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COMPARATIVE STATICS UNDER RISK AVERSION AND PRUDENCE

By

Suyeol Ryu

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Economics

2001

 

COMPARATIVE STATICS UNDER RISK AVERSION AND PRUDENCE

By

Suyeol Ryu

A DISSERTATION

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

 

DOCTOR OF PHILOSOPHY
Department of Economics

2001

 

 

 

 

 

ABSTRACT

COMPARATIVE STATICS UNDER RISK AVERSION AND PRUDENCE
By

Suyeol Ryu

An economic decision model with randomness consists of the following four
components: a set of decision makers, an objective function, random exogenous
parameters and choice variables. An important comparative static question in the study of
decision under uncertainty is how to predict the direction of change for a choice variable
selected by the decision maker when a given random parameter changes. This general
comparative static analysis is carried out by restricting the following three components:
(i) the changes in probability distribution function (PDF) or cumulative distribution
function (CDF) of the random parameter, and/or (ii) the set of decision makers, and/or
(iii) the structure of the economic decision model.

Our study focuses on finding sufficient conditions (or a necessary condition) on the
change in distribution of the random parameter that cause risk averse decision makers
with u" 2 O to adjust their choice variable in the same direction in a general decision
model. Therefore all the comparative statics results obtained in this dissertation are
associated with the set of risk averse individuals with a non-negative third derivative of
their utility fimction. This set includes utility functions representing quite plausible

preferences, such as the ones exhibiting decreasing absolute risk aversion (DARA)

 

generally accepted as a reasonable attitude toward risk. This class of utility functions also

includes the concept of ‘prudence’ (77 = — u’"/u") introduced by Kimball (1992), which
denotes a precautionary saving motive.

This study deals with two particular types of R-S increases in risk with single
crossing and three types of R-S increases in risk with multiple crossing in chapter 3, and
three types of K-L increases in risk in chapter 4. For each given type of change in the
random parameter, we developed conditions on the class of decision makers and the
structure of the decision model that are sufficient for making a general comparative static
statement. In these chapters, we use the traditional approach that restricts separately the
changes in PDF or CDF and the structure of the given decision model for comparative
static purposes. However Gollier (1995) restricts two components jointly with a single
restriction to obtain a general comparative static statement. In chapter 5, following his
technique, we deal with the problem of determining the conditions under which a change
in distribution of the random parameter increases the optimal value of a decision variable
for the set of risk averse individuals with u" > 0 , which was done before by Gollier

(1995) for all risk averse individuals.

 

~ ”was? -

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4.

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Cepyright by
Suyeol

 

 

Dedicated to My Parents

 

 

 

 

 

ACKNOWLEDGEMENTS

First, and foremost, I would like to express my sincere appreciation to Professor
Jack Meyer, my dissertation chairperson, for his careful and patient guidance. Without
his constant support, understanding and insights, this work could not have been
completed. I also want to thank Professor Carl Davidson and Professor Lawrence Martin
for serving on my Ph. D guidance committee and their valuable cements.

I certainly could not have made it through the program without my fellow
graduate students. Thanks to Soojong Kim, Younghoon Seo, Taejung Kim, Jaeboong
Hwang and the rest of the graduate students I worked with.

I wish to thank my parents, parents-in—law, brothers and sister for their love,
support and encouragement. I would like to express my deepest thanks to my wife,
Hyekyung, for her love, understanding and support, and for always believing that I could
succeed at my goals. I also thank my daughters, Mina and Hwarim, for so much joy they

bring to me.

vi

 

 

 

TABLE OF CONTENTS

List of Figures ....................................................................................... vii
CHAPTER 1
INTRODUCTION .................................................................................... 1
CHAPTER 2
LITERATURE REVIEW ........................................................................... 8
2.] Stochastic Dominance Criteria ...................................................... 8
2.2 Decision Model ....................................................................... 16
2.3 Comparative Static Analysis ........................................................ 20
2.3.1 An Overview .................................................................. 20
2.3.2 Comparative Statics Results with Subsets of FSD Shifts ............... 22
2.3.3 Comparative Statics Results with Subsets of R-S increases in risk. . .26
2.3.4 Gollier’s Work ............................................................... 32
CHAPTER 3
COMPARATIVE STATIC ANALYSIS FOR SUBSETS OF R-S INCREASES IN
RISK .................................................................................................. 34
3.1 Subsets of R-S Increases in Risk with Single Crossing ......................... 36
3.1.1 Comparative Static Analysis ............................................ 40
3.2 Subsets of R-S Increases in Risk with Multiple Crossing ...................... 45
3.2.1 Other Characterizations of the MSIR Order and Comparison with
the ESIR Order ............................................................... 51
3.2.2 Comparative Static Analysis ............................................... 58
3.3 Concluding Remarks ................................................................. 63
CHAPTER 4
COMPARATIVE STATIC ANALYSIS FOR SUBSETS OF K-L INCREASES IN
RISK .................................................................................................. 65
4.1 Subsets of Increases in Risk in the K-L sense .................................... 66
4.2 Comparative Static Analysis ....................................................... 72
CHAPTER 5
THE COMPARATIVE STATICS OF CHANGES IN DISTRIBUTION OF THE
RANDOM VARIABLES .......................................................................... 92
5.] Comparative Statics of Change in Distribution of the Random
Parameters ............................................................................ 93

 

 

5.2 Concluding Remarks ............................................................... 102

REFERENCES .................................................................................... 105

viii

Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4

Figure 3.5

Figure 4.1
Figure 4.2

Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7

Figure 4.8

Figure 5.1

Figure 5.2

 

LIST OF FIGURES

G L-SIR F .............................................................................. 37
G L-RWIR F ........................................................................... 39
G MSIR F .............................................................................. 47
G OSIRF .............................................................................. 49
G ORSIR F ............................................................................ 50
G RSIRK F ............................................................................. 69
G SIRK (r) F ........................................................................... 70
k,=x',k2=aandk3=bwhenx,5x'Sml .................................. 8O
k,=a,k2=x’andk3=bwhenm15x'5m° ................................. 80
k1=a,k2=x'andk_.=bwhenm°$x'$m2 ................................. 81
kl=a,k2=bandk3=x'wheanSx‘Sx4 ................................. 81
ml S x' S m0 .......................................................................... 87
m0 s x' 5 m2 .......................................................................... 88
G MPS F ............................................................................... 94
FSSD G ............................................................................... 94

 

 

 

 

Chapter 1

INTRODUCTION

The effects of uncertainty on an individual’s choices are theoretical interesting
and have significant policy implications. In fact, the attention paid to this aspect of
economic decision-making has a long tradition in the history of economics. Since its
introduction by von Neumann and Morgenstem (1944), expected utility theory has been
the dominant framework for the economic analysis of uncertainty and there has been
much progress in the theoretical and applied analysis of choice under uncertainty. This
risk analysis is included in many economic fields of study such as insurance, futures
markets, stock markets, international trade and finance.

For the decision maker who is faced with a specific pair of risky prospects and
whose utility function is a member of a specified class U, the concept of stochastic
dominance (SD) or stochastic ordering of risky prospects is an important tool to identify
efficient or undominated prospects among a given choice set. The stochastic dominance
(SD) rules that have been developed to date include rules for decision makers who prefer
more wealth to less, those who are also risk averse, and those who also have a non-
negative third derivative of their utility function (u”' 2 0). These rules are the ‘first degree

stochastic dominance’ (FSD) rule, the ‘second degree stochastic dominance’ (SSD) rule,

 

and the ‘third degree stochastic dominance’ (TSD) rule, respectively.

Since the seminal and simultaneous publications of Hadar and Russell (1969) and
Hanoch and Levy (1969) there has been a virtual explosion of papers investigating
implications of stochastic dominance rules for decisions under uncertainty. Dominance
principles have important applications to portfolio choice, capital budgeting and financial
intermediation decisions. Stochastic dominance can also be applied to investment and
production decision problems and others under uncertainty. In these areas, overall
impacts of uncertainty have received wide attention.

An important comparative static question in the study of decisions under
uncertainty is how to predict the direction of change for a choice variable selected by the
decision maker when a given random parameter changes. This general comparative static
analysis is usually carried out by restricting the following components; (i) the changes in
probability distribution function (PDF) or cumulative distribution function (CDF) of the
random parameter, and/or (ii) the set of decision makers, and/or (iii) the structure of the
given economic decision model. When the first and the third components are restricted
separately or jointly, many authors obtain the comparative statics results that are
associated with either the set of all individuals with non-decreasing utility functions or
the set of all risk averse agents. These examples are included in Meyer and Orrniston
(1985), Black and Bulkley (1989), Landsberger and Meilijson (1990), Dionne, Eeckhoudt
and Gollier (1993), Eeckhoudt and Gollier (1995), Gollier (1995) and others. Note that
relatively strong restrictions on one component are usually related to relatively weak
restrictions on the other two components. In this study we impose somewhat stronger

restrictions on the risk preference of decision makers.

 

 

 

 

 

All the comparative statics results obtained in this dissertation are associated with

the set of risk averse individuals with a non-negative third derivative of their utility
function. This set includes utility functions representing quite plausible preferences, such
as the ones exhibiting decreasing absolute risk aversion (DARA) generally accepted as a
reasonable attitude toward risk. This class of utility functions also includes the concept of

‘prudence’ (7] = — u'"/u") introduced by Kimball (1990), which denotes a precautionary

saving motive. Note that the term ‘prudence’ is meant to suggest the propensity to
prepare and forearm oneself in the face of uncertainty. Therefore our study focuses on
finding sufficient conditions (or a necessary condition) on the change in distribution of
the random parameter that cause risk averse decision makers with u'" _>. 0 to adjust their
choice variable in the same direction in a general decision model.

In order to generate interesting comparative statics results, the common
restrictions to impose on the changes in PDF or CDF are general stochastic dominance
orders. It implies that these SD rules play an important role in this comparative static
analysis. There are three approaches specifying these restrictions. First, the CDF
difference approach directly imposes restrictions on the difference between the initial and
the final CDFs or PDFs. This approach is used to define a ‘mean-preserving truncation’
in Eeckhoudt and Hansen (1981) and a ‘strong increase in risk’ (SIR) in Meyer and
Ormiston (1985). Second, the ratio approach imposes restrictions on the ratio of a pair of
PDFs or of a pair of CDFs. Examples using this approach are included in Black and
Bulkley (1989) who introduce the concept of a ‘relatively strong increase in risk’ (RSIR),
Landsberger and Meilij son (1990) who define a ‘montone likelihood ratio’ (MLR) and

Eeckhoudt and Gollier (1995) who consider a ‘monotone probability ratio’ (MPR) among

 

 

 

 

others. Finally, the deterministic transformation approach also restricts the change in the

distribution by placing conditions on the transformation, which transforms an initial
random variable into another random variable. This approach is popularized by Sandmo
(1971) and extended by Meyer and Ormiston (1989) who define a ‘simple increase in
risk’ (SIR). These three approaches are used in this study to specify restrictions imposed
on the changes in PDF or CDF.

This dissertation is organized as follows. In the next chapter, we will give a short
review of the literature concerning the SD selection rules and the previous work
concerning comparative static analysis under uncertainty. We introduce a special
‘increase in risk’ defined by Kroll et. a1 (1995) and call it a ‘K-L increase in risk’, that is,
third degree stochastic dominance (TSD) change with equal means. Since SSD implies a
TSD, the set of K-L increases in risk includes the set of R-S increases in risk. This
implies that K-L increases in risk extend the R-S definition of risk to a larger set of CDFs
that could not be classified as ‘more risky’ before. We also provide terminology,
notation, definitions and the decision model used in this study. The terminology and
notation used here follows that in the established literature. When defining an increase in
the riskiness of a random variable in the R-S sense or the K-L sense, we use F (x) and
G(x) to denote the initial, less risky, and the final, more risky, CDFs. We use the general
decision model previously employed by Kraus (1979) and Katz (1981) in their work.

Chapter 3 treats several subsets of R-S increases in risk. In addition to the existing
subsets of R-S increases in risk, we define three subsets of R-S increases in risk called a
‘lefi-side relatively weak increase in risk’ (L-RWIR), a ‘left-side strong increase in risk’

(L-SIR) and a ‘monotone strong increase in risk’ (MSIR). Each subset generalizes a

 

 

 

 

 

definition in the published literature. This implies that a ‘strong increase in risk’ (SIR) in
Meyer and Ormiston (1985), a ‘relatively weak increase in risk’ (RWIR) in Dionne,
Eeckhoudt and Gollier (1993) and an ‘extended strong increase in risk’ (ESIR) in Kim
(1998) are extended to a L-SIR, a L-RWIR and a MSIR, respectively. The restrictions
used to define these subsets are discussed and graphical examples are given to illustrate
the basic relations among the subsets. For basic relations, it is shown that the set of L—SIR
shifts is included in the set of ‘lefi-side relatively strong increases in risk’ (L-RSIR) shifts
in Kim (1998), which is, in turn, included in the set of L-RWIR shifts. We propose two
more subsets of R—S increases in risk with multiple crossing called an ‘outside strong
increase in risk’ (OSIR) and an ‘outside relatively strong increase in risk’ (ORSIR). The
conditions in these two shifts imply that there exists a R-S decrease in risk in the given
interval and a R-S increase in risk outside this interval. We also provide other
characterizations of the MSIR ranking in order to provide an interpretation and compare
the MSIR order with the ESIR one. We show that the MSIR order implies that the
conditional expectation of a random variable under F is greater than and equal to that
under G, and the converse is also true. Restricting the payoff function to be linear in the
random variable, we show that the effects of these shifts can be determined for all risk
averse decision makers with non-negative third derivative of utility functions. We also
show that the MSIR condition is less restrictive than the Gollier (1995) condition
presented in chapter 2 when the payoff function is linear in the random variable.
Chapter 4 provides general comparative static statements regarding several

subsets of K-L increases in risk. In order to obtain comparative statics results, many

researchers have specified particular types of CDF changes which are subsets of F SD

 

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shifts, SSD shifts, or R-S increases in risk. So far, no one has examined subsets of TSD

shifts for comparative static purposes. We introduce three subsets of K-L increases in
risk. A ‘strong increase in risk in the K-L sense’ (SIRK) is defined by imposing
restrictions on the difference between two cumulative of CDFs (C-CDFs). These
restrictions replace the restrictions on the difference between two CDFs used by Meyer
and Ormiston (1985) who define the SIR order. A ‘simple increase in risk across r in the
K-L sense’ (sIRK(r)) is formally defined in this chapter. This concept also can be
characterized by using a deterministic transformation of a random variable. A ‘relatively
strong increase in risk in the K-L sense’ (RSIRK) consists of three parts that are a ‘lefi-
side monotone probability ratio’ (L-MPR), a ‘right-side monotone probability ratio’ (R-
MPR) and FSD shifis, or two parts which are ‘left-side extended strong increases in risk’
(L-ESIR) and ‘right-side extended strong decreases in risk’ (R-ESDR). A L-ESIR (R-
ESDR) requires the monotonicity of the probability ratio only on the lefl-side (right-side)
of the given pair of CDFs. These restrictions on the ratio for CDFs replace the restrictions
on the ratio for PDFs imposed by Black and Bulkley (1989) who introduce the RSIR
order. We have the basic relationship between the SIRK and the RSIRK orders as: the
RSIRK ranking includes the SIRK one. All the comparative statics results obtained in this
chapter are concerned with the set of risk averse individuals with u'" 2 0.

In chapter 5, we deal with the problem of determining the conditions under which
a change in distribution of the random parameter increases the optimal value of a decision
variable for the set of risk-averse individuals with a positive third derivative of their
utility function (u" > 0), which was done before by Gollier (1995) for all risk averse

individuals. Given the set of decision makers, the traditional approach restricts separately

 

 

 

 

 

 

 

 

 

 

 

 

 

the changes in PDF or CDF and the structure of the given decision model. However he

restricts two components jointly with a single restriction to obtain a general comparative
static statement. His condition is quite simple and generalizes all previous restrictions on
changes in risk such as a SIR and a RSIR imposed for all concave utility functions.
Following the technique used in Gollier (1995), we obtain a necessary condition under
which all risk-averse individuals with u" > 0 react in the same direction when facing a
given shift in distribution. Our condition is less restrictive than the Gollier one, but is
difficult to interpret. We also obtain a sufficient condition for the same economic
problem. We define a different subset of SSD that satisfies our sufficient condition and

includes a R-S increase in risk as a special case.

 

 

Chapter 2

LITERATURE REVIEW

For many years the expected utility approach to decision making under
uncertainty has gained increasing attention among economic researchers. An important
question in the study of economic decision models involving randomness is how a
particular type of a change in the random variable affects the level of the choice variable
selected by a decision maker. General stochastic dominance (SD) orders play an
important role in this comparative static analysis. Section 2.1 provides a brief review of
general stochastic dominance rules in the literature. Section 2.2 gives a general decision
model formulation used in this study. Section 2.3 presents a review of the literature
concerning comparative static analysis under uncertainty. For notational convenience, the
random variable x is assumed to have the initial and the final distribution characterized by
CDFs F (x) and G(x), respectively and if not stated otherwise, the supports of these

CDFs are assumed to be in the finite interval [a,b] where F (a) = G(a) = 0 and

F(b) = G(b) = 1 .

2.1 Stochastic Dominance Criteria

 

When decision makers compare uncertain prospects, it is assumed here they

follow the rules of expected utility maximization. This section provides various
formulations of SD rules and various ways of defining preference categories. Each
formulation has advantage in solving certain issues. These rules apply to pairs of random
variables, and indicate when one is to be ordered higher than the other by specifying a
condition which the difference between their CDFs must satisfy. Given two risky
prospects with CDFs F and G, the establishment of an order of preference between F and
G is determined.

There are the following three main SD rules for ordering a pair of uncertain
prospects F and G: first, second and third-degree stochastic dominance, denoted
respectively by FSD, SSD, and TSD which are introduced by Quirk and Saposnik (1962),
Schneeweiss (1969), Hanoch and Levy (1969), Hadar and Russell (1969) and Whitrnore

(1970). Each SD rule is formally defined as follows:

Definition 2.1. The following three main rules are introduced:
(a) 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 0, for all x 6 [a,b],
(b) F (x) is said to stochastically dominate G(x) in the second-degree (denoted by F

SSD G) if and only if
II [G(x)— F(x)]dx 2 0, for all t e la, b],

(c) F (x) is said to stochastically dominate G(x) in the third~degree (denoted by F TSD

G) if and only if

 

 

 

 

 

 

 

 

 

j: jg [G(x)— F(x)]dxdt 2 o for all s e [a, b] and

flat.)— F(x)]dx 2 0.

Each rule defines a partial ranking on the set of all probability distributions such
that the set of CDFs can be ordered by F SD is included in the set that can be ordered by
SSD, and the latter set is included in the set that can be ordered by TSD. Each stochastic
dominance ranking has the property of transitivity such that considering CDFs FI , F2
and F3 , if Fl FSD (SSD or TSD) F2 and F2 FSD (SSD or TSD) F3 , then Fl FSD (SSD
or TSD) F3-

Rothschild and Stiglitz (R-S) (1970) propose several definitions of “more risky”

or “more variable” random variables and show that they are equivalent to one another:

1. A random variable y is riskier than a random variable x if y is equal to x plus a
noise term a:

=x+£
y d
where “j ” means “has the same as distribution as” and a is a random variable

with the property that E (3] x) = 0 for all x.
2. A random variable y is riskier than a random variable x if and only if all risk averse
individuals prefer x over y, namely
E U (x) 2 E U (y) for all concave utility functions.

3. A random variable y is riskier if it has more weight in the tails than a random

 

 

variable x.

Rothschild and Stiglitz analyze these three definitions and conclude that the
three definitions are equivalent and lead to a single definition of ‘increased risk’. An
important aspect of R-S risk analysis is included in their third definition which leads to
the definition of a mean preserving spread (MPS). This definition is well known as
‘integral conditions’ being the restrictions imposed on the difference between two CDFs.
Assuming x and y are random variables with CDFs F and G, respectively, R-S give a

general definition of an ‘increase in risk’ in the following:

Definition 2.2. G(x) is said to be riskier than F(x) in the Rothschild-Stiglitz

sense (denoted by G MPS F) if and only if
b
(a) I. [G(x)—F(x)]dx=0

(b) j [G(x)— F(x)]dx 2 o, for all r e [a,b].

The MP8 (or R-S increases in risk) is a shift of probability mass when probability
is moved from the center of the distribution to the tails without affecting the mean. This
implies that probability mass is taken from a certain set of points and redistributed to
points to the lefi and the right in such a way that the mean value of the random variable is
kept unchanged. Note that a R-S increase in risk is an SSD change with equal means and
also gives a partial ranking with the property of transitivity on a set of probability

distributions like the three previous rules.

 

 

 

 

By analogy to the R-S definition of MPS, Kroll et al. (1995) introduce a new

concept of probability mass shifts and call it a ‘mean preserving spread-anitspread’
(MPSA). It is assumed that Y and Z are two random variables with a cumulative
distribution function of F (x) and G(x), respectively. S(x) and A(x) denote the mean
preserving spread (MPS) function and the mean preserving antispread (MPA) function,
respectively. The antispread shifts in probability mass function are the exact opposite to
those imposed by the well-known MPS suggested by R-S.

The ‘mean preserving spread-antispread’ (MPSA) function SA is defined as
follows:

SA(x) = S(x)+ A(x).
Similar to R-S’s MPS, if F(x) = G(x)+ SA(x), F(x) differs from G(x) by a single

MPSA step.

Definition 2.3. A MPSA function SA = S + A is said to satisfy the TSD criterion

1: I:S(t)dtdv 2 — j jA(t)dtdv

for all x in [(1, b] and with a strict strong inequality for at least one x.

Note that the MPA improves a distribution while the MPS does the opposite.
Observe that the MPA is smaller but located to the right of the MPS in Definition 2.3.

According to Kroll et al., they specify the conditions on the MPSA functions that
enable the classification of one random variable as ‘more risky’ than another random

variable for different sets of utility functions. In their first theorem, ‘more risky’ random

 

variables are defined by employing MPS in the same way as R-S. For all risk averters’

utility functions Theorem 2.1 is established.

Theorem 2.1 (SSD). Let Y and Z be two random variables with equal means with

support bounded by [a, b]. Let F (x) and G(x) be the CDFs of Y and Z, respectively.
Then F (x) dominates G(x) by the SSD criterion if and only if there exists a sequence of

{SA, }: of MPSA such that

l
F+Z:ISA,=G,

where for each i = 1, 2,. . ., A, = 0 (in SA, ) and the convergence is in the weak sense.

Note that this is the original theorem of R-S (1970).

For the TSD rule, R—S’s original MPS is not sufficient and the antispread function
is needed as well (that is, A at 0 in SA). In their second theorem, they specify the
conditions that ensure that G can only be constructed from F using MPSA when F
dominates G by the TSD criterion. For all risk averse individuals with DARA utility

functions Theorem 2.2 is established.

Theorem 2.2 (TSD). Let F and G be the CDFs of two equal mean random
variables, Y and Z respectively, with support bounded by [a, b]. Then F (x) dominates
G(x) by TSD if and only if there exists a sequence of {5A, } 7:, of MPSA satisfying the
TSD criterion such that

F+Z:ISA,=G.

 

 

ulili:

mid

 

The MPSA that satisfies the third degree Stochastic dominance (TSD) criterion

generates a riskier distribution for all DARA utilities but not all risk averters. This new
definition extends R-S’s definition of risk to a larger set of CDFs that could not be
classified as ‘more risky’ before. We call it an ‘increase in risk in the K-L sense’ and give

its formal definition as:

Definition 2.4. G(x) is said to be riskier than F (x) in the K-L sense if and only if
b
(a) L[G(x)—F(x)ldx=0

(b) j j’ [G(x)—F(x)]dxdt 2 o for all s 6 [a,b].

Condition (a) implies that two distributions have equal means. Condition (b)
implies that a MPSA function SA = S + A satisfies the TSD criterion. These conditions
imply that an increase in risk in the K-L sense is a TSD change with equal means. Since
SSD implies TSD, the set of K-L increases in risk includes the set of R-S increases in
risk. Observe that, for random variables with equal means, Definition 2.4 is equivalent to
the TSD rule.

Assume that all risk averse individuals have DARA utility functions. As analyzed
in Vickson (1975) and Bawa (1975), in the case of E (Y ) = E (Z ), a necessary and
sufficient condition for all such individuals to prefer Y to Z is for Y to dominate Z

according to the TSD rule. That is, if E(Y) = E(Z), Z is riskier than Y for all DARA
utility functions if and only if f j"[G(t)— F(t)]dtdv 2 o for all x in [a, b] and with a

strict strong inequality for at least one x.

 

 

 

 

Each of the above stochastic dominance rules for unanimous choice is associated

with a well-defined set of utility functions. These utility filnction classes are defined as

follows:

Definition2.5. Assuming that a utility function u(-) is continuous, bounded and
three times differentiable for the support [0, b], five utility function classes are defined in

the following ways:

(a) U. = {u(-)lu’20}

(b) U2 = {u(-)|u' 20and u" so}

to) U. = {u(-)lu”S 0}

(d) U, = {u(-)|u' 2o,u' s Oand u'" 2 o}

(e) U 5 = {u() | u(-)displays decreasing absolute risk aversion (DARA)}.

Note that generally a utility function that satisfies the DARA condition
(A'(x) = (— u"(x)/u’(x)) < 0, Vx e R) implies that u’ 2 0 , u" S 0 and u" 2 0; however
vice versa this is not always valid. We assume that, for a utility function u, E U F and

E U 0 represent the expected utilities when the CDFs are given by F and G, respectively.

The SD rules and the relevant class of preferences are related in the following way:

Theorem 2.3. Let F and G be the cumulative distribution of two distinct uncertain

prospects. The followings are established:

 

 

Each of the above stochastic dominance rules for unanimous choice is associated

with a well-defined set of utility functions. These utility function classes are defined as

follows:

Definition2.5. Assuming that a utility function u(-) is continuous, bounded and
three times differentiable for the support [a, b], five utility function classes are defined in
the following ways:

(a) U1 = {u(-) u' 2 0}

 

(b) U2 ={u(~)|u'20andu'SO}

 

(c) U. = {u(-) u" s 0}
((1) U4 = {u(-)|u' Z 0,u" S 0and u" 2 0}
(e) U 5 = {u(-) I u(-)displays decreasing absolute risk aversion (DARA)}.

Note that generally a utility function that satisfies the DARA condition
(A'(x) = (— u"(x)/u'(x)) < 0, Vx e R) implies that u' 2 0 , u” S 0 and u” 2 0; however
vice versa this is not always valid. We assume that, for a utility function u, E U ,, and

E U G represent the expected utilities when the CDFs are given by F and G, respectively.

The SD rules and the relevant class of preferences are related in the following way:

Theorem 2.3. Let F and G be the cumulative distribution of two distinct uncertain

prospects. The followings are established:

 

 

(a) EU,.- 2 EU(,. for every u in Ul ifand only ifFFSD G

(b) EU, 2 EU“. for every u in U2 ifand only ifFSSD G
(c) EU F 2 E U“. for every u in U3 if and only if G is riskier in the R-S sense than F
(d) EU,, 2 EU“ for every u in U4 ifand only ifFTSD G

(e) E U F 2 E U“ for every u in U 5 if and only if G is riskier in the K-L sense than F.

The proofs of Theorem 2.3 are found in the literature: Quirk and Saposnik (1962)
and Schneeweiss (1969) for the FSD rule, Hanoch and Levy (1969) and Hadar and
Russell (1969) for the SSD rule, Rothschild and Stiglitz (1970) for the MPS rule,
Whitrnore (1970) for the TSD rule and Vickson (1975), Bawa (1975) and Kroll et al.

(1995) for the condition (e) in Theorem 2.3.

2.2 Decision Model

An economic decision model with randomness consists of the following four
components: a set of decision makers, an objective function, random exogenous
parameters and choice variables. Many economic decision models including randomness
can be usually divided into two types; specific decision models and general decision
models. While the former are constructed to represent specific economic situations where
the model structure and variables have specific interpretations, the latter are formulated to
include many specific models as special cases.

We use the general decision model in this study introduced by Kraus (1979) and

 

 

Katz (1981) in their work. The decision maker is assumed to choose the optimal value for

a choice variable a taking the random variable x as given. He chooses a so as to
maximize expected utility, where utility u depends on a scalar valued filnction of the
choice variable and the random variable, z(x, a). Formally, the economic agent’s
decision problem is to select a to maximize E [u(z(x,a))]. That is,

mtax E [u(z(x, a))]. (2.1)
In this decision framework, utility depends only on the outcome variable 2, that is, the
objective function is single dimensional. Thus, problems involving multidimensionality
are avoided.

In this study, we assume that utility function u(z) is thrice differentiable with
respect to its argument with u'(z) 2 0 , u'(z) S 0 and u"(z) 2 0; thus, the decision maker
is a risk averter with u"(z) 2 0. The function z(x, a) is assumed three times differentiable
with zaa (x, a) < 0. This condition insures that the second order condition for the
maximization problem is satisfied. To simplify the discussion, we follow the literature
and focus on the case where 2‘ (x, b) 2 0. This assumption, combined with u'(z) 2 0 ,
indicates that higher values of the random variable are preferred to lower values. The
case where z, (x,a) S 0 can be handled with appropriate modifications. To focus on
interior solutions to the maximization problem, it is assumed that za (x,a) = 0 is satisfied
for some finite or for all relevant values of x.

Many researchers have used this general decision model in the study of choice
under uncertainty and it includes a variety of economic decision problems. When we

assume that the outcome variable is linear in the random variable, the simple form of

 

 

lesa) may be expressed as z(x, a) -=- (z(x -— c)+ 20 where 20 and c are exogenous

constants. As analyzed by Sandmo (1971), Rothschild and Stiglitz (1971), F ishbum and
Porter (1976), Dionne, Eeckhoudt and Gollier (1993) and Eeckhoudt and Gollier (1995),
the applications of this simple form of a decision model are numerous: the standard
portfolio problem, the problem of the competitive firm with constant marginal cost under
output price uncertainty, the coinsurance problem and others.

In the standard portfolio model, the payoff function can be written as
z(x,a) = 20 + bWO (x — c) where b is the fraction of the initial wealth Wo allocated to the
risky asset, x the random rate of return of the risky asset and 20 5 W0 (1 + c) with c being
the sure interest rate. This payoff function is equivalent to the simple form of z(x,a)
when a E bW0 . For the competitive firm, the linear function is z(x,a) = (z(x -— c)+ zo ,
where x is the uncertain output price, c marginal cost, — 20 the fixed cost and or the output
level. In the standard coinsurance problem, the payoff function is given by the final
wealth z(x, a) = W0 — 11p — (l - b) (x — /l,u) where x is the amount of random loss, [1 the
expected loss, b coinsurance rate, bill the insurance premium, and Wo the initial wealth.
This payoff function is equivalent to the simple form of z(x,a) when 20 5 W0 — [Lu ,
a a —(1 — b) and c E [in . If we limit the discussion to private insurance contracts, the
coinsurance rate b belongs to the interval [0,1]. Then, by definition, or is non-positive and
belongs to the interval [— 1, 0]. Other examples of this simple form with appropriate
modifications are included in Fedar (1977) who examines the problem of hiring workers

and in Paroush and Kahana (1980) who investigate the cooperative firm model.

While most decision models include only one random and one choice variable,

 

some include more than one random and one choice variable. For decision models with

one random and more than one choice variable, it is more difficult to make determinate
comparative static statements than for the case with one random and one choice variable
because of the interactions among choice variables. There are some examples of specific
decision models with one random and two choice variables. Batra and Ullah (1974)
investigate a competitive firm’s input decisions with two inputs under output price
uncertainty. Feder, Just and Schmitz (1977) examine an international trade model. Katz,
Paroush and Kahana (1982) deal with the optimal policy of a price discriminating firm
which operates under price uncertainty in one of two markets. Eeckhoudt, Meyer and
Ormiston (1997) investigate the decision to insure and the portfolio composition decision
and analyze the interaction between the demands for insurance and for the risky asset. All
of these cited papers deal with a specific form of a decision model. Considering a general
form of a decision model with one random and two choice variables, Choi (1992)
investigates a special case of a comer solution.

For a decision model with more than one source of randomness, it is generally
difficult to do the comparative static analysis for a change in any one random variable
without restricting the correlations among random variables. To solve this problem, some
papers assume that the random variables are independent of one another, allowing one to
Change while the others are held fixed. Hadar and Sec (1990) investigate the optimal
Proportions of the assets when the distribution of one of the assets undergoes some
general type of shifi which uses the general SD orders as restrictions on the change in the
random variable. When random parameters are not independently distributed, we face the

difficulty of defining a shift in risk of one random parameter without altering the

 

 

riskiness of the other random parameters; that is, without altering the marginal

distribution of the other parameters. Meyer (1992) proposes the use of deterministic
transformations to solve this problem. Other examples with multiple sources of risk are
analyzed in the articles of Meyer and Ormiston (1994), Dionne and Gollier (1992, 1996)
and others. For independently distributed other risks, referred to as background risk,
Meyer and Meyer (1998) examine the effect of changes in the distribution function for
this background risk on the decision to insure. Other examples for a change in
background risk are included in Eeckhoudt, Gollier and Schlesinger (1996), Eeckhoudt

and Kimball (1992) and others.

2.3 Comparative Static Analysis

Faced with uncertainty concerning the economic environment, an interesting
question for comparative statics is to investigate necessary and/or sufficient conditions
for determining the direction of change in the decision variable when a given random
parameter changes. Our study focuses on a general decision model z(x, a) that includes
one random, one choice and one outcome variable. Therefore, we review the comparative
statics results in the literature concerning a general one-argument decision model in more

detail.

2.3 .1 An Overview

It is generally known that the standard SD rules such as F SD, SSD and R-S

 

 

 

increases in risk are not sufficient to allow one to make general comparative static

statements concerning the effect of a change in random variable on the choice made by

an arbitrary decision maker with a non-decreasing and/or concave utility function. Using
a general decision model Meyer and Ormiston (1983) demonstrate that the class of SSD
shifls in distribution or R-S increases in risk causes all risk averse decision makers to
adjust their level of the choice variable in the same direction if and only if its optimal
level under certainty is independent of the value of the random exogenous variable,
clearly not an interesting economic problem. Therefore, general comparative static
analysis is carried out by imposing restrictions on the following components; the set of
decision makers, the structure of the decision model and the set of changes in the random
variable. Generally, when relative strong restrictions are imposed on one component, the
derived comparative static statements are usually related to relatively weak restrictions on
the other components. Many examples showing the above relationship are included in the
following literature; Rothschild and Stiglitz (1971), Hadar and Russell (1978), Hadar and
Sec (1990), Meyer and Ormiston (1983, 1985, 1989), Dionne, Eeckhoudt and Gollier
(1993), Black and Bulkley (1989), Ormiston and Schlee (1993) and others.

In order to generate interesting comparative statics results, the common
restrictions to impose on the changes in the random variable are general stochastic
dominance orders. It implies that these SD rules play an important role in this
comparative static analysis. There are three approaches specifying these restrictions. The
CDF difference approach imposes restrictions on the initial and the final CDFs F and G,
that is, restrictions on the difference between two CDFs G — F . Some examples using

this approach are included in Sandmo (1971), Kraus (1979), Katz (1981), Eeckhoudt and

21

 

 

Hansen (1981) and Meyer and Ormiston (1985). The ratio approach imposes restrictions

on the ratio for PDFs or the ratio for CDFs. Some examples using this approach are
included in Black and Bulkley (1989), Landsberger and Meilijson (1990), Dionne,
Eeckhoudt and Gollier (1993) and Kim (1998).

The deterministic transformation approach proposed by Meyer and Ormiston
(1989) imposes some specific restrictions on the transformation function, which
transforms an initial random variable into another random variable. The emphasis in this
approach is on pairs of random variables which are related one another by means of a
transformation, or on sets of random variables where each are related to a common
random variable by means of a transformation. Assuming that the initial random variable
x is characterized by CDF F (x) , the transformed random variable y = t(x) is obtained
from x by means of a transformation. The transformation t(x) is assumed to be non-
decreasing, continuous and piecewise differentiable. Combined with the monotonic
preferences for outcomes, the non-decreasing assumption ensures that the transformation
does not reverse the preference ranking over the various possible outcomes of the original
random variable. This assumption is necessary to make interesting statements about the
effects of transformations of random variables on expected utility. Sandmo (1971),
Meyer (1989), Meyer and Ormiston (1989) and Ormiston (1992) give examples on this

approach.
2.3.2 Comparative Statics Results with Subsets of FSD Shifis
Before we give a short review for comparative statics results with subsets of FSD

shifis, we assume that the support of G(x) is a finite interval [xl , x3] and the support of

22

 

 

 

 

 

F (x) is another finite interval [x2,x4] where xl S x2 and x3 S x4.

Fishburn and Porter (1976) demonstrate that the F SD order does not allow a
determinate general comparative static statement for all risk averse agents, even in the
simplest case of portfolio problem. Landsberger and Meilijson (1990) introduce the
concept of a ‘monotone likelihood ratio’ (MLR) order that is defined by imposing a
monotonicity restriction on the ratio of a pair of PDFs. This restriction is widely used in

the statistical literature.

Definition 2.6. F (x) represents a monotone likelihood ratio FSD shift from G(x)
(denoted by F MLR G) if there exists a non-decreasing function I : [x2,x3] —> [0,00) such

that f(x) = l(x)g(x) for all x 6 [x2, x3].

Definition 2.6 implies that g(x) 2 f (x) when 1 S1, g(x)S f (x) when l 2 l, and
the PDFs f and g cross only once. An MLR order is a FSD shifi. Using the standard
portfolio choice problem, a MLR shift induces all individuals with non-decreasing utility
fimctions to increase their demand for the risky asset. Landsberger and Meilij son’s
analysis depends on a weak optimality condition of the economic decision problem where
the optimal level of the decision variable is determinable, including an unbounded or a
corner solution.

Eeckhoudt and Gollier (1995) introduce the concept of a ‘monotone probability
ratio’ (MPR) order that is defined by imposing monotonicity restriction on the ratio of a
pair of CDFs. This restriction replaces the restriction on the ratio of a pair of PDFs used

by Landsberger and Meilij son who define a MLR order.

23

 

 

Definition 2.7. F (x) represents a monotone probability ratio F SD shift from

G(x) (denoted by F MPR G) if there exists a non-decreasing function h : [x2, x3 ] —> [0,1]

such that F(x) = h(x) G(x) for all x 6 [x2, x3].

A MPR order is a FSD shift from Definition 2.7. A MPR order is less restrictive
than a MLR order since the former does not restrict the number of times of crossing
between the PDFs f and g. Note that the MLR ranking implies the MPR one. All risk
averse decision makers are considered in their paper.

Kim (1998) finds more subsets of FSD shifts by weakening the restrictions
imposed on changes in PDF or CDF. Among them, he considers the concept of a ‘left-
side monotone likelihood ratio’ (L-MLR) order that extends the MLR order. The L-MLR
shifts are obtained from the relaxing the monotonicity requirement for points to the right

of the crossing point.

Definition 2.8. F (x) represents a left-side monotone likelihood ratio FSD shift
from G(x) (denoted by F L-MLR G) if there exists a point m 6 [x2 , x3] and a non-
decreasing function I : [x2,m] —> [0,1] such that f (x): [(x) g(x) for all x e [x2,m) and

g(x) S f(x) for all x e [m,x3].
The L-MLR condition requires that two PDFs cross only once at the point m and

that g(x) 2 f (x) for all points to the lefi of m and g(x) S f (x) for all points to the right

Ofm. Compared with the result in MLR shifis, the comparative statics result in L-MLR

24

 

 

shifts includes a larger set of FSD changes and a smaller set of decision makers.

That is, all risk averse decision makers are considered in his analysis. He also obtains the
property such that the L-MLR order lies between the MLR and the MPR one. This
implies that F MLR G :> F L-MLR G => F MPR G.

Note that the above three papers use the ratio approach in the comparative static
analysis under uncertainty. Turning to the deterministic transformation approach,
Ormiston (1992) defines a simple FSD transformation as a class of FSD shifts and

provides a general comparative static statement.

Definition 2.9. The random variable described by a transformation z(x)
represents a simple FSD shift from the initial random variable x given by F (x) if

k(x)at(x)—x 20 and k'(x)SO, for all x 6 [a,b].

Theorem 2.4. The optimal value of the choice variable increases for any simple
F SD shift if

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

(b) 2x20,zxxS0 and zaVZO.

Remember that generally a utility fimction satisfying the DARA condition implies
that u’ 2 0 , u" S 0 and u’" 2 0; however vice versa this is not always valid. Theorem 2.4
Contains the Sandmo (1971) result concerning the effect of a linear risk-altering
transformation on the competitive firm’s choice of output level as a special case. The

transformation function used in his analysis is a linear form as:

25

 

 

t(x)=y(x—x)+t9+x

where 7 is a multiplicative shift parameter, 0 an additive shift parameter and f the
mean of the random variable x. An increase in 0 (at y = 1 and 0 = 0) is equivalent to
moving the probability distribution to the right without changing its shape. That is, it

defines a type of FSD shift.

2.3.3 Comparative Statics Results with Subsets of R-S increases in risk

This subsection reviews some important subsets of R-S increases in risk which
provide general comparative static statements. Examination of literature shows that many
authors have investigated an increase in risk from an initial nonrandom situation to yield
interesting comparative static theorems for the set of risk averse decision makers. Some
call it an ‘introduction of risk’ and others a ‘global increase in risk’, which is a particular
type of R—S increases in risk. Examples of this concept are included in Sandmo (1971),
Leland (1972), Batra and Ullah (1974), Kraus (1979), Katz (1981) and others.

While a global increase in risk yields interesting comparative statics results for all
risk averse agents, its restriction is rather severe and limits significantly the situation to
which those results can be applied. Eeckhoudt and Hansen (1981) propose an alternative
definition of marginal change in randomness, that is, a ‘mean-preserving truncation’

which is less restrictive than a global increase in risk.

Definition 2.10. F (x) represents a mean-preserving truncation from G(x) if their

difference, G(x) — F (x) , satisfies

26

 

 

(a) Lj‘icto-thimo

(b) f[G(x)— F(x)]dx 2 0 for all y e [x,,x4]
(c) F(x)=0for all xe [x,,x2), F(x)=1 for all xe(x3,x4] and F(x)=G(x) for all

xe[x,,x,].

Definition 2.10 is a R-S increase in risk and includes a global increase in risk as
a special case. Compared with the Sandmo result, their theorem is rather robust because
the same comparative statics result is obtained for a more general set of R-S increases in
risk without additional assumptions required. A mean-preserving truncation can be
applied to real economic phenomena such as the existence of guaranteed minimum and/or
imposed maximum prices.

Meyer and Ormiston (1985) introduce the concept of a ‘strong increase in risk’
(SIR) as a subset of R-S increases in risk, which is a direct generalization of probability

mass transfers involved in the introduction of risk. A SIR is formally defined as:

Definition 2.11. G(x) represents a strong increase in risk from F (x) (denoted by

G SIR F) if their difference, G(x) — F (x) , satisfies
(a) j: [G(x)— F(x)]dx = o
(b) J:[G(x)— F(x)]dx Z 0 for all te [xl,x4]

(c) G(x)— F (x) is non-increasing on (x2 , x3 ) , where the support of F is contained in

[x2,x3], the support of G is contained in [xl ,x4 ].

27

 

 

A SIR is carried out by transferring probability mass from the interval (x2,x3) to

the left tail interval [x2 , x3] and the right tail interval [x3 , x4 ]. Compared with the result in

Eeckhoudt and Hansen (1981), the comparative statics result in SIR shifts includes a
larger set of R-S increases in risk and a general decision model. As a result, the SIR order
represents a net improvement over the mean-preserving truncation order without any cost
of additional assumptions. Note that the above subsets of R-S increases in risk are
defined by imposing restrictions on the difference between the initial and the final CDFs.
That is, the CDF difference approach is used as restrictions on changes in CDF.
A further generalization of a SIR is given by Black and Bulkley (1989) who

introduce the concept of a ‘relatively strong increase in risk’ (RSIR). They use a ratio
approach as restrictions on changes in PDF and their comparative static analysis is

carried out for the set of risk averse agents.

Definition 2.12. G(x) represents a relatively strong increase in risk from F (x)
(denoted by G RSIR F) if
(a) j;‘[G(x)—F(x)]dx=0
(b) For all points in the interval {x}, x, l, f (x) 2 g(x) and for all points outside this
interval f (x) S g(x) where xI S x2 S x3 S x4 S x5 S x6 , [x1 ,x6] being the supports of
x under G(x) and [x2 , x5] being the supports under F (x)
(c) f (x)/ g(x) is non-decreasing in the interval [x2,x3)

(d) f (x)/ g(x) is non-increasing in the interval (x,,x5].

28

 

 

 

 

Conditions (a) and (b) are sufficient for G(x) to represent a R-S increase in risk

from F (x) . That is, these conditions impose the restrictions that the two distributions
have the equal mean, two PDFs cross only twice and probability mass is transferred from
points within the interval (x3 , x4) to points lying outside this interval. Conditions (c) and
(d) restrict the extent to which probability mass can be transferred to any one value in the
tails of F (x) relative to any other.

According to Black and Bulkley, the RSIR conditions are satisfied in many
decision models if f(x) and g(x) are both normal distributions with f(x) = N(;z, 0,2)
and g(x) = N(/1, 0'22) where 0',2 > 0'12. Conditions (a) and (b) are obviously satisfied.

Since

 

a[f(x)/g(x)] _ _ _ L _ _1_ f(x)
ax ' (x “’ 2 0: g(x)
f (x)/ g(x) is increasing for x < ,u and decreasing for x > ,u. Hence (c) and (d) are also

satisfied. Note also that the conditions for a RSIR order are met if f (x) and g(x) are
both gamma distributions with the same mean.

By relaxing the restrictions imposed to the right of the point m, Kim (1998)
defines a ‘lefi-side relatively strong increases in risk’ (L-RSIR) that is a less stringent
type of R-S increases in risk than a RSIR order. We assume that the support of G(x) is a

finite interval [xl,x5] and the support of F (x) is another finite interval [x2,x,] where

x| szSx4Sx5.

Definition 2.13. G(x) represents a left-side relatively strong increase in risk from

29

 

 

 

F(x) (denoted by G L-RSIR r) if

(a) l:[6(x)— F(x)]dx = 0
(b) I: [G(x)— F(x)]dx Z 0 for all y e [x1,x5]

(c) There exists a point m 6 [x2 ,x,] such that F(x) S G(x) for all x e [x2,m) and
F(x) 2 G(x) for all x e [m,x4]
(d) There exists a point x3 6 [x2,m] such that f (x)/ g(x) is non-decreasing for all

x e [x2,x3) and f(x)2g(x) for all xe[x3,m].

Conditions (a) and (b) define R-S increases in risk. Condition (c) imposes the
restriction that the two CDFs cross only once at the point m. Condition ((1) implies that, to
the left of the point m, a L-RSIR order requires the same restriction used by Black and
Bulkley who define a RSIR one. When we assume that the payoff fimction is linear in the
random variable, the following result shows a trade—off between the restrictions on the set
of decision makers and the set of changes in distribution. Compared with the result in
RSIR shifts, the comparative statics result in L-RSIR shifis contains a larger set of
changes in distribution and a smaller set of decision maker with an additional assumption
such as u" 2 0.

Let us turn to a deterministic transformation approach. Meyer and Ormiston
(1989) provide a fourth characterization of a R-S increase in risk and introduce the

concept of a “simple increase in risk” which is obtained by further restricting the k(x)

functions to be monotonic.

 

 

Definition 2.14. The transformation t(x) represents a simple increase in risk for a

random variable given by F (x) if the function k(x) E t(x)— x satisfies
(a) fk(x)dF(x) = o
(b) f‘k(x)dF(x) s o, for all is [a,b]

(c) k'(x) 2 0.

Condition (a) guarantees that the mean of the random variable is preserved.
Condition (b) guarantees that for an increase in risk, the initial random variable
dominates the transformed random variable in the second-degree. These conditions
provide a fourth characterization of a R-S increase in risk. Condition (c) is the added
condition which identifies this particular type of an increase in risk and allows general

comparative static statements to be made.

Theorem 2.5. Facing a simple R-S increase in risk, a decision maker will
decrease the optimal value of a if
(a) u(z) displays DARA

(b) 2,20,2nS0, 211,20 and wa0.

Theorem 2.5 generalizes the Sandmo-Ishii result in some aspects: while the latter
uses a linear transformation of the random variable and a specific form of a decision
model such as the competitive firm model, the former adopts a more general class of

transformation and the general decision model which includes a specific form as a special

3l

 

 

 

case .

2.3.4 Gollier’s Work

Examining the restrictions used to do comparative static analysis gives a general
notion that given the set of decision makers, many authors restrict separately the changes
in distribution of the random variable (PDFs or CDFs) and the structure of the given
decision model to obtain the intuitively appealing comparative statics results. Gollier
(1995), however, restricts them jointly with a single restriction to obtain the necessary
and sufficient condition under which all risk averse individuals adjust the decision
variable in the same direction when faced with a given shift in a random parameter. This
implies that he obtains the least constraining condition on changes in risk that yields the
general comparative static statements for a given economic model and for the class of
risk averse agents. He presents the condition using the location-weighted probability
mass functions as follows:

T(x,or;G,z)S yT(x,0L;F,z), Vx 6 [a,b]
where 7 > 0, T(x,a;F,z)E rzadF(s) and T(x,a;G,z)E rzadG(s).

He restates the necessary and sufficient condition as requiring that

. T(x,a;G,z) T(x,a;G,z)
lnf —— 2 —.
{*leaJ’JFO} T(x,a;F,z) {.rrrunreho} T(x,a;F,z)

The above condition shows the importance of the ratios for two T functions. This implies
that the infimum of the left-side ratio (when T (x,a; F ,2) < O) is greater than or equal to
the supremum of the right-side ratio (when T(x,a; F, 2) > 0 ). The reverse case of his

condition yields ambiguous comparative statics.

 

 

He also introduces the concept of the greater central riskiness order when the

payoff function is linear in the random variable, z(x,a) = a x + 20 , where 20 is an

exogenous parameter. His condition is quite simple and generalizes all previous
restrictions on changes in risk such as a SIR and a RSIR imposed for all concave utility
functions. His another result shows that the SSD order is neither sufficient nor necessary
to get the result. By using the same technique in Gollier, we deal with this economic
problem for the class of risk averse individuals with a positive third derivative of their

utility functions that will be presented in chapter 5.

33

 

 

 

Chapter 3

COMPARATIVE STATIC ANALYSIS

FOR SUBSETS OF R-S INCREASES RISK

This chapter provides definitions and general comparative static statements
regarding several subsets of R—S increases in risk. We define two subsets of R—S increases
in risk called a ‘lefi-side strong increase in risk’ (L-SIR) and a ‘left-side relatively weak
increase in risk’ (L-RWIR). Each subset generalizes a definition in the established
literature. Specifically, a ‘strong increase in risk’ (SIR) in Meyer and Ormiston (1985)
and a ‘relatively weak increase in risk’ (RWIR) in Dionne, Eeckhoudt and Gollier (1993)
are extended to a L-SIR and a L-RWIR, respectively. For basic relations among these
subsets, the set of L-SIR shifts is included in the set of ‘lefi-side relatively strong
increases in risk’ (L-RSIR) shifts in Kim (1998), and the latter set is included in the set of
L-RWIR shifts. Whereas the L-SIR and the L-RSIR orders impose monotonicity
restrictions on the difference between two CDFs and on the ratio of a pair of PDFs,
respectively, the L-RWIR condition imposes a bound on the likelihood ratio in a specific
interval of the support under the initial distribution. These shifts allow only single
crossing between two CDFs for comparative static purposes.

When multiple crossing between two CDFs is allowed, we define a ‘monotone

 

 

 

strong increase in risk’ (MSIR) that imposes monotonicity restrictions on the ratio of the

two cumulative of cumulative distribution functions (C-CDFs). This shifi also generalizes
a definition in the literature. That is, an ‘extended strong increase in risk’ (ESIR) in Kim
(1998) is extended to a MSIR. The MSIR order implies that the conditional expectation
of a random variable under F is greater than or equal to that under G in the interval under
the initial distribution.

We also define two more subsets of R-S increases in risk with multiple crossing
called an ‘outside strong increase in risk’ (OSIR) and an ‘outside relatively strong
increase in risk’ (ORSIR). The conditions in these two shifts imply that there exists a R-S
decrease in risk in the given interval and a R-S increase in risk outside this interval. Note
that the OSIR ranking can be decomposed into two L-SIR shifts, and the ORSIR ranking
can be decomposed into a L-SIR shifi and a ‘left-side extended strong increase in risk’
(L-ESIR) shift. The basic relationship between these two shifts shows that the set of
OSIR shifts is included in the set of ORSIR shifis.

Restricting the payoff function to be linear in the random variable (2“ = 0), we

show that the effect of these shifts can be determined for all risk averse decision makers
with non-negative third derivative of utility functions u'” 2 0. This implies that we extend
the subsets of R-S increases in risk, but use somewhat stronger restrictions on the
structure of the decision model and the set of decision makers.
Section 3.1 provides definitions and comparative statics results for two subsets of
R-S increases in risk with single crossing. Section 3.2 gives definitions and comparative
statics results for several subsets of R-S increases in risk with multiple crossing. We also

provide other characterizations of the MSIR ranking in order to provide an interpretation

35

 

 

and to compare the MSIR ranking with the ESIR ranking. Section 3.3 contains some

remarks specific to this chapter.

3.1 Subsets of R-S Increases in Risk with Single Crossing

In this section, we begin by identifying a special category of risk increases that is
a subset of R-S increases in risk, called a ‘left-side strong increase in risk’ (L-SIR). The
L-SIR order is a less stringent type of R-S increases in risk than the SIR order proposed
by Meyer and Ormiston (1985), who imposes the restriction on the difference between
the two CDFs. We assume that the supports of x under G(x) are [x| , x4] and under F (x)

are [x2,x3] where xl sz Sx3 Sx4.

Definition 3.1. G(x) represents a left-side strong increase in risk from F (x)

F(x) (denoted by G L-SIR F) if
(a) I: [G(x)— F(x)]dx 2 0 for all y e [x,,x4]
(b) I: [G(x)— F(x)]dx = 0

(c) There exists a point m 6 [x2 ,x,] such that G(x)— F (x) is non-increasing on

xe(x2,m) and F(x)2G(x) for all xe [m,x4].

Conditions (a) and (b) imply that the L-SIR order is a R-S increase in risk. That is,

F dominates G in the second degree and the mean of the random variable is kept

 

 

fit

 

constant. Condition (c) imposes the restriction that the two CDFs cross only once at a
point m. This condition implies that, to the lefi of the point m, the L-SIR order requires
the same restriction used by Meyer and Ormiston to define the SIR order. Note that, to
the right of the point m, restrictions (F 2 G) imposed on L-SIR shifts and L-RWIR shifts
are same, and they are less restrictive than those on SIR shifts. Therefore the set of SIR

shifts is a subset of the set of L-SIR shifts.

 

 

 

0 xl x2 m x3 x4

Figure 3.1. G L-SIR F.

Figure 3.1 illustrates an example of a left-side strong increase in risk and a case
where restrictions on the difference between the two CDFs in the interval x e [m, x3) to
obtain a strong increase in risk are not met. Note that the L-SIR order can be obtained
from the SIR one by relaxing the restrictions imposed to the right of the point m.

Dionne, Eeckhoudt and Gollier (1993) introduce the concept of a ‘relatively weak

37

 

 

 

increases in risk’ (RWIR). In the second definition, we propose the ‘lefi-side relatively

weak increase in risk’ (L-RWIR) order that extends the RWIR order.

Definition 3.2. G(x) represents a left-side relatively weak increase in risk from

F(x) (denoted by G L-RWIR F) if

(a) I: [G(x)— F(x)]dx Z O for all y e [xl,x5]

(b) fflG(x)— F(x)]dx = 0

(c) There exists a point m e [x2,x4] such that F(x) S G(x) for all x e [x2,m) and
F(x) 2 G(x) for all x e [m,x,]

((1) There exists a point x3 6 [x2,m] such that f (x) S g(x) for all x e [x2,x3) and
f(x)2 g(x) for all x e [x3,m]

(e) When 1:. e [x2,x3), the following condition is satisfied:

g(x)>gx' x <x<x’
f(x)’fx" " ‘ ’
Afl<gx. x°<x<x
f(x)—fx" ‘ ‘ 3’

where x. is the value of x satisfying za (x,a,, ) = 0 in the interval [x2 , x4 ].

Conditions (a) and (b) define R—S increases in risk. Condition (c) imposes the
restriction that the two CDFs cross only once at a point m. Conditions (d) and (e) imply
that, to the left of the point m, the L-RWIR order requires the same restriction used by

Dionne, Eeckhoudt and Gollier to define the RWIR order. Note that, to the right of the

38

 

 

 

point m, there is no restriction on the number of times of crossing between the two PDFs,

nor is there any restriction on the ratio of a pair of PDFs. The restriction on the right of
the point m is that F S G. Therefore the set of L-RWIR shifis includes the set of RWIR
shifts. Note also that the above condition (e) is less restrictive than that proposed by Kim
(1998), who imposes a restriction on the sign of the derivative of the likelihood ratio

when defining the L-RSIR order.

f (x) g(x)

 

 

 

 

 

 

 

 

x1 x2 x x3 m x4 x5

Figure 3.2. G L-RWIR F.

Figure 3.2 illustrates an example of a left-side relatively weak increase in risk and

a case where a monotonicity restriction on the interval x 6 [x2 , x3) to obtain a left-side

relatively strong increase in risk is not met. Compared with the RWIR order, the L-RWIR

order is obtained by relaxing the restrictions imposed to the right of the point m.

39

 

Observe finally that the set of L-SIR shifts is included in the set of L-RSIR shifts

that, in turn, is included in the set of L-RWIR shifts. That is, G L-SIR F :> G L-RSIR F

=> G L-RWIR F.

3.1.1 Comparative Static Analysis

In this subsection, we provide general comparative static statements concerning
the L-SIR and the L-RWIR orders. In what follows, the payoff function is restricted to be
linear in the random variable. Using the simple form of the general decision model such

as z(x,a) E 20 + a x , the necessary and sufficient condition for a,, is written as
fu'(zo + an) xdF(x) = 0. (3.1)
It is well known that a, has the same sign as E ,, (b) = bedF (x) (see Dionne, Eeckhoudt

and Gollier (1993)). Therefore, we assume that E F (b) is positive, as it is the case in the
applications presented in chapter 2. In order to prove a, 2 do for a specified change in

PDF (or CDF ) from f (or F ) to g (or G ), it is sufficient to show that for all x 6 [a,b],

b
Q(a,,. ) = Lu’(zo + a,,x)xd[F(x)— G(x)] 2 O. (3.2)
The following comparative statics results indicate that, when 2“ = 0 , one can

further extend the subset of R-S increases in risk with the cost of adding additional
restriction on the risk preferences of decision makers. Since the set of L-SIR shifts is

included in the set of L-RWIR shifts, we only consider the L-RWIR shifts.

Theorem 3.1. If G L-RWIR F and z” = 0 , then a,, 2 a“. for all risk averse

40

 

 

decision makers with u’" 2 0.

Proof: Using the general decision model z(x,a) in (3.1), let x. be the value of x
satisfying 2,, (x,a,,. ) = 0 , and then x' exists in the interval [x2, x4 ]. We consider the

following three cases:

Case (i): x2 S x' S x3.
We consider the sign of the expression 1:2,, (f — g)dx. First, assume that
J: (f—g ).dx Rewriting Q(a, ) m (3.2) as
Q(ar)=j:31’(2..—(f g)dx+j: u’)2(z (f-:ug)dX+l '(Z)Z (f-g)dx-
Using the given assumptions and the L-RWIR condition, we have
Q(a. ) 2 u’lz(x1ar )1 L72. (f — g)dx + u'[z(m.a. )1}: 2.. (f — g)dx
+ I: u'(z)z.. (f — g)dx. (3.3)
Adding and subtracting u '([z m, a, )sz" (f— g)dx in the RHS of (3. 3) gives
Q(a,. ) 2 {u'[z(x' , (2,, )]— u'[z(m, (1,, )] ”:2“ (f — g)dx
+u’l2(m ar )ljza (f- g)dx +j;'u'( 2..(-f g)dx (34)

Since u ’(2) IS non-increasing and [:20 (f— g)dx, the first term in the RHS of (3. 4) ls

non-negative. Integrating the second term in (3.4) by parts and using the assumption

41

 

Add,

 

z” = 0, we obtain

u'[z(m,aF )H: z,er [G(x)— F(x)]dx = u'[z(m, 01,, )] 2m I:[G(x)— F(x)]dx. (3.5)
Also using integration by parts, the third term in (3.4) is equal to

g(a, ) = J: u"(z)z_,z,, [G(x)— F(x)]dx + I“ u'(z)zm [G(x)— F(x)]dx. (3.6)
Since 2,, is positive and [G(x)— F (x)] is non-positive for all x e [m,x5], the first term in

(3.6) is non-negative. Thus, we have
(1(a); lfu'(z)z...[G(x)—F(x)]dx- (3.7)
Because u’(z) 20“ is non-increasing and [G(x)— F (x)] is non-positive for all x e [m,x5],

we have

q(a..< ) 2 u'[z(m,a. )12. I,” [G(x)— F(x)]dx- (3.8)

Hence, from (3.5) and (3.8), the second and third term in (3.4) can be written as
u'l2(msar life. (f - g)dx + l. u'(Z)Z.. (f - g)dx
2 u'[z(m,a,_. )]za, I:[G(x)— F(x)]dx = o. (3.9)
Therefore, Q(a,) is non-negative.
Second, assuming that 1:112 (f — g)dx s o, let’s rewrite Q(aF) in (3.2) as
Q(a..~)2 {guided-gm”u'lztm.a..-)1L“,z.(r—g)dx
+ j;’u'(z)z,(f—g)dx. (3.10)

Adding and subtracting u’[z(m,a[, )Hxiz,z (f — g)dx in the RHS of (3.10) gives

42

 

 

bilhg ‘

 

 

Q(-ar)>I:u u’)2.(z (f- g)dx- quzmarllIL-ZU g)dx

+u'([2m,a,)]I: 2 (f— g)dx+J:u( ’(asz)z (f—g)dx.
From (3.9) and the assumption £32“ (f — g)dx S 0 ,

Q(a.~)2Lx‘u’(2)2.(f-g)dx=-L" u’(2)2.gdx+L:’u’( )2.(f- g)dx (3.11)

The first term with minus sign in the RHS of (3.1 1) is non-negative, and by applying the

condition (d) in Definition 3.2, the second term can be written as

Iju u'zz() ..—(f g)=dx If}: '(.z)z[1—7)fdx>Il— §::]I:U'(2)2.fdr.

Since g(x‘)/ f ( )>1 and I:u'( )2 af dx S O by the first-order condition, Q(a,) ls

non-negative.

Case (ii): x3 S x. S m.

Let’s rewrite Q(a,,) in (3.2) as

Q(a.)= I (2)2 (f g)dx+I.u '.(z)z (f- g)dx+ I )2.(f- g)dx

"I

Using the given assumptions and the L-RWIR condition, we have
Q(a )>u’(l2x..a. )lI:2 .(—f g)dx+u'(lzm a.)lL.2. (f— g)dx
+ I;5u'(z)za(f—g)dx. (3.12)

Adding and subtracting u '([2 m, a, )IIjza (f— g)dx 1n the RHS of (3. 12) gives

43

 

 

 

Q(ar)2{u'l2(x.,ar)l-u’l2(m.ar)l}L:'2.(f—g)dr

+ u'[z(m,a.~ )lez. (f —g)dx + Ij’u'(z)z.(f—g)dx. (3.13)

From (3.9), the second and third term in the RHS of (3.13) is non-negative. Thus, we

have
Q(a.~ ) 2 man... )]— u'lz(m.a. )1 if 2.(f — g)...
Integrating by parts,
I:2.(f-g)dx = fields-Fiona: = 2.. chbhrbimo
because 2“, is non-negative and does not depend on x, and I: [G(x)— F(x)]dx 2 0 for all

(e [x,, x5]. Thus, by the assumption u”(2) S 0 , Q(aF) is non-negative.

an. O
Case (111): m S x S x,.

Integrating by parts, Q(a,..) can be written as
9(a): Ijlu"(z)z.z. +u’(z)z-l[G(x)—F(x)]dx-

Note that u"(z)zxza + u’(z)z is positive and non-increasing in x in the interval [xI , x'l,

(IX
and it has its maximum at x = x. in the interval [x‘, x5] because u”(z)zxza is always

non-positive and u’(2)zax is non-increasing in x. Since m S x' , this implies that

lu'(2)2.2. + u’(2)2..l

 

 

 

m" 2 [u'(z)z,za + u'(2)za,]

Z lu”(2)2.2. + u'(2)2..l

x2”! '
By the L-RWIR condition which implies G(x)— F (x) Z 0 for all x e [xl ,m] and

G(x)— F (x) S O for all x e [m, x5], we have the following inequality,

44

 

 

 

pr(

 

Q(a,.~) 2 lu”(2)2.2. + u'(2)2..l

 

2., L100) - F (20] db: = 0- Q.E.D.

Remember that the L-SIR order implies the L-RWIR order. Therefore we obtain

the following result.

Corollary 3.1. If G L-SIR F and z” = 0 , then (IF 2 ac for all risk averse

decision makers with u’" _>_ 0.

While 2,“ = 0 restricts the set of decision problems, linear payoffs prevail in
many economic environments; as indicated in chapter 2, these include the standard
portfolio model, the optimal behavior of a competitive firm with constant marginal costs,
and the coinsurance problem. Compared with the comparative statics results in L-SIR
shifts and L-RSIR shifts in Kim (1998), the result in Theorem 3.1 includes a larger set of
R-S increases in risk. As a result, the L-RWIR order represents a net improvement over
the L-SIR and the L-RSIR orders without any cost of additional assumptions. That is,
Theorem 3.1 improves the robustness of the results in Corollary 3.1 and Theorem in Kim
(1998). Compared with the comparative statics results in SIR shifts and RWIR shifts in
the published literature, the results in Corollary 3.1 and Theorem 3.1 contain a larger set
of changes in distribution, a smaller set of decision makers and a smaller set of decision

problem.

3.2 Subsets of R-S Increases in Risk with Multiple Crossing

45

 

 

Many researchers have obtained intuitively appealing comparative statics results

by restricting the changes in distribution of the random parameter when two CDFs cross
only once. When multiple crossing between the initial and the final CDF F and G is
allowed, this section provides definitions and general comparative static statements
regarding several subsets of R-S increases in risk with multiple crossing.

Kim (1998) defines an ‘extended strong increase in risk’ (ESIR) that imposes
monotonicity restrictions on the ratio of a pair of CDFs. We introduce the less stringent
type of R—S increases in risk and name it a ‘monotone strong increase in risk’ (MSIR).
Instead of using monotonicity restrictions on the ratio for CDFs, the MSIR order is
defined by imposing monotonicity restrictions on the ratio of the two cumulative of CDFs

(C—CDFs).

Definition 3.3. G(x) represents a monotone strong increase in risk from F (x)

(denoted by G MSIR F) if
(a) j’ [G(x)— F(x)]dx 2 0 for all y 6 [a,b]
b
(b) L [G(x)— F(x)]dx = 0
(c) There exists a non-negative and non-increasing function H (x): (c,b] —) [1, 00) such
that F(x) = F(x) = 0 for all x e [a,c] and G(x)/F(x) = H(x) for all x e (c,b] where

F(x): J:F(t)dt = fF(t)dt, G(x): J:G(t)dt and a s c s b.

The first two conditions guarantee that G represents a R-S increase in risk from F.

Condition (c) does not restrict the number of times of crossing between the two CDFs.

46

 

 

 

This comes from replacing a monotonicity restriction on the ratio of a pair of CDFs with

a monotonicity restriction on the ratio of a pair of cumulative of CDFs (C-CDFs). Note
that the ESIR order defined in Kim (1998) requires that two CDFs cross only once.
Therefore the MSIR ranking is less demanding than the ESIR ranking. In next subsection,
we will provide an equivalent definition of the MSIR order with a nice interpretation, and
show that an ESIR order implies a MSIR one. Figure 3.3 shows an example of a

monotone strong increase in risk that is not an ESIR.

F(x), G(x)

G(x) F(x)

 

 

a c b

Figure 3.3. G MSIR F.
We also propose two more subsets of R-S increases in risk called an ‘outside

strong increase in risk’ (OSIR) and an ‘outside relatively strong increase in risk’

(ORSIR), where the two CDFs have multiple crossing. We assume that the support of

47

 

 

 

F (x) is contained in some finite interval [x2 ,x6] and the support of G(x) is contained in

another finite interval [x,,x7] where xl S x2 S x3 S x4 S x5 S x6 S x7. First, we introduce

an OSIR shifi that is defined as:

Definition 3.4. G(x) represents an outside strong increase in risk from F (x)

(denoted by G OSIR F) if
(a) I: [G(X)— F (x)ldx = 0
(b) Iy [G(x)— F(x)]dx > 0 for all y e[x1,x7) and I: [G(x)— F(x)]dx < 0 for all
y e (x. , x7]
(c) G(x)— F(x) is non-increasing on x e (x,,x,), F(x) 2 G(x) for all x e [x,, x,],
F(x) 3 G(x) for all x e [x,,x,], I:[F(x)— G(x)]dx = 0 and F(x) 2 G(x) for all

xe(x5,x7].

Conditions (a) and (b) define a R-S increase in risk. Condition (c) implies that the
two CDFs have multiple crossing and to the left of the point x3 , the OSIR ranking
requires the same restriction used by Meyer and Ormiston to define the SIR order. These
conditions imply that there exists a R-S decrease in risk in the interval [x3, x5] and a R-S
increase in risk outside this interval. Thus, we name this increase in risk the OSIR order.
Figure 3.4 illustrates an example of the OSIR order. The OSIR ranking can be
decomposed into two left-side strong increases in risk (L-SIR). That is, if G L-SIR GI

and GI L-SIR F, then G OSIR F.

48

 

 

 

 

 

 

 

0 xl x2 x3 x4 x5 x6 x7

Figure 3.4. G OSIR F.

We also define an ‘outside relatively strong increase in risk’ (ORSIR) that relaxes

the restriction imposed to the left of the point x3 in OSIR shifis.

Definition 3.5. G(x) represents an outside relatively strong increase in risk from

F(x) (denoted by G ORSIR r) if
(a) J;’[G(x)-F(x)1¢v=o

(b) I:[G(x)— F(x)]dx > 0 for all y e [x| ,x7) and I: [G(x)— F(x)]dx < 0 for all

49

 

 

 

 

y e (x19x7l

(c) G(x)/F(x) is non-increasing on x 6 (x2 ,x3) , F(x) 2 G(x) for all x e [x3,x,,],
F(x)S G(x) for all x e [x4,x5], I:5[F(x)— G(x)]dx = 0 and F(x) 2 G(x) for all

xe(x5,x7].

 

 

 

 

 

 

0 x1 x2 x3 x4 x5 x6 x7

Figure 3.5. G ORSIR F.

The conditions in ORSIR shifts are same as those in OSIR shifts except that, to
the left of the point x3 , G(x)/ F (x) is non-increasing. Note that the OSIR order implies

the ORSIR order. Figure 3.5 illustrates an example of the ORSIR order that cannot be the

OSIR order. The ORSIR ranking can be decomposed into a ‘left-side strong increases in

50

 

 

 

PI

(3.]

This

 

risk’ (L-SIR) and a ‘left-side extended strong increase in risk’ (L-ESIR). That is, if G L-

SIR GI and GI L-ESIR F, then G ORSIR F.

3.2.1 Other Characterizations of the MSIR Order and Comparison with the ESIR Order
In this subsection, we provide other characterizations of the MSIR ranking in
order to provide an interpretation and to compare the MSIR order with the ESIR one.

First, we demonstrate that the following three conditions are equivalent.

Lemma 3.1. The following three conditions are equivalent.

(a) G MSIR F for all x e (c,b].

)

(x) < G(x
F (x) _ G(x)

V

for all x e (c,b]

 

 

(b)
where F(x) = I:F(t)dt = fF(t)dt since F(x): 0 for x S c and G(x): I:G(t)dt.

(c) EFIIIthIZEGIIIISx] forall xe (c,b].

Proof: We rewrite a MSIR condition as: For all x e (c,bI
G(x)/F(x) = H(x) (3.14)
where H (x)is a non-negative and non-increasing function.

(3.14) is equivalent to

 

i x :1 am =G<x>fi<x)—F(x)é<x)<
H” I I m.) ‘0'

This is in turn equivalent to

51

 

 

the!

latte

d 10g

Ihaul)

 

 

 

G(x)F(x)S F(x)G(x) 4: mos G(x) (3.15)

F(x) G(x)
Thus, conditions (a) and (b) are equivalent.

Condition (c) can be written as follows: since E ,, [t] t S x] is not defined for

xSc,forall x>c,

EtdF(t) I:th(t)
W 2 m— . (3.16)

By integrating the numerator of (3.16) by parts, we obtain

xF(x)— I:F(t)dt > xG(x)— I:G(t)dt
F (x) _ G(x)

 

’

or, equivalently,

 

 

F(x) < G(x)

F(x) _ G(x)'
Thus, conditions (b) and (c) are equivalent. Since condition (a) is equivalent to condition
(b) which is also equivalent to condition (c), condition (a) is equivalent to condition (c).

This concludes the proof. Q.E.D.

Condition (a) implies that the MSIR order imposes a monotonicity restriction on

the ratio of a pair of cumulative of CDFs (C-CDFs). Since condition (b) can be written as
F (x)/ F (x) 2 G(x)/G(x) , it implies that F (x) is smaller than G(x), but the percentage
rate of increase for F (x) is always greater than or equal to that for G(x). That is,

d log F(x)/dx is always greater than or equal to d log G(x)/ dx . Condition (c) implies

that the conditional expectation of a random variable under F is greater than or equal to

52

 

 

 

 

that under G for all x e (c,b]. Note that in the condition (c) f((t) = f (t)/ F (x) is a

dF(t) is a mean value of the given
F (x)

 

conditional PDF with its support Ia, x] and Ixt

interval.

When the payoff function is linear in the random variable, we compare the MSIR
condition with the Gollier (1995) condition. As indicated in chapter 2, the Gollier
condition can be written as

y I‘tdF(t) 2 I’szQ) for all x
where y is a scalar. The MSIR condition in (3.16) indicates that a scalar y is replaced
with G(x)/F(x) that is a function ofthe random variable x. The Gollier condition can be

written as 7 = max%. If 7 is big enough (for example 7 ——) co) and IxtdF(t) is
x a

negative for some x, the above condition is violated. Therefore the MSIR condition is less
restrictive than the Gollier condition.
In order to compare a MSIR order with an ESIR one, recall that Kim (1998)

formally defines an ESIR order as:

Definition 3.6. G(x) represents an extended strong increase in risk from F (x)

(denoted by G ESIR P) if

(a) I:[G(x)— F (x)]dx 2 0 for all y 6 [a,b]

o) flan-Follow

(c) There exists a non-negative and non-increasing function h(x): (c, b] —> ( 0, 00) such

53

 

 

 

 

that F (x) = 0 for all x e [a,c] and G(x) = h(x) F (x) for all x e (c,b] where

aSch.

Conditions (a) and (b) simply require that an extended strong increase in risk be a
R-S increase in risk. Condition (c) implies that the two CDFs cross only once, but it does
not restrict the number of times of crossing between the two PDFs.

Whereas the ESIR order is linked to the derivative of the ratio of a pair of CDFs,
the MSIR order is linked to the derivative of the ratio of a pair of C-CDFs. The assertion
that the ESIR order implies the MSIR order can be made by showing that the condition
that the ratio of a pair of CDFs, h, is non-increasing implies the condition that the ratio of

a pair of C-CDFs, H, is non-increasing. We prove the following lemma.

Lemma 3.2. The ESIR order implies the MSIR order.

Proof: Before proving this lemma, we define I:l(t)dt = i (x) where I = F or G. First,
notice that by the definition of ESIR, we have for m 2 c

G(m) = G(c) + If h(x) F(x)dx
with h(x) being non-negative and non-increasing. Define function H (m) as

H(m)=L;IEf:))Nfl forall m2c.

Combining the above two equations yields G(m) = H (m)F (m) for all m > c. Therefore,

we have to prove that H (m’) S H (m) for all c < m < m'. We have

 

 

 

llll'

 

G(c)+ f’a(x)p(x)ax
F(m')

H(m')=

_ G(c)+ Lmh(x)F(x)dx + I:'h(x)p(x)ax

F(m’)

 

= H(m)F(m)+ I:'h(x)p(x)ax
F(m’) '

 

Because h(x) is non-increasing, we obtain

 

. < H(m)fi(m)+h(m)lfi(m')—F(m)l
”M“ Na)

Adding and subtracting H (m)F (m’) in the above inequality gives

H(m.) S H (”OF (M)- H ("M (M')+ H((m)f‘(m')+ h(m)[F“(m')- 13" (m)] _
F m’

This implies in turn that

, < [F(m')-F(m)][h(m)—H(m)] + m
H(m)- F(m') H( )-

 

It follows that H (m’) S H (m) if [F(m’) — F(m)][h(m)— H (m)] is non-positive. Since m'

is larger than m, this is equivalent to h(m) S H (m), or

h(m) S G(c)+ [2:3 F(x)dx

This is always true since

G(c)+ I”h(x)F(x)dx > fh(x)F(x)dx _h(m)_ I: h’(x)I:F(t)dtdx
F(m) ‘ F(m) ‘ F(m)

 

2 h(m).

55

 

 

Since h'(x) S 0 for all x e (c,b], this concludes the proof. Q.E.D.

Observe that a MSIR shift imposes less stringent restrictions on the change in the
random variable than an ESIR shifi does. Lemma 3.2 states that as subsets of R-S
increases in risk, the set of all CDF changes satisfying the MSIR order includes the set of
all CDF changes satisfying the ESIR order.

Kim (1998) also introduces the concept of a ‘lefi-side extended strong increase in
risk’ (L-ESIR) that relaxes the restrictions imposed on ESIR shifts. That is, this class of
R-S increases in risk includes, as a special case, the ESIR order. The L-ESIR order is

formally defined as:

Definition 3.7. G(x) represents a left-side extended strong increase in risk from

F(x) (denoted by G L~ESIR F) if
(a) Iy[G(x)— F(x)]dx 2 0 for all y 6 [a,b]
b
(b) I, [G(x)— F(x)]dx = 0
(c) There exists a point m e [ab] and a non-increasing function h(x): (c, m] —> [1, 00)

such that G(x) = h(x) F(x) for all x e (c,m] and G(x) S F(x) for all x e (m,b] where

aSch.

The L-ESIR order has the same restriction as the ESIR one in that the two CDFs
cross only once at the point m, but it is less restrictive than the ESIR order since the L-

ESIR order requires only the left-side monotonicity of the CDF ratio.

56

 

 

 

 

The following lemma shows that the set of L-ESIR shifts is also included in the

set of MSIR shifts.

Lemma 3.3. G L-ESIR F implies E, [I] I S x12 EGIII t S x] for all x e (c,b].

That is, the L-ESIR order implies the MSIR order.

 

Proof: Since E ,, [t | t S x] is not defined for x S c , we have to prove that for all x > c ,

I:tdF(t) > I:th(t)
F(x) _ G(x)

 

 

By integrating the numerator of the above equation by parts, we obtain

I:F(t)dt < IaxG(t)dt

F(x) ' G(x)

 

 

, (3.17)

or, equivalently,

h(x)F(x)I:F(t)dt s F(x)I:h(t)F(t)dt.
This is in turn equivalent to
h(x)S Ifh(t)a)(l)dt ,
where (00) = F (t)/ ILXF (y) dy. For all x e (c,m], the fact that h(t) is non-increasing

establishes this lemma. For all x e (m,b], from (3.17),

f I:G(t)dt > su G(x)

IaxF(t)dt _ pF(x)

=1.

 

 

This concludes the proof. Q.E.D.

57

 

 

 

 

 

 

 

Pro

 

 

3.2.2 Comparative Static Analysis

In this subsection, we provide general comparative static statements regarding the
MSIR, the OSIR, and the ORSIR orders. Before we present the comparative statics result

in a MSIR shift, Kim (1998) obtains the following result.

Theorem 3.2. For all risk averse decision makers, aF 2 an. if
(a) G ESIR F

(b) 2,20,2m20,zm,so and zauSO.

Kim shows that if G ESIR F, then there exists a series of RSIR shifts. That is, any
ESIR shift can be decomposed into a series of RSIR shifts. Remember that Theorem 3.2
shows a net improvement over the result in Black and Bulkley’s analysis.

Before proving Theorem 3.3, we need the following lemmas. First, we define

frdFQ) _ ftdFO)

F‘" 7W to)

 

for all x > c since IxtdF(t) = xF(x)— F(x) and F(x) = O

A

for all x S c _ Note that k;(x): W Z 0 for all x > C . We assume that (2,. =1 and
x

20 = 0 for notational convenience.
Lemma 3.4. 6 (r) = Iru"(x) k,,. (x)F(x)dx is always non-negative for all r e [c,b].

Proof: Note that 6(c) = 0. From (3.1), integrating by parts of 6 (r) gives

58

 

 

 

 

5 (r) = u’(r) kF (r)F(r)— Icru'(x) xdF (x) We know that [f(x) xdF (x) S 0 for all

r e [6, b]. Let x0 denote the point where k ,, (x) changes sign from negative to positive.
For all r 2 x0 , 6 (r) 2 0 since the first part is positive and the second one with minus is
also. For all r S x0 , 6 (r) is an increasing function since u”(x) k,,. (x)F (x) 2 0. Given that

6(c)= 0 , 6(r) 2 0 for all r S x0 . This concludes the proof. Q.E.D.

Lemma 3.5. (p(r) = — I: [u'(x) k}, (x)+ u"(x) k,.. (x)] F(x)dx is always non-

negative for all r e [ab].

Proof: Note that (p(c) = 0. From (3.1), integrating by parts of (p(r) gives
¢(r) = u’(r) kF (r)F (r) — u"(r)k,, (r)F(r)— Iru’(x) xdF (x) We know that
Iru’(x) xdF (x) S 0 for all r e Ic,b]. Let x0 denote the point where k ,, (x) changes sign

from negative to positive. For all r 2 x0 , (0(r) 2 0 since the first part is positive, and the

second and the third one with minus are also. For all r S x0 , (0(r) is an increasing
function since — [u”(x) k} (x) + u'"(x) k). (x)] F (x) 2 0. Given that (p(c) = 0 , ¢(r) 2 0 for

all r S x0 . This concludes the proof. Q.E.D.

Theorem 3.3. If G MSIR F, 210r = 0 and zaa = 0, then 01,, 2 a0 for all risk

averse decision makers with u'" 2 0.

59

 

 

Proof : The case with an unbounded a, yields an obvious comparative statics property

(see Dionne, Eeckhoudt and Gollier (1993)). Therefore. without loss of generality, we

assume that (1,, =1 and 20 : 0 (for notational convenience). We define E, (x): rid/(I)

and k,(x)= E, (x)/F(x) where I = F or G. From (3.2).

Q(a,,- ) = u'(b)I:xdF(x) — Iju'(x)LrldF(t)dx

_ u(z))fxaob). [fawnfiaobm
=u'(E>1E,.(b>— E.,.(E>l— Ifu~lx>1E.(x)- E.(x)]dx-

(3.18)
Since from = foam, E,(b)= E (a). Therefore, (3.1s) becomes
Q<a.)= -1,”u~(x)1E,..(x)— E (x)]dx
= fu"(x)k.(x)G(x)dx— fu~(x)[k,,(x)E(x)_k,,(x)o(x)].a. (3.19)

Since aF is finite, the lower bound of the support of F (x)- which is denoted c- must be

negative and k6 (x) S (c S)O for all x S c . Therefore, the first term in the RHS of (3.19)
is non-negative. Adding and subtracting Ihu'(x) k,, (x)G(x)dx in the RHS of (3.19) gives

Q<a.>2—1,,”u~(x>k.(x)1E(x)-G<x>1dx—l”u'<x>lk..(x)-h(x)]abldx. (3.20)

Since, by the condition (c) in Lemma 3.1, kG (x) is less than or equal to k ,, (x) for all x in

the support of F (x), the second term with minus sign in the RHS of (3.20) is non-

negative. Therefore, (3.20) implies that

on.» - futon. (x)1E<x)- G(x)]dx.

(3.21)

60

 

Integrating the RHS of (3.21) by parts yields

Q(aF)2 —u”(b)kr (b) I " [F(x)- G(x)]dx
+ I." [u'(x)k}(x)+ u~(x)k,,(x)]I I:F(t)dt— I:G(t)dtIdx. (3.22)

I:tdF(t)
F(c)

u"(b)kF (b) I G(x)dx = I‘ G(t)dt I" [u"(x)k;, (x)+ u”(x)k,.(x)]dx and

We assume that k,,. (c) = 0 since k,,.(c) = , F (c) = 0 and F (c) = 0. Since

I:F(x)dx = I"F(x)dx, adding and subtracting u"(b)k,, (b) I‘ G(x)dx in the RHS of
(3.22) implies that
Q0212)Z -u”(b) kr (b) I: [F (x) - G(x)]dx
+ I." [u~(x)k;,(x)+ u~(x)k,(x)]I from _ fo(r)atI.a. (3.23)
Since I" [F(x)— G(x)]dx = 0, the first term in the RHS of (3.23) is zero. Since

IXF(t)dt = IXF(t)dt = F (x), integrating the second term in the RHS of (3.23) by parts

yields
Q(ar)211- H(b)1f 1u~(x>k;.(x)+ u"(x)k.(x)]fi(x)dx
+ 13mm:1u~(z)k;..(t>+u~(r)k.(r)lE(t)drEx. (3.24)
Since H(b)=1, the first term in the RHS of(3.24) is zero. Since H'(x) s o for all
x e (c,b] and (p(x) = I " [u"(t)k;,. (t)+ u"(t)k,. (I)]F(t)dt s o for all x e [c,b] by Lemma

3.5, the second term in the RHS of (3.24) is non-negative. Therefore, Q(a, ) 2 0. This

61

 

 

 

concludes the proof. Q.E.D.

Remember that from Lemma 3.3, the L-ESIR order implies the MSIR order.

Therefore we obtain the following result.

Corollary 3.2. If G L-ESIR F, z” = 0 and zaa = 0, then aF 2 do. for all risk

averse decision makers with u" 2 0.

Since the class of MSIR shifts contains the class of L-ESIR shifts, the result in
Theorem 3.3 improves the robustness of the comparative statics result in L-ESIR shifts.
The efficiency gain is obtained from extending the admissible set of R-S increases in risk
yielding the result, without any cost of additional restrictions. Note also that, compared
with the result in ESIR shifts, our result in Theorem 3.3 extends the set of R-S increases
in risk with the cost of adding additional restrictions (er = 0 and u’" 2 0) on the
structure of the decision model and the risk preferences of decision makers.

Next theorems concern the OSIR and the ORSIR orders. As indicated in the
beginning of this chapter, the OSIR ranking can be decomposed into two L-SIR shifts
and the ORSIR ranking can be decomposed into a L-SIR shift and a L-ESIR shift. We

use these properties to prove the following theorems.

Theorem 3.4. If G OSIR F and z“ = 0 , then a). 2 ac. for all risk averse

decision makers with u" 2 0.

62

 

 

 

 

 

 

det

inc

COI

35$

 

513'.

and

 

Proof: It is clear by Corollary 3.1 in L-SIR shifts.

Theorem 3.5. If G ORSIR F and 2n = 0 , then 05,, 2 at“ for all risk averse

decision makers with u" 2 0.

Proof: It is clear by Corollary 3.1 in L-SIR shifts and the proof in L-ESIR shifts in Kim

(1998).

The basic relationship between the comparative statics results in these two shifts
shows that the ORSIR ranking represents a net improvement over the OSIR one. That is,
when the same restrictions on the structure of the decision model and on the set of the
decision makers are used in the analysis of these two shifts, the set of ORSIR shifts

includes the set of OSIR shifts.

3.3 Concluding Remarks

While linear payoffs turn out to be a special case in the set of decision problems
under uncertainty, they appear to prevail in many economic environments. Reviewing the
comparative static analysis in this chapter shows that, when z” = 0 and u" > O are
assumed, more general subsets of R—S increases in risk are allowed in this comparative
static analysis. Notice that our study in this chapter follows the CDF difference approach

and the ratio approach in order to generate interesting comparative static statements.

63

 

From the results in this chapter, it seems that the effect of adding u" > 0 is to focus on

imposing restrictions on the left-hand side of two PDFs or two CDFs. These properties
are found in the analysis of the L-SIR, the L-RSIR, the L-RWIR, the OSIR and the
ORSIR orders.

Most comparative statics results in the published literature and this study are

 

associated with whether the two CDFs cross only once. This single crossing property is
not a restrictive assumption since members of some of well-known families of
distributions have this property. There is a list of these distributions; uniform, normal, log
normal, binomial, exponential and others. This list only applies if two random variables
are each from the same family; for example both must be normally distributed or both
must be uniformly distributed. Note also that a pair of random variables, each from a

different family may not guarantee single crossing.

64

 

Chapter 4

COMPARATIVE STATIC ANALYSIS

FOR SUBSETS 0F K-L INCREASES IN RISK

This chapter provides general comparative static statements regarding several
subsets of increases in risk in the K-L sense. In order to obtain comparative statics
results, many researchers have specified particular types of CDF changes which are
subsets of FSD shifts, SSD shifts, or R-S increases in risk. So far, no one has examined
subsets of TSD shifts for comparative Static purposes. Kroll et al. (1995) define a special
‘increase in risk’ as a TSD change with equal means. Note that whereas R-S increases in
risk are SSD changes with equal means, K-L increases in risk are TSD changes with
equal means. Since SSD implies TSD, the set of K-L increases in risk includes the set of
R-S increases in risk.

In this chapter, we consider three subsets of K-L increases in risk. These are a
‘strong increase in risk in the K-L sense’ (SIRK), a ‘simple increase in risk across r in the
K-L sense’ (sIRK (r)) and a ‘relatively strong increase in risk in the K-L sense’ (RSIRK).
Notice that, since a K-L increase in risk is a particular type of TSD change, the derived
comparative static statements are associated with a risk averse decision maker with

u'" 2 0. In section 4.1, we provide definitions for subsets of K-L increases in risk used in

65

 

 

 

 

 

 

 

 

this study. In section 4.2, we do comparative static analysis concerning the three types of

K-L increases in risk defined in section 4.1.

4.1 Subsets of Increases in Risk in the K-L Sense

This section gives the definitions for three subsets of increases in risk in the K-L

 

sense. Meyer and Ormiston (1985) consider the concept of a ‘strong increase in risk’
(SIR), a particular type of R-S increases in risk defined by additional restriction on the
difference between the two CDFs. Using the similar restriction, we introduce a ‘strong
increases in risk in the K-L sense’ (SIRK) which is defined by imposing restrictions on
the difference between the two cumulative of CDFs (C-CDFs). We assume that the

supports of F (x) and G(x) are located in the interval (x2,x3) and (x,,x4), respectively,

wherex Sx Sx Sx4.WedefinethatFx = IFI dt ande = th dt.
1 2 3 I] x.

Definition 4.1. G(x) represents a strong increase in risk in the K-L sense with

respect to F (x) (denoted by G SIRK F) if
(a) 1: [G(x)—F(x)]dx=0
(b) I: [G(x)— F(x)]dx 2 0 for all s e[x1,x4]

(c) G(x)— F (x) is non-increasing on (x2, x3).

66

 

 

 

 

 

Condition (a) implies that the two distributions have the equal mean. Condition

(b) expresses the third-degree stochastic dominance (TSD) between F (x) and G(x).
These conditions are sufficient for G(x) to represent an increase in risk in the K-L sense
from F (x) . Condition (c) states that G(x) never exceeds F (x) in the interval [x2 , x3].
This implies that G S F . This is the added condition which allows general statements to
be made concerning the effect of a SIRK on the choice made by a risk averse decision
maker with u'” 2 0. These three conditions transfer area under F (x) from locations
where it is initially distributed, to areas at or to the left and to the right of the endpoints of
the interval over which the original distribution is defined; that is, this is to areas to the
right of and to the left of the interval containing the support of F (x) These restrictions
imposed on the difference between two C-CDFs to define the SIRK order are similar to
the restrictions on the difference between two CDFs used by Meyer and Ormiston (1985)
who define the SIR order.

Black and Bulkley (1989) use the concept of a ‘relatively strong increase in risk’
(RSIR) that is a particular type of R-S increases in risk by imposing the restriction on the
ratio of a pair of PDFs. By replacing the restriction on the ratio of a pair of PDFs with the
restriction on the ratio of a pair of CDFs, we introduce the second definition of K-L
increases in risk and call it a ‘relatively strong increase in risk in the K-L sense’ (RSIRK).
Note that, as the SIR order implies the RSIR order, the SIRK ranking also implies the

RSIRK ranking. That is, the set of SIRK shifts is included in the set of RSIRK shifts.

Definition 4.2. G(x) represents a relatively strong increase in risk in the K-L

sense with respect to F (x) ( denoted by G RSIRK F ) if

67

 

 

(a) Ifl‘lG(x)—F(x)ldx =0

(b) I; IG(x)— F(x)]dx 2 O for all s e [x,,x,]

(c) There exists a point mo satisfying I:0 [G(x)— F (x)]dx = I; [G(x)— F (x)]dx = 0

(d) There exists a pair of points m, , m2 6 [x2 , x,] where m, S m2 , such that F (x) 2 G(x)
for all x e [m,,m2] and F(x)S G(x) for allx in [x2,m,] and [m,,x3]

(e) For all x e [x2,m, I, there exists a non-decreasing function H, :[x2 , m, ) —> [0,1] such
that F(x) = H, (x)G(x)

(f) For all x e [m,,x3], there exists a non-increasing function H 2 :(m2,x3] —> [0,1] such

that F(x) = H2 (x) G(x).

Conditions (a) and (b) define a K—L increase in risk. Condition (c) means that a
relative strong increase in risk in the K—L sense (RSIRK) satisfies second-degree

stochastic dominance (SSD) on both side of m0 with equal means. Condition ((1) imposes
the restriction that the two CDFs cross only twice. Conditions (6) and (f) restrict the
extent to which cumulative probability mass can be transferred to any one value in the
tails of F (x) relative to any other and impose monotonicity restrictions on the ratio
between a pair of CDFs in the intervals x e [x,,m,] and x e [m2,x3].

Figure 4.1 satisfies the RSIRK conditions given in Definition 4.2. H, is non-
decreasing and H 2 is non-increasing for each corresponding interval x e [x,,m,] and

x 6 [m2 , x3 I , respectively. If, for any point in the interval [x2 , m, )( (m2 , x3] ), the tangent

68

 

 

 

 

   

G(X)=g(x)

 

 

 

 

 

 

0 x1 x2 m; m m2 x3 x4

Figure 4.1. G RSIRK F.

lines f (x) and g(x) (2 G(x)) do not meet to the left (right) direction in the probability
space between zero and one, the ratio of a pair of CDFs H , (H 2) is non-decreasing (non-
increasing). A RSIRK order consists of the sum of three shifts, which are a ‘left-side
monotone probability ratio’ (L-MPR) in the interval x e Ix, ,m, I, a ‘right-side monotone
probability ratio’ (R-MPR) in the interval x e Im2 , x ,I and a first-order stochastically
dominated shift (the opposite of a FSD shift) in the interval x 6 Im, , m2 I. Or, it consists of
the sum of two shifts, which are ‘left-side extended strong increases in risk’ (L-ESIR)

and ‘right-side extended strong decreases in risk’ (R—ESDR) for each corresponding
interval x e Ix, , m°I and Im",x4 I, respectively. A L-ESIR (R-ESDR) requires the
monotonicity of the ratio only on the left side (right side) of the given pair of CDFs.

We introduce the last specialized concept of K-L increases in risk and name it a

69

 

 

 

‘simple increase in risk across r in the K-L sense’ (isK (r)) that is formally defined as:

Definition 4.3. G(x) represents a simple increase in risk across r in the K-L sense

with respect to F (x) (denoted by G isK (r) F ) if
(a) I: [G(x)— F(x)]dr = 0
(b) I IG(x)— F(x)Idx 2 0 for all s e Ix, ,x4I

(c) (F(x)—G(x))(x—r)2 O for all x e Ix,.x,I,

where r is the point in the interval Ix,,x,,I.

C-CDFs

 

 

 

 

x, x2 r x3 = x,

Figure 4.2. G sIRK (r) F.

70

 

 

 

 

Me

Pal

aPl

inc

if]

 

Conditions (a) and (b) imply that the isK(r) order is a K—L increase in risk.

Condition (c) means that the sIRK (r) ranking satisfies the second-degree stochastic
dominance (SSD) on both side of r. That is, this condition expresses the SSD of F (x)
over G(x) in the interval x e Ix,,r) and of G(x) over F (x) in the interval x 6 Ir, x4].
This implies that F (x) is less than G(x) whenever x is less than r, and F (x) is larger
than G(x) whenever x is larger than r. A sIRK (r) is a particular case of TSD. Its
particularity comes from the fact that the curves F (x) and G(x) are forced to intersect at
a single point x = r. Such sIRK (r) is presented in Figure 4.2. Note that a simple increase
in risk in the K-L sense allows for cumulative probability mass transfers to points inside
the initial support, but a strong increase in risk in the K-L sense limits the redistribution
of cumulative probability mass to points outside the initial support.

It can also be convenient to represent a SIRK (r) by using a deterministic
transformation of returns for asset x. As popularized by Sandmo (1971) and extended by
Meyer and Ormiston (1989), a deterministic transformation of a lottery replaces every

payoff x of the lottery by a new payoff t(x). A so-called deterministic transformation

approach imposes some specific restrictions on the function t(x).

Definition 4.4. The function f(x) represents a simple deterministic (risk-
increasing in the K-L sense) transformation across r for a random variable given by F (x)

if the function k(x) E t(x)— x satisfies

(a) I:k(x)dF(x)= o

71

 

 

Co

the

C011

IIOI'.

 

(b) I: I, k(s)dF(s)dI S 0 for x e Ix,.x,I

(C) Ix k(s)dF(s) S 0 whenever x < r, and I" k(")dF(S)2 0 whenever x > r.

Condition (a) guarantees that the mean of the random variable is preserved.
Condition (b) implies that the initial random variable third-order stochastically dominates
the transformed random variable. These conditions characterize increases in risk in the K-
L sense. Condition (0) implies that a simple deterministic transformation across r satisfies
the second-order stochastic dominance (SSD) transformation on both sides of r. That is,
this condition expresses the SSD transformation of x over t(x) in the interval x e Ix, ,r)
and of t(x) over x in the interval x 6 Ir, x4 I. A simple deterministic transformation across
r differs from the notion of simple deterministic transformation by Meyer and Ormiston

(1989) who impose k(x) = t(x) — x to be monotonically increasing. Recall that we do not
consider any other restrictions on Ix k(s) dF(s) than nonpositivity for x < r and

nonnegativity for x > r.

4.2 Comparative Static Analysis
In this section, we provide general comparative static statements regarding the

SIRK, the isK (r), and the RSIRK order defined in section 4.1. Doing comparative static

analysis, we assume that the first and the second-order condition are satisfied to

72

 

 

 

guarantee a unique interior solution. Faced by the CDF of the random variable, F (x) , the

first-order condition defining the optimal value for a is
I " u'Iz(x,a)I 2,, (x,a)dF(x) = 0 (4.1)

The optimal solution satisfying (4.1) is guaranteed to be a global optimum by the second-

order condition that

 

Ix" Iu"Iz(x, a)Iz: + u’Iz(x, a)Izaa IdF(x) < 0. (4.2)
Under risk aversion, restriction z,m < 0 insures that the condition (4.2) is satisfied. Thus,

in order to see the result that a, 2 a“. for a specified change in CDF from F to G, it is

sufficient to show that
Q(ar ) = I u'lz(x, at )1 2. (m1: )d[G(x)- F (16)] S 0- (4.3)

Before proving Theorem 4.1, the following result from Yitzhaki (1983) is needed.

Lemma 4.1. Let s(x) be any function and s(y) = Iys(x)dx Z O for all y e Ia,bI.
If another function I (x) is non-negative and non-increasing for all x e Ia, b], then

Lys(x)l(x)dx20 for all yeIa,bI_

Theorem 4.1. For a class of risk-averse decision makers with u’" 2 0 , aF 2 (16 if
(a) G SIRK F

(b) ZXZO, zxxSO, ZMZO, z,m S0 and ZWZO.

73

 

 

Si

 

Proof: From (4.3), Q(a,.-) can be written as

Q(ar)= I: u'(z)z..(x,a.)dH(x) (4.4)
where H(x) = G(x)— F(x).
Since H (x, ) = H (x4 ) = 0 , integration by parts of (4.4) yields

Q(ar) = - I {u’(z)z..(x,ar )+ u”(2)2.z. (w.- )}H(x)dx

=_ I: “(2)2... (x, 51,. )H(x)dx — Ix" u”(z) 2,2,, (x, (1F )H(x)dx . (4.5)

First, let’s consider the first term in (4.5).

Integration by parts of the first term in (4.5) yields

— u'(z)za, (x, a, )H (x)

 

L] + I: {u'(z)zm (x, aF )+ u”(z)zxzm (x, (1,. )}H(x)dx. (4.6)

The first term of (4.6) is zero because of the condition (a) in Definition 4.1. Integration

by parts of the second term in (4.6) yields

{u’(z)z,m + u"(z) zxzm II: H(t) dt

 

X4
x=x,

— Ix" {u(z) 2m + 2u"(z) zxz,m + u"(z) znzm + u"(z)z:zm IIX H(t)dtdx. (4.7)
(4.7) is non-positive because of the condition (b) in Definition 4.1 and assumptions made
about 2“.

Second, let’s consider the second term in (4.5).

Integration by parts of the second term in (4.5) yields

— u'(z) 2,, I: 2,, (x,a,, )H(x)dx

 

‘4
y=xl

+ 1;; {u”(z)z.. +u~<z>zi}I.fz.(x,a,..>E<x)dxdy. <43)

74

 

 

A sufficient condition for (4.8) is

Iy 20, (x, (1,. )H(x)dx S 0 for all y e Ix, ,x, I. (4.9)
Integration by parts of (4.9) yields

z (x, a, )H(x)

 

. ., ‘Ijza (x at )H (-x)dx (4.10)
Using Lemma 4.1 shows that the integral in (4.10) is non-negative because of the
condition (b) in Definition 4.1 and assumptions made about 2,1. The first term in (4.10) is
equal to

2.. (x, at )1? (.V) (4.11)
which is zero when y = x F . Hence (4.9) is satisfied at x,.. . Taking into account
Definition 4.1, (4.11) is non-positive for y belonging to Ix, , x2] and because of
2,1 (y, 01,) is non-positive for y S x,,. , (4.9) is also satisfied for y belonging to (x,, xF).
Indeed,

I: z (x, a, )H(x)dx= I"za (,x a)H (x)dx~ I:za (,x a, )H(x)dx (4.12)

Clearly, the first term on the right hand Side of (4.12) is non-positive, as shown above,
and the second integral is positive because of the condition (c) in Definition 4.1 and the

assumptions made about zu.

For y in (x,,,x_,) one gets
Iiz (x, a, )H(x)dx= I:za (,x a, )H(x)dx+ I z (z(x, a, )H(x)dx. (4.13)

Because of the condition (c) in Definition 4.1, (4.13) is non-positive. Finally, for

y e Ix3, x, I , (4.10) is non-positive from Definition 4.1. Hence (4.9) holds for all

75

v- 6“"

 

 

 

 

 

 

 

 

 

 

 

y e [x,,x,]. Q.E.D.

Our result in Theorem 4.1 uses somewhat stronger restrictions on the structure of
the decision model and the risk preferences of decision makers than does Meyer and
Ormiston (1985) for SIR.

Next theorem concerns the class of a simple increase in risk across r in the K-L

sense (sIRK (r)) presented in section 4.1. Using the simple form of the general decision

 

model z(x,a) E 2,, + a x , we assume that the random variable x changes sign from

negative to positive at the point r. Because a sIRK (r) is a particular case of TSD changes,
we know that such a change in risk in the K-L sense reduces the expected utility of

decision makers. Therefore, it is sufficient to show that

Q(a,)= I:u z()(x r)dIG(x)— F(x)]SO. (4.14)

Theorem 4.2. If G sIRK (r) F and z,“ = 0 , then a,,. 2 a6 for all risk averse

decision makers with u" 2 0.

Proof: The case with an unbounded aF yields an unambiguous comparative static
property (see Dionne and Gollier (1992) and Dionne, Eeckhoudt and Gollier (1993)).
Therefore, we limit the analysis to the case with a bounded solution. Without loss of
generality, we assume that 01,, = 1 . We have to prove that (4.14) is true. Integrating

(4.14) by parts yields

Q(a.)= u’(z(xoa ))I (x- r ))-le(x F(-x)] I:’u( '( z):I (t-r)d[G(t)- F(t)]dx

76

 

 

 

 

Integrating the first term of Q(aF) by parts and using the sIRK (r) conditions in

Definition 4.3, the first term of Q(a,..) is non-positive. Thus, we have
Q(a,.) s — I u'(z)I: (z — r)d[G(t)- F(t)]dx. (4.15)
Integrating (4.15) by parts yields
Q(a,.. ) s —u"[z(x, , a)”: I‘ (t — r)d[G(t) — F(t)]dx
+ I: u"(z) I I (z —r)d[G(t)- F(t)]dsdx. (4.16)
We denote that W(x): I II (t — r)d[G(t)— F(t)]ds for all x e [x,,x,]. Given the

assumption on u, the left-hand side of equality in (4.16) would be negative if W(x) is
non-positive for all x e [xI ,x4 1. Integrating W(x) by parts two times for all x e [xl,x4]
yields
W(x) = (x _ r)[c‘;(x)_ F(x)]— 2 I: [G(s)_ F(s)]ds. (4.17)

Observe that the first term in the right-hand side of (4.17) is negative by the condition (c)
in Definition 4.3 presented in section 4.1, since (x — r) and [G(x)— F(x)] always exhibit
opposite signs. The second term in (4.17) is always positive since it satisfies TSD.
Therefore, the difference of these two terms is always negative, that is, W(x) S 0 for all

x e [xI , x4 I. This concludes the proof. Q.E.D.

As the last result developed in this section, the set of K-L increases in risk for the
comparative statics result in Theorem 4.1 is extended to the set of RSIRK shifis which is

larger than the set of SIRK shifis. We assume that 2x > 0 , z” 2 0 , and z“ = 0. Given the

77

 

 

 

 

assumptions about z(x,0t) to prove a,.- Z a“ it is sufficient to show

Q(a,.)= I“ u' (2)2, (x, a,, )d(F(x)— G(x)) 2 o. (4.18)
Integration by parts of the right-hand side (RHS) of Q(01,, ) in (4.18) yields
Q(aF ) = u' (z(xi , at ))[T(xt,a,.~; F, z)— T(xi,ai-;G, 2)]
— I u" (2)2, [T(x,a,. ; F, z)— T(x,a,, ;G, 2)]dx 2 0 (4.19)
where T(-,-;I,z): Ix,,x4]x R —+ R is defined as
T(x,a,,;1, z) s I 2, (1,01,. )dl(!) where 1 = F, G.

where Tdenotes location-weighted probability mass function used in Gollier (1995).

In order to prove Theorem 4.3 we need first to prove the following lemma.

.V

Lemma 4.2. If z(u 2 0 , then (p(s) 2: I u"(z)sz(x,a; F,z)dx 2 0 for all s in

[352,354].

Proof: Note that (p(x2 ) = 0. By integrating the first-order condition by parts we get

(p(s) = u' (z)T(s, a; F, 2) — Iu u' (z) zadF (x) From the first-order condition

I: u'(z)zadF(x) S 0 for all s in Ix,,x4]. Since the sign of g(x a; F, 2) is equal to the

sign of z(I , T must alternate in sign. Let x“ denote the value of x where Tchanges sign
from negative to positive. For all s 2 x‘ , (p(s) 2 0 because the first term is positive and

the second one with minus sign is also. For all s S x“ , (p(s) is an increasing function

 

 

 

 

because u" (z) sz(x,a; F, 2) 2 0. Given that (()(x2 ) = 0 , (0(s) 2 O for all s S x‘. Q.E.D.

Before proving Lemma 4.3, we need to know the following things. The integrand
in expression (4.19), u" (2) 2I [T(x, (1,, ; F ,z)— T(x,aF;G, 2)], has a sign depending on the
difference between two T fimctions.

T(x,0t,.;F,z)— T(x,a,,.;G,z)

= z, [F(x)— G(x)]— I 2,, [F(t)— G(t)]dt. (4.20)
When zu has a sign change over the intervals Ix,,ml], [mpmo], [m°,m2] and [m2,x,],
(4.20) changes sign at most two times from positive to negative. When (4.20) has a sign

change only once, it is trivial. Over four intervals let x' denote the point where (4.20)

changes sign when zu has a sign change for each interval. Let us define that
S(x, a) = 2a [F(x) — G(x)] and I(x, a) = IX za, [F(t)— G(t)]dt. We consider the following

four cases in (4.20); (i) kl = x', k2 = a and k3 = b (ii-a) and (ii-b) k1 = a, k2 = x° and
k3 = b , and (iii) k1 = a , k2 = b and k3 = x' , where k2 is the end point when the first
sign change occurs for each interval, and k1 and k3 are the points which the first and the

second sign change occurs from positive to negative, respectively. Now, we draw the

figures to get a clear demonstration for each case (see Figure 4.3-Figure 4.6).

Lemma 4.3. The value of on which maximizes E [u(z(x, 00)] is lower for a class of
risk-averse decision makers with u’" 2 0 under G(x) than F (x) where G(x) and F (x)

satisfy conditions (a), (b) and (c) for G(x) to represent a RSIRK presented in Definition

 

 

 

 

because u" (z) sz(x,a;F,z) 2 0. Given that (0(x2): 0, ¢(s) 2 0 for all s S x‘. Q.E.D.

Before proving Lemma 4.3, we need to know the following things. The integrand
in expression (4.19), u" (z)zx [T(x,a,. ; F, z)— T(x,a,. ;G, 2)], has a sign depending on the
difference between two T functions.

T(x,a,.;F,z)— T(x,0t,;;G,z)

= z, [F(x) — G(x)]- I 2,, [F(t)— G(t)]dt. (4.20)
When zu has a sign change over the intervals [x1,m,], [m,,m°], [m°,m2] and [m2,x4],
(4.20) changes sign at most two times from positive to negative. When (4.20) has a sign

change only once, it is trivial. Over four intervals let x' denote the point where (4.20)

changes sign when za has a sign change for each interval. Let us define that

S(x, a) = 20 [F (x) — G(x)] and I (x, a) = I: 2a, [F (t)—— G(t)]dt . We consider the following
four cases in (4.20); (i) kI = x‘, k2 = a and k3 = b (ii-a) and (ii-b) kl = a, k2 = x' and
k3 = b , and (iii) k1 = a , k2 = b and k3 = x‘ , where k2 is the end point when the first
sign change occurs for each interval, and kI and k3 are the points which the first and the

second sign change occurs from positive to negative, respectively. Now, we draw the

figures to get a clear demonstration for each case (see Figure 4.3-Figure 4.6).

Lemma 4.3. The value of a which maximizes E [u(z(x, 00)] is lower for a class of
risk-averse decision makers with u" 2 0 under G(x) than F (x) where G(x) and F (x)

satisfy conditions (a), (b) and (c) for G(x) to represent a RSIRK presented in Definition

79

 

 

 

 

Case(i): kl =x', k2=a and k3=b when x] Sx'Sml.

 

 

Figure4.3. kl =x', k2 =a and k3 =b when x] Sx' Sml.

Case (ii-a): kl =a, k2 =x' and k3 =b when ml Sx’ Sm°.

 

 

Figure4.4. kl =a, k2 =x' and k3=b when ml Sx' Sm°.

80

 

 

Case (ii-b): kl =a, k2 =x' and k3 =b when m0 Sx' sz.

 

 

Figure 4.5. k1 =a, k2 =x. and k3 =b when m0 Sx' sz.

Case(iii): kl =a, k2 =b and k3 =x' when m2 Sx' Sx4.

 

 

Figure4.6. kl =a, k2 =b and k3=x. when m2 Sx' Sx,.

 

 

 

 

 

 

4.2 if

I2 [T(x,aF;F,z)— T(x,a,,;G,z)]dx > 0

and if the given assumptions about z(x, on) are satisfied, where k2 is the end point when
the first sign change occurs for each interval, and k1 and k3 are the points which the first

and the second sign change occurs from positive to negative, respectively in (4.20).

Proof: For notational convenience, we define that T(x, a). ; F, 2) = TF (x, aF) and
T(x,a,.s;G,z) = TG(x,a,,.).
Q(aF ) = u’(2(xt , at )) [T is (xi , an ) - To (x4 , at )l
— I: u"(z) zx [T,.- (x, a). )— TG (x, aF )Idx.
Using the given assumptions about z(x, a) and the RSIRK conditions, the first term of

Q(0t I,.) is non-negative. Therefore,
k, ”
g(aF)Z "L u (Z)zx[TF(x’ar)- TG(x’aF)]dx
- li‘u"(z)z.1r.<x,a.—) z(x,a,.t)]dx.
Since — u"(z)zJr is non-negative and decreasing, we have the following four cases:
(i) k1 = x’, k2 = a and k3 =b (ii-a) and (ii-b) kl = a, k2 = x’ and k3 = b, and (iii)

kl =a, k2 =b and k3 =x‘.
(2011:)Z ‘u'(z(k1sals ))2x I: [TI-(Lat)- To (x, aF )ldx

_ u"(z(k3,a,,))zx I: [Tr'(x9ar)— TG(x’aF )]dx-

82

 

 

 

Adding and subtracting — u”(z(k3,aF )) z,‘ I [(1 [T F (x, aF)— T G (x, a,. flair in the RHS of the

above inequality gives
Q(ar‘ ) 2 {—u'(z(kl ’ a]? )) 21' + u"(z(k3 a aF )) zx if: [TF (x, aF ) _ To (xa aF ”dx
‘ u"(z(k3,aF )) zx I: [Tr (x’alt')— Tc; (x, aF )Idx.

Since - u”(z) z, is non-negative and decreasing, the bracket of the first term in the RHS

of the above inequality is non-negative.

If Ikz [TF(x’aF)_ T(;(x’aF )Idx Z 0’

g(aF ) 2 —u"(z(k3 ’ aF )) zx I: [TF (x, (2,; )_ To (x, 0F )]dx-
The given assumptions about z(x, 0t) and the RSIRK conditions imply

I: [TF (x,aF)— T(,.(x,a,.-)]dx Z 0 and hence Q(01,,.)2 0. Q.E.D.

Lemmas 4.2 and 4.3 are required in the proof of Theorem 4.3.
Theorem 4.3. For a class of risk-averse decision makers with u" 2 O , a,. 2 a0 if

(a) G RSIRK F

(b) 21‘ 20, szO, zmSO and zu=0.

Proof: Let k1, k2, k3 be the ordered values of x' , a, b. With the points ml , m° and m2

used in Definition 4.3 where x2 S mI S m° S m2 S x4 , we consider the following four

83

 

 

 

 

cases:

Case (i): k1 = x', k2 = a and k3 = b when xl S x' S ml (see Figure 4.3).
Let’s rewrite Q(et,,) in (4. 19) as
Q(ar.- ) = 14' (2(xt , at: )) [Tr (x4 , at: )- TO (M , an )1
— I:u'(z )z[T,( x, a,)— T G(x,aF)]dx

—J;u ~(.z>z [r (W)- T (x 0:. >14
Using the given assumptions about z(x, 0t) and the RSIRK conditions, the first term of

Q(0L,,) is non-negative. Therefore,
91a.» — If, u~(z)z.1r,.-(x.a.)— T..(x.a,..)idx
_I‘u (2)2 [T x (1,) z,(x,a,.)]dr.
Consider the sign of the expression I: [T,, (x, aF)— To (x, a, )Idx . First, if
I: [T,,. (x, aF)— TO. (x,a,, )]dx 2 0 , then Theorem 4.3 follows from Lemma 4.3.
Second, assuming that £[TF (x, 01,, )— TO (x, a,, )Idx S 0 , Q(01 1:) becomes
gap - If, u"(z». 1T... (x, (1,)- r. (x. a. )14x
—u'(z(b,a,.~))z. I,“ Wm.)—T..(x,a,.r)]dx.
Adding and subtracting — u”(z(b, 01,, )) 2x I: [T,,. (x,aF )— T0. (x,aF )Idx in the RHS of the

above inequality gives

84

 

 

 

 

 

 

 

 

Q(al)> ‘Eu (2)2 x[TI«( x aF)_ T( ”(x al- )ldx

+u (z(,ba,» zIZIT,( (,xa,)— T(xa,)]dx
— u”(z(b, a, )) z, I: [T ,, (x, a, )— 7}, (x,a,.. )Idx.
From the RSIRK conditions and I: IT, (x,a,, )- 7}, (x, 0:, )]dx S 0 ,
Q(a,)2 — I: u”(z)z,IT,,(x,a,.- )— T(,.(x,a,.~)]dx , (4.21)
where — I:u'(z)zx [T,- (x, a, )— T(,.(x,a,, )Idx
— -I:u ")(z z T (x,,a,) — dxIiu”(z)z,IT,(x,a,_~)-TG(x,a,)]dx.
Let xobe the value ofx that satisfies z,(x,a,. ) = o. T,,(x,,a,)= zaG(x2 )— I’ sz(t)dt

is non-positive since Za S 0 for x e Ix1 ,xz]. Therefore, (4.21) becomes

>—I:u ”(2)2 [T,( x, (1,) T(,.,(x a, )Idx. (4.22)
Consider the term IT, (x,a, )— T“. (x, a, )I for x e Ix2 , a).

Tlt‘(x’alt')— 717(xaah)

=z.[F(x)—G(x)]— I.,[r(.)_e(.)]a

=21.1411 Ihm (rah (-.)

= z, [1 — h, (x)]F(x)— [1 — h, (x)]I sz(t)dt — I h((t) I z,F(k)dkdt .

The last equality of the above equation can be obtained by integration by parts. Since,

85

 

 

 

 

according to the condition (e) in Definition 4.2, H, (x) is non-decreasing and H, (x) S 1

for all x e Ix2 , m,), h1 (x) is non-increasing and h1 (x) 2 1 . Therefore,
T F (x. a.)- Tu (w. ) 2 z. [1 - h. (X)] F (x)— [1 - h. 00]]; Z.F(t)dt

Z [1 _ h1(x)]TF (xaalr)'

Thus (4.22) becomes
Q(aI‘) Z " I: u”(z)zx [1 _ hl (3‘)] TF (x,a, )dx 2 _[1_ h] (xv )II: u”(z)szF (x’aI')dx -
Since h, (x) is non-increasing, h, (x”)2 1 and I0 u"(z) z], (x, a,,)dx 2 0 from Lemma

4.2, Q(0t,,.)2 0.

Case (ii): k1 = a , k2 = x. and k3 = b when m, S x' S m2 (see Figures 4.4 and 4.5).
Before proving this case, we need to know the sign of 2,1 (x',a, )I: IF(x)— G(x)]dx. Let
x0 be the value of x satisfying 20L (x, 0t, ) = 0 and m0 be the point satisfying

I: IF(x)— G(x)]dx = 0. From the definition of RSIRK I: IF(t)— G(t)] d! changes sign
from negative to positive at the point m” in the interval (m, , m2 ). We define that

I; F (t) d! = F (x) and I: G(t)dt = G(x). Let x' denote the point where

IT, (x, a, )— T0 (x, a, )] changes its sign when 2,,l has a sign change for each interval, that

. O
is, for x

z, [F(x) — G(x)] = I 2“, [F(t) — 6(1)] dt . (4.23)

86

 

 

 

 

 

 

 

 

 

 

 

 

We consider the following two sub-cases where S (x, a) = z‘ll [F (x) — G(x)] and

I(x,a)= Ilzm[F(t)—G(t)]dt.

Case (ii-a): k1 = a , k2 = x. and k3 = b when ml S x' S m° (see Figures 4.4 and 4.7).

S(x,a)

FM].

 

I(x,a)

Figure 4.7. m, S x' S m0.

In Figure 4.7 za (x°,a,,.)= O and za (x',aF)< 0 since [F(x' )— G(x')] is positive from

Definition 4.2 and S(x‘ , a) is negative. From Lemma 4.3,
J: [T]: (x, a1" )_ To (x, (1,, )]dx
= [[F(x)—G(x)]dx— 1:1; zm[F(r)—c<t)1drdx

= 2,,(xzaF)[F‘(x')—G‘(x‘)l— Ij’zmlflxmxldx

87

 

 

 

-I: I [-F(t) G(t)]dtdx (4.24)

The RHS of (4.24) is non-negative since 211 (x',a,, )< 0 , [F(x' )— C(x' )] is non-positive

and the condition (b) in Definition 4.2 is satisfied. Therefore,
I: [E(xfll: )_ To (La/4' )]dx 2 0 -

This is sufficient for Q((XF ) 2 0.

Case (ii-b): k1 = a , k2 = x. and k3 = b when m0 S x. S m2 (see Figures 4.5 and 4.8).

I(x,a)

 

 

Figure 4.8. m° S x. S m.

In Figure 4.8 za (x°,a,;)= 0 and Zn (x',a,,)> 0 since [F(x' )— G(x' )] is positive from
Definition 4.2 and S (x. ,a) in positive. The RHS of (4.24) is non-negative since
2,, (x',a,,. )> 0 and [F(x' )— G(x' )] is non-negative and the condition (b) in Definition 4.2

is satisfied. Therefore,

88

 

 

 

 

 

I: [7", (x, a, )— To. (x, 01,)Idx 2 0.

This is sufficient for Q(01, ) 2 0.

Case (iii): k1 = a , k2 = b and k3 = x' when m2 S x. S x4 (see Figure 4.6).

From the definition of 01 ,

I:u (z)za (,x a,)dF(x)=0.

Integrating by parts gives
I:‘u'(z)zadF(x)= u'(z(x4,a,))T, (,—x4,a,) I:u"( )z T, (x, a, )dx
3 u'(z(x4 ’alr )) Tr (x49aF)_ u”(z(xc, aF ))zx I:TF (x2 a!" )dx.

The above inequality holds since — u" (z) z,r is non-negative and decreasing in x, and

T, (x, (1,) changes sign from negative to positive at the point x”. Therefore,

rnuflnazm
but this implies IX'T, (,x (1, )dx > 0 for all s 6 [x2, x4]. From Lemma 4. 3 wherek2 - —,b
I: [TF (x2 aF )- To (x2 a!" )]dx

=,,"I {2 [F (x)— G(x)]— I [F(t)— G(tdx)ldt} (4.25)

Consider the second term with minus sign in the bracket in the RHS of (4.25).
— I ‘ z, [F(t)— 0(1)] dt = - I" z, [F(t)— G(t)]dt — I:za, [F(t)— G(t)]dt . (4.26)

The first term with minus sign in the RHS of (4.26) is non-positive since b is located at

89

 

 

 

 

the right side of m0 , where m0 is the point satisfying IMO [F (x)— G(x)]dx = 0.

Therefore,
— I:z mz[(Ft)— G(t)]dtS— —Iz Il—H :(t—jIF(t)dt ILetH—21()=h,(t)I

s —[1 — h2(x)]I:sz(t)dt — I’h;(t)I: z,F(k)dkdt

s —[1 — h, (x)]I:za,F(t)dt . (4.27)
The last inequality in (4.27) can be obtained by integration by parts. Since, according to
the condition (1) in Definition 4.2, H 2 (x) is non-increasing and H 2 (x) S 1 for all

x e (m2,x3], h2 (x) is non-decreasing and h2(x)21 for all x e (m2,x3].

Therefore, adding and subtracting — [1 — h2 (x)]I: sz(t)dt in the RHS of (4.27) gives
— I x zm [F(t)— G(t)] d1 S —[1 — h2 (x)]I: za, F(t)dt +[1— h2 (x)]I: sz(t)dt
s —[1 — h, (x)] I: sz(t)dt. (4.28)

Since [1 — h2 (x)] is non-positive and I: sz (t) dt is non-negative, the last inequality in
(4.28) holds. Thus,
T, (x, a, ) — T“ (x, (1,.)
s z, [1 — h2 (x)] F(x) — [1 — h, (x)] I sz(t)dt = [1 — h, (x)] T, (x, a, ). (4.29)
Therefore (4.25) becomes

I: [TF (x2 a!" ) _ Ta (x, aF )Idx S I: [1 — hz (x )1 TI" (x, ar )dx -

90

 

 

s [1 — h, (x‘ )] I “ T, (x, a, )dx 5 0. (4.30)

The second inequality in (4.30) holds since h2 (x) is non-decreasing and T, (x, (1,)
changes sign from negative to positive at x”. Thus, the left-hand side of (4.30) is non-

positive because [1 — h2 (x“ I] is non-positive and I: T, (x, 0:, )dx is non-negative. Since

I: [TF(xaa/-‘)_ To (x,a,)]dx Z 0 a
this implies

k, =h)
I( [Tr(xaar~)-Tg(x,a,,)]dx20,

XI

and hence from Lemma 4.3, Q(a, ) z 0. Q.E.D.

Since the SIRK ranking implies the RSIRK ranking, the comparative statics result
in Theorem 4.3 includes a larger set of K-L increases in risk but shows somewhat
stronger restriction on the structure of the decision model than that in Theorem 4.1. When
the structural restriction z“ = 0 is added, the relationship between the results in Black
and Bulkley’s analysis and in Theorem 4.3 shows a trade-off between the restrictions on

the set of decision makers and the set of changes in distribution.

91

 

 

Chapter 5

THE COMPARATIVE STATICS OF CHANGES IN

DISTRIBUTION OF THE RANDOM VARIABLES

This chapter deals with the problem of determining the conditions under which a
change in distribution of the random parameter decreases the optimal value of a decision
variable for the set of risk averse individuals with a positive third derivative of their
utility function (u' > 0), which was done before by Gollier (1995) for all risk averse
decision makers. Unlike the traditional approach that restricts separately the changes in
PDFs or CDFs and the structure of the given decision model, he restricts these two
components jointly with a single restriction to obtain a general comparative statement.
Following the technique used in Gollier (1995), we obtain the least constraining
conditions under which all risk averse individuals with u” > 0 react in the same direction
when facing a given shift in distribution. We also define a different subset of SSD that
satisfies our sufficient condition and includes a R-S increase in risk as a special case. In
section 5.1, we provide the least constraining conditions on the change in distribution of
random parameter for signing its effect on utility function with u’ 2 0 , u” S 0 and

u" > 0. Section 5.2 contains some concluding remarks specific to this study.

92

 

 

5.1 Comparative Statics of Change in Distribution of the Random Parameter

Since the seminal papers by Rothschild and Stiglitz (R-S) (1970, 1972), many
researchers have investigated the comparative statics effect of change in distribution of a
random variable such as subsets of FSD shifts, SSD shifis or R-S increases in risk. When
they use the stochastic dominance orders (SD) to get the desired comparative statics
results, they impose some restrictions on a pair of PDFs or CDFs. These examples are
included in Meyer and Ormiston (1985), Black and Bulkley (1989), Landsberger and
Meilijson (1990), Dionne, Eeckhoudt, and Gollier (1993), Eeckhoudt and Gollier (1995),
Kim (1998) and others. As indicated in chapter 2, as an alternative to this traditional
approach, Gollier (1995) obtains a quite simple condition for risk averse decision makers
by restricting the changes in PDFs or CDFs and the structure of the decision model
jointly. His condition entails all existing sufficient conditions as particular cases,
including a SIR and a RSIR, imposed for all concave utility functions. We follow his
approach to obtain a general comparative static statement for the case where u” > 0 is
also assumed.

Before presenting a sufficient condition in Theorem 5.2, we introduce the

following definition.

Definition 5.1. G(x) is said to be riskier than F (x) if there exists a scalar Ii. 2 1

such that

(a) I:[G(x)—/1F(x)]dx=0

93

 

 

G(y)
F (y) I

Figure 5.1. G MPS F.

 

 

 

 

 

 

 

Figure 5.2. F SSD G.

94

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b) I:IG(x)— 1F(x)]dx 2 o, for all y 6 [a,b].

This definition introduces another subset of SSD that includes a R-S increase in
risk as a special case. Note that ,1 adjusts a probability so that in the interval Ia,bI the

areas under two distribution functions, F and G, are equal. This implies that ,1 is a
scaling factor that makes 1 I:F(x)dx and I:G(x)dx equal. From condition (b) in
Definition 5.1,

Ly[G(x) — F(x)+ F(x)— 1F(x)]dx Z 0,
which is equivalent to

Iy[G(x)— F(x)]dx 2 (2 — 1) [gram 2 0.

Since A 2 1, SSD condition, Iy[G(x)— F (x)]dx 2 O, is always satisfied. That is, this
condition is stronger than SSD condition but not MPS condition. If ,1 = l , Definition 5.1

is exactly same as R-S increases in risk presented in chapter 2. If 2. > 1 , Definition 5.1

defines a different subset of SSD. To explain more clearly, we define IyF(x)dx = F (y),

I:G(x)dx = 602) and draw Figures 5.1 and 5.2. Figure 5.1 (A = I) shows the definition

of R-S increases in risk from F (x) to G(x). Figure 5.2 does not satisfy R-S increases in
risk from F (x) to G(x) but satisfies Definition 5.1.

Remember that a decision maker maximizes the expected utility of his final
payoff. The economic decision model in (2.1) can be rewritten as

u(u, F, 2) e arg mgvc H(a; u, F, 2) = E , [u(z(x, 00)]

95

 

 

= I”u(z(x,a))dp(x), (5.1)

a

where the subscript F for the expectation symbol E indicates that random variable x is
distributed according to CDF F whose support is in Ia,bI. The first-order condition is

H'(a(u, F, z); u, F, 2) = I:u'(z(x, a(u, F, 2») z,z (x, (z(u, F, 2)) dF(x) = O (5.2)
It is assumed that Zea is non-positive so that condition (5.2) is necessary and sufficient
condition for maximization problem (5.1), since it makes H concave in a. Note that
zu (x,a(u, F 2)) must alternate in sign in order to satisfy condition (5.2).

An integration by parts of the first-order condition yields

H' (a;u, F, 2) = u‘ (z(b,a)) T(b,a; F, z)— Iqbu" (z) zXT(x,a;F, z)dx , (5.3)
where T(-,~;F,z): [a,b]x R —) R is defined as

T(x,a; F,z) a I z, (t,a)dF(t). (5.4)

According to Gollier (1995), this location-weighed probability mass function has some
properties which are worth noting:

T(a,(x; F, 2) = o (5.5)
sgnaExT—(x, a; F, z) = sgn za (x,01). (5.6)
The sign of the derivative of T with respect to x is independent of the distribution

function. Note also that T is right continuous in x. Since 20, can be either positive or

negative, T has an ambiguous sign. However, from (5.2) and (5.3), it is apparent that
T(x, a(u, F, 2); F, 2) must alternate in sign.

Multiplying (5.2) and (5.3) by any scalar ,1 does not change the first-order

96

 

 

condition. To prove a, 2 01,, , it is sufficient to show that

Q(a, ) = u'(z(b, a» [21T(b, a; F, 2) — T(b, a; G, 2)]
— I"u'(z) z, [H(x, a; F, z) — T(x, a; G, z)]dx. (5.7)

Gollier (1995) presents the necessary and sufficient condition under which all risk

averse individuals react in the same direction when facing a given shift in distribution.

Theorem 5.1. Take the economic model represented by the payoff function 2 and
the initial and final distribution functions F and G as given. All risk-averse individuals
selecting or under F and z reduce their optimal level of on afier the change in CDF from
F to G if and only if there exists a scalar y e R such that

T(x,01;G,z) S yT(x,01;F,z), Vx 6 [a,b]. (5.8)

The Gollier’s condition generalizes all previous restrictions on changes in risk
such as ‘strong increases in risk” (SIR) defined by Meyer and Ormiston (1985) and
‘relatively strong increases in risk’ (RSIR) introduced by Black and Bulkley (1989).

When the payoff function is linear in the random variable, z(x,a) = z0 + a x , where 20 is

an exogenous parameter, he introduces the concept of the greater central riskiness (CR)
order by using his condition. He also shows that second stochastic dominance (SSD) is
neither sufficient nor necessary for centrally riskier dominance.

Before proving Theorem 5.2 for risk averse individuals with a positive third

derivative of their utility function, we need the following lemma.

97

 

 

 

 

Lemma 5.1. If zm > 0 and 20,“ S 0 , Definition 5.1 implies

2T(b.a; F,z) 2 T(b.a:G.z).

Proof: 1T (b,a;F , z) 2 T(b,a;G,z) can be written as

2 I:za(x,a)dF(x)2 Ifz,(x,a)dc;(x). (5.9)
Integration by parts of (5.9) yields

2, (b,a)(,1 — 1)— I:zm(x,a)(AF(x)— G(x))dx 2 o. (5.10)
Since 2,, (x,a) changes sign from negative to positive, z,z (b, a) is positive. Note that

from Definition 5.1, ,1 2 1 . Therefore, the first term in (5.10) is positive. Integrating the

second term in (5.10) by parts gives
_ za,(b,a)Ib(/1F(x)— G(x))dx + ;G,, Iy(/l.F(x)— G(x))dxdy 2 0. (5.11)
Since conditions (a) and (b) in Definition 5.1 are satisfied, and 2,, is concave in x, (5.11)

is positive. This concludes the proof. Q.E.D.

Theorem 5.2. Let F (x) and G(x) be the initial and the final distribution function.
All risk-averse individuals with u'" > 0 reduce their optimal level of or afier the change
in CDF from F (x) to G(x) if there exists a scalar x1 2 1 such that

M(x,a; F, z)—‘I’(x,a;G, z)2 0 with 2T(b,a;F,z)—T(b,a;G,z)2 o, (5.12)
where lI’(x,a;F,z)= Ez,(y, a) T(y,a;F,z)dy and

LI’(x,0:;G,z)= I:zx(y,a) T(y,a;G,z)dy for all x e Ia,bI.

98

 

 

Proof: First, from Lemma 5.1, the first term of the right-hand side in (5.7) is non-

negative. Second, consider the second term of the right-hand side in (5.7). Its integration

by parts gives

—u'(z(b,a)) Ifz, [1T(x, a; F,z)—T(x,a;G,z)]dx

+ Ifu"(2)2. Ijz.(y,a)[2T(y,a;F,z)— T(y,a;G,z)]dydx.

Since u” > 0 and (5.12) is assumed to hold, Q(a,)2 O. Q.E.D.

Note that if ,1 =1 or ,1 > 1 , our condition in (5.12) must be true for the subsets of
R-S increases in risk or the subsets of SSD shifis, respectively.

Now, consider our condition in (5.12) to compare with the Gollier’s condition.
It is obvious that the Gollier’s condition in (5.8) implies our condition. Therefore, our
condition is less restrictive than the Gollier one. A ‘strong increase in risk’ (SIR) and a
‘simple increase in risk’ (SIR) satisfy the Gollier’s condition with y=1, and a ‘relatively
strong increase in risk’ (RSIR) also satisfies his condition, but with a 7 that may differ
from unity. Since SIR, sIR and RSIR are the subsets of R-S increases in risk with k=l in
Definition 5.1, they also satisfy our condition.

We have found a sufficient condition for a risk-averse individual with u" > 0
when facing the change in CDF from F to G. If so, what is the necessary condition for
this economic problem? Integration by parts of the first-order condition in (5.3) yields

H'(a; u, F, 2) = u'(z(b, a» T(b,a; F, z)— u”(z(b,a))‘l’(b,a; F, 2)
+ I’u"(z)z,lr(x,a;F,z)dx = 0 (5.13)

where W(x,a;F,z) is defined in (5.12).

99

 

 

Similar to T in (5.5) and (5.6), ‘I’ has the following properties:

LNew; F, 2) = 0 (5.14)
6‘?
58” E(x,a;F,2)= sgn2.(y,a)T(y,a;F,2). (5.15)
Since T can be either positive or negative, ‘I’ has an ambiguous sign. From (5.13), if

u’" = 0, W(b,a;F,z) must be negative. But if u” > 0, W(x,a; F,z) must altematein sign

under risk aversion with u" > 0.

Theorem 5.3. If all risk-averse individuals with u" > O selecting a under F and 2
reduce their optimal level of a after the change in CDF from F (x) to G(x), then there
exists a positive scalar A such that

zl‘l’(x,a; F,z) 2 W(x,a;G,z) for all x 6 [a,b].

Proof: Define X_ = {x e Ia,bII lI’(x,oz;F,z) < 0} and X+ = {x e Ia,bII W(x,a;F,z)> 0}.
Neither X _ nor X + is empty. Suppose by contradiction that for all It , there exists

y 6 [a,b] such that ,1‘1’(x,a;F,z) < lI’(x,a;G, 2). Because ‘1’ is right continuous in x,
this is possible only if there exist non-empty subsets Q_ and Q, in X _ and X +

respectively such that

LI’(x_,a;G,z) W(x+,a;G,z)
< 9
lI’(x_,or;F,z) lI’(x+,a.r;F,z)

 

Vx_ 6 Q_, Vx+ 69+, (5.16)

or equivalently,

z,(x_,a)‘l’(x_,a;G,z) zx(x+,a)‘l’(x+,a;G,z)
zr,(x_,a)‘P(x_,a;F,z) zx(x+,a)‘l’(x,,a;F,z),

 

Vx_ EQ_, ‘t/x+ 69+. (5.17)

100

 

 

 

Define

a)_, = IQ z,(x,a)‘l’(x,a;F,z)dx, (0+, = IQ zx(x,a)‘l’(x,a;F,z)dx,
(04, = IQ z,(x,a)‘l’(x,a;G,z)dx, (0+0 = I2 z,(x,a)‘I’(x,a;G,z)dx.

Since 2, > 0 and W(x,a;F,z)<O in Q_, we have a)_, < 0. Similarly, (0,, > 0. After
some algebraic manipulations, (5.17) implies that

w—(}

< %.
a) I" w+F

(5.18)

Consider an individual with utility v , where v(z(x,a)) = vl (z(x,a))— la(x,a) and k is a

positive scalar, defined as

1 if x 6 Q_
v'”(z(x,a)) = 7} if x 6 (2+
0 otherwise,

where 77 is some positive scalar. Function v is fully specified by assuming that
v,’ (z(b,a)) = k and vl”(z(b,a)) = O. This is possible since z(x,a) S z(b,a) for all
x 6 [a,b]. In order to force the optimal solution for individual v to be a , we select

77 = —a)_, /a),, > 0. Therefore, we can verify that
H’(a; v, F, z) = Ibv’"(z(x, 01)) z,‘ (x, a) LF(x, a; F, z)dx
=w_, + 7760,,» = 0.

We now verify that an individual with v increases the optimal value of a when facing

the change in CDF:

a)_, a) ,.
H'(a;v,G,z)=a)_G+na),G =a, —‘——i >0.
0) , a)

— +I"

l01

 

 

 

The last inequality is the consequence of a)_,.- < 0 and inequality in (5.18). Therefore,

contradiction is established. Q.E.D.

5.2 Concluding Remarks

All the comparative statics results obtained in this dissertation are associated with
the set of risk averse decision makers with a non-negative third derivative of their utility
function (u' 2 0). Using the simple form of the general one-argument decision model,
general comparative static statements are developed for the subsets of R-S increases in
risk presented in chapter 3 and for the subsets of K-L increases in risk given in chapter 4.
In this comparative analysis, we use three approaches found in the published literature,
which specify restrictions between the initial and the final CDFs or PDFs: the CDF
difference approach, the ratio approach and the deterministic transformation approach. As
an alternative to these traditional approaches, we follow the Gollier’s technique to obtain
a general comparative static statement presented in section 5.1.

For the subsets of R-S increases in risk or K-L increases in risk, each definition
specifies a particular type of change from a CDF F (x) to another G(x) or,
correspondingly, from a PDF f (x) to another g(x). Therefore there exist basic
relationships among subsets. For the subsets of R-S increases in risk, we have (i) G L-
SIR F 2 G L-RSIR F => G L-RWIR F, (ii) G ESIR F :> G L-ESIR F => G MSIR F,
and (iii) G OSIR F :> G ORSIR F. Another relationship between the subsets of K-L

increases in risk shows that G SIRK F :> G RSIRK F.

l02

 

 

 

When we assume that the payoff function is linear in the random variable x, we

discuss the sufficient conditions obtained in this dissertation. Assuming that a, = l and
z0 = 0 for notational convenience, we consider our sufficient condition in (5.12) which
can be rewritten as

,1 I I' de(y)dt 2 I I' de(y)dt. (5.19)
Remember that this condition is less restrictive than the Gollier condition in Theorem 5.1.
Integration by parts of (5.19) yields

,1 I tF(t)dt — I tG(t)dt + I G(t)— 213(1)](1: 2 0. (5.20)
The L-SIR, the L-RWIR and the MSIR orders presented in chapter 3 satisfy Definition
5.1 with ,1 = 1 . Therefore (5.20) becomes

I’:F(:)dz — I‘rG(z)dz 2 0. (5.21)

The sufficient condition for L-SIR shifts or L-RWIR shifis satisfies (5.21) that implies
our condition in (5.12). Note that, after some algebraic manipulation of (5.12), if (5.21) is
satisfied, the MSIR condition also implies our condition. As indicated in chapter 3, the
Gollier condition is violated for the MSIR order.

Definition 5.1 can be modified for the subsets of TSD that include the subsets of

K-L increases in risk as special cases.

Definition 5.2. G(x) is said to be greater riskier than F (x) if there exists 1 2 1

such that

(a) I"[G(x)- lF(x)]dx = 0

103

 

 

 

 

(b) I'V[G(x)— lfi(x)]dx 2 0 , for all y 6 [a,b].

This definition introduces another subset of TSD with a scaling factor ’1 where ,1

makes the areas under F and G equal. From condition (b) in Definition 5.2,
I’[c‘(x)_ my. F(x)— lfi(x)]dx 2 0,

which is equivalent to
max)— fi(x)]dx 2 (,1 _1)I"fi(x)dx 20.

Since A 2 l, TSD condition, Ly[G(x)— 1:“ (x)]dx Z O, is always satisfied. That is, this

condition is stronger than TSD condition but not a K-L increase in risk. If A = 1 ,
Definition 5.2 is exactly same as K-L increases in risk presented in chapter 2. If ,1 > 1 ,
Definition 5.2 defines a different subset of TSD. Thus, with a linearity of the payoff
function, Lemma 5.1 can also be applied for Definition 5.2.

The SIRK, the RSIRK and the sIRK (r) orders presented in chapter 4 satisfy
Definition 5.2 with 2. = 1. The sufficient condition for SIRK shifts or RSIRK shifts
satisfies (5.21) that implies our condition in (5.12). Integration by parts of (5.21) and a
TSD condition give a sufficient condition for the isK (r) order that also implies our

condition.

104

 

 

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