal37 distribution G is assumed to be always riskier than an initial distribution F in the
R—S sense, that is, G is a mean-preserving spread from F (denoted by G MPS F). Assume
that both the CDF’s F and G have their points of increase in bounded intervals. For
notational convenience, let the support of F (x) be a ﬁnite interval [x, , x,] and the
support of G(x) be another ﬁnite interval [x, , x,] where x, S x, S x, S x,. Deﬁning the
supports in this way, we also assume that: for continuous case, G(x) > 0 for all
x e(x,, oo), l-G(x)> 0 for all x e(-oo, x,), F(x)> 0 for all x e(x,, co), and

1— F(x) > 0 for all x e(—oo, x,); and for discrete case, G(x) > 0 for all x e [x,, do),

 

37 In order to keep the notation for F to be stochastically dominant over G, as in chapters 3 and 4, the
CDF’s F and G are regarded as an initial and a ﬁnal distribution of the random variable x, respectively.

81

82

l—G(x)>0 for all x e(—oo, x4], F(x)>0 for all x e[x,, co), and l— F(x)>0 for all

x e(—oo, x,].38

5.1 Important Subsets of R-S Increases in Risk

This section gives the deﬁnitions for several subsets of R-S increases in risk.
Including the concept of a ‘relatively strong increase in risk’ (RSIR) introduced by Black
and Bulkley (1989), we add three concepts of R-S increases in risk. Being more general
than an RSIR, they are a ‘left-side relatively strong increase in risk’ (L-RSIR), an
‘extended strong increase in risk’ (ESIR), and a ‘left-side extended strong increase in
risk’ (L-ESIR). In sub-section 5.1.1, basic deﬁnitions of these four concepts are given.
Graphical examples in sub-section 5.1.2 are presented to help the reader understand the

basic relations among the concepts.

5.1.1 Basic Deﬁnitions of Subsets of R-S Increases in Risk

In order to make a desirable comparative static statement, Meyer and Orrniston
(1985) introduce the concept of a ‘strong increase in risk’ (SIR). Deﬁning a shift from
F (x) to G(x), an SIR speciﬁes a transfer of probability mass from points in the interval
[x,, x,] to points in the intervals [x,, x,) and (x,, x,]. It is the case where the initially
possible outcomes are less likely to occur, and the previously impossible outcomes which
were outside the support of the initial CDF F (x) now have a non-zero probability mass.

As we saw in chapter 2, the set of SIR shifts includes, as special cases, ‘mean-preserving

 

38 Similar to F SD cases, sets of R-S increases in risk used in this chapter are deﬁned by imposing
monotonicity restrictions on the ratio of f to g or the ratio of F to G in the overlapped interval [x,, x, ].

Then this assumption allows both the ratios to be strictly positive in the interval [x,, x,] for discrete case.
and in the interval (x,, x,) for continuous case, which are often used to simplify the proofs of the results
developed in chapters 5 and 6.

truncations’ deﬁned by Eeckhoudt and Hansen (1980) and ‘global increases in risk’
analyzed in Kraus (1979), Katz (1981). Recently, as a more general concept than an SIR.
Black and Bulkley (1989) use an RSIR in their comparative static analysis. This is

deﬁned as:

Deﬁnition 5.1. G(x) represents a relatively strong increase in risk from F (x)

(denoted by G RSIR F) if39
(a) J: [G(x) - F(x)]dx = 0

(b) there exists a pair of points k,, k, e[x,, x,] where k, S k, , such that f (x)2 g(x) for
all x e[k,, k,] and f(x)Sg(x) for allx in [x,, k,) and (k,, x,]

(c) for all x e [x,, k, ), there exists a non-decreasing function h,:[x,, k,) a [0, 1] such
that f(x)= h(x)g(x)

(d) for all x e(k,, x,], there exists a non-decreasing function h,:(k,, x,] —> [0, I] such

that f(x) = (h(x)g(x)-4°

Condition (a) implies the same mean between the two distributions. Condition (b)
imposes the restriction that the two pdf’s cross only twice and speciﬁes a probability
transformation from points within a centric interval [k,, k,] to points lying outside the
interval. These two conditions are sufficient for G(x) to represent an R-S increase in risk
from F (x), satisfying the integral conditions in Deﬁnition 2.3. It is obvious that an SIR,

given in Deﬁnition 2.8, is a special case of an RSIR where k, = x, and k, = x,.

 

39 We deﬁned an RSIR in somewhat different way from the original one. This is to include not only the
continuous case but the discrete case where g(x) and/orﬂx) is not always positive in the given supports.

‘0 The two crossings at points k,,k, e[x,, x,] imply that f(x)S g(x) for all x in [x,, k,) and (k,, x,] and
f(x)2g(x) for all x e(k,, k, ). At the points k,, 1': l, 2, it is usual that f(k,)= g(k), but ifthe pdfg orf
is discontinuous at the points, it is possible that f (k,)< g(k,) and f (k,)> g(k,). Deﬁnition 5.1, thus,
includes the cases where f(x)Sg(x) for allx in [x,, k,] and/or [k,, x,], and f(x)2 g(x) for allx in

[k,, k,) or (k, , k,]. This discussion is applied for all the other deﬁnitions given in this section.

84

Conditions (c) and (d) restrict the transformation from f(x) to g(x) to be such that.
for both the intervals [x,, k,) and (k,, x,], the ratio of f to g is non-decreasing as x goes
farther from the end points x, and x,. In other words, considering an R—S decrease in risk
from G to F , the condition G RSIR F implies that a non-increasing proportion of left-tail
probability mass and a non-decreasing proportion of right-tail probability mass are
transferred to the centric interval.

The RSIR order is closely related to the L-MLR and the R-MLR orders deﬁned in
section 3.1. Conditions (a) and (b) in Deﬁnition 5.1 imply that the CDF’s F and G cross
only once, and there exists a point k e [k,, k,] such that F (x) S G(x) when x S k and
F (x) 2 G(x) when x 2 k. Given an RSIR, it can be divided into two shifts: one for the
left and the other for the right-side of the point k. In each side of k, there is only one
crossing between the two pdf 5, and for each end interval such that f ( x) S g(x), the ratio
between the pdf s should be monotone. Applying Deﬁnition 3.3 and 3.4, it is obvious
that the shift for the left-side is understood as ‘F dominant over G in the L-MLR sense.’
and the shift for the right-side is understood, in the opposite direction, as ‘G dominant
over F in the R-MLR sense.’ Thus an RSIR shift can be viewed as the sum of two shifts.
keeping the mean same, an L-MLR dominated shift and an R-MLR dominant Shift.

In the next deﬁnition of an R-S increase in risk, the restriction on the right-side of
the point k is relaxed, and we name it a ‘left-side relatively strong increase in risk’ (L-

RSIR).

Deﬁnition 5.2. G(x) represents a left-side relatively strong increase in risk from

F (x) (denoted by G L-RSIR F) if
(a) J [G(x) - F(x)]dx = 0

(b) there exists a point k e [x,, x,] such that

F(x)S G(x) for all x e[x,, k) and F(x)2G(x) for all x E[k, x,]

85

(c) there exist a point k’ e[x,, k] and a non-decreasing function h: [x,, kI ) -—>[O, 1] such

that f(x) = h(x)g(x) for all x e[x,, k’) and g(x)Sf(x) for all x G(k’, k].

Condition (b) imposes the restriction that the two CDF’S cross only once at a point k, and
combined with condition (a), this obviously implies that an L-RSIR is an R-S increase in
risk. Condition (c) implies that, for the left-side of the point k, an L-RSIR requires the
same restriction used to deﬁne an RSIR. Now for the right-side of the point k, there is no
restriction on the number of times of crossing between the pdf 5 f and g, nor is there any
monotonicity restriction on the ratio between the pdf 5. Only the required restriction on
the right-side of k is that F (x) 2 G(x). Therefore the set of RSIR shifts are a subset of the
set of L-RSIR shifts.

Similar to the case of an RSIR shift, an L-RSIR shift can be divided into two
shifts: one for the left and the other for the right-side of the point k. The shift for the left-
side is understood as ‘F dominant over G in the L-MLR sense’ and the shift for the right-
side is understood, in the opposite direction, as ‘G dominant over F in the FSD sense.’
Thus an L-RSIR shift can be viewed as the sum of two shifts, an L-MLR dominated shift
and an FSD dominant shift.

" To see a special case of L-RSIR shifts, consider a random variable which has only
two outcome values. If the smaller outcome becomes riskier (mean-preserving) than
before, then it is the case of an L-RSIR, while an increase in risk in the higher outcome
does not lead to an L-RSIR. Generally, given an initial pdfﬂx) with its support [x,. x, ],
let’s consider a conditional pdf f ' (t) = f (t)/ F (T) with its support [x,, T] where
T S x,. Then an RSIR shift in f ’ leads to an L-RSIR shift in the initial f(x).

Compared with an RSIR, while an L-RSIR is deﬁned by relaxing the restrictions
imposed on the right-side of the point k, another less stringent type of R-S increase in risk
is introduced and it is named as an ‘extended strong increase in risk’ (ESIR). Instead of

using the monotonicity restrictions on the pdf ratio of f to g, an ESIR is deﬁned by

86

imposing monotonicity restrictions on the ratio of an initial C DF F to a ﬁnal C DF G and

on the ratio of (l-F) to (l—G) as:

Deﬁnition 5.3. G(x) represents an extended strong increase in risk from F (x)

(denoted by G ESIR F) if
(a) J: [G(x) — F(x)]dx = 0

(b) there exists a point k e [x,, x,] such that .
F(x)S G(x) for all x e[x,, k) and F(x)2 G(x) for all x E[k, x,]

(c) for all x e [x,, k), there exists a non-decreasing function H,:[x,, k)—>[0, I] such that
F(x) = H,(x)G(x)

(d) for all x e[k, x,], there exists a non-increasing function H ,:[k, x,] —> [0. 1] such

that 1— F(x) = H,(x)[l — G(x)].

The same conditions (a) and (b) in Deﬁnition 5.2 are used and they guarantee that G
represents an R-S increase in risk from F. An ESIR requires the restriction that the two
CDF’s cross only once at a point k, but does not restrict the number of times of crossing
between the pdf” 3. There is no direct relationship between an ESIR and an L-RSIR. For
the left-side of the point k, while an ESIR has no restriction on the number of times of
crossing between the pdf’ 8, an L-RSIR requires only one crossing between them. For the
right-side of k, while an ESIR requires that the ratio of (l — F) to (1— G) should be
monotone, an L-RSIR requires only the condition that F (x) 2 G(x). Thus, for the interval
[x,, k), an ESIR is less demanding than an L-RSIR, and vice versa for the interval
[k, x,].

However an RSIR implies an ESIR. In section 5.2, we provide a formal proof for
this assertion, showing that the monotonicity restrictions imposed on the ratio of f to g

imply that the ratio of F to G in the interval [x,, k) and the ratio of (l - F) to (l — G) in

87

the interval [k, x,] are monotone. The steps used in the proof is similar to the ones used

in the proof of the assertion that an L-MLR implies an MPR (Property 3.1). This implies
that, compared with an RSIR, an ESIR uses less stringent restrictions on the changes in

C DF. It does not use the restrictions on both the number of times of crossing between the
pair of pdf’ 5 f and g and the monotonicity of the likelihood ratio between the pdf’ 5.

The ESIR order is closely related to the MPR order deﬁned in section 3.1. Given
a shift from F to G satisfying G ESIR F, it can be divided into two shifts: one for the left
and the other for the right-side of the point k. Then, applying Deﬁnition 3.2, the shift for
the left-side can be understood as ‘F dominant over G in the MPR sense.’ The shift for
the right-side is a special FSD shift. If we deﬁne a new random variable y as y = —x and
the initial and the ﬁnal pdf’s of y, f" (y) and g( y), are given as f (x) = f (— y) = f’( y)
and g(x) = g(— y) = g( y) , then the shift for the right-side is understood as F MPR G.
Thus an ESIR shift can be viewed as the sum of two shifts, one MPR dominated shift and
the other specialized F SD dominant shift.

As the last specialized concept of an R-S increase in risk presented in this section,
the next one is obtained from further relaxing the restrictions imposed on either an ESIR
or an L-RSIR. Now the condition ((1) used in Deﬁnition 5.3 is dropped. Thus in the
following deﬁnition, the right-side monotonicity restriction on the CDF ratio is relaxed

and we name it a ‘left-side extended strong increase in risk’ (L-ESIR).

Deﬁnition 5.4. G(x) represents a left-side extended strong increase in risk from

F (x) (denoted by G L-ESIR F) if
(a) J: [G(x) - F(x)]dx = o

(b) there exist a point k e[x,, x,] and a non-decreasing function H:[x,, k)—>[0, 1] such

that F(x)= H(x)G(x) for all x e[x,, k) and G(x)SF(x) for all x E[k, x,].

88

An L-ESIR still has the restriction that the two CDF’s cross only once at a point k, which
is implied by condition (b). Combined with condition (a), this guarantees that an L-ESIR
is an R-S increase in risk. Since an L-ESIR requires only the left-side monotonicity of
the CDF ratio, it is" more general concept than an ESIR. The L-ESIR order is also more
general than the L-RSIR order because, with the same restrictions on the points to the
right-side of the point k, an L-ESIR is less demanding than an L-RSIR for the points to
the left-side of k. This can be proved by showing that the monotonicity of the pdf ratio
implies the monotonicity of the CDF ratio, which is given in the next section. Similar to
the above three cases of R-S increases in risk, an L-ESIR shift from F to G can also be
understood as the sum of two shifts: one for the left-side of the point k is F MPR G, and
the other for the right-side of the point k is, in the opposite direction, G FSD F.

As in the case of an L-RSIR, a special L-ESIR is as follows. Given an initial pdf
f(x) with its support [x,, x,], let’s consider a conditional pdf f 1(t) = f (t)/ F (T) with its
support [x,, T] where T S x,. Then any ESIR shift in f ’ results in an L-ESIR shift in

the initial f(x).

5.1.2 Graphical Examples

Each of the above four concepts of R-S increases in risk speciﬁes a particular
subset of the set of all R-S increases in risk. They all have a common restriction that the
CDF’s F and G cross only once, but each type of shifts is deﬁned by imposing additional
restrictions on the changes in distribution functions. In this sub-section, using graphical
examples including seven continuous cases and two discrete cases, we show the basic
relationships among those four types of R-S increases in risk discussed in sub-section
5.1.1. In what follows, we examine each example to see the differences among the
restrictions used to deﬁne each type of R-S increase in risk. Thus every example is
assumed to preserve the mean same before and after the given shift in pdf or CDF. First

two examples are special cases of RSIR shifts: example 5.1 shows a mean-preserving

89

 

 

 

x, x, :kl x, =k2 x,,

Figure 5.1. Example 5.1: Mean—preserving truncation from G to F.

.—

ﬂx). g(x)

\ 4% ﬁx)

gIX)

 

 

 

 

x, x,=k, X,=k, x,

Figure 5.2. Example 5.2: G SIR F.

90

XX). 80‘)

 

 

x, x, c, k, k k, c, xx,

Figure 5.3. Example 5.3: G RSIR F.

1‘00. 190‘)

 

 

 

1 II I
x, x, k k x, x4

Figure 5.4. Example 5.4: G L-RSIR F.

9l

g(X)
f(x)

 

 

l

x]

tIIIIIIIIJ .................

IIIIIIIIIIIIIIIIIII

I‘IIIIIIIIII‘I‘II.‘

_I

x,

['I'II"

 

x3 =x4

Figure 5.6. Example 5.6: G L-RSIR F.

 

Figure 5.5. Example 5.5: G L-RSIR F (a discrete case).

 

 

 

f(x). g(X)

 

 

 

k

3‘2

 

I'-II"|I1

llllllll

30‘)
XX)

llllllll

Figure 5.7. Example 5.7: G ESIR F.

 

IIIIIIII

 

 

x4

3

x
Figure 5.8. Example 5.8: G ESIR F (a discrete case).

2

C

k

 

93

 

 

 

 

 

 

 

 

 

 

 

 

 

|.......J
I
r----
I
,_-_-l
F(X) l '
I
I
I
I
b----+
G(X) ,
r..---J
I
----4
I I
I’""J ‘
I
I
x, x, c, k c, x, x,

Figure 5.9. CDF representation of example 5.8.

 

 

 

Figure 5.10. Example 5.9: G L-ESIR F.

x,=x4

94

truncation, and example 5.2 gives a case of a strong increase in risk (SIR).

(1) Example 5.1: Mean-preserving truncation from G to F in Figure 5.1. This
type of R-S increase in risk is deﬁned by imposing somewhat severe restrictions on the
changes in CDF. It speciﬁes a probability transformation such that truncated both tail
probability mass is transferred to each end point. As Figure 5.1 shows, all the probability
of occurring for points less than x, (larger than x,) is moved to the point x, (x,). This
transformation is a special case of an RSIR. Since x, = k, and x, = k,, the intervals
[x,, k,) and (k,, x,] shown in Deﬁnition 5.1 are empty, thus the mean-preserving

truncation satisﬁes the RSIR condition.

(2) Example 5.2: G SIR F in Figure 5.2. The restriction that F (x) = G(x) for the
interval [x, , x, ], which is required for a mean-preserving truncation, is relaxed. Instead,
an SIR restricts the CDF difference G — F to be non-increasing for the interval [x,, x,].
Equivalently, the restriction means that f 2 g for all x e [x,, x,], as shown in Figure 5.2.
It speciﬁes a probability transformation from points in the interval [x,, x,] to points
outside the interval. An SIR is also an RSIR, specifying a special case that x, = k, and
x, = k,. Note that both shifts given in examples 3.1 and 3.2 satisfy the condition that the

pdfs f and g cross only twice (at points k, and k, ), which is required for an RSIR.

(3) Example 5.3: G RSIR F in Figure 5.3. This example shows that the pdf’ 3 f
and g cross only at points k, and k,. Let’s deﬁne the likelihood ratios h, and h, as
h, = f / g for the interval [x,, k,), and h, = f / g for the interval (k,, x,]. Then, for
each given interval, the ratio function h, is non-decreasing and h, is non-increasing,
respectively. Thus example 5.3 satisﬁes the RSIR conditions given in Deﬁnition 5.1. To
see this monotonicity in the graphic example, we can follow the logic used in the

discussion of the MLR example given in Figure 3.1. Since g 2 f for all x e [x,, k, ), the

95

ratio f / g is non-decreasing in the interval when g' > 0 (g' < 0) and the tangent lines

I, and I, in Figure 5.3 do not meet to the left (meet to the right) direction in the space

above x axis. Similarly, since g 2 f for all x G(k,, x,], the ratio f / g is non-

increasing in the interval if g' < 0 (g' > 0) and the tangent lines I, and I, do not meet to

the right (meet to the left) direction in the space above x axis.

(4) Example 5.4: G L-RSIR F in Figure 5.4. In this example, the point k is
selected such that G(k) = F (k), and it is assumed that G 2 F for all x e [x,, k), and
G S F for all x e [k, x,]. For the left-side of the point k, the pdf 5 f and g cross only
once at the point k’ . Deﬁne the likelihood ratio h as h = f / g for the interval [x,, k’ ).
Since the ratio is non-decreasing in the interval and f is larger than g for all x e (k’ , k],
the shift in the example satisﬁes the L-RSIR condition in Deﬁnition 5.2. The pdf 8 f and

g cross four times, and thus the shift cannot be an RSIR change.

(5) Example 5.5: G L-RSIR F in Figure 5.5. This is a special case of an L-RSIR
in which the initial random variable has only two outcomes, x, and x,, with equal
probability of occurring. In this example, an R-S increase in risk occurs only in the
smaller outcome x,. Since it is such a case that x, = k’ = k, the non-decreasing ratio
function h shown in Deﬁnition 5.2 has, as its domain, an empty interval and F (x) 2 G(x)
for all x e [k, x,]. Thus the shift satisﬁes the L-RSIR condition. However the example
does not satisfy the RSIR condition. Since, for the interval including the points c and x,
satisfying f S g, the ratio of f to g is zero at c and one at x,, the ratio likelihood function
h, should be increasing at least one point in the interval [c, x, ]. The shift cannot be an
ESIR either. For the interval [k, x,], the CDF ratio (1 — F) / (l — G) should be increasing
at least one point in the interval, which contradicts the ESIR condition in Deﬁnition 5.3.

In this instance, if an increase in risk occurs in the higher outcome x, with the

lower outcome unchanged, then it is the case where k’ = k = x,. This does not satisfy the

96

L-RSIR condition because the pdf ratio function h should be decreasing at least one point
in the interval [x,, k1 ). The shift is not an L-ESIR either because the CDF ratio H
deﬁned by H = F / G should be decreasing at least one point in the interval [x,, k).
Consider the case where R-S increases in risk occur in both the outcome values. Then it
is the case where the CDF’s F and G cross three times, and thus it does not satisfy any of

the four types of R-S increases in risk deﬁned in this section.

(6) Example 5.6: G L-RSIR F in Figure 5.6. As another special case of L-RSIR,
example 5.6 shows that, given an initial pdff(x) with its support [x,, x,], an RSIR type
of change occurred only on a left-side sub-interval [x,, T] where T S x, results in an L-
RSIR shift in the initial pdf f. In Figure 5.6, let’s deﬁne a conditional pdf
f’(t) = f(t)/ F(T) with its support [x,, T], then the shift shown in the ﬁgure can be
understood as an RSIR change from the pdf f’. The shift satisﬁes that F S G for all
x e [x,, k) and F 2 G for all x e [k, x,]. Since the ratio f to g is constant and less than

one for the interval [x,, k’ ), and f 2 g for all x E[k’, k], it is obvious that F L-RSIR G.

(7) Example 5.7: G ESIR F in Figure 5.7. While the shift is represented by CDF
change, it is easy to see that the pdf’s f and g cross more than two times in the interval
[x,, k]. This implies that the shift is neither an RSIR nor an L-RSIR. The CDF’s F and
G cross only once at the point k. The probability ratio functions HI and H, in Deﬁnition
5.3 can be deﬁned by H, = F/G for the interval [x,, k), and by H, =(1— F)/ (1- G)
for the interval [k, x,], and the ﬁgure shows that H, is non-decreasing and H, is non-
increasing for each corresponding interval, respectively. This guarantees that G ESIR F.
The way to see the monotonicity follows the discussion about the example of an MPR
shift given in Figure 3.7. If, for any point in the interval [x,, k) ([k, x,]), the tangent
lines I, and IG(= G(x) in example 5.7) do not meet to the left (right) direction in the

space between zero and one, the CDF ratio H, (H,) is non-decreasing (non-increasing).

97

(8) Example 5.8: G ESIR F in Figure 5.8 and 5.9. This is an example ofan ESIR,
for a discrete random variable case. To see that G ESIR F, consider the CDF
representation of the shift given in Figure 5.9. Since there is only one crossing between
the CDF’s and the monotonicity restrictions on the two CDF ratios are satisﬁed, it is
obvious that G ESIR F. Figure 5.8 shows that the difference g — f changes sign more
than two times for the left-side of the point k. This implies that the shift cannot be an
RSIR or an L-RSIR change.

(9) Example 5.9: G L-ESIR F in Figure 5.10. As the last example of single-
crossing between the CDF’s F and G, example 5.9 gives an L-ESIR change. The shift
satisﬁes that F S G for all x e[x,, k) and F 2 G for all x e[k, x,]. For the left-side of
the point k, the CDF ratio F / G is non-decreasing, which guarantees that G L-ESIR F.
The example is also a special case of an L-ESIR change. Figure 5.10 shows that G = F
for all x e [T, x,]. Similar to the example 5.6, let’s deﬁne a conditional pdf
f’(t) = f(t)/ F(T) with its support [x,, T], then the shift shown in Figure 5.10 can be
understood as an ESIR change from the pdf f ’. Thus an ESIR type of change occurred
only on a left-side sub-interval [x, , T] where T S x, results in an L-ESIR change in the

original pdf.

5.2 Some Properties among Subsets of R-S Increases in Risk

This section examines several relationships among the four concepts of R-S
increases in risk, deﬁned in section 5.1. First, two properties clarify the basic
relationships among the orders generated by these concepts. One relationship is that the

RSIR order implies the ESIR order which, in turn, implies the L-ESIR order, and the

98

other is that the RSIR order implies the L-RSIR order which, in turn, implies the L-ESIR
order. We provide formal proofs for these relationships. .

In addition, there are important relationships between an RSIR and an ESIR, and
between an L-RSIR and an L-ESIR. In chapter 4, we have shown that the property that
an MPR shift can be decomposed into a series of L-MLR shifts gives an important
generalization of a comparative static result, extending admissible set of CDF changes
from the L-MLR shifts to the MPR shifts. Similarly, this section presents two assertions
that: (i) “an ESIR shift can be decomposed into a series of RSIR shifts,” and (ii) “an L-
ESIR shift can be decomposed into a series of L-RSIR shifts.” They are the key
relationships that allow us to make general comparative static statements in chapter 6.
Let’s begin with the statement that the ESIR order lies between the RSIR and the L-ESIR

orders.
Property 5.1. G RSIR F :> G ESIR F => G L-ESIR F.

Proof. It is obvious that “if G ESIR F then G L-ESIR F” because only the
conditions (a), (b) and (c) in Deﬁnition 5.3 are sufﬁcient for the L-ESIR order. The proof
for the claim that “if G RSIR F then G ESIR F” follows the same steps used in the proof
of Property 3.1. The condition (a) in Deﬁnition 5.1 and 5.3 is just the same mean
condition. The condition (b) in Deﬁnition 5.1 implies that there exists a point k e [k,, k,]
such that F(x) S G(x) for all x e[x,, k), and F(x) 2 G(x) for all x E[k, x,], which is the
condition (b) in Deﬁnition 5.3. Since G(x) > 0 for all x e(x,, co) and l—G(x) >0 for all
x e(—oo, x,), we can deﬁne a pair of functions H, for the interval [x,, k) and H, for the

interval [k, x,] as,41

 

4' If x is continuous and x, = x, or x, = x, , then G(x,) = 0 or G(x,) = 0, and the proof is also true for this
case with the function deﬁned as,

H,(x) ={ F(x)/G(x). forx E[k, x,)

0, when x = x,.

0, when x = x,

F(x)/G(x), forx e(x,, k) °' ”2“) =l

99

H105): F(x)/G(x), forx e[x,, k)
and
H200: [1’ F(JC)I/l1-G(x)], forx E[k, x,]

respectively. Given G RSIR F, since H, EU), 1] and H, e[0, 1], it is sufﬁcient to show
that H, is non-decreasing in x e[x,, k) and H, is non-increasing in x E[k, x,]. These

monotonicity conditions are equivalent to,
f(x)G(x) — g(x)F(x) 2 0, for all x e[x,, k) (5.1)
and
f(x)[1— G(x)] — g(x)[l — F(x)] 2 o, for all x e [k, x,], (5.2)

respectively. For the interval [k,, k), the condition (5.1) is satisﬁed because G RSIR F
implies g(x) S f (x) and G(x) 2 F (x), and for the interval [k, [(2], the condition (5.2) is
satisﬁed because G RSIR F implies g(x) S f (x) and l— G(x) 2 l - F (x). For the

intervals [x,, k,) and (k,, x,], the RSIR condition implies that

(i) if g(x) = 0, then f(x) = O

Itix)=f(x>/ g(x), forx elxz. k.)

(ii) if g(x) at 0, then {
112(x)=f(x)/ g(x), forx 6(k2, x3l°

Thus, for all values of x such that g(x) = 0, the conditions (5.1) and (5.2) are satisﬁed.
For ease (ii), the condition (5.1) and (5.2) can be written as,

h,(x)G(x) - F(x) 2 0, for all x e [x,, k,) such that g(x) at 0 (5.3)

and

100

h,(x)[l - G(x)]- [l — F(x)] 2 O, for all x e(k,, x,] such that g(x) = 0, (5.4)

respectively. Since h,(x,)G(x,)— F (x,) 2 0 and the condition h, non-decreasing implies

that the LHS of (5.3) is non-decreasing, i.e.,
€1h,(x)G(x) - F(x)]/ 0‘): = h,’G + h,g - f = h,'G 2 O,

the condition (5.3) is satisﬁed. Since h,(x,)[l - G(x,)] — [1 — F (x,)] 2 0 and the condition

h, non-increasing implies that the LHS of (5.4) is non-increasing, i.e.,
&{h,(x)[l — G(x)]-[1— F(x)]} / dc = h,'[l — G]- h,g+ f = h§[l - G] 2 0,

the condition (5.4) is satisﬁed. Hence the conditions (5.1) and (5.2) are satisﬁed for all

x e[x,, k,) and x e(k,, x,], respectively. This completes the proof. Q.E.D.

Next property gives the relationship that the RSIR order implies the L-RSIR order
and the L-‘RSIR order implies the L-ESIR order.

Property 5.2. G RSIR F :> G L-RSIR F 2:» G L-ESIR F.

Proof. According to Deﬁnition 5.1, 5.2 and 5.4, three types of shifts G RSIR F,
G L-RSIR F and G L-ESIR F require a common restriction that the CDF’s G and F cross
only once. Given such a single-crossing R-S increase in risk from F to G, let k be the
point at which the crossing occurs. Then the ﬁrst relationship that “if G RSIR F then G
L-RSIR F” is assured by comparing the restriction used in Deﬁnition 5.1 and 5.2. For the
interval [x,, k], both G RSIR F and G L-RSIR F require the same condition, but for the

interval (k, x,], the L-RSIR order requires only that G(x) S F (x) which is less stringent

than the conditions required for the RSIR order.

lOl

Consider the second relationship that “if G L-RSIR F then G L-ESIR F.” For the
interval (k, x,], both the orders L-RSIR and L-ESIR require a common restriction that

G(x) S F (x). For the interval [x,, k], the proof follows the same method used in the

proof of Property 5.1. Q.E.D.

Let’s deﬁne four subsets of R-S increases in risk, each being composed of pairs of

CDF’S (F, G), such as:

[2,, = {(F, G)| G RSIR F}
0,4,, = {(F, G)| G L- RSIR F}
{2,, = {(F, G)| G ESIR F}
(2,,, = {(F, G)| G L- ESIR F}.

Now we have, from Property 5.1 and 5.2, the following two formal relationships among

the subsets of R-S increases in risk:

(1) 0R5 C 011s C 01.453
(2) 0R8 C QL—RS C QL-l-JS'

‘ There is a notable thing before going into the next property. While both the MLR
and the MPR orders, discussed in section 3.2, have the property of transitivity, the RSIR
and the ESIR orders do not have this property. That is, the relationships G, ESIR G,
and G, ESIR G, do not necessarily imply that G, ESIR G, , and the same is applied for
the RSIR order. This implies that an R-S increase in risk that can be decomposed into a
series of RSIR or ESIR shifts is not always an RSIR or an ESIR shift. In Figure 5.11, we
give an example of this case. It shows that G ESIR (RSIR) G, and G, ESIR F, but it is
not true that G ESIR F because the CDF’s F and G cross three times.

In general, a series of RSIR (ESIR) shifts does not necessarily imply an RSIR

(ESIR) Shift. However, as a special case, if a series of ESIR shifts have a single common

102

 

 

 

x, x, k x, x4

Figure 5.11. G ESIR G, ESIR Fbut not G ESIR F.

crossing between every pair of CDF’s, the summation of all the shifts results in an ESIR.
That is, if a series of shifts G ESIR G, ESIR Gm, ESIR G" ESIR F, where all
CDF’s G, F, and G,'s for all i =1, m, n have a single common crossing, the shift from
F to G is an ESIR. The reason is as follows. Let k be the single point of crossing, then
the series of ESIR shifts implies that G 2 G, 2 2 G" 2 F for all points to the left-
side of k, and G S G, S S G" S F for all points to the right-side of k. Deﬁne a new
random variable y = -x and let f(x) = f(— y) = f( y), g(x) = g(—y) = g( y) and

g, (x) = g,(—y) = g, (y) for all i = 1, ---, n. Then, as we saw in section 5.1, every ESIR
shift can be understood as the sum of two specialized F SD shifts. The series of shifts are
expressed as: for the left-side of the point k, F MPR G" MPR G, MPR G, and for the
right-side of k, F MPR G" MPR G, MPR G. Since the MPR order is transitive
(Property 3.3), the shift from F to G can beunderstood as the sum of two FSD shifts such
that F MPR G for the left-side of k, and F MPR G for the right-side of k. These two
shifts sum up to imply a shift satisfying G ESIR F.

103

A special family of C DF ’5 holds the transitivity of the ESIR order. If the initial
and ﬁnal CDF’s F and G are restricted to be symmetric42 from the mean, every ESIR shift
from F to G has a common crossing at the point of mean and thus the ESIR order is
transitive. The family of symmetric distributions include normal, triangular and uniform
distributions.

While the RSIR and the ESIR orders, in general, do not display transitivity, the
non-transitivity gives important implications with regard to comparative static analysis
under uncertainty. As we have discussed in section 3.2, if a given CDF order is sufﬁcient
for making a general comparative static statement and the order is non-transitive, the
admissible set of changes in CDF includes every shift that can be decomposed into a
series of shifts generated by the given order. The CDF change given in Figure 5.11 is an
example of this case. That is, the comparative static result made for the set of ESIR
changes is also obtained for the shift given in the ﬁgure.

The above discussion leads to another implication of the non—transitivity with
regard to the comparative static analysis. Assume that, given two CDF orders, an
arbitrary change in CDF generated by one CDF order can always be decomposed into a
series of shifts that satisfy the other CDF order. Then, if the latter order gives a sufﬁcient
condition for a general comparative static result, then all the shifts that satisfy the former
order are also admissible for the result. Therefore, examining the presence of this kind of
relationship between CDF orders is meaningful for making a general comparative static
statement. In what follows, two such relationships are presented: one between the RSIR
and the ESIR orders, and the other between the L-RSIR and the L-ESIR orders which are
also non-transitive. They are: (i) “an ESIR shift can always be decomposed into a series
of RSIR shifts,” and (ii) “an L-ESIR shift can always be decomposed into a series of L-
RSIR shifts.”

 

42 A CDF F or a corresponding pdf f, with its mean f, is said to be ‘symmetric from the mean’ if, for any
numberc, F(f-c)= l — F(f+c) or f(f—c)=f(f+c).

104

Property 5.3. Any ESIR shift can be decomposed into a series of RSIR shifts,
that is, if G ESIR F, then there exists a series of CDF’s G, ,---, G" such that G RSIR G,

RSIR G RSIR G" RSIR F.

"-1

Proof. Given an arbitrary pair of pdf’ 5 f and g satisfying G ESIR F , let’s deﬁne a

function h:R —> [0, 00) as

“f(x) / g(x), when g(x) 2 0
h(x) = J the deﬁned last value off(x) / g(x), when g(x) = f(x) = 0

 

\00, when g(x) = 0 andf(x) :4 0.

Note that, according to Deﬁnition 5.3, there is a point k e [x,, x,] such that F (x) S G(x)
for all x e[x,, k) and F(x) 2 G(x) for all x E[k, x,], and there is no restriction on the
number of times of crossing between the pdf 5 f and g. This implies that the function h is
generally not monotone and thus there are points at which h changes from non-increasing

to non-decreasing. Assume that there are n(= n, + 11,) number of such points and denote

them as: for the left-side of the point k, c: for i, =1, 2,-~, n, where n, 20 and

I. I.
C», (Ch, -I

< ~--<c,"; and for the right-side of k, c: for i, =1, 2,-~-, n, where n, 2043 and

c” <c,R< w<c". For every 0" and c", deﬁne CDF’s G‘(x) and G."(x) as,
l - n, I, I, I, I,

,
I. I.
A,lG(x), when x < c,l

G,(‘(x) = < F(x), when cf; 3 x 3 df (5.5)

 

l— Aﬂl — G(x)], when x > d:

and

 

43 When n, = n, = 0, it is the case where the ESIR shift is also an RSIR shift.

105

"if; G(x), when x < .1,:

G,:‘(x) = < F(x), when d: S x S c: (5.6)

 

Ll — A: [l — G(x)], when x > cf,

where if; = F(c,f )/ G(cf ). if = [1 - F(df )]/ [1 — G(df )1, if; = F(dJ )/ G(dg) and
if: = [1 — F (cf: )]/ [1— G(c: )], and d: (2 k) and d,’;(S k) are the selected points such that
G,(‘(x) and G: (x) have the same mean as F or G, respectively. Since G ESIR F implies
that the order of n values of It: and A: is same as the order of the corresponding n values
of 2,: and if: , the order of n points c: and d: (left-side of k) is same as the reversed
order of the corresponding n points cJ‘ and d: (right-side of k). Let (sf, 3,”) for

i = 1, 2,- - -, n, be n pairs of points where sf<s,f’_,< m<s," are the ordered n values of

c: and d,’2‘, and .9," < sf < ---< sf are the ordered n values of c: and d,:, respectively.

Then the deﬁned n CDF’S can be rewritten, for each i-th pair of points (5,". s," ), as

,
AJ'G(x), when x < s,"

G,(x) =J F(x), when s,‘ S x S s,” (5.7)

 

k1_ ’l'rRll — G(x)], When x > s,”
where if. = F(sf)/ G(sf‘) and if = [1—F(s,R)]/[1—G(s,R)], and it, conesponding pdf is

,
xlf‘g(x), when x < s,"

g(x) = i f(x), when s,‘ S x S s," (5.8)

 

,xif‘g(x), when x > 5,".

Now all CDF’s F, G and GI ,- - C, have the same mean and have a common crossing at

the point k. This implies that, if we let G = Go and F = G

n+l’

i=1, 2,---, n+1, G,(x)SG,_,(x) for all x e[x,, k) and G,(x)2G,_,(x) for all

then for all

x E[k, x,,], and we have an order that G0 MPS G, MPS G, MPS GM. In addition,

106

=AR

n+1

set 11,; = x1?" =1, 21"

,I. _ R _ I. _ R
m, = 0, 5,, — 3,, — k, 5",, — x, and Sm, =x,, then from (5.8) we

see that, for every i=1, 2,---, n+1, g, is generally related to g,_, as

I(,1',‘ //if‘_, )g,_,(x), when x < s,"
(h / xif_,)g,_,(x)(= f(x)), when s," S x < s,’;,
g(x) = < g,_,(x)(= f(x)), when s,’;, S x S sf, (5.9)

(h / if..)g.-.(x><= f(x)). when slit < x s s"

I

 

K(2’: //t',‘_,)g,_,(x), when x > 5,”.

By the condition G ESIR F, if /Af‘_, <1 and It,” / xii, <1, for every i=1, 2,---, n +1.
From (5.7), the fact that G,(sf) = F (3,") and G,(x) S F (x) for all x e[x,, 3,") implies that
at every point s,”, g, meets f from the below, and the fact that G,(s," ) = F (5,” ) and

G,(x) S F (x) for all x e(s,", x,,] implies that at every point s,", g, meets f from the above.
Thus if? S h(sf‘) and [if S h(s,R). Now for each of the intervals [sfg s,"_,) and (3,5,, sf],
there is no point at which the function h changes from non-increasing to non-decreasing.
and (5.7) and (5.9) imply that g,(x) S g,_,(x) for all x e[x,, 3,") and (sf, x,], and

G,(s,’;,) = G,_,(s’;,) and G,(s’f, ) = G._,(s,’:, ). Accordingly, there exists a pair of points

k," e[s,", s,"_,) and k," e(s,’:,, s,"‘]44 such that g,(x)2 g,_,(x) for all x e[k,", k,"], and there
exist a non-decreasing function h,‘ :[x, , k,‘ ) —> [0, 1] and a non-increasing function
h,":(k,", x,] —> [0, I] such that g,(x) = h,"(x)g,_,(x) for all x e[x,, k,") and
g(x) = h,R(x)g,_,(x) for all x E(k,R, x,] where

 

4" If s," is one ofcf's and g,(sf)2 g,,,(sf’ ), or s," is one ofcf's and g,(s,") 2 g,_,(sf‘ ), then s," = k," or
s,” = k,R and from (5.7),

g,(x) = (if //I,l'_[ )g,_,(x), forx e[x,, 5,")

g,(x) 2 g,_,(x), forx E[sf, s,"]

g,(x) = (11.” / 15032.0). forx em". x.]
which imply G,_, RSIR G,. This is also the case where G,_, RSIR G“, , and more generally for every

." ), or s"

I+j 1+}

j=0. 1, m, if s.‘ . is one ofcf's and g,,,(SﬁﬂZgr—KS

If}

g,.,(s,r’i, ) 2 g,_,(s,.’:, ), then G,._I RSIR G

is one of of 's and

“n+1 '

107

{if //if_,, forx e[x,, 5,")
hrl'lx) =

h(x)/11.1,, forx e[s,", k,.")

l

and
h(x)/1,2,, forx e(k,", s,"]
h:R(x) =
17R / 1,6,, forx e(s,", x,],
Hence, by Deﬁnition 5.1, G,,, RSIR G, for all i=1, 2,”: n+1. Q.E.D.

Property 5.4. Any L-ESIR shift can be decomposed into a series of L-RSIR
shifts, that is, if F L-ESIR G, then there exists a series of CDF’s G, , - - . , G,, such that G L-

RSIR G, L-RSIR G L-RSIR G,, L-RSIR F.

"—1
Proof. For a given CDF change from F to G, if it satisﬁes the condition G L-
ESIR F, then there is only one crossing between the two CDF’S. As we noted, an L-ESIR
shift can be understood as the sum of two FSD shifts: an MPR dominated shift for the
left-side of the crossing point, and an FSD dominant shift for the right-side of the point.
This implies that the proof of the assertion that “any MPR shift can be decomposed into a

series of L-MLR shifts” (Property 3.6) can be used for the proof of Property of 5.4.

To illustrate the result in Property 5.3, consider an example of an R-S increase in
risk given in Figure 5.12. Since the pdf representation (while it does not provided) for
the shift shows more than two crossing between the pdf’ 3 f and g, it cannot be an RSIR
shift. Following the method given in the proof of Property 5.3, three intermediate CDF’s
G,'s, for i =1, 2, 3, are deﬁned. The shift from F to G is divided into four shifts, from F

to G,, ---, and from G, to G. By closely examining the shifts, we know that each shift

satisﬁes the RSIR condition in Deﬁnition 5.1. Thus the example shows that a shift from

108

 

 

mi: ------------ ; - T I 1 i

x,c" cl" d,‘ CIR deR
. I. . R

(53 ) (32) (51 ) (3| 52 33 )

 

x3:x4

Figure 5.12. G ESIR F that is the sum of four RSIR shifts.

G to F, satisfying G ESIR F, can be decomposed into four RSIR shifts such that:

G RSIR G, RSIR G, RSIR G, RSIR F.

These last two properties have important implications with regard to providing
general comparative static statements. By Property 5.3, the comparative static statement
made for the set of RSIR changes can be generalized to the set of ESIR changes. By
Property 5.4, the comparative static statement made for the set of L-RSIR shifts can be

generalized to the set of L-ESIR shifts. These are shown in chapter 6.

Chapter 6

COMPARATIVE STATIC RESULTS
WITH SUBSETS OF R-S INCREASES IN RISK

This chapter provides general comparative static statements regarding the subsets
of R-S increases in risk presented in chapter 5. The admissible set of R-S increases in
risk for the comparative static result in Black and Bulkley (1995) is extended to the set of
ESIR shifts which is larger than the set of RSIR shifts. This generalization is directly
obtained from the relationship that an ESIR shift can always be decomposed into a series
of RSIR shifts (Property 5.3), without any additional cost of assumptions. Another
determinate comparative static statement is made for the set of L-RSIR shifts. Restricting
the concerned payoff function to be linear in the random variable, we show that the effect
of an L-RSIR shift is determinable for all risk averse decision makers with non-negative
third derivative of utility functions. Another relationship established in chapter 5 gives a
further generalization of this result. Using the relationship that an L-ESIR shift can
always be decomposed into a series of L-RSIR shifts, the admissible set of R-S increases
in risk is extended to the set of L-ESIR shifts without imposing additional assumptions.

Section 6.1 gives some preliminary discussions about the results in Black and
Bulkley (1989), and Dionne, Eeckhoudt and Gollier (1993). Given the general decision
model (1.1), section 6.2 examines the properties regarding the marginal utility with
respect to the choice variable, measured at the Optimal level. These properties are

required for developing the comparative static results in section 6.3.

109

110

6.1 Literature Review and Preliminary Discussions

Rothschild and Stiglitz (1971) have developed a sufﬁcient condition for signing
the effect of an arbitrary R-S increase in risk on the optimal choice made by an economic
agent. Given the general one-argument decision problem (1.1), assume that the solution
of the problem satisﬁes the ﬁrst and the second-order conditions. Then the R-S condition
implies that the optimal value of the choice variable b does not increase (decrease) for
any R-S increase in risk if u’(z)z,, is concave (convex) in x, or the following expression is

signed as non-positive (non-negative):

I

u -z,,,,, + 2n" - z,r oz,“ + z,,(u" ~zu + 11'" ~23). (6.1)

However, as Kraus (1979) noted, the RS sufﬁcient condition cannot generally be
satisﬁed for the general decision problem (1.1). This is because the ﬁrst-order condition
that E [u’(z)z,,(x, b)] = 0 at the optimal level of b implies that 2,, must change its sign as
x varies and thus the sign of the last term in (6.1) is generally ambiguous. In addition,
since (6.1) includes the third derivative of the utility function, it is also obvious that the
effect of an arbitrary R-S increase in risk in the random variable on the choice variable
made by a risk averse agent is generally not determinable.

Instead, many authors have examined subsets of R-S increases in risk when
analyzing the choice behavior of an arbitrary risk averse decision maker. Examples
include the cases of a global increase in risk analyzed in Kraus (1979) and Katz (1981), a
mean-preserving truncation in Eeckhoudt and Hansen (1981), and a strong increase in
risk (SIR) in Meyer and Orrniston (1985). Being a subset of R-S increases in risk, SIR
shifts include all reverse cases of mean-preserving truncations which in turn include all
global increases in risk. As we saw in chapter 2, Meyer and Onniston generalize the

comparative static results developed in the other cited papers. Using the general decision

Ill

model (1.1), they show that an SIR in the random variable x decreases the choice variable
b made by an arbitrary risk averse decision maker.

A further generalization is given by Black and Bulkley (1989) who introduce the
concept of a relatively strong increase in risk (RSIR). As shown in Deﬁnition 5.1, RSIR
shifts include, as a special case, the set of SIR shifts. They prove that the same desirable

comparative static result holds for the set of RSIR shifts.

Theorem 6.1. For all risk averse decision makers, b, 2 b,, if

(a) G RSIR45 F

(b) 2, 2 0, 2,,, 2 0, 2,,, < 0 and 2,,“ S 0.

Accordingly, the admissible set of changes in risk for the comparative static result is
improved as: global changes in risk —> mean-preserving truncations -> strong increases in
risk —> relatively strong increases in risk.

All these results contain the same set of decision makers (all risk averse
individuals) and the same restrictions on the structure of the concerned decision model. as
the ones in condition (b) in Theorem 6.1. The structural restrictions on the payoff
function such as, 2,, 2 0, 2,,,r 2 0 and 2,,, S 0, are often assumed by many authors and
they are not very harmful for the comparative static analysis because many economic
problems already satisfy these restrictions. However, another common restriction that the
concerned payoff should be strictly concave in the choice variable (2,,, < O) is quite
restrictive. This excludes the decision models in which payoffs are linear in the choice

variable (2,,, = 0) such as the standard portfolio problem, the model of a competitive ﬁrm

 

‘5 An RSIR deﬁned in Deﬁnition 5.1 is slightly modiﬁed from the original one. Similarly, the proof of
Theorem 6.] needs a modiﬁcation of the original proof, substituting the pdf ratio function h for the ratio

g/f.

112

under price uncertainty with a constant marginal costs, and the coinsurance demand
model.

Many authors have assumed that, doing their comparative static analysis, the ﬁrst
and the second-order conditions are satisﬁed to guarantee an interior unique solution. Let
f(x) be a given pdf of the random variable x, then the ﬁrst-order condition which

determines the optimal value of b is,
J::ll’[z(x, b)Izh(-x’ b)f(x)dx = 0 (62)

The solution satisfying (6.2) is guaranteed to be a global optimum by the second-order

condition that,
ﬂuids or: + u'lztx.b)12..1f<x)dx < o. (6.3)

The condition (6.3) is saitisﬁed46 with the restriction 2,,, < 0 when risk aversion (u" S O),
or with the restriction 2,,, S 0 when strict risk aversion (u" < 0). Assume that an initial

optimal solution satisﬁes both the ﬁrst and second-order conditions. Then, to prove the

result that b, 2 b,, for a speciﬁed change in pdf from f to g, it is sufﬁcient to show that
db.) = J: u'iztx. b,.~)12.(x. b,.-)[f(x) — now 2 0. (6.4)

This is an often used technique to prove comparative static theorems. However, if the
initial solution does not satisfy the ﬁrst or the second-order condition, the method using
(6.4) is generally not enough for proving a comparative static result.

When 2,,, = 0, since 2,, does not depend on the choice variable b, there are cases
where an interior solution does not exist. If a decision maker is risk neutral, his choice
depends only on the expectation of 2,, and there is no interior unique solution. In this
case, both the ﬁrst and the second-order conditions are not satisﬁed. Given the

assumption 2,,, 2 0, if E(z,,) > (<)0 then b,, = +00 (—oo), and if E(z,,) = 0 then the,

 

4" The non-decreasing utility function is implicitly assumed to be increasing at least some .1: e [x,, x, J.

I|3

solution is indeterminate. For a strictly risk averse individual, E ( 2,,) S 0 implies
b,. = —oo, in which the solution satisfying (6.2) does not exist. Another problem caused
by the restriction 2,,, = 0 is that one cannot examine the effect of a change from no risk to
risky situation because there is no interior unique solution under certainty. Thus,
considering the decision models with payoffs being linear in the choice variable, the case
of an unbounded or a corner solution should be examined for making a general
comparative static statement.

Dionne, Eeckhoudt and Gollier (1993) raise this issue and show that, when the
concerned payoff is linear in the choice variable b, Black and Bulkley’s comparative
static result can be extended to include more shifts47 that cannot be generated by the

RSIR order. They consider a linear decision model of the form,48

z(x, b) = 2,, + b(x — c) " (6.5)

where 2,, and c are exogenous parameters. This is the case where the payoff function is
linear in both the choice and the random variable. The constant 2,, can be interpreted as
an initial endowment in the portfolio problem and the coinsurance problem, or a ﬁxed
cost or an initial wealth in the competitive ﬁrm model, and c represents either a sure
interest rate, a marginal cost of insurance, or a marginal cost of production. Restricting
the choice variable to be non-negative, this linear model satisﬁes the above structural

restrictions as, 2, = b 2 0, 2,,, =1 and 2,,,“ = z = 2,,, = 0.

In order to make a general comparative static statement, Dionne et. al. examine
the cases of an unbounded and a comer solution for the decision problem (6.5). Given an
initial pdff(x), every possible optimal choice of a risk averse decision maker is

summarized in Table 6.1. When E (x) > c, b, is strictly positive because the marginal

expected utility, E[u'(z)(x — 0)], is strictly positive at b = 0, and when E(x) < c, b, = 0

 

‘7 They introduce the concept of a ‘relatively weak increase in risk’ which is not related to our deﬁnitions
of R-S increases in risk. Thus this dissertation do not reproduce the deﬁnition of the concept.
‘8 This linear model is already shown in (4.10).

114

Table 6.1. Different solutions when b 2 0.

 

 

Cases u"(z) < 0 u"(z) = 0
i) c S x, < E (x) < x, unbounded solution unbounded solution
bl; = +00 b,, = +w
.. in ' ' '
11) x, < c < E(x) < x, tenor or unbounded solution unbounded solution
- - 0<bFS+oo b,,-=44”
comer solution undetermined
111) x, < c = E(x) < x,
bF = 0 b}; = 0
iv) x, < E (x) < c S x, comer solution comer solutlon
b, = 0 b,,. = 0
v) x < E (x) < x < c comer solution comer solution
2 3
b, = 0 b,,. = 0
. * unbounded solution unbounded solution
v1) c < x, = x,
' ‘ b, = +00 b,,. = +00
vii) x _ c _ x corner solution undetermined
2 — _ 3
b, = 0 b, = 0
viii) x _ x < c comer solution comer solution
2 - 3
b, = 0 b,. = 0

 

* In Dionne et. al., only the ﬁrst ﬁve cases are considered, but in order to examine the no risk to risk case,
as Meyer and Orrniston (1985) note, we add the case of initially certain.

115

because E[u'(z)(x — c)] S u'[z(c, b)][E(x) — c] < 0 for all positive b. Consider the case,
E(x) = c. If u" < 0, then b,. = 0 because E[u’(z)(x — c)] < u’[z(c, b)][E(x) — c] = 0 for
all positive b, and if u" = 0, then the solution is indeterminate because, for all possible b,
E[u’(z)(x — c)] = u'(z)[E(x) - c)] = 0. It is easy to see that when a corner or an
unbounded solution prevails, any R-S increase in risk will not increase the optimal level
of choice variable. First consider the risk neutral case. The solution depends only on the
expectation of the random variable x. If E(x) > (<) c then b, = +00 (0), and if E(x) = c
then the solution is indeterminate. Since an R-S increase in risk does not affect the mean,
E(x), it does not change the solution.49 Consider the case of strict risk aversion. When
b, = +00, the solution cannot increase any more, and when b,, = 0 due to E(x) S c, the
solution will not change because E(x) remains unchanged after an R-S increase in risk.
Thus, for both the cases of a comer or an unbounded solution, we obtain the
desired comparative static results that any R-S increase in risk does not increase the initial
optimal solution. Now, in order to make a general comparative static statement for a
particular type of R-S increase in risk, only the ease of an interior solution is left to be
considered. Since an interior solution is possible only when u"(z) < 0, the solution
satisﬁes the second-order condition and the solution satisfying the ﬁrst-order condition is
unique. This allows Dionne et. al. to use the sufficient condition (6.4) to prove their
comparative static result. With the above discussion, the result in Black and Bulkley’s
analysis can be applied for decision models with linear payoffs. That is, if the concerned
decision model with the structural restriction 2,,, = 0 follows the form (6.5) in which the

payoff is also linear in the random variable, Theorem (6.1) can be generalized as follows.

Theorem 6.1'. For all risk averse decision makers, b,.. 2 b,, if

(a) G RSIR F

 

‘9 This assertion is not valid for the case where the solution is indeterminate, but even in this case a risk
neutral individual has no reason to change his original choice after the change in risk.

ll6

(b) 2, 2 0, 2,,, 2 0, 2,,, S O, 2,,,u S 0, and z,“ = 0 when 2,,, = 0.

The types of changes in CDF such as, a global increase in risk, a mean-preserving
truncation, an SIR, or an RSIR, have a common restriction that the pdf 5 f and g cross
only twice50 and the comparative static results developed for these shifts contain the same
set of decision makers (all risk averse individuals). In this chapter, we extend the set of
admissible R-S increases in risk allowing the desirable comparative static properties.
First, the results in Theorem 6.1' is generalized for the set of ESIR shifts in which the
pdf’ 5 f and g are not restricted to cross only twice. Another extension is given by using a
trade-off between the restrictions on the set of decision makers and the set of changes in
CDF. If a risk averse individual is restricted to have a utility function with its non-
negative third derivative, a desirable comparative static statement is obtained for the set
of L-RSIR and this result is further generalized to the set of L-ESIR shifts.

Before going into the main body of this chapter, the next section examines several
properties regarding the marginal utility with respect to the choice variable b, u'(z)z,,.

They are used to develop a comparative static result in section 6.3.

6.2 Properties of the Function u'(z)z,,

Given a CDF F (x), this section investigates several properties regarding the
function u'[z(x, b, )]z,(x, b,.) which is the marginal utility of b, measured at the optimal
level b,,. The properties are derived from the structural restrictions on the payoff
function such as, 2,, 2 0, 2,,, 2 O and/or z,” S 0, and the assumptions on utility function
such as u’ 2 0, u" S 0, and/or u’" 2 0. Some of the properties have already been used to

prove the comparative static results obtained in the literature. Others are required for the

 

50 For the case of a global change in risk, the two crossings occur at the point of mean.

117

results in section 6.3. In this section, assume that the support of the CDF F and its
corresponding pdf f lie in a bounded interval [0, l], and let b,. be the interior bounded

solution satisfying the ﬁrst and the second-order conditions for the maximum problem

(1.1). The condition 2,,, 2 0 implies that 2,,(x, b,.) has a single crossing of the x axis,

changing its sign from negative to positive, and if we let x' be the value of x satisfying

z,,(x, b,,.) = 0 then x’ e [0, l]. The ﬁrst two properties are easily derived and often

encountered in the previous comparative static analysis.

Property 6.1. Assume that u’ 2 0 and 2,,, 2 0, then for all y e [0, 1],
(i) J'u'lztx. b.)1z.<x. br)f(x)dx s 0 and (ii) J u'[z(x. b.)1z.(x, b,.-)f(X)dx 2 0.

Proof. For a given CDF F (x), the ﬁrst-order condition for the problem (1.1)

implies that
Ju'(z>z.(x, br)f(x)dx = 0.

and thus the assumptions u’ 2 0 and 2,,, 2 0 complete the proof. Q.E.D.

" Property 6.2. Assume that u' 2 O and z,,,, 2 0, then (i) u'(z)z,, S 0 for all
x E[O, x'], and (ii) u'(z)z,, 2 O for all x E[x', 1].

Proof. Since the assumption 2,,, 2 0 implies that 2,, S 0 when x S x', and 2,, 2 0

when x 2 x', the assumption u’ 2 0 completes the proof. Q.E.D.

Property 6.3. Assume that u’ 2 0, u” S 0, z, 2 0, 2,,,r 2 0 and 2,,.“ S 0, then (i) for
the interval [0, x'], u'(z)z,, is non-positive, non-decreasing in x and its derivative with
respect to x is minimized at x = x' ; and (ii) for the interval [x', l], u'(z)z,, is non--

negative and its derivative with respect to x is maximized at x = x’.

118

Proof. By Property 6.2, u'(z)z,, S 0 when x S x', and u'(z)z,, 2 0 when x 2 x'.

The derivative of u'(z)z,, with respect to x is
u"(z)zxz,, + u'(z)z,,,. (6.6)

(1) When x S x': The given assumptions imply that both terms in (6.6) are non-

negative and thus u'(z)z,, is non-decreasing in x. Also we know that the ﬁrst term is zero
at x = x’ and the second term is non-increasing in x. This implies that (6.6) has its
minimum value at x = x' within the interval [0, x'].

(ii) When x 2 x’: The given assumptions imply that the ﬁrst term in (6.6) is
always non-positive and the second term is non-increasing in x. Thus (6.6) has its

maximum at x = x' within the interval [x', l]. _Q.E.D.

A portion of the assertion in Property 6.3 can be restated as,

a[(2)211
&

2 o'lt'(z)z,,
dc

2 étt'(z)z,,
dc

 

 

 

xSx' x=x° .t2.t'

Property 6.4. Assume that u' 2 0, u" S 0, z, 2 0, 2,,, 2 O and Z... = 0, then

G'lt'(z)z,, > u’(z)z,,

 

 

' _ , f all 0, '
(1) 03: x—x' or xe[ x)
(ii) at: (")2” S u (2)2.” , for allx E(x’, 1].

dc x—x

Proof. Since we assume that z is linear in the random variable x, that is 2,, = 0,

the payoff function can be expressed as
z(x, b) = r,(b)x + r,(b)

where r, and r, are arbitrary functions of b, satisfying z,r = r,(b) 2 0 and 2,, = r,'(b) 2 0.

119

With the linearity, we have x’ = —r,'(b,.- ) / r,(b, ). 5' This implies that, for all x a x',

z, =,r,’(b. )x+r,’,(b )_ r,'(b, )x+r,'(b,. )
x—x' x- x x+[r,',(b )/r,'(b,,. )]

 

 

=l(bl-' ):’ Zhr’ (67)

and thus.

cit'(z)z,,
dc

         

= u"(z)z,z,, + u'(z)z,, = u"(z)z,z,, (6.8)

The fact that u”(z)z,z,, is non-negative for the interval [0, x') and non-positive for the

interval (x’, 1] completes the proof. Q.E.D.

The term

 

u’(z)z,, J: u'[z(x, b,,.)]z,,(x, b,)-u’[z(x', b,..)]z,,(x°, b,)] is the

x — x x — x'
average change rate of the marginal utility u'(z)z,, from any x to the point x'. Property

6.4 is same as the assertion that the average change rate is non-increasing in x, because

the restriction 2,,, = 0 implies that (6.7) and thus,

 

(Jug?) /ar= 0'14(z)z,,,/éc<0, forallxe[0, l]andx¢x.

Property 6.5. Assume that u' 2 0, u" S 0, z, 2 0, 2,,, 2 0 and z,“ = 0, then (i) if
u'" 2 0, u’(z)z,, is non-positive, non-decreasing and concave in x e [0, x’]; and (ii) if

u'" S 0, u’(z)z,, is non-negative and concave in x e [x’, 1].

Proof. Form Property 6.2 and 6.3, we know that u’(z)z,, is non-positive and non-
decreasing in x e [0, x’], and non-negative for x e [x‘, I]. Since 2 is assumed to be

linear in x, differentiating (6.6) with respect to x gives

u"'(z)z_fz,, + 2u"(z)z,z,,x. (6.9)

 

5' Since 2,,(x, b,,) = r,'(bF )x + r,'(b,.) is assumed to have a single crossing at x' e [0, 1], r,'(b,,) at 0.

120

Hence with the given assumptions, (1) for the interval [0, x'], if u'" 2 0 then (6.9) is non-
positive and thus u’(z)z,, is concave; and for the interval [x’, 1], if u'" S 0 then (6.9) is

non-positive and thus u'(z)z,, is concave. Q.E.D.

6.3 Comparative Static Analysis

Using the general one-argument decision model (1.1), this section provides
general comparative static statements concerning the three types of R-S increases in risk,
an ESIR, an L-RSIR, and an L-ESIR, presented in chapter 5. Again, let the supports of
CDF’s G and F (assuming that G is riskier than F in the R-S sense) be the ﬁnite intervals
[x,, x,,] and [x,, x,], where x, S x, S x, S x,, respectively. First, consider the set of
ESIR shifts deﬁned in Deﬁnition 5.3. This class of R-S increases in risk includes, as a
special case, the set of RSIR shifts (Property 5.1). Since every ESIR change can be
decomposed into a sequence of RSIR shifts (Property 5.3), the result in Theorem 6.1’ is

generalized as:

Theorem 6.2. For all risk averse decision makers, b,,. 2 b,,. if

(a) G ESIR F

(b) zI 2 O, 2,,,r 2 O, 2,,, S O, 2,,,“ S 0, and z,“ = 0 when 2,,, = 0.

Proof. By Property 5.3, if G ESIR F then there exists a series of CDF’s
G,,---,G,, such that G RSIR G, RSIR G,, RSIR F. Combined with the result in

Theorem 6.1', it completes the proof. Q.E.D.

Theorem 6.2 improves the robustness of Black and Bulkley’s results without any

cost of additional assumptions. Deﬁning an ESIR, the restriction that the pdf’s f and g

cross only twice is dropped. Instead, it requires that the pair of CDF’s F and G cross only
once and that, for the left side of the crossing point, the ratio of F to G should be non-
decreasing and, for the right side of the point, the ratio of (l — F) to (l — G) should be
non-increasing.

One notable thing is that, as we saw in section 5.2, the ESIR order does not have
the property of transitivity. Consider the example of an R-S increase in risk given in
Figure 5.11, which shows that G ESIR G, and G, ESIR F, but not G ESIR F because the
CDF’s F and G cross three times. However, by Theorem 6.2, we know that
b,, S b,,l S b,. In general, it is obvious that an R-S increase in risk that can be
decomposed into a series of ESIR shifts allows one to make the desired comparative
static statement provided in Theorem 6.2. This implies that the ESIR order is only a
sufﬁcient (but not necessary) condition for obtaining the result in Theorem 6.2. Allowing
multiple crossing between the initial and the ﬁnal CDF’s F and G, it is an open question
to ﬁnd the set of admissible R-S increases in risk yielding the comparative static result.

In. order to make another general comparative static statement, we investigate a
trade-off between the restrictions on the set of decision makers and the set of changes in
the random variable. In what follows, the concerned payoff is restricted to be linear in
the random variable. We show that, when 2,,, = 0, one can further extend the class of
admissible R-S increases in risk with the cost of an additional restriction on the set of
decision makers. In the next comparative static result, the set of L-RSIR shifts which
includes all RSIR shifts is admissible by adding the restriction that risk averse decision
makers have positive third derivative of utility functions. When 2,,, = 0, we have shown
in section 6.1 that, for the cases of a comer or an unbounded solution, an arbitrary R-S
increase in risk will not increase the choice variable b. In the following theorem, we
consider only the case of an interior solution, and thus, to prove that b,,. 2 b,, for a

speciﬁed change in pdf from f to g, it is sufﬁcient to show that

W.) = J: u'iztx, b.)1z.<x. b,.)iftx1— g(x)ldx 2 o. (6.4)

Theorem 6.3. For all decision makers with utility functions u’s such that u’ 2 0,
u" S O and u'" 20, b,, 2 b,, if
(a) G L-RSIR F

(b) 2:,r 2 0, 2,,r 2 0, 2,,, S 0 and 2,,, = 0.

Proof. Let x’ be the value of x satisfying 2,,(x, b,.) = 0, then x’ exists in the
interval [x,, x,]. With the points k' and k used in Deﬁnition 5.2, where x, S kI S k S x,,
we consider the following three cases:

Case (1): when k S x’ S x,. Integrating by parts, Q(b,.) in (6.4) can be written as
Q(b,.) = J‘ [u"(z )zxz, + u'(z)z,,](G - F)dx

With the given assumptions, we know that for the interval [x,, x'], u"(z)z,z,, + u'(z)z,, is
non-increasing in x by Property 6.5, and for the interval [x’, x ,J, it has its maximum at

x = x’ by Property 6.3. Since k S x', this implies

 

 

[u"(z)z,z,, + u'(z)z,,,]

 

It 2- [“"(Z)Z.rzb ‘I' “'(Z)Zhrl l 2 [u"(z)zrzh '1‘ u'(Z)Z,,,]

.rS .t= .rzk’

and by the condition G L-RSIR F which implies G — F 2 0 for all x e [x, , k], and
G — F S 0 for all x e [k, x,], we have the following inequality,

 

Q02.) 2 iu"(z)z.z. + u'(z)zr,.l,=,, J ‘ (G — F>dx = 0.
Case (ii): when k’ S x' S k. Let’s rewrite Q(b,,.) in (6.4) as

Q(br.-) = J "u’(2)2.(f - g1dx + J. u'(z)zt(f — g1dx + J ‘ u'(z)zt(f - g)dx.

With the given assumptions and by the condition G L-RSIR F,

Q(b,.~) 2 u’[Z(k’.b,.-)1J.zt(f - g)dx +u’[Z(k.b,. )1 J. Z.(f -g)dx + J u'(:):,,( f - g)dx.
Adding and subtracting u’[z(k, b,)] J .z,( f - g)dx in the RHS gives.
ch.) 2 {u'[z(k', b,. )1 - u'[z(k, b,.- )1} J .2,,(f - g)dx
+ u'[z(k, b,.. )1 J z,( f — g)dx + J ‘ u'(z)z,,( f — g)dx. (6.10)
Integrating by parts,

Jz,(f—g)rtr = Jz,,(G— F)dt =z,,,, J(G— F)dt 20
because 2,,, does not depend on x and non-negative, and J (G — F )dt 2 0 for all

x e [x, , x,,]. Thus, by the assumptions u" S 0 and z, 2 0, the ﬁrst term in (6.10) is non-

negative. Integrating the second term in (6.10) by parts and using the assumption

2,, = 0, we have

u’[z(k,b,. )] J 2,,,(G - F)dx = u’[z(k,b,, )]sz J (G - F)dx. (6.11)
Also using integrating by parts, the third term in (6.10) is equal to
J ‘ lu"(z)z.z. + u'tz)z..1(G — F)dx.

From Property 6.4, it is known that

0‘14’(z)z,, < u'(z)z,,
03: x - x'

, for allx E(x', co)

and since u'(z)z,, / (x — x') is non-increasing for all x E(x', 00), we have

u'[z(k, b, )]za(k, b,,.)

_x.

 

I, [“"(lerzt + u’(Z)zt.l(G - F W 2 J (G - F)dx. (6.12)

From (6.7), it is shown that the linearity assumption 2,,, = 0 implies, for all x ¢ x’,

zh

 

, =z,,,,. (6.13)

x—x

Hence, from (6.11), (6.12) and (6.13),
u’[z(k,b,.)] J 2,,(f— g)dx + J l/(z)z,,(f— g)dx 2 t/[z(lr,b,,)]z,,r J ‘ (G— F)dr = 0.. (6.14)
Therefore we have the desired result that
db.) 2 {u'iztx,,b,.)1— u'[z(hb. )1} J2.(f - gm 2 0.

Case (iii): when x, S x' S k' . Consider the sign of the expression J z,,( f — g)dx.

First, assume that J 'z,( f — g)dx 2 o. Rewriting Q(b,,) in (6.4) as
th.) = J ’u'(2)z,,(f — g)dx + J u’(z)z.(f — g1dx + J" u'(2)z.(f - g)dx.
Using the given assumptions and the L-RSIR condition, we have
9G.) 2 u'[z(x‘.b.)1Jz.(f—g)dx+wizthbalf, z.(f—g>asc + J‘ u’(z)z,,(f-g)cbc.
Adding and subtracting u'[z(k, b,.. )1 J Iz,,( f — g)dx in the RHS gives.
db.) 2 {u'lztx'. (3.11-u'[z(k, b,.. )1} J 'z.(f— g1dx
+ u’(z(k, b,)) J 2,,( f — g)dx + J ‘ u’(z)z,,( f — g)dx. (6.15)

Thus (6.14) and the assumption J z,,( f - g)dx 2 0 imply Q(b,,.) 2 0. Now assuming

that J Iz,( f — g)dx s 0, let’s rewrite Q(b,) in (6.4) as
Q(br)—>— Jili(2)zt(f-g)dX+11[Z(k,b,.)lJ21(f-g)dX+ J‘u'tz)z.(f—g)dx.
Adding and subtracting u’[z(k, b,. )1 J Iz,( f - g)dx in the RHS gives,
th.) 2 J [u’(z)z.(f - g14x — u'lzthb. )1 J I2.(f — g1dx

+ mac. 12.1)]: z.(f — g14x + J ‘ u'(2)zt(f - g)dx.

125

I

From (6.14) and the assumption J z,,(f — g)ch .<_ O,

Q(b/-‘) 2- Lu'(z)zh(f — g)dx = ‘J: “(3)236ng + J: u'(3)zh(f " g)dx- (6-16)

The ﬁrst term in the RHS of (6.16) is non-negative, and since, according to Deﬁnition

5.2, the function h is non-decreasing and strictly positive for all x 6 [x2, k’ ]. the second
term can be written as

I , I I w 1 1 I ' w l.
J: u (z)zr(f-g)dx = J2u(2)zh(1—-}1-)fdx 2 [l —m]£ u (déhfdx.

Since h(x') $1 and Ju’(z)z,,gdx s O by Property 6.1, Q(b,.) 2 0. Q.E.D.

Adding the structural restriction Zn = 0 is not very harmful for comparative static
analysis because all the economic decision problems introduced in chapter 1 follow the
restriction. Compared with the comparative static result in Black and Bulkley’s analysis,
the result in Theorem 6.3 is an example of a trade-off between the restrictions on the set
of decision makers and the set of admissible set of changes in distribution. Our result
extends the class of admissible R-S increases in risk with the cost of adding additional
restriction on the risk preferences of decision makers, such as u'" 2 0. Following the
economic literature about the analysis under uncertainty, this set of decision makers
includes utility functions representing quite plausible preferences, such as those
exhibiting decreasing absolute risk aversion (DARA) which is a widely accepted property
of a risk averse decision maker. -

As the last result developed in this section, the admissible R-S increases in risk for
the comparative static result in Theorem 6.3 is extended to the set of L-ESIR shifts which
is larger than the set of L-RSIR shifts (Property 5.2). PrOperty 5.4 gives an easy and

direct generalization of the result in Theorem 6.3.

126

Theorem 6.4. For all decision makers with utility functions u's such that u’ 2 0.
u" S 0 and u’" 2 O, b,,. 2 (I. if
(a) G L-ESIR F

(b) 2, 2. O, z,” 2 O, z,,,, _<. O and zJr = 0.

Proof. It is clear by Theorem 6.3 and Property 5.4.

Accordingly, Theorem 6.4 improves the robustness of the result in Theorem 6.3,
and the efﬁciency gain is obtained from extending the set of admissible R-S increases in
risk yielding the result, without any cost of additional assumptions. With the restriction
of linearity in the payoff function, the result also shows a trade-off between the
restrictions on the set of decision makers and the set of admissible set of changes in
distribution. Compared with the result in Theorem 6.2, it is known that the result in
Theorem 6.4 contains a larger set of changes in distribution and a smaller set of decision
makers. This is another example that supports the general statement made in chapter 1
that, when relatively strong restrictions are imposed on one of the components composing
a decision model, the derived comparative static results are usually associated with
relatively weak restrictions on the other components.

In this chapter, comparative static theorems for subsets of the set of R-S increases
in risk are developed. Following the ratio approach, all the subsets are deﬁned by
imposing restrictions on the ratio between a pair of pdf 5 or between a pair of CDF’s.
Those developed comparative static results are associated with relatively weak
restrictions on the risk preferences of decision makers, i.e., the set of all risk averse agents
or the set of risk averse agents with non-negative third derivative of utility function. In
this sense, we refer all the subsets CDF changes concerned in this chapter to “R-S

increases in risk in the strong sense.”

Chapter 7

SUMMARY AND SOME REMARKS FOR
COMPARATIVE STATICS WITH SUBSETS OF SSD SHIFTS

In this dissertation, the analysis of comparative statics under uncertainty is
continued. Using the general one-argument decision model of the form (1.1), general
comparative static statements are made for the subsets of FSD changes presented in
chapter 3, and for the subsets of R-S increases in risk given in chapter 5. As a special
feature in deﬁning these subsets of changes in distribution, the ratio between an initial
and a ﬁnal pdt‘s or CDF’s is often restricted to be monotonic increasing or decreasing.
This chapter shows that the results obtained in previous chapters can be applied for cases
of second-degree stochastic dominance (SSD) shifts. Following the ratio approach,
special subsets of general SSD changes are deﬁned and general comparative static
statements are made for these subsets. We start this chapter by giving a summary for the

results developed in the previous chapters.

7.1 Summary

Within the context of ratio approach, most subsets of CDF changes concerned in
this dissertation are deﬁned by imposing restrictions on the ratio between an initial and a
ﬁnal pdf’s or CDF’s. They include six subsets of FSD shifts and four subsets of R—S

increases in risk. The general comparative static statement derived for each of these

127

subsets contains a relatively large class of decision makers, such as ‘all individuals with
non-decreasing utility functions,’ ‘all risk averse agents,’ or ‘all risk averse agents with
non-negative third derivative of utility functions.’ In sub-section 7.1.1, the results
provided in chapters 3 and 5 are summarized and the comparative static results developed

in chapters 4 and 6 are summarized in sub-section 7.1.2.

7.1.1 Relationships among Subsets of CDF Changes
First, consider the six types of FSD changes deﬁned in section 3.]. Each
deﬁnition speciﬁes a particular type of shiﬁ from a CDF G(x) to another F (x) or,

correspondingly, from a pdf g(x) to another f(x) such as:

- F MLR G (Deﬁnition 3.1): the ratio of f to g is monotonic increasing.

- F L-MLR G (Deﬁnition 3.3): decreasing proportion of the' probability
mass on points to the leﬁ of a given point is transferred to points to the
right of that point.

- F R-MLR G (Deﬁnition 3.4): probability mass on points to the left of a
given point is transferred to points to the right of that point, requiring the
ratio of f to g to be monotonic increasing for the right-side of the point.

- F MPR G (Deﬁnition 3.2): the ratio of F to G is monotonic increasing.

- F k-FSD G (Deﬁnition 3.5): given a point k, f(x) .>. g(x) for all x .>. k.

~ F k-MPR G (Deﬁnition 3.6): given a point k, the ratio of F to G is
monotonic increasing for x 2 k, and the ratio of F to G is maximized at k

for x .<. k.

Section 3.2 examines several relationships and properties among these subsets.

As shown in Figure 7.1, basic relationships among the subsets are:

- MLR :> L-MLR 2 MPR :> c-MPR => FSD (Property 3.1).

 

J Monotone Likelihood Ratio (MLR) Shifts t
J Landsberger and Meilijson (1990) J

\ lRight-Side Monotone Likelihood

J Ratio (R-MLR) Shifts

 

 

 

 

l 1
J Left-Side Monotone Likelihood Ratio (L-MLR) Shifts J

 

An MPR shiﬁ can be decomposed pdf’s cross only once I
into a series of L-MLR shiﬁs.

 

 

Jr

 

J Monotone Probability Ratio (MPR) Shifts
J Eeckhoudt and Gollier (1995)

 

 

 

J 1
J MPR with respect to a point k (k—MPR) J

 

 

A k-MPR shiﬁ can be decomposed J , . 3
into two shifts: one k-FSD and the . FSD Wlth respect to a 130mt k

other k-MPR. / J (k—FSD)

l . . . .
; First-Degree Stochastic Dominance (FSD) Shifts
Quirk and Saposnik (1962), Schneiweiss (1969),
Hadar and Russell (1969), and Hanoch and Levy (1969)

 

 

 

 

 

Figure 7.1. Relationships among subsets of FSD shifts.

- MLR => R-MLR => FSD.
. k—FSD :> FSD.

Regarding the property of transitivity, it is shown that each of the four C DF

orders, MLR, MPR, k-F SD, and k-MPR are transitive:

- The MLR order is transitive (Property 3.2).

- The MPR order is transitive (Property 3.3).

- The k-FSD order is transitive (Property 3.4).
- The k—MPR order is transitive (Property 3.5).

Transitivity does not hold for the L-MLR order. Instead, since an L-MLR shift is
also an MPR shift, a series of L-MLR shifts leads to an MPR shift. In addition to this, an

important relationship exists between these two CDF orders.

- An MPR shift can always be decomposed into a series of L-MLR shifts
(Property 3.6).

This allows the comparative statics for the set of L-MLR shifts to be also admissible for
the set of MPR shifts.
As the last property concerned in section 3.2, a relationship exists between the k-

MPR and k—F SD orders.

- A k-MPR shift can always be decomposed into an MPR and a k-F SD
shifts (Property 3.7).

Given the general comparative static theorems for the set of MPR and for the set of k-
FSD shifts, the last two properties extend the allowable set of changes in CDF without

any additional assumption.

131

Next, consider the four types of R-S increases in risk deﬁned in section 5.1. Each
deﬁnition speciﬁes a particular type of shift from a CDF F (x) to another G(x) or,

correspondingly, from a pdfﬂx) to another g(x) such as:

- G RSIR F (Deﬁnition 5.1): a probability mass from a centric interval is
moved to both tails, requiring the ratio of f to g to be monotonic
increasing for the left-tail interval and monotonic decreasing for the
right-tail interval.

- G L-RSIR F (Deﬁnition 5.2): requires only the left-side monotonicity
restriction on the pdf ratio in an RSIR.

- G ESIR F (Deﬁnition 5.3): when G 2 F, the ratio of F to G is monotonic
increasing, and when GS F, the ratio of 1— F to l— G is monotonic
decreasing.

- G L-ESIR F (Deﬁnition 5.4): requires only the left-side monotonicity

restriction on the CDF ratio in an ESIR.

Section 5.2 gives formal proofs for the basic relationships among these subsets,

and they are given as:

- RSIR => ESIR :> L-ESIR 2 MPS (Property 5.1).
- RSIR :> L—RSIR :> L-ESIR :> MPS (Property 5.2).

Figure 7.2 shows these relationships including the concepts of a ‘global increase in risk,”
a ‘mean-preserving truncation,’ and a ‘strong increase in risk.’

In general, the above four CDF orders do not have the property of transitivity.
However there are important relationships among the orders and section 5.2 also provides
the following two properties which play the key role for developing general comparative

static statements in chapter 6.

132

 

I

Global Increases in Risk J
LSandmo (1971), Ishii (1977), Kraus (1979) and Katz (1981) l

l

Mean-Preserving Truncations ;
(Deﬁning R-S decreases in risk): Eeckhoudt and Hansen (1980) J

l

J Strong Increases in Risk (SIR)
J Meyer and Onniston (1985)

l

J Relatively Strong Increases in Risk (RSIR)
J Black and Bulkley (1989)

An ESIR can be decomposed .
into a series of RSIR shifts / \ pdfg cross only tW1ce T

l . . Left-Side Relatively Strong Increases in i
?E t d d St I R1 k ESIR
J x en e rong ncreases in s ( ) Risk (L-RSIR) .

An L-ESIR can be decomposed
into a series of L-RSIR shifts.

‘I

Left-Side Extended Strong Increases in Risk (L-ESIR) J

 

ﬂ

J
l
J
l

 

 

 

 

 

 

 

 

 

 

 

l

 

 

 

J CDF’s cross only once I

J Mean Preserving Spreads (MPS )
Rothschild and Stiglitz (1970)

 

 

Figure 7.2. Relationships among subsets of R-S increases in risk.

133

- An ESIR can always be decomposed into a series of RSIR (Property 5.6).
- An L-ESIR can always be decomposed into a series of L-RSIR

(Property 5.7).

With these properties, the general comparative static statement made for the set of RSIR
(L-RSIR) shifts is generalized into the set of all ESIR (L-ESIR) shifts, without further

assumptions required.

7.1.2 Comparative Static Analysis

Regarding the subsets of FSD shifts, the comparative static results developed in
chapter 4 are summarized in Table 7.1. Each of the results contains restrictions on the
following three components: (1) the set of changes in randomness, (2) the set of decision
makers, and (3) the structure of the concerned decision model. It is obvious from the
table that, as the restrictions on the type of CDF change become less demanding, the
required restrictions on the risk preferences of decision makers and the structure of the

concerned decision model are more demanding.

- Result (i) in Theorem 4.1 contains all MLR shifts, all the individuals with
non-decreasing utility functions, and can be applied for decision models
with linear or non-linear payoff functions — a generalization of the

comparative static result in Landsberger and Meilijson (1990).

- Result (ii) in Theorem 4.2 contains, compared with the result (i), the
same set of decision makers and a larger set of changes in CDF, and adds
a structural restriction that the payoff is linear in the choice variable b - a

trade-off between the restrictions on the components (1) and (3).

- Result (iii) in Theorem 4.3 contains, compared with the result (i), a larger

set of changes in CDF and a smaller set of decision makers, and can be

Table 7.1. Comparative static results regarding subsets of F SD shifts. ’

 

 

Types of shifts between CDF’s F and G Sufﬁcient conditions for b“. _<. b,,.
i) FMLRG u'_>.0,z,,,20
.. FL-MLR G with k E[C, x,] u’ 2 O, z,u 2 O
11) FR-MLR G with k E[xz, c] z is linear in b and 2,,(c, b) = 0
u'ZO, u"SO
111) FL-MLRG 2 >0 2 >0
_ , h, _
2 O,u "s 0,
iv) F MPR G ’2‘ > 02 > o
v) Fc-FSD G u' 2 O, u" _<. O, z, 2 O, 2,,, 2 O,
F C'MPR G 2 is linear in b and 2,,(c, b) = O

 

applied for decision models with linear or non-linear payoff functions — a

trade-off between the restrictions on the components ( I) and (2).

- Result (iv) in Theorem 4.4 contains, compared with the result (iii), a
larger set of changes in CDF and the same restrictions on the other
components — a generalization of the result (iii), which is directly

obtained from Property 3.6.

- Result (v) in Theorem 4.5 contains, compared with the result (iv), the
same set of decision makers and a larger set of changes in CDF, and adds
a structural restriction that the concerned payoff is linear in the choice
variable - a trade-off between the restrictions on the components ( l) and

(3), which is obtained from Property 3.7.

135

The comparative static results developed in chapter 6 are summarized in Table
7.2. These results concern the subsets of R-S increases in risk, and as in the cases of
subsets of FSD shifts, each of the results contains restrictions on the three components
(1), (2). and (3). The ﬁrst four52 comparative static results are from the literature and the
last three are provided in chapter 6. As the table shows, all the results (i) - (v) contain the
same restrictions on the components (2) and (3). Only the set of admissible changes in
CDF is generalized in the order: global increases in risk —) mean-preserving truncations
—-) strong increases in risk -> relatively strong increases in risk -> extended strong
increases in risk. The restrictions contained in the comparative static results (iv) - (vii)

are compared as:

- Result (iv) in Theorem 61' contains all RSIR changes, all risk-averse
agents, and can be applied for decision models with linear or non-linear

payoff functions — a generalization of the results (i) - (iii).

- Result (v) in Theorem 6.2 contains, compared with the result (iv), a
larger set of changes in CDF and the same restrictions on the other
components — a generalization of the result (iv), which is directly

obtained from Property 5.3.

- Result (vi) in Theorem 6.3 contains, compared with the result (iv), a
larger set of changes in CDF and a smaller set of decision makers, and
adds a structural restriction that the concerned payoff is linear in the
random variable — a trade-off between the restrictions on the component

(1), and the components (2) and (3).

 

’2 The previous papers producing the results (i) - (iv) are: Sandmo (1971), Ishii (I977), Kraus (1979), and
Katz (1981) for the result (i); Eeckhoudt and Hansen (1980) for the result (ii); Meyer and Onniston (1985)
for the result (iii); and Black and Bulkley (1989).

136

Table 7.2. Comparative static results regarding subsets of R-S increases in risk.

 

 

Types of shifts between CDF’s F and G Sufﬁcient conditions for b“. S b,,.
. Global changes in risk from u' 2 0, u" g 0,
') a certainty (F) to a risky G zx 2 0, z,” 2 0, 2M 3 0, * gm 3 0
.. Mean-Preserving Truncations u' _>_ 0, u" g 0,
11) from G to F 2! 2 0, z,“ 2 0, 2b,, S O, 2,,” S 0

u'ZO, u"SO,
111) G SIR F

zx 20, 2,,, 20, z,,,, 50. z,”ur SO
. GRIR u'20,u"SO,
1V) . S F 2, 2 0, 2,,, 2 0, 2,,, s o, 2,,” s 0
G ESIR F u, 2 0’ u" S 0’
V) 2, 2 0, 2,,, 2 0, 2,,, s 0, 2,,“ s. 0
. u’ZO, u"SO, u"'20
V’) G L'RSIR F 2, 2 0, Zn = 0, 2,,,r 2 O, z,,,, _<_ O
.. u'ZO, u"SO, u"'20
v11) G L-ESIR F

2, 20,2M =0, 2,,, 20,z,,,, SO

 

* In Table 7.2, when the concerned payoff is linear in the choice variable, the decision model is assumed to
follow the form (6.5), that is, 2:: = 0 when z,,,, = 0. Decision models of the linear form can be included in

the first four comparative static results with the analysis given by Dionne, Eeckhoudt and Gollier (1993) —
see section 6.1.

 

137

- Result (vii) in Theorem 6.4 contains, compared with the result (vi), a
larger set of changes in CDF and the same restrictions on the other
components — a generalization of the result (vi), which is directly

obtained from Property 5.4.

7.2 Remarks for Comparative Statics Regarding Subsets of SSD Shifts

The second-degree stochastic dominance (SSD) order speciﬁes a more general
type of changes in distribution of a random variable than the FSD order or the concept of
an R-S increase in risk deﬁnes. In section 2.1, we indicated that an FSD shift is also an
SSD shift, and that an R-S increase in risk is a special case of an SSD. It is generally
known that an SSD shift can be decomposed into a combination of an FSD shift and an
R-S decrease in risk (mean-preserving contract (MPC)). This is formally stated by Hadar
and Seo (1990) who prove the statement that “a pair of CDF’s F and G satisﬁes F SSD G
if and only if there exists an intermediate CDF F” such that F MPC F 0 and F 0 F SD G.
In other words, any SSD change can be decomposed into an FSD shift and an MPC shift.
Using the results just summarized for subsets of general F SD and MPC changes, this can
be used for deﬁning subsets of the set of SSD shifts which are admissible for making
general comparative static statements. Two examples follow.

Considering SSD shifts, this section assumes that the ﬁnal distribution F
dominates the initial distribution G in the sense of second-degree (denoted by F SSD G).
Both the CDF’s G and F are assumed to have their points of increase in bounded
intervals. Let the support of G(x) be a ﬁnite interval [xl , x4] and the support of F (x) be
another ﬁnite interval [x,, x3] where x1 3 x,. We drop the restriction that x3 3 x4 which
is required in section 5.1. Similar to the case of an R-S increase in risk, we assume that:

for continuous case, G(x) > 0 for all x E(xl, oo), 1- G(x) > O for all x 6 (—oo, x,),

F(x) > 0 for all x E(xz, co), and 1— F(x) > O for allx e(—oo, x3) ; and for discrete case,
G(x)> O for all x E[xl, co), 1-G(x)> 0 for all x e(—oo, x4], F(x) > O for all
x E[xz, co), and 1- F(x)>0 for all x e(—oo, x3].

. In what follows, ﬁrst we deﬁne subsets of SSD shifts which can be understood as
the combinations of FSD shifts and R-S increases in risk, that are generated by the CDF
orders deﬁned in sections 3.1 and 5.1, respectively. Then the comparative static

statements for these subsets are derived in sub-section 7.2.2.

7.2.1 Subsets of SSD Shifts
In this sub-section two concepts of SSD shifts are introduced. Relaxing the equal
means condition, the conditions used in Deﬁnition 5.3 (for an ESIR) and Deﬁnition 5.4

(for an L-ESIR) are modiﬁed to give the following two types of SSD shifts.

Definition 7.1. F (x) represents an extended strong SSD shift from G(x) (denoted
by F ESSSD G) if
(a) J[G(x) — F(x)]dx 2 o for all y e [x,, 00)

(b) there exists a point k E[xz, x3] such that
F(x)S G(x) for all x E[xz, k) and F(x)2G(x) for all x E[k, x3]

(c) for all x E[xz, k), there exists a non-decreasing function H,:[x2, k) —-> [0, 1] such that
Hn=mowo)

(d) for all x e [k, x,], there exists a non-increasing function H,:[k, x3] -> [0, 1] such that

1- F(x) = H.(x)11— G(x)].

Definition 7.2. F(x) represents a left-side extended strong SSD shift from G(x)
(denoted by F L-ESSSD G) if

(a) J [G(x) — F(x)]dx 2 0 for all y e[x,, oo)

139

(b) there exist a point k E[xz, x3] and a non-decreasing function H:[xz, k)—->[O, 1] such

that F(x) = H(x)G(x) for an x e [x,, k) and G(x)SF(x) for all x E[k, x, 1:

Condition (a) in both deﬁnitions is just the SSD condition shown in Deﬁnition 2.2. The
other conditions in Deﬁnition 7.1 and 7.2 are same as the ones in Deﬁnition 5.3 and 5.4,
respectively.

As a special case, if the CDF ’s F and G have a same mean, then the relationship F
ESSSD G just implies G ESIR F, and F L-ESSSD G means G L-ESIR F. In addition,
both the concepts include, as a special case, an MPR shift. Consider the case where
x4 < x3. Condition (b) in both deﬁnitions implies that the point k must be equal to x3 , to
satisfy the condition that G(x) S F (x) for all x e [k, x,]. In this case, both an ESSSD and
an L—ESSSD are reduced to MPR shifts.

The properties associated with the ESIR and the L-ESIR orders can be adopted for
these two SSD orders. When x3 S x,, the CDF’s F and G cross only once at the point
k 6 [x2 , x3], and an ESSSD (L-ESSSD) shift can be understood as the combination of an
MPR dominant shift for the left-side of the point k and another dominated F SD shift for
the right-side of the point. More importantly, there are properties which allow us to
provide general comparative static statements for the two subsets of SSD shifts, which are
given in the next sub-section. The properties are: (i) an ESSSD shift can always be
decomposed into an MPR shift and an ESIR; and (ii) an L-ESSSD shift can always be
decomposed into an MPR shift and an L-ESIR.

Property 7.1. If F ESSSD G, there always exists a CDF GI such that F MPR
G,, and G ESIR GI.

Proof. If F ESSSD G and x4 < x,, then F and G meets at the point x3 and this

implies that an ESSSD shift is just an MPR shift. In this case, the claim in Property 7.1 is

140

satisﬁed by letting G, = G. For the case where x3 S x,, , let’s deﬁne an intermediate

CDF G, (x) such that,

AG(x), when x < s
G,(x) = (7.1)

F(x), when x 2 s

where xi = F (s)/ G(s) and s is a selected point in the interval [x2 , k] such that G, has a

same mean as G. Then, with the conditions in Deﬁnition 7.1, it is easy to see that F MPR

G,, and G ESIR G,. Q.E.D.

Property 7.2. If F L-ESSSD G, there always exists a CDF G, such that F MPR
G,, and G L-ESIR G,.

Proof. Similar to the case of ESSSD in Property 7.1, if F L-ESSSD G and
x4 < x,, then this is just an MPR shift and the claim in Property 7.2 is satisﬁed by letting
G, = G. When x3 S x,, deﬁne an intermediate CDF G, (x) as in (7.1). Then, with the
conditions in Deﬁnition 7.2, it is easy to see that F MPR G,, and G L-ESIR G,. Q.E.D.

Two graphical examples are given to illustrate the results in these two properties.
The shift from a CDF G to F shown in Figure 7.3 satisﬁes F ESSSD G, and the one
shown in Figure 7.4 satisﬁes F L-ESSSD G. The ﬁgures also show that there exists an
intermediate G, such that F MPR G, in both examples and G ESIR G, in Figure 7.3 and
G L-ESIR G, in Figure 7.4. ’

7.2.2 Comparative Statics for Subsets of SSD Shifts
For each of the subsets of ESSSD shifts and L-ESSSD shifts, the general
comparative static statements are obtained from the properties given in sub-section 7.2.1.

Since a shift from G to F satisfying F ESSSD G can always be decomposed into two

l4l

 

F (x)

G'(X)

G(x

 

x, x2 3 x3 x,

Figure 7.3. An ESSSD shift that is the sum ofan MPR and an ESIR.

 

 

 

x, x2 3 x3 =x4

Figure 7.4. An L-ESSSD shift that is the sum of an MPR and an L-ESIR.

 

142

dominant shifts, one MPR and the other ESIR,53 a general comparative static statement
regarding an ESSSD shift can be derived from combining the two results in Theorem 4.4
and 6.2.

Theorem 7.1. For all risk averse decision makers, b,,. 2 b,,. if

(a) FESSSD G

(b) 2,,r 2 O, 2,,, Z O, 2,,, S O, 2,,” S O, and Zn = 0 when 2,,, = 0.

Proof. Given Property 7.1, the comparative static results in Theorem 4.4 and 6.2
complete the proof. Q.E.D.

Similarly, for the subset of L-ESSSD shifts, we combine the results in Theorem
4.4 and 6.4. That is, adding the restriction that the concerned payoff is linear in the
random variable, a general comparative static statement regarding an L-ESSSD shift can

be derived from Property 7.2.

Theorem 7.2. For all decision makers with utility functions u’s such that u’ .>. 0,
u" S O and u’" 2 O, b,. 2 b,, if
(a) F L-ESSSD G

(b) 2,, 20, 2,,, 20, 2,,, SO and 2,“ = 0.

Proof. Given Property 7.2, the comparative static results in Theorem 4.4 and 6.4

complete the proof. , Q.E.D.

We ﬁnish this section by adding Some remarks on the comparative static analysis
regarding the subsets of SSD shifts. Transitivity does not hold for both the ESSSD and
the L-ESSSD order. That is, the relationship F; ESSSD (L-ESSSD) F2 ESSSD (L‘-

 

5’ Here an ESIR dominant shift means an R-S decrease in risk.

I43

ESSSD) F, does not necessarily implies F; ESSSD (L-ESSSD) F, . This implies that the
admissible set of changes in CDF can be extended to other shifts that are not generated by
the given two CDF orders. Figure 7.5 gives an example of this case, showing that the
shift from G to F, while it does not satisfy F ESSSD G, allow the desirable comparative

static result in Theorem 7.1 because the shift can be decomposed into two shifts such that
FESSSD G, ESSSD G.

 

 

 

x, x2 x3 x4

Figure 7.5. A shift showing that F ESSSD G, ESSSD G but not F ESSSD G.

Another extension for the set of admissible SSD shifts is possible when we restrict
the concerned payoff to be linear in the choice variable. In section 4.2, we showed that,
regarding a decision model of the form (4.10) in which a point c satisfying 2,, (c, b) = 0
exists, the desirable comparative static property holds for the combined set of all c-FSD
shifts and c-MPR shifts (Theorem 4.5). This can be shown by the fact that, given a point
c, a c-MPR shift can always be decomposed into an MPR shift and a c-FSD shift
(Property 3.7). In the same way, if the concerned decision model follows the form (4.10),
the result in Theorem 7.1 (7.2) also holds for any shift that can be decomposed into two
shifts, one ESSSD (L-ESSSD) and the other c-FSD. Graphical examples showing these

144

 

 

 

x, x2 c x3 x4

Figure 7.6. A shift that is the sum of an ESSSD and a c-FSD.

 

 

 

x, x2 c x,=x,

Figure 7.7. A shift that is the sum of an L-ESSSD and a c-FSD.

145

cases are given in Figure 7.6 and 7.7. While both the examples do not satisfy any CDF
order concerned in this dissertation, each shift can be admissible for the comparative
static results in Theorem 7.1 and 7.2, respectively. The former gives an example of a
shift that can be decomposed into two shifts, one c-F SD and the other ESSSD, and the

latter shows an example that can be decomposed into two shifts, one c-FSD and the other

L-ESSSD.

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