‘ x

, ll .
\.-wl. ('9‘:

.A‘ I
I 13cc
Ck:CXACDLrL(C$

 

 

bViESI.) RETURNING MATERIflggz
Place in book drop to
LJBRARJES remove this checkout from

-__—_ your record. FINES will
be charged if book is
returned after the date
stamped below.

 

 

 

 

 

I
(«A-w " ’” I? “z /‘

. . x . A
, "6531"”,9'lc“

 

 

 

 

 

A NEH MEASURE OF ATTITUDE TOUARDS RISK

By

Beverly Fleisher

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Depart-ent of Agricultural Economics

1985

Copyright by
BEVERLY FLEISHER
1985

In loving memory
of my father,
Oscar T. Fleisher, Jr.

ACKNOWLEDGEMENTS

Completion of a project such as this requires intellectual, financial
and emotional support. I gratefully acknowledge the support I have received
in all three areas.

The Michigan Agricultural Experiment Station provided the research
funds required for this project since its inception.

Intellectual support and guidance came from the members of my disserta-
tion and guidance committees and numerous colleagues at Michigan State
University and elsewhere. Dr. Lindon Robison, who served not only as my
dissertation supervisor but also as my thesis supervisor and major profes-
sor during my Ph.D. program, has truely been a mentor extraordinaire,
encouraging me to takle new areas of research and publish the results. The
members of my dissertation committee each contributed to the completion of
this project in invaluable ways. Dr. Lester Manderscheid provided thought-
ful insights into many of the issues I confronted. Dr. Jack Mchwen's
assistance was invaluable in the design and administration of the question-
naire. His immediate response to questions was most appreciated. Dr. Jack
Meyer provided helpful assistance in clarifying theoretical issues. My
thanks also go to Dr. James Bonnen and Dr. Harold Riley who, as members of
my guidance committee, helped steer my through my doctoral program, and to
Dr. Glenn Johnson who has provided intellectual challenges throughOUt my

graduate career.

iii

Technical assistance was also provided by Steve Koenig and Patricia
Crowley Brown who served as my agricultural experts and responded to
numerous questions about corn varieties, grain dryers and loan financing.
Bruce Koenig made many house calls to coax my microcomputer to do things
that, alas, I alone could not make it do.

Sally Hittler, Harry Holzer, and Hillard Caldwell responded with buck-
etloads of patience and encouragement during the many times when I was
convinced that all hope was lost.

Extra special thanks go to my mother, Ruth Fleisher, who has responded
with empathy and levity to many a late night long distance call, and who
helped finance my microcomputer, without which I would still be typing the
first draft of Chapter 1.

iv

TABLE OF CONTENTS

CHAPTER I: INTRODUCTION: HHY 00 NE NEED A NEH MEASURE OF

ATTITUDE TOHARDS RISK? .................................... 1
Some Basic Concepts and Definitions .................... 2
Currently Used Measures of Attitude Towards Risk ........ 4
Objectives of this Study ................................ 12
Overview of the Dissertation ............................ 13

CHAPTER 2: HHY DOES THE ARROH-PRATT COEFFICIENT OF ABSOLUTE RISK

AVERSION INDICATE UNSTABLE ATTITUDES TOHARDS RISK? ........ 15 '
Hhat Does the Utility Function Actually Represent? ...... 16
A Brief Digression into the Foundations of Utility ...... 18
The Importance of Heterogeneity of Preferences .......... 26

CHAPTER 3: THE CONCEPT OF INTRINSIC RISK ATTITUDE: SEPARATING PREFERENCE

FOR RISKLESS OUTCOMES AND ATTITUDE TOHARDS RISK ........... 36
How Can He Measure Pure or Intrinsic
Attitude Towards Risk? ................................ 37
Intrinsic Risk Aversion ................................. 38
Comparing Value and Utility Functions ................... 48
Testing the Hypothesis of Constant
Intrinsic Risk Attitude ............................... 49
The Adequacy of the Present Theory of
Intrinsic Risk Attitude ............................... 51

CHAPTER 4: A NEH MEASURE OF ATTITUDE TOHARDS RISK .......... A .......... 55
Independence Conditions for Additive and

Multiplicative Value and Utility Functions ............ $6
Multiattribute Value and Utility Functions .............. 58
Reversals in Preference and Sign Conditions ............. 59
Economic Interpretation of Parameters k and w ........... 61
Allowable Transformations Between Value and

Utility Functions ..................................... 64
Synthesizing and Applying the Results ................... 67
Appendix 4A ............................................. 80
Appendix 48 ............................................. 82
Appendix 4C ............................................. 84
Appendix 40 ............................................. 86

CHAPTER 5: EXAMINATION OF THE THEORY IN AN EMPIRICAL CONTEXT ........ 88
Holistic vs. Decomposed Assessment of

Multiattribute Functions ............................. 68
Methods for Estimating Value and Utility Functions ..... 91
The Number of Attributes Used .......................... 94
Determining and Interpreting Scaling Factors ........... 95
The Respondents ........................................ 102
The Questionnaire ...................................... 102
Problems with the Questionnaire ........................ 105
Satisfaction of Independence Conditions ................ 107
The Issue of Consistency of Responses .................. 109
Appendix SA ............................................ 115
Appendix 58 ............................................ 146

CHAPTER 6: SUMMARY, CONCLUSIONS, AND ISSUES REQUIRING

FURTHER RESEARCH ......................................... 148
Summary ............................................... 149
Placing the Research in Context ....................... 153
Issues Requiring Futher Research ...................... 159
Nhat Conclusions Can Be Drawn? ........................ 164

BIBLIOGRAPHY ............................................. 167

vi

Figure 1:

Figure 2:

Figure 3:

Figure 4:

Figure 5:

Figure 5:

Figure 7:

Figure 8:

Figure 9:

Figure 10:

Figure 11:

Figure 12:

Figure 13:

LIST OF FIGURES

A von NeumanneMorgenstern Utility Function and the Rela-
tionship between Arrow-Pratt, Risk Premium and Certainty
Equivalent Measures of Atttitude Towards Risk ........... 10

The von Neumann-Morgenstern Utility Function
as a Homomorphic Mapping ................................ 17

Value Functions of Two Individuals with
Constant Marginal Value of Money ........................ 27

Value Functions of Two Individuals without
Constant Marginal Value of Money ........................ 29

Value Functions and Indifference Curves of Two
Individuals who Have Constant Marginal Value of

Money and Homogeneous Preferences ....................... 31
Value Functions and Indifference Curves of Two

Individuals who Have Constant Marginal Value of

Money and Heterogeneous Preferences .................... 32
Value Functions and Indifference Curves of One

Individual in Two Situations with Constant Marginal

Value of Money and Heterogeneous Preferences ............ 34

Pure Risk Atttitude as a Function Transforming Value
Functions to Utility Functions on the Real Line ......... 39

Value and Utility Functions for the Case of
Intrinsic Risk Neutrality ............................... 42

Value and Utility Functions for the Case of
Intrinsic Risk Aversion ................................. 46

Value and Utility Functions for the Case of
Intrinsic Risk Pronenss ................................. 47

Indifference Curves for Two Attributes which Meet the
Independence Conditions Required for Additivity ......... 53

Relationships Among Independence Concepts ................. 60

vii

Figure 14:

Figure 15:

Figure 16:

Figure 17:

Figure 18:

Figure 19:

Using Lotteries to Interpret k in a Multiplicative
Multiattribute Utility Function ......................... 62

v(x), u(x) and u(v) for an Intrinsically Risk Prone
Individual when v(x) = E ln x ........................... 6B

v(x), u(x) and u(v) for an Intrinsically Risk Prone
Individualwhen v(x) = I x2 .............................. 71

v(x), u(x) and u(v) for an Intrinsically Risk Averse
Individualwhen v(x) = I ln x ............................ 72

v(x), u(x) and u(v) for an Intrinsically Risk Averse
Individualwhen v(x) = E x2 .............................. 74

v(x), u(x) and u(v) for an Intrinsically Risk Averse
Individual when v(x) = fl ln x ........................... 77

viii

Table
Table

Table
Table

Table

LIST OF TABLES

Methods for Estimating Value or Utility Functions ........ 92

Number of Individuals Nho Had Difficulty Answering
Specific Types of Questions ............................ 107

Number of Responses Satisfying Independence Conditions ... 107

Frequency of Additive and Multiplicative Value and
Utility Functions by Decision Context .................. 108

Frequency of Combinations of Different Types of
Value and Utility Functions by Decision Context ........ 109

ix

CHAPTER 1

INTROUJCTION: WY 00 NE NEED A NEH MEASURE OF ATTITUDE TOHARDS RISK?

The concept of an attitude towards risk which is unique to the in-
dividual is used both formally and informally by agricultural economists
concerned with a wide variety of problem settings in which uncertainty is
present. Attitudes towards risk are used formally in predictive models of
agricultural sector participants' responses to changes in the economic,
institutional or technological environments and in formulating extension
reco-endations. Perhaps more commonly, the truism that "all farmers are
risk averse“ is used by agricultural economists and policy makers in less
formal analysis.

Young, et. al. (1979) state that I'knowledge of risk preferences of
individual agricultural producers is necessary for many useful private
managerial and public policy analyses of decision making under risk.“
Agricultural economists have perceived the need to incorporate the effects
of uncertainty into their analysis and have, over the past thirty years,
engaged in numerous endeavors to quantify attitudes towards risk. Despite
this accumulated body of research, the profession still has little consis-
tent evidence supporting the assuqation of risk aversion on the part of
agricultural sector participants. Nevertheless, this assumption continues
to be the cornerstone of much of the disciplinary and problem solving
research done today.

Young, et. al. expressed their primary reservations with the elicita-
tion and use of risk preferences as twofold: (1) the errors inherent in
previously used measurement techniques, and (2) the possible temporal
instability of preferences. It should also be noted that there is little
evidence to support the common assumption that risk preferences elicited in
one decision context can be generalized to other contexts, even at the same
point in time.

This dissertation focuses on the problem of the instability of at-
titudes towards risk indicated by current measures. The research is
motivated by the assumption that individuals‘ attitudes towards risk are
stable, and that the perceived instability is due to the measures used. He
begin with an examination of why currently used measures of attitudes
towards risk result in the appearance of instability. A new measure of
attitude towards risk is then developed which is hypothesized to be stable

globally within one decision context, and also both across decision con-
texts and intertemporally.

Sggg,§g§15_Conggpts and Definitions

The study of attitudes towards risk and their formal use in analysis is
part of a broader field of inquiry known as decision theory or decision
analysis. Although used extensively by economists and agricultural
economists, decision theory finds its home in no one discipline. Develop-
ments in decision theory are drawn from and used in many disciplines in-
cluding economics, management science, mathematical psychology, finance,
anthropology, sociology, statistics and systems science. Hhile this conflux
of numerous, seemingly unrelated, disciplines often creates definitional

problems (Robison and Fleisher, 1983), it, at the same time, provides a

varied and fertile set of conceptual frameworks from which to draw. In
fact, the foundations for much of the development of the new measure of
attitude towards risk presented in this dissertation come not only from
economics, but also from recent theoretical work in other disciplines
including mathematical psychology, management science and systems science.

An example of the definitional problems inherent in work in the field
of decision theory is the term relative risk aversion; economists encoun-
tering this term immediately think of the Arrow-Pratt coefficient of rela-
tive risk aversion, described later in this chapter, while mathematical
psychologists think of an entirely different concept, to be discussed in
detail in Chapter 3. To avoid confusion, the mathematical psychologists
concept of relative risk aversion is referred to as intrinsic risk aversion
in this document.

Hhile many terms will be defined as they are needed, several concepts
used throughout the dissertation are defined here.

Risk and uncertainty, two of the most commonly used terms in decision
theory, are probably the most difficult to define. In fact, Stiglitz has
commented that, “Risk is like love; we have a good idea of what it is, but
we can't define it precisely“ (Roumasset, 1979, p. 4). Fleisher and
Robison‘s definitions will be used here. They state (Fleisher and Robison,
1985, pp. 3-4):

“Certain and uncertain are adjectives used to describe
events. Events with only one possible outcome are defined as
certain: the single outcome has a probability of one of
occurring. Uncertain events are those with more than one
possible outcome: each possible outcome has a probability
between zero and one of occurring. Since events are either
certain or uncertain, we cannot say that one event is more

or less uncertain than another. Each event is either uncer-
tain or certain.

A, class of uncertain events which alter the gellrbeing
of either a weTT'defifiEd class of decision makers 9;, a
angle—decision gaker ag @le risRy events. Riskiness,
because it depends on the decision makers‘ attitudes, likes
and dislikes, cannot be made more precise without first
defining whose well-being is used to give meaning to the
concept. Once we define the class of decision makers, we may
be able to make comparative statements like action choice
A's outcomes are more or less risky than 8'3. The important
point to remember is that an event's riskiness depends on
the preference of an individual or a class of individual
decision makers. Riskiness cannot and should not be used
interchangeably with the word uncertainty. Riskiness should
also not be confused with the dictionary definition of the
noun 'risk.‘ Risk is defined in the dictionary as the pos-
sibility a: loss or injury. Lg the context gfi decision
anal sis wever, a risk event ma result in favorable or
unfavorable cogggguénces _gg____,d§%ision mafier. ‘—

 

 

 

 

Another distinction which needs to be made is between action choices,
outcomes and attributes. Action choices are defined as the set of alterna-
tives available to the decision maker, including the alternative of doing
nothing. In a tractor purchase decision, a set of action choices might
include buying brand A, buying brand 8, or not purchasing a tractor. An
outcome can be thought of as the result or consequence of an action choice.
In the case of a tractor purchase, the outcome may be buying brand A trac-
tor and having it live up to the purchasers' expectations. An attribute, on
the other hand, is a specific characteristic of an action choice or an
outcome. In the case of the tractor purchase decision, attributes of each
tractor could include fuel efficiency, PTO and draw bar power.

Currently Used Measures gfi Attitude Towards Risk
Models of decision making and their application are as diverse as
decision theory's bases. Some models of decision making are descriptive,

attempting to capture and explain the cognitive processes used in making
actual decisions. Others are prescriptive and are used to help decision

makers make l'better" decisions. The third group can be classified as
predictive. These models make no attempt to formally model decision making
processes. Instead, they treat the cognitive decision process as a black
box; decision makers' preferred action choices are predicted as, if they
followed a certain set of procedures without assuming that these are the
procedures actually used.

Although increasing use is being made of prescriptive models by
agricultural economists working in farm management and related fields,
economists, in general, tend to be primarily concerned with predictive
models of decision making under uncertainty. The research presented here
also falls into this category.

Hhile there are numerous models which can be used to predict preferred
action choices under uncertainty, Hey (1979) estimates that the expected
utility hypothesis provides the basis for over 90$ of the disciplinary work
(in economics) on individual or firm behavior under uncertainty. The ex-
pected utility hypothesis was first formally deduced from a set of axioms
by Ramsey (1931) and later developed more fully by von Neumann and Mor-
genstern (1944). The expected utility hypothesis asserts that if a decision
maker's preferences satisfy certain conditions of “rationality“, their
utility function elicited over risky prospects can be used to predict their
preferred action choices in other risky situations. Specifically, von
Neumann and Morgenstern proved that their four basic axioms imply the
existence of numerical utilities for outcomes such that the individuals'
expectations for lotteries over these outcomes preserve their preference
orderings, i.e. greater expected utility corresponds to higher preference.
In von Neumann and Morgenstern's own words, they “practically defined
numerical utility as being that thing for which a calculus of expectations

is legitimate' (von Neumann and Morgenstern, from Schoemaker, 1982, p.
531). Because of their role in the formalization of the expected utility
hypothesis, a utility function elicited over risky prospects is usually
referred to as a von Neumann-Morgenstern utility function.

The four axioms of 'rational behavior” which expected utility maxi-

Inzers are assumed to follow are:

1. Ordering: If an individual confronts two risky prospective
action choices, A and 8 each with more than one potential

outcome or with a Egobabilitz distribution of outcomes, he
will prefer one of t two ris y prospects or will be indif-
ferent between them.

2. Transitivit : If the individual confronts three risky
prospEEES,A, B and C and prefers A to B and B to C, then he
will also prefer A to C.

3. Continuity: If an individual prefers A to 8 to C, then there

exists a unique probability, p, such that he will be indif-
ferent between B and a lottery of the form pA + (1 - p)C.

4. Independence: If action choice A is preferred to 8 and C is
some 0 r ottery, then the individual will prefer a lot-

tery of pA + (1-p)C to the lottery p8 + (1-p)C.

The four axioms of rational behavior provide both the mathematical
foundations for expected utility theory and the behavioral conditions that
an individual's preferences must meet to be an expected utility maximizer.
Satisfattion of the behavioral requirements can be viewed as a precondition
for expected utility maximization. If a decision maker's preferences con-
form to these axioms, a utility function u(x) can be formulated which
reflects the preferences of the decision maker. A utility function u(x)
derived for an expected utility maximizer has the following properties:

1. If x1 is preferred to x2 then u(xl) > u(xz).

2. The utility of a risky prospect is equal to the expected
utility of its possible outcomes.

3. The scale on which utility is measured is arbitrary; utility
functions are unique up to a positive linear transformation.

In addition to the four formal conditions of “rational behavior“ listed
above, many applications of the expected utility hypothesis implicitly
assume that several other conditions are met. These include:

1. That outcomes can be denominated in terms of a common unit.

2. That there is homogeneity of preferences for attributes of
riskless outcomesl.

3. That individuals can accurately educe their preferences.

4. That individuals subjective probabilities are the same as
stated “objective” probabilities.2

5. That utility functions elicited usin hypothetical decision
contexts are applicable to actual dec sions.

Different variants exist within the general expected utility framework
depending on what type of probability transformations are allowed, how the
outcomes are measured and how utility is measured.

Hhile the axiomatization of the expected utility hypothesis was a major
step in the development of decision theory, a suitable numerical measure of
attitude towards risk was needed to arrive at quantifiable theories and
predictions of decision makers' behavior. The application of the expected
utility hypothesis to empirical problems was given impetus when Pratt
(1964) and Arrow (1965) set forth a means of characterizing individuals on
the basis of a numerical measure of their attitude towards risk. Pratt

 

1 Homogeneity of preferences implies that, at equilibrium, each in-
dividual values equally, or derives the same amount of utility from,
the last unit of each commodity purchased and that this value or
utility is identical with the value or utility received from an amount
of money identical to the goods' price. This assumption ignores the
existance of consumer surplus.

2 Several variants of the expected utility hypothesis, such as the sub-
jective expected utility and subject ve y weighted utility models do
not assume that this condition holds. Readers are referred to
Schoemaker (1982) for an interesting exposition on different schools of
probability theory related to assumption number four.

showed that a numerical measure of risk aversion, based on the rate of
bending of an individual's von Neumann-Morgenstern utility function, is
unique and invariant under positive linear transformations of the utility
function. The measure he proposed, now known as the Arrow-Pratt coefficient
of absolute risk aversion, is defined as:

(1.1) r(x) = -u'(x)/u'(x).
where x is income or wealth.

The ArrowbPratt coefficient of absolute risk aversion can be taken at
any arbitrary point on an individual's utility function. Two other measures
of attitude towards risk can be shown to be related to the Arrow-Pratt
measure. A risk premium measure of attitudes towards risk can be derived
from the same individual's utility function by asking “for a small gamble
with variance 02 and mean xt, what risk premium, I, would the individual be
willing to pay to eliminate the uncertainty?“ Pratt showed that the ap-
proximate relationship between these two measures, when the ArrowePratt
coefficient is taken at the mean of the gamble, is:

(1.2) s = r(x*)63/2
or that the risk premium, I, is equal to the value of the coefficient of
absolute risk aversion at the mean of the gamble times the variance of the
action choice divided by two. The certainty equivalent, CE, of the same
gamble can be found by replacing I, the risk premium, with x* - CE. This
can be expressed as:

(1.3) CE = x. - r(x*)01/2 = x* - m
Pratt then inferred that the more risk averse the individual, the larger
the risk premium he would require. Furthermore, he asserted that, at a
point, or 'in the small,” individuals could be ordered according to their

attitude towards risk measured in terms of a risk premium, a certainty

equivalent, or the Arrow-Pratt coefficient of absolute risk aversion. A von
'Neumann-Morgenstern utility function and the three related measures of
attitude towards risk are shown in Figure 1.

Note that these are all local measures of attitude towards risk. If the
attitude towards risk is constant over the range of outcomes associated
with one decision, Pratt showed that the utility function must take one of
the forms:

u(x) is a positive linear transform of x if r(x) = 0
u(x) is a positive linear transform of -e‘°x if r(x) = c > 0

u(x) is a positive linear transform of e-cx if r(x) = c < 0

Pratt also defined a local measure of relative risk aversion defined

(1.4) rr(x) = -(x * u'(x))/u'(x)
This coefficient measures the elasticity of the marginal utility of wealth
and is invariant not only with respect to changes in units of measurement
of utility but also with respect to changes in units of wealth.

Numerous empirical studies have measured farmers' attitudes towards
risk using one of the three related measures discussed above. The measured
attitudes towards risk have often been used to order farmers according to
their risk attitude, predict preferred action choices, or attempt to estab~
lish correlations between socio-economic attributes of individuals and
their risk attitudes. Readers interested in a review and discussion of this
literature are referred to Fleisher and Robison (1985).

Although the large body of literature on measurement of attitudes
towards risk in agricultural settings will not be reviewed here, several of
the points made by Fleisher and Robison must be noted because of their

10

 

u U(X)
u(8)-8.3 //
u(5)=6.4

 

 

 

 

 

 

 

.5u(8)
+.5u(2) ,
-5.66 g
' I
E I
u(2)=2.8 }
l Hi
i :
n l
.I .I L 4'
2 4.1 5 8 x
x*
u(x) = 4 ln x CE = 4.1
r(xs) = 1/xs I = .9

Hhen presented with and equal chance lottery between 2 and 8, or 5 for
sure, the individual selects 5, since:

CE = x - r(xs) 01/2 = x* - m
which, in this case yields:
4.1 = 5 - .2 * 9/2 = 5 — .9

Figure 1: A von Neumann-Morgenstern Utility Function and the Relationship
between Arrow-Pra t, Risk Premium and Certainty Equivalent
Measures of Attitudes Towards Risk.

11

direct bearing on the present study. The first is the caveat that measures
of attitudes towards risk taken at a point on the individuals' utility
function are local measures and cannot justifiably be used to order in-
dividuals according to their attitudes towards risk "in the large“. The
second is that when utility functions are elicited using hypothetical money
gambles, they have not proven very useful in predicting preferred action
choices in ''real world' situations. Third, when attitudes towards risk for
the same individuals are taken at two different points in time, they are
not stable (Halter and Mason, 1971: Hhittaker and Winter, 1980: Love and
Robison, 1984; Mosteller and Nogee, 1951). In addition, Hildreth and
Knowles (1982) cite several empirical studies of individuals‘ risk
preferences that produced apparently risk neutral or risk loving responses
in cases where the decision makers' true preferences would be expected to
be risk averse. These factors support the contention that something other
than attitude towards risk influence preference rankings.

In addition, several studies have examined the conformance of decision
makers' preferences with the behavioral conditions specified by the axioms
of expected utility theory. Of particular concern is the independence axiom
which assumes invariance of preference between certainty and uncertainty
when other things are equal. According to Machina (1982, 1983),
empirical/experimental research on the independence axiom has uncovered
four types of systematic violations of the axiom: the common ratio effect,

the common consequence effect, oversensitivity to changes in small

12

probabilities and the utility evaluation effect. Only one type of the

common consequence effect will be discussed here.3

Kahneman and Tversky (1979) observed a certainty effect by which deci-
sion makers viewed outcomes obtained with certainty as being dispropor-
tionately larger than those which are uncertain. A related experiment on
the effect of certainty versus uncertainty was performed by Ellsberg (1961)
showing systematic violation of the independence axiom. Other authors (e.g.
Masson, 1974) have shown that violations of the independence axiom create
discontinuities in the utility function at certainty. Heiss (1984) has
demonstrated that the certainty effect can be modeled by allowing decision
makers to have 'two-rule' type measurable utility functions reflecting
dichotomous behavior towards certainty and uncertainty. He also
demonstrates (Heiss, unpublished) that predicted behavior under uncertainty
when one allows for dichotomous behavior differs from the predictions made
when this dichotomy is not allowed.

Despite' these findings, most empirical applications of the expected
utility hypothesis and other models of decision making under uncertainty
which derive local measures of attitude towards risk treat them as true
indicators of preferences and employ them in generalized conclusions about
the risk attitudes of a population or the ordering of individuals within
the population.

 

3 For a discussion of oversensitivity to changes in small probabilities,
the reader is referred to Kuenreuther and Slovic (1978). A discussion
of the utility evaluation effect may be found in McCord and de Neuf-
ville (1983) and a discussion of the common consequence effect can be
found in Raiffa (1961) and Slovic and Tversky (1974).

13

Objectives gfi,th1§_§tggy,

This goal of this undertaking is to contribute to the improvement of
the application of decision theory in agricultural teaching, research and
extension by developing a new measure of attitude towards risk which
separates the influence of preferences for attributes of outcomes under
certainty from like or dislike of Uncertainty. The first step towards
attainment of this goal is to show why the Arrow-Pratt coefficient of
absolute risk aversion is not stable across decision contexts and therefore
why agricultural economists can not continue to use coefficients derived in
one decision context to predict preferred action choices in other decision
contexts or at other points in time. The second step is to develop a new
measure of attitude towards risk which is hypothesized to be stable and
therefore can be used to generalize across decision contexts and compare
individual's attitudes towards risk. The third step is to examine the new

measure more closely through empirical application.

Overview 9: the Dissertation

This chapter has provided a brief introduction to the problem addressed
in the dissertation. Chapter 2 examines the question of why risk attitudes
appear to be unstable when measured using the Arrow-Pratt coefficient of
absolute risk aversion. This includes a brief digression into the founda-
tions of utility theory and some of the controversies, questions, an am-
biguities in terminology which have arisen and often obscured our under-
standing of utility functions and what they actually represent. Chapter 3

14

introduces the concept of intrinsic risk aversion4. These chapters provide
the foundation for Chapter 4, in which a new measure of attitude towards
risk is developed and discussed. Chapter 5 discusses the procedures used to
empirically examine the new measure and presents the limited conclusions
that can be drawn from this exercise. Chapter 6 contains a summary of the
research and observations regarding application of the results as well as a
discussion of issues requiring further research.

The seasoned reader of dissertations will note that there is no chapter
which is explicitly devoted to a review of the pertinent literature. Be-
cause the literature/ which provides the background for this study is so
diverse, ranging from measurement theory to management science, individual

sources are cited in the text where they are most applicable.

 

4 This concept is referred to as relative risk aversion in the mathemati-
cal psychology and much of the management science literature. Since
this concept of relative risk aversion bears no relationship to the
Arrow-Pratt coefficient of relative risk aversion confusion will be
reduced by referring to this concept by a new, if less descriptively
accurate, term.

CHAPTER 2

HHY DOES THE ARROH-PRATT COEFFICIENT OF ABSOLUTE RISK AVERSION
INDICATE UNSTABLE ATTITUDES TOHARDS RISK?

Various authors have suggested that the instability in empirically
measured Arrow-Pratt coefficients of absolute risk aversion is due to a
wide variety of problems including measurement error, subjectively held
probability estimates which differ from those given in the elicitation
procedure, discontinuities in the utility function and ”other factors".
Hhile it is recognized that all of these arguments may have validity, they
will not be examined closely in this chapter. Instead, we will focus on
what the von Neumann-Morgenstern utility function actually represents and
how this affects what the Arrow-Pratt coefficient of absolute risk aversion
derived from the utility function actually characterizes.

He begin by using a measurement theoretic approach to examine what the
utility function actually represents. This is followed by a brief digres-
sion into the development of utility theory and the expected utility
hypothesis' place in this context. Special attention will be paid to two
assumptions: constancy of the marginal utility of money, and homogeneity of
preferences. This discussion is then used to examine what effect the viola-
tion of these assumptions has on the use of Arrow-Pratt coefficients as a

numerical measure of attitudes towards risk.

15

16

Haas Q9g§,thg_Utility Function Actually Represent?

Measurement theory starts with an observed or empirical relational
system (e.g. someones likes or dislikes for outcomes), called the domain. A
mapping to a numerical relational system (e.g. a utility function), called
the range, is sought. This mapping preserves all of the relations and
operations ( e.g. preferences) in the original observed or empirical sys-
tem. Axioms of the expected utility hypothesis can be conceived of as the
representation theorem which does the mapping. Hhile many questions raised
in examining the axioms as a representation theorem are of concern to those
interested in decision theory, they are not the major focus of this chap-
ter. Instead, our present concern is with what the representation theorem
is actually mapping.

The expected utility hypothesis takes as its domain preferences over
probability distributions over outcomes and maps this to a utility function
whose range is the real line from which attitudes towards risk are derived.
This mapping is shown in Figure 2.

The conceptualization of the domain and range for the utility function
under uncertainty results from von Neumann and Morgenstern's use of risky
prospects in developing utility functions. This utility function differs
from the utility function in standard consumer theory in that the latter
involves only certain outcomes. These utility functions map from the domain
of preferences over riskless outcomes to a range on the real line. This
standard consumer theory preference function will be referred to as a value

function to distinguish it from a von Neumann-Morgenstern utility function
Cover risky prospects.

17

 

 

preferences over utility function whose

over outcomes

 

 

A
X
I
probability distributions --- g ---> range is the real line
S

[domain] [representation theorem] [range]

Figure 2: The von Neumann-Morgenstern Utility Function as a
Homomorphic Mapping

18

Note that the expected utility hypothesis takes as its domain
preferences over probability distributions over outcoggs. If the repre-
sentation theorem provides a mapping which preserves all relations and
orderings in the domain (a homomorphic mapping), the resultant utility
function incorporates both preferences over probability distributions and
preferences over outcomes. The former may be viewed as containing informa-
tion about attitudes towards risk while the latter contains information
about preferences for riskless outcomes.

The obvious question, then, is if the utility function whose range is
the real line incorporates information about both preference for riskless
outcomes and attitude towards risk, can we use the rate of bending of the
utility function [the Arrow-Pratt coefficient] as a measure of attitude
towards risk? Before this question can be answered, we must have a better

understanding of value and utility functions and the assumptions each
embodies.

A Brief Digression into the Foundations g§_Utility_

This digression into the foundations of utility is being made in the
hope that it will clarify some of the confusion surrounding what the
utility function actually represents. At the same time it will highlight
the assumptions in the expected utility hypothesis which are a direct
carryover from the earlier cardinalists and which give rise to the present
concern over the use of the Arrow-Pratt coefficient of absolute risk aver-
sion as a measure of pure attitude towards risk. Many attacks on the ex-
pected utility hypothesis are based on earlier criticisms of cardinality as
used by the material welfare school [Baumol, 1951: Majumdar, 1958]. mile
some of these criticisms remain valid when examining the expected utility

19

hypothesis, many others ignore the important distinctions which can be made
between the early and new cardinalists.

Discussion of the historical foundations and concepts of utility are
made more complex by the fact that the distinctions between what we shall
call the early cardinalists, the ordinalists, and the new cardinalists
cannot be made purely on the basis of debate over the l'measurability" of
utility. Cardinalists and ordinalists also disagree among themselves and
with each other as to whether utility is an introspective or a behavioral
concept, on its importance in welfare economics, and its use in determining
preference orderings among risky prospects.

Early cardinalists viewed the fundamental natural operation in the
measurement of utility as taking place within the mind of the subject:
utility measurement was a process of introspectively weighing the amounts
of satisfaction associated with different outcomes. Elicitation of utility
was simply a matter of mapping these internal measures of satisfaction onto
the real number line. It was argued that since individuals could not only
order degrees of satisfaction but also judge their strength of preference
for each outcome in a cardinal fashion (e.g. outcome A provides twice as
much satisfaction as outcome 8), the utility function derived from the
mapping process was cardinal. In other words, not only was the mathematical
representation viewed as cardinal, but the extra-mathematical interpreta-
tion was also cardinal in nature.

Three important assumptions about the nature of utility are implied by
the early cardinalists. First, an increment of utility is not only quan-
tifiable in principal, but also is measurable in terms of money. Secondly,
utility is a measure of worth to the individual and does not necessarily

imply any preference ordering; it is simply a measure of the amount of

20

satisfaction an individual could obtain from a given outcome. Furthermore,
this satisfaction refers directly to the meeting of physical needs, not the
preference for more abstract desires. Third, utility is additive and inter-
personally comparable.

It was argued that if we look upon money as a measure of general pur-
chasing power, then there can be no objection to regarding it as an in-
dicator of alternative welfare-yielding situations. Therefore, the amount
of sacrifice, in terms of money, which a person would make rather than go
without a particular thing is a measure of his welfare derived from it. The
difficulty with this conclusion rests with a further assumption. If money
provides the measuring rod of utility then, as with all measuring rods, its
unit must be invariant; it must measure the same amount of utility in all
circumstances. The first dollar must mean the same to the individual as the
hundredth or millionth. This means that one has to assume constant marginal
utility of money. The importance of this assumption will be discussed
below.

Jevons was one of a group of economists who shifted the focus of
utility from a purely individualistic approach to that of explaining be-
havior and conditions of exchange in a market economy. Hhile Jevons still
believed (albeit ambiguously) in the cardinality of utility, he did not
view "total utility” for any good as being intrinsically measurable or
interpersonally comparable. Instead, he hypothesized that interpersonal
comparisons of welfare could be obtained through the use of the concept of
marginal utility. He states in his second edition of The Ihggry_g§_flg]1tir
gal Eggnggy (1879, from Stigler, 1950, p. 317) that

'It is from the quantitative effects of the feelings that we
must estimate their comparative amounts. I never attempt to

21

estimate the whole pleasure gained by purchasing a com-

modity; the theory merely expressed that, when a man has

purchased enough, he would derive equal pleasure from the

possession of a small quantity more as he would from the

money price of it.‘
Jevons uses this to argue that, at equilibrium, welfare is maximized be-
cause each individual's marginal utility is equal, and thus, all in-
dividuals would obtain the same degree of satisfaction of needs from the
last unit purchased. From this argument comes the assumption of homogeneity
of preferences across individuals, implying that the last unit purchased by
each individual is valued by each individual at the market price of that
unit.

Marshall extended Jevons' concept of utility to formulate a hypothesis
of diminishing marginal utility for goods. Marshall argued that an incre-
ment of utility (marginal utility) is quantifiable in principle and is
actually measurable in terms of money. He also asserted the law of nega-
tively sloping demand curves in all generality (Marshall, 1890, pp. 159-
160, from Stigler,1950, p. 326):

“There is then one law and only one law that is common to
all demand schedules, viz. tha the greater the amount to be
sold, the lower will be the price at which it will find
purchasers.“

It is important to note that the Marshallian demand theorem cannot be
derived from the hypothesis of diminishing marginal utility except in a one
commodity world without contradicting the assumption of constant marginal
utility of money with respect to changes in prices and income. If this
difficulty is avoided by giving up the assumption of constant marginal

utility of money, then money can no longer provide the measuring rod for

22

utility and we can no longer express the marginal utility of a commodity in
units of money.

Hith the advent of an ordinal concept of utility, utility was no longer
viewed as a psychological entity measurable in its own right. Instead,
utility was viewed as a convenient label for the explicit value of a func-
tion which described consumer behavior. Not all ordinalists argued that
utility was not quantifiable in principle. Their fundamental argument
centered on the fact that a basic understanding of the behavior of par-
ticipants in an exchange economy did not depend on the assumption of car-
dinally measured utility. Assumptions regarding cardinality in extra-
mathematical interpretation were viewed as extraneous to the issues at
hand; since equally strong results for price theory could be obtained
without this assumption, there was no reason to suppose that anything more
than ordinal utility was required. The move to an ordinal concept of
utility was not necessarily a total rejection of the mathematical concept
of measurable utility; its extra-mathematical interpretation was simply
unnecessary.

Hith the publication of von Neumann and Morgenstern's Ihgggy,gf_§a___
agg_Economic Behaylgg in 1944, the debate over the cardinality of utility
was rekindled. From a measurement theoretic perspective, von Neumann-
Morgenstern utility is cardinal because the utility scale has interval
properties (it is unique up to a positive linear transformation) However,
from a preference perspective, von Neumann-Morgenstern utility is ordinal
in that it provides no more than ordinal rankings of lotteries. It differs
from earlier cardinal utility in that utility is viewed not as I'measurable"
in the sense that it is correlated with any significant economic quantity
such as the quantity of feeling or satisfaction or intensity. It is

23

cardinally measurable only to the extent that the numerical operation of
forming mathematical expectations on the basis of these numbers would be
related to observable behavior, so as to be emprically meaningful.

Von Neumann and Morgenstern's utility also differed from early car-
dinalists in that they use risk choices to define utility differences, not
to estimate introspectively ascertained differences. Therefore, von
Neumann-Morgenstern utility should not be interpreted as cardinally measur-
ing strength of preference for uncertain outcomes; as a preference measure
it is strictly ordinal.

The distinction between this new cardinality in the purely measurement
theoretic sense, and the earlier cardinalists is made more confusing by the
strong similarities in notation and methods used to determine the outcome
which maximizes utility. Ellsberg (1954) illustrates the source of this
confusion with an example. Suppose an individual who prefers A to B, B to C
and A to C must choose between having a lottery ticket offering A with
probability p or C with probability 1-p. Both early cardinalists and von
Neumann and Morgenstern would assign a triplet of numbers, UA, U3, and Uc,

to the three outcomes with the property that UA > U3 > Uc. This triplet is

a utility index for the three outcomes since their order of magnitude
reflects the order of preference.

Next, both would form the expression:

(2.1) U3 is greater than, less than, or equal to pUA + (1-p)Uc

where p and l-p are the respective probabilities of A and C.

In each case the individual would choose the prospect with the highest
mathematical expectation of utility. Furthermore, for both the early car—
dinalists and von Neumann and Morgenstern, utility would be said to be

24

measurable. The necessity of this assumption is seen if the expression is

rewritten:
(2.2) p(UA-UB) - (1-p)(U3-Uc) is less than, greater than, or equal to 0

The rewritten expression shows clearly that the rules rely on comparing
differences in utility. In order to retain consistency in results, it is
necessary to find some operation that gives meaning to differences in
utility numbers, and hence to the numerical operations implied by the rule.
A utility index which is ordinal in the measurement theoretic sense (unique
up to a monotone transformation) would not satisfy this requirement. There-
fore, a more restricted set of indices, which summarizes the results of the
additional operations as well as preferences, must be used. This require-
ment is satisfied by the set of indices which are invariant up to a posi-
tive linear transformation, or, in measurement theoretical terms, cardinal.

The similarity between von Neumann Morgenstern and early cardinalist
indices is that both require a utility index determined up to a positive
linear transformation. However, the differences between utility points
could be different for the two indices. Moreover, the indices would, in
general, but not always, be monotonic and not linear transformations of
each other; preservation of orderings requires a positive linear transfor-
mation of utility but only a positive monotonic transformation of expected
utility. Therefore, predictions based on the rule of maximizing expected
utility could differ for the two approaches.

A more fundamental difference exists between the two approaches; early
cardinalists would come to their conclusions regarding an individuals
choice between the options, one of which is a risky prospect, without ever
observing that individual's behavior in other risk situations. For von
Neumann and Morgenstern, it would be necessary to observe the individual's

25

behavior in other risk situations, involving different outcomes or the same
outcomes with different probabilities. Utility is defined by an
individual's choices among risky prospects. Hhere early cardinalists postu-
lated a type of consistency between individual's risk choices and their
feelings of relative differences in desirability of the outcomes, von
Neumann and Morgenstern hypothesize simply a consistency between risk
choices and other risk choices.

However, von Neumann and Morgenstern's break with the early car-
dinalists is not as complete as it may seem from the discussion above. They
state (von Neumann and Morgenstern, 1944, p. 8):

'..He shall therefore assume that the aim of all par-
ticipants in the economic system, consumers as well as
entrepreneurs, is money, or equivalently, a single monetary
commodity. This is supposed to be unrestrictedly dividible
(sic) and substitutable, freely transferrable and identical,
even in the quantitative sense with whatever 'satisfaction'
or 'utility' is desired by each participant.“

In addition, as Schoemaker (1982, p. 533) points out,

'..(P)references among lotteries are determined by at least
two separate factors: namely (1) strength of preference for
the consequences under certainty, and (2) attitude towards
risk. The NM utility function is a compound mixture of these
two without direct resort to interval comparisons or
strength of preference measures. As a preference theory, it
is wholly ordinal. Nevertheless, it implicitly assumes that
a neoclassical type of utility exists, otherwise it would
not be possible psychologically to determine the certainty
equivalence of a lottery."

26

[he Igpgrtance gfi_Heterogeneity of Preferences
In describing the environment in which the expected utility hypothesis
may be applied, von Neumann and Morgenstern employ a very restrictive set

of initial conditions and auxiliary assumptions.l For money to serve as the

measuring rod of "satisfaction'' or I'utility" each individual must have

constant marginal value2 for money. For money as a measuring unit to be
”...unrestrictedly dividable and substitutable, freely transferrable and
identical for whatever satisfaction..." (von Neumann and Morgenstern, 1944,
p. 8) preferences and the utility function must be homogeneous. In other
words, indifference curves for any two goods, or one good and money, must
be straight lines with slope of -1 and each individual must value goods at
their market price. .

Unfortunately, empirical evidence does not lend credence to the use of
these initial conditions and auxiliary assumptions. The following examples
will demonstrate the importance of violation of these conditions and as-
sumptions when the von Neumann-Morgenstern utility function and Arrow-Pratt
coefficients of attitudes towards risk are used to predict preferred action
choices in the real world. As a base for comparison, Figure 3 shows the
value functions for two individuals who conform to von Neumann-
Morgenstern's set of initial conditions and auxiliary assumptions. The
reader will recall that the independence axiom of the expected utility

 

1 Giere (1979, pp.87-88) defines initial conditions as “...either the
occurrence of a state of the system or a statement describing this

occurrence.‘ and auxiliary assumptions as “...additional assumptions
about the system being stud ed and about its immediate environmen .'

2 In the literature, this phenomenon is usually referred to as constant
marginal utility for money. Since we are distinguishing between utility
as preference for risky outcomes and value as preference for riskless
outcomes, the term constant marginal value for money is used here.

27

v /<s)
6 —— _

___\

 

 

 

 

 

 

 

 

 

 

 

 

4 u
I
I
I
I
2 ---- /:
l I
I
l l
l
2 4 6 5
Individual A
v V($)
6 I
I
I
I
4 i
I I
l
l
2 l l
l
I
: l-
2 4 6 5
Individual 8

Figure 3: Value Functions of Two Individuals with Constant Marginal
Value of Money

28

hypothesis assumes invariance of preference between certainty and uncer-
tainty. Under this assumption, the individual's utility function is a
linear transform of their value function.

Both individual A and 8 exist in a world which consists of one monetary
commodity. They have identical constant marginal value for money. He will
assume for this case and the subsequent ones that individuals are actually
risk neutral. This means that their risk premium will be zero, or in other
words, their certainty equivalent will be equal to the mean of the lottery.
If we present our two individuals with a choice of an equal probability
gamble between $2 and $6, or $4 with certainty (which is also the mean of
the lottery), .each will be indifferent between the lottery and the sure
thing because their certainty equivalent is equal to the mean value of the
lottery. For both individuals, the value of the lottery is:

(2.3) .Sav($2) + .5ev($6) = .5m(2) + .5(6) = 4
which is identical to the certain offer of $4 which also has a value of 4.
Any response other than indifference would be an accurate indicator of risk
aversion or risk proneness.

Figure 4 presents the value functions for two individuals. In this
case, individual 8 does not have constant marginal value for money. The two
individuals are presented with the same choice between a gamble and a sure
thing and individual A is indifferent between the two. But individual 8 is
not; he would take the sure thing over the gamble. Using our current no-
tions of attitude towards risk, we would say that this indicated that
individual 8 was more risk averse than individual A with regard to lot-
teries with monetary outcomes. But in this case, that conclusion is not
justified. The value of the lottery for individual 8 is:

(2.4) .5*(v($2) + .5*v($6) = .5*(4) + .5*(6.25) = 5.125

29

v /(s)
6

 

\_.

 

 

 

N
_——————--—

 

 

N
4.\
mP.-

0)

Individual A

 

 

V
6.25- /fl$)
l
5.50 l'
l
4.00 ~---

 

 

 

qF—------

 

N

b

0‘
(D

Individual 8

Figure 4: Value Functions of Two Individuals without Constant Marginal
Value of Money

30

which is less than the value of the sure consequence to individual 8 since,
for him, v($4) = 5.5. In this case the apparent risk averseness of in-
dividual 8 is due to his non-constant marginal value for money, not the
requirement for a risk premium.

Figures 5 and 6 present cases where we further relax the assumptions
made by von Neumann and Morgenstern. In these cases individuals are assumed
to not live in a world which consists of just one monetary commodity;
individuals A and 8 use the money to buy many different commodities. The
individuals whose value functions and indifference curves are shown in
Figures 5 and 6 are concerned with making a decision about tractor services
which can be purchased for $1 per unit of service. In Figure 5, individuals
A and 8 conform to von Neumann and Morgenstern's assumptions and have
constant marginal value for money and homogeneous preferences and have
indifference curves for money and tractor services of slope -1. He can
present the two individuals with our standard choice between $4 for sure
and an equal chance gamble of receiving $2 and $6. In this case both in-
dividuals are indifferent between the gamble and the sure thing, indicating
risk neutrality.

This result does not hold in Figure 6 where individuals A and 8 do not
have homogeneous preferences. Specifically, individual 8 does not have
homogeneous preferences for money and tractor services; examination of the
figure shows that individual 8 does not have a linear transformation be-
tween money and tractor services. The value the individual derives from $2
is 2 while the value the individual derives from $2 of tractor services is
3, the value he derives from $4 is 4 while the vlaue he derives from $4 of
tractor services is 5 and the value he derives from $6 is 6 while the value
he derives from $6 of tractor services is 8. If the individuals are

31

Individual A Individual B

V /r($> " /’<$’
e 6 -—

 

 

 

 

 

 

 

 

- P—-- -

 

 

2 z, 6 $ 2 4 6 5

(tractor v(tractor
/ services) 6" services)

o~<

\
_\___

 

 

 

 

 

 

 

 

 

 

 

 

I I
I I
I I
I I
2 I i 2 —---- .
I I I I' I
I I I l
I I I
I I I
_1 I I
4 6 2 4 6
tractor services tractor services
0 $1 per unit 0 $1 per unit
$ $
tractor services tractor services
0 $1 per unit 0 $1 per unit

Figure 5: Value Functions and Indifference Curves of Two Individuals Hho

Have Constant Marginal Value of Money and Homogeneous
Preferences

32

Individual A Individual 8

v($) V /($>
6

 

 

  

 

 

N
I
I
I
\\\\\\|
N b

 

 

 

 

 

 

 

b ---

6 $ 2 4 6 $

8 'Iv(tractor

v ////v(tractor services)

6 services)
‘////// 5
4 __ :

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
l
I I
I 3 F"" I i
2 I . I I
. : I
I I i
2 '4 6 2 4 6 .
tractor services tractor services
0 $1 per unit 0 $1 per unit
$ $
tractor services tractor services
Q $1 per unit 8 $1 per unit

Figure 6: Value Functions and Indifference Curves of Two Individuals Hho
Have Constant Marginal Value of Money and Heterogeneous
Preferences ‘

33

primarily concerned with receiving tractor services and we present them
with the standard choice between the gamble and the sure thing, individual
A will be indifferent between the gamble and the sure thing while in-
dividual B will choose the gamble since:

(2.5) .5*v($2) + .5*v($6)

.59v(2 units tractor services) + .S*v(6 units tractor services)

.5*(3) + .s:(a) = 5.5

while the value of $4 of tractor services is 5. Therefore, the value of the
lottery to the individual is greater than the value of the sure thing. Our
current notions of attitude towards risk would indicate that individual 8
was risk prone relative to individual A. But again, if we examine in-
dividual 8's preferences for riskless outcomes, we find that these explain
apparently risk loving behavior when tractor services are the real concern
of the individual. The value of the lottery to the individual is 5.5 while
the value of the sure thing is only 5.

The examples used to this point all concern comparisons of risk at-
titudes of two individuals. Often, however, we use von Neumann-Morgenstern
utility functions and coefficients of attitudes towards risk derived in one
situation to predict the individuals preferred action choice in another
situation. In other cases we determine attitudes towards risk in two dif-
ferent situations, compare them, and, if they are different, assert that
the individual has an unstable attitude towards risk. Figure 7 shows the
case where we have one individual faced with two different decisions. In
the first case it is receiving tractor services, in the second case it is
receiving services from a car. Note that the individual has heterogeneous
preferences; the transformation between car services and money is non—

linear while the transformation between tractor services and money is

v /v<s>
6 __

 

2 -....-

 

 

 

 

:_____\\

 

 

 

 

 

 

 

 

 

 

 

 

 

$
8 _
v I
/v(tractor v(ear .
. V serv1ces)
| serv1ces) i
I s --------,
l
4 . I I
I l I
I I 3 ~-——- I
-_---/ i I i
I I I I
l I I I
I I l I
- I I I
I g I I
2 4 6 2 4 6
tractor services car services
0 $1 per unit @ $1 per unit
$ $
tractor services car services
@ $1 per unit @ $1 per unit

Figure 7: Value Functions and Indifference Curves of One Individual in
Two Situations with Constant Marginal Value of Money and
Heterogeneous Preferences

35

linear. Hhen presenting the individual with the standard choice bwteen $4
for sure and an equal chance gamble between $2 and $6 when he is concerned
with car services, we find that he prefers the gamble to the sure thing
since the results of equation (2.5) hold; when presenting him with standard
choice between $4 for sure and an equal chance gamble between $2 and $6
when he is concerned with tractor services we find that he is indifferent
since the results of equation (2.3) hold. Does this mean that the
individual‘s risk attitudes have changed? No, the "instability“ in risk
attitude only reflects heterogeneity of preferences for riskless outcomes.

More complex examples could be developed for combinations of the cases
where the marginal value for money is not constant and individuals do not
have heterogeneous preferences. However, these would serve only to make the
same point as was made in the examples presented in Figures 3 through 7:
for the Arrow-Pratt coefficient to measure only attitudes towards risk for
outcomes having more than one attribute, the individual's preferences for
riskless outcomes would have to conform to the initial conditions that (1)
the marginal value of money is constant, (2) preferences for goods are
homogeneous, and (3) there is a linear transformation between money and
goods. In other words, we would have to be able to justify the assumption
that the value functions were linear and of identical slope for each in-
dividual across commodities and that they were, in addition, of identical
slope for all individuals.

Chapter 3

THE CONCEPT OF INTRINSIC RISK ATTITUDE:
SEPARATING PREFERENCE FOR RISKLESS OUTCOMES AND ATTITUDE TOHARDS RISK

Recent theoretical work by mathematical psychologists suggests, as we
do, that the perceived instability of Arrow-Pratt coefficient of risk
aversion is due to the fact that the von Neumann-Morgenstern utility func-
tion is a confounding of attitudes towards risk and preferences under
certainty. It has been hypothesized that if the two could be separated, we
would find that individuals do have a stable pure or intrinsic attitude
towards risk and that only preferences under certainty change across situa-
tions because of non-linear transformations between the actual attributes
of outcomes and money. Hhile a measure of pure attitude towards risk or a
measure of strength of preference for certain outcomes cannot replace the
expected utility hypothesis as a means of predicting preferred action
choices under uncertainty, pure attitude towards risk, if stable, can
provide a justifiable means of generalizing across situations or comparing
individuals. Schoemaker (1982, pp. 534-535) cites five reasons to separate
risk attitudes from strength of preference for certain outcomes, namely:

1. Emphasizing the point that there exist different types of
cardinal utility, even within the categories of preferences
for riskless and risky outcomes, that need only be related

monotonically.

2. Providing a means for obtaining a measure of ”intrinsic risk
attitude“.

36

37

3. If intrinsic risk attitude is stable, construction of u(x)
can be simplified by building on v(x), strength of
preference for riskless outcomes.

4. Group decision making can be simplified by aggregating
individuals' v(x) function and transforming this to a von
Neumann-Morgenstern utility function using a group measure
of risk attitude.

5. Cardinal utility under uncertainty is useful for welfare
economics.

This chapter will review the work that has been done on developing a
measure of pure or intrinsic attitudes towards risk by mathematical
psychologists and management scientists. Several studies which have
measured individuals' utility and value functions and shown them to be non-
identical will also be briefly reviewed. Several items are important to
note at the outset. First, although various authors have expressed the
belief that instability in Arrow-Pratt coefficients is due to the confound-
ing of strength of preference for riskless outcomes and attitudes towards
risk, no exposition of this problem analogous to the one presented in
Chapter 2 has been published. Secondly, studies which have ascertained that
the utility and value functions are not identical have not attempted to
develop a relationship between the utility and value function. Thirdly, of
the few studies which have developed a theoretical relationship between
value and utility functions, only one has actually empirically tested this
relationship.

Hgy,§ag,H§,Measure Pure 95 Intrinsic Attitude Igyaygg Risk?

Unfortunately, pure or intrinsic attitude towards risk is not easy to
measure. However, it can be obtained indirectly from knowledge of the
decision makers' von Neumann-Morgenstern utility functions over risky

prospects and their strength of preference or value functions over riskless

3B

outcomes. From a measurement theoretic standpoint, pure or intrinsic at-
titude towards risk can then be seen as the mapping from the value function
over certain outcomes to the utility function for risky prospects as shown
in Figure 8.

This is essentially a two stage process. First a von Neumann-
Morgenstern utility function is obtained using the representation theorem
contained in the expected utility hypothesis. This maps preferences over
probability distribution functions over prospects to a utility function
whose range is the real number line. A value function for the same certain
outcomes is also mapped to the real number line using one of several avail-
able representation theorems. The second, and critical step, is to develop
a transformation between the value function over certain outcomes to the
von Neumann-Morgenstern utility function over risky prospects. This trans-
formation encapsulates the individual's pure or intrinsic attitude towards

risk, or the preference over probability distributions.

Intrinsic Risk Aversion 1

Bell and Raiffa (1979) first introduced the concept of a pure measure
of attitude towards risk by reference to lotteries whose outcomes had been
specified in a particular manner. Specifically, the outcomes of the lot-
teries were first transformed to eliminate what, in our terms, are the
problems of non-constant marginal value for money and heterogeneity of

preference. To do this, they transformed both money and the pertinent

 

1 To avoid confusion with the concept represented by the Arrow-Pratt
coefficient of relative risk aversion, the term intrinsic risk aversion
is used to represent the concept often referred to by mathematical
psychologists and management scientists as relative risk aversion.

 

 

 

 

 

 

 

 

 

 

 

 

 

ST 1 -----
A
X
preferences over I utility function whose
probability distributions ---- O ----- > range is the real line
over outcomes g
A
X
preferences over -------- I ----- > value function whose
riskless outcomes H range is the real line
5
[domains] [representation theorems] [ranges]
5T_ .2.
value function whose utility function whose
range is the real line > range is the real line
transformation

encapsulating pure
attitude towards risk

Figure 8: Pure Risk Attitude as a Function Transforming Value Functions
to Utility Functions on the Real Line

41

attribute of the outcome to strength of preference scales. These scales
were then further manipulated so that money and the attribute of concern
have constant marginal values and a constant marginal rate of substitution.
They hypothesized that the von Neumann-Morgenstern utility function
measured over these transformed outcomes would always take the negative
exponential form and that an individual's basic psychological attitude
towards risk could be captured by the value of the negative exponent.

Hhile Bell and Raiffa's work was an important step in the development
of the concept of intrinsic risk aversion, the development of the theory
must be attributed to Dyer and Sarin (1981). They propose a method by which
preferences for risky alternatives can be determined relative to strength
of preference for outcomes over the same range. The following is a summary
of the most important aspects of their theory.

It is assumed that both the measurable value function v(x) and the von
Neumann-Morgenstern utility function u(x) are monotonically increasing in x
and are continuous and twice differentiable. Using the ArrowePratt measure
of absolute risk aversion, defined as r(x) = -u'(x)lu'(x), as a model, a
local measure of the strength of preference of an individual for an asset
at level x can be defined as m(x) = -v“(x)lv'(x). Just as r(x) >,=,< 0
respectively indicate 'risk aversion“, “risk neutrality“, or “risk
preference", m(x) >,=,< 0 respectively indicate decreasing marginal value
at x, constant marginal value at x, and increasing marginal value at x.

If the individual's value function and utility function exhibit the
same rate of bending at one level of the attribute, x, as shown in Figure
9, it can be argued that the riskiness of the lottery has no effect on
their preferences. Notice that the ' risk premium“, which represents the
curvature of the utility function, and incorporates information about both

42

u(x)

    
    

I

FTT— v(x)

 

 

 

 

 

Pn——--—-——

 

 

x1 x x*

n

I‘D
>4

N

A = risk premium = x* - Xce
B = satiation sacrifice = xa - xev
C = risk premium due to only attitude towards risk = Xev'xce = 0

Figure 9: Value and Utility Functions for the Case of Intrinsic Risk
Neutrality.

43

attitude towards risk and preferences for the outcomes under certainty, is
identical to the satiation sacrifice, which represents the curvature of the
value function sacrifice and incorporates information about preferences for
the outcomes under certainty. Since the utility function incorporates
information about preferences for outcomes under certainty and attitude
towards risk, the curvature of the utility function which can be inter-
preted as the ”risk premium“ is actually attributable to the individual's
preferences for outcomes under certainty, not an aversion or proneness to
risk taking. At any point x, m(x) = r(x) and the individual can be said to
be intrinsically risk neutral. Note that at this point, x, both m(x) and
r(x) could be >,=,< 0. The important point in the measurement of intrinsic
risk aversion is that they are equal. Hhen v(x) is not equal to u(x), the
relationship between m(x) and r(x) can be used to define a local measure of
intrinsic risk aversion.

An individual is said to be intrinsically risk averse if m(x)<r(x),
intrinsically risk neutral if m(x)=r(x), and intrinsically risk prone if
m(x))r(x). This indicates that the rate of bending of r(x) is respectively
greater than, equal to, or less than m(x) at point x.

Hhen a von Neumann-Morgenstern utility function is elicited, it incor-
porates both the individual's preferences for riskless outcomes and their
attitude towards risk. A value function incorporates only preference for
riskless outcomes. In principle, we can also define a utility function
which maps only the individualfs preference for the riskiness of outcomes
independent of his preferences for the outcomes themselves. This is equiv-
alent to eliciting u(v) rather than u(x). He can then define a measure of

44

the rate of bending of this utility function as b(x) = -h"[v(x)]/h'[v(x)],

where the differentiation is taken with respect to v.2

Unfortunately .b(x) cannot be measured directly. Instead, it must be
derived from knowledge of the individual's value and utility functions.
Using a method parallel to that used by Pratt (1964) in his proof of the
coefficient of absolute risk aversion, it can be shown that an individual
is intrinsically risk averse if and only if b(x))0, intrinsically risk

neutral if and only if b(x)=0 and intrinsically risk prone if and only if
b(x)<0. Given:

(3-1) "(X) = hIVIXll
Then:

(3.2) 6u(x)l6x = h'[v(x)]v'(x)

(3.3) 6'u(x)/6xz = h'[v(x)]v'(x) + h'[v(x)](v'(x))2
Remembering that r(x)=-u'(x)/u'(x), m(x)=-v“(x)/v'(x), and b(x) =
-h'[v(x])/h‘[v(x)], we can divide equation (3.3) by equation (3.2) and
rearrange terms to obtain:

(3.4) r(x) - m(x) = v'(x)*b(x)
or

(3.5) (r(x) - m(x))lv'(x) = b(x).
He know that v'(x)>0 from the assumptions of monotonic increases in v(x)

and continuity. Therefore, we are primarily concerned with the relationship
between r(x) and m(x) in the signing of b(x).

 

2 It should be noted that the notation used in the literature to refer to
the rate of bending of the function u(v) is rv(v) rather than b(x). It

has been changed here to avoid confusion with the notation used for the
Arrow-Pratt coefficient of absolute risk aversion, r(x).

45

If m(x)=r(x), indicating that both the value and utility functions bend
at the same rate, r(x)-m(x)=0 and b(x)=0. This indicates that the rate of
bending of the utility function over risky prospects is due solely to the
non-linearity of the value function for the outcomes in riskless situa-
tions. Similarly, if m(x)<r(x), indicating that the rate of bending of the
utility function is greater than that of the value function, r(x)-m(x))0,
and b(x)>0. This implies that the rate of bending of the utility function
is due, in part, to the non-linearity of the value function and, in part,
to some other factor which is assumed to be attitude towards risk. Notice
that we do not treat r(x) as attitude towards risk: instead, attitude
towards risk is only that part of r(x) not attributable to m(x). This is
illustrated in Figure 10. The case of intrinsic risk proneness, when
r(x)<m(x) and b(x)<0 is illustrated in Figure 11. In both of these cases,
in contrast to the case of intrinsic risk neutrality illustrated in Figure
9, the entire curvature of the utility function can not be explained by the
individuals' preferences for outcomes under certainty measured via the
satiation sacrifice. Since the utility function incorporates information
about preferences for outcomes under certainty and attitude towards risk,
the degree of curvature not attributable to preferences for outcomes under
certainty can be attributed to intrinsic risk attitude.

Note that like the Arrow-Pratt coefficient of absolute risk aversion,
Dyer and Sari n's measure of intrinsic risk aversion is a local measure. If
Bell and Raiffa's hypothesis that rational individuals exhibit constant
intrinsic risk attitude in risky situations is true, then, Dyer and Sarin
argue, the hypothetical utility function for riskiness of outcomes should
take one of the following forms:

46

 
    

1.

 

f v(x)

I

 

 

 

O

x -——-—

 

-———

pg .
N-

x
ev 2

024
ID I-

x1

A = risk premium = x* - Xce
B = satiation sacrifice = x* - Xev
C = risk premium due to only attitude towards risk = Xev'xce > 0

Figure 10: Value and Utility Functions for the Case of Intrinsic Risk
Aversion

47

u(x)

 

v(x)

    

 

 

 

 

 

 

 

x x xce x* x2 X

A = risk premium = x* - Xce
B = satiation sacrifice = x* - xev
C = risk premium due to only attitude towards risk = Xev’xce < 0

Figure 11: Value and Utility Functions for the Case of Intrinsic Risk
Pronenss.

4B

u(x) is a linear transform of -e‘bV(X) iff b(x) = b > 0
u(x) is a linear transform of v(x) iff b(x) = 0
u(x) is a linear transform of ebVIX) iff b(x) = b < 0.

These can be viewed as the function that transforms the value function
into the utility function. These three functional forms are nearly identi-
cal to the forms of the utility function u(x) which Pratt proves are re-
quired in the cases of constant absolute risk aversion, constant absolute
risk neutrality and constant absolute risk proneness. However, these trans-

formations are from v(x) to u(x) rather than from x to u(x).

Qggpayigg_yalge,agg_Utility Functions

Numerous studies can be found in the marketing and management litera-
ture in which both utility and value functions are assessed over the same
attribute and then compared. These studies have been concerned primarily
with measurement techniques and have tested for differences between utility
and value functions rather than trying to establish a relationship between
the two. A very small sample of these studies is reviewed here to give a
flavor of their results.

Tversky (1967) assessed value and utility functions for eleven subjects
and reported that risky and riskless preference functions are not linearly
related, even for fairly trivial outcomes. Fischer (1977) used ten subjects
to test assessment procedures for value and utility functions. In the
context of a job selection problem, he found that for five out of the ten
subjects, the hypothesis that u(x) is a linear transformation of v(x) was
not supported. Although Fischer hypothesizes that there exists some
monotonic transformation between value and utility functions where they are

not the same, he does not develop this concept further.

49

Allais (1979) reports utility and value functions for four subjects who
took part in a large experiment on the assessment of subjective preferences
conducted in 1952. u(x) was not equal to v(x) for any of the subjects.
Currim and Sarin (1984) tested assessment procedures for riskless value and
risky utility functions within the context of a job selection problem. They
found that u(x) was not equal to v(x) for 30 of the 43 subjects involved in
the study.

Ig§51n3,§he_Hypgthesis 9f Constant Intrinsic Risk Attitude

Krzysztofowicz (1983) presents the results of the only study which
assesses u and v, tests the hypothesis of constant intrinsic risk attitude
for all x e X, and compares the intrinsic risk attitudes of decision makers
for two decisions. The experiment he conducted supplies two utility func-
tions and one value function for the same subject, attribute and decision
context, but for two different decisions. Krzysztofowicz elicited responses
fromrsix water resource experts regarding announcement of the level of
controlled releases from a multipurpose water reservoir given an uncertain
supply of water. The individuals were presented with three scenarios. In
the first scenario, the actual water supply was equal to the planned
supply. Thus, no uncertainty was present. A value function was obtained to
quantify the relative value of water through the strength of preference of
the decision makers. The second scenario was termed the pessimistic deci-
sion environment and the decision makers were instructed to determine the
utility of increments of water received during the season if, at the begin-
ning of the season, the predicted water supply was at a very low level and
thus, the announced planned release was low. The third scenario was termed

the optimistic decision environment and the decision makers were instructed

50

to determine the utility of increments of water received during the season
if, at the beginning of the season, the predicted water supply was at a
very high level and the announced planned release was high.

The hypothesis of constant intrinsic risk attitude was supported for
both the “pessimistic“ and “optimistic“ decision scenarios. However, only
three of the six subjects had the same intrinsic risk attitude for both
decisions. The remaining three subjects changed their intrinsic risk at-
titudes from intrinsically risk averse to intrinsically risk seeking or
vice versa. This result, although obtained from limited testing, points out
the important distinction which must be made between constant intrinsic
risk attitude for one decision (that the transformation between u(x) and
v(x) is constant), and the idea of an intrinsic attitude towards risk which
is constant across situations.

The reversal of intrinsic risk attitudes between decisions in
Krzysztofowicz's study may be due, in part, to the construction of his
experiment. Kahneman and Tversky (1979) have observed that individuals are
generally risk seeking in the domain of losses and risk averse in the
domain of gains. They argue that the utilities elicited will be influenced
by how the questions are framed. The two scenarios analyzed in
Krzysztofowicz's experiment can be viewed as two framings of the same
outcome space: any outcome above the extremely pessimistic decision could
be viewed by the decision maker as a gain while any outcome below the
extremely optimistic decision could be viewed as a loss. Slovic, Fischhoff,
and Lichtenstein (1982) report that if an outcome representing a monetary
loss is framed in terms of an insurance premium rather than a sure loss, a

reversal in risk attitude is often observed.

51

Because of the limited nature of the experiment and the possible in-
fluence of framing, Krzysztofowicz's results cannot be interpreted as
conclusive evidence supporting or rejecting the hypothesis that risk at-
titude is independent of the decision context. Therefore, we continue with
our hypotheses that: (1) there is some transformation between u(x) and v(x)
for a given decision, which encapsulates risk attitude and (2) intrinsic
risk attitude is constant across situations and may be termed the
individual's intrinsic attitude towards risk. The first hypothesis ad-
dresses the question of whether the transformations between utility and
value functions reflect actual preferences of decision makers. The second
hypothesis addresses the question of whether individuals' intrinsic risk
attitudes are independent of the attributes over which their preferences
are assessed. This hypothesis seems intuitively plausible since any non-
linear transformations between an actual attribute and the value of that
attribute are incorporated in v(x) and, therefore, do not enter into the
relationship between v(x) and h[v(x)].

[he_Agegyagy_gf the Present Theory 2: Intrinsic Risk Attitude

Stiglitz (1969, pp. 664-666) has shown that for constant absolute risk
attitudes to be exhibited, all indifference curves must be homothetic and
all Engel curves must be straight lines through the origin. Dyer and Sarin,
by explicitly separating the effect of non-linear value functions from
attitude towards risk have obviated the need for homothetic indifference
curves in the development of the concept of intrinsic risk attitude.
However, Dyer and Sarin's development of constant intrinsic risk attitude
implicitly contains another assumption which is equally troublesome. By
using holistic preference judgements, Dyer and Sarin assume that one of the

52

following two conditions hold: (1) that all decision problems are simple
value problems, i.e. that outcomes refer to only one attribute, or (2) that
preferences for attributes in multiattributed outcomes are independent and
additive.

Extension agents, farm management specialists and others concerned with
decision making in the food and fiber sector know that decision makers are
often concerned with multiple attributes of any outcome. A farmer may be
concerned not only with maximizing income, but also with his asset/debt
ratio, quality of life, maintaining soil productivity, or implications of
current decision outcomes for future flexibility in organization of the
farm firm. Therefore, if our goal is to develop a measure of attitude
towards risk which can be used in predicting preferred action choices for
realistic decisions, we cannot assume that each possible outcome of a
decision has only one attribute.

To do this, we must reconsider the assumption that preferences for
attributes in multiattributed outcomes are independent and hence, additive.
The assumptions of independence required for additivity are quite strong

and easily violated.3 Consider an outcome X with attributes (x1,x2). The
assumption of strict independence of attributes required for additivity is,
loosely speaking, that preferences for attribute x1 are not influenced at
all by the level of attribute x2. The indifference map for this case is
shown in Figure 12.

Keeney and Raiffa (1976, pp.330-332) show that if preferences over an

outcome space for three attributes are compatible with an additive value

 

3 The independence conditions required for additive and multiplicative
value and utility functions will be discussed in detail in Chapter 4.

53

x2

 

 

x1

Figure 12: Indifference Curves for Two Attributes which Meet the
Independence Conditions Required for Additivity.

54

function and conditions for a multiplicative utility function are met, then
the utility function must have one of the following three forms:

u(x) is a linear transform of -e'CVix), c > 0

u(x) is a linear transform of v(x)
u(x) is a linear transform of eCVixl, c > 0

The second case, analogous to Dyer and Sarin's constant intrinsic risk
neutrality, implies that both the value and utility functions are additive.
The first and third cases, analogous to Dyer and Sarin's constant intrinsic
risk aversion and constant intrinsic risk proneness, imply that the value
function is additive and the utility function is multiplicative. Again, the
case of a multiplicative value function is not addressed.

Development of a more general set of transformations between value and
utility functions which allow for cases of additivity or multiplicativity
in both value and utility functions is the goal of Chapter 4. It will also
be shown that the results obtained for the case when the value function is
additive and the utility function is multiplicative are equivalent to those
given by Dyer and Sarin and Keeney and Raiffa.

CHAPTER 4

A NEH MEASURE OF ATTITUDE TOHARDS RISK

Impetus for development of a new measure of attitude towards risk has
come from the observations that marginal value of money is not constant,
preferences are not homogeneous, and preferences for multiple attributes of
outcomes do not necessarily meet the conditions required to assume ad-
ditivity of the value or utility function. The first two observations are
addressed by the concept of intrinsic risk attitude. The third must be
addressed by the use of multi-attribute utility theory as a basis for a new
measure of attitude towards risk.

The chapter begins with an examination of the independence conditions
which are required to be met in order to assume additivity or multi-
plicativity of value and utility functions. This is followed by a brief
exposition of the aspects of multiattribute utility theory which are per-
tinent to the development of the measure of attitude towards risk. This
discussion includes some special problems which arise in the current ap-
plication and which necessitate the weakening of some of the conditions of
independence. The allowable transformations between value and utility
functions which form the basis for measuring attitude towards risk are then
developed. These results are used in a sample application and then compared
to those obtained by Dyer and Sarin in their development of the concept of
constant intrinsic risk attitude.

55

56

Ingependence Conditions £2; Additive agg Multiplicative
a ue a__ ilit'y Fufitions
Using one of several representation theorems, value functions can be
constructed which preserve orderings among multiattributed riskless out-
comes. The form of the value function (additive or multiplicative) depends
upon the degree to which preferences meet certain conditions of independ-
ence. Both additive and multiplicative value functions require the condi-
tion of mutual preferential independence.
Eyeferential Inde ndence: Attribute Y, where Y e X is
preierentially pe nt of its complement Y' if the
preference order of consequences involving only changes in

the levels in Y does not depend on the levels at which
attributes in Y' are held fixed.

 

This definition implies that the conditional indifference curves over Y do
not depend on the attributes Y'.

Mutual Preferential Inde ndence: Preferential independence

must also h6ld for attriBuEes in Y' with respect to Y.

For a value function to be multiplicative, it is required only that
mutual preference independence and weak difference independence hold.

Heak Difference Inde ndence: Y is weak difference independ-

efit of €53 remaining attributes if the ordering of

preferences on Y does not depend on the fixed level of the

other attributes.

For a value function to be additive, a stronger condition must be met.
In addition to mutual preference independence, strong difference independ-
ence must hold.

57

Stro Difference Inde ndence: Attribute Y is stron dif-
erence IfidEiEfidEfit of ESE remaining attributes i the

preference difference between two levels of Y is not af-
ected by fixed levels of the other attributes. _

Similarly, additive and multiplicative utility functions can be distin-
guished on the basis of the restrictiveness of the independence conditions
which preferences meet. Both additive and multiplicative utility functions
require that the condition of utility independence is met.

Utilit Independence: Attribute Y where Y e X, is utilit
ifidEpenden 0 ts complement Y' if the conditiona
preference order for lotteries involving only changes in the

levels of attributes in Y does not depend on the levels at
which the attributes Y' are fixed.

For a utility function to be multiplicative, only utility independence and

mutual preference independence must h0ld. Additivity of a utility function

requires that an additional condition, known as the marginality assumption,
holds.

Marginality Assu tion: Attributes Y where Y e X are addi-
ve ndependen preferences over iotteries on Y depend
only on their marginal probability distributions and not on
their joint probability distributions.
The marginality assumption is sometimes difficult to intuit, but can be
made clearer using the following example. The marginality assumption re-

quires that for any a 6 Y1 and b 6 Y2, the decision maker is indifferent
between lotteries l1 and l2 shown below.

/(a’b.)
z ‘2 \
(8*,b‘“) (8*Ib)

(8.b)

/\

SB

Multiettribute Velue egg geilley Functions
If preferences meet the independence conditions required for additivity
discussed above, an additive multiattribute value function is written as:
(4.1) v(X) = {Vivmil
where v1 is a single attribute value function and w1 is a scaling constant

between zero and one.1
If preferences meet the independence conditions required for multi-
plicativity, a multiplicative multiattribute value function is written as:
(4.2) 1 + wv(x) = “[1 + ww1v1(x1)]
where w is a constant reflecting the type and degree of non-additivity
present.2
Similarly, if preferences meet the independence conditions required for
additivity, an additive multiattribute utility function is written as:
(4.3) u(x) = {kiu1(x1)
where “i is a single attribute utility function and k1 is a scaling con-

stant between zero and one.

If preferences meet the independence conditions required for multi-
plicativity, a multiplicative multiattribute utility function is written
as:

(4.4) 1 + ku(x) = fl[1 + kk1u1(x1)]
where k is a constant reflecting the type and degree of non-additivity
present and must meet the conditions specified for w in footnote 2.

 

1 Calculation of scaling constants is discussed in Chapter 5.

2 Calculation of w is also discussed in Chapter 5. w must be either
-1Sw<0 or w>0 and solve the condition 1 + w = “(I + HHi).

59

Reversals ip_Preferences egg Sign Conditions .

The multiplicative value and utility functions defined above do not
allow for reversals in preferences over subsets of the attributes. In many
situations, we would not expect reversals to occur in an individual's
preferences. However, there are important cases where they do seem
reasonable. For example, in an agricultural situation, one could expect
that preferences for rain to increase or decrease depending on the level of
sunshine available. If this occurs, the conditions of preference and
utility independence required for multiplicative value and utility func-
tions will be violated. It is impossible to have reversals in preference
with the additive form of the value and utility functions. The violation of
the independence conditions required for multiplicative functions can be
avoided by weakening the independence conditions to those of generalized

preference independence and generalized utility independence (Fishburn and
Keeney, 1974, 1975) .

Generalized Preference Indepengence: Y1 is generalized
preferentially independent of Y2 if, given any two levels of
y2, the two orderings of Y1 are either identical, reversals
of each other, or indifference exists among the Y1.

Generalized Utility Independence: Attribute Y1, is general-
ized utility independent of attribute Y2 if the conditional

preference order for lotteries involvin only changes in the
evels of attributes in Y1, given any wo levels of y2, are

either identical, reversals of each other, or indifference
exists among the Y1.

Generalized preference independence is also known as sign dependence
(Krantz, et al, 1971, pp. 329-332) or the sign axiom (Roskies, 1965). The
relationships among independence concepts is shown in Figure 13.

60

MARGINALITY ASSUMPTION

UTILITY INDEPENDENCE

/

GENERALIZED UTILITY INDEPENDENCE PREFERENCE INDEPENDENCE

GENERALIZED PREFERENCE INDEPENDENCE

STRONG DIFFERENCE INDEPENDENCE

I

HEAK DIFFERENCE INDEPENDENCE

1. Given y,y',x,x'cX, x preferred to y and x' preferred to y'
implies

2. Given X = X1 x X2 x..x Xn for some positive integer n, each
X1 is a convex subset of a finite dimensional Euclidean
space and:

Za(x)u(x) > Xb(x)u(x) iff a is preferred to b for all a,beA

holds for some u: X -> Re which is continuous in the rela-
tive usual product topology for X.

Figure 13: Relationships Among Independence Concepts3

 

3 Adapted from Fishburn and Keeney (1974, p. 300)

61

It has been noted that either w>0 or -1$w<0. If w is greater than zero
and 1 + wv is a multiplicative value function with terms 1 + ww1v1(xi) then
v' = a(1 + wv) where a>0 is also a multiplicative value function with terms
a1(1 + ww1v1(x1)) where Ia] = a. But if -1Sw<0 and v is the original value
function, 1 + wv produces an inverse ordering. However, -(1 + wv) = -fl(1 +
wwIV1(Xi)). or:

(4.2a) (sgn w)(1 + wv) = (sgn w)n(1 + ww1v1(x1))
Analogous results apply the the multiplicative utility functions so that:
(4.4a) (sgn k)(1 + ku) = (sgn k)l( 1 + kk1u1(x1))
These results are used in the proofs of the transformations between value
and utility functions. Reference to (4.2) and(4.4) in the following sec-

tions will, unless otherwise stated, refer to the forms shown in (4.2a) and
(4.4a).

Economic Interpretation pf Parameters 5 egg 5 4

Consider the two equal probability lotteries <A,C> and <8,D> shown in
Figure 14 and assume that preferences are increasing in both Y and 2. It
will be assumed that the individuals value functions are homogeneous. It
can be shown that if <A,C> is preferred to <8,D>, k)0 and Y and 2 can be
viewed as complements, if <A,C> is indifferent to <8,D>, k=0, indicating an
additive utility function and that preferences for Y and Z are independent,

while if (8,0) is preferred to <A,C>, k<0 and Y and 2 can be viewed as
substitutes.

 

4 gzgtg4lgr this section are adapted from Keeney and Raiffa (1976, pp.

62

 

 

 

 

 

 

 

 

Z I I I
I I l
I
I
I
22 -------- B C
I I
l
21 -------- A D
I I
Y1 Y2 Y

Figure 14: Using Lotteries to Interpret k in a Multiplicative
Multiattribute Utility Function.

63

The lotteries are structured so that for <A,C>, the individual will
either receive a high level of both Y and Z or a low level of both. For
lottery <8,D), the individual will receive either a high level of Y or 2,
but not a high level of both. If <A,C> is preferred, it is as if the in-
dividual needs an increase in Y to complement and increase in Z in going
from A to C. Otherwise, the full worth of the increase in 2 could not be
exploited. On the other hand, to prefer <8,D> implies that is important to
receive a high level of at least one attribute and, given a high level of,
say, Y, the increased preference due to an increase in 2 is not as much.
Thus Y and 2 can be thought of as substitutes for each other.

To illustrate a complementary case, consider a farm manager who is
thinking of investing in the use of a new, pest sensitive, high yielding
variety of grain and a new, more expensive pest manageaent system. At-
tribute Y can be viewed as the productivity of the new variety on his
field, given existing soil types, drainage, etc. Attribute 2 can be viewed
as the successfulness of the new pest management system. If both the high
yielding variety and the new pest management system are successful, the
farm manager is very happy. However, if either fails (the HYV does not
flourish on his soil type, or the pest management strategy does not keep
away the targeted pests), the financial consequences may be as bad as if
both had failed.

To illustrate a case of substitution, consider a farm manager who is
thinking of using a new variety of corn on two separate fields which have
very different soil and drainage characteristics. Attribute Y, in this
case, is the yield from one field while attribute Z is the yield from the
other. Hhile the farm manager would like it if both fields flourished,
achievement of attributes Y and 2 would, most likely, be substitutes.

54

Fischer (1976) sets forth another interpretation for k. He argues that
a k>0 could be indicative of complementary attributes, or the simple fact
that the individual is willing to give up a sure thing in either attribute
for the chance of receiving something which is better with regard to both
attributes, i.e. risk proneness. Similarly, a k<0 could indicate attributes
which are substitutes or simply an aversion to high variance outcomes.

These two interpretations of k are not necessarily contradictory: both
can add insight into the formation of attitudes towards risk. If Keeney and
Raiffa's interpretation of k is applied to the interpretation of w, then if
w>0, the attributes can be said to be complements, and if w<0, the at-
tributes can be said to be substitutes. Because a multiattribute utility
function, like a von Neumann-Morgenstern utility function, incorporates
both attitudes towards risk and information about the individual's value
function, examination of k without information about w does not allow us to
determine what portion of k is due to the complementarity or sub-
stitutability of the attributes and what portion is due to attitude towards
risk.

Allowable [Lensfggeetions Between Value egg UELLLEy Functions

Results which relate value and utility functions can be found scattered
throughout the recent literature in measurement theory (e.g. Dyer and
Sarin, 1979; Keeney and Raiffa, 1976; von Hinterfeldt, 1979). Further work
is needed to exactly define the class of allowable transformations between
utility and value functions. In this section we will present proofs of the
relationships between measurable value and utility functions. A common way
to relate v(x) and u(x) is through the uniqueness theorems of their respec—

tive measurement theoretic representations. The proofs presented here

65

differ from many earlier proofs in that we will rely on the use of Cauchy's
four fundamental functional equations: this greatly simplifies the proofs
and allows us to depart from the standard assumptions about the additivity
of the value function.

Cauchy was not concerned with decision theory, utility functions, or
value functions. Instead, he introduced the first systematic treatment of
some basic functional equations to prove the powerful and practical nature
of some rather abstract and original concepts he introduced such as the
concept of limit and the concept of a continuous function. Cauchy's func-
tional equations are applicable to the solution of many problems in
economics (Gehrig, 1984). They are used here in a new application.

He begin with the general functional equation:

(4.5) F[x1,x2,...,xn] = h{G[x1,x2,...,xn]}
This general functional equation can be rewritten for the specific cases of
additive and multiplicative value and utility functions.

For the case where the value and the utility function are additive, a
general functional equation can be written as:

(4-6) v(xl = luiIXi) = hIIViIXIII = h[VIXl]

This is of the same form as the general Cauchy equation:

(4-7) h(X) + bi!) = h(x + y)
thich, if h is continuous, has the general solution:

(4.8) h(r) = ar

Similarly, for the case where the value function is additive and the
utility function is multiplicative, the functional equation, Cauchy equa-
tion, and general solution for the Cauchy equation can be written, respec-

tively, as:

66

(4.9) 1 + ku(x) = fl( 1 + kk1u1(x1)) = h[2v1(x1)] = h[v(x)]
(4.10) h(x)h(y) = h(x + y)
(4.11) h(r) = car.

If the utility function is additive and the value function is multi-
plicative, the functional equation, Cauchy equation, and general solution
to the Cauchy equation can be written, respectively, as:

(4-12) "(XI = zIIIIXi) = h[II(1+ WIVIIXilll = h[1+ WOO]
(4-13) h(X) + h(r) = h(XY)
(4.14) h(r) = a In r

If both the utility and value functions are multiplicative, the func-
tional equation, Cauchy equation, and general solution to the Cauchy equa-
tion can be written, respectively, as:

(4.15) 1 + ku(x) = "(1 + kk1u1(x1)) = h[u(1 + ww1v1(xi))] = h[l + wv(x)]

(4-16) h(XIhm = h(xv)

(4.17) h(r) = r3.

The Cauchy equations and their general solutions can be used to solve
the general functional equations for the cases of additive and multiplica-
tive value and utility functions. The solutions, which can be viewed as the
transformation between the utility and value functions, provide the basis
for a new measure of attitude towards risk. Each possible case and its

solution are given below. Proofs are contained in appendices 4A, 4B, 4C and
4D.

Qeee_;: If the value and utility functions are additive, then:

(4A.1) u = v
Ceee, 2: If the value function is additive and the utility function is
multiplicative, then:

(48.1) 1+ ku = (1 + k)V

67

Case 3: If the value function is multiplicative and the utility function is
additive, then:

(4C.1) u = (ln(1 + wv))/(ln(1 + w))
Case g; If the value and utility functions are multiplicative, then:
(40.1) 1 + ku = (1+ wv)(I"(1+ k))/(In(1+ w))

Synthesizing end Applying ghe 31133131;

It has been shown that there exists a specific set of allowable trans-
formations between value functions and utility functions. To envision how
these results can be applied in determining an individual's pure or intrin-
sic attitude towards risk, consider the individual whose value and utility
functions are shown in Figure 15. Assume that the individual's value func-
tion is additive and that:

(4.18) v1(x1) = ln x1
and that the individual's utility function is multiplicative and that:
(4.19) "1(in = (x: -1)(1/<e ~1n
Note from panel A of Figure 15 that the utility function is linear. The
ArrowePratt coefficient of absolute risk aversion, taken at any xeX will be
zero, indicating risk neutrality.

But our concern is not with the utility function alone; we are inter-
ested in separating pure attitude towards risk from the effects of non-
constant marginal value. He may proceed in several directions, depending on
what information is available about the individual's value and utility
functions.

If we had actually elicited the individual's multiattribute utility
function and determined k, which in this example is assumed to be e-1, we
could test for the validity of the hypothesized transformation between

68

 

 

 

 

Panel A
U
V v(x)
u(x)
X
Panel 8
u(X)
u(v)
v(x)

Figure 15: v(x), u(x) and u(v) for an Intrinsically Risk Prone
Individual when v(x) = E In x.

69

additive value functions and multiplicative utility functions developed in
the previous section. Taking the transformation:

(4A.1) 1 + ku = (1 + k)v
and substituting In (4.18), (4.19) and the value for k we obtain:
(4.20) 1 + (e-1)[(x - 1)(1/(e-1))] = (1 + e - 1)ln x
which has the solution:
(4.21) x = x
and thus, the transformation is valid.
Similarly, the value of k could be predicted from (4A.1), (4.18) and
(4.19) and tested against the estimated k. In this case:
mzml+kux-nuuenn=(1+nmx
which has the solution:
(4.23) k = e - 1
How do we interpret the k value? In this case, k )0, which indicates

risk prone behavior. This can be seen more intuitively by examining the
function u(v) in panel 8 of Figure 15 which is of the form:

(4.24) u(v) = (ev - 1)l(e - 1)

The function is convex to the origin. A measure of the rate of bending of
this function, analogous to the Arrow-Pratt coefficient, is:

(4.25) -(83u16v2)/(6u/6v) = -1 at all v
which indicates constant risk proneness.

This information about the individual's pure attitude towards risk can
be used in predicting preferred action choices in other situations. Assume
that in a different decision situation, the decision maker's value function
is again additive with:

(4-25) Vi(Xil = Xi"

78

Using our knowledge of the transformation between additive value functions
and multiplicative utility functions, and the value of k, the individuals
von Neumann~Morgenstern utility function is easily obtained:

(4.27) u(x) = (ex2 - 1)l(e - 1)

The ArrowePratt coefficient at a point x on this function is:

(4.20) r(x) = [2eX‘(2x2-1))/(e-1))[(eI1-X’)/2x)-(1/(2xeX‘)))
which, when evaluated at x = 1.5, is -3.67, which indicates risk proneness
as would be expected from the convex curvature of u(x) shown in panel A of
Figure 16 .

But again, if we are concerned with separating the pure attitude
towards risk from the effects of non-linearity in the value function, we
want to know the rate of bending of u(v), not u(x). Because u(v) is:

(4.29) u(v) = (ev - 1)l(e - 1) P '
the function, shown in panel 8 of Figure 16, is convex to the origin.
Measuring the rate of bending of this function yields:

(4.30) -(6*u/6vz)/(6u/6v) = -1 at all v
which indicates constant risk proneness.

If only the Arrow-Pratt coefficient were to be examined, we would
surmise that the individual's attitude towards risk had changed from risk
neutrality to risk proneness between the two situations. But we can see
from this example that the individual's pure attitude towards risk has not
changed; the change in the rate of bending of u(x) was due to a change in
the value function, not a change in pure attitude towards risk.

Now, let us look at another individual making decisions in the same
context. This second individual is identical to the one above except that
his k value is equal to -.8. Panel A of Figure 17 shows his value and

71

 

 

 

 

Panel A
u
v
v(x)
u(x)
x
Panel B
"(X)
u(v)
v(x)

Figure 16: v(x), u(x) and u(v) for an Intrinsically Risk Prone
Individual when v(x) = 2 x3.

72

 

 

 

 

Panel A
II
V u(x)
v(x)
x
Panel 8
am
u(v)
V(X)

Figure 17: v(x), u(x) and u(v) for an Intrinsically Risk Averse
ndividual when v(x) = I In x.

73

utility functions when v = In x. This individual's Arrow-Pratt coefficient

is:

(4.31) r(x) = (ln 5 + 1)/x
which, when evaluated at x = 1.5 is 1.74, indicating aversion to risk. The
function u(v) for this individual is shown in panel B of Figure 17. Measur-
ing the rate of bending of this function yields:

(4.32) -(6‘ul&v3)l(6u/6v) = ln 5 = 1.61 at all v
which, like the Arrow-Pratt coefficient, indicates risk aversion, but a
lower level than would have been indicated by examining only the utility
function over x.

This individual is also faced with a second decision in which, like for
the first individual, v = x3. Panel A of Figure 18 shows this individual's
value and utility functions for this situation. In this second situation,
the individual's ArrowbPratt coefficient is:

(4.33) r(x) = 2x ln 5 - (1Ix)
which, when evaluated at x = 1.5, is 4.16, indicating a very high level of
aversion to risk. Is it possible that the individual's attitude towards
risk has changed that much between the two situations? As before, if we
examine the individual's utility function over v, shown in panel B of
Figure 18, we see that it is concave to the origin, indicating some degree
of risk aversion. Measuring the rate of bending of this function yields:

(4.34) -(63u/6v3)/(6u/8v) = ln 5 = 1.61 at all v
which, like the Arrow-Pratt coefficient, indicates risk aversion, but at a
much lower level than would have been indicated by examining only the
utility function over x.

The examples discussed so far all involve cases where the value func-

tion is additive and the utility function is multiplicative. For these

74

 

 

 

 

Panel A
u
v
u(x)
v(x)
I x
Panel B
u(x)
u(v)
v(x)

Figure 18: v(x), u(x) and u(v) for 2an Intrinsically Risk Averse
ndividuai when v(x) = 2x1.

75

examples, the rate of bending of an individual's utility function over v,
u(v), has been constant. Does the same hold for cases in which the value
function is multiplicative and the utility function is either additive or

multiplicative?5 Preliminary results suggest that it does not.

Consider the case of an individual with a multiplicative value function
where:

(4.35) V1(X1) = ln x1
and a multiplicative utility function. The transformation between the value
and utility function takes the form:

(40.1) 1 + ku = (1 + uv)(ln(1 + k))/(1n(1 + HI)
Assume that k = -.5 and w = .9. Interpretation of these k and w values a la
Keeney and Raiffa would suggest that for the multiattribute value function,
the attributes were highly complementary while for the multiattribute
utility function, the same attributes were substitutes. Fischer's inter-
pretation of k would lead us to conclude that the individual is risk
averse.

Given these values for k and w, the individual's von Neumann-
Morgenstern utility function can be written:

(4.36) 1 - .5u(x) = (1 + .9 In x)(l" (1 ' ~5))/(I" (1 + ~9II

OY‘

(4.37) u(x) = ((1 + .9 In x)-1-08 - 1)/-.s

 

5 Actual cases where the utility function is additive and the value
function is multiplicative are rare. From a strictly measurement
theoretic point of view, this condition is impossible since, as was
shown in Figure 13, the independence conditions for additive utility
imply the conditions for additive value. However, in some cases,
economic theory or a priori knowledge about the attributes of outcomes
involved may dictate that a multiplicative value function be used in
conjunction with an additive utility function.

76

The value and utility functions for this individual are shown in panel A of
Figure 19. The Arrow-Pratt coefficient, taken at x is:

(4.38) (234/(125 x + 112.5 x In x)) + (11x)
which, when evaluated at x = 2, is 1.076, indicating risk aversion.

The function u(v) for this same individual is shown in panel B of
Figure 19. The rate of bending of this function is:

(4.39) -(6¢u/6v¢)l(6ul&v) = 234/(125 + 112.5 x)
which, when evaluated at x = 2, is 1.153, also indicating risk aversion,
but at a higher level than that indicated by the Arrow-Pratt coefficient.

Some interesting points can be made if we examine the same individual's
ArrowbPratt coefficient and rate of bending of the function u(v) when

v1(x() = xii. In this case the ArrowePratt coefficient is:

(4.40) r(x) = ((936 x3)l(250 x + 225 x3)) - (11x)
which, when evaluated at x = 2 is 1.128. The rate of bending of the func-
tion u(v) when x = 2 is .407. Note that at a given x, the rate of bending
of u(v) when v1(x1) = ln x1 is not the same as the rate of bending of u(v)
when v1(xi) = x13. However, when the rate of bending of u(v) is evaluated
at the same value of v1(x1), it is the same for the two cases. Hhen x = 2
and v1(x1) = In x), v1(x1) = .693. Hhen x = .8326 and v1(x() = x13, v1(xI)
= .693. The Arrow-Pratt coefficient, evaluated at this x is .718 and the
rate of bending of the function u(v) is 1.153, identical to the case when
v1(x1) = ln x1.

Therefore, we can conclude that when the value function is multiplica-
tive, the rate of bending of the function u(v) will not be constant at all

levels of v within one decision context. It will, however, be constant at a

given level of v across decision contexts. This differs from the case where

 

 

 

 

Panel A
u
v
u(x
v(x)
x
Panel B
u(x)
u(v)
v(x)

Figure 19: v(x), u(x) and u(v) for an Intrinsically Risk Averse
Individual when v(x) = N In x.

78

the value function is additive and the rate of bending of u(v) is constant
for all levels of v within one decision context as well as across decision
contexts.

Hhat relationship do these results bear to Dyer and Sarin's measure of
intrinsic risk attitude? Recall from Chapter 3 that Dyer and Sarin estab-
lish a measure of intrinsic risk attitude, b(x), such that:

(4.41) b(x) = (r(x) - m(x))lv'(x)
where b(x) is the rate of bending of the function u(v) and m(x) is the rate
of bending of the value function. This measure remains valid when the value
function is additive and the utility function is either additive or multi-
plicative and when the value function is either additive or multiplicative.

Given this result, we can now re-examine Dyer and Sarin's hypothesis

that the hypothetical utility function for riskiness of outcomes will take
the forms:

u is a linear transform of -e'bV(X) iff b(x) = b > 0

u is a linear transform of v iff b(x) = 0

u is a linear transform of ebVIX) iff b(x) = b < 0

The second case, where b(x) = 0 follows directly from (4A.1). The first and
third cases will be examined first for the situation where the value func-
tion is additive and the utility function is multiplicative. From Appendix
48, this case can be written as:

(48.1) 1 + ku = (1 + k)v
or, before standardization, as:

(48.7) (sgn k)(1 + ku) = e3(39n k)v

79

Hhen k < 0 and b(x) > 0, indicating risk aversion, (48.7) can be rewritten
as: t

(4.42) -(1 + ku) = e‘3V
or

(4.43) u = (-e'av - 1)lk
which corresponds to Dyer and Sarin's hypothesis. Similarly, when k > 0 and
b(x) < 0, indicating risk proneness, (48.7) can be rewritten as:

(4.44) u = (eav - 1)/k
which also conforms to Dyer and Sarin's hypothesis.

Hhen the value function is multiplicative and the utility function is
either additive or multiplicative, however, Dyer and Sarin's hypotheses
regarding u are rejected. Therefore, we can conclude that Dyer and Sarin’s

hypothesis of constant intrinsic risk aversion hold for the cases when the
value function is additive, but not when it is multiplicative.

APPENDICES TO CHAPTER 4

80

APPENDIX 4A

geee_1: If the value and utility functions are additive, then:
(4A.1) u = v
Proof:
Let Viviixi) = Vi' (Xi)
kiUiIXi) = Ui'(Xi)-
Assume that for the additive value function (4.1) and the additive utility

function (4.3) there exists some continuous function h, such that, from
(4.6):

(44-2) {Ui'IXIl = hIZVI'IXIII
Let x3 = x30 = 0 for all j not equal to i. (4A.2) is reduced to:
(48-3) ui'(Xi) = h[Vi'IXii]
Substituting the right hand side of (4A.3) into (4A.2) yields:
(44 4) ZhIVi'IXill = h[EVi'Otil]
Now let x) = x)” = 0 for all i = 3,4,...,n. (44.4) can now be written:
(44-5) h[V1'IX1l] + h[V2'(X2)l = h[VI'IXII + v2'(X2)]
which is Cauchy's fundamental functional equation from (4.7) with general
solution:
(4.8) h(r) = ar
Substituting (4.8) into (4A.2) yields:
(44.6) ZUi'IXI) = aXVI'IXI)
Since Eu1'(x)) = u from (4.3) and Ev1'(xi) = v from (4.1),
(4A.7) u = av
This solution of the functional equation is unique up to the specifica-

tion of one parameter, a. To make the solution unique, the free parameter

81

must be “used up“. To do this, choose elements x0, x1 c X such that x1>x0
and defining u(xo) = v(xo) = 0 and u(xl) = v(xl) = 1. x0 and x1 can be
viewed, respectively, as the worst and best alternatives available. Solving
for a in (4A.6) yields:

(4A.8) a =1
Therefore, the standardized equation takes the form:

(4A.1) u = v

82

APPENDIX 48

Case 2: If the value function is additive and the utility function is
multiplicative, then:

(48.1) 1 + ku = (1+k)V
Proof:
Let H1V1(X1) = V1'(X1)
1 + kkiUiIXi) = UI'IXI)
Assume that for the additive value function (4.1) and the multiplicative

utility function (4.4) there exists some continuous function h, such that,
EI‘OI (4. 9):

(48.2) (sgn kIIUI'IXII = h[(sgn k) {Vi'IXIII
(48.2) is the compact form of the two functional equations u' = h[v'] for
k>0 and -u' = h[-v'] for k<0.
Let x5 = x30 = 0 for all j not equal to i. (48.2) is reduced to:
(4B 3) (sgn k)Ui'IXI) = h[(sgn k) Vi'IXIil
Substitution of the right hand side of (48.3) into (48.2) yields:
(48-4) "h[(sgn k) vi'(Xi)] = h[(sgn k) VI'IXIIJ
Now let x1 = x19 = 0 for all i = 3,4,...n. (48.4) can now be written:
(48-5) h[(sgn k) V1'(X1)]h[(39n kl V2'(X2)]
= h[(sgn k) v1'(x1) + (sgn k) v2'(x2)]

which is Cauchy's fundamental functional equation from (4.10) with the

general solution:

(4.11) h(r) = ear

83

Substituting (4.11) into (48.2) yields:

(48.6) (sgn k)lu('(x1) = e159" kIXV'IX)
Since v1'(x() = v from (4.3) and (sgn k)nu('(x1) = (sgn k)(1 + ku) from
(4.4),

(45.7) (sgn k)(1 + ku) = e8 (89" kI v
Using the standardization conventions adopted in the proof of case 1, and
solving for a yields:

(48.8) a = ln(1 + k)
Therefore, the standardized equation can be written:

(45.1) 1 + ku = (1 + k)V

84

APPENDIX 4C

Case 3: If the value function is multiplicative and the utility function is
additive, then:

(4C.1) u = (ln(1 + wv))/(ln(1 + w))
PI‘OOF:
Let 1 + ww1v1(x1) = v1'(x1)
kiui(XI) = Ui'IXI)
Assume that for the multiplicative value function (4.2) and the additive

utility function (4.3) there exists some continuous function h, such that,
from (4.12):

(40-2) (89n V) XUi'IXi) = h[(sgn HIIVI'IXill
(4C.2) is the compact form of the two functional equations u' = h[v'] for
w>0 and -u' = h[-v'] for w<0.
Let x1: x30 = 0 for all j not equal to i. (4C.2) is reduced to:
(4c.3) (sgn w) ui'Ixi) = h[(sgn wivi'IxIII
Substituting the right hand side of (4C.3) into (4C.2) yields:
(40-4) {h[(sgn VIVI'IXIII = h[(sgn w)IIVi'(Xi)]
Now let x, = x10 = o for all i = 3,4,...,n. (4c.4) can now be written:
(40-5) h[(sgn w)V1'(XI)] + h[(sgn w)V2'(X2)I
= h[(sgn w)(v1‘(x1))(69n w)(vz'(xziil
which is Cauchy's fundamental functional equation from (4.13) with the

general solution:

(4.14) h(r) = a In r

85

Substituting (4.14) into (4C.2) yields:

(4C.6) (sgn w) Eu1'(x1) = a ln((sgn w) flv1'(x1))
Since (sgn w)uv1'(x1) = (sgn w)(1 + wv) from (4.2) and 2u1'(x1) = u from
(4.3),

(4C.7) (sgn w) u = a ln((sgn w )(1 + wv))
Using the standardization conventions adopted in the proof of case 1,
solving for a yields:

(4C.8) a = 1/(ln(1 + w))
and the standardized equation takes the form:

(4C.1) u = (ln(1 + wv))l(ln(1 + w))

86

APPENDIX 40

Case 5: If the value function and utility function are multiplicative,
then:

(ma)1+m=(1+wHMU+kHMM(1+w)
Proof: .
Let 1 + wiVi(XII = vi'IXi)
1 + kkiUiIXi) = ui'(Xi)
Assume that for the multiplicative value function (4.2) and multiplicative

utility function (4.4) there exists some continuous function h, such that,
from (4.15):

(4D.2)(89n H)(ssn k) Ini'<Xi) = h[(sgn k)(39n H) IVi'(Xi)]

(4D.2) uses the same compact forms of the functional equations described in
Appendices 48 and 4C.

Let x1 = x39 = 0 for all j not equal to i. (40.2) is reduced to:
(40-3) (89" w)(sgn k) Ui'IXI) = h[(sgn k)(sgn w) Vi'IXiil
Substituting the right hand side of (40.3) into (40.2) yields:
(40 4) Ih[(59n k)(ssn v) Vi'IXI) = h[(sgn k)(ssn w) IVi’(Xi)l
hwiax1=m0=0mrm1i=amnqn(mA)mnmwanumt
(40.5) "h[(sgn k)(sgn w) V1'(XI)]h[(39n k)(sgn w) V2’(xz)l
= hI((89n k)(ssn w)V1'(XI)) ((sgn k)(ssn w)vz'(42))]

which is Cauchy's fundamental functional equation from (4.16) with the

general solution:

(4.17) h(r) = r8

87

Substituting (4.17) into (4D.2) yields:

(40.6) (sgn w)(sgn k) lu1'(x1) = (sgn k)(sgn w) Ilv1'(x1)a
Since (sgn w)v1'(x1) = (sgn w)(1 + wv) from (4.2) and (sgn k)u1'(xi) = (sgn
k)(1 + ku) from (4.4),

(40.7) (sgn w)(sgn k)(1 + ku) = (sgn k)(sgn w)(1 + wv(x))a
Using the standardization conventions adopted in the proof of case 1 and
solving for a yields:

(40.8) a = (ln (1 + k))/(ln(1 + w)
Therefore, the standardized equation is:

(4D.1) 1 + ku = (1 + w)(Ini1 + k))/(ln(1 + w))

CHAPTER 5

EXAMINATION OF THE THEORY IN AN EMPIRICAL CONTEXT

This chapter introduces and explains the procedures used to collect
data for use in an empirical examination of the model developed in Chapter
4 and draws some preliminary conclusions from the collected data. Before
launching into a discussion of the specifics of the data collection and the
conclusions which can be drawn from the data, it is important to examine

some issues surrounding the assessment of multiattribute value and utility

functions. It is to this topic that we now turn.

Holistic ye, Decomposed Assessment g: Multiattribute Functions

During the 1970's the management science literature was filled with
debate over the relative merits of various methods for assessing multiat-
tribute value and utility functions. The two methods under consideration
were that of holistic evaluation, also known as statistical estimation, and
decomposed assessment, also referred to as algebraic composition.

Holistic evaluation of a multiattribute utility function determines the
univariate utility functions and scaling constants from a utility function
over holistic outcomes. In other words, the individual whose utility func-
tion is being determined is asked a series of questions about outcomes
specified in terms of all of the relevant attributes. Assuming that the
researcher knows the form of the individual's utility function, the

88

89

multiattribute function can be fit using standard OLS procedures. The
utility functions and scaling constants for individual attributes can be
determined from the holistic utility function. These estimates are derived:
they are inferences which are completely based on the appropriateness of
the underlying preference structure.

A decomposed assessment procedure, while relying on the appropriateness
of the same preference structure as the holistic evaluation procedure,
differs from it primarily in that a univariate utility function is directly
elicited for each each attribute and scaling constants are determined
directly by questioning the respondent. These utility functions and scaling
constants are then algebraically combined to determine the individual's
multiattribute utility function.

Under the assumption of riskless choice, the over-all evaluation func-
tion used in the statistical approach need only satisfy the properties of
an ordinal scale. However, if a decomposed value function is to be used,
the value functions defined over each attribute must satisfy the properties
of an interval scale, even though the over-all values generated must only
be ordinal.

Numerous studies of holistic evaluation of multiattribute utility
functions have generally indicated that people can assign values to multi-
attribute outcomes in a meaningful fashion. According to Fischer (1977),
however, these studies have also revealed important shortcomings in this
approach. These include:

1. Limited numbers of attributes can be used.

2. Significant random error is introduced, with the amount of
error increasing with the number of attributes considered.
This error has been shown to be an important source of

suboptimality in both simulated and real world decision
making.

90
3. Holistic evaluation is difficult to apply in situations
where there are a large number of possible outcomes.

Several studies have been conducted to compare the efficacy of the
holistic and decomposed approaches. They have *almost exclusively been
concerned with job selection problems where each outcome is comprised of
attributes related to salary, location, and job task (Currim and Sarin,
1984: Huber, 1974: Huber, Daneshgar, and Ford, 1971; Barron and Person,
1979; Fischer, 1977). Hhile some of the studies have been concerned with
choices under certainty and some with choices under uncertainty, almost all
have assumed, a priori, that preferences met the conditions required for
additivity. The authors differ in their assessment of the predictive ac-
curacy of the holistic and decomposed models.

Despite the inconclusive evidence in support of the superiority of
either approach, there is one factor which is of overriding importance for
this study. In statistical (holistic) approaches, the diagnosticity of the
estimated function is influenced by many factors and represents a combina-
tion of the underlying reasons. The algebraic (decomposed) approaches, if
carefully implemented, provide the direct means for interpretation of
preferences for specific attributes, interactions between the attributes,
and attitudes towards risk. Therefore, they provide better insight into the
decision making process. For this reason, we will use the decomposed ap-
proach in developing the value and utility functions in our empirical test.
However, there is one striking disadvantage of the algebraic approach which
has been noted by several authors including Hauser and Urban (1979); the
measurement techniques employed by the algebraic approach lack an error
theory and thus prevent one from making statistical statements about

91

parameter estimates. Therefore, consistency must be assumed or checked by

repeated assessments.

Methods fen Estimating Value eng Utility Functions

There exists a plethora of methods for deriving value and utility
functions, each with its attendant advantages and disadvantages. The
breadth of methods will not be explored here, but a feel for the number of
possibilities can be obtained by examining Table 1.

Numerous articles have been written on the presence of assessment
errors in single and multiattribute utility functions (e.g. Hershey, Kun-
reuther and Schoemaker, 1982: Krzysztofowicz and Duckstein, 1980, Tversky
and Kahneman, 1981; Keeney and Sicherman, 1976; Fishburn and Kochenberger,
1979). Sources of error cited by Hershey, et. al. include:

1. Response Mode Bias: Certainty equivalence methods generally
yield greater risk-seeking that probability equivalence
methods.

2. Dimensions of Lotteries: Probability and outcome levels used
in reference lotteries induce systematic bias.

3. Domain of Lotteries: Combining gain and loss domains yields
different utility measures than separate examinations of the
two domains.

4. Inertia Bias: Responses to questions framed in terms of
assumption of risk, e.g. the reservation price for entering
into a speculative venture, may lead to an inertia or status
quo bias.

5. Context Bias: Also known as framing, the way way information
is presented to individuals regarding choices under uncer-
tainty affects their final choice.

Krzysztofowicz and Duckstein focus on the errors which may result in a
single attribute utility function and in the type of multivariate risk

attitude obtained from the effects of range of outcomes. Their discussion

9.
18.
11.

12.

13.
14.

P N 9999P?

92

METHOD CLASSIFICATION METHOD CLASSIFICATION
Ranking N/</llD 15. Ordered Metric N/</3/B
Direct Rating N/=/1IE 16. Probabilistic
Standard Gamble 1 PIE/llE Ordered Metric P/</3IB
Standard Gamble 2 P/2/1/C 17. Single Trade-Off NI=I2IC
Direct Midpoint N/=/1/C 18. Double Trade-Off N/=/3/B
Probabilistic 19. Single Transfor-

Midpoint P/Z/l/C mation N/=/2/C
Direct Ordered
Metric N/<l1/D 20. Double Transfor-
Probabilistic mation NI=I2IC
Ordered Metric P/</1ID 21. Discrete
Ranking N/</2/8 Trade-Off NI=I2/DC
Direct Ranking NI=I2/B 22. Discrete Transfor-
Probabilistic mation NI=I2/DC
Rating P/2/2/B 23. Discrete Ad-
Sucessive jancency N/=/2/DC
Comparison NI=l2/B 24. Saw-Tooth N/=/2/CE
Half-Value Sum NI=I3/B
Direct-Ordered
Metric N/(l3/B

The 24 methods are classified by four aspects, a/b/c/d, defined as follows:

Aspect a
Aspect b

Aspect c
Aspect d

the method does not use probabilities
the method uses probabil ties in scaling utilities
the method is based on preference judgements
the method is based on I'direct" inequality judgements
the method is based on indifference judgements
-the method is based on 'direct' equality judgements
The number denotes the number of factors involved in
any one judgement
B: the method is most applicable with binary factors or
is applicable for estimating scale-transformation
parameters
C: the method is best used with continuous factors
D: the method is best used with discrete factors
E: the method is usable with discrete or continuous
factors
DE or
CE: the method is applicable using one discrete and one
continuous factor or one continuous and another
factor.

."E9WP¥

Table 1. methods for Estimating Value or Utility Functionsl

 

1 Adapted from Fishburn (1967, pp. 438-440)

93

is particularly important in the determination of scaling factors and will
be discussed under that heading.

Hhile these sources of error are often referred to as assessment error,
implying that they are largely due to either the instrument used or the
inability of respondents to accurately educe their preferences, they are
all directly related to the behavioral content of the axioms of the ex-
pected utility hypothesis. Thus, many "errors in assessment" may actually
reflect deviation of preference behavior from that assumed in the axioms.

The method used in this study to determine value functions for in-
dividual attributes is the mid-value splitting technique. To use this
technique we first assign the worst possible amount of an attribute the
designation x' and the best possible amount attribute the designation x‘.
v(x’) = 0 and v(xe) = 1. The respondent is then asked to answer the follow-

ing series of questions:
1. Give me an amount x' such that you would give up the same

amount of another attribute, called Y, to go from x' to x'
as you would to go from x' to x». v(x') = .5

2. Give me an amount x“ such that you would give up the same
aaount of another attribute, called Y, to go from x' to x“
as you would to go from x' to x'. v(x') = .25
3. Give me an amount x" such that you would give up the same
amount of another attribute, called Y, to go from x“ to x“'
as you would to go from x" to x9. v(x") = .75
The method used in this study to determine utility functions for the
individual attributes is the certainty equivalent method. The worst pos-
sible amount of an attribute is designated x’ and the best possible amount
of an attribute is designated xa. u(x') = 0 and u(xr) = 1. All other at-
tributes are assigned their worst possible amounts. The decision maker is

asked to respond to the following series of questions:

94

1. Give me the certain amount of attribute x, called x', such
that you would be indifferent between receiving (x', y°, z')
for certain or a lottery with a 501 chance of receiving (x°,
y;, z') and a 50% chance of receiving (xr, y°, 2’). u(x') =

2. Give me the certain amount of attribute x, called x', such
that you would be indifferent between receiving (x', y°, z‘)
for certain or a lottery with a 50% chance of receiving (x',
y', 2’) and a 50% chance of receiving (x', y’, 2'). u(x') =
.25

3. Give me the certain amount of attribute x, called x", such
that you would be indifferent between receiving (x", y',
2') for certain or a lottery with a 501 chance of receiving
(x', y°, z’) and a 50% chance of receiving (x4, y°, 2').
u(x") = .75

4. To check for consistency, we may ask the decision maker to
specify the certain amount of attribute x, called x, such
that they would be indifferent between receiving (x, y', z’)
for certain or a lottery with a 50% chance of receiving (x',

y°, 2’) and a 50% chance of receiving (x“', y°, 2'). x
should be the same as x'.

T_he Hum 91 Attributes Usfi

One of Fischer's criticisms of the holistic assessment procedure is
that it is limited in the number of attributes which can be simultaneously
assessed. Hhile the approach taken here will be that of decomposed assess-
ment, the question of the number of attributes of any outcome which should
be included remains relevant.

Eliashberg (1980) argues that empirical studies on consumer preference
judgements suggest that consumers are engaged in explicit tradeoffs between
the two most important attributes of outcomes once the less important
attributes pass their acceptable levels. He attributes this to the limited
nature of human information processing capacity. Eliashberg's argument is
supported by other researchers. Hansen (1969) chose the three most impor-

tant attributes for each individual in a consumer choice problem and

95

compared the model's performance against one which used all relevant at-
tributes and noted that the predictive power of the model was not
diminished with the restricted attribute set. Hilkie and Heinreich (1972)
argue that, in accord with related results in choice theory and information
processing, it appears that attitudes can be efficiently described (in
predictive terms) with fewer attributes than are typically gathered in
marketing research; the typical practice of using all attributes is likely
to significantly understate the predictive power of the multiattribute
model in marketing. In a study of consumer choice of cereal brands, Pekel-
man and Sen (1974) found that over 851 of their sample used just one or two
attributes in making decisions. None used all five attributes listed.
Shoemaker and Haid (1982) found that using the three most important at-
tributes yielded better estimates than a model using all six attributes
presented.

Detennining eng Interpreting Seeling Eectors
Much of the discussion of attribute importance in the marketing litera-
ture is based on the assumption that the weighting or scaling factor w) or

k( in the multiattribute value or utility function is an indicator of the

attribute's importance to the decision maker. In fact, the studies dis-
cussed above defined the most important attributes to be those with the
highest scaling factors. Keeney and Raiffa (1976, pp. 271-272) argue that
this interpretation of the scaling factors is incorrect. The scaling fac-
tors are not easy to interpret since they depend on the choices of the
maximum and minimum values of all of the attributes considered which, in

turn, are dependent on the possible consequences of the decision problem at
hand.

96

If, for example, we have assessed the scaling factors for a two at-
tribute outcome decision problem and found one to be .75 and the other to
be .25, we cannot say that the first attribute is three times as important
as the second. In fact, it cannot be concluded that the first attribute is
more important than the second. All we can say is that, starting from the
minimum possible levels of both attributes, the decision maker would rather
change the first factor from its lowest to its highest value than change
the second factor from its lowest to its highest value. If, for example,
the range of the second attribute is small, then the scaling factor for
that attribute will be small, going to zero as the range shrinks to zero.
The attribute may be very important, even though its range, and thus, its
scaling factor, is very small.

Because of the misinterpretation of the scaling factors, several tech-
niques for determining scaling factors have been proposed to circumvent the
lengthy calculations required to determine them within the decomposed
approach presented by Keeney and Raiffa (1976). The client-explicated
technique is representative of these (Eliashberg, 1980). The client is
asked to assign the most important attribute a value of one and then to
indicate the relative importance of the remaining attributes, using as a
reference the most important attribute. Variations of this technique give
the individual 100 reference points and ask them to distribute them among
the attributes on the basis of their importance.

The scaling factors basically serve to compress or stretch the scales
of the value or utility functions over individual attributes so that they
are consistent (Fishburn, 1967). They can be determined using one of

several methods.

97

Keeney and Raiffa (1976) propose a method for determining the scaling
constants for an additive value function. The method outlined below is a

generalized form of the Keeney-Raiffa procedure which is applicable also to
multiplicative value functions.

Given:
(5.1) v(x1r, x24, x3*) = 1
(5.2) v(xl', x2', xg’) = 0
(5.3) v1(x(*) = 1 and v1(x(') = 0 for all i

(5.4) v(xl, x2, x3) = Iw1v1(x1) + possible other terns2
the decision maker can be asked to specify an x1' such that they are indif-
ferent between receiving:
(5.5) (xl', x2°, x3°) and (xl’, x24, xa')
Using the conditions set forth in (5.3) and (5.4):
(5-6) w1V1IX1') = w2
Analogous choices can be made to determine that:
(5 7) ~2V2IX2') = H3
(5.8) w3V3(X3') = wl

 

2 This formulation becomes clearer if we recognize that both the additive
and multiplicative value and utility functions are specific cases of a
multilinear function when the conditions of mutual preference and

utility preference, respectively, hold. The multilinear value function
can be written:

v(x1,x2) = v(x1,x2°) + v(x1°,x2) + wv(x1,x2°)v(x1°,x2)
If w = 0, this becomes:

V(XI.X2)= v(X1.X2’) + VIXI'.X2)
If w is not equal to 0:

let v'(x1,x2) wv(x1,x2) + 1
wv(x1°,x2) + wv(x1,x2°) + w1v(x1°,x2)v(x1,x2°) + 1
[VV(XI.X2°) + l][HV(X1’.X2) + 1]
v'(x1,x2°)v'(x1°,x2)
Therefore, for an additive value function, there are no possible other

terms while for a multiplicative value function, 1 + w is the remaining
term.

98

v1(x1') can be determined from the individual's value function over at-
tribute i.

If the value function is additive with n attributes, only n-l such
indifference choices must be made because the scaling factors are con-
strained to meet the condition that:

(5.9) w1 + w2 + ... + w" = 1
If the value function is multiplicative with n attributes, n such indif-
ference choices must be made. In addition, w must be determined such that:
(5.10) 1 + w = n (1 + wwI)
For the case of three attributes described above, (5.10) becomes:
(5.11) 1 + w = (1 + ww1)(1 + ww2)(1 + ww3)
The resulting set of four equations, (5.6), (5.7), (5.8) and (5.11), can be
solved as a simultaneous equation system to determine the four unknowns, w,
w1, w2, and w3.

This technique is known as certainty scaling. A similar technique,
known as probabilistic scaling, is proposed by Keeney and Raiffa for deter-
mining the scaling constants for multiattribute utility functions.

Given:
(5.12) u(x1*, x2e, x39) = 1
(5.13) u(xl°, xz', xg') = 0
(5.14) u((x(*) = 1 and u1(x(°) = 0 for all i
(5.15) u(x1, x2, X3) = Ek1u1(x1) + possible other terms
then:
(5.16) u(xie, xj°) = k) for all j not equal to i.
The decision maker may then be asked, for what probability, p, are you
indifferent between:

99

1. The lottery giving a p chance at (x1*, x2*, X3*) and a (1-p)
chance at (x1°, x2°, x3°) and

2. The consequence (x1e, x2°, X3°)
If the decision maker's answer is defined as p1, then using (5.15), the
expected utility of the lottery is p1, and from (5.16), the utility of the
consequence is k1. Equating the expected utilities of the lottery and
consequence, it is found that:
(5.17) k1 = p1

The values of each of the ki's can be generated in an analogous fashion so

that:
(5.18) k2 = 92
(5.19) k3 = p3

If the utility function is additive with n attributes, only n-l such
probabilities must be specified because the scaling factors are constrained
to meet the condition that:

(5.20) k1 + k2 + ... + kn = 1

If the utility function is multiplicative with n attributes, n such prob-
abilities must be specified. In addition, k must be determined such that:
(5.21) 1 + k = I (1 + kki)
For the case of three attributes described above, (5.21) becomes:
(5.22) 1 + k = (1 + kk1)(1 + kk2)(1 + kk3)
The resulting set of four equations, (5.17), (5.18), (5.19) and (5.22) can
be solved as a simultaneous equation system to determine the four unknowns,
k, k1, kg, and k3.
Keeney and Raiffa also suggest that only one of the ki's need be

elicited using the probabilistic scaling procedure; the remainder can be
determined using a variation of certainty scaling. The decision maker is

108

asked to select a level of x1, x1', and xj, XJ', such that, for any fixed

levels of all of the other attributes, they are indifferent between:
1. A consequence yielding (xI', xj‘) and

2. A consequence yielding (x(°, Xj')
The utilities of these two indifferent consequences can be equated to
yield:
(5.23) k(u((x1') = kjuj(xj')
The major criticism of the procedures set forth by Keeney and Raiffa

are their reliance on extreme levels of the attributes. Hhile the range

from x1' to x(9 must, by definition, cover the entire range of the at-
tribute, X1, the implications of, and preferences for, extreme levels are

often difficult for the decision maker to assess. Although some researchers
(e.g. Keeney and Sicherman, 1976; Krzysztofowicz and Duckstein, 1980) have
realized that asking extreme value questions may be a shortcoming of the
existing scaling techniques, the effects of the range of gamble outcomes on
the individual's measured risk attitude has not received much attention in
the literature. Studies of variance preferences in laboratory experiments
where the range of gamble payoffs were very small have resulted in skep-
ticism about range effects (Slovic and Lichtenstein, 1968). However, Myers
and Sadler (1960) have shown a negative correlation between the size of the
range of outcomes and the predictive power of the resulting function.

For the case of certainty scaling this problem is easily ameliorated by
asking the decision maker a set of indifference questions at non-extreme
values. For example, the decision maker may be asked to specify a level of
x1' such that they are indifferent between receiving:

(5.24) (xl', x2', X3') and (xl', x2”, X3.)

181

Si nce:
(5.4) v(x1,x2,x3) = {w1v1(x1) + possible other terms

For the additive value function, there are no other terms. Equating the

values of the two consequences yields:

(5.25) V1V1IX1') + wzvzlxz') = HIVI(XI") + vzvzlxz")
Rearranging the terms yields:

(5 26) HI = H2IIV2IX2") - vzixz'))/(VI(XI' - V1IX1“))]
Since the v((x1)'s are easily obtainable from the individual's univariate
value functions:

(5.27) wl = 8 wz where 8 = [-]
For the multiplicative value function there is another term, and the solu-
tion can be written:

(5 28) HIIV1(X1'l-V1IX1')]=vz[V2IX2'l-V2IX2'II

+ "V1V2[V1(x1')V2(x2') - VIIXI'IV2IX2'II
Similarly, for probabilistic scaling the decision maker may be asked,

for what probability, p, are you indifferent between:

1. The lottery giving a p chance at (x1',x2',x3°) and a (1-p)
chance at (xl',x2',x3°) and

2. The consequence (x1', x2",x3°)
For the additive utility function, the utility of the lottery is:
(5 29) pk1U1(41') + pk202IX2') + (l-P) kIUIIXI') + (l-p)k202(xz”)
and the utility of the sure consequence is:
(5.30) k1u1(x1') + k2u2(x2')
Equating the utilities of the lottery and the sure consequence, and rear-
ranging terms yields:

(5 31) kIIIP-IIUIIXI') - (l-pIUIIXI")) = k2PIU2<X2”) - uzlxz'II

182

For the multiplicative utility function there is another term and the
solution can be written:
(5-32l k1[(P-1)01(X1') - (1~P)UI(X1”)) = k2PIU2IX2') - Uzixz'll
+kk1k2IU1IXI'IU2(X2')
-kk1k2IPUIIXI')uz(xz')+(1-p)uIIXI')uzixz')l
where u1(x1'), u2(x2')=.75 and u1(x1”), u2(x2”)=.25.

Before leaving the subject of scaling procedures it should also be
noted that while the value and utility functions are elicited attribute by
attribute, determination of scaling factors requires holistic judgements

(or at least consideration of n>1 attributes at a time).

InehRespondents

For this preliminary examination of the theory, subjects were recruited
from the agricultural technology program at Michigan State University. The
agricultural technology program offers a certificate to students after two
years of study in technical agriculture. All of the individuals participat-
ing in the completion of the questionnaire had farm backgrounds and were
enrolled in agricultural economics type courses within the agricultural
technology program.

This group was selected because of their familiarity with decision
making in agricultural settings and the relative ease with which the ques-
tionnaire could be administered. Fifty-nine students participated in the
study.

Ine,Qneetionnaire

Participants were given two hours to complete a questionnaire designed

to elicit preferences within the context of different agricultural

183

decisions. A total of three decision contexts were used; each participant
responded to questions about two of the decisions. Each participant was
presented with a booklet containing instructions, descriptions of each
decision context and a questionnaire for each decision context.

A copy of the complete questionnaire for one decision context, that of
selecting a variety of feed corn, can be found in Appendix 5A. The format
of the questionnaire has been changed slightly due to page width

restrictions.3 The content is identical to that presented to the respon-
dents. Descriptions of the other two decision contexts are contained in
Appendix 58. The questions for these decision contexts are functionally
equivalent to those contained in Appendix 5A.

The questionnaire was designed to elicit several types of information
including preference conformance with independence conditions, single
attribute value and utility functions, and the scaling constants required
to convert the single attribute functions into multiattribute functions.

After reading the instructions, each student was presented with a
description of their first decision context.

The respondents were first asked to respond to a series of questions
(ll-I4) designed to test whether preferences met the condition of general-
ized preference independence required for either an additive or a multi-
plicative value function. Preference for one variety over another In each
case constitutes satisfaction of the condition of generalized preference

independence.

 

3 Specifically, much of the text which was double spaced in the actual
questionnaire is single spaced in the Appendix. In addition, scales

used to elicit value functions In questions 12-14 have been compressed
in the Appendix.

184

The next series of questions (IS-I7) checked for the presence of strong
difference independence required for an additive value function. In each
case, if the response for part (a) is identical to the response to part (c)
and the response to part (b) is identical to the response to part (d), the
individual is said to exhibit strong difference independence between the
two attributes whose values change in the question.

Since the existence of an additive or multiplicative value function
does not necessarily imply that the utility function meets the same condi-
tions, generalized utility independence and the additional condition for an
additive utility function, known as the marginality condition must also be
checked.

Questions 8-10 check for the existence of generalized utility independ-
ence. If the condition holds, responses to parts (b) and (c) are Ino“.
Question 11 tests for satisfaction of the marginality condition required
for an additive utility function. Respondents should be indifferent for
(a), (b), and (c) if this condition holds.

The next step is to elicit value functions for each of the attributes
using a mid-value splitting technique. Because this technique may be dif-
ficult to understand, these questions are prefaced with an example. In the
example and the representative question, the first response yields v(x)=.5,
where x is the attribute in question, the second response yields v(x)=.25
and the third response yields v(x)=.75. Questions 12—14 elicit the value
functions for the individual attributes.

Univariate value functions are only one of the two types of information
needed to determine a multiattribute value function. The second type,
information needed to determine scaling factors, is elicited using ques-

tions 15(a), (b) and (c). The first question in each part elicits

185

information needed to determine the scaling constants using extreme values:
the second question elicits the information needed to determine the scaling
constants using non-extreme values.

Univariate utility functions are then elicited for each attribute in
questions 16-18. In each question, the first response yields u(x)=.5, the
second response yields u(x)=.25, the third response yields u(x)=.75 and the
fourth response is a consistency check for u(x)=.5.

The final series of questions elicit the information needed to deter-
mine the scaling factors to convert the univariate utility functions into a
multiattribute utility function. Because this technique also may be dif-
ficult to understand, the questions are prefaced with an example. As with
the scaling constants for the value function, the first questions elicits
probabilities using extreme values while the second question uses non-
extreme values.

After responding to a functionally equivalent set of questions for the
second decision context, respondents are asked to indicate their agricul-
tural background and experience in making decisions in contexts similar to
those presented in the questionnaire.

Most respondents took from one hour to one and a half hours to answer
the set of questions for the first decision context and about one-half hour

to answer the set of questions for the second decision context.

Problems H1§n_tne_9nestionneire

Decision theory has often be criticized by those interested in applied
problem solving because of the cost, time and difficulties involved in
obtaining reliable information with which to determine an individual's
utility function. This study was not exempt from these difficulties.

186

Previous published studies which have obtained either an individual's
multiattribute value or utility function for one decision have stressed the
time involved to familiarize the respondent with the parameters of the
decision and the techniques to be used in the elicitation of their
utilities. Then the researcher or ”decision analyst“ works through the
series of questions with the respondent, giving him or her constant feed-
back on the consistency and implications of their responses. If mathemati—
cal consistency is not obtained with their original responses, respondents
are asked to reconsider and modify their responses until consistency is
reached. For instance, when Keeney (1972) elicited the multiattribute
utility function of a blood bank administrator for two attributes, shortage
and outdating of supplies, he spent ”several afternoons" completing these
steps.

In contrast to these previous studies, the participants involved in
this study were asked to complete a questionnaire which elicited informa-
tion required to determine two multiattribute value functions and two
multiattribute utility functions in only two hours. Although written and
verbal explanations were given for the questions which were anticipated to
present problems (the mid-value splitting technique for determining single
attribute value functions and the probabilistic questions for determining
scaling constants for multiattribute utility functions), they completed the
questionnaires without feedback from the researcher. In addition, there
were few formal or informal consistency checks.

Table 2 shows that many of the individuals had difficulty understanding
some of the questions and hence, their questionnaires were not usable for

further analysis.

187

lable 2: Number of Individuals Hho Had Difficulty Answering Specific
Types of Questions.

 

 

I'Iype pf Question Number g£_Individuals I
I I
I Mid-Value Splitting 25 I
l Certainty Equivalent 11 l
I Certainty Scaling4 1 I
I Probabilistic Scaling4 23 I

l

 

Satisfaction pf Independence Congjtions

Although many of the responses were not usable for a complete analysis
because of problems in responding to the questions noted in Table 2, almost
all were usable to determine the degree to which independence conditions
required for either additive or multiplicative multiattribute utility and
value functions were met. The summary statistics for those individuals
whose responses met the independence conditions are shown in Table 3. The
statistics are compiled by decision context, not by individual. Since some
individuals only completed the first half of the questionnaire, the same

number of responses were not available for each decision context.

Table 3: Number of Responses Satisfying Independence Conditions

 

 

 

| Decision Context

I Independence Condition Variety Dryer Loan

I
Generalized Preference Independence 39 39 34 I
Strong Difference Independence 4 12 5
Generalized Utility Independence 18 17 17
Marginality Condition5 9 10 a i

 

 

 

4 The first part of these questions do not use responses to earlier
questions. If these were answered reasonably, the individual was
regarded to have understood the question.

5 Only those responses for which generalized preference independence was
met are included in statistics for the marginality condition.

188

Those responses for which generalized utility independence was not met
are not usable as this preference condition is required for either an
additive or a multiplicative utility function.

Several interesting points can be made by observing the frequency of
additive or multiplicative value and utility functions in those decision
context responses in which the condition of generalized utility independ-
ence was met. Tables 4 and 5 contain the summary statistics for the usable
responses where the condition of generalized utility independence was met.
Again, these summary statistics are by decision context, not by numbers of
individuals.

The reader will recall that the use of Dyer and Sarin's transformations
between value and utility functions implicitly requires, in all cases, that
the value function is additive. The utility function may be either additive
or multiplicative. It can be noted from Table 4 that only nine out of the
fifty value functions examined meet this requirement.

Table 4: Frequency of Additive and Multiplicative Value and Utility
Functions by Decision Context.

 

Decision Context

Combination Rele, Variety Dryer Lgen
Value Function
Additive 2 4 3
Multiplicative 15 12 14
Utility Function
I Additive 8 11 8
I Multiplicative 9 5 9

 

 

 

For intrinsic risk neutrality to hold within Dyer and Sarin's
framework, both the value and utility functions must be additive. Table 5
shows that only four of the cases examined meet this requirement. Five of

the cases meet Dyer and Sarin's implicit condition for intrinsic risk

189

proneness or aversion, that the value function is additive and the utility
function is multiplicative. The remaining 41 cases can not be analyzed
within Dyer and Sarin's framework. However, they should be capable of
analysis using the transformations developed in Chapter 4.

Table 5: Frequency of Combinations of Different Types of Value and
Utility Functions by Decision Context.

 

| Decision Context |

 

 

 

Combination Variety Dryer Lgen
Additive Value and

Additive Utility 1 1 2
Additive Value and

Multiplicative Utility 1 3 1
Multiplicative Value and

Additive Utility 7 10 6
Multiplicative Value and |

Multiplicative Utility 8 2 8 I

l I

Ine_Ieene,gfi Consistency g£_Respgnses

The only formal consistency test conducted in the questionnaire was an
additional indifference question for each single attribute utility function
where the respondent was asked to specify u(x)=.5 as a function of the
extreme levels of the attribute and as a function of the levels of the
attribute where u(x')=.75 and u(x")=.25. If the responses to the two
questions were identical, the respondent was assumed to be giving consis-
tent responses to that utility function elicitation question.

An additional, informal consistency test can be made by examining the
responses to the two sets of questions used to determine each scaling
constant. The reader will recall from the earlier discussion of the ques-
tionnaire that each respondent was asked functionally equivalent sets of
indifference questions using extreme values and then non-extreme values of

the attributes. In none of the cases examined were the responses identical

118

for the extreme and non—extreme questions. This may be due, in part, to the
aforementioned difficulty that individuals have in responding to extreme
value type questions.

However, the issue of consistency is not so easily dismissed. It is, in
fact, quite important in reviewing the responses to the questionnaires. To
determine the scaling constants needed to aggregate the single attribute
value functions into an additive value function, an overidentified system
of simultaneous equations was used. This system of equations can be writ-

ten:

(5.6) w1v1(x1') = wz

(5.7) w2v2(x2') = w3

(5.8) wavaixa') = v:
where x(' is the individuals response to the indifference question and
vi(x1') can be determined from the individual's value function over at-
tribute i.
The fourth equation for an additive value function constrains the values of
the wi's so that:

(5.9) w1 + w2 + w3 = 1

For a solution to be found for the subsystem of equations (5.6), (5.7)
and (5.8), the following must hold:

(5.33) v1(x1') * v2(x2') * V3(X3') = 1
If this condition does not hold, a convergent solution for the system of
equations can not be found. Analogous arguments hold for the systems of

equations needed to find the scaling factors for multiplicative value
functions and both additive and multiplicative utility functions.

111

For the cases examined, the value of equation (5.33) ranged from .125
to 1.225. No set of responses met the condition set forth in (5.33), that
lvi(x(') = 1.

The initial conclusion which can be reached from this finding is that
the subjects responses are, in some way, 'inconsistent'. However, with the
data available from the questionnaires it is not possible to determine the
cause of this inconsistency. The central issue is whether the inconsis-
tencies in responses are due to fuzzyness in human cognition, e.g. the
inability of respondents to accurately educe their preferences, or whether
the preference model used is not appropriate.

If the apparent inconsistency is due to fuzzyness in human cognition,
the problem can be remedied by continuous feedback from the researcher .to
the respondent causing the respondent to modify their answers until mathe-
matical consistency is obtained. An alternative approach would be to ask
the same series of questions in numerous guises, determine the distribution

of responses for each xi', and then take the mean as the 'true' value. Hith

the data available from the questionnaire, it is impossible to determine
the type of distribution of possible 'fuzzy' responses or whether the
responses are even near the mean or in either the right or the left tail of
the distribution.

However, before this approach can be justified in another data collec-
tion effort, several other questions must be answered. Perhaps most dis-
turbing is the question of how far the researcher can go in forcing mathe-
matical consistency and still reflect the true preferences of the respon-
dent. Although respondents apparently met the independence conditions for
additivity or multiplicativity of their multiattribute value and utility
functions (determined through responses to questions 1-11), their

112

preferences may not, in fact, be accurately reflected by either. the addi-
tive or multiplicative model.

The problem of 'inconsistency' of responses has been discussed by
several researchers. Interesting work has been done in this area by Kah-
neman and Tversky (1979, 1982), who have demonstrated that apparent incon-
sistencies in responses are due to violations of the behavioral content of
the axioms of the expected utility hypothesis stemming from human cognitive
processes. Perhaps the most well known of these factors is framing, the
effects arising when the same alternatives are evaluated in relation to
different points of reference. Ellsberg (1961), in a test of the Savage
axioms of expected utility, asked several eminent decision theorists to
answer a series of questions to determine whether their preferences vio-
lated the axioms. He found that the respondents fell into three groups:
those who were careful to calculate their responses in accordance with the
axioms of the expected utility hypothesis and, thus, gave consistent
responses; those who claim to be expected utility maximizers but whose
responses were inconsistent, and: those who gleefully gave inconsistent
responses. Ellsberg's report (1961, pp. 655-656) of their responses brings
levity to this important issue:

“...You might now pause to reconsider your replies. If you
should repent of your violations -- if you should decide
that your choices implying conflicts with the axioms were
“mistakes“ and that your 'real' preferences, upon reflec-
tion, involve no such inconsistencies - you confirm that the
Savage postulates are, if not descriptive rules for you,
your normative criteria in these situations. But this is by
no means a universal reaction; on the contrary, it would be
exceptional.

Responses do vary. There are those who do not violate
the axioms, or say they won't, even in these situations
(e.g. G. Debreu, R. Schlaiffer, P. Samuelson); such subjects
tend to a ply the axioms rather than their intuition, and
when in dou t, to apply some form of the Principle of

113

Insufficient Reason. Some violate the axioms cheerfully,
even with gusto (J. Marschak, N.Dalkey); others sadly but
persistently, having looked into their hearts, found con-
flicts with the axioms and decided, in Samuelson's phrase,
to satisfy their preferences and let the axioms satisfy
themselves. Still others (H. Raiffa) tend, intuitively, to
violate the axioms but feel guilty about it and go back into
further analysis.

The Important finding is that, after rethinking all
their 'offendinp' decisions in the light of the axioms a
number of peop e who are not only sophisticated but
reasonable decide that they with to persist in their
choices. This includes people who previously felt a “first-
order commitment“ to the axioms, many of them surprised and
some dismayed to find that they wished, in these situations,
to violate the Sure-thing principle. Since this group in-
cluded L.J. Savage, when last tested by me (I have been

m

reluctant to try n again), it seems to deserve respectful
consideration.“

Ellsberg goes on to state, in his conclusions (ibid., p. 669):

"...Are they foolish? It is not the object of this paper to
judge that. I have been concerned rather to advance the
testable propositions : (1) certain information states can
be meaningfully identified as highly ambiguous; (2) in these
states, many reasonable people tend to violate the Savage
axioms with respect to certain choices; (3) their behavior
is deliberate and not readily reversed upon reflection: (4)
certain patterns of 'violating' behavior can be distin-

guished and described in terms of a specified decision
rule.“

Hithout knowledge of the source of the apparent inconsistency in
responses, a formal empirical test of the theory presented in Chapter 4 can
not be completed. However, several points have been brought to light which
can provide the basis for further research in this area. First is the
finding that, although previous work on multiattribute value functions has
assumed the existence of an additive value function, only nine of the fifty
value functions examined meet this requirement. Second is the finding that,
even though forty—nine of the fifty respondents have preferences which
appear to satisfy the weaker independence conditions required for a

114

multiplicative multiattribute value function, none of the responses were
able to be analyzed fully because of inconsistent responses to questions
designed to provide the information necessary to determine scaling con-
stants. Because consistency checks were not incorporated throughout the
questionnaire, it is not possible, at this juncture, to determine whether
the inconsistencies are due to the inability of respondents to accurately
educe their preferences or whether the preference model used is not ap-
propriate. Resolution of these issues must precede an empirical test of the

theory. They are discussed in greater detail in Chapter 6.

APPEMDICES TO CHAPTER 5

115

APPENDIX 5A
INSTRUCTIONS

As you know, agricultural economists are often called upon to make exten-
sion recommendat ons, analyze the impact of farm policy and predict the
impact of future events on the agricultural sector. One of the pieces of
information which is often used in carrying out these tasks is some measure
of peoples' attitudes towards risk.

Unfortunately, the methods we now use to gather this information are not
very good. Therefore, our analyses are often not as good as we would like
the: to be. Hith better information, agricultural economists can do better
wor .

As part of the research project required for a Ph.D. in agricultural
economics, I have developed a new measure of attitudes towards r sk which I
think is better than the ones that are now used. But before I can convince
anyone else that the new measure is better, I have to test it. By consent-
ing to participate in this study by completing a questionnaire, you will
help me in doing a pilot test of the new measure.

In this questionnaire, you will be asked to answer the same series of
questions about two different hypothetical decisions. In each case, you
will first be given a description of the decision which you are being asked
to make. This is followed by a series of questions relating to that deci-
sion. Many of the questions may appear to be repetitive, but each is in-
cluded for a specific reason. You will be able to answer some of the ques-
tions without much thought, while others will be more difficult. This is
not an exam, so take your time and think about the questions; there are no
right or wrong answers. If there is something you do not understand, please
ask me and I will try to explain.

The questionnaire packet will take about two hours to complete. You will
not be penalized in any way if you decide during the two hours that you
cannot complete the entire questionnaire. If, before you start, you feel
that you will not be able to complete the entire questionnaire because of
the length of time involved, I would appreciate it if you would let me
know. To test the theory, complete questionnaires are needed.

To confirm that you freely consent to participate in the study, please sign
you name on the line below. Your name will not be used in any part of the
study. All of your answers will be treated with strictest confidence and
you will remain-anonymous.

116

If you are curious as to what your responses indicate about your attitude
towards risk, I would be happy to send you the information which results
from your questionnaire packet. If you would like to receive this informa-
tign, please write your address under your name below. This is purely
op ona .

I freely consent to participate in this study

 

(please sign here)

Thank you very much for agreeing to participate in this study.

Beverly Fleisher

For this decision situation, imagine that you have just been to talk with
the local seed dealer about the variety of feed corn to plant this year.
You have already decided that you want to use a 101-107 day long maturing
variety. He has told you about several varieties which fit this require-
ment, but differ with regard to three other important characteristics. Seed

117

SELECTING A VARIETY OF CORN

for all of the varieties costs the same amount.

1.

Potential Yield which, for the varieties you are consider-

ing, ranges from 100 to 120 bushels per acre, depending on
the variety selected.

Field Losses which are due to pests (both insect and dis-
ease) an nd Iod dging. For the varieties you are considering,

field losses range from 0 to 20$ of potential yield, depend-
ing on the variety selected.

Nitrogen Reguirements which, in the form of anhydrous am-

monia, range from 150 to 200 pounds per acre, depending on
the variety selected.

118

Suppose you must make choices between two varieties at a time. The five

varieties the seed dealer told you about have the characteristics shown
below.

Variety Variety Variety Variety Variety
Characteristics 1 g g 4, §

Potential Yield

(in bushels/acre) 100 110 110 110 100
Field Losses

(as I of yield) 51 51 15$ 15$ 15:
Nitrogen Requirement

(as lb./acre) 175 175 175 200 200

(1) In a choice between variety 1 and variety 2, would you (check one):
______ prefer variety 1 to variety 2?

______ prefer variety 2 to variety 1?

______ be indifferent between the two varieties?

(2) In a choice between variety 2 and variety 3, would you (check one):
_ ____ prefer variety 2 to variety 3?

______ prefer variety 3 to variety 2?

______ be indifferent between the two varieties?

(3) In a choice between variety 3 and variety 4, would you (check one);
______ prefer variety 3 to variety 4?

,______ prefer variety 4 to variety 3?

_______be indifferent between the two varieties?

(4) In a choice between variety 4 and variety 5, would you (check one):
______ prefer variety 4 to variety 5?

__*____prefer variety 5 to variety 4?

be indifferent between the two varieties?

119

(5) Now suppose that you are aware of a variet with the characteristics
shown under the heading variety 1 below and that here exists another group
of varieties which have different potential yields and different percentage
of field loss. In each of the four cases described below, please indicate
theiegiegd loss you would accept to have the higher potential yield of
var y .

Characteristic Veeiety L Veglety a
Potential Yield 100 110
Field Losses 5% ?
Nitrogen Requirement 175 175
(a) I would accept a i field loss from variety 2.
Characteristic Vegiety I Vegiety_g
Potential Yield 100 110
Field Losses 10! ?
Nitrogen Requirement 175 175
(b) I would accept a 1 field loss from variety 2.
Characteristic !§21221.1 Vegleiy g
Potential Yield 110 120
Field Losses 5% ?
Nitrogen Requirement 175 175
(c) I would accept a 1 field loss from variety 2.
Characteristic !§£1££!.L yeelegy g
Potential Yield 110 120
Field Losses 10$ ?
Nitrogen Requirement 175 175

(d) I would accept a : field loss from variety 2.

120

(6) Now suppose that you are aware of a variety with the characteristics
shown under the heading variety 1 below and that there exists another group
of varieties which have different percentage of field loss and different
nitrogen requirements. In each of the four cases described below, please
indicate the required nitrogen per acre you would accept to have the lower
field loss of variety 2.

Characteristic Vegiety L Veglety,g
Potential Yield 110 110
Field Losses 15$ 5%
Nitrogen Requirement 150 ?
(a) I would accept a ___“_____lb/acre nitrogen requirement for variety 2.
Characteristic 2m ;_ V_a_r;i_e_i_:y_ _2_
Potential Yield 110 110
Field Losses 15$ 5%
Nitrogen Requirement 175 ?
(b) I would accept a lblacre nitrogen requirement for variety 2.
Characteristic yee1e31,1 Veeiety_2
Potential Yield 110 110
Field Losses 10$ 51
Nitrogen Requirement 150 ?
(c) I would accept a lblacre nitrogen requirement for variety 2.
Characteristic Vegiety_1 Vegiety g
Potential Yield 110 110
Field Losses 10$ 5%
Nitrogen Requirement 175 ?

(d) I would accept a lblacre nitrogen requirement for variety 2.

121

(7) Now suppose that you are aware of a variety with the characteristics
shown under the heading variety 1 below and that there exists another group
of varieties which have different potential yield and different nitrogen
requirements. In each of the four cases described below, please indicate
the potential yield you would require to to have the increased nitrogen
requirement of variety 2.

 

 

Characteristic Veeiety,1 Vegiety g
Potential Yield 100 ?
Field Losses 10$ 10%
Nitrogen Requirement 175 200

(a) I would need a bu/acre potential yield from variety 2.
Characteristic Men 1 ' V_ai_1ei_:y_ g
Potential Yield 110 ?
Field Losses 10$ 10%
Nitrogen Requirement 175 200

(b) I would need a bu/acre potential yield from variety 2.
Characteristic EerieEy 1 Y§£1££1.Z
Potential Yield 100 ?
Field Losses 10$ 10%
Nitrogen Requirement 150 175

(c) I would need a _____"__ bu/acre potential yield from variety 2.
Charecteristic Veeiety_1 Veeieey_g
Potential Yield 110 ?
Field Losses 10% 101
Nitrogen Requirement 150 175

(d) I would need a bu/acre potential yield from variety 2.

122

(8) Now imagine that you are choosing between two varieties, but that you
do not have complete information about them. You know the potential yield
and field losses for variety 1, but you do not know its nitrogen require-
ments. The seed dealer does not know the exact nitrogen requirements of
variety 2, but can tell you that that it has a 50% chance of having the
characteristics shown in the first column under variety 2, and a 50$ chance
of having the characteristics shown in the second column under variety 2.
Nhat level of nitrogen requirement for variety 1 would make you indifferent
between the two varieties?

Variety 1 Variety 2
Characteristiee 59} chance §§$ chance
Potential Yield 110 110 110
Field Losses 10$ 10$ 10%
Nitrogen Requirement ? 200 150
(a) A lblacre nitrogen requirement for variety 1 would make me

indifferent between the two varieties.

(b) Hould your answer change if you knew that the potential yield for both
varieties was 120 bu/acre?
no
______"__yes
(c) Nould your answer change if you knew that the field losses for both
varieties was 20%?

no

yes

123

(9) Now imagine that you are choosing between two varieties, but that you
do not have complete information about them. You know the potential yield
and nitrogen requirements for variety 1, but you do not know its field
losses. The seed dealer does not know the exact field losses for variety 2,
but can tell you that that it has a 50% chance of having the characteris-
tics shown in the first column under variety 2, and a 50% chance of having
the characteristics shown in the second column under variety 2.

Hhat level of field losses from variety 1 would make you indifferent be-
tween the two varieties?

Variety 1 Variety 2
Characteristics §9§ chance 5e; chance
Potential Yield 110 110 110
Field Losses ? 10$ 20%
Nitrogen Requirement 175 175 175
(a) A i field loss from variety 1 would make me indifferent be-

tween the two varieties.

(b) Hould your answer change if you knew that the potential yield for both
varieties was 120 bu/acre?
no

yes

(c) Vould your answer change if you knew that the nitrogen requirement for
both varieties was 200 lblacre?
no

yes

124

(10) Now imagine that you are choosing between two varieties, but that you
do not have complete information about them. You know the nitrogen require-
ments and field losses for variety 1, but you do not know its potential
yield. The seed dealer does not know the exact potential yield requirements
of variety 2, but can tell you that that it has a 50% chance of having the
characteristics shown in the first column under variety 2, and a 501 chance
of having the characteristics shown in the second column under variety 2.

Nhat level of potential yield for variety 1 would make you indifferent
between the two varieties?

Variety 1 Variety 2

Characteristics SQ} chance §Qfi chance
Potential Yield ? 100 120
Field Losses 101 ‘ 10$ 10%
Nitrogen Requirement 175 175 175

(a) A bu/acre potential yield for variety 1.would make me indif-

ferent between the two varieties.

(b) Hould your answer change if you knew that the field losses for both

varieties was 201?
no

yes

(c) Vould your answer change if you knew that the nitrogen requirement for
both varieties was 200 lblacre?

no

we--.—

yes

-“—--_

125

(11) Now imagine that you are choosing between two varieties; you are not

sure of the

characteristics of each variety, but know that it has a 501

chance of having the characteristics listed in the first column under that

variety,

and a 501 chance of having the characteristics listed in the

second column under that variety. For each of the three pairs of varieties

below, you will
indifferent between them.

Variety 1
Characteristics §g$ chance §ex chance
Potential Yield 120 100
Field Losses 20$ 10%
Nitrogen Requirement 175 175

(a) Nould you (check one):
prefer variety 1 to variety 2?

prefer variety 2 to variety 1?

be indifferent between the two varieties?

 

Variety 1
Characteristics 59} chance 59} chance
Potential Yield 110 110
Field Losses 20$ 10!
Nitrogen Requirement 200 150

(b) Hould you (check one):
prefer variety 1 to variety 2?

___.__ prefer variety 2 to variety 1?

be indifferent between the two varieties?

be asked if you prefer one to the other, or if you are

Variety 2
§9$ chance, Se} chance
120 100
10$ 20%
175 175
Variety 2
§Qfi ehance §Qfi chance
110 110
10$ 20%
150 200

126

 

Variety 1 Variety 2
Qneeacteristics 59} chance 59? chance SQ} chance SQ} chance
Potential Yield 120 100 120 100
Field Losses 10$ 10$ 10$ 10%
Nitrogen Requirement 200 150 150 200

(c) Nould you (check one):
prefer variety 1 to variety 2?
prefer variety 2 to variety 1?

be indifferent between the two varieties?

127

An Example to Help in Answering Questions 12, 13, and 14
Suppose I am to be given a fruit basket with bananas, apples and oranges. I
know that the number of bananas in the basket is constant, but that there
is some flexibility in the number of oranges and the number of apples in
the basket. I am asked to specify a number of apples, A, such that I would
give up the same number or amount of oranges to move from 0 apples to A
apples as from A apples to 10 apples.
1 really like apples and am not too fond of oranges, although I occas-
sionally like to eat them. I decide that I would give up alot of oranges,
say 8 of them, to move from 0 apples to 4 apples and another 8 oranges to
move from 4 apples to 10 apples.
For me, then, A would be 4. I would mark this on the line below as:

0 1 2 3 4 5 6 7 8 9 10

I also put my answer, A, on the line below. Now, I am asked to specify
another amount of apples, B, such that I would give up the same number of
oranges to move from 0 apples to B apples as from B apples to A apples. I
decide that I would give up 3 oranges to move from 0 apples to 3 apples and
another 3 oranges to move from 3 apples to A, which is 4 apples. Note that
the number of oranges I used in answering to the second question did not
have to be the same I used in answering the first. I mark my answer on the
line below as B.

128

Now I am asked to specify another amount of apples, C, such that I would
give up the same number of oranges to move from A apples to C apples as
from C apples to. 10 apples. I begin by marking my response, A, from the
first question on the line below. I then decide that my response to this
question is 8 apples. I also mark this on the line below.

0 1 2 3 4 5 6 7 B 9 10

I then record my answers at the bottom of the page in the space provided.

(a) A = apples
(b) B = ________ apples
(c) C = apples

Questions 12, 13, and 14 ask you to answer questions about the corn
varieties that are like this question. Feel free to refer back to this

example if you have trouble understanding how to answer.

129

(12) Suppose the nitrogen requirement is the same for all of the varieties
you are considering. Please specify the potential yield (in bulacre), A,
such that you would accept the same increases in field losses to move from
100 bulacre to A bulacre as from A bulacre to 120 bulacre.

Please mark your response on all three of the lines below as A.

1?0 1?1 1?2 1?3 1?4 1?5 1?6....1}3 114 115 116 117 118 119 120
___—___ _____________|_

POTENTIAL YIELD (in bulacre)

Now please specify the potential yield, B, such that you would accept the
same increases n field losses to move from 100 u/acre to B as from B
bulacre to A bulacre (which you have marked on the line below). Mark your
response to this question on the line below as B.

1?0 1?1 1?2 1?3 1?4 105 1?6....1}3 114 115 116 117 118 119 120
___ ___ ___ ___ ___|___ .... ___ ___ ___ ___ ___ ___ ___

POTENTIAL YIELD (in bulacre)

Check to make sure that your response, A, is also marked on the line below.
Now, please specify the potential yield, C, such that you would accept the
same increases in field losses to move from A bulacre (which you have

marked on the line below) to C bulacre as from C bulacre to 120 bulacre.
Hark your response to this question on the line below as C.

1?0 1?1 1?2 1?3 1?4 1?5 1?6....1}3 114 115 1:6 117 118 119 120

POTENTIAL YIELD (in bulacre)

Please write your responses below:

(a) A = bulacre potential yield.
(b) B = bulacre potential yield.
(c) C = bulacre potential yield.

130

(13) Now suppose that the field losses are the same for all of the
varieties you are considering. Please specify the pounds of nitrogen per
acre, 0, such that you would require the same increase in potential yield
to move from applying 150 lblacre to applying D as you would to move from
applying 0 lblacre to 200 lblacre.

Please mark your response on all three of the lines below as D.

150 155 160 165 ..... 185 190 195 200
I___l___i___l___l___|___l ----- |___l___|___l___|___|___l

NITROGEN REQUIREMENT (in lblacre)

Now please specify the pounds of nitrogen per acre, E, such that you would
require the same increase in potential yield to move from applying 150
lblacre to applying E as you would to move from applying E lblacre o D
lblacre (which you have marked on the line below). Mark your response to
this question on the line below as E.

150 155 160 165 ..... 185 190 195 200
l___l___l__.l___l___l__.l ----- l___|__Hl___|___l___I___I

NITROGEN REQUIREMENT (in lblacre)

Check to make sure that your response, 0, is also marked on the line below.
Now, please specify the pounds of nitrogen per acre, F, such that you would
require the same increase in potential yield to move from applying D
lblacre (which you have marked on the line below) to F lblacre as to move
from applying F lblacre to applying 200 lblacre. Mark your response to this
question on the line below as F.

150 155 160 165 ..... 185 190 195 200

NITROGEN REQUIREMENT (in lblacre)

Please write your responses below:
(a) D = ____ lblacre nitrogen requirement.
(b) E = lblacre nitrogen requirement.

(c) F = _____lb/acre nitrogen requirement.

131

(14) Now suppose that all of the varieties you are considering require the
same application of nitrogen per acre. Please specify the field losses, 6,
such that you would require the same increase in potential yield to move
from 0i field losses to 6% field losses as you would to move from 6! field
losses to 20% field losses.

Please mark your response on all three lines below as G.

o 1 2 3 4 5 6....13141516171819 20
I_i__l__i_l_l_I-.-.l___l__l_l_l_l_l___l

FIELD LOSSES (in 1)

Now please specify the field losses, H, such that you would require the
same increase in potential yield to move from 01 field losses go H! field
losses as you would to move from Ht field losses to 61 field losses (which
you have marked on the line below). Mark your response to this question on
the line below as H.

0 1 2 3 4 5 6....13 14 15 16 17 18 19 20
|__|_|_|__|__|._._l---- |_|__l__l_|_l__.l__l

FIELD LOSSES (in i)

Check to make sure that your response, 6, is also marked on the line below.
Now, please specify the field losses, I, such that you would require the
same increase in potential yield to move from 61 field losses (which you
have marked on the line below) to It field losses as you would to move from

1% field losses to 20$ field losses. Mark your response to this question on
the line below as I.

0 1 2 3 4 5 6....13 14 15 16 17 18 19 20
|_I_I___|____i___l_l---- |__|___l___|_|___l_._l_._.l

FIELD LOSSES (in 1)

Please write your responses below:
(a) G = ____"__, 1 field losses.

(b) H
(C) I

1 field losses.

t field losses.

132

Some of your responses from questions 12, 13, and 14 will be needed to
answer question 15. Please transfer them from the previous pages to the
spaces provided below.

B = potential yield C = potential yield
E = _, nitrogen required F = _.__ nitrogen required
H = 1 field losses I = 1 field losses

(15a) Hhat level of potential yield for variety 1 would make you indif-
ferent between variety 1 and variety 2 whose characteristics are shown

below?

 

Cheeecteristics Veeiety_1 Vegiegy,2
Potential Yield ? 100
Field Losses 201 01
Nitrogen Requirement 200 200

bulacre potential yield for variety 1 would make me indifferent

between the two varieties.

Please write your answers for B, H, and I in the appropriate spaces below.
Then answer the question: what level of potential yield for variety 1 would
make you indifferent between the varieties shown below?

Characteristics Vegje§y_1 Vegiety,2
Potential Yield ? B = ______
Field Losses I = __1 H = ___1
Nitrogen Requirement 200 200

bulacre potential yield for variety 1 would make me indifferent

between the two varieties.

133

(15b) Hhat level of nitrogen per acre required for variety 1 would make you
indifferent between variety 1 and variety 2 whose characteristics are shown

below?

Characteristics Va_riei_:y_ 1_ V__a_r_i_et_y_ 2
Potential Yield 100 120
Field Losses 201 201
Nitrogen Requirement ? 200

lblacre nitrogen required for variety 1 would make me indifferent

between the two varieties.

Please write your answers for B, C, and F in the appropriate Spaces below.
Then answer the question: what level of nitrogen per acre required for
variety 1 would make me indifferent between the two varieties?

Mastmeristics 1:21:11 1 11119.21 2.
Potential Yield 8 = ____, C = ______
Field Losses 201 201
Nitrogen Requirement ? F =

 

lblacre nitrogen required for variety 1 would make me indifferent

between the two varieties.

134

(15c) Hhat level of field losses for variety 1 would make you indifferent

between variety 1 and variety 2 whose characteristics are shown below?

Characteristics VegieEy 1 Veeiety 2
Potential Yield 100 100
Field Losses ? 201
Nitrogen Requirement 200 150

1 field losses for variety 1 would make me indifferent between the

two varieties.

Please.write your responses for E, F, and I in the appropriate spaces
below. Then answer the question: what level of field losses for variety 1
would make me indifferent between the two varieties?

Characteristics Vegiety,1 Vegjety,2
Potential Yield 100 100

Field Losses ? I = _____1
Nitrogen Requirement F = E = _____

 

1 field losses for variety 1 would make me indifferent between the

two varieties.

135

In questions 16, 17, and 18, notice that variety 2 has a 501 chance of
having the characteristics shown in column 1 and a 501 chance of having the
characteristics shown in column 2 under the heading variety 2.

(16a) Please specify the sure amount of field losses, J, for variety 1,
that would make you indifferent between variety 1 and variety 2 with the

characteristics shown below

Characteristics Variety 1 591 QEQEEEFY 2 §e1 EhéflSQ.
Potential Yield 100 100 100
Field Losses ? = J1 01 201
Nitrogen Requirement 200 200 200

J = , 1 field losses

(16b) Please also fill in your answer for J in the table below. Then
specify the sure amount of field losses, T, for variety 1, that would make
you indifferent between variety 1 and variety 2 with the characteristics
shown below.

Variety 1 Variety 2
Characteristics 591 chance 591 chance
Potential Yield 100 100 100
Field Losses ? = T1 01 J =
Nitrogen Requirement 200 200 200

T = 1 field losses

136

(16c) Please fill in your answer for J in the table below. Then specify the
sure amount of field losses, L, for variety 1, that would make you indif—
ferent between variety 1 and variety 2 with the characteristics shown

below.

Variety 1 Variety 2
Characteristics 591 chance 591 chance
Potential Yield 100 100 100
Field Losses ? = L1 201 J = _
Nitrogen Requirement 200 200 200

L = 1 field losses

(16d) Please fill in your answers for T and L in the table below. Then
specify the sure amount of field losses for variety 1 that would make you
indifferent between variety 1 and variety 2 with the characteristics shown
below.

 

Variety 1 Variety 2
99eracteristics 591 ehance 591-chance
Potential Yield 100 100 100
Field Losses ? T = 1 L = $
Nitrogen Requirement 200 200 200

? = 1 field losses

Please write your responses below.

J = 1 field losses
T = 1 field losses
L = 1 field losses

? = 1 field losses

137

(17a) Please specify the potential yield, M, for variety 1, that would make
you indifferent between variety 1 and variety 2 with the characteristics
shown below.

Characteristics Variety 1 591 egggegfy 2 591 QDQESQ
Potential Yield M = ? 100 120
Field Losses 201 201 201
Nitrogen Requirement 200 200 200

M = bulacre potential yield

(17b) Please also write your answer for M in the table below. Then specify
the potential yield, N, for variety 1, that would make you indifferent
between variety 1 and variety 2 with the characteristics shown below.

 

Variety 1 Variety 2
Characteristics 591 chance 591 chance
Potential Yield N = ? M = 100
Field Losses 201 201 201
Nitrogen Requirement 200 200 200

N = bulacre potential yield

138

(17c) Please write your answer for M in the table below. Then specify the
potential yield, 0, for variety 1, that would make you indifferent between
variety 1 and variety 2 with the characteristics shown below.

Characteristice Variety l 591 QEEEEEFY 2 591 eeeeee
Potential Yield 0 = ? M = __”_,__ 120
Field Losses 201 201 201
Nitrogen Requirement 200 200 200

O = bulacre potential yield

(17d) Please write your answers for N and O in the table below. Then
specify the potential yield for variety 1 that would make you indifferent
between variety 1 and variety 2 with the characteristics shown below.

 

 

Variety 1 Variety 2
gheracteristics 591 egeeee, 591 eneeee_
Potential Yield ? N = O = _______
Field Losses 201 201 201
Nitrogen Requirement 200 200 200

? = bulacre potential yield

Please write your responses below.

M = bulacre potential yield
N = bulacre potential yield
0 = bulacre potential yield

? = bulacre potential yield

139

(18a) Please Specify the nitrogen requirement, P, for variety 1, that would
make you indifferent between variety 1 and variety 2 with the characteris-
tics shown below.

Variety 1 Variety 2
Characteristiee 591 eflegee, 591 eneeee
Potential Yield 100 100 100
Field Losses 201 201 201
Nitrogen Requirement ? = P 150 200
P = lblacre nitrogen requirement

(18b) Please write your answer for P in the table below. Then specify the
nitrogen requirement, Q, for variety 1, that would make you indifferent
between variety 1 and variety 2 with the characteristics shown below.

 

Variety 1 Variety 2
Characteristics 591 chance 591 chance
Potential Yield 100 100 100
Field Losses 201 201 201
Nitrogen Requirement ? = Q P = 150

Q = lblacre nitrogen requirement

140

(18c) Please write your answer for P in the table below. Then specify the

nitrogen requirement, R, for variety 1, that would make you indifferent

between variety 1 and variety 2 with the characteristics shown below.

Variety 1 Variety 2
Characteristics 591 chance 591 ehance
Potential Yield 100 100 100
Field Losses 201 201 201
Nitrogen Requirement ? = R P = 200

lblacre nitrogen requirement

(18d) Please write your answers for Q and R in the table below. Then

specify the nitrogen requirement for variety 1 that would make you indif-

ferent between variety 1 and variety 2 with the characteristics shown

be]

ow.

Variety 1 Variety 2
Characteristics 591 EDQEQQ. 591 egeeee
Potential Yield 100 100 100
Field Losses 201 201 201
Nitrogen Requirement ? Q = R =

lblacre nitrogen requirement

Please write your responses below.

p =

Q = ._______

R:

?

lblacre nitrogen requirement
lblacre nitrogen requirement
lblacre nitrogen requirement

“....

,lb/acre nitrogen requirement

141
An Example to Help in Answering Question 19

(19) Suppose you are considering two varieties. You know, for certain, that
variety 1 has the the characteristics shown under variety 1. You also know
that variety two has some probability of having the characteristics shown
in the first column under variety 2 and some probability of having the
characteristics shown in the second column under variety 2. What probabil-
ity, p, of variety two having the characteristics in column 1 under variety
two and probability (l-p) of variety two having the characteristics in

column 2 under variety 2 would make you indifferent between the two
varieties?

To answer these questions, it may be easier to think of variety 1 as a sure
thing, and variety 2 as a gamble between the two columns. Since the prob~
abilities on the two columns must add up to one, the probability for the
second column occurring is just one minus the probability of the second

column occurring. You do not need to specify the probability for the second
co umn.

The following example might help to explain how to answer the questions:
Let's say I am presented with a choice between two fruit baskets, the
contents of which are shown below.

Fruit Basket 1 Fruit Basket 2

Characteristics 2 ehance (153) chance
Apples 10 10 2
Oranges 5 15 5
Bananas 0 0 0

= .7 would make me indifferent between the two fruit baskets. This says
that I am indifferent between receiving 10 apples, 5 oranges and no bananas
and playing a gamble in which I have a 7 out of 10 chance of getting 10
apples, 15 oranges and no bananas and a 3 out of 10 chance of getting only
2 apples, 5 oranges and no bananas.

142

Some of your responses from questions 16, 17, and 18 will be needed to
answer question 19. Please transfer them from the previous pages to the
spaces provided below.

1 field losses

L 1 field losses
potential yield 0
R

potential yield

0 z -i
II

= nitrogen required

= nitrogen required

(a) Hhat p would make you indifferent between variety 1 and variety 2 shown
below?

Variety 1 Variety 2
Characteristics a_ehegee, (1—2) engage
Potential Yield 120 120 100
Field Losses 201 01 201
Nitrogen Requirement 200 150 200
p = _______ would make me indifferent between variety 1 and variety 2.

(b) Please write your responses for L, N, O, and T in the appropriate
spaces below. Then answer the question: what p would make you indifferent

between variety 1 and variety 2 shown below?

Variety 1 Variety 2
Characteristics 9 chance (172) engage
Potential Yield 0 = O = N = _____
Field Losses L = _____ T = L =
Nitrogen Requirement 200 200 200

p = would make me indifferent between variety 1 and variety 2.

143

(c) Hhat p would make you indifferent between variety 1 and variety 2 shown
below?

Variety 1 Variety 2
Characteristics 9 eneeee, (172) eeeeee
Potential Yield 100 120 100
Field Losses 201 01 201
Nitrogen Requirement 150 150 200
p = ________would make me indifferent between variety 1 and variety 2.

(d) Please write your answers for N, O, Q, and R in the appropriate spaces
below. Then answer the question: what p would make you indifferent between
variety 1 and variety 2 shown below?

Variety 1 Variety 2
Characteristics 2 chance (172) chance
Potential Yield N = 0 = N =
Field Losses 201 201 201
Nitrogen Requirement Q = Q = R =

 

 

p = would make me indifferent between variety 1 and variety 2.

144

(e) Hhat p would make you indifferent between variety 1 and variety 2 shown

below?

Variety 1 Variety 2
gamma; 2 change (1:2) 91111122
Potential Yield 100 120 100
Field Losses 01 01 201
Nitrogen Requirement 200 150 200
p = _______ would make me indifferent between variety 1 and variety 2.

(f) Please write your answers for L, R, Q, and T in the appropriate spaces
below. Then answer the question: what p would make you indifferent between

variety 1 and variety 2 shown below?

Variety 1 Variety 2

Characteristics 2 chance (179) chance
Potential Yield 100 100 100
Field Losses T = T = __““_ L =
Nitrogen Requirement R = Q = R =

 

 

 

p = would make me indifferent between variety 1 and variety 2.

145

Thank you very much for participating in this study and completing the
questionnaire. You have been a big help to me! If you are curious as to
what your responses indicate about your attitudes towards risk, please be
sure to write your address on the second page of the instructions. I an-

ticipate having the results in about a month and would be happy to send
them you.

By this time, you are probably tired of answering questions. But I have one
last one for you. would you take a moment and briefly tell me what your
background in farming is. For example, do your parents farm or do you
anticipate farming or working in an agricultural industry when you
graduate? In addition, I would like to know if you have been involved in

making decisions similar to the ones you answered questions about here.

THANKS AGAIN !!!

For this decision situation,

dried at the elevator. You normally harvest corn at 251 moisture content
and want to dry it down to 151 (grade 2). You have made a trip to the farm
machinery dealer to discuss dryers, but have not yet decided which one to
purchase.

automatic batch dryers and some which are continuous flow. The dryers which

146

APPENDIX 58

PURCHASING A CORN DRYER

The dealer has told you about several dryers, some which are

you are considering vary with respect to:

1.

9519; Ca acit which, for the dryers you are considering,
ranges rom 150 to 200 bushels per hour, depending on the
dryer selected.

gee; of Drying E ui ment which, for the dryers you are
considErTfig, ranges rom $10,000 t0 $20,000.

Operating Cost which is comprised mainly of the cost of
electricity or LP to run the dryer and, for the dryers your
are considering, ranges from 9 to 13 cents per bushel.

imagine that you own a 40,000 bushel corn
operation and have decided to buy a grain dryer rather than have your grain

147

FINANCING A COMBINE

For this decision situation, imagine that you have decided to purchase a
new combine. The self propelled combine, with headers, is going to cost you
$75,000. You have been shopping around for a loan to finance the combine.
Several sources are willing to loan you the money, with at least the com-
bine as collateral, but the loans differ with regard to:

1. Interest Rate which ranges from 111 to 151

2. Years tg,Eey_Back 59e_Lgee, also known as years to amortize
the loan, which range from 7 to 10 years

3. Collateral Reguireg which, in addition to the combine it-
self, ranges from none to an additional $20,000 worth of
machinery and equipment.

CHAPTER 6

SUMMARY, CONCLUSIONS, AND ISSUES REQUIRING FURTHER RESEARCH

The research presented in this dissertation was motivated by the ap-
parent instability of the ArrowbPratt coefficient of absolute risk aversion
across situations. Despite this perceived instability, the Arrow-Pratt
coefficient has continued to be one of the major measures used to describe
individuals' attitudes towards risk, order individuals according to their
attitude towards risk, and generalize their responses to risky events
across situations. Recent work by mathematical psychologists suggests the
need to separate an individuals attitude towards risk from their strength
of preference for riskless outcomes in order to examine the stability of
each component across situations and over time. In essence, this approach
calls for a re—examination of the behavioral assumptions underlying ex-
pected utility theory. In the dissertation, particular attention was given
to three behavioral assumptions and the implications of their violation on
the stability of the Arrow-Pratt coefficient of absolute risk aversion. The
three assumptions are:

1. Individuals have constant marginal value for money.
2. Preferences are homogeneous.

3. Preferences remain invariant between certainty and uncer-
tainty, other things being equal.

148

149

In response to evidence of consistent violation of these three assump—
tions, a new measure of attitudes towards risk was developed whose assump-
tions are believed to more accurately reflect individual's behavior.

This chapter briefly summarizes the research conducted, explores in
further depth the methodological questions which have arisen, discusses
issues requiring further research and examines what can be concluded at

this juncture.

m:

Chapter 2 begins with a brief review of the historical development of
utility theory in order to discover the source of the three assumptions
listed above. It was found that the assumption of constant marginal value
for money stems from the early cardinalists' argument that money could be
used as a measuring rod for utility and, as such, must have constant units
of value just as each inch on a standard ruler must represent the same unit
of measure. The assumption of homogeneity of preferences stems from Jevons
and Marshallian demand theory which argues that the last unit of a good
purchased by any individual gives him equal value or satisfaction as the
monetary equivalent of that unit of good. Since, in a perfectly competitive
market at equilibrium, each individual faces identical prices, it is argued
that individuals' marginal utility is equal. Thus, all individuals would
obtain the same degree of satisfaction of needs from the last unit pur-
chased which is, in turn, the same degree of satisfaction they would obtain
from having an amount of money equivalent to the price of that good. This
assumption ignores the concept of consumer surplus.

If an individual's preferences meet the conditions implied by these two

assumptions, their indifference curves for any two goods, or any one good

158

and money, are straight lines with a slope of negative one. Indifference
curves of this type can be obtained when an individual's value and utility
functions over multiattributed outcomes are additive. If both the utility
and value functions meet these conditions, the utility function must be a
positive linear transformation of the value function. This positive linear
transformation mathematically ensures that preferences are invariant under
certainty and uncertainty.

The implications of violations of the assumptions of constant marginal
value for money and homogeneous preferences for our ability to measure and
compare attitudes towards risk using the Arrow-Pratt coefficient of ab-
solute risk aversion were demonstrated graphically. Specifically, it was
shown that, even if an individual actually has stable attitudes towards
risk, if his preferences for riskless outcomes do not conform to these
assumptions, his risk attitude, measured using the Arrow-Pratt coefficient,
will appear unstable.

Chapter 3 begins with the conceptualization of a means whereby intrin-
sic attitudes towards risk can be determined by separating the effect of
attitude towards risk on a von Neumann-Morgenstern utility function from
the effect of preferences for riskless outcomes. Recent work on intrinsic
risk attitude by Dyer and Sarin (1981) is discussed and illustrated graphi-
cally. (Note that what is referred to as intrinsic risk attitude is called
relative risk attitude by Dyer and Sarin and others in the mathematical
psychology and management science literature. The terminology was modified
by this author to avoid confusion with an entirely different concept, the
Arrow-Pratt coefficient of relative risk aversion.) It was shown that Dyer
and Sarin's local measure of intrinsic risk attitude effectively compen-

sates for violations of the assumptions of constant marginal value for

151

money and homogeneous preferences. However, the global measures of intrin-
sic risk attitude developed by Dyer and Sarin place severe restrictions on
the form which the individual's value function and utility function may
take. Their measures require that one of the following two conditions hold:

1. All problems are simple value problems, i.e., that outcomes
refer to only one attribute, or

2. Preferences for attributes in multiattributed outcomes are
independent and additive.

The first condition does not allow for cases where, for example, a farmer
is concerned with not only profits, but also his asset/debt ratio, quality
of life, soil productivity, or implications of current decision outcomes
for future flexibility in the organization of the farm firm. The second
condition allows for consideration of several attributes, but requires that
value of attainment of any level of one attribute is completely independent
of the initial levels or changes in the levels of other attributes. In
other words, the individual's multiattribute value function must be addi-
tive.

In Chapter 4, a new measure of attitude towards risk was developed
which builds upon the work done by Dyer and Sarin. It begins with a closer
examination of the independence conditions which preferences for attributes
in a multiattributed outcome must meet in order to assume their additive or
multiplicative combination. The new method of separating attitudes towards
risk from preferences for riskless outcomes is more general than previously
developed methods because it allows for consideration of a larger class of
value functions. Specifically, while previous work requires that
preferences meet strong independence conditions required for additive
combination of preferences for different attributes of an outcome, the new

theory only requires that preferences meet the weaker conditions of

152

independence required for their multiplicative combination. The need for
this generalization was confirmed by the finding that fifty out of fifty-
nine respondents to a questionnaire administered by the researcher had
preferences which met the conditions for multiplicativity while only nine
respondents had preferences which also met the stricter conditions for
additivity.

The new measure of intrinsic attitude towards risk was developed using
Cauchy's four fundamental functional equations to ascertain the transforma-
tions between additive or multiplicative multiattribute value and utility
functions. These transformations are then to determine intrinsic attitudes
towards risk. These transformations allow one to do three things:

1. Given a utility function, u(x), and a value function, v(x),
determine u(v), the utility for the riskiness of outcomes.

2. If u(v) is stable across situations, determine v(x) from
u(x) or vice versa.

3. In modeling behavior, separate the influence of strength of
prizerence for riskless outcomes from attitudes towards

Use of the new measure is illustrated with hypothetical additive and
multiplicative value and utility functions. It is shown that, while Dyer
and Sarin‘s local measure of intrinsic attitude towards risk holds when the
value function is either additive or multiplicative, the new, more general
measure which allows for multiplicative value functions is needed for a
global measure of intrinsic risk attitude.

The procedures used to collect data to examine the empirical validity
of the new measure and the and the stability of intrinsic attitude towards
risk across situations are discussed in the beginning of Chapter 5. This
includes a description of the basic methods used in multiattribute utility

theory as well as discussion of some of the current debate over certain

153

aspects of these methods. The questionnaire used in this study is then
described and discussed. ’
Attempts to use the data collected with the described instrument to
examine the theory were beset by difficulties; the data could not be used
to clearly support or refute the new measure's predictive power because of
mathematical inconsistencies in the responses. The question which remains
to be answered is whether these (lnconsistencies reflect 'fuzzyness' in
human cognition which prevents individual's from accurately educing their
preferences or whether these inconsistencies reflect the fact that in-
dividuals preferences do not conform to the behavioral assumptions underly-

ing the methods used. This issue is discussed in the sections below.

Eleeleg,§9e_Research 19 Qgeeege

This research has been primarily concerned with re-examining a subset
of the behavioral assumptions of expected utility theory and their implica-
tions for measures of attitudes towards risk such as the Arrow-Pratt coef-
ficient of absolute risk aversion. The reader will recall that the expected
utility hypothesis assumes, among other things, invariance of preferences
between certainty and uncertainty when other things are equal, homogeneity
of preferences and constant marginal value of money. It has been argued
that because these behavioral assumptions diverge from actual preferences,
the Arrow-Pratt coefficient of absolute risk aversion does not reflect a
decision makers' true attitude towards risk. Therefore, our methods of
determining attitudes towards risk must be modified to more accurately
reflect decision makers' behavior.

The test of a hypothesis is usually preceded by the formalization of an

if, then statement such as, if the global measure of intrinsic risk

154

attitude accurately reflects decision maker's preferences, then a utility
function indirectly obtained using the individual's multiattribute value
function and the appropriate transformation described by (4A.1), (48.1),
(4C.1) or (40.1) will be a positive linear transformation of the
individual's directly elicited von Neumann-Morgenstern utility function.

In a discussion of the basic elements of a good test of a hypothesis,
Giere (1979, pp. 86-94) argues that this simple if, then statement is not
sufficient; one needs to specify not only the hypothesis, H, and the
prediction, P, but also the initial conditions, IC, and auxiliary assump-
tions, AA, which are expected to hold, such that the following two condi-
tions are true.

Condition 1: If [H and IC and AA], then P

Condition 2: If [not H and IC and AA], then very probably, not P
Note that the if statement contains not only the hypothesis, but also the
initial conditions and auxiliary assumptions.

It must be remembered that only a few behavioral assumptions of the
axioms of expected utility theory were examined: the remaining content of
the theory was assumed to reflect accurately decision maker behavior. (For
summaries of recent research done on the behavioral and mathematical con-
tent of the independence and other axioms of expected utility theory, the
reader is referred to Schoemaker (1982) and Machina (1983).) This strong
assumption created somewhat of a methodological paradox in the collection
and analysis of data in a study designed to test the new measure of at-
titudes towards risk: while the behavioral content of expected utility
theory was being examined, the representation theorem underlying expected
utility theory, including the independence axiom, was needed to collect the

155

data. Several aspects of these assumptions are relevant in light of the
inconsistencies pervading the collected data.

More formally stated, the initial conditions used in the study dis—
cussed in Chapter 5 were that all of the assumptions underlying the axioms
of expected utility theory and multiattribute utility theory except the
assumptions of constant marginal value of money, homogeneity of preferences
and invariance of preferences between certainty and uncertainty are mathe-
matically correct and accurately reflect decision maker behavior. The
auxiliary assumptions employed relate to the methods used for measurement
and include the assumptions that individuals can accurately educe their
preferences and that individuals can understand and effectively work with
probability concepts.

Assuming, for the purpose of argument, that the inability to complete
the analysis implies an incorrect prediction, we might reason through a
test of the hypothesis using the following logic:

First Premise: If [H and IC and AA], then P
Second Premise: Not P

Preliminary Conclusion: Not [H and IC and AA]
Preliminary Conclusion: Not H or not IC or not AA

At this point, we must make explicit our commitments to the truth of
both the initial conditions and auxiliary assumptions. The emphasis above
on the fact that only a few of the assumptions underlying the expected
utility hypothesis were examined has alluded to the lack of commitment to
the truth of the remaining assumptions. The remainder of this section and
the ones following bring to light more issues which prevent us from con-

tinuing with the deduction that:

156

Additional Premise: IC and AA (are true)

Conclusion: Not H
Instead, we are left with the second preliminary conclusion:

Preliminary Conclusion: Not H or not IC or not AA

Most axiomatizations in measurement theory try to capture an important

property of real numbers called the Archimedian property. Roberts (1979, p.
123) defines this property as follows:

”If a and b are real numbers and a > 0, then there is a

positive integer n such that na > b. That is, no matter how

small a might be or how large b might be, if a is positive,

then sufficiently many copies of a will turn out to be

larger than b.”
This property of the real number system is what makes measurement of any
type, including utility, possible by making it possible to roughly compare
the relative magnitudes of any two quantities by seeing how many copies of
the first quantity are needed to exceed the second, larger quantity. This
condition is necessary to achieve a homomorphic mapping from an abstract
relational system, such as preferences, to the real line, yielding a
utility function. The Archimedian property is embodied in the continuity
axiom of expected utility theory which states that if an individual prefers
A to B to C, then there exists an unique probability, p, such that he will
be indifferent between B and a lottery of the form pA + (1-p)C. Note that
this is the exact form of the questions used in the questionnaire to elicit
information needed to determine the scaling factors for multiattribute
utility functions.

It may be argued that the Archimedian axiom requires standards of
accuracy which are unrealistic. For the interpolations involved in decision

analysis to be valid, decision makers must not only have a good

157

understanding of probability, but their responses must also be accurate and
precise. However, the decision makers responses may often vary because of
the inability to accurately educe preferences, changes in preferences over
time, changes due to introspection about preferences evoked by closer
examination, or in response to feedback about mathematical inconsistencies
which may cause the decision maker to alter responses to the extent that
they may or may not reflect true preferences.

Another set of assumptions which come into play in extending expected
utility theory to multiattribute utility and value theory are mutual
preference and mutual utility independence. These conditions require that
the weight that the individual gives to each attribute will not vary with
changes in the levels of other attributes. Mutual independence only holds
if the weight given to one attribute when other attributes are at a
specified level does not change when the attributes are at another level.
Evidence of failure of preferences to meet this condition can be found if
the individual has weights which are not constant. Hhile generalized
preference and utility independence allow for symmetric reversals in
preferences to occur when moving from the domain of gains to the domain of
losses, or vice versa, generalized independence does not allow for asym-
metrical reversals in preferences. Violation of generalized independence
can easily occur if the individual attaches a different weight to improving
the level of an attribute than he does to maintaining or preventing a
decrease in the level of the attribute. In the questionnaire, generalized
independence was only checked over one domain (either gains or losses)
rather than over both. Therefore, the apparent satisfaction of the condi-
tions of generalized independence may be misleading.

158

Kahneman and Tversky (1979) describe this reversal of preferences about
the status quo as the reflection effect and argue that preferences in the
domain of losses are a mirror image of (symmetric with) preferences in the
domain of gains. However, Friedman and Savage (1948) argue that preferences
in the domain of losses are not symmetric with those in the domain of
gains. Instead, they argue that the utility function in the domain of
losses is much “steeper" than that in the domain of gains. They support
this argument with the observation that individuals will, given the chance
at significant gains, participate in unfair odds and, for the prevention of
much smaller losses, accept equally unfair odds to preserve the resources
which they hold. Although there is much disagreement with the specifics of
the 'everymans‘ utility function“ Friedman and Savage developed in response
to this observation, the observation has been reconfirmed by subsequent
studies.

It is important to remember that the axioms of utility theory not only
define ”rational behavior“ but go much further and define behavior which
must be described both precisely and with mathematical consistency. This
presents a dilemma to both the decision analyst and the decision maker: if
the decision makers' preferences meet the conditions set forth in the
axioms, the theory may be interpreted as a good descriptor of behavior and
used to predict behavior while if the decision makers' preferences do not
meet conditions set forth, the theory becomes prescriptive. In the latter
case, the decision analyst may appeal to the apparent reasonableness of the
axioms and the rationality of the decision maker in asking him to revise
his responses. But a decision maker can be reasonable and have sound
reasons for particular preference structures without conceding to the

axioms: in this case, the decision analyst must reflect on how far they can

159

push for mathematical consistency and still accurately reflect the
preferences of the decision makers. Efstanthiou (1984, p. 310), in a dis-
cussion of this issue, argues that:

"..in obtaining utility curves for single attributes and

imposing the ax oms of multiattribute utility theory, a

dangerous mixture of descriptive and normative procedures
will be produced. "

Ieegee_8egglgigg Further Reseegen

The myriad issues which require further research can be classified into
three major areas: mathematical content of the axioms, behavioral content
of the axioms and the educement of preferences. The third area, although
formally a subset of the behavioral content of the axioms, has enormous
impact on the issue of mathematical consistency. Currently, research is
being done in all three areas by researchers from many different dis-
ciplines. Three particularly promising areas of research directly related
to the problems discussed above are described briefly below.

The issue of 'fuzzyness' in human cognition has plagued decision
analysis since its inception. In contrast to the uncertainty due to random-
ness of events external to the decision maker, this fuzzyness results from
a form of uncertainty due to vagueness and imprecision internal to the
decision maker. According to the proponents of fuzzy set theory, these
distinct qualities must be modeled in different ways, the former using
probability theory and the latter using fuzzy set theory. Fuzzy set theory
provides a means of quantifying the degree of imprecision associated with
any input into the decision process through the use of membership func-
tions. Hatson, Heiss and Donnell (1979) argue that probabilities and
utilities can inherently only be represented by somewhat rough sets of

160

numbers. Their “fuzzy decision analysis“ method is motivated by the need to
handle the imprecision accompanying the judgemental inputs into decision
analysis in a systematic and consistent manner.

Zadeh (1965), one of the first to argue for a new fuzzy approach to
systems analysis and decision making under uncertainty, holds that fuzzy
set theory is not an alternative to probability theory and expected utility
theory but is, instead, a parallel calculus to be used to handle the im-
precision inherent in human cognitive processes. The central concept in
fuzzy set theory is the membership function which numerically represents
the degree to which an element belongs to a set. The function is valued
between zero and one and is assessed subjectively with small values repre-
senting a low degree of membership in the set and high values representing
a high degree of membership. For example, the statement “it will probably
rain tomorrow“ would have a higher degree of membership in a set regarding
likelihood of rain than the statement ”it might rain tomorrow". In applica-
tions of fuzzy set theory, the values used to represent degrees of member-
ship are not elicited directly. Instead, they are taken from curves drawn
by individuals to represent their degrees of belief that an event will
occur.

The calculus of fuzzy sets is based on three propositions to which
numbers indicating membership should conform. These propositions are
analogous to those used in conventional set theory and include:

1. The degree to which X belongs to set A and set 8 is equal to
the smaller of the individual degrees of membership.

2. The degree to which X belongs to either set A or set 8 is
equal to the larger of the individual degrees of membership.

3. The degree to which X belongs to (not A) is one minus the
degree to which X belongs to A.

161

The calculations involved in the decision analysis can be considered to
be a functional relationship between the inputs regarding probabilities and
utilities and the output of the analysis in the form of the expected

utility of an action choice. The three relationships cited above are used

to deduce the 'fuzz' on the output given the fuzzyness of the inputs.1 As
with conventional utility analysis, probability distributions may be gener-
ated which characterize the range of possible outcomes for each action
choice. Hhereas the distributions obtained from conventional analysis are
taken to be the true distributions, in fuzzy set theory the extent to which
the distribution of inputs, probabilities and utilities implies a preferred
action choice is only as large as the least level of implication for each
set. Unless one distribution clearly dominates another, it cannot be said
to indicate the preferred action choice. To determine the preferred action
choice when two sets overlap, one must determine the extent to which one
set is preferred over the other through the use of Zadeh's fuzzy calculus.

There remain questions regarding the axiomatization of a fuzzy set
calculus which can be used to elicit membership functions. Experimental
evidence does show, however, that individuals are able to draw curves or
probability distributions to represent their perceived imprecision regard~
ing degrees of belief such as "better than ever”, “pretty likely", or
”about X1". The precise shapes of these distributions are somewhat ar-
bitrary, but this fact does not affect the inferences which can be drawn
from fuzzy set analysis as it is the general shape of the distributions
that matter.

 

1 For particulars of the mathematical methods used, see Hatson, Weiss and
Donnell (1979) and Zimmermann, Zadeh and Gaines (1984).

162

Machina (1982), in examining the mathematical content of the independ-
ence axiom of expected utility theory starts with the observation from
elementary calculus that for a function to be differentiable, it must be
locally linearly approximable. The independence axiom is like a linearity
assumption for utility functions; it implies that the utility function may
be represented as the expectation with respect to the given distribution of
a fixed function defined over the set of possible outcomes. In other words,
the utility function is constrained to be a linear function over the set of
distributions of outcomes or, as commonly phrased, "linear in the
probabilities”. However, empirically determined utility functions do not
always conform to the linearity condition because of the certainty effect
or other phenomena.

Machina argues that, despite these violations, the basic concepts,
tools and results of expected utility analysis are still applicable because
they are not dependent upon the independence axiom. They can also be
derived from a weaker assumption of local smoothness of preferences over
alternative probability distributions. Machina shows that if you can assume
that preference orderings underlying the utility function in a region are a
differentiable functional, then the utility function is locally linear.
Hence, the utility function is differentiable and the results of expected
utility theory hold, at least locally since, at each point, a locally
linear utility function can be obtained. Hith his generalized utility
analysis, Machina elegantly proves that the assumption of smoothness of
preferences and consistency in the shape of the utility function in a given
region, in essence, rescues the independence axiom. Machina does not prove

the local applicability of the other axioms of expected utility theory.

163

Hhile Machina examines the mathematical content of the independence
axiom, Heiss (1983, 1984), with equal mathematical rigor, examines the
behavioral content of the axioms. Neiss begins with the observation that
lotteries, which can be expressed as cumulative distribution functions, can
take several forms including degenerate lotteries, where the probability of
the outcome occurring is one, and continuous lotteries, where the probabil-
ities of outcomes sums to one. Degenerate lotteries coincide with the case
of certainty while continuous lotteries coincide with the case of uncer-
tainty.

In Heiss' approach, decision makers compare lotteries according to
their measurable utility and are allowed to use a “two-rule“ measurable
utility function which allows, but does not require, them to use one
preference rule for choosing between degenerate lotteries and another for
choosing between continuous lotteries. In a model of producer response to
price uncertainty, Heiss finds that, when a decision maker is allowed to
use a “two-rule“ measurable utility function, he generally will choose a
utility maximizing output level different from the one he would choose if
he followed a more restrictive expected utility maximizing approach. In
particular, if his marginal utility of money is greater under certainty
than under uncertainty and both the expected utility and measurable utility
models have interior solutions, then the latter model will generate a lower
optimal production level than the former. Furthermore, it will prove pos-
sible to have a zero production level and a positive production level being
simultaneously optimal (Weiss, 1983, p. 16). A particular pricing situation
to which his analysis applies is that of random price that is truncated
below the mean through the introduction of a support price.

164

These three examples illustrate the breadth of possibilities which can
be explored by researchers in decision theory, all of which are useful in
moving towards resolution of the common problems encountered in this dis-
sertation. The first priority of this researcher is to obtain the mathe-
matically consistent responses required to complete a test of the new
measure of attitudes towards risk. This will 'require careful, lengthy
procedures to elicit the multiattribute value and utility functions for
actual decisions faced by a small group of decision makers. Particular
attention needs to be paid to the issue of how far responses can be altered
to meet the consistency required for analysis and still reflect decision
makers' preferences. In the longer run, Neiss' suggestion (personal com-
munication) that localized behavioral theorems based on preference order-
ings, analogous to the work done by Machina on the mathematical content of
the axioms of expected utility theory, be developed seems promising and in
keeping with the concerns which originally motivated this dissertation

research.

There is strong evidence to refute many of the behavioral assumptions
underlying the axioms of expected utility theory. It has been demonstrated
that if decision makers' preferences violate any ohne of the assumptions of
constant marginal utility of money, homogeneity, or invariance of
preferences between certainty and uncertainty when other things are equal,
the Arrow-Pratt coefficient of absolute risk aversion is a confounding of
information about attitude towards risk and preferences for the attributes

of outcomes under certainty. Conformance of decision makers' preferences

165

with each of the three assumptions is necessary, but not sufficient, for
the Arrow-Pratt coefficient to measure only attitude towards risk.

It has been shown that, when any one of these assumptions is violated,
measurement of attitude towards risk requires consideration of the
individual's value function as well as his von Neumann-Morgenstern utility
function. Dyer and Sarin's measure of intrinsic risk attitude (known in the
mathematical psychology and management science literature as relative risk
attitude) introduces a useful approach to this problem. However, it was
demonstrated that their method unduely constrains analysis to those cases
where individuals' preferences for the attributes of outcomes under cer-
tainty are completely independent and additive.

A new measure of attitude towards risk that is more general than Dyer
and Sarin's measure has been developed. This new measure is applicable to
individuals whose preferences for attributes of outcomes under certainty or
uncertainty are either additive or multiplicative.

It was found that determining individual's multiattribute value and
utility functions without forcing mathematical consistency results in data
that does not meet the level of consistency required to determine scaling
factors for either the additive or multiplicative multiattribute models. At
this Juncture, it can not be determined whether the inconsistencies in
responses to the survey instrument are due to problems with the new measure
of intrinsic risk attitude, fuzzyness in human cognition, mis-specification
of initial conditions and auxiliary assumptions, or poor design of the
survey instrument.

The issue of mathematical consistency is particularly important for
this new measure of attitudes towards risk. Hhile many of the studies of

multiattribute utility or value theory are concerned with either an

166

individual's value function or their utility function, the new measure of
attitudes towards risk is concerned with the relationship between the
individual's value and utility functions. Therefore, both functions must be
measured accurately and shown to reflect decision makers’ preferences
before the new measure of intrinsic attitude towards risk can be tested.
Resolution of this question requires that other assumptions of expected
utility theory and multiattribute utility theory undergo scrutiny similar
to that received by the assumptions examined in this dissertation.

Despite the problems, caveats and unresolved issues discussed above,
the new measure of intrinsic attitude towards risk may bring us one step
closer to a means of accurately predicting preferred action choices and
enable us to uniquely characterize individuals or classes of individuals by

their attitude towards risk both across situations and over time.

BIBLIOGRAPHY

BIBLIOGRAPHY

Aczel, J. editor. Functional E uations: Histor , Applications and
Theory. DordreEfit, Holla : D. Reide ub ishing Company,_I984.

 

Aczel, J. Lectures 99 Functional Eggatioge egg_1heir A lications.
Matheiitics in SEience and EfigIneering,_'Volume 19. NEw York:
Academic Press, 1966.

 

Aczel, J. On A iieeiions egg Theory 59 Eggetional gggations. New York:
AcadeiTc ress,

Allais, M. “The so Called Allais Paradox and Rational Decisions Under
Uncertainty,“ pp. 437-481 in M. Allais and O. Hagen, editors,
Eygeeiee_ Qtility Hypotneeie egg gee Allais Paradox. London: D.
Reidel Publishing Company, 1979.

 

Allais, M. and O. Hagen, editors. Expected Utility H thesis and EDQ.
Allais Eeradox. London: D. Reidel Publishing Company, I979?“

Arrow, K. Aepects 9f 59e_Theor 99 Risk Bearing. Helsinki: Yrjo
Jahhssonin Saatio, 1

Arrow, K. "Behavior Under Uncertainty and Its Implications for Policy,"
Stanford Institute for Mathematical Studies in the Social
Sciences, Technical Report No. 399, 1983.

Barron, F. and H. Person. ''Assessment of Multiplicative Utility Functions
via Holistic Judgements,” Organizational Behavior 599_ 99559
Eerformance, 24(1979): 147-166.

Baumol, H. ”The Neumann-Morgenstern Utility Index - An Ordinalist View,"
[be Journal 99 Egiiiical Economy, 59(1951): 61-66.

167

168

Bell, D. and H. Raiffa. ”Marginal Value and Intrinsic Risk Aversion,”
Harvard Business School Working Paper No. 79-65, 1979.

Borchb K. Economics 9f Uncertainty. Princeton. Princeton University
ress,

Chipman, J. ”The Foundations of Utility," Econometrica, 28(1960): 193-224.

 

Carmen, G. 'An Application of Stochastic Dominance with Respect to A
Function: Measuring the Relationship of Producer Attributes to
Risk Preferences, Unpublished Ph. D. Dissertation, Department of
Agricultural Economics, Michigan State University, 1982.

Cooter, R., and P. Rappoport. "Here the Ordinalists Hrong About Nelfare
Economics?“ Journal 99 Economie Literature, 22(1984): 507-530.

Currim, I. and R. Sarin. 'A Comparative Evaluation of Multi-attribute
Consumer Preference Models,I Managgggnt Science, 30(1984): 543-
561.

 

Currim, I. and R. Sarin. 'A Procedure for Measuring and Estimating
Consumer Preferences Under Uncertainty," Journal of Marketing
Resear___ch, 20( 1983): 249-256.

Delbeke, L. and J. Fauville. 'An Empirical Test of Fishburn' s Additivity
Axiom,“ Acta E_ychologica, 38(1974): 1- 20.

Diamond, P. and J. Stiglitz. "Increases in Risk and Risk Aversion,
Journal of Economic Theor, 7(l974): 337-360.

Debreu,G Mor of _ei_fi nAxiomatic Anal sis of Economic
EguilLbrium.$ WY %hn H ley and Sans, 195 9%

Dillon,J . 'An ExBository Review of Bernoullian Decision Theory in Agri-
culture: Is tility Futility?" Review of Marketing and Agricul-
tural geggomics, 39(1971):

Dobb, M. Theories of Value and Distribution Since Adam Smith: Ideolo
and Ecom mic Tfieorye. CaiBr1dge: Cambr1dge‘Uhiversity Press,

169

Dyer, J. and R. Sarin. I'Relative Risk Aversion," Management Science,
28(1982): 875-886.

 

Dyer, J. and R. Sarin. "Group Preference Aggregation Rules Based on
Strength of Preference,“ Managgggnt ngencg, 25(1979): 822-832.

Dyer, J. and R. Sarin. “Measurable Multiattribute Value Functions,”
Operations Research, 27(1979): 810-822.

Efstathiou, J. ”Practical Multiattribute Decision Making and Fuzzy Set
Theory,“ pp. 307-320 in H. Zimmermann, L. Zadeh and B. Gaines,
editors, Fuzzy Sets and Decision Analysis. TIMS/Studies in the
Management Sciences, “Volume 20. Amsterdam: Elsevier Science
Publishing Company, Inc. , 1984.

Eliashberg, J. “Consumer Preference Judgements: An Exposition with
Empirical Applications," Management Science, 26(1980): 60-77.

Ellsberg, D. 'Classic and Current Notions of “Measurable Utility“, '
Economic Journal, 64(1954): 528-556.

Ellsberg, D. ”Risk, Ambiguity and the Savage Axioms, ' Quarterly Journal
of Economics 75(1961): 643- 669.

Fama, E. “Ordinal and Measurable Utility,“ pp. 125-145 in M. Jensen,
editor, Studies in the, Theory Lf Capital Markets. New York:
Praeqer Publishers, 1972.

Fischer, G. “Convergent Validation of Decomposed Multiattribute Utility
Assessment Procedures for Risky and Riskless Decisions, ' 0r-
ganizational Behavior and Human Performance, 18(1977): 295- 315.

Fischer, G. “Multidimensional Utility Models for Risky and Riskless
Choice,” rganizational Behavior and Human Performance, 17(1976):
127-146.

 

Fishburn, P. The Foundations Lf Ex ted Utilit, Theory and Decision
Library, volume 31, Boston: .‘REidel Fub1shing Company, 1982.

Fishburn, P. 'Lexicographic Additive Differences,” Journalo Lf Mathe—
gatical PsychoLQSI. 21(1980): 191-218.

 

170

Fishburn, P. “Cardinal Utility: An Interpretive Essay,“ Rivista Inter-
nazionale d§_Scienze Econonighg g Connerciali, 23(1976): 1l02-

 

 

 

Fishburn, P. Mathelatics of Decision Theor . Methods and Models in the
Social iences,'V3lune 3, Paris: 0, 1972.

 

Fishburn, P. Utility Theory for Decision Making. New York: John Riley
and Sons, 1970.

Fishburn, P. “Methods of Estinating Additive Utilities,“ anagglent
Science, 13(1967): 435-453.

Fishburn, P. and R. Keeney. ”Generalized Utility Independence and Some
Ilplications,' ngrations Research, 23(1975): 928-940.

Fishburn, P. and R. Keeney. “Seven Independence Concepts and Continuous
Multiattribute Utility Functions,” Journal, of Mgthggatical
Psychology, 11(1974): 294-327.

Fishburn, P. and G. Kochenberger. “Two-Piece von Neumann-Morgenstern
Utility Functions,” Decision Science, 10(1979): 503-518.

Fleisher, B. and L. Robison. “Applications of Decision Theory and the
Measure-ant of Attitudes Towards Risk in Farm Management Research
in Industrialized and Third world Settings," Michigan State
Daggersity International Develop-ant Paper Series, Paper No. 6,

Frieduan, M. and L. Savage. ”The Expected Utility Hypothesis and the
Measurability of Utility,” The. gourngl of Political Economy,
60(1952): 463-474.

 

Fried-an, M. and L. Savage. I'The Utility Analysis of Choices Involving
Risk,“ Journal of Political Econogy, 56(1948): 279-304.

Gehrig, U. I'Functional Equation Methods Applied to Economic Problems:
Sole Examples,“ pp. 33—52 in J. Aczel, editor. Functional Egua-

tions: distgry, Applications agg_Theory. Dordrecht, Holland: D.
Reidel Publishing Company, 1984.

171

Giere, R. Understanding Scientific Reasoning. New York: Holt, Rinehart
and Winston, 1979.

Grandmont, J. ”Continuity Properties of a yon Neumann-Morgenstern
Utility,“ Jgggggl g: Eeonomic Theor , 4(1972):45-57.

Halter, A., and R. Mason. “Utility Measurement for Those Hho Need to
fngw,‘ Hestern Journal g: Agricultural Economics, 3(1974): 99-
0 .

Hansen, F. “Consumer Choice Behavior: An Experimental Approach,“
Journal 9: Marketing Reseeech, 6(1969): 436-443.

Hauser,J. and G. Urban. “Assessment of Attribute Importances and Con-
sumer Utility Functions -- von Neumann-Morgenstern Theory Applied

gnguman Behavior," Journal gf Consumer Research, 5(1979): 251-

Hershey, J., H. Kunreuther and P. Schoemaker. ”Sources of Biases in
Assessment Procedures for Utility Functions,” Managegent Science.
28(1982): 936-954.

Herstein, I., and J. Milnor. “An Axiomatic Approach to Measurable
Utility,” Econometreee, 21(1953): 291-297.

Hey, g. Uncertainty 1g Microeconomics. New York: New York University
ress,

Hildreth, C. and G. Knowles. “Some Estimates of Farmers' Utility
Functions,“ University of Minnesota Agricultural Experiment
Station Technical Bulletin No. 335, 1982.

Hogarth, R., editor. Ne! Directions for Methodology gfi Social egg
Behavioral Sgiegge: Quesfiion'Egiilgg_ egg Resggnee, ansistency.
San Francisco: Jossey-Bass, Inc., 1982.

 

Huber, G. “Multiattribute Utility Models: A Review of Field and
Field-like Studies," Menagement Science, 20(1974): 1393-1402.

172

Huber, 6., R. Daneshgar and D. Ford. "An Empirical Comparison of Five
Utility Models for Predicting Job Preferences, Organizational
Behavior and Human Performance 6(1971): 267 282.

 

Kahneman, D. and A. Tversky. "Prospect Theory: An Analysis of Deci-
sions Under Risk, ' Econometrica, 47(1979): 263~291.

Keeney, R. ''An Illustrated Procedure for Assessing Multi-attributed
Utility Functions,‘I Sloan Manegemegt Review, 14(1972): 37-58.

Keeney, R. “Multiplicative Utility Functions, ' Qperations Research,
22(1974): 22- 34.

 

Keeney, R. , and H. Raiffa. Decisions with Multi le 0b ectives:
Preferences and Value Tradeoffs. New York: John ey and Sons,
1976.

Keeney, R. and A. Sicherman. ''Assessing and Analyzing Preferences Con-
cerning Multiple Objectives: An Interactive Computer Program,“
Behavioral Science, 21(1976): 173-182.

Kihlstrom, R., and L. Mirman. ”Risk Aversion with Many Commodities,"
Jgggggl gfi Economic Theory. 7(1974): 381-388.

Krantz, D, H. Luce, P. Suppes, and A. Tversky. Foundations of
Measurement, Volume I, New York: Academic Press, 1971.

 

Krzysztofowicz, R. “Strength of Preference and Risk Attitude in
Utility Measurement,” Qrganizational Behavior egg Human
Performance, 31(1983): 88-113.

Krzysztofowicz, R. and L. Duckstein. ”Assessment Errors in Multi—
attribute Utility Functions, “ rganizational Behavior and Human
Performance, 26(1980): 326-348.

Kunreuther, H. and P. Slovic. “Economics, Psychology and Protective
Behavior, ' American Economic Review, 68(1978): 64-77.

Lange, 0., editor. Studies in Mathematical Economics and Econometrics.
Chicago. University of Chicago Press, 1942.

 

173

Love, R., and L. Robison. ”An Empirical Analysis of the Intertem ral
Stability of Risk Preferences,” Sggthggn_ggugnal of Aggicu tural
Econogi§§. 16(1984): 159-166.

Machina, M. "The Economic Theory of Individual Behavior Toward Risk:
Theory, Evidence, and New Directions,“ Stanford University In-

stitute for Mathematical Studies in the Social Sciences Technical
Report No. 433, 1983.

Machina, M. “Generalized Expected Utility Analysis and the Nature of
Observed Violations of the Independence Axiom," University of

California, San Diego, Department of Economics Discussion Paper
No. 82-32, 1982.

Machina, M. "Expected Utility' Analysis Without the
Independence Axiom," Econometrica, 58(1982): 277-323.

Majumdar, T. The Measurement of Utility. London: Macmillan and
Company, Ctd., 1958.

Masson, R. "Utility Functions with Jump Discontinuities: Some Evidence
and Implications from Peasant Agriculture," Economic Inquiry,
12(1974): 559-566.

 

McCord, M. and R. de Neufville. I'Empirical Demonstration that Expected
Utility Analysis is Not Operational," pp. 181-199 in B. Stigum
and F. Nenstop, editors, Foundations gf_Utility and_ Risk Theory
with Applications. Dordrecht, Holland: ,D. Reidel Publishing
Company, 1983.

 

Mosteller, F. and P. Nogee. “An Experimental Measurement of Utility,"
Journal gfi_Political Economy, 59(1951): 371-404.

Myers, J. and M. Alpert. "Semantic Confusion in Attitude Research:
Salience vs. Importance vs. Determinance," pp. 106-118 in N.
Perreault, Jr., editor, Adyances_in Consumer Research, Volume IV.
Association for Consumer Research, 1976.

Myers, J. and E. Sadler. I'Effects of Range of Payoffs as a Variable in
Risk Taking,” JournaL of Experimental Psychology, 68(1968): 306—
309.

 

174

Pekelman, D. and S. Sen. “Mathematical Programming Models for the

Determination of Attribute Heights,” Maaagagaat Saience,
28(1974): 1217-1229.

Perreault, R., Jr., editor. AQvances in Consume: Raaearch, Volume IV.
Association for Consumer Research:

 

Pratt, J. ”Risk Aversion in the Small and in the Large," Econometriga,
32(1964): 122-136.

Raiffa, H. ”Risk, Ambiguity, and the Savage Axioms," Quarterlngurnal
a: Egonomics, 75(1961): 698-694.

 

Ramseyési. Ina Foundaaions 9f Mathematics. New York: Harcourt, Brace,

Roberts, F. Measuremegt Theory with Applications ta Decisionmaking,
Utility éflfl.§h§.§2§1§l Sciencaa, Encyclopedia of Mathematics and
Its Applications, Volume 7, Reading, Mass.: Addison-Nelsey Pub-
lishing Company, 1979.

 

Robison, L. I'An Explanation of Apparently Risk Preferring Behavior by
Risk Averse Decision Makers,“ paper presented at a meeting of the
North Central Research Committee NC—113, held in Saint Louis,
Missouri, October, 1982.

Robison, L. and B. Fleisher. “Risk: Can Ne Model Nhat He Can‘t Define
or Measure?“ pp. 34-51 in, "Risk Management Strategies for
Agricultural Production Firms,” proceedings of a seminar spon-
sored by the Southern Regional Research Project S-180, held in
San Antonio, Texas, March, 1983.

Roskies, R. “A Measurement Axiomatization for an Essentially Multiplica-
tive Representation of Two Factors," Jagrnal at Matheaatical
P_§yg_ho_1<_>gy, 2(1965): 266-276.

 

Roumasset, J. ”Modeling Decision Making Under Uncertainty: Introduction
and State of the Arts,“ pp. 1-22 in J. Roumasset, J. Boussard and
1. Singh, editors, Riak, Uagartainty BBQ. Agricultural
Development. New York: Agricultural Development Council, 1979.

 

175

Roumasset, J., J. Boussard and I. Singh, editors. Riak, Uncertainty aag
ggaicultural Develogaent. New York: Agricultural Development
ouncil, 1979.

 

Sakawa, M. "Interactive Multiobjective Decision Making by the Fuzzy
Sequential Proxy Optimization Technique -- F Spot,“ pp. 241-268
in H. Zimmermann, L. Zadeh and B. Gaines, editors, Fuzz 5€t§.§fl9
Daaision Analysia. TIMSIStudies in the Management SETEnces,
gaggle 28. Amsterdam: Elsevier Science Publishing Company, Inc.,

Samuelson, P. “Constancy of the Marginal Utility of Income," pp. 75-91
in 0. Lange, editor, Studies {Q Mathematical Eggagalaa aaa
Econometrics, Chicago: University of Chicago Press, 1942.

Sarin, R. “Strength of Preference and Risky Choice,“ Oparationa
Rasearch, 38(1982): 982-997.

Schoemaker, P. “The Expected Utility Model: Its Variants, Purposes,
Evidence and Limitations,‘I Journal afi Economic_ Literature,
28( 1982): 529-563.

Schoemaker, P. and C. Naid. 'An Experimental Comparison of Different
Approaches to Determining Heights in Additive Utility Models,"
Maaagaaaat,5cienca, 28(1982): 182-197.

Slovic, P., B. Fischhoff and S. Lichtenstein. "Response Mode, Framing, and
Information Processing Effects in Risk Assessment,” pp. 21-36 in
R. Hogarth, editor. Na!_Directions faa_Methodology_of Social aaa
Behavioral Scienca: Questiaa Framing aag Response _ConsisEency.
San Francisco: Jossey-Bass, Inc., 1982.

 

Slovic, P. and S. Lichtenstein. “Importance of Variance Preferences in
Gambling Decisions,“ Jaagaal 9: Experimental anchology,
78(1968):646-654.

 

Slovic, P. and A. Tversky. "who Accepts Savage's Axiom? Bah_yjoral
Saience, 19(1974): 368-373.

Stigler, G. ”The Development of Utility Theory," Ina Journal afi Poli-
tiaai Egonomy, 58(1958): 387—327: 373-396.

 

176

Stiglitz, J. ”Behavior Towards Risk with Many Commodities," E9939-
aetrica, 37(1969): 668-667.

Strotz, R. “Cardinal Utility,” American Economic Review, 43(1953):
384-397.

Stigum, B. and F. Henstop, editors. Foundations 9: Utility and Risk
Theory with Appligations. Dordrecht, Holland: D. RéTdel Pub-
lishing Company, 1983.

Tversky, A. 'Additive Utility and Subjective Probability,” JournaL 9:
Mathegatical Psycholggy, 4(1967): 175-281.

Tversky, A., and D. Kahneman. “The Framing of Decisions and the
Psychology of Choice,” pp. 3-36 in R. Hogarth, editor, May_Qiaaa-
tions fag_Methodology a: Social aag Baflavioral Science: Question
Framing aaafRespgnse Consistengy, San Francisco: Jossey-Bass,
Inc., 1982.

 

Venkatensan, M., editor. Proceedin 3 9f tag Third Annual anferenga a:
the Association far Caaaagaa Researafl. New York: ASsociation for
CEESumer Research, 1972.

 

von Neumann, J., and 0. Morgenstern. [Ma_Theory af_6ames, Princeton:
Princeton University Press, 1944.

 

von Hinterfeldt, D. I'Functional Relationships Between Risky and Riskless
Multiattribute Utility Functions,” University of Southern
California Social Science Research Institute, Research Report No.
79-3, 1979.

Natson,S., J. Weiss and M. Donnell. “Fuzzy Decision Analysis," IEEE Trans-
actions aa Syatems, Mag aag Cyernetics, SMC-9(1979): 1-9.

Heiss, M. :Conceptual Foundations of Risk Theory,” unpublished manu-
scrip .

Neiss, M. "Risk Concepts in Agriculture: A Closer Look," paper presented
at the American Agricultural Economics Association meetings in
Ithaca, New York, August, 1984.

177

Weiss, M. “Discontinuous Utility in Peasant Agriculture: A Re-
examination,“ paper presented at the Western Economic Association
Meetings in Las Vegas, Nevada, June, 1984.

Weiss, M. ”Decomposition of Measurable Utility Functions," paper
ressgged agaghe Econometric Society meetings in San Francisco,
e r, .

Wendt, D., and C. Vlek, editors. Utilit , Probability aaa Human
Decision Making, Theory and_ Cision Library VoTume II, Boston:
D. Reidel Publishing Company, 1975.

Whittaker, J., and J. Winter. “Risk Preferences of Farmers on Empirical
Examples: Some Questions and Suggestions for Future Research,“
pp. 217-244 in “Risk Analysis in Agriculture: Research and Educa-
tional Developments,“ proceedings of a seminar sponsored by the
American Agricultural Economics Association Western Regional
Research Project W-149, in Tuscon, Arizona, January, 1988.

Wilkie, W. and R. Weinreich. I'Effects of the Number and Type of
Attributes Included in an Attribute Model: More is not Better,”
pp. 325-348 in M. Venkatesan, editor, Proceedings of tha, Third
Annual Conference a: tag Association fag Consumer—Research. New
York: Association for Consumer Research, 197 .

Yaari, M. “Some Remarks on Measures of Risk Aversion and Their Uses,“
Journal at Econoaia IMagay, 1(1969): 315—329.

Young, D., W. Lin, R. Pope, L. Robison and R. Selley. ”Risk Preferences
of Agricultural Producers: Their Measurement and Use,” pp. 1-28
in “Risk Management in Agriculture: Behavioral, Managerial, and
Policy Issues," proceedings of a seminar sponsored by the Western
Regional Research Project W-149 in San Francisco, California,
January, 1979.

Zadeh, L. ”Fuzzy Sets," Laformation ayg Control, 8(1965): 338-353.

Zimmermann, H., L. Zadeh B. Gaines, editors. Egaay_§ata aag Qaaiaion
Analysis. TIMS/Studies in the Management Sciences, Voluméfl28.
Amsterdam: Elsevier Science Publishing Company, Inc., 1984.