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OPERATIONAL TECHNIQUES FOR APPLIED DECISION
ANALYSIS UNDER UNCERTAINTY

By

Robert P. King

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Agricultural Economics

1979

ABSTRACT

OPERATIONAL TECHNIQUES FOR APPLIED DECISION
ANALYSIS UNDER UNCERTAINTY

By
Robert P. King

The techniques developed in this study are designed for use
during four phases of an applied decision analysis: problem formula-
tion, the determination of subjective probability distributions, the
measurement of decision maker preferences, and the identification of
preferred choices. When considered together, they represent an
integrated set of techniques which facilitate the application of
decision theory based on the expected utility hypothesis.

Problem formulation is an important first step in any applied
decision analysis. Two important considerations related to problem
formulation are emphasized in this study. First, the need to identify
and classify the factors which have an important impact on the outcome
of the decision to be made is noted, and a classificatory scheme based
on systems identification is presented. Second, the need to give
careful attention to the specification of what is to be decided is
stressed. The desirability of flexible decision strategies in many
situations is noted, and the use of feedback control rules to introduce
flexibility into a strategy is described.

Direct probability assessments of exogenous stochastic factors
and the modelling of more complex stochastic processes are combined
in the procedure for the determination of the distribution of outcomes

associated with any choice which is presented in this study. Under

Robert P. King

this approach, the decision maker's expectations concerning future
levels of critical environmental variables are elicited directly.
Monte Carlo simulation techniques are then used to determine the
effect of these factors on the distribution of outcomes associated
with any particular strategy. The value of this approach is greatly
enhanced by the generalized procedure for the generation of sample
vectors from multivariate distributions with non-normal marginals,
which was developed as part of this study.

With regard to the measurement of decision maker preferences,
shortcomings of both single-valued utility functions and commonly used
efficiency criteria such as first and second degree stochastic
dominance are identified, and a new approach to the measurement of
decision maker preferences is presented. This new procedure permits
the construction of interval measurements of a decision maker's absolute
risk aversion.

Unlike other preference measurement function procedures, it allows
the direct specification of the degree of precision with which pre-
ferences are measured, since the absolute risk aversion interval can
be of any desired width. Interval measurements of this sort can be
used in conjunction with the evaluative criterion of stochastic
dominance with respect to a function to order alternative choices.

The final methodological contribution of this study is the formu-
lation of a generalized risk efficient Monte Carlo programming model,
which combines random search procedures, Monte Carlo simulation, and
evaluation by the criterion of stochastic dominance with respect to

a function within a single analytical framework for the identification

Robert P. King

of preferred choices. This model is flexible and computationally
efficient, and it is well-suited for use in the analysis of a wide
range of practical decision problems.

The methodological tools developed in this study are applied to the
analysis of two related problems. The first is concerned with land
rental and crop production decisions on a small cash grain farm under
conditions of uncertainty with respect to prices, yields, and time
available for fieldwork. In the second problem analyzed, these same
decisions are considered in conjunction with the selection of a
flexible marketing strategy which evaluates forward contracting

strategies over a seven-month period.

For Jane

ACKNOWLEDGEMENTS

For their guidance and assistance, I wish to thank the members
of my thesis committee: Roy Black, Ralph Hepp, Glenn Johnson, and
Lindon Robison. I am especially grateful to Lindon Robison, my
thesis supervisor, whose questions prompted this study and whose
willingness to explore new ideas was a continuing source of help.

I also want to give particular thanks to Warren Vincent, who was my
major professor throughout most of my graduate career. Without the
freedom and the good advice he gave me this study might not have been
undertaken.

Funding for this study was provided by the Michigan Agricultural
Experiment Station. I gratefully acknowledge their support.

Finally, I thank my wife, Jane, who shared all the good and bad
days that come with an undertaking like this one. Her encouragement

and sense of perspective were a constant source of help.

iii

TABLE OF CONTENTS

LIST OF TABLES ................... . . . . .
LIST OF FIGURES ............... . .......
Chapter
I. INTRODUCTION
1.1 Background and Need for the Study .........
1.2 Problem Statement .................
1.3 Objectives of the Study ..............
1.4 Plan for the Remainder of the Study ........

II. PROBLEM FORMULATION IN THE ANALYSIS OF
DECISIONS MADE UNDER UNCERTAINTY
2.1 Introduction . . ........... . ......
2.2 The Identification and Classification .......
of Factors Relevant to the Decision
2.3 The Specification of the Decision to be Made . . . .
2.3.1 The Specification of Feedback Control Rules .

2.3.2 Plans, Decisions, and Actions ..... . . .
2.4 A Formal Statement of the Decision Problem .....
2.5 An Application . . ............ . . . . .

III. THE DETERMINATION OF SUBJECTIVE PROBABILITIES
3.1 Introduction ........... . ........
3.2 Probability Encoding Procedures ..........
3.3 Modelling Stochastic Processes ...........
3.4 An Application ...................

IV. THE MEASUREMENT OF DECISION MAKER PREFERENCES

4.1 Introduction ...................

4.2 The Use of Single Valued Utility Functions to . . .
Represent Decision Maker Preferences

4.3 Efficiency Criteria and the Representation . . . . .
of Decision Maker Preferences

4.4 Stochastic Dominance with Respect .........
to a Function

4.5 An Interval Approach to the Measurement ......
of Decision Maker Preferences

4.6 Implementation of the Procedure . . . . . . . . . .

4.7 An Empirical Test .................

4.8 An Application . . ........... . .....

12
14

21
23
28
33
35

Chapter Page

V. COMPUTATIONAL PROCEDURES FOR THE IDENTIFICATION
OF PREFERRED CHOICES
5.1 Introduction . . . . ............... 130
5. 2 A Review of Existing Computational Procedures . . . 131
5.3 A Generalized Procedure for the Identification . . . 137
of Preferred Choices Under Uncertainty

5.3.1 Generation of a Feasible Management Strategy. 139

5.3.2 Determination of the Distribution of . . . . 146
System Output Levels

5.3.3 The Evaluation of Alternative Strategies . . 147

5.3.4 General Comments on the GREMP Model ..... 151

5.4 An Application .............. . . . . . 156

VI. COMBINED PRODUCTION AND MARKETING DECISIONS BY
CASH GRAIN FARMERS: AN EXTENDED APPLICATION

6.1 Introduction ............ . ..... . . 162
6.2 Problem Formulation . . . ........... . . 166
6.3 The Determination of Subjective .......... 172
Probability Distributions
6.4 Decision Maker Preferences .......... . . 179
6.5 The Identification of Preferred Choices ...... 180
6.6 Further Discussion of the Results . . . . . . . . . 191
6.7 Implications for Further Research . . . ...... 194

VII. SUMMARY AND CONCLUSIONS
7.1 A Review of the Methodological Tools . . . . . . . . 197
Developed in this Study

7.2 Empirical Findings ............... . . 201
7.3 Implications for Future Research .......... 203
APPENDIX A ........................... 207

A GENERALIZED MULTIVARIATE PROCESS GENERATOR

APPENDIX B. . . ................. 240
IMPLEMENTATION OF THE INTERVAL APPROACH TO THE
MEASUREMENT OF DECISION MAKER PREFERENCES

APPENDIX C .................... . . . . . 278
IMPLEMENTATION OF THE GREMP MODEL

LIST OF TABLES

Table

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(00000

as 01 01 01 O1 01 010101
. o o o o o o o

by 01 J: to re

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(ON

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—l

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u—l

Standard Crop Enterprise Budgets ..........
Planting and Harvest Periods and Possible .......

Crops for Each Combination

Average Corn Yield and Moisture Content ....... . .

by Planting and Harvest Date
Average Soybean Yield by Planting and Harvest Date

Percent of Time Available for Fieldwork by . . . . :

Calendar Period for Well Drained Sandy
Loom Soils in the Lenawee, Monroe,
Livingston County Area

Beta Distribution Parameters for Time Available . . . .

for Fieldwork by Calendar Period

Beta Distribution Parameters for Corn . . . .....

and Soybean Price Distributions

Two Possible Management Strategies .........
System Performance under Strategy 1 ..........
System Performance under Strategy 2 ..........

Probability Distribution Associated with .......

Two Alternative Action Strategies

Sample Distribution from a Normal Distribution . . . .

With u = 3000 and o = 1000

Boundary Intervals for Pairs .............

of Sample Distributions

Performance Indicators for Alternative ........

Preference Measures

Efficient Strategies for Decision Maker A .......
Efficient Strategies for Decision Maker B ...... . .
Efficient Strategies for Decision Maker C .......

System Performance under a Sample ...........

Management Strategy

Representative Strategies from the Efficient .....

Sets of Four Decision Makers

System Performance under the Preferred Strategy of.

a Risk Neutral Decision Maker

System Performance under the Representative ......

Strategy for Decision Maker A

System Performance under the Representative ......

Strategy for Decision Maker B

System Performance under the Representative ......

Strategy for Decision Maker C
vi

Lass
37

39
67

68
71

158
159
161
178
182
183
186
188

190

P335:

Parameters of Marginal Beta Distributions ........ 236
Specified Correlation Matrix .............. 236
Sample Correlation Matrix for ......... . . . . . 238
Multivariate Normal Data

Sample Correlation Matrix for . . . . . ......... 238
Multivariate Beta Data

An Example of Preference Data Input ........... 270

for Program UFUNC

vii

LIST OF FIGURES

Figure Page

2.1 A General Scheme for System Identification ....... 16

3.1 A Cumulative Distribution Function . . . . . . ..... 51

3.2 A Cumulative Distribution Function . .......... 62
Based on Five Sample Observations

3.3 Cumulative Distribution Functions for ....... . . . 82
Two Alternative Strategies

4.1 Illustrations of First and Second ..... . . . . . . . 93
Degree Stochastic Dominance

4.2 Cumulative Distribution Functions F(y) and G(y) ..... 102

4.3 A Graph of the Function [G(y)-F(y)] ...... . . . . . 103

4.4 A Sequence of Interval Preference Measurements ..... 110

4.5 Upper and Lower Bound Absolute Risk Aversion ...... 111
Functions Based on Three Interval Measurements

4.6 An Absolute Risk Aversion ................ 113
Measurement Scale

4.7 A Sample Questionnaire ............... . . 119

4.8 Interval Measurements of Absolute Risk ......... 128
Aversion for Three Decision Makers

5.1 A Flow Chart of the GREMP Model ......... . . . . 140

5.2 Sample Observations from the Distribution ........ 148
of Outcomes Associated with Strategy One

5.3 Sample Observations from the Distribution . . . . . . . . 150
of Outcomes Associated with Strategy Two

5.4 Sample Observations from the Distribution ........ 152
of Outcomes Associated with Strategy Three

6.1 Dates for the Application of Forward .......... 173
Contracting Rules

6.2 Price Forecast Line and Observed ......... . . . 175
Control Price Levels for One
State of Nature

6.3 The Expected Net Return Maximizing Strategy . . ..... 193

viii

Figure _9_Pa

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01th

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'—'LD (”N01

d

“HOCDNO‘U'l-bw N

Generation of Random Variates by the . . . . . . . . . . 212
Inverse Transformation Method

A Program for Generating Exponential .......... 214
Random Variates
A Program for Generating Gamma ............. 216
Random Variates
A Program for Generating Beta .............. 219
Random Variates
An Approximate Representation of a ........... 221

Cumulative Distribution Function
Based on Six Known Points

A Table Look-up Function Subprogram ........... 222
A Generalized Univariate Process Generator ....... 224
A Program for Generating Multivariate . . . . ...... 227
Normal Random Variates

A Generalized Multivariate Process Generator . . . . . . 231
A Multivariate Beta Process Generator .......... 234

Sequence of Choices for a Measurement of Absolute . . . . 242
Risk Aversion Based on Three Questions

A Suggested Absolute Risk ................ 244
Aversion Measurement Scale

A Listing of Program NORGEN ............... 245
Sample Output from Program NORGEN ........ . . . . 248
A Listing of Program INTID ............ . . . 250
An Eight Element Measurement Scale ....... . . . . 255
Sample Output for Program INTID ....... . ..... 256
A Three-Stage Hierarchy of Questions .......... 258
Sample Questionnaire .................. 261
A Listing of Program UFUNC ........ . . . . . . . 264
Representation of a Utility Function by ...... . . . 266
Interpolation Between Known Points

An Interval Preference Measurement ........... 268
Utility Functions Associated with Upper and ....... 271
Lower Bound Absolute Risk Aversion Functions

A Listing of Program NSTDO ............... 272
A Sample Output from Program NSTDO ........... 276
General Flow Chart of Program GREMP ........... 279
Statements in GREMP Which Generate ........... 286
Feasible Strategies

A Simple Version of Subroutine CHECK .......... 288
A Simple Version of Subroutine DISGEN .......... 291

ix

CHAPTER I
INTRODUCTION

1.1 Background and Need for the Stugy

 

Uncertainty is that state of knowledge in which the consequences
of actions being considered cannot be specified exactly; it is that
condition in which knowledge is to some degree imperfect or incomplete.1
As such, the presence of uncertainty is a basic fact in nearly all
decision situations.

Though almost always present to some degree, uncertainty is not a
factor which must be considered explicitly in every instance. Often the
consequences of ignoring imperfectionsin knowledge are judged to be
minimal, and decisions can be made as though all relevant factors were
known with certainty. In many other instances, however, when the
outcome of an important choice is highly uncertain, the failure to
consider uncertainty explicitly may not be justifiable.

Uncertainty can have an important impact both on the process by
which decisions are made and on the character of decisions themselves.

Learning, which has no value when knowledge is perfect, becomes a

 

1Knight's (1921) distinction between risk and uncertainty is not
made in this study. General acceptance of Ramsey's (1931) observation
that knowledge of the true probability distribution of a random variable
is not possible obviates the need for this distinction. It is important
to recognize, however, that degrees of uncertainty can vary and that a
decision maker's state of knowledge has an important impact on his
actions (Wald, 1947; Johnson and Lard, 1961).

2
potentially worthwhile activity under uncertainty. As a result, the
decision maker's attention may be focused primarily on the decision of
whether or not to continue learning rather than on the actual choice of
an action to be undertaken.1 When learning stops and a choice of
actions is made, the character of that choice may also differ from that
of one made under certainty. For example, decisions made under uncer-
tainty often take the form of flexible strategies which make forth-
coming actions contingent upon future events that the decision maker
can observe but cannot control.2 Such strategies would be of little
value if knowledge were perfect and all future occurrences could be
known in advance. The presence of uncertainty also affects the character
of the decision rules used to identify a preferred choice. Decision
rules which give explicit consideration to uncertainty generally
require more specific information about decision maker preferences
than is required under certainty, and they must permit the synthesis of
this normative information with probabilistic information about the
possible outcomes of any choice being considered.

Despite these and other impacts of uncertainty in the decision
process, uncertainty is often not considered explicitly in the analysis
of decisions upon which it may have a profound effect. In many instances
the failure to give proper attention to such an important factor is not

due to a lack of recognition of the impacts of uncertainty or to

 

1The role of learning in the decision process is discussed in
Bradford and Johnson (1953) and in Johnson and Lard (1961). More
recently, the work of Aoki (1975), MacRae (1975) and others have dealt
with the "dual-control" problem of learning and setting policies
simultaneously.

2Massé (1962), Cocks (1953), Rae (1971), and Day (1975) have all
stressed the importance of such adaptive decision strategies.

3
inadequacies in the theory of decision making under uncertainty. Rather,
this failure can often be attributed to a lack of workable analytical
techniques which permit the explicit consideration of uncertainty in
the analysis of practical decision problems, techniques which are
flexible enough to allow the application of powerful theoretical results
in a wide range of complex situations.

This study is concerned with the development of an integrated set
of techniques which facilitate the incorporation of explicit considera-
tions of uncertainty into a decision analysis. Particular emphasis is
placed on the development of methodological tools which make the applica-
tion of decision theory based on the expected utility hypothesis more
feasible in a wide range of practical problem solving contexts. The
expected utility hypothesis has been the basis for much of the body of
theory concerned with decision making under uncertainty and has been
used to explain a diverse range of behavioral patterns.1 It is also a
potentially powerful tool for the analysis of decision problems in a
practical context. But for a few notable exceptions such as Grayson
(1960), Howard, Matheson, and North (1972), and Keeney (1973), however,
this body of theory has rarely been applied successfully in the solution
of practical decision problems.

A number of difficulties have limited the usefulness of decision
theory based on the expected utility hypothesis. The expectations and

preferences of decision makers have proved to be difficult to determine

 

1Notable among the theoretical applications of the expected utility
hypothesis are the important early articles by Friedman and Savage (1948),
the more recent work of Samuelson (1967) and Ehrlich and Becker (1972),
and the extensive literature concerned with portfolio selection they
based on the work of Tobin (1958), and Markowitz (1959) and Baumol
(1970), among others.

4
and represent accurately, and, as Zadeh (1973) and Watson, Weiss, and
Donnell (1979) note, it is somewhat paradoxical that extremely precise
decision rules based on the expected utility hypothesis are applied in
situations where relevant information is highly imprecise. Computa-
tional problems associated with the implementation of expected utility
maximizing decision rules are also a source of difficulty. Often they
force the imposition of restrictive assumptions on the way expectations
and preferences are represented and so further limit the theory's
usefulness in an applied context (Anderson, 1975). Finally, as Johnson
(1976), Day (1964), Cyert and March (1963), and others have observed,
much more than the appreciation of a decision rule is involved in the
choice process. Even when other methodological difficulties can be
resolved, decision theory based on the expected utility hypothesis can
be successfully applied only if a better understanding of problems and
the process by which they are resolved is attained.

These difficulties are serious ones, but they stem from problems
with the way decision theory has been applied rather than from the
theory itself. They point to a need to develop methodological tools
which can be used to make decision theory based on the expected utility

hypothesis truly operational in a practical context.

1.2 Problem Statement

In response to this need, this study focuses on the problem of
formulating an integrated set of operational techniques for the analysis
of decision making under uncertainty, techniques which are consistent
with theory based on the expected utility hypothesis and which overcome

a number of the problems encountered in earlier attempts to apply that

5

theory. Within this broad problem, four more specific areas of diffi-
culty can be identified: problem formulation, the determination and
representation of expectations, the measurement of decision maker
preferences, and the identification of preferred choices.

A problem is said to exist when "a condition, situation, or thing
is not as good or is worse (more bad) than it could be," (Johnson,
1976, p. 270). Before the information needed to resolve a problem can
be collected and analyzed, before a course of action can be selected
and implemented, the problem itself must be clearly defined. Problem
formulation is the process by which such a problem definition is
developed. It requires that performance criteria, choice variables, and
relevant factors in the decision situation which are beyond the control
of the decision maker must be defined and that the nature of the decision
to be made must be clearly specified. Despite its importance, problem
formulation is often given relatively little attention. Frequently,
for example, the definition of a problem under consideration is dictated
by the computational tools to be used as aids in its resolution. As a
result, important sources of uncertainty may not be considered and the
special character of decisions made under uncertainty may be ignored.
Despite insights provided by Johnson (l961aL Cyert and March (1963),
Churchman (1968), Day (1971, 1975) and others, then, problem formulation
continues to be a problem in the analysis of decisions made under
uncertainty.

The process of selecting a course of action which will best resolve
a particular problem requires the synthesis of two types of knowledge:

(1) positive knowledge, which pertains to beliefs about what is, what

6

will be, and what can be done; and (2) normative knowledge, which pertains
to beliefs about the goodness or badness of particular conditions,
situations, and things. The product of such a synthesis is prescriptive
knowledge, which allows the decision maker to prescribe or specify the
right strategy or set of actions.1

Expectations concerning the relative likelihood of alternative
events occurring in the future represent an important part of the posi-
tive information required in any applied decision analysis. A number
of methodological problems arise in connection with the determination
and representation of expectations. Frequently, they are not well
formulated in the mind of the decision maker, who may not be accustomed
to thinking in probabilistic terms or may simply not be familiar with
a particular factor which may have a significant impact on the conse-
quences of his choice. Even when the decision maker's expectations are
well formulated, problems may arise because he is unable to express them
in a form which is useful analytically. Other difficulties may stem
from the fact that the process by which the outcomes of particular
actions are determined may be so complex that it cannot be comprehended
as a unified whole. Given these problems, there is a need for proce-
dures which help the decision maker structure his own thoughts and
help him to use information from more expert sources, a need for tech-

niques which allow the decision maker to break down complex processes

 

1The distinctions between "right" and "wrong" and "good" and “bad"
made by Lewis (1955), is an important one. The adjectives "right" and
"wrong" refer to the nature of an act, while "good“ and "bad" refer to
the conditions existing prior to an act or to its consequences. To say
a condition or consequence is good is the statement of a normative
belief. To say that an act is right is the statement of a prescriptive
belief which is founded both on positive information concerning the con-
sequences of the act and on normative information pertaining to the
quality of goodness of the consequences.

 

_. A

7
into more comprehensible sub-processes about which expectations can be
more easily formulated and then reintegrate that information for use
in the decision analysis.

Information on decision maker preferences is the primary normative
input in any decision analysis. Problems associated with the measure-
ment and representation of preferences also cause serious difficulties
in the analysis of decisions made under uncertainty. Currently avail-
able measurement techniques are used to construct single valued utility
functions, which are precise but often inaccurate representations of
preference, and many place little faith in them. Efficiency criteria
based on stochastic dominance, on the other hand, require little
specific information about the decision maker's preferences, but they
often fail to order choices and may not eliminate enough alternatives
when a large number must be evaluated. These difficulties indicate
that there is a need for preference measurement techniques which are
more reliable and easier to use in an applied context.

The fourth major area of difficulty is that of identifying a pre—
ferred choice or set of choices from what may be an infinitely large
number of alternatives. This requires the synthesis of both positive
and normative information, the simultaneous consideration of both
expectations and preferences. Mathematical programing techniques are
commonly used as computational aids in the solution of complex quanti-
tative problems. They are best suited, however, for use in situations
where uncertainty is not a major factor. As Anderson (1975) notes,
the use of mathematical programming in the analysis of decisions made
under uncertainty often requires that rather severe restrictions be

placed on the manner in which a decision problem is posed and in the way

8

information on expectations and preferences is represented. These

difficulties point to the need for more flexible computational procedures.

1.3 Objectives of the Study

The problems which motivate this study are primarily methodological.

They have important implications, however, for the analysis of decision

problems in a practical context. In response to them, the objectives of

the study are:

1.

To present a framework for problem formulation which can
serve as a guide in the identification and structuring of
information required for the analysis of decisions made
under uncertainty.

To review procedures used in the elicitation of information
on decision maker preferences and to refine techniques for
the determination of probability distribution for outcomes
which are the result of complex processes affected by a
variety of stochastic factors.

To develop and test a technique for the measurement of
decision maker preferences which is well suited for use

in an applied context and which overcomes some of the
difficulties associated with other preference measurement
procedures.

To formulate and make operational a computational proce-
dure for the identification of preferred choices which is
flexible enough to be used in the analysis of a wide

range of practical problems and which imposes few restric—
tions on problem formulation or on the representation of

decision maker expectations and preferences.

9
The fulfillment of each of these four objectives contributes to
the primary purpose of this study, which is to develop an integrated
set of techniques for use in the analysis of decisions made under
uncertainty. Emphasis should be placed on the fact that the usefulness
of the techniques presented below is greatly enhanced by their having

been combined within a single methodological framework.

1.4 Plan for the Remainder of the Study

The principle objectives of the study are addressed in the next
four chapters. Problem formulation is the subject of Chapter II. Par-
ticular emphasis is given to the usefulness of system identification
(Manetsch and Park, 1977a) as an aid in structuring information in a
practical decision context. The need to recognize the importance of
opportunities for learning and adaptive behavior and the impact such
opportunities have on the character of decisions is also stressed.

Techniques for determining decision maker expectations are described
in Chapter III. Procedures for eliciting information on subjective
probability distributions are reviewed, and the use of Monte Carlo
simulation techniques to model the performance of complex stochastic
systems for which outcome distributions cannot be determined analytically
is discussed. The value of this approach is greatly enhanced by the
generalized multivariate process generator developed as part of this
study. Described in detail in Appendix A, this analytical tool can be
used to model multivariate probability distributions defined by marginals

of any form and by any positive definite correlation matrix.

10

The measurement of decision maker preferences is the subject of
Chapter IV.1 Procedures used to derive single valued utility functions
are first reviewed, and several commonly used efficiency criteria are
discussed as possible alternatives to the use of single valued utility
functions in applied decision analyses.2 A more general and more
powerful efficiency criterion, stochastic dominance with respect to a
function (Meyer, 1977a), is then described and a new approach to the
measurement of decision maker preferences, developed as part of this
study for use in conjunction with this criterion, is introduced. This
new approach allows the analyst to specify the degree of precision with
which decision maker preferences are measured. Results of an experi-
mental test of this technique are also presented. They demonstrate its
flexibility and its predictive power.

Computational procedures for the identification of preferred
choices are the subject of Chapter V. Uses of mathematical programming
techniques in the analysis of decisions made under uncertainty are
reviewed first, and the major shortcomings of these techniques are
identified. A new procedure for the identification of preferred choices
which combines random search methods, simulation techniques, and evalua-

tion by the criterion of stochastic dominance with respect to a function

 

1Though the importance of multiple objectives in many decision
situations is recognized, attention in this study focuses entirely on
preference relationships which depend only on the level of a single
performance criterion.

2Computer programs used in the implementation of this technique
are listed in Appendix B.

 

11
is introduced and described in detail.1 This computational tool is
remarkably flexible, placing few restrictions on the way probabilities
and preferences are represented or on the general form of the problem
to be solved.

Two related examples are used to illustrate the techniques developed
in this study. Both are concerned with decisions affecting the opera-
tion of a southeastern Michigan cash grain farm. The simpler of the two
examples, which focuses on land rental and crop mix decisions under price
and yield uncertainty, is discussed at the end of the four methodological
chapters. It is used to demonstrate how techniques for problem formula-
tion, the determination of expectations, the measurement of preferences,
and the identification of preferred choices can actually be applied.

The second example, which considers the selection of a marketing strategy
in conjunction with production and land rental decisions, is the subject
of Chapter VI. Again prices and yields are uncertain. The marketing
strategies considered may include cash sales at harvest, forward
contracting,or any combination of these. Emphasis is placed on the
adaptive nature of such strategies and on the impact of preferences on
the combined production-marketing strategy selected.

Finally, in Chapter VII the strengths and weaknesses of the inte-
grated set of techniques developed in this study will be discussed.
Particular attention will be given to an evaluation of the range of
applications for which these techniques can be of use and to the identi-

fication of areas where further methodological improvements are needed.

 

1The computer program which implements this decision model is
presented in Appendix C.

 

CHAPTER II

PROBLEM FORMULATION IN THE ANALYSIS OF
DECISIONS MADE UNDER UNCERTAINTY

2.1 Introduction

 

A problem exists "when an indeterminate situation, present or
projected, is regarded as unsatisfactory and a more satisfactory alter-
native situation is sought" (Johnson and Zerby, 1973, p. 3). Management
is the process by which problems of a practical nature are resolved.1
In describing the managerial process, Johnson (1976) has identified six
major types of activities: problem definition, observation, analysis,
decision, execution, and responsibility bearing. This study is concerned
primarily with the development of analytical tools which can aid the
decision maker during the analysis and decision phases of the management
process--tools which facilitate the determination of distributions of
outcomes associated with alternative actions, the measurement of decision
maker preferences, and the application of decision rules used to identify
preferred choices. These tools can be of little use in a practical

context, however, if the problems to which they are applied have not

been clearly and correctly specified.

 

1As Johnson and Zerby (1973) note, problems can be practical or
theoretical in nature. Practical problems are those which are related
to the choice of an action and so demand some form of resolution.
Theoretical problems, on the other hand, are not linked to a definite
action ghich can be fixed in space and time and so may never be fully
reso ve .

12

 

13

Problem formulation is the process by which a problem is defined
and structured for analysis. It is the process by which features of the
problematic situation judged to have an important impact on the choice
to be made are identified and classified and the process by which the
nature of the decision to be made is specified. The product of this
process should serve as a guide in the collection of additional infor-
mation, should provide a framework for the organization of that infor-
mation, and should help structure the analysis which leads to a decision.
As such, problem formulation is a critical activity within any applied
decision analysis. Major expenditures of resources for information
gathering and analysis may be required before a choice can be made in
some decision situations. These resources can be used effectively only
if the problem under consideration is clearly and correctly defined. A
carefully determined solution to an irrelevant problem is of little use.

Problem formulation is the subject of this chapter. The discussion
in subsequent sections focuses on two important aspects of the process
of problem formulation: the identification and classification of
variables relevant to the analysis of a particular problem and the
actual specification of what is to be decided. With respect to both of
these activities, emphasis is placed on the need to recognize the
dynamic character of many decision situations and the impact it has on
the choice process. The need to recognize the role of learning and
the effect it has both on the way problems are formulated and the manner
in which decisions are made is also stressed. The purpose of this
discussion is not to introduce new concepts or to develop a comprehensive

procedure for problem formulation. Rather, it is to restate some

 

14
valuable observations made by others and to present a general view of
decision problems and a working vocabulary for the discussion of them
that can provide insights into the process of problem formulation.

2.2 The Identification and Classification
of Factors Relevant to the Decision

 

At the outset of a decision analysis the problem under consideration
may be only vaguely defined in the mind of the decision maker. To better
understand the nature of the problem, one of the first tasks usually
undertaken is the identification of factors judged to have an important
impact on the choice to be made. Not only does this help to clarify the
problem, but it also establishes a set of variables which can be the
focus of observations and analysis. As more is learned about the pro-
blem, this set of relevant factors is, of course, repeatedly revised.

Efforts to identify the important factors in a particular decision
situation are facilitated by the presence of a general classificatory
framework which suggests broad types of variables that should be con-
sidered. Such a framework is presented by Manetsch and Park (1977a) in
their discussion of system identification. A system can be defined as
a collection of objects or processes which interact to perform a given
function or set of functions. System identification is a generalized
scheme for structuring information about the characteristics of par-
ticular systems. It is a particularly valuable classificatory frame-
work in a practical problem solving context because it is well suited
for the description of static as well as dynamic decision problems and
because it encourages the explicit consideration of the sources of

uncertainty in any particular situation.

 

15

Manetsch and Park identify five broad classes of variables which
should be considered in any decision situation: system outputs, con-
trollable system inputs, exogenous system inputs, system state variables,
and system design parameters. These are the major elements in Figure 2.1.
Before discussing each of these categories it should be noted that a
"system" is a concept rather than an actual entity. The definition of
a system-~the specification of its functions and component processes--
depends on the purposes for which it is being considered. In the analysis
of a particular decision, the function of the system considered should
be to determine or at least affect the situation or condition which is
judged to be problematic, and the system's component processes should
include all those which have an important impact on that situation. For
example, the problem facing a farm family may be that its standard of
living is unacceptably low. In trying to improve this unsatisfactory
situation they will want to consider the system whose functions it is to
provide them with the resources for obtaining food, clothing, shelter,
and other necessities. The component processes within that system
would be that set of processes by which such resources are generated--a
set which might include farm production and marketing, off-farm work,
and public assistance.

System identification begins with the specification of system
output variables. System outputs are the products of the processes
which comprise a particular system; and system output variables, which
measure levels of system outputs, should serve as indicators of the
degree to which the system under consideration performs its designated

functions. They should be the basis for a reliable representation of

16

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17
all relevant features of system performance, providing information
about both the desired and undesired impacts of any choice being con-
sidered. For this reason, considerable care should be taken in identifying
system output variables. In some situations all relevant information may
be conveyed by a single output measure, but in other instances more than
one system output variable is needed to adequately represent system
performance. For the family in our example, net annual household income
may be an adequate measure of system performance; but it may be necessary
to specify other system output variables if, for example, there are
important costs associated with the acceptance of public assistance
which are not reflected by the level of net annual income.

System output levels are determined by inputs to the system and by
its structure. System inputs are factors or stimuli emanating from
outside the system which affect its performance. They can be classified
as controllable or exogenous. Controllable system inputs are those for
which a level can, to some degree, be specified by the decision maker.
The level of a controllable system input may represent an amount of some
physical factor of production flowing into one of the processes within
the system or it may specify a level of some well defined activity. In
our example the set of controllable system input variables might include
designations of levels for each farm production enterprise and for
hours of off-farm work activity and a binary variable indicating parti-
cipation or non-participation in public assistance programs.

The levels of exogenous system inputs cannot be determined by the
decision maker. Rather, they are determined by the system environment,

a set of processes which affect system performance but are not, in turn,

18
significantly affected by the system's behavior. The set of exogenous
system inputs in our example might include farm product prices, levels
of rainfall, wage rates, and levels of public assistance. All of these
have a potentially important impact on the family's standard of living
but are beyond its control. The distinction between the system and
its environment is not always evident, nor is it necessarily fixed.
It depends on the problem under consideration and on the power of the
decision maker. The distinction is an important one, however, especially
in the analysis of decisions made under uncertainty, since stochastic
factors in the environment can be viewed as the primary source of uncer-
tainty in most decision situations.

System structure determines the relationship between system inputs
and system outputs. The structure of a system is described by system
state variables and by system design parameters. State variables are
descriptors of the state or condition of a system at any point in time.
In general system outputs can be viewed as functions of the system's
state at some specified time or as functions of the system's state
through time. In addition to determining system output levels, the
state of the system may also affect the range of allowable levels for
controllable system inputs. Therefore, it is important to give careful
consideration to the specification of system state variables. In our
example the set of system state variables related to crop production
processes might include current levels of acreage planted to each crop
grown, current amount of each crop harvested, current crop production
expenses incurred, and current crop sales receipts.

System design parameters define the relationship between inputs to

the system and its resultant state. As such they describe the processes

 

19
which comprise the system. With regard to the crop production pro-
cesses in our example, the set of system design parameters could include
variable production costs per acre of each crop grown, time required to
plant or harvest an acre of each crop grown, and parameters indicating
tillage practices and standard procedures concerning the order of pro-
cedures for the planting and harvesting of each crop. System design
parameters have an important impact on system performance, and in some
instances they can be altered by the decision maker. For example, a
change in tillage practices may significantly affect both timeliness and
crop yields for the farm in our example. Such a change, however, may be
costly. As a result, alterations in system design are usually under-
taken only in response to serious problems which cannot be resolved by
other means.

In any decision analysis variables in each of these categories
should be identified. The analysis itself focuses on the specification
of a strategy to be undertaken to resolve the problem being considered.
A strategy is defined by desired levels for controllable system input
variables and by the new values of any system design parameters that
are to be changed. Manetsch and Park (1977a) define “management" or
"control" as the process by which desired levels for controllable system
input levels and "design" or "planning" as the process by which specifi-
cations for system structure are made.1 In general there are limits on
the range of strategies that can be undertaken in any particular situa-

tion. As indicated earlier in Figure 2.1, the state of the system,

 

1This concept of management is a much more narrow one than that used
in this study. The management of a system is but one activity within the
broader managerial process.

20
system design parameters, and factors in the environment all affect
the control process. They impose constraints which restrict the
allowable range of values for controllable system input variables.
Though not shown explicitly in Figure 2.1, similar constraints also
limit design changes. As the process of system identification proceeds,
then, it is also important to identify those factors which restrict the
range of available choices.

To this point in the discussion systems have been viewed as
essentially unidirectional processes which convert inputs to outputs.
This view is limited because it fails to recognize the fact that as
strategies are implemented there may be opportunities for learning and
for revising chosen plans of action on the basis of newly acquired
information. Introduction of the concept of feedback into our view of

system identification helps to overcome this limitation. Feedback is
the return flow of information (both positive and normative) on the
s tate of the system and into environment to the central process unit.
Recognition of the feedback loop between the set of system state
Va riables and the control unit changes our conceptualization of the
process by which inputs are transformed to outputs from one which is
es sentially unidirectional and disjointed to one in which this process
'3 S viewed as a continuous cycle or closed loop. This is a more
r‘ea listic way of representing the context in which decisions are made,
especially in situations where the impact of uncertainty is important.
“081: decisions are not made in isolation, nor are they implemented
instantaneously. Rather, they are made sequentially, and the outcome
01: One decision affects the opportunity set from which future choices

can be made. Furthermore, because decisions are implemented over a

 

21
period of time, there are often opportunities to revise them. In such
a context, learning is of considerable importance, and feedback in the
medium through which learning takes place. The recognition of such
informational flows can have an important impact on the specification
of what is to be decided. Therefore, there is a need to consider such
factors during the process of problem formulation.

Finally, it should be noted that the identification and classifica-
tion of the important factors in a particular decision situation is, in
itself, a learning process. A decision maker's view of a problem and of
the set of processes by which that problem can be resolved is repeatedly
revised and made more complete. This kind of learning continues until
1Further efforts are judged not to be worthwhile or until the decision

lwlaker is forced to take an action.1 The degree of detail included in
the description of a particular system, then, depends on the usefulness
of that detail in helping the decision maker determine his preferred

course of action.

2? —. :3 The Specification of the Decision to be Made
As was noted in Chapter I, the presence of uncertainty can have
Ell”! important impact on the character of decisions. A11 situations
‘3 r‘I\rolving uncertainty are, in a sense, dynamic, since they are charac-
tel"‘ized by changes in the decision maker's knowledge through time.
One makes a choice and begins to act in the present, but only later do
the consequences of one's actions come to be known. Often the outcomes

\

1Johnson and Lard (1961) relate the decision of whether or not to
COHt‘inue learning to five more formally defined knowledge situations:
eawning, forced learning, forced action, inaction, and risk.

 

22
associated with a particular strategy unfold over an extended period of
time, and there are opportunities for learning as the strategy is
implemented. The existence of such opportunities may make it desirable
to introduce flexibility into the specification of a decision strategy.
The strategy becomes a conditional plan--a set of contingency rules
which direct actions on the basis of currently available information.
In this way the importance of future opportunities for learning is
recognized explicitly when a choice is made. Though not all decisions
made under uncertainty take this form, many do. The discussion in this
section will focus, in part, on the specification of flexible strategies
based on feedback control rules.

Another important consideration which affects the character of
decisions is the length of the planning horizon. Because current
choices have an impact on future opportunities, it is often necessary

to formulate strategies which extend into the future. When knowledge
‘5 s perfect,it is possible to specify future actions extending over an
infinite planning horizon. When knowledge is not perfect and reliable
in formation about future events can be attained only at considerable
Cos t, on the other hand, the time horizon for which it is worthwhile
to formulate a plan may be shortened considerably (Modigliani and
CO hen, 1961; Kleindorfer and Kunreuther, 1978). Specification of the
re‘! evant planning horizon and the distinctions among a plan, a decision,
and an action, then, will also be considered in this section.

\

t _ 1Dreyfus (1968) demonstrates that flexible strategies are superior
° ‘Inflexible ones in multistage decision problems and uncertainty.

 

23
Before beginning the discussion it should be noted that the

specification of what is to be decided, like the identification and
classification of the important factors in a decision situation, is a
process which continues throughout a decision analysis. As more is
learned about the problem at hand, as observation and analysis continue,
one's conceptualization of what is to be decided is repeatedly revised,
and refinements in this aspect of problem formulation continue until

further changes are not worthwhile.

2.3.1 The Specification of Feedback Control Rules
A feedback control rule is a rule for processing current informa-
‘tion on the state of a system and its environment in order to repeatedly
thdate desired controllable system input levels. Feedback control rules
1c:an take a variety of forms. They can be as simple as the statement,

' JIf the forward contract price of corn is below $2.00 on May 1, I'll
comply with the federal set aside program requirements; if it's above
$2-00, I won't participate." Alternatively, they can be complex
‘f’c.llnctions of several state variables. In specifying a feedback control

Pu 1e to determine the level of some controllable system input variable
31‘12» any point in time, one must be concerned with the identification of
State variables which can be expected to have a significant impact on
the desired level of the controllable system input being considered,
“‘3 th the form of the rule, and with the actual parameters of the rule.
The simple feedback control rule stated above determines the level

01" a binary controllable system input variable which has a value of 0

if the operator chooses not to participate and 1 if he chooses to par-

t1C‘1 pate. The only state variable affecting the choice of a level for

24
this variable is the forward contract price of corn. The form of the
rule is that of an "if-then” statement, and the parameters of the rule
are a forward contract price of corn—-$2.00--and a date when a decision
will be made--May 1. It should be noted that the effectiveness of the
rule is affected by the variables considered, by its form, and by its
parameters. All affect its impact on system performance, and in
selecting a preferred management one may be concerned with the specifi-
cation of all three factors. It should also be noted that this rule is
but one component of a management strategy which might include other
feedback control rules and direct specifications of some controllable
system inputs.
The specification of feedback control rules can be a difficult
'task in more complex situations. In some special cases optimal control
rrnethods can be used to derive feedback control rules which optimize
.ssjrstem performance, but the presence of uncertainty greatly complicates
the application of these analytical tools.2 Often, then, it is necessary
to specify a general form of a control rule and perform experiments to
determine its parameters. A simple example related to forward contracting
551‘t2rrategies by cash grain farmers should help to explain how a reasonable
1":<>rm for a feedback control rule can be determined.
Let v(t) be a controllable system input which specifies the number
(3‘7: bushels of corn which, at time t, the operator contracts to deliver
-———_._2
1In reality, of course, other variables may affect this decision.
2Optimal control techniques are discussed in detail in Aoki (1967),

Karreman (1968), Sage (1968), and Kirk (1970). The first two are con-
CeY‘ned with optimal control decisions under uncertainty.

 

 

25
at harvest. Let x(t) be the total number of bushels contracted prior
to time t--i.e. x(t) = :5: v(t)--and let d(t) be the total number of
bushels the operator desires to have contracted at time t. The level of
v(t) is defined by the following expression:
v(t) = d(t)-x(t) if d(t)>x(t) 2.1
0 otherwise
This is a feedback control rule which specifies the level of v(t) at
any point in time. That level is equal to the difference between
desired and actual contracting levels to date. Since a contract, once
Inade. cannot be dissolved, however, levels of v(t) are restricted to
rwon-negative values.
Actual contracting levels can be observed, but desired contracting
‘1 evels cannot be.1 In order to implement this rule, then, a more
complete specification is needed, a specification which defines desired
¢::<311tracting levels as a function of observable variables.
The projected size of the operator's corn harvest, h(t), is one
f‘a ctor upon which the desired level of contracting is expected to
depend. Initially, then, d(t) might be defined by the expression
d(t) = chit) 2.2
Nb ‘3 ch implies that the desired level of contracting is some specified
fEr‘actionus, of the projected corn harvest. This is not a very satis-
Fa(:‘lzory specification, however, because the operator's estimate of how
ma ny bushels of corn he expects to harvest may not change much over the
per“Tod during which the rule is to be applied. Furthermore, this
Specification ignores the impact of prices on desired contracting levels.

X

1If the operator knew his desired contracting level, he would have
“0 need for this rule.

 

 

26

An additional factor which should be considered, then, is the
sign and magnitude of the difference between the current forward contract
price, c(t), and the farmer's current estimate of the expected cash
price at harvest, e(t).1 The more the contract price is above the
expected cash price,the more the farmer will wish to have contracted;
the more it is below the expected cash price,the less he will wish to
have contracted. Therefore, a revised specification of d(t) could be:

d(t) = h(t)[a(c(t)-e(t))] 2.3
where a is a positive constant.

The desired level of contracting may also depend on movements of
the contract price. If it is rising rapidly the farmer may wish to
cielay the commitment of an additional portion of his crop to a forward
contract. On the other hand if the contract price is falling he may

wish to lock in a relatively high price. To reflect this, the specifi-
cation of d(t) can be further revised so that

d(t) = h(t)[a(c(t)-e(t))+s “—3911 2.4

Where dc(t)/dt is the rate of change in the contract price and B is a
negative constant. The interaction between the two terms in brackets
Should be noted. If the contract price is falling but is less than the
expected cash price, the first term should override the second, and no
new contracting will be desired. On the other hand, if the contract
pr-i ce is both above the expected cash price and falling, the two terms
rei nforce each other to raise the desired contract level.

\

1The expected cash price at harvest, e(t), can be determined by
3“ EXpectations model, the complexity of which can be determined by
t e requirements of the particular decision situation.

27
A third factor which could influence a forward contracting strategy
is the percentage of desired corn acreage actually planted at time t,
p(t). Fearing unusually bad weather which could delay or prevent the
planting of some of the specified acreage, some operators may hesitate
to contract much of their projected harvest until planting is nearly
complete. Similarly, many farm operators, fearing the consequences of
a sharp downturn in prices, desire toihave some of their crop contracted
in nearly all situations. Therefore, the specification of d(t) can be
revised once again to become
d(t) = h(t)[a(c(t)-e(t))+8 d—gitfl mun 2.5
vvhere y is a positive constant.
One other restriction on d(t) should be noted. In some situations
'tzhe desired contracting level implied by the specification above may be
unacceptably high either for the farm operator or for the manager of
the local elevator. Therefore, it may be advisable to establish an
Upper bound on d(t). This can be considered to be a prespecified con-
ES‘t:rr~aint or it can be treated as a parameter. In this case the upper
130 und on d(t) will be set at 1.5 h(t), which implies that a maximum of
1 5 0 percent if the projected crop can be contracted.1
By substituting the right hand side of equation 2.5 for d(t) in
equation 2.1, the feedback control rule can be expressed in a form
Wh‘i ch contains only observable variables. This rule has been presented
On'ly as a relatively simple example of the types of rules that can be
Specified. In some situations it may be desirable to consider more

\

1Due to the form of the feedback control rule no lower bound on
d(ll) is needed. Such a lower bound could be specified, however, if
“acessary.

28

factors and to experiment further with functional form. The power of

such a rule should be evident, however. Once a functional form has been

specified and values for the three parameters have been selected, an
adaptive marketing strategy for an entire year has been established, a
strategy which only requires information which is readily available

to any farm operator and which allows the decision maker to take advan-

tage of opportunities to learn. In many decision situations such a

ru] e may be preferable to a management strategy which specifies an

1"1’1 exible marketing strategy prior to the planting season.

2.3.2 Plans, Decisions, and Actions

The preceding discussion has shown how the incorporation of feed-
back control rules into the specification of an action strategy intro-

dUCes flexibility into the concept of what is to be decided in a

Particular situation. In attempting to gain a better understanding of

the general characteristics of decision problems involving uncertainty
and in giving further consideration to the basic question of what con-
st‘i tutes an action choice, it is also important to draw clear distinc-
ti Ons among the three related concepts of a plan, a decision, and an
acti on. The distinctions made here parallel those made by Modigliani
and Cohen (1951) and by Day (1971).

A plan is a strategy for controlling system performance through
ma"Iagement or design which extends into an uncertain future--a strategy

has«ed on information about the current situation and on expectations

torIcerning future events. In general plans can be altered. Such

a] terations may be costly, however, because resources must be expended

to gather and analyze new information and to reformulate the plan

 

29
itsel f and because actions undertaken to implement the initial phases
of a p‘l an may restrict the opportunities open to a decision maker.1

A decision, on the other hand, is a choice which is essentially
irreversible. It is that part of a plan which is to be implemented
before further planning is undertaken. As such, decisions are the
desired output of the analytical process which is of primary concern in
this study.

Fi nally, an action is the realization of a decision. In a world
freer <31F uncertainty, decisions and actions would be effectively identical.
IO'DCJSS‘t: instances, however, events that cannot be known with certainty
at tt‘ia' 'time when decisions are made affect the extent to which they can
be i"'IID‘I emented and, of course, the outcomes associated with them. What
15 "EEEIT1 ized may not be what was chosen. It is necessary, then, to
dIStVi rIguish between decisions and actions.

'Trlnere is a crucial interplay among these three activities, an
inte"D'lay which must be recognized during the formulation of decision
prob1 ems. Though the primary focus of a decision analysis is on the

“‘0" Ce of actions to be undertaken, one must be aware that the outcomes
of“ (itarrent decisions affect the opportunity set which circumscribes
futdare decisions. Therefore, it is often necessary to formulate plans
ft’T‘ periods extending beyond the immediate period in which decisions are
tc’ be implemented. In the multiperiod model of decision making under

conditions of perfect knowledge developed by Hicks in Value and Capital

 

\

1This second point is demonstrated by Johnson (l961b)and Johnson
and Quance (1972) in their discussions of asset fixity.

 

30
(1946) the planning horizon is of infinite length, since an event at
any time can have an effect on the total flow of system outputs. In
most cases, however, information is not perfect, and Modigliani and
Cohen (1961) have observed that in an uncertain context forecasts are
subject to error and reliable information about future events can be
attained only at a cost--a cost which is directly related to the degree
of uncertainty and inversely related to the time proximity of the future
event. In such situations it is not worthwhile to formulate plans over
an infinite horizon. Rather, the planning horizon should be extended

1 Fur-

only to the point where current decisions cease to be affected.
thermore, actions at the end of the planning horizon need not be planned
in as great a detail as those at the beginning. In formulating decision
problems, then, it is necessary to determine the appropriate length of
the planning horizon.

The act of planning, which involves the collection and analysis
of information, is expensive, however, and it is also important to
consider how often plans should be reformulated. If the costs of
planning are high or if the benefits from it are comparatively low,
frequent revision of plans may not be worthwhile. Therefore, when
formulating decision problems, it is also necessary to consider the
length of what might be termed the decision horizon-—the length of

time over which decisions apply and replanning does not take place.

The length of the decision horizon has an important impact on the

 

1Kleindorfer and Kunreuther (1978) note that the cost of planning
and the length of the planning horizon depend on the costs of fore-
casting future events, the degree of uncertainty, and the computational
costs associated with the determination of an optimal plan.

 

31
character of decisions themselves. When it is short,the importance of
learning through feedback may be minimal, and a valid decision strategy
would specify levels for all controllable system inputs over the entire
decision horizon. When the decision horizon is long, however, imple-
mentation of a decision strategy may involve a series of actions which
are affected by factors that cannot be known with certainty at the
time when decisions are made. In such instances opportunities for
learning may exist, but the reformulation of plans on the basis of new
information is not worthwhile. As a result, the decision may be more
concerned with the choice of adaptive decision rules to be followed
over the entire decision horizon than with the determination of desired
levels for all choice variables over that period. Relating this to the
scheme of system identification, the management process unit is viewed
as a controller which directs system performance through the applica-
tion of feedback control rules.

In formulating rules of this sort, one must also consider what can
be termed an action horizon--the length of time between successive
reassessments of the current situation and expectations for the future
and reapplications of the adaptive decision rule. The action horizon
corresponds to the time increment embodied in a feedback loop. Its
desired length will depend on the cost of monitoring the state of the
system and the environment and on the relative costs and returns of
applying the feedback control rule. As such, the length of the action
horizon also has an impact on the nature of decisions, and attention

should be given to its specification in the process of problem formulation.

 

 

32

The following example should help to clarify the distinctions among
plans, decisions, and actions. It should also help to demonstrate the
importance of making these distinctions when formulating decision pro-
blems.

Production and investment decisions made by farmers in any given
year have an impact on their operation for a number of subsequent years.
They affect future cropping patterns, levels of available resources, and
cash flow requirements, though their exact impact cannot be known in
advance due to uncertainties with respect to a number of environmental
factors beyond the control of the individual decision maker. Farmers
find it desirable to formulate production and investment plans, then,
but the high degree of uncertainty they face may cause them to limit
their planning horizon to, perhaps, three years. Once a farmer has
begun to implement his p1an--once he has purchased seed, fertilizer,
and other inputs needed to grow the first year's crops and, perhaps,
new land or machinery--a decision has been made and in most cases exten-
sive revision of his plan will not be worthwhile until the end of the
crop year. It can be said, then, that the decision horizon is approxi-
mately one year. The farmer's decision should not be considered to be
a rigid strategy which defines his actions for each day of that year,
however. Rather, it is a set of specified levels for major controllable
system inputs and a set of adaptive decision rules which structure

future efforts to collect and analyze information and direct his actions

in response to changes in the state of his operation and the environment.

For example, a management strategy could be comprised in part of a set
of desired acreage levels for each cropping activity and a set of adap-

tive rules which automatically adjust the crop plan in response to

33
information on current acreage planted and changes in relative prices.
A simple adaptive rule might be to shift unplanted corn acreage to
soybeans if the forward contract price for corn is below $2.10 and the
contract price for beans is higher than $6.50. If such a rule were
applied weekly, the action horizon would be said to be one week.

It is important to recognize the impact of each of these activities--
planning, decision making, and action--on the conceptualization of the
decision problem. Failure to recognize the need to plan may lead to
decisions which, while beneficial in the short run, have a harmful long-
run impact on the system. Similarly, it must also be recognized that
planning itself is expensive and that in most cases a period of time
exists over which extensive plan revision is not worthwhile. A portion
of any plan, then, can be viewed as a decision which will not be altered.
Finally, recognition of the fact that uncertain aspects of the environ-
ment are likely to affect the implementation of any decision and that
opportunities for learning and adaptive behavior exist when the decision
horizon is relatively long leads to the conclusion that in many instances

decision makers should choose flexible rather than inflexible strategies.

2.4 A Formal Statement of the Decision Problem

 

In the analysis of complex decision problems it is often
desirable to make a formal statement of the problem to be resolved.
When mathematical decision aids such as those developed in this study
are used in the decision analysis, this may be a necessity. In the
most general terms, the basic decision problem under uncertainty can be
stated in the following manner: identify a feasible action strategy

which results in system performance over a specified time horizon that

34
can be considered optimal according to some relevant evaluative cri-
terion. System performance is measured by system output variables,
at least some of which are stochastic. An action strategy is defined
by a set of controllable system input levels and/or by a set of control
rules which determine the levels.1 This is, in essence, the statement
of a stochastic optimal control problem. Though it may be impossible
or prohibitively expensive to find the truly optimal solution to such
a problem in most decision situations, this is a generalized formal
statement of the problems decision makers seek to resolve. It is a
problem which requires information on the physical, human, and institu-
tional realities of the context in which choices are made, assessments
of probabilities associated with stochastic events beyond the central
of the decision maker, and an understanding of the decision maker's
normative beliefs if it is to be resolved successfully. As the process
of problem formulation continues, a more comprehensive understanding
of the problem being analyzed should be gained. This understanding of

the problem should serve as a guide throughout the decision analysis.

 

1Symbolically, the problem is

max u = h(y<1).1> + 73 g(y(t),t)dt
st

y(t)=f(X(t).t)
x(t)=a(X(t).v(t).e(t).t)
r(X(t),x(t).t)Ss
5(U(t).tlfv

where y(t) is a vector of system output variables, x(t) is a vector of
system state variables, v(t) is a vector of controllable system inputs,
e(t) is a vector of exogenous system inputs, and a is a function com-
prised of system design parameters. The final two constraints limit
the set of allowable states and the set of allowable controllable input
levels. The elements of the vector v(t) are the choice variables in
this problem.

35
2.5 An Application

In this final section,the concepts related to problem formulation
presented earlier are applied to a more concrete decision problem. The
example introduced here will also be used to illustrate the techniques
developed in subsequent chapters. This is not an actual case example.
Rather, it might be termed a synthetic case study, since it synthesizes
circumstances and concerns common to many individuals' situations.

The decision maker in our example is the operator of a relatively
small cash grain farm in southeastern Michigan. He owns 240 acres of
tillable land on which he grows corn and soybeans. He is heavily in
debt, with interest and principle repayment commitments on long and
intermediate term debts of $35,000 per year. Except for approximately
$6,000 income from off farm work by the operator and his wife, all of
the family's income is derived from the farming operation. If the
family's level of income is insufficient to meet debt repayment commit-
ments and family living expenses, they face the prospect of refinancing
some loans or of being forced out of farming altogether. In 1978 their
income was low, and they relied in part on savings to cover expenses.

The operator views his current situation as uncertain and unsatis-
factory. He feels a strong need for a higher, less uncertain level of
income in the year to come. Though he has other needs and desires,
this is his primary concern. The problem to be analyzed, then, is that
of identifying a strategy which best provides a level of income adequate
to meet debt repayment obligations and family living expenses. The
operator believes this problem is a serious one, and he is willing to
expend the resources required to undertake a careful analysis of his

alternatives and their consequences.

our
010‘
the
ces:
11' 1'
of '
fan
off
nei
far
far

15$

C0?)
C0”,

36

Having identified the decision maker and the problem which motivates
our analysis, it is necessary to look more carefully at the kinds of
choices the farm operator can make and at the factors which will affect
the outcome of these choices. One might begin by considering what pro-
cesses affect the family's level of income--by defining the system that
will be the focus of the analysis. In this case the system is comprised
”ofuthe set Of Production and marketing processes which constitute the \
farm operation and the set of processes associated with engaging in
off farm work. In order to simplify this example, it is assumed that
neither the operator nor his wife is willing to take a permanent off-
farm job. As a result, opportunities for affecting the pattern of off;
farm earnings are limited, and the level of off-farm income will be
assumed given.

The optput Of this system is measured by a single variable, y, which
is defined as apppalmgggpwincome available for family living expenses,
ipgpme tax, and investment after all debt repayment commitments and

othgphbusiness expenses have been met.1

The level of income realized
depends on the structure of the system, on exogenous inputs to the
system, and on the choices made by the operator.

The structure of the system defines more exactly the set of pro-
cesses by which the system output, net cash income, is generated. In

this example the conceptualization of the system will be kept as simple

as possible. Standard crop budgets, which are given in Table 2.1, define

 

1Other performance measures could be identified, but the cost of
considering them is deemed excessive in this case, since it greatly
complicates the analysis.

37

 

 

 

Table 2.1 Standard Crop Enterprise Budgets
Corn Soybeans

Seed, bu. (.23) 9.70 (.83) 8.30
Fertilizer

Nitrogen, lb. (120) 16.80 (10) 1.40

Phosphorous (P205), lb. (75) 13.50 (50) 9.00

Potassium (K20), 1b. (100) 9.00 (25) 2.20
Lime .80 1.10
Herbicide, other chemicals 10.80 13.00
Fuel and repair 14.40 9.60
Utilities 2.00 2.10
Miscellaneous 2.20 2.20
Total Selected Cash Expenses 79.20 48.90
Drying cost, per point per bu. .01 O
Hauling cost, per bu. .10 .14
Time required for

Planting, hours per acre .757 .757

Harvest, hours per acre .418 .502

 

Source: Nott, et a1. (1977).

38

the basig production processes for gorp_and soybeans. They specify
not only physical inputs such as seed, fertilizer herbicides and fuel,
but also the time required for planting and harvesting, the two critical‘
fieldwork operations. These budgets alone are not considered sufficient
to adequately represent crop production, however. The timeliness of
plapting and harvesting also affects crop yields and, ultimately, the
level of income realized. Therefore, the acreage planted in each crop
is classified according to when it is planted and harvested. Six
planting periods and five harvest periods are defined in Table 2.2, and
possible planting-harvest combinations are specified for each crop. A
fjpgl characteristic of the crop production process which should be
noted is the rule of thumb that all corn is planted before soybeans are
planted and that all soybeans are harvested before corn is harvested.
Other relevant system design features include the fact that the farmer
does all the fieldwork himself, though he gets some help from his wife
who hauls grain at harvest. Again to simplify the example, it is
assumed that all production is sold at harveston the cash market.
Marketing alternatives such as forward contracting, hedging, storage,
and participation in government set aside programs will not be considered.

These system design characteristics define the processes by which
income is generated. System state variables are also useful in under-
standing the system's structure, since they serve as descriptors which
represent fully the system's performance through time. In this example
the set of stgtpflggpipples includes an indicator ofccgppentmpet_cash
ipgpme which is repeatedly updated as costs are incurred and crop sale

a.
receipts taken in, an indicator of tptal the acreage of each crop

39

 

 

 

 

 

 

 

.Amumc ocv .Fm pm .xumpm "moczom

-11- mammazom\ccoo mcmonzom\ccoo mcmmnzom\ccou mcmmnxom mplmp «can

u--- mcmmnAom\ccou mcwoaaom\ccoo mammnxom\ccou mcmmaaom FFue menu

:cou mcooaxom\:cou mcmmazom\cgou mammazom\=gou mammaAom m mcacium an:

ccou mcmmnxom\ccou memoaxom\ccoo mcmmnaom\:cou mammaxom mmimp an:

atom ccoo cgoo :coo ccou mpuop an:

:cou ccoo :Lou :cou ccou o. xmzimm F.ca<

mm-m Loasasoz -m..wm”wwwu ..-.. casebuo o.-. .mno.uo -.chmmumwmwm no.28a
m=.p=a.a

wowcma umm>cmz

 

 

cowumc.anu summ cow maocu

m_n.mmoa ecu meowcma umo>gmz ucm mcwucmpa N.N mpamh

4O
EIEEEEP,EDP harvested to date, indicators of theflnumber 9f agres of
feaghcrop_which remain unplanted or Dnharvested, and indicators of the
EDEERSFLQI bushels of each crop harvested to date. In additioniitjme
itself is monitored, as are the number of acresgof eachflcrop planted in
eagh of the six planting periods and the number of acres of each crop
planted in a particular planting period which are harvested in each of
the five harvest periods.

Inputs to the system, as well as its structure, affect the level
of income realized by the farmer. A number of exogenous system inputs
can be identified in this example. Those considered to be stochastic
include: the price at harvest of each crop, the number of days avail-
able for fieldwork in any particular planting or harvest period, and
Subjective assessments of the probability distributions for these
variables will be required for the analysis.

Relatively few controllable system inputs will be considered in
this example. Since the analysis focuses on decisions related to the
farming operation, those of primary concern are the number of acres
rented (land rental opportunities do exist) and the number of available
acres planted in each of the two crops grown, corn and soybeans.
Several factors limit the range of possible values for the three con-
trollable system inputs of primary concern. Due to the characteristics
of the local land market, the number of acres rented, v], can be assumed
to take only five values: 0, 80, 160, 240, 320. Limits on available
land imply that total crop acreage must be less than or equal to that

which is owned plus that which is rented. If v2 is acreage planted in

41

\ corn and v3 is acreage planted in soybeans, then the following relation-

' ship must hold:
<

v2+v3-240+v

1 2.6

Thgffiarm‘operator not only has control over the several system

 

 

 

i 'TPUES discussed above, but he also may be able to affect the performance

0 1“ tint-{system through design changes such as the purchase of new

mé chinery or the alteration of cultural practices. Given the operator's
ra ther precarious financial position and the fact that his crop yields
ha ve not been notably low, however, it seems best to focus the analysis
on

the specification of controllable system input levels. Therefore,

des ign changes will not be considered.

What is the nelevant planning horizon in this example? If herbicide

Carry-over problems are not considered to be of major importance, and if
Ions term leases are not required for rental of any of the 80 acre
t"‘acts, then the choices to be made in this example have an impact on
533 tern performance and on the opportunities open to the farmer only in

tfi'le year they are made. Therefore the relevant planning horizon is a

5“ hgle year.1 Because the costs of the analysis to be undertaken are

"(51: insignificant, the farm operator does not wish tozconsider major

Q"‘tanges in his strategy unless conditions change so dramatically that
l this is deemed worthwhile.

 

The decision horizon, then, is also one
year. It is recognized, however, that there will be opportunities to

1 earn over the course of the year and that new knowledge may lead to a

\

'I

If major design changes such as investment in new machinery were
t>eeing considered, the planning horizon would need to be longer.

 

42
desire for some minor revisions in the strategy. Of particular concern
are possible losses in yields due to a lack of timeliness in planting
corn. Therefore, a simple feedback control rule will also be considered
in the analysis. The rule takes the following form: "Regardless of
specified crop acreage levels, soybeans will be planted on all unplanted
acreage after v4 (a parameter indicating a specific date)." Soybeans
will not be planted before May l9, nor will corn be planted after June 3,
and the Operator wishes to check the feedback control rule at the end
of each planting period between these dates. Given these restrictions,
the possible values for the parameter, v4, are May l8, May 26, and
June 3. The action horizon for this rule, then, is eight days during
the period when it is operative.

The management strategy in this example is defined by values of
the three controllable input variables and by the single feedback
control rule parameter. Our problem is to find the strategy, v*, which
best satisfies the farm operator's need for a higher and more stable
level of income. Constraints on allowable control variable levels and
the structural characteristics of the system which determine the
relationship between system inputs and outputs must be considered when
the choice is made. The fact that the choice must be made under condi-
tions of uncertainty with respect to product prices, crop yields, and
time available for fieldwork must also be considered, since this means
that the outcome of any strategy can be specified only in probabilistic
terms. To make such a choice requires the integration of positive
knowledge of what is and what may be with normative knowledge concerning

the goodness or badness of particular outcomes. The expected utility

{

43
hypothesis, which states that the preferred choice of a decision maker
is that which has the highest expected utility, will serve as the basis
for this integration. Our problem, then, is to identify the management
strategy v* for which the associated distribution of realized net cash

income, f(y), maximizes the expected utility of the decision maker.

CHAPTER III
THE DETERMINATION OF SUBJECTIVE PROBABILITIES

3.l Introduction

 

Choices made under uncertainty are affected by a decision maker's
preferences for alternative outcomes and by his expectations concerning
the likelihood of each possible outcome associated with the action
strategies under consideration. Both of these factors are subjective
and vary from decision maker to decision maker, and information on both
is a critical input in the analysis of any decision problem in which
the impact of uncertainty is of major importance. In this chapter,
techniques for eliciting information on expectations and methods of
structuring that information for use in a formal decision analysis will
be examined.

Expectations are reflected in a decision maker's beliefs about
the probabilities of different events occurring. These beliefs may be
based in part on logical deductions, on inferences drawn from empirical
observations, on intuition, or on a combination of all three types of
information. In general, however, probabilities must be viewed as
personal or subjective and, as such, cannot be judged to be correct or

incorrect.1 The problem in a decision analysis is one of representing

 

1This personalistic view of probabilities whereby they are con-
sidered to be "degrees of belief" rather than objective facts has its
origins in Ramsey's (193l) discussion of probability in the essay
"Truth and Probability."

44

45
subjective probabilities in a manner which is consistent with the
decision maker's actual beliefs and in a form which facilitates the use
of this information in the evaluation of alternative choices. In many
situations, this already difficult problem is made more so by the fact
that the decision maker's beliefs may be poorly defined and may be
based on quite limited information. As Hogarth (l975, p. 273) notes:

. . man is a selective, stepwise information processing

system with limited capacity, and, as I shall argue, he is

ill-equipped for assessing subjective probability distribu-

tions. Furthermore, man frequently just ignores uncertainty.

The assessment of subjective probabilities, then, can be a difficult,
complex task. Elicitation procedures should be designed to help the
decision maker think in probabilistic terms. Furthermore, they should
serve as an aid in structuring information from a wide range of sources,
including that provided by experts more knowledgeable than the decision
maker himself.

Of primary concern.in the analysis of choices made under uncer-
tainty are the subjective probability distributions of the outcomes
associated with each management strategy under consideration. In
general, such distributions cannot be assessed directly by the decision
maker, however, since they are usually dependent both on specified
levels of system control inputs and on a number of stochastic and
non-stochastic environmental factors. Rather, their assessment requires
both the encoding, or direct elicitation, of subjective probability
distribution for important stochastic exogenous system inputs and the
modelling of the relationships between system inputs--controllable and
exogenous--and system outputs. The combined use of encoding and

modelling allows the decision maker to break down what may be a complex

46
stochastic process into more manageable sub-units about which he can
formulate expectations directly. It also encourages the decision maker
to think explicitly about how controllable and uncontrollable factors
interact to determine the outcome of any choice.

The value of careful system identification should be evident here,
since it is the process by which controllable and exogenous system inputs
having an important impact on system performance are identified. The
number of stochastic factors considered and the complexity of the model
used to represent their effort on the outcome of a particular choice
should be determined by the nature of the problem being analyzed. In
many instances the stochastic process under consideration will be
modelled more than once with new exogenous system inputs and a more
refined view of the system itself being considered at each stage. As
Spetzler and Stael von Holstein (1975, p. 341-2) note:

Modelling efforts tend to be most effective and most economical

if they begin with a gross model that is successively refined.

A model should be refined only as long as the cost of each

additional refinement provides at least comparable improvement

in information. The criterion for how much information is

needed depends on how significantly the information bears on

the decision at hand.

The elicitation and structuring of information on subjective probability
distributiong,then, can be viewed as a learning process which extends
the knowledge gained during system identification. It is a process
which should lead to an improved understanding both of the system under
consideration in a decision analysis and of the decision problem itself.

In the remainder of this chapter, procedures for encoding subjec-
tive probabilities will first be reviewed. Methods for modelling

stochastic processes will then be examined with particular attention

being given to the use of Monte Carlo methods and simulation techniques

 

47
to represent the performance of complex systems. Finally, these tech-
niques are applied to the analysis of the decision problem introduced

in the concluding section of the preceding chapter.

3.2 Probability Encoding Procedures

 

During the system identification phase of problem formulation,
stochastic exogenous system inputs which are expected to have an
important impact on system performance are specified. In many instances,
preliminary models may be constructed and sensitivity tests performed
in order to better determine which environmental factors affect system
performance most significantly. Once this has been done, the encoding
of subjective probabilities associated with important exogenous
variables can begin.

Encoding is the process by which a decision maker's beliefs about
the relative likelihood of different events are elicited and used to
represent a subjective probability distribution. It is one means by
which information from a range of sources is structured for use in a
decision analysis. It should be emphasized once again that a single,
correct subjective probability distribution does not exist. Furthermore,
it should be noted that the decision maker may not even think in
probabilistic terms. As Winkler (l967, p. 778) notes:

. there is no 'true' prior distribution. Rather, the
assessor has certain prior knowledge which is not easy to

express quantitatively without careful thought. An elici-

tation technique used by the statistician does not elicit

a 'true' prior distribution, but in a sense helps to draw

out an assessment of a prior distribution from the prior

knowledge.

Finally, it should be noted that the encoding process may also involve

the consideration of information from outside expert sources or the use

48

of historical data. This may be particularly important when the
decision maker's own knowledge about a particular factor is limited.
In such instances he may be willing to accept the assessments of others
or to base his expectations for the future entirely on past occurrences.

Before actual encoding begins, the stochastic variable under con-
sideration should be clearly defined,and its importance in the decision
analysis should be recognized by the decision maker. The variable
should be viewed as truly exogenous to the system under consideration.
If its level will be affected by the decision maker's actions, it cannot
be considered to be exogenous. Finally, the variable's relationship to
other random factors should be considered carefully. If its level is
conditional upon that of other exogenous variables, this should be
recognized and dealt with explicitly during the encoding process.

Possible sources of bias should also be considered before encoding
begins. Biases are said to exist when an encoded subjective probability
distribution does not conform with the decision maker's actual beliefs.
As such, they are impossible to measure, since the elicited information
is the only available indicator of the decision maker's beliefs. Evi-
dence from controlled experiments in which subjects use sample observa-
tions to assess probability distributions known to the experimentor,
however, indicates that interview procedures can have on impact responses
(Hogarth, 1975). Furthermore, experiments designed to reveal how sub-
jects assess probabilities rather than how well they assess them indicate

that several of common perceptual heuristics can cause problems in the

 

49
assessment of probabilities.1 The introduction of biases into the
encoding process can be minimized through the careful design of inter-
view procedures.

At the outset of the encoding interview, efforts should be made to
gather and make note of all available information which may help the
decision maker formulate his expectations. If a substantial data base
of past values of the random variable being considered exists, and if
the subject believes these data accurately reflect his expectations
concerning future events, he may choose to let the historical data
define his subjective probability distribution. This may be a reasonable
procedure, for example, in the case of rainfall patterns. In other
instances, this cataloging of available information may reveal that the
decision maker knows little about the variable being encoded. A decision
must be made, then, as to whether more information should be sought from
expert sources or whether encoding should proceed. This decision will
depend on the cost of new information and on the sensitivity of system
performance to fluctuations in the variable under consideration.

The purpose of an encoding interview is to construct a quantitative
representation of the decision maker's subjective probability distribu-
tion for a particular variable. In general, this representation takes

the form of a cumulative distribution function, such as that in

 

1Of these, the most notable are availability (Kahneman and Tversky,
1972), representativeness (Kahneman and Tversky, l973), and anchoring
(Tversky and Kahneman, l974).

50
Figure 3.l.1 Moments are sometimes used to describe subjective distri—
butions, but most subjects find it difficult to translate probabilistic
beliefs directly into statements of the moments of a particular distri-
bution.

Several types of questions can be used to elicit information on a
decision maker's expectations. Spetzler and Stael van Holstein (l975)
classify elicitation techniques according to encoding method and response
mode. They identify three encoding methods:

l. Those which require the assessor to specify a probability

level while the value of the random variable is fixed--i.e.,
the respondent indicates the probability that the variable X
will fall below x*.

2. Those which require the assessor to specify a value of the
random variable while the probability level is fixed-~i.e.,
the respondent indicates a value of the variable X, x*, such
that the probability X will fall below x* is equal to a
specified probability level.

3. Those in which the assessor specifies both a value of the
random variable and a probability value associated with it.

In effect, this is done when historical data are said to
reflect the decision maker's beliefs concerning future events.
Two response modes are identified: direct and indirect. Under the

direct response mode the assessor is asked to explicitly specify values

 

1For any value x* of the random variable X, the corresponding
value of the cumulative distribution function, F(x*), is the probability
that a sample observation of X will have a value less than or equal
to x*.

F(X)
1.0}

 

51

 

Figure 3.l

A Cumulative Distribution Function

52
of the random variable or probability levels which define points on
the cumulative distribution function. Under the indirect mode the
assessor is asked to indicate which of two or more bets he prefers.
One of the bets serves as a reference which allows a probability
assessment to be inferred from the assessor's response.

Several types of elicitation techniques may be used in a single
encoding interview. Questioning might begin with direct response mode
questioning to determine extreme values of the variable to be encoded.
Indirect response mode questioning can then be used to determine a
number of points on the cumulative distribution function. Finally, direct
response mode questions can be asked to determine probability quartiles
of the distribution which can be used as a consistency check.

At the completion of the encoding process a number of points on the
cumulative distribution function of the decision maker's subjective
probability distribution for the variable under consideration have been
identified. Similarly, if historical data are used in lieu of subjec-
tive assessments to define the distribution, each observation can be
assigned a position on a cumulative distribution function according to
the following rule: "If a sample of n observations is drawn from some
distribution and arranged in order of size, the kth observation is a
reasonable estimate of the k/ (n + l) fractile of the distribution"
(Schlaiffer, 1959, p. l04). The cumulative distribution function must
be defined for all possible values of the random variable, however, not
simply at selected points.‘ This is usually accomplished by sketching
a smooth curve through the observed points, though if there are good

reasons to believe the distribution of a random variable is from a

53
particular family of distributions with cumulatives of a known form,
regression techniques can be used. In either case the reliability of
such a representation may be questioned, especially when only a few data
points have been identified.

Anderson (l974b)has investigated the estimational reliability of
handbsketched smoothed cumulative distribution functions based on
sparse empirical data. The questions he poses are also relevant in
connection with the construction of smoothed cumulative distribution
functions from a small number of subjectively assessed points. His
results indicate that, as expected, estimational reliability increases
when a large number of observations are available, but they also show
that in a surprisingly large number of cases a fairly good estimate of
an underlying distribution can be made on the basis of only three to
five observations. These results are somewhat encouraging, but they
should also serve as a warning of the need in some cases to consider
explicitly the inexactness of assessed probability distributions, as is
done by Watson, Weiss and Donnell (1979).

The encoding interview should end with the decision maker's verifi-
cation of the quantitative representation of his beliefs. This can be
done by asking him to examine either the cumulative distribution function
which has been constructed or a random sample drawn from the distribu-
tion it defines. More formal verification procedures involve the use

of scoring rules (Minkler, 1969; Stael von Holstein, l970; Savage, l97l),

54

but as Hogarth (1975) notes, the usefulness of such rules is questionable
in many situations.1

Finally, it should be noted that the encoding procedures discussed
above are designed for the determination of subjective probability
distributionscflirandom variables judged to be statistically independent
of all other random factors under consideration. In many instances,
however, random factors are not independent. On a particular farm,
for example, yields for two crops such as corn and soybeans are likely
to be highly correlated. Similarly, prices received for these two
crops would not be expected to be statistically independent. In such
a situation one of two alternative procedures can be followed: the
process by which the correlated random variables are determined can be
modelled back to the point where all stochastic exogenous factors can
be assumed to be independent or the decision maker's joint probability
distribution for the correlated variables can be assessed directly.
Neither alternative is particularly attractive. Modelling can be
costly, and as the model becomes more complex the number of variables
to be encoded may increase rapidly. Furthermore, while the decision
maker may have well formulated expectations about many of the non-
independent random variables, he may have little or no knowledge of the

statistically independent underlying variables in the more extensive

 

1A scoring rule is a payoff function with the vector of stated
probabilities for each of a set of mutually exclusive and expansive
events and a vector of probabilities representing the decision maker's
true beliefs being the arguments. If a scoring rule is strictly
proper, it will be maximized when the stated probabilities coincide
directly with the assessor's true beliefs.

55
model. Encoding of joint probability functions, on the other hand, is
a difficult, time-consuming process. Experimental results indicate
that many subjects are ill-equipped for the assessment of correlations
between random variables (Chapman, l967; Tversky and Kahneman, 1973).
Therefore, elicited distributions may simply reflect poorly formulated
expectations.

When joint specification rather than more extensive modelling is
the preferred course of action, it is advisable to rely on historical
data whenever possible. For example, if distributions for rainfall
and daylight hours without cloud cover over a particular two-week period
are to be assessed, past weather data could probably be used to repre-
sent most decision makers' expectations. Similarly, yield data for
several crops over an extended time period, if corrected for time
trends and other identifiable factors, may provide adequate information
to construct a marginal distribution for each crop and to estimate
correlation coefficients between crops. Even in the case of crop
prices for which past experience may not be relevant in the formulation
of each marginal distribution, it may be possible to use historical
data to estimate correlation coefficients which could be used in con-
junction with marginal distributions determined by other methods.

In cases where historical data are not available or are considered
to be irrelevant, joint specification of bivariate subjective probability
distributions can be accomplished by encoding one of the marginal distri-
butions and then encoding conditional distributions for the second
variable at several values of the first. Anderson, Dillon, and Hardaker
(1977) describe this procedure in some detail and explain how it can be
extended to cases where more than two correlated variables are to be

encoded.

56
3.3 Modelling Stochastic Processes

 

The encoding techniques reviewed in the preceding section are used
to elicit direct assessments of subjective probability distributions.

In most decision situations, however, such direct probability assessments
can be made only for the distributionscfi’exogenous system input variables.
They usually cannot be made for the distributionscfl'system output
variables, the distributionscn’primary concern in the evaluation of
alternative choices. Rather, these distributions, which depend on
complex interactions among a number of factors, must be determined
indirectly by modelling system performance. A model is a deterministic
mathematical representation of the set of processes by which controllable
and exogenous system inputs determine system output levels. As will be
demonstrated below, given information about the levels of controllable
system inputs which define a particular strategy and information about
the probability distributions of exogenous system inputs, a model can be
used to determine the associated probability distribution of system out-
put levels.

Even in situations where the distributions of system output
variables associated with alternative strategies can be assessed
directly, a model of the system under consideration can be useful for
several reasons. First, it can increase the decision maker's under-
standing of the set of processes which determine the outcome of any
strategy, since modelling can be viewed as a learning activity. Second,
by providing a logical representation of the processes which comprise a
system, a model allows the decision maker to focus his attention on the

formulation and representation of expectations about future levels of

57

individual exogenous system input variables. He need not consider all
such factors and their interactions with other determinants of system
performance simultaneously. Finally, if the number of alternatives
being considered is large relative to the number of exogenous system
input variables, the use of a model to determine system output variable
distributions can significantly reduce the number of probability dis-
tributions which must be encoded, since only the distributions of
exogenous system input variables must be assessed.1

The exact nature and complexity of the model used in any particular
decision analysis will depend on the characteristics of the problem
under consideration. In some instances the appropriate model may be
quite simple. For example, if the set of controllable inputs is defined
by v, a column vector of acreage levels for each of several crops; if
the set of exogenous system inputs is defined by e, a column vector of
net returns per acre for each crop activity; and if the total net return
for all crop activities, y, is the only system output variable of
concern, the appropriate model of this system may simply be the
following linear equation:

y = e'v 3.1
In other cases, much more complex models may be required to adequately
represent the relationships among system inputs and system outputs.
This is true, especially, when the system under consideration is

dynamic and when strategies are defined by feedback control rules as

 

1The same probability assessments for these variables are used in
the determination of system output distributions for each strategy
considered. Therefore, exogenous system inputs must be truly exogenous--
i.e., their levels must not be significantly affected by system per-
formance.

58

well as by fixed specifications of controllable system input levels.1
Specific conceptual and quantitative tools used in systems modelling
will not be discussed in this study. Excellent discussions of such
techniques can be found in Forrester (1961), Manetsch and Park (1977a),
and Manetsch (1978).

As defined above, a model is a deterministic representation of the
relationship between a set of system inputs and a set of system outputs.
Given specified levels of all system inputs, controllable and exogenous,
the set of system outputs can be calculated exactly. In decision
situations involving uncertainty, however, levels of stochastic exo—
genous system inputs cannot be known exactly prior to their occurrence.
This implies, of course, that the system output levels associated with
a particular strategy cannot be determined exactly either. Rather, they
can be specified only in probabilistic terms. In such instances a
system modeL.despite its deterministic character, can be of use in
describing system performance.

In some special cases. a system model can be the basis for the
analytical determination of the distribution of system outputs associated
with any strategy being considered. Returning to the simple linear
model specified in equation 3.1, for example, if each random factor in
the vector of exogenous system inputs, e, is normally distributed, the

distribution of total net revenue, y, is also normal with mean, u, and

variance, 02, defined by the following expressions:

 

1See Forrester (1961), Cyert and March (1963), and Dent and
Anderson (1971) for examples of more complex models.

59
p = m'v

02 = v'av 3.2
where m is a column vector of the expected net revenues for each crop
activity and Q is the variance-covariance matrix for net returns.
Anderson (1975) has shown that the distribution of y can also be
determined analytically when each element of e has a Beta distribution.

In situations where subjective probability distributions for exo-
genous system inputs are not all members of the same family of distri-
butions or where a more complex model is required to represent system
performance, it may not be possible to analytically derive distributions
for system outputs. In fact, when a model is particularly complex, it
may not even be possible to calculate system outputs analytically for
the special case when levels for all controllable and exogenous system
inputs are known with certainty. In such instances, numerical simula-
tion techniques and Monte Carlo methods are required to determine sys-
tem output distributions.

Manetsch and Park (1977b, p. 8-1) define simulation as "a tech-
nique for obtaining particular time solutions of a mathematical model
corresponding to specific assumptions regarding model inputs and values
assigned to parameters." The model specified in equation 3.1 can provide
the basis for a simple example of simulation. Consider the case in
which 200 acres of land are to be planted and only two crop activities
are possible. Let corn be crop 1 and soybeans be crap 2. If the
controllable and exogenous system input vectors, v and e respectively,

are defined as follows,

_ 150 _ 95.00
V ‘E50 9 'Eioopo 3'3

60
then simulation of system performance for this particular case requires

only that the following matrix multiplication be carried out:

y = [ 95.00 100.00 J ['38 3.4

In this case, with 150 acres of corn and 50 acres of soybeans planted,
total net revenue, y, is equal to $19,250.

Systems of concern in most practical decision situations are
larger and more complex than this one, and their simulation is generally
more involved. Frequently numerical solution techniques are required.
and many simulation models are computerized. One of the distinct advan-
tages of simulation, however, is that it is a remarkably flexible pro-
cedure which allows complex processes to be represented realistically.
Naylor, et al. (1966), Schmidt and Taylor (1970), and Manetsch and
Park (1977b) all provide excellent discussions of simulation techniques.

Monte Carlo methods are commonly used in combination with simula-
tion to model the performance of complex stochastic systems. Under this
approach, numerical procedures are employed to generate sample observa-
tions from the decision maker's subjective probability distributions
for exogenous system input variables. Each sample vector, e*, specifies
a level for each exogenous system input and, as such, defines a state
of the system's environment. By constructing a large number of sample
states of the environment and simulating the system performance associated
with a particular strategy for each of these environmental states, a set
of sample observations from the distribution of system outputs associated

with that strategy is generated. These observations can be used to

61

define a cumulative distribution function according to the procedures
described in the discussion of encoding.1

Returning once again to the simple example discussed above, let
the joint subjective probability distribution of the two elements of
the exogenous systems input vector, e, be defined in the following
manner. The marginal distribution of revenues for corn is normal with
mean $115.00 and standard deviation $35.00. That for soybeans is a
member of the gamma family of distributions with a mean of $135.00, a
standard deviation of $40.00 and a minimum value of $55.00. The
correlation coefficient for the two net revenues is .75. Using Monte

Carlo techniques described in detail in Appendix A, the following five

sample vectors from this joint probability distribution were generated:

1 [107.87 2 _ [155.77 3
e = e - e = 80.08
107.43 128.99 92.23
4 = 158.18 5 = 152.05 3.5
9 147.11 9 210.51

They represent five sample states of the environment. Simulating system
performance under the strategy defined by the controllable system input
vector v' = [150 50], five sample observations from the distribution

of net income levels associated with this strategy are determined:

y1 = 521552.00, y2 = 529955.00, y3 = 515523.50, y4 = 531082.50.

y5 = $33334.50. These five sample observations were used to construct

the cumulative distribution function shown in Figure 3.2 In general,

 

1In addition to Schlaiffer (1959), Mood and Graybill (1953) and
Barnett (1975) also discuss the validity of the rule which, when N
observations of a random variable are arrayed in increasing order,
states that the Kth observation can be used as an estimate of the
K/(N+l) fractile.

62

F(y)
1.0

5/6

2/3

1/2

1/3

1/6

 

 

l l I l 1

15000 20000 25000 30000 35000

X

Figure 3.2 A Cumulative Distribution Function
Based on Five Sample Observations

63

at least 20 sample environmental states should be simulated, and for
many problems it may be desirable to simulate as many as 50 to 100
sample states.

Even this simple example demonstrates the power of this approach.
Because the marginal distributions of the two net return variables are
not of the same family, the distribution of net income levels cannot
be derived analytically. The combinations of Monte Carlo sampling tech-
niques and simulation, however, permits the representation of the dis-
tribution. This same approach can be easily extended for use in the
analysis of much more complex systems in which the interactions between
controllable and exogenous system inputs are not so straightforward.1
It should also be noted that this method imposes no restrictions on the
nature of the system inputs or system outputs. Subjective probability
distributions for environmental factors can take a form which most
closely reflects the decision maker's encoded beliefs. System output“
distributions are determined by the structure of the model, by the
subjective probability distributions for exogenous system inputs, and
by the management control strategy. When used in a decision analysis,
these distributions can be described by their moments or by their
cumulative distribution functions.

One serious criticism of this approach to modelling stochastic
processes is that statistical dependence between random environmental
factors is often ignored (Anderson, l974a) This may be due, in part,

to difficulties associated with the joint specification of subjective

 

1The example in the final section of this chapter demonstrates
how this approach can be applied in the analysis of a more complex
system.

64
probability distributions. Even when statistical dependence has been
recognized in the encoding process, however, it is often ignored in
stochastic system models due to a lack of available techniques for
generating sample observations from multivariate probability distri-
butions.

Procedures have been developed for the generation of random variates
from a wide range of univariate probability distributions (Naylor, et al..
1966; Schmidt and Taylor, 1970). Process generators have also been
formulated for several multivariate distributions, most notably the
multivariate normal and Wishart distributions (Naylor, et al.. 1966;
Newman and Odell, 1971). More recently, Coleman and Saipe (1977) have
developed a procedure for generating serially correlated lognormal
variates, which is a special case of more general procedures for
modelling bivariate random variables with prescribed marginals and
correlations (Coleman and Saipe, 1976). A need remains, however, for
a generalized multivariate process generator which permits greater
flexibility concerning the specification of marginal distributions and
which has the capacity to be easily extended beyond the bivariate case.
Such a procedure has been developed as part of this study and is
described in detail in Appendix A.1 This generalized multivariate pro-
cess generator can be used to generate sample observations from multi-

2

W
variate distributions comprised of up to fifty random variables. The )‘

 

1Procedures used to generate sample observations from univariate
distributions are also reviewed.

2The program can easily be expanded to model processes with
still more individual variates.

65

marginals of the distribution modelled can be of any form and they need
not all belong to the same family of distributions. It is assumed only
that enough information is available on each marginal to construct its
cumulative distribution function and that the correlation coefficient
between each pair of variates within the distribution can be specified.
The only restriction placed on the matrix of correlation coefficients:
is that it be positive-definite and symmetrical, a condition required 1
for feasibility and internal consistency. I)
The existence of such a procedure greatly enhances the power of
the approach to the modelling of stochastic processes described in this
section. It permits greater realism in the representation of underlying
probability distribution without requiring that the stochastic dependence

between some random factors, which has an important impact on choices

in many decision situations, be ignored.

3.4 An Application

 

The techniques introduced in this chapter can be applied to the
cash grain farm example formulated in Chapter II. Crop yields for
each planting-harvest period combination, product prices, and time
available for fieldwork have been identified as stochastic factors
which have an important impact on the level of income realized by the
farm operator. In this section the specification of subjective pro-
bability distributions for each of these exogenous system inputs and the
use of simulation to determine the impact of these random factors on
the outcomes associated with particular management strategies are

discussed.

66

Because the case farm used in this example is a synthetic one, no
actual decision maker has been identified. Therefore, experts in the
Department of Agricultural Economics at Michigan State University were
relied on for the assessment of probability distribution for the stochastic
factors. This is not altogether unrealistic, since in many cases actual
decision makers choose to rely heavily on the opinions of experts in
the formulation of their expectations.

The assessment of probability distributions for yields was based
in part on historical data and in part on more subjective information.
Estimates of expected corn and soybean yields for each planting-harvest 1)
combination are given in Tables:3£land 3.2.1 These estimates are based
on figures ”5291'" Telplan Program #18, a commonly used decision aid
which focuses on choices similar to those being analyzed in this example.
They are the product of a group assessment of experimental data and the
personal observations of experts. No estimates of_variances or other
features of these probability distributions were made by this group; nor
did they assess the degree of correlation between yields for different
erops and different,planting-harvest combinations. Therefore, the
following subjective assessments were made. ‘All yield distributions
were assumed towbegngrmal, having means equal to those specified in
Tables 3.1 and 3.2. Specification of the variances of these distribu-

tions was‘based on the assumption that the coefficient of variation for

all corn yields is 11 percent and that for all soybean yields is

 

1Base yields of 100 bu/acre for corn and 33 bu/acre for soybeans
are assumed.

67

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wovcma umm>cmx

 

mcoz mcoz acoz mcoz mzoz mpump mesa
wcoz mcoz mcoz mcoz mcoz FF-¢ mesa
gum e an we gem e an em “mm 9 an mm xmm e an on mcoz m mczwunm an:
nmm e an Fm Num 9 an mm aom e an mm Num 9 an em «:02 mmump >dz
mmm e an om Nam e an om Nwm e an Pm Rom e an mm Rom e an mm mFIFF an:
Rpm e an mm Rmm @ an mm Row 9 an mm uwm e an cop umm e an om op >.mznmm nga<

 

 

mum: pmm>cmz ucm mappcmpa an
pcmucoo mczpmwoz ucm upmw> cgou mmoLm><

p.m mpnmh

68

.Amumu ocv .pm pm .xumpm "mugzom

 

 

 

 

 

 

 

mcoz an up an mm an mm an RN mpimF mcau

acoz an om an mm an mm an Fm Fpue mean

mcoz an —N an mm an on an Nm m mczcuum mm:

5:52 an NN an mm an pm an mm m~-m_ a”:

mcoz mcoz mcoz mcoz mcoz mpupp an:

mcoz mcoz mcoz mcoz wcoz op xmzumw Frga<

m~-m conso>oz -chwmmwwmu N_-F_ conoooo o_-¢ conoooo -Nwmcumumwmmm 55,258
acco=m_a

cowgma umm>cmx

 

 

 

mama umm>cmz use mcwucmpa an upmv> cmmnzom mmmgm>< ~.m mpnu»

69

.Z.5 percent.1 Concerning yield correlations, thgflcorrelation coefficient
betweenféfiyltwo corn yield distributions oriafiyltwo soybean distributions
wassetequal to .90, while that between anylpair of distributions com—
prised of one corn yield distribution and one soybean yield distribution
was set at .80.2 These parameters--means, coefficients of variation,

and correlation coefficients--define a multivariate normal distribution.
Since there are thirty-four individual yield distributions, eighteen

for corn and sixteen for soybeans, this multivariate distribution is
comprised of_thirty-four random variables. One additional characteristic
was specified for this set of distributions. It was felt that the
multivariate normal distribution, as specified, did not adequately
account for the possibility of extremely low yields due to serious
drought. Such conditions occur in southeastern Michigan about one year
in twenty. Therefore it was specified that in any year there is a .05
probability that drought conditions will prevail and that corn yields

will be one-half and soybeans two-thirds of what they would have been

under more normal conditions.

 

1The coefficient of variation, 0, is defined by the expression

C=9-

1.1
where o is the standard deviation of a distribution and u is its mean.
If the mean and coefficient of variation are known, the variance of that
distribution, 52, is defined by the expression 52 = (cn)2.

2The correlation coefficient for any two random variables x only is
defined by the expression
0
r =._£X_
O')(O"y
where °xy is the covariance between x and y and Ox and 0y are the
respective standard deviations of x and y.

7O
A§§§§§mgflt§wgfflgrgbability.distributions of time available for
fieldwgrk in each planting or harvest period were based on the informa-

tion presented in Table 3.3,_whieh isalsobased on figures used in

 

m..
“‘J—Or ‘1

Telplan Program #18. Alludistributions of time_available for fieldwork
were assumed to be membersof the Beta family of distributions. This
assumption was made because of the flexibility of Beta distributions
and because, like the amount of time available for fieldwork, Beta
distributions are bounded from above and below. The choice of para-
meters for each distribution was based solely on the information in
Table 3.3, which is adequate to determine upper and lower bounds and
one intermediate point on the cumulative distribution function. With
the aid of tables of twentiles of the standard Beta distribution given
in Pratt, Raiffa, and Schlaiffer (1965), parameters were selected for
each period according to the simple criterion that the cumulative
distribution function should pass as close as possible to the single 0
observed data point. These parameters are given in Table 3.4. No
information on correlations between time available for fieldwork in
different periods was available, but it was felt that correlations do
exist between levels observed in adjacent or nearly adjacent time
periods. Therefore the following aesumptions were made. The correla-
tion coefficient for the time available in any two adjacent periods was
set at .5; that for periods separated by a single period was set at .3;

and thatgfor periods separated by two periods was set at .1. All othere>i}(”(,

)1V1\|{'.

I r

.\‘J‘

correlation coefficients were set at 0, including those between any 2
planting period and harvest period.
Encoding procedures such as those outlined in Section 3.2 above

were used to elicit the author's own subjective probability distributions

71

Table 3.3 Percent of Time Available for Fieldwork by
Calendar Period for Well Drained Sandy
Loom Soils in the Lenawee, Monroe,
Livingston County Area

 

 

 

Period Casingar Percentagea
April 25-May 10 16 50
May ll-18 8 37
May l9-26 8 65
May 27-June 3 8 70
June 4-11 8 70
June 12-19 8 70
September 27-0ctober 3 7 53
October 4-10 7 53
October 11—17 7 53
October lB-November 7 21 33
November 8-28 21 14

 

aThe probability that the percentage of days available for fieldwork
will be less than this value is .3.

Sources Black, et a1. (no date).

72

Table 3.4 ‘Beta Distribution Parametersa for Time Available
for Fieldwork by Calendar Period

 

 

 

Period 83:6; ggfifidb a B Meanc 0:3?giign
April 23-May 10 0 192 9 7 1081»: v 21.1 “
May 11-18 0 96 7 9 42 11.5
May 19-26 0 96 14 6 67 9.6
May 27-June 3 O 96 12 4 72 10.1
June 4-11 0 96 12 4 72 10.1
June 12-19 0 96 12 4 72 10.1
Sept. 27-Oct. 3 O 84 10 7 49 9.7
October 4-10 0 84 10 7 49 9.7
October 11-17 0 84 10 7 49 9.7
Oct. l7-Nov. 7 O 252 6 9 101 30.9
November 8-28 0 252 3 12 50 25.2

 

aThe density function of the standard Beta distribution is given by
the expression:

f(x)== ; 3+? 8 xa'] (l-x)8'1 for O<x<l

0

bTime is measured in hours under the assumption of twelve-hour
work days.

cThe mean of the Beta distribution is given by the expression:

_ a
“ ’ T8187 (b2'b1) * b1

where b2 and b1 are the upper and lower bound values.

dThe standard deviation of the standard Beta distribution is given
by the expression:

a: a8 -b 2
“(0+B)2(a+8+1) “’2 l)

where b2 and b1 are again the upper and lower bound values.

 

 

73

for corn and soybean prices at harvest in 1979. These assessments,
which are representative of expectations in early January 1979, were
reviewed by outlook specialists for these crops and revised to incor-
porate their opinions. Upper and lower bounds beyond which each price
level could not reasonably be expected to fall and three intermediate
points as each distribution were specified. As was the case with days
available for fieldwork distribution, it was deemed reasonable to
represent these probability assessments with Beta distributions, and
Beta tables were again used to select appropriate parameters. The price
distributionparametersare given in Table 3.5. Historical data series
for November prices of corn and soybeans over the period 1958-1972 were

1 The

used to assess the degree of correlation between the two prices.
sample correlation coefficient on de-trended prices for this period is
.56. Because some experts feel there is less basis for correlation
between corn and soybean prices now than in the past, the subjective
estimate was set at :45. Finally, prgduction of corn and soybeans in
the region surrounding southeastern Michigan does not represent a major
_portion of national production of either crop. Therefore, both prices
were assumed to be statistically independent of yields and days avail-
able for fieldwork.

Clearly the determination of subjective probability assessments
can, in practice, be an imprecise undertaking, though the resulting

distributions in this instance are not unreasonable. This high degree

of imprecision leads us to two important observations concerning the

 

1The period after 1972 was considered to be somewhat atypical of
the current situation due to the lack of government stocks from l973-76.

74

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FN.a ¢_.N» m N om.ma om.pm csou
copumw>wo com: m a venom Loan: ucsom Logo; :cou mo
ucmccepm moped amm>cmz

 

 

mmcowuanwcumpo moped cmmnxom one
atom cow msmpmsmsoa copaznwcumwo mumm m.m mpamh

75
analysis of decisions made under uncertainty. First, we note that
there is a lack of available material which can serve as an aid in the
assessment of probability distribution. The need for such information
is generally not considered when agronomic experiments are designed or
when outlook information is reported. When one considers the degree of
uncertainty experienced by agricultural producers, however, and the
impact of this uncertainty in the decision process, the need for more
expert assessments of probability distribution is evident. Second. it
should be noted that decisions must be made even in the absence of
reliable information upon which to base probability assessments. The
costs of obtaining additional information must be weighed against the
possible benefits. Further refinements should not be made in subjective
probability assessments beyond the point at which the decision maker and
the analyst believe the farmer's expectations are reasonably well
represented.

To this point probabilitydistributions for exogenous system inputs
jgdged to haveja_significant impact on system performance, as measured
by net cash income, have been specified. The task of actually deter-
mining the effect these factors have on the distribution of net cash
income levels remains. This requires the application of the Monte
Carlo simulation techniques outlined above in Section 3.3. For any
management strategy under consideration, net cash income levels are
determined for a number of randomly selected states of nature. These
levels are viewed as sample observations from the distribution of net
cash income levels under the particular management strategy. When
arrayed in order of increasing magnitude they serve as the basis for the

construction of the cumulative distribution function of that distribution.

76

In this example, it will be recalled..flflméfléggmgntestrategy is
defined_bylevels ofcontrol variables indicating acreage rented,
aereggemtgmbe planted in corn, and acreage planted to be planted in soy-
beans and by_a single feedback control role parameter indicating the
date after which all unplanted acreage is to be planted in soybeans.

A statewof the environment is defined by a set of specified values for
all exogenous system inputs-~by values for all non-stochastic environ-
mental factors and by one sample observation from the multivariatefl
probability distribution comprised of crop yields, time available for
fieldwork, and product prices.

A simple simulation model was specified to determine the level of
net income realized under any particular managerial strategy in a given
state of the environment. The simulation begins with the computation
of charges for land rental, if any. Snbject to time availablefor
fieldwork,themodel then simulates the planting of corn until the
specified corn acreage is attained or until the date after which all
remaining acreage is to be planted in soybeans. Planting of soybeans
then proceeds until all acreage is planted or until June_l9, the final
day of the last planting period. Throughout the planting process,
hsystem state variables indicating the acreage of each crop planted in
each planting period are repeatedly updated. There is no assurance
that all available acres will be planted in a particular state of
nature; this depends on levels of time available for fieldwork. Costs
for seed, fertilizer, herbicides, and fuel are incurred for each acre
actually planted.

Harvesting is simulated in a similar manner. Subject to time

available, soybeans are harvested as quickly as possible, with acreage

77
planted first being harvested first. This continues until all planted

soybeans acreage is harvested or until November 8, the date after which
unharvested acreage is judged to be a total loss. Theharvest of corn
then begins, again with acreage planted first being harvested first.
Thisprocess continues until all planted corn acreage is harvested or
until November 28, the last day of the final harvest period. Again,
there is no assurance that all acres planted will be harvested.1 All
harvested acreage is classified according to crop, planting period, and
harvest period, and system state variables indicating the number of acres
in each category are repeatedly updated. The values of these variables
are multiplied by corresponding crop yields for each planting-harvest
combination to determine the total number of bushels of each crop
harvested. Drying and hauling costs are assessed for each bushel har-
vested. Finally, receipts from crop sales are determined by multiplying
the number of buShels of each crop harvested by the relevant price,
and net cash income is computed by subtracting costs incurred and debt
repayment commitments from the sum of crop receipts and off-farm income.
This model was used to determine net cash income levels realized
in twenty randomly selected states of nature for each of the two manage-
ment strategies defined in Table 3.6. Levels for stochastic factors
in each state of nature were generated using the Monte Carlo procedures
described in Appendix A. In effect each state can be viewed as a sample
observation from the combined multivariate probability distribution of
prices, yields, and days available for fieldwork. Net income levels

realized under Strategy 1 are given in the first column of Table 3.7.

 

1Nothing is recovered from unharvested soybean acreage, but yields
equal to one-half those realized for corn planted in the fourth planting
period and harvested in the final harvest period are assumed to be
recoverable on unharvested corn acreage.

78

Table 3.6 Two Possible Management Strategies

 

 

 

 

Central Strategy

Variable 1 2
Land Rental 0 240
Corn Acreage O 180
Soybean Acreage 240 300
Stopping Date for June 3 May 26

Corn Planting

 

 

 

79

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o..o 00.0m 00.00N 00.0q~ 0-.~ . 00.0 00.0 00.0npq
00.0 .0..m 00.00m 00.00N ~0.N . 00.0 00.0 «0.0000
0~.0 0~.m~ 00.00N 00.00N 0..N . 00.0 00.0 0N.0000-
.0.m .0.Nn 00.00N 00.00N o..~ . 00.0 00.0 «0.0.;n
00.0 00.Nm 00.00m 00.00N N0.N o. 00.0 00.0 00..noo
00.00 0.0.. 0oumo>cuz 00000.0 00.00 0.0.» 0oumo>gaz 00000.0 0:000.
00000>t 00000 0000‘. 00000>< 0000< 0000< 0000
0022260 0:00

 

 

. .mmumeum 0000: «000000.000 smumxm N.m 0.00.

 

80

Information on soybean prices and average yields per acre is also given.
If this strategy is selected by the operator of the farm in our example
the expected level of net cash income is rather low--only $3,816. The

probability is .3 that net cash income will be negative, but there is

only a probability of :95 that losses will exceed $10,000. On the other A

-—~——-—"

 

hand, there is a probability of .25 that net cash income will exceed
$10,000 and a probability of .05 that it will exceed $20,000. Net
income levels realized under Strategy 2 are given in Table 3.8. Under
this strategy, which calls for the rental of 240 acres and a more
balanced crop mix, the expected level of net cash income is much
higher--$10,798. The probability is .6 that net cash income will
exceed $10,000 under this strategy and .3 that it will exceed $20,000.
The probability of realizing a negative net income level is .25, which
is less than under Strategy 1. When losses occur, however, they tend
to be substantial and there is a :12 probability that net cash income
will be less than $19,900. The other information in Table 3.8 demon-
strates how the management strategy is revised by the feedback controll
rule.) In;ten;of_the twenty states of nature less than 130 acres of corn
are planted because of the stipulation that all unplanted acreage be
planted in soybeans.after May 26. The information in this table also
demonstrates that when the number of acres cultivated reaches this high
a level, there is no assurance that all available acreage will be
planted or that all planted acres will be harvested.

The figures given in Tables 3.7 and 3.8 can be used to construct

cumulative distribution functions of net cash income levels associated

with each of the two strategies. These are shown in Figure 3.3. The

_._R_ ‘.

81

 

 

 

a... .m.00 ~0.~.n 00.N.m Nn.N 00.00 a...0. 0...0. 0.....0N
00.. 0.... 00.00. 00.00. 00.~ .0.00. 00.0.. 00.0.. «0.0000.
0m.m .0.o~ 00.00n 00400n m~.~ o0.m0. 00.0.. 00.00. 0..ac~n.:
.0.m ~0.s~ 00.00m 00.00m .o.. 00.n0 00.0.. 00.00. oo.~00.u
no.0 ...0~ 0~.00n 0~.00n 0~.~ .0.00 00.0.. 00.0.. 00.0000
.... .0.m~ 0m.~0~ 00.00N .0.N mm... 00.0.. 00.00. N..000.
00.0 n0.o. 00.00. 00.00» om.~ 0~.50 . 00.0.. 00.00. m .m-0.-
00.0 00.m~ 00..0m 00..0m m0.. 00... um... ner «n.0r. 0....N..-
mm.. 00.0n 00.00n 00.00m .m. .0.No 00.00. 00.00. c .u0sow
00.. No.0N 00.00. 00.00. .n.N 00.00 00.00. 00.00. .s.~mma~
.u.h ~0..n 00.000 00.000 n0.~ 00.00. 00.00. 00.0.. 00.00.0n.
mu. 00.0N 00.00n 00.00» .N.. 0...0 00.00. 00.00. .0.=.0n.o.
00.0 m..0~ 00.00. 00.000 ~N.~ w...o 00.00. 00.00. .0.00m~.
0N.0 m~.0~ 0..NNm o..-n 00.N 00.00 .~.sm. .~.~m. 0....00.
00.0 .N.0~ 00.000 00.000 n..~ 00.00 00.00. 00.00. .0.000~.
00.0 .0.0~ 00.00r 00..~n 0m.0 m~.00 .<.rx -m 0v 0m.wr. 00.000.
00.0 no.o~ 00.00. 00.000 .0.N o...0. 00.0.. 00.00. m0.0000N
0~.0 N...~ 00.0mm 0N.mmn 0..N 00.0. .m.m..-n;-¥ 0~.00. 00.0.00-
.0.0 00.0. 00.00. 00.00. 0..N .m.mo 00.0.. 00.0.. 00.Nm.n.
...0 n~..n No.00. 00.00. .0.0 .c.m.. 00.0.. 00.0.. 00.000NN
00.00 0.0.. 00000>00= 00000.0 00.00 0.0.. 00000>00= 00000.0 00000.
00000>0 00000 0000: . 00000»:. 00000 00000 0000
0000000» 0000 00:

 

 

 

 

N .000000m 00000 0000000.000 Empmxm m.m 0.00.

82

00.00.000m m>.umcgm..< 03.
00. m:0..00:u 00..00.0.m.0 m>..0.=sao m. m 0.00..

 

008. 000$ 000$. 800w 0000. 08.... 0000 000. 0 000.- 0000- 008.- 0000.-
d d d d I d d 1 d u d d
\ \
\ \ \
\\ \
........ w 0.32.... \
IIII . 33:0.
\
\
\
\
\

 

0..

 

....

83
procedure for ordering alternative choices which is introduced in the
next chapter requires that such a function be constructed for each

strategy considered.

CHAPTER IV
THE MEASUREMENT OF DECISION MAKER PREFERENCES

4.1 Introduction

 

Choices made under uncertainty are affected by decision maker pre-
ferences for alternative outcomes as well as by subjective assessments
of probability distributions of system outputs. When confronted with
the choice between participation in two uncertain activities for which
all possible outcomes and their probabilities are specified exactly,
one decision maker may choose the first alternative while another may
choose the second. This divergence in behavior cannot be attributed
to a difference in subjective probabilities, since all relevant pro-
babilities are specified prior to the time when a choice must be made.
Rather, it must be attributed to a difference in the preferences of the
two decision makers. Preferences, like assessments of probabilities in
situations less highly structured than this example, are personal in
nature, and some determination must be made of them in any applied
decision analysis. This chapter examines procedures for eliciting
information on decision maker preferences and techniques for combining
this information with subjective probability assessments to identify
preferred choices.

A decision maker's preferences can be represented quantitatively
by a utility function,

U = u(y) 4.1

84

85
This is simply a relationship between the outcome of a choice as
represented by a vector of system output levels, y, and an index of
its desirability, U. It is a relationship which assigns values to
alternative situations or conditions. When combined with a decision
rule, such as utility maximization, a utility function becomes the
basis for the identification of a preferred course of action.

If the system output levels associated with a particular strategy
can be known with certainty, and if the decision maker's utility
function is also known, calculation of the utility level of this
choice is a relatively simple, direct matter. In uncertain decision
situations, however, levels of system outputs realized under a specified
strategy cannot be known exactly at the time when a choice is made, and
the associated level of utility cannot be determined directly. In
such situations the expected utility hypothesis provides a way of
assigning assessments of value to alternative choices. First proposed
in the eighteenth century by mathematician Daniel Bernoulli to explain
the gambling behavior of some decision makers in uncertain situations,
and derived more formally nearly 200 years later by Ramsey (l931) and
by von Neumann and Morgenstern (1944), the expected utility hypothesis
states that for any decision maker whose preferences are complete,
transitive, continuous, and independent:

(1) An ordinal utility function, u(y), can be constructed

such that u(y) is defined for all system output levels
and u(y])>u(y2) if the outcome defined by y1 is preferred
to that defined by ya.
(2) The utility of an uncertain prospect is equal to the expected

utility of its possible outcomes.

86

(3) The scale of x(y) is arbitrary up to a positive linear
transformation-~i.e., rankings of the utility function
V(y) = a+bu(y) are identical to those according to u(y)

if a and b are constants and b>0.1

The second result is the key one, since it is the basis for the

commonly used decision rule which states that a decision maker's pre-

ferred choice is that which maximizes his expected utility.

If the expected utility hypothesis is to be applied in a practical
context, the decision maker's utility function must be represented
accurately enough to serve as a reliable aid in the identification of
a preferred course of action. As is true in the determination of sub-
jective probability distributions, however, the degree of accuracy
sought in the measurement of preferences is dependent largely upon the
characteristics of the decision problem under consideration. In some
instances a very precise measure of preferences may be required, while
in others nearly all feasible alternatives can be eliminated from con-
sideration on the basis of only an approximate measure of preferences.

In the remaining sections of this chapter several alternative
approaches to the measurement and representation of decision maker
preferences will be examined along with the evaluative criteria used in

conjunction with each type of measurement to order action choices.2

 

1See Fishburn (1970) and Hirshleifer (l970) for more extensive dis-
cussions of the derivation of the expected utility hypothesis and for a
complete explanation of the axioms which underlie it.

2Though the importance of preference measures based on more than one
system output variable is recognized, the difficulty and cost of deter-
mining such measures preclude their use in most practical decision situa-
tions. The discussion below focuses entirely, then, on the measurement
of preferences for outcomes which are adequately described by a single
system output variable. See Keeney and Raiffa (l977)for an excellent dis-
cussion of the measurement of multidimensional preference relationships.

87
Each of the approaches considered is based on the expected utility
hypothesis, but each requires a different level of precision in the mea-
surement of preferences. Techniques for the representation of decision
maker preferences and for the identification of preferred choices based
on the use of single valued utility functions are reviewed first. The
concept of an efficiency criterion, which allows the partial ordering
of possible alternatives on the basis of relatively unrestrictive
assumptions about decision maker preferences, is then introduced and
several commonly used criteria are examined. A more recently developed
efficiency criterion, stochastic dominance with respect to a function
(Meyer, l977a) is then described, and a procedure for determining the
interval measurements of decision maker preferences required for the
application of this criterion is presented. This new measurement
technique, developed as part of this study, permits the construction of
a representation of decision maker preferences which is only as precise
as the decision problem under consideration requires. Results of an
empirical test of this procedure, which are also presented below,
demonstrate that it is both accurate and flexible. In the final
section of the chapter, the incorporation of information on preferences
into the sample problem discussed in the preceding two chapters is
examined.

4.2 The Use of Single Valued Utility Functions to
Represent Decision Maker Preferences

 

 

Perhaps the most direct approach to the measurement of preferences
is to actually derive the decision maker's utility function. This re-

quires that a number of points in the utility function be determined by

88

direct elicitation. A curve is then fitted through these points, and
that curve is said to be the decision maker's utility function.

A utility function is a highly structured representation of a
decision maker's preferences. Like probabilities, however, preferences
are often not clearly formulated in the mind of the decision maker.
Therefore, the interview procedures used to elicit information on
preferences should be designed both to clarify and to structure the
decision maker's assessments of value. Several procedures have been
developed for the elicitation of information on preferences. The most
commonly used are reviewed in Officer and Halter (1968) and Anderson,
Dillon, and Hardaker (1977). Each procedure requires that a series of
choices be made between pairs of uncertain alternatives or between
certain and uncertain alternatives. If these choices are properly
structured, each should reveal enough information about the decision
maker's preferences to determine one point on his utility function.

Once a set of data points has been elicited, a curve is fitted
through them to obtain an explicit relationship between levels of
utility and all relevant levels of the system output variable. In
choosing a functional form, a number of factors should be considered.
Goodness of fit is, of course, important, since the estimated utility
function should conform as closely as possible to the information ob-
tained in the elicitation interview. Ease of estimation and the tracta-
bility of a function in the calculation of expected utilities should
also be considered. Polynomial specifications are, perhaps, the most
commonly used in empirical work, but a number of other alternative forms

have also been prepared (Lin and Chang, 1978).

89

Having estimated a decision maker's utility function, action
choices can be ordered by calculating the expected utility of each.
If the outcome associated with a particular management strategy, v*,
is described by a single discrete system output variable, y, the

expected utility of the strategy, EU(ylv*), is given by the expression:

n
EU(ylv*) = Z f(yilv*)u(y
i=1

where f(yilv*) is the probability of the it

i) 4.2

h possible outcome under
strategy v* and ”(yi) is the utility of that outcome. When the system
output variable, y, is continuous, the expected utility is defined by I

the expression:

EU(ylv*) = If; f(ylv*)u(y)dy 4.3
where f(ylv*) is the probability density function of y under strategy v*.
Consider, for example, the case in which the decision maker's utility
function is of the form

U(y) = MM 4.4
and the probability distributions associated with two strategies, A and
B, are as specified in Table 4.l. Since the system output variable is
discrete, equation 4.2 can be used to calculate the expected utilities
for these two alternatives. That for strategy A is

EU = .5'In(500)+.1 'ln(1000)+.l '|n('|500)+.3 'ln(2000) 4.5

A
= 6.81
while that for strategy B is

.2 ln(500)+.6 1n(1000)+.l ln(1500)+.l ln(2000) 4.6

EUB

= 6.88

For this decision maker, then, strategy A is preferred to strategy B.

90

Table 4.1 Probability Distribution Associated with
Two Alternative Action Strategies

 

 

 

 

System Output Probability
Level Strategy A Strategy B
500 .5 .2
l000 .l .6
1500 .l .l
2000 .3 .l

 

Representation of a decision maker's preferences with a single-
valued utility function has several serious shortcomings in an applied
decision analysis. With regard to preference measurement procedures,
the hypothetical choices posed in the elicitation interviews are, in
general, less complex and less interesting than those actually facing
a decision maker. As a result, it may be difficult to hold the full
attention of the respondent through a series of similar questions.
Furthermore, it can be argued that, because the types of choices made
during the elicitation interview bear little resemblance to those made
in real life, the value of the interview itself as a learning process
whereby the respondent can gain a better understanding of how he makes
decisions is limited. Other problems encountered in the elicitation
of the information required to construct single valued utility functions
are discussed in Officer and Halter (1968).

Still more serious problems arise as a result of the way empirically
estimated utility functions are generally used in a decision analysis.
Once a set of data points from a decision maker's utility function has
been elicited, a curve is fitted through its elements. Rare indeed is

the case in which the fit is perfect so that the parameter values of

91

the utility function can be known with certainty. Even if the fit were
perfect, shortcomings of the preference elicitation procedures make it
likely that the data points themselves include measurement errors.
Therefore, an empirically estimated utility function cannot be con—
sidered to be an exact representation of decision maker preferences.
Despite the possible sources of imprecision, however, a utility function,
once estimated, is usually treated as though it were an exact representa-
tion of preferences when alternative choices are ordered, and any absolute
difference in the expected utilities associated with two choices is

taken as a clear indication that one is preferred to the other. If a
utility function does not accurately reflect a decision maker's actual
preferences, this can result in the recommendation of a choice which is
not actually the preferred choice of the decision maker. When empirically
estimated single valued utility functions are used to order alternative
choices, then, there is a high likelihood that errors of this sort will

be made.

4.3 Efficiency Criteria and the Representation
of Decision Maker Preferences

 

The difficulties associated with the use of single valued utility
functions to order choices in a practical context have been the impetus
for the development of several efficiency criteria which overcome some
of the shortcomings identified above. An efficiency criterion is a
preference relationship which provides a partial ordering of feasible
action choices for decision makers whose preferences conform to certain
rather general specifications. As such, an efficiency criterion can be

used to eliminate some feasible choices from consideration without

92

requiring detailed information about the decision maker's preferences.
In many instances, the use of such a criterion may greatly reduce the
number of alternatives to be considered. If enough alternatives can be
eliminated, it may be possible for a final choice to be made on the
basis of direct comparisons of the distributions of outcomes associated
with each of the remaining alternatives.

First and second degree stochastic dominance are among the simplest
and most commonly used efficiency criteria. Both were formulated
independently by Hadar and Russell (l969) and Hanoch and Levy (l969).
First degree stochastic dominance holds for all decision makers who
prefer more of the system output to less--i.e., for all decision makers
having positive marginal utility with respect to the system output
variable. An alternative for which the associated distribution of the
system output variable is described by the cumulative distribution
function F(y) is preferred to a second alternative with associated
cumulative distribution G(y) by the criterion of first degree stochastic
dominance of

F(y)SG(y) 4.7
for all possible levels of y and if the inequality in 4.7 is a strict
inequality for at least some value of y. In Figure 4.1, for example,
F(y) dominates G(y) by this criterion, since it is always below and to
the right. Neither F(y) nor G(y) can be ordered with respect to H(y)
according to this criterion, since both are strictly greater than H(y)
for some system output levels.

While first degree stochastic dominance holds, in effect, for

all decision makers, second degree stochastic dominance places an

P(y)

 

93

G(y)

F(y)

 

Figure 4.l

Illustrations of First and Second
Degree Stochastic Dominance

H(y)

94

additional restriction on preferences. It requires that the marginal
utility of the system output variable be both positive and decreasing--
i.e., it requires that the decision maker's utility function be concave.
Given two alternatives having system output distributions defined by the
cumulative distribution functions F(y) and G(y), respectively, the first
alternative is preferred to the second under the criterion of second
degree stochastic dominance if

13’” F(x)dx51¥w G(x)dx 4.8
for all possible values of y and if the inequality in 4.8 is a strict
inequality for at least some value of y. In effect, this means that
the first alternative dominates the second if the area under cumulative
F(y) is always less than or equal to that under G(y). In Figure 4.1,
for example, F(y) dominates both G(y) and H(y) by this criterion, since
the area order this cumulative is less than that order either of the
others at all values of y. G(y) and H(y) cannot be ordered by this
criterion, however, since the area order H(y) is at times less than
that order G(y) and vice versa.

Other efficiency criteria depend on additional restrictions in the
maker's preferences or in the nature of the probability distributions
of system outputs. The mean-variance efficiency criterion (Markowitz,
l959) is simply a special case of second degree stochastic dominance in
which all probability distributions are normal. Third degree stochastic
dominance (Whitmore, 1970) is similar to first and second degree stochas-
tic dominance, but it requires the additional assumption that the
decision maker's utility function have a positive third derivative with

respect to the system output variable.

95

Once a particular criterion with its associated restrictions on
preferences has been specified, an ordering of any two alternatives can
be made strictly on the basis of properties of the two associated
probability distributions of the system output variable. Under such
an ordering, one alternative will dominate the other, or the criterion
will not be able to order the two alternatives and both will be con-
sidered efficient. If one alternative does dominate the other, it is
unanimously preferred by the class of decision makers for whom the
criterion applies. By making a series of pair-wise comparisons of all
alternatives under consideration and eliminating from consideration any
alternative which has been dominated, an efficient set of choices can
be determined for any finite set of alternatives. This set will contain
the preferred choice of any member of the class of decision makers for
whom the criterion applies.

The use of an efficiency criterion to order alternative choices is,
in many respects, preferable to the use of a single valued utility
function. No direct measurements of preferences need be made. Rather,
relatively easily accepted restrictions are simply imposed on the
decision maker's preferences. Unfortunately, however, none of the
efficiency criteria mentioned above is a particularly discriminating
evaluative tool. In an application of second degree stochastic
dominance by Anderson (1975), for example, twenty of forty-eight
randomly generated farm plans were in the efficient set. Furthermore,
though the restrictions on preferences required by most efficiency
criteria do not appear to be unduly strict, they often run counter to

empirical evidence. Again focusing attention on second degree stochastic

96
dominance, despite the fact that strong theoretical arguments have been
made for the near universality of concave utility functions (Arrow, 1971),
the weight of empirical evidence indicates that decision makers do at
times exhibit increasing marginal utility (Officer and Halter, 1968;
Conklin, Baquet, and Halter, 1977).

While the concept of an efficiency criterion is an attractive one,
then, efficiency criteria have not proved to be useful tools in practice.
There is a need for efficiency criteria which are both more flexible
and more discriminating than those described above. Furthermore, there
is a need for techniques for obtaining measures of decision maker
preferences which, though less precise than those used to construct a
single-valued utility function, facilitate the empirical determination
of whether or not a particular efficiency criterion adequately repre-
sents the preferences of a decision maker. In the sections which
follow, a more powerful efficiency criterion, stochastic dominance with
respect to a function (Meyer, 1977a), is introduced, and a method for
measuring decision maker preferences designed to be used in conjunction
with this criterion is presented.

4.4 Stochastic Dominance with Respect
to a Function

 

Stochastic dominance with respect to a function is an evaluative
criterion which orders uncertain action choices for classes of decision
makers defined by specified lower and upper bounds, r1(y) and r2(y), on
the absolute risk aversion function. The absolute risk aversion

function (Arrow, 1971; Pratt, 1964), r(y), is defined by the expression:

r(y) = -U" (y)/U'(y) 4.9

97
where u'(y) and u" (y) are the first and second derivatives of a
van Neumann-Morgenstern utility function u(y). In the most abstract
terms, values of the absolute risk aversion function are simply local
measures of the degree of concavity or convexity exhibited by a decision
maker's utility function. Since u'(y) is assumed to be positive, a
positive value of r(y) implies a negative value of u" (y) which in turn
implies a concave utility function. Similarly, the utility function
is convex at y if r(y) is negative. As such, the absolute risk aversion
also serves as a local indicator of the extent to which a decision maker
is risk averse or risk loving. Following Arrow's (1971) definition,
an individual is risk averse (loving) if, from a position of uncertainty,
he is unwilling (willing) to take a bet which is actuarially fair
(unfair).1 Concavity of the utility function and risk aversion are
synonymous under this definition, and both are implied by a positive
value of r(y). A negative value of r(y) implies both local convexity
of the utility function and risk loving behavior. Perhaps the most
important property of the absolute risk aversion function, however, is

that it is a unique measure of preferences, while a utility function is

 

1Arrow's definition of risk aversion has been the source of some
confusion, since risk aversion and risk preference have often been
equated with an aversion to and a love for gambling. Unless some
measure of the degree of gambling associated with a particular choice
is identified as a system output and included as an argument in a
decision maker's utility function, however, his choices are, by the
omission of this factor, assumed to be unaffected by the degree of
gambling involved. Arrow's concept or risk aversion refers only to
the characteristics of a utility function with a single argument. As
Friedman and Savage (1948) demonstrate, such a utility function can be
used to explain why gambling has utility or disutility in certain
situations without requiring that preferences for gambling per se be
measured.

98

unique only to a positive linear transformation.‘ In effect, then,
upper and lower bounds on a decision maker's absolute risk aversion
function define an interval measurement in his preferences. Stochastic
dominance with respect to a function orders choices on the basis of
such a measurement.

The major advantage of this criterion is that it imposes no re-
strictions on the width or shape of the relevant region of risk aversion
shape. The interval measurement can be as precise or imprecise as is
deemed necessary for a particular decision analysis. Negative as well
as positive levels of absolute risk aversion can lie within the risk
aversion interval at some or all levels of system output. Less flexible
efficiency criteria such as first and second degree stochastic dominance
can be viewed as special cases of this more general criterion. The
requirement under first degree stochastic dominance that the decision
maker have positive marginal utility places no restrictions on the
decision maker's absolute risk aversion function--i.e., r1(y) = -w and
r2(y) = m for all possible values of y. The requirement under second
degree stochastic dominance that marginal utility be decreasing as well
as positive, on the other hand, implies that r](y) = O and r2(y) = w for
all values of y.

More formally stated, stochastic dominance with respect to a

function is a criterion which establishes necessary and sufficient

 

1Because a utility function is unique only to a positive linear
transformation, u(y) and

u*(y) = a+bu(y). b>0
are strategically equivalent, though perhaps highly dissimilar, utility
functions. The absolute risk aversion functions of these two utility
functions are identical, however:

r(y) = -U" (y)/u'(y)

99

conditions for the distribution of system outputs defined by the cumu-
lative distribution function F(y) to be preferred to that defined by
the cumulative distribution function G(y) by all agents whose absolute
risk aversion functions lie everywhere between lower and upper bounds
r](y) and r2(y). As developed by Meyer (l977a), the solution procedure
requires the identification of a utility function u0(y) which minimizes

1
6 [G(y)-F(y)JU'(y)dy 4.9

subject to the constraint

r1(y)s -u"(y)/u'(y)2r2(y> y zto. 11 ' 4.10
The expression in equation 4.9 is equal to the difference between the
expected utilities of system output distributions F(y) and G(y).2 If,
for a given class of decision makers, the minimum of this difference is
positive, F(y) is unanimously preferred to G(y). If the minimum is zero,
it is possible for an agent in the relevant class of decision makers to
be indifferent between the two alternatives and they cannot be ordered.
Should the minimum be negative, F(y) cannot be said to be unanimously

preferred to G(y). In this case, the expression

J}, [F(y)-G(y)]u'(y)dy 4.11

 

1The range of system outputs is normalized so that all values of
y fall on the bounded interval [0, 1].

2This can be demonstrated in the following manner. Let f(y) and
g(y) be the probability density functions associated with F(y) and G(y)

:3 f(y)u(y)dy - x3 g(y)u(y)dy = I; [f(y)-g(y)]uly)dy

is the difference between the expected utilities associated with the two
distributions. Integrating by parts,

I; [i(y)-g(y)1u(y)dy = [F(y)-G(y)1u<y) ug-x; [F(y)-G(y)]u'<y>dy =

I; [G(y)-F(y)]u'<y)dy
since [F(O)-G(O)] and [F(l)-G(l)] are both equal to zero.

100

must then be minimized subject to 4.10 to determine whether G(y) is
unanimously preferred to F(y). It should be noted that a complete
ordering is not ensured by the criterion. It is possible for the
minimum of both (4.9) and (4.11) to be negative, which implies that
neither distribution is unanimously preferred by the class of decision
makers being considered.

Meyer uses optional control techniques outlined in Arrow and Kurz
(1970) to derive the necessary and sufficient conditions for the solu-
tion of this problem. These conditions do not represent a closed form
solution. Rather, they define a rule for determining the absolute risk
aversion function of the utility function which minimizes 4.9--a rule
can be applied if the relatively unrestrictive assumption that [G(y)-F(y)]
changes a sign finite number of times over the interval [0, l] is met.
The following theorem (Meyer, l977a, p. 333) is the basis for that rule:

Theorem An optimal control -u6(y)/ub(y) which minimizes

I; [G(y)—F(y)1u'(y)dy subject to r](y)s[u3(y)/u5<y)12r2<y)
and ub(0) = 1 is given by:

-UOU) = {7'1 (Y) if I; [G(X)-F(x)]u6(x)dx<0

“OIY7 r2(y) if ;; [G(x)-F(x)]ub(x)dx20

This theorem implies that the value of the absolute risk aversion
function which minimizes the difference in the expected utilities
associated with F(y) and G(y) is determined at any point y* by the
sign of the objective function integrated from y* forward to 1 using
the optimal control. Furthermore, it implies that the value of the

absolute risk aversion function is always either r1(y) or r2(y).

101

Application of the rule requires that the solution procedure work
from back to front as is commonly done in dynamic programming. A simple
example should help to illustrate how this can be done. Consider the
two cumulative distribution functions shown in Figure 4.2. Neither
dominates the other by first or second degree stochastic dominance. Let
the lower and upper bounds on the absolute risk aversion functions both

1 The utility function associated

be constants--r1 = .001 and r2 = .002.
with each of these can be shown to be of the negative exponential form
(Pratt, 1964), so that
-r.y

ui(y) =-e ' i=1, 2, 4.12
where ri is the upper or lower bound absolute risk aversion level.
The function [G(y)-F(y)] is graphed in Figure 4.3. Between y=5000 and
y=7000 its value is negative, and above y=7000 its value is zero.
According to the theorem above, this implies that for values greater
than 5000 ri(y) = .001 is the optimal control. Calculating the value

of the objective function from y=5000 and upward,

1:000 [G(y)-F(y)JU5(y)dy 4.13

= f7ooo
5000

-.00194

(-1/3)(.001)e"°°'ydy

we see that it is negative.2 The solution rule indicates that the

optimal control remains at r(y)=.OOl. The procedure continues back

 

1There is no requirement that they be constant. This assumption
is made here to facilitate calculation.

2The value of the integral is zero over regions where [G(y)-F(y)]=0.

102

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103

G(y)-F(y)

 

l—l
l I
| l
I l
. l
L I % I I i I 1
l
2000 4000l l 6000 I 8000 10000“v
' I 1 l
I l l l
l 2 I I
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Figure 4.3 A Graph of the Function [G(Y)-F(Y)]

104

until the point where y = 4000. At this point

’2000 [G(y)-F(y)JU6(y)dy 4.14

= [4382
4000

= 0.

(-1/3)(.001)e"°°'ydy-.00194

Therefore, the optimal control switches to y = .002 for values of y less

than 4000. The procedure continues back with the same optimal control until

If, [G(y)-F(y)]ub(y)dy 4.15

3 -.0
= 12388 (l/3)(.002)e 02ydy

.00528

Since the value of the objective function is positive, distribution F(y)
is preferred to G(y) by all decision makers whose absolute risk aversion
functions lie everywhere between r1 = .001 and r2 = .002. The utility
function which minimizes the objective function has an absolute risk
aversion function such that:

r(y) = .002 when yS4000 4.16

.001 when y>4000
Note that this utility function does not have constant absolute risk
aversion.

Stochastic dominance with respect to a function is a remarkably flex-
ible evaluative criterion which has considerable potential for use in the
analysis of practical decision problems. Unlike other efficiency cri-
teria, it does not require that fixed restrictions be imposed on the
representation of the decision makers' preferences: and, because the
bounds on absolute risk aversion can be as close or as far apart as desired,
stochastic dominance with respect to a function can be used to order more

choices than can be done with other criteria. Unlike a single valued

105
utility function, it does not require that an exact representation of
the decision maker's preferences be specified. Furthermore, stochastic
dominance with respect to a function is relatively easy to apply. A com-
puter program which implements the solution procedure defined above has
been developed by Meyer, and a modified version of that program is pre-
sented in Appendix B.

4.5 An Interval Approach to the Measurement
of Decision Maker Preferences

 

 

Stochastic dominance with respect to a function is a powerful
analytical tool. Before it can be used in an applied context, however,
an operational procedure must be developed for the determination of
lower and upper bounds on a decision maker's absolute risk aversion
function. A technique for making such interval measurements of decision
maker preferences is introduced in this section. This procedure uses
information revealed by a series of choices between carefully selected
distributions to establish lower and upper bounds on an individual's
absolute risk aversion function. The degree of precision with which
preferences are measured--i.e., the size of the interval between the
lower and upper bound functions--can be specified directly in accordance
with the characteristics of the problem under consideration. At one
extreme the interval can be of infinite width, and at the other extreme
it can converge to a single line.

The procedure for constructing interval measurements of decision
maker preferences is based on the fact that under certain conditions a
choice between two outcome distributions defined over a relatively narrow
range of system output levels divides absolute risk aversion space over

that range into two regions: one consistent with the choice and one

106

inconsistent with it. The level of absolute risk aversion at which the
division is made depends solely on the two distributions-—i.e., their
properties define the two regions. The decision maker's preferences, as
revealed by his ordering of the two distributions, however, determine

into which of these two regions his level of absolute risk aversion is
said to fall. By confronting the decision maker with a series of choices
between carefully selected pairs of distributions, the region of absolute
risk aversion space which is consistent with the decision maker's prefer-
ences can repeatedly be divided. With each choice a portion of that region
is shown to be inconsistent with the decision maker's preferences, and the
interval measurement for the level of absolute risk aversion is narrowed.
The procedure continues until a desired level of accuracy is attained.
Upper and lower limits for the level of absolute risk aversion are deter-
mined at a number of income levels. These values are used to estimate
upper and lower limits for the absolute risk aversion function over the
relevant range of incomes.

The validity of the statement that a choice between two distributions
is, under certain conditions, the basis for a division of absolute risk
aversion space into regions consistent and inconsistent with a decision
maker's revealed preferences can be demonstrated using concepts developed
by Meyer in "Second Degree Stochastic Dominance with Respect to a Function."
In that paper Meyer (l977b, p. 483) proves the following theorem:

Theorem For cumulative distributions F(y) and G(y)

1% [G(x)-F(x)]dk(x)zo yz[0’ 1] and
I; [G(x)-F(x)]dk(x)=0 only if

I; [G(x)-F(x)]dk(x)SO yz[0, l]

107
The theorem states that F(y) is preferred to G(y) by all decision makers

more risk averse than the utility function k(y) and that decision makers
having utility function k(y) are indifferent between the two distributions
only if G(y) is preferred to F(y) by decision makers less risk averse than
k(y).1 The function k(y), then, can be considered to be a boundary func-
tion, since it separates a class of decision makers who prefer F(y) from
a class who prefer G(y).

If the distributions F(y) and G(y) are defined over a narrow range
of system output levels and if the decision maker's absolute risk aversion
function can be approximated by a constant value A over that range, pre-
ference for F(y) implies that A is greater than or equal to the minimum
value of the absolute risk aversion associated with k(y). Otherwise, the
decision maker would be less risk averse than k(y) and his choice would
be inconsistent with expected utility maximization. Preference for G(y),
on the other hand, implies that A is less than or equal to the maximum
value of the absolute risk aversion function associated with k(y), since

F(y) is preferred by all decision makers more risk averse than k(y). It

 

1Using Pratt's definition of risk aversion in the large, a decision
maker with utility function u(y) is more risk averse than k(y) if

_kll < _ull Uy’
k'éy) ' U'éy)

while he is less risk averse than k(y) if

-k"§y; 2 -u"{x; uy
k y u' y
Meyer (l977b) shows that F(y) is preferred to G(y) by all decision makers
more risk averse than k(y) if
1% [G(x)-F(x)]dk(x)20 Uye[O, 1]

and if the inequality is strict for some value of y. He also shows that
G(y) is preferred to F(y) by all decision makers less risk averse than k(y) if

r; [G(x)-F(x)]dk(x)50 Uye[0, 1]
and if the inequality is strict for some value of y.

108
should be noted that the assumption that a decision maker's absolute

risk aversion function can be adequately approximated by a constant value
over a narrow range of system output levels is critical here. The theorem
stated above does not imply that decision makers who prefer F(y) to G(y)
are more risk averse than k(y); nor does it imply that decision makers

who prefer G(y) to F(y) are less risk averse than k(y). With the assump-
tion of constant absolute risk aversion in the neighborhood of a given
system output level, however, it can be inferred that decision makers who
prefer F(y) to G(y) are not less risk averse than k(y) and those who pre-
fer G(y) to F(y) are not more risk averse than k(y).

The properties of a utility function which serves as a boundary func-
tion between two distributions are dependent upon the two distributions.1
By careful selection of distributions, a boundary function can be placed
anywhere in risk aversion space. A series of questions can be devised,
then, which allows the repeated reduction of region of risk aversion

space consistent with the revealed preferences of a decision maker, thereby,

narrowing the interval measurement of absolute risk aversion.

A simple example should help to illustrate how the procedure works.
Let the boundary function for two distributions, k1(y), have an absolute
risk aversion function which lies everywhere on the interval (A],A2),
and let the first distribution be preferred by decision makers more risk
averse than k(y) while the second is preferred by those less risk averse
than k](y). In this case the decision maker prefers the first distribu-
tion. Given the assumption of constant absolute risk aversion over

this range of system output levels, this implies that his level of

 

1A boundary function does not exist for each pair of distributions.
One would not exist, for example, if one distribution dominates the other
by first degree stochastic dominance. Similarly, the existence of one
boundary function does not preclude the existence of others.

109
absolute risk aversion between y1 and y2 lies everywhere above A], as
is shown in part (a) of Figure 4.4. If choices between two additional
pairs of distributions indicate first that the level is greater than

A2 and second that it is less than A it can be inferred that the

3.
decision maker's level of absolute risk aversion over this range of
system outputs lies between A2 and A3, as is shown in part (c) of

Figure 4.4. With each choice, then, a portion of the region of absolute
risk aversion space consistent with prior choices is shown to be incon-
sistent with the decision maker's preferences and the interval measure-
ment of absolute risk aversion is narrowed. Choices are presented to
the decision maker until a desired level of accuracy is attained. An
interval measurement of a decision maker's absolute risk aversion
function can be constructed over a much broader range of system outputs
by making interval measurements in the neighborhood of several system
output levels and connecting known portions of the upper and lower

bound absolute risk aversion function with linear segments, as is done
in Figure 4.5. In this case direct interval measurements have been made

in the neighborhood of three system output levels: 3,000; 10,000; and
17,000.

4.6 Implementation of the Procedure

 

The discussion above describes an iterative approach to the con-
struction of interval measurements of a decision maker's absolute risk
aversion function. It does little, however, to answer the basic opera-
tional questions of how appropriate distributions can be selected and
of how the boundary interval for any pair of distributions can be

identified. The techniques used to implement the interval approach to

110

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Figure 4.5 Upper and Lower Bound Absolute Risk Aversion
Functions Based on Three Interval Measurements

112
the measurement of decision maker preferences are described briefly in
this section. A more technical explanation of them is given in Appendix B.

The first step in the implementation of the procedure is to estab-
lish a measurement scale, which is defined by a number of reference
levels for absolute risk aversion. In Figure 4.6, for example, four
reference levels are specified: -.0001, .0001, .0005, and .0010. This
scale or grid determines the accuracy with which absolute risk aversion
can be measured. Any number of reference levels can be specified. The
intervals between them can be as wide or narrow as is deemed necessary,
and they need not all be of equal size. In many cases it may be desirable
to put more fineness or detail in the measurement scale in regions of
risk aversion space where it is believed a priori that the decision
maker's level of absolute risk aversion is likely to fall.

Next, a set of distributions which will serve as the basis for the
choices made by the decision maker must be constructed. These distri-
butions should be defined over a relatively narrow range of system
output levels, since the decision maker's level of absolute risk aver-

] As described in

sion is assumed to be constant over that range.
Appendix B, they are constructed in a random manner by generating
several hundred random numbers from a specified distribution and
grouping them into sets of six observations each. Each is a distribu-
tion of outcomes, and each element is said to have a 1/6 probability of
occurrence. Only six elements are included in each distribution because

more complex distributions might make decision makers' choices unduly

 

1Experience to date indicates that a range of five to ten percent
the size of the entire range of system output levels over which pre-
ferences are to be measured is adequate.

113

r(y)

 

 

 

 

r(y) = .0010
r(y) = .0005
r(y) = .0001

‘ y

 

 

Figure 4.6 An Absolute Risk Aversion
Measurement Scale

r(y) = -.0001

114
difficult. Distributions with fewer elements, on the other hand, may
not be rich enough to make the choices interesting. The use of six
element distributions also facilitates explanation of the choice situa-
tion to the decision maker, since the probability of any one element
occurring can be equated directly to the probability of obtaining a
specified number of dots on a single role of a die. Three distributions
constructed in this manner are shown in Table 4.2.

Once a measurement scale has been specified and sample distribu-
tions have been constructed, the interval on the measurement scale
which contains the risk aversion function associated with the boundary
function for each pair of distributions must be identified. The proce-
dure by which this is done is described in detail in Appendix B. In
essence, criteria identified by Meyer (1977b) are used to identify the
highest reference level on the measurement scale, A], such that all
decision makers 1g§§_risk averse than A1 prefer one distribution and the
lgwg§t_reference level in the measurement scale, A2, such that all
decision makers more risk averse than A2 prefer the other distribution.
It follows that the absolute risk aversion function associated with the
boundary function for the two distributions lies everywhere within the
interval (A],A2), which is called a boundary interval. The boundary
intervals for each pair of distributions from Table 4.2 are given in
Table 4.3. In this case each pair has a different boundary interval.
It should be noted that more narrow boundary intervals could have been
identified if a more detailed measurement scale had been specified.

After the boundary interval has been identified for each pair of

distributions, a series of questions is formulated. Each question asks

115

Table 4.2 Sample Distribution from a Normal Distribution

 

 

 

 

 

With u = 3000 and o = 1000
Distributiona
l 2 3
2100 1000 1750
2400 2050 1950
2550 2650 2500
3100 3800 2750
3250 3900 3950
3450 5200 4000
u = 2808 u = 3100 u = 2817
o = o = 1370 o = 883

 

aThe elements of each distribution are rounded to the nearest
50 units.

116

 

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117

the decision maker to indicate which of two selected distributions he
prefers. His responses serve as the basis for the interval measurement
of absolute risk aversion. Each question focuses on a particular
interval of risk aversion space--an interval which corresponds to the
boundary interval for the two distributions the decision maker is asked
to rank. Let the first question in the example being developed here be:

Compare distributions 1 and 2 and indicate which

you prefer.
This question focuses on the interval (.0001, .0005).1 If distribution 1
is preferred, the information in Table 4.2 indicates that the decision
maker's level of absolute risk aversion is greater than .0001. If
distribution 2 is preferred, his level of absolute risk aversion is
below .0005.

The choice of a second question will depend on the respondent's
answer to the first. If the respondent prefers distribution 1, for
example, it makes little sense to ask him to rank distributions 1 and 3.
Such a question indicates whether his level of absolute risk aversion
is greater than -.0001 or less than .0001. Since his level of absolute
risk aversion is already said to be greater than .0001, this information
would be of value only as a consistency check. A choice between distri-
butions 2 and 3, on the other hand, would add to our kn0wledge. If
distribution 2 is preferred, the decision maker's level of absolute
risk aversion can be said to fall on the interval (.0001, .0010), while

if distribution 3 is preferred, the level must be greaterlthan .0005.

 

1It is advisable to focus the first question on a boundary interval
at or near the middle of the measurement scale.

118

Because the order of questioning is contingent upon the decision maker's
responses, the interview schedule takes a form similar to that of a
programmed learning text, as is exemplified in Figure 4.7.

Clearly, the set of questions specified in this example does not
lead to an accurate measure of a decision maker's preferences. By
specifying a finer measurement scale, however, and by asking a series
of three or four questions rather than only two, much more accurate
measurements can be made. It should also be noted that these questions
serve as the basis for a measurement of absolute risk aversion only for
system output values between 1000 and 5000. Similar sets of questions
must also be constructed to measure preferences in the neighborhood of
several other system output levels so that upper and lower bound
absolute risk aversion functions can be constructed for a wider range
of possible system output levels.

Figure 4.5 above shows upper and lower bounds on an absolute risk
aversion function based on interval measurements at three system output
levels. Note that the slope of the absolute risk aversion function is
not restricted. In this example it rises, then falls, and at lower
levels of system output the measurement interval includes negative as
well as positive values. When absolute risk aversion functions are
derived from empirically estimated utility functions, on the other hand,
their form is often severely limited by the functional form used to
estimate the utility function. It should also be noted that the inter-
val approach to the measurement of preferences also avoids‘another common
problem encountered in the estimation of single-valued utility functions.

Because all questions posed require a choice between two uncertain

119

1. Compare distributions 1 and 2 and circle
the one you prefer.

If you prefer distribution 1 go to question 3,
otherwise go to question 2.

2. Compare distributions 1 and 3 and circle
the one you prefer.

3. Compare distributions 2 and 3 and circle
the one you prefer.

Figure 4.7 A Sample Questionnaire

120

prospects, biases due to preference for an aversion to gambling pg§_§g
are eliminated.

One final step is required for the implementation of the interval
approach to the measurement of preferences. Although Meyer's (1977a)
analytical development of stochastic dominance with respect to a
function depends only on absolute risk aversion functions, the computer
program he has developed to implement the criterion requires that
utility functions having absolute risk aversion functions corresponding
to lower and upper bounds, r1(y) and r2(y), be specified by the user.
Given the definition of the absolute risk aversion function,

- - ull
r(y) - u y 4.17
the following system of differential equations, which relates levels

of absolute risk aversion to values of u(y) and u'(y) can be derived:

d My) - 1 0 HM
337 [WM] ’ [4‘01 0][U'(y):l 4'18

Once initial values of u(y) and u'(y) have been specified, recursive
numerical integration techniques can be used to solve for u(y) and

u'(y) at any level of system output.1

Utility functions associated
with the lower and upper absolute risk aversion functions are repre-
sented by a table look-up routine in the computer algorithm which

implements the stochastic dominance with respect to a function criterion.
Corresponding values of y and u(y) determined by numerical integration

serve as data points for each table look-up function.

 

1The initial values of u(y) and u'(y) correspond to the arbitrary
scale factors of a van Neumann-Morgenstern utility function. See
Appendix B for a listing of the computer program which performs the
numerical integration.

121

4.7 ‘An_Empirical Test

 

A simple experiment was designed and conducted to test the efficacy

of the interval approach to the measurement of decision maker preferences."

Three questionnaires were administered to a group of graduate research
assistants from the Department of Agricultural Economics at Michigan
State University. The first questionnaire employed the procedure
described above to obtain an interval measurement of each subject's
absolute risk aversion function. The second questionnaire was used to
elicit information required for the construction of a single-valued
utility function for each subject.1 Finally, the third questionnaire
asked the respondent to make a series of six choices between pairs of
distributions, each distribution being comprised of six elements and
each being defined on the interval over which preferences had been
measured. Information from the first two questionnaires was used to
predict the choices made by each respondent in the third questionnaire,
and these predictions were compared to the actual responses. In this
way the accuracy of each of the two approaches to the measurement of
preferences was tested.

In evaluating each approach, two criteria were considered: the
number of correct predictions and the number of choices for which a
definite ordering was made. A prediction was said to be correct if the
respondents' actual choice was not excluded from the efficient set of
choices and incorrect if it was excluded. The preference measure having

the highest proportion of correct predictions was said to be the more

 

1Because it is the most commonly used and most easily implemented
elicitation technique, the ELCE method (Anderson, Dillon, and Hardaker,
1977) was used to identify points on the decision maker's utility
unctions.

122

accurate according to this criterion. Concern with the proportion of
correct predictions is analogous to concern with the probability of

Type I error in a statistical test, the latter being the probability
that a true statement will be judged to be false and be rejected. This
measure of accuracy is not a good indicator of the relative discriminatory
power of preference measurements based on these two approaches. The
criterion of first degree stochastic dominance, which holds for all
decision makers who prefer more of the system output to less, should
never excludeaapreferred choice from the efficient set and so should

be perfectly accurate according to the criterion defined above. Often,
however, it also fails to exclude many choices from the efficient set.

A single-valued utility function, on the other hand, is the basis for

a complete ordering of choices--i.e., it always leads to an efficient

set having a single element. Therefore, the number of choices actually
ordered was also considered. Concern with this measure of discriminatory
power is analogous to concern with the Type 11 error associated with a
statistical test, which is the probability that a false statement will

be judged to be true and not rejected.

Clearly there are trade-offs between the accuracy and the discrimi-
nating power of a preference measurement. Unlike other measurement
techniques and evaluative criteria, the combined use of interval pre-
ference measurements and stochastic dominance with respect to a function
permits explicit consideration of these trade-offs. As the precision
of the interval measurement increases, it becomes a more discriminating
basis for the ordering of choices; but the probability of excluding

preferred choices from the efficient set also increases.

123

Such trade-offs between accuracy and discriminatory power were
also analyzed in the experimental test of the interval approach to the
measurement of preferences. Direct interval measurements of absolute
risk aversion were made at three levels of income--the relevant system
output variable in this instance. These measurements were based on a
sequence of four questions at each income level. By constructing
interval measurements on the basis of information available at the end
of each question, however, four preference measurements-~each more pre-
cise than the one which preceded it--were made for each subject.

Nine of ten subjects correctly completed all three questionnaires.
Since each subject made six choices on the third questionnaire, each
preference measurement was used to predict a total of fifty-four
choices. The results of the experiment are presented in Table 4.4.

They show that there is a clear trade-off between accuracy and discrimi-
natory power. First degree stochastic dominance and the single-valued
utility function are at opposite extremes in this trade-off relation-
ship, and the interval measurements are arrayed between the two.
Several factors should be noted. With regard to the accuracy of the
interval measurements, it falls at a relatively constant rate as the
number of questions posed increases, but even at the higher levels of
precision it exceeds that realized with the single-valued utility function.
The discriminating power of the interval measurements, on the other hand,
increases dramatically as the number of questions asked at each income
level increases. First and second degree stochastic dominance, on the
other hand, clearly do not discriminate well among the distributions

which were the basis for the decision makers' choices.

124

 

 

 

 

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125

It should be noted that these results represent but one test of
the interval approach to the measurement of decision maker preferences.
Results presented in the next section and in Chapter V provide addi-
tional evidence of the power of this approach, but further experimenta-
tion is needed. It should also be noted that few attempts have been
made to apply this technique. With more applied experience may come
refinements that will improve the discriminatory power of preference
measurements based on this approach without leading to increases in the

probability of excluding the preferred choice from the efficient set.

4.8 An Application

 

In order to test the interval approach to the measurement of
decision maker preferences in a more practical setting, questionnaires
implementing the procedure were administered to seventeen farmers who
were all participants in an extension workshop on cash grain marketing
strategies. The questionnaires were viewed as an exercise in the
workshop--an exercise designed to help individuals think systematically
about how they make decisions. Interval measurements of absolute risk
aversion were made in the neighborhood of four income levels: ~$3,000,
$7,000, $17,000, and $27,000. Each measurement was based on a series
of three questions. In addition, the respondents were also asked to
make a series of choices between distributions,as was done in the experi-
ment described in the preceding section.

The farmers had little difficulty in completing the questionnaires,
and they seemed to find the choices to be interesting. The range of
responses was quite broad. Individuals within the sample of seventeen

ranged from the extremely risk averse to the extremely risk loving.

126

Several discernable patterns did emerge, however. Most decision makers
exhibited increasing absolute risk aversion over the lower income
levels and decreasing absolute risk aversion at higher levels. For
most, the interval measurement of absolute risk aversion included
negative values at some level of income. In fact, only four of the
seventeen decision makers had lower level absolute risk aversion
functions which were everywhere non-negative. This casts serious

doubt upon the applicability of a criterion such as second degree
stochastic dominance which is valid only for decision makers who are
risk averse at all system output levels.

Choices made in the final section of the questionnaire were pre-
dicted remarkably well by the preference measures. Ninety-one out of
102 possible choices were predicted correctly for a success rate of
.892. This compares quite favorably with that obtained in the more
carefully controlled experiment with student subjects. To test the
discriminatory power of the preference measures, that derived for each
decision maker was used to order a set of thirty-three distributions,
none of which was dominated by any other by the criterion of first
degree stochastic dominance. The resultant efficient sets ranged in
size from one to twenty-three, with the average size being 10.5. Clearly
this represents a sizeable reduction in the size of the efficient set
over that attained with first degree stochastic dominance. The second
degree stochastic dominance efficient set for these thirty-three dis-
tributions had only five elements. It must be remembered, however, that
this criterion is valid only for four of the seventeen farmers whose
preferences were measured. For these four individuals the size of the

efficient set averaged only 2.5.

127

Upper and lower bound absolute risk aversion functions for
three representative decision makers are shown in parts a, b, and c
of Figure 4.8. The interval measurement for decision maker A declines
across the relevant range of net income levels. That for decision
maker B rises and then falls, while that for decision maker C is con-
stant and then rises at the highest income level. Decision maker C is
one of the four individuals in the sample who is everywhere risk averse.

The distributions of net income levels associated with each of
the two strategies defined in Section 3.3 of Chapter III were ordered
for each of these three decision makers using the criterion of stochastic
dominance with respect to a function. It will be recalled that Strategy 1
calls for no land rental and for the planting of all acreage in soybeans.
Strategy 2 calls for the rental of 240 acres and for a more balanced
crop mix. Strategy 1 is preferred to Strategy 2 by decision maker A.
Neither distribution dominates the other given the preferences of
decision makers B and C.

In themselves these results are not particularly interesting. They
do demonstrate, however, that the interval measurements of decision
maker preferences can lead to different efficient sets for different
decision makers. They also show that remarkably dissimilar distributions
may be included in an efficient set. The two strategies considered here
bear little resemblance to each other. One could be called extremely
cautious and the other moderately risky. For decision maker A it is
clear that the cautious approach is preferred. For the other two
decision makers the preferred choice is not so clear-cut. If these two

strategies were but two of many being considered and if a single-valued

128

 

 

 

 

 

 

 

r(y)
.0008~————
\
.0006— \
\
.0004— \
\
.00021— \ \ \\
\3 12000 \ -— — —
J l y a ‘ g L a 1 L
-.4000 o 4000 8000 _ \ \ 16000 21m0_ 2_4000 28000 1!
”0001—
aInterval Measurement of Decision Maker A
r(y)
.0008—
.0006—
,/""“""
.m— / \ \
\ \——-—-
.0002— /
p- —— — — — ——f
1 1 a a J/ ’ L ‘1‘ \g L
-‘19‘2— 1 _ .3202 __fl92/ ’12000 16000 20000 24000 28000
:1
-.0002-—
bInterval Measurement for Decision Maker 8
r(y)
em- //I_
.0006- /
.0004 — /
..______.____ ___ _______- //
OOOZF- a’ ”
° /
. /
a A A l l l ’1 l 1.—
-4000 0 4000 8000 12000 16000 20000 24000 28000
.1!
-.0002-

 

Figure 4.8 Interval Measurements of Absolute Risk

cInterval Measurement for Decision Maker C

Aversion for Three Decision Makers

129

utility function were being used to identify a preferred choice, one
would have been eliminated from consideration. As the experimental

results reported in the preceding section indicate, however, there is
a relatively high probability that the preferred strategy would have

been the one eliminated.

CHAPTER V

COMPUTATIONAL PROCEDURES FOR THE IDENTIFICATION
OF PREFERRED CHOICES

5.1 Introduction

 

Techniques developed in the preceding chaptensfor the determination
and representation of subjective probability distributions and for the
measurement of decision maker preferences provide the information
required to order any two specified action choices. In most decision
situations, however, a large if not infinite range of choices is open to
the decision maker. As a result, some systematic technique for the
identification and evaluation of a large number of possible strategies
is needed in many applied decision analyses. Such a technique should
be flexible enough to be applicable in a wide range of decision situa-
tions without requiring that important simplifying assumptions be made
concerning preferences, probabilities, and the nature of the problem
itself. It should serve as an aid in solving practical problems,
without forcing the decision maker to alter his conceptualization of
the problem at hand.

In this chapter a computational procedure designed to meet these
needs is formulated, and its implementation is discussed. This proce-
dure integrates concepts and operational techniques related to problem
formulation, the determination of subjective probability distributions,
and the measurement of decision maker preferences developed in the pre-
ceding chapters. It is a decision aid which is both powerful and

highly flexible.
130

131
In subsequent sections of this chapter, existing computational
procedures for the identification of preferred choices are first
reviewed, and their strengths and weaknesses are identified. The pro-
cedure developed for this study is then introduced and described in
detail. Finally, the procedure is applied to the sample problem dis-

cussed in the preceding three chapters.

5.2 A Review of Existing Computational Procedures

Mathematical programming models are commonly used in the analysis
of complex decision problems when the assumption of perfect knowledge
can reasonably be made. They are analytically elegant, computationally
efficient, and easily adapted for use in a wide range of decision situa-
tions. A number of difficulties are encountered, however, when mathe-
matical programming models are employed in the analysis of decisions
made under uncertainty--difficulties related to problem formulation, to
the determination of the probability distributions for system output
variables associated with alternative strategies, and to the representa-
tion of decision maker preferences. Despite such difficulties, mathe-
matical programming techniques are the basis for the most commonly
used computational procedures for the identification of preferred
choices under uncertainty. I

Quadratic programming (Markowitz, 1959; Freund, 1956) is
perhaps the most familiar and the most widely accepted mathematical
programming technique for the analysis of decisions made under uncer-
tainty. Conceptually it is an attractive tool because it is so closely

linked with mean-variance analysis, which has been the basis for a wide

132
range of theoretical developments, and because when used in a para-
metric programming mode it can, under certain conditions, be used to
identify an efficient set of strategies which includes the preferred
choice of any risk averse decision maker.1 With respect to practical
considerations, it is an attractive technique because the formulation
of quadratic programming problems is only slightly more complex than
that of a standard linear programming problem and because quadratic
programming packages are available in most computer systems. The use-
fulness of quadratic programming in an applied decision analysis is
severely limited, however, by a number of other factors.

With regard to problem formulation, standard quadratic programming
models require that input-output relationships be linear and additive
and that all controllable system input levels be perfectly divisible.
In many practical decision situations these assumptions simply do not
correspond closely with reality. Equally serious are the limitations
imposed by the standard quadratic programming model on the definition
of a management strategy. Decisions are analyzed as though they were
inflexible, though, as was noted in Chapter II, one of the most important
characteristics of choices under uncertainty is that they are often
adaptive in nature. Finally, though computer codes which implement
quadratic programming algorithms are readily available, many limit
the size of the problem which can be considered.

With regard to the determination of probability distributions,

quadratic programming requires that they be determined analytically,

 

1As was noted in Section 4.3 of Chapter IV, the criterion of
mean-variance efficiency is a special case of second degree stochastic
dominance. The mean-variance efficient set is identified by parametric
quadratic programming.

133

which may greatly limit the types of stochastic factors which can be
considered in an analysis and may also limit the complexity of the model
used to represent a particular stochastic process. Furthermore, within
a quadratic programming framework, the distributions of exogenous sys-
tem inputs and of system output variables are described only by means,
variances, and covariances. Implicitly, then, all distributions are
assumed to be normal. When this assumption does not hold, quadratic
programming may eliminate from consideration the preferred choices of
some decision makers (Tsiang, 1972; Robison and King, 1978).

Finally, with respect to the representation of decision maker
preferences, quadratic programming requires that the decision maker's
utility function be of the quadratic or negative exponential form if
a single preferred choice is to be identified. When parametric quadratic
programming is used to identify a mean-variance efficient set, on the
other hand, the efficient set holds only for risk averse decision
makers. It is possible that a decision maker's preferences cannot be
adequately represented under any of these assumptions.

In response to some of the shortcomings, several linear programming
alternatives to quadratic programming have been proposed. These include
the game theoretic (McInerney, 1969; Hazell, 1970; Low, 1974) and focus-
loss (Boussard and Petit, 1967) approaches and the MOTAD model developed
by Hazell (1971). All these models can be solved using standard linear
programming algorithms, and none requires that stochastic returns be
normally distributed. They impose fewer limitations on the size of_
problems which can be considered and all are more amenable to the
relaxation of restrictions on linear constraints and production pro-

cesses and on divisibilities of choice variables through the use of

134

separable programming and mixed integer programming techniques. Each
does have serious shortcomings, however. Covariance between returns
for different activities is ignored in both the MOTAD and focus-loss
models. This can lead to a serious misrepresentation of the distribution
of outcomes associated with any particular action choice. More impor-
tant, however, the links between the decision criteria used in these
approaches and the expected utility hypothesis are much weaker than is
the case with quadratic programming. The safety-first behavioral
assumption implied by the focus-loss approach and employed in many
applications of the game theoretic model is especially difficult to
reconcile with the axioms underlying the expected utility hypothesis.
The Risk Efficient Monte Carlo Programming (REMP) model developed
by Anderson (1975, 1976) is in nearly all respects a more attractive
alternative to quadratic programming as a decision aid in practical
situations. The REMP model employs Monte Carlo programming techniques
(Donaldson and Webster, 1968) to construct a large number of feasible
management strategies in a random fashion. The distribution of total
net returns associated with each strategy under consideration is deter-
mined analytically under the assumption that distributions of net
returns for each activity and distributions of total net returns for
each strategy are members of the beta family. Covariance between
returns for each activity is considered explicitly. Under this proce-
dure, a cumulative distribution function is constructed for total net
returns associated with each strategy and the criterion of second
degree stochastic dominance is used to identify an efficient set of

choices. The REMP model allows considerable flexibility in the

135
representation of probability distributions, since the beta distribution

can take a variety of forms. The model also places few restrictions on
decision maker preferences. The criterion of second degree stochastic
dominance requires only that the decision maker be risk averse at all
levels of system output. With regard to problem formulation, the REMP
model also has several distinct advantages. Most notably, the use of
Monte Carlo programming allows, at least partially, the relaxation of
restrictive assumptions of linearity, additivity, and perfect divisibility
required by most mathematical programming models. Despite these advan-
tages, the usefulness of the REMP model is greatly limited by the fact
that second degree stochastic dominance is not a very discriminatory
criterion. In many instances the size of the efficient set identified
with the REMP model is so large that the task of selecting a single
preferred strategy may be prohibitively difficult.

None of these alternatives to linear programming resolves the
problems associated with the incorporation of explicit consideration
of flexibility into the decision analysis. Recursive programming
techniques (Day, 1963; Heidhues,l966) do permit the consideration of
such factors within a linear or quadratic programming framework. The
behavioral constraints, feedback rules, and repeated optimization which
characterize this approach are used to model the process of planning,
decision making, and action. The constraints and rules of thumb which
drive the model through this process--the desired output of a decision
analysis--must be determined exogenously, however. The value of the
recursive programming as an aid to decision makers, then, is limited.

Stochastic programming (Cocks, 1968; Rae, 1971) is conceptually a more

136

attractive approach to the resolution of this problem. Adaptive
decision strategies are determined endogenously under this procedure,
which can be used with a standard linear or quadratic programming
algorithm. If the problem under consideration is even moderately com-
plex, however, the size of the input-output matrix quickly expands to
an unmanageable level, so the usefulness of this approach as an applied
decision analysis is also limited. Linear decision models, as developed
by Holt, Modiglioni, Muth, and Simon (1966), represent a third alterna-
tive. Under this approach, which has been further refined by Zellner
(1971), Chow (1973), and McRae (1975), dynamic programming techniques
are used to determine analytical solutions for a special class of
optimal control problems. Having assumed quadratic cost (utility)
functions and linear relations between state and control variables,
linear decision rules are derived which can be used to determine optimal
levels for control variables on the basis of forecasts of key factors.
The parameters of these rules remain invariant for as long as the system
design parameters are unchanged. Models of this sort are subject to the
same limitations imposed on preferences and probability distribution
under quadratic programming. Furthermore, the complexity of such
models and the high cost performing the initial analysis needed to
determine optimal decision rules for a particular process limit applica-
bility of this approach to situations where similar decision are made
repeatedly.

The models described above also fail to completely resolve the
problem that complex stochastic processes can often not be adequately

represented in a quadratic programming framework. Of particular

l37

importance is the fact that the impact of uncertainty concerning fixed
resource availability levels cannot be satisfactorily analyzed in the
models discussed above. Such factors can be important in some situa-
tions. In the decision problem discussed in the preceding three chap-
ters, for example, the number of days available for fieldwork during
any planting or harvest period is highly uncertain and has a major impact
on crop yields and net income levels. Consideration of this type of
uncertainty is commonly incorporated into programming models using
chance constrained programming techniques (Charnes and Cooper, 1959).
Though relatively simple to implement, however, this approach does not
permit explicit consideration of the cost associated with violating a
constraint and so does not actually facilitate the explicit determination
of the impact of stochastic fixed resource levels on the distribution
of system output levels realized under any particular management strategy.

Finally, none of these models, as specified, permits the representa-
tion of preferences by an empirically determined interval measurement
of absolute risk aversion in the ordering of choices by the criterion
of stochastic dominance with respect to a function, the advantages of
which were demonstrated in Chapter IV. It should be noted, however,
that stochastic dominance with respect to a function can be rather
easily incorporated into the REMP model, as will be demonstrated in the
next section.
5.3 A Generalized Procedure for the Identification

of Preferred Choices Under Uncertainty

While each of the decision models discussed above is attractive

in light of at least one theoretical or practical consideration and

138

while each may be appropriate for use in some decision situations,

none can be said to be generally applicable in a wide range of prac-
tical contexts. Furthermore, many rather common types of decision
problems cannot be adequately analyzed with any of the procedures
discussed above. A more general approach is needed--one which permits
greater flexibility with respect to problem formulation, the determina-
tion of probability distributions, and the representation of decision
maker preferences without sacrificing the power of decision theory

based on the expected utility hypothesis. Such an approach is presented
in this section.

The generalized procedure for the identification of preferred
choices described here is in many respects an extension of Anderson's
(1975, 1976) REMP model. Feasible strategies are generated using a
modified form of the Monte Carlo programming model developed by Donaldson
and Webster (1968), which also serves as a basis for Anderson's model.
Under the more generalized procedure, however, a strategy may be
defined by specific controllable system input levels, by a set of
adaptive decision rules, or by some combination of the two. Distri-
butions of system output levels associated with particular strategies
are not determined analytically. Rather, they are determined by
simulating system performance in a number of sample states of nature,
using techniques described in Chapter III. This facilitates the con—
sideration of theinunct of a wide range of stochastic exogenous system
inputs and permits greater flexibility in the representation of complex
stochastic processes. Finally, strategies are evaluated using interval
measurements of decision maker preferences and the evaluative criterion

of stochastic dominance with respect to a function.

139

Like the REMP model this procedure is an iterative one. A large
number of strategies are generated and evaluated sequentially. The
determination of a truly optimal choice is not ensured. If a sufficiently
large number of plans is examined, however, it is reasonable that the
efficient set will contain a nearly optimal choice for a decision maker.

Because of its similarity to the REMP model, this generalized
procedure for the identification of preferred choices under uncertainty
can be called the generalized risk efficient Monte Carlo programming
model (GREMP). Interrelationships among the three major processes
within the mode1--strategy generation, system output distribution
determination, and evaluation--are illustrated by the flow chart in
Figure 5.1. Each of these processes will be discussed in greater

detail in the remainder of this section.

5.3.1 Generation of a Feasible Management Strategy

 

At the outset of each iteration of the GREMP model a management
strategy is constructed. As defined in Chapter II, a management strategy
is a set of controllable system input levels, a set of feedback control
rules for determining controllable system input levels over the duration
of the planning horizon, or some combination of the two. The nature of
the problem under consideration determines the types of choices which
must be made, and the nature of the decision situation determines the
range of choices open to the decision maker.

Regardless of how decisions are defined, the presence of constraints
makes some management strategies impossible. Limits on available
resources may restrict the set of admissible values for some controllable

system inputs. Similarly, some controllable system inputs may be

140

 

Generate a
Feasible
Management
Strategy

 

 

 

 

 

Use Monte Carlo
Silulation to
Determine System

Output Distribution

 

 

 

 

Evaluate New
Strategy Relative
to Current Members

of Efficient Set

 

 

 

 

 

Update
Efficient
Set

 

 

 

 

    
 

 

Examined?

 

    

Print Information
on Efficient Set

 

Figure 5.1 A Flow Chart of the GREMP Model

141

indivisible and so must take integer values. In addition, logic or
common sense may dictate that a particular parameter of a feedback
control rule should be positive or that one parameter should always

be greater than another. Constraints may also be of a form which
renders two activities mutually exclusive--e.g., the choice of a par-
ticular functional form for a feedback control rule precludes the use
of an alternative form. Given the definition of a management strategy
in a particular decision situation and the constraints on the range of
available choices, some method of identifying feasible strategies for
consideration in the decision analysis is needed. When the number of
alternatives is small, each can be explicitly specified and evaluated.
When the number of alternatives is large, Monte Carlo programming tech-
niques can be a valuable tool for the identification of feasible
strategies.

Monte Carlo programming is a search procedure which constructs sample
management strategies at random from the set of feasible strategies.

In determining the values for controllable system inputs and/or feedback
control rule parameters, techniques similar to those introduced in the
discussion of the simulation of stochastic processes are used. Monte
Carlo programming is a remarkably flexible tool which can be relatively
easily and inexpensively implemented and so is well suited for use in an
applied decision analysis.

Monte Carlo programming techniques are explained in detail by
Donaldson and Webster (1968). A technical discussion of their applica-
tion is also presented in Appendix C of this study along with a listing
of the computer program used to implement the GREMP model. In the context

of current discussion an example is, perhaps, the most effective medium

142
for the explanation of the process by which Monte Carlo programming is
used to construct feasible strategies. The management strategy specified
in relation to the decision problem introduced in Chapter II will be
the basis for this example. It will be recalled that this strategy is
defined by three controllable system input levels and by one simple
feedback control rule which has a single parameter. The three controllable

system input variables are:

v1 = number of acres rented
v2 = number of acres planted in corn
v3 = number of acres planted in soybeans

The feedback control rule is: "Regardless of specified crop
acreage levels, soybeans will be planted on all unplanted acreage after
v4 (a parameter indicating a specific date)."

Four indivisible 80 acre tracts of land are available for rental.
Therefore, the only admissible values for v1 are 0, BO, 160, 240, and
320. Two types of constraints are imposed on crop acreage levels.
First, the farmer states that if he grows a crop at all, he wishes to
plant at least fifty acres of that crop, i.e.

v1 = O or vi 2 50 i = 2, 3 5.1
Second, total crop acreage is restricted to that which is owned by
the farmer, 240 acres, plus that which is rented, v]. The farmer
wishes to plant all available acreage, if possible. Therefore:
= 240 + v 5.2

v
2 3 1
With regard to the control rule parameter v4, three possible values will

+V

be considered: May 18, May 26, and June 3. As was noted in Chapter II,

these are the ending dates of the last three possible corn planting

143

periods. Let v4 = 1 if May 18 is chosen, 2 if May 26 is chosen, and 3
if June 3 is chosen.

The construction of a management strategy is a sequential process,
and in some cases it may be necessary to set a value for one choice
variable before levels of others can be established. In the example
being discussed here, for example, a value of v1 must be determined
before crop acreage levels can be specified. Therefore, after control
variables have been identified and constraints on them have been
specified, the control variables must be classified according to the
sequence in which they should be considered. In our example, land
rental can be classified as a resource acquisition activity, and v1 is
the first variable for which a value is determined. Crop acreage levels,
v2 and v3, refer to resource using activities and are specified next.
Finally, the value of the control rule of parameter, v4, is set.1

In constructing a management strategy, the value of each controllable
system input or feedback control rule parameter is treated as a random
variable. In our example, v1 and v4 are clearly discrete random
variables since each has only a few possible values. Unless it is
desirable to assign greater probability weight to some particular value,
both can be treated as discrete uniform random variables. In the case
of land rental, then, there is probability of 1/5 that no land will be
rented, 1/5 that 80 acres will be rented, 1/5 that 160 acres will be
rented, 1/5 that 240 acres will be rented, and 1/5 that 320 acres will

 

1The order in which control variable levels are specified depends,
in general,on the characteristics of the problem under consideration and
on computational convenience. All that is actually required in this
example is that a value of v1 be set before v2 and v3 are considered.

144

be rented. Similarly, a probability of 1/3 can be assigned to each of
the three possible values of the control rule parameter.

Acreage levels for each crop, v2 and v3, can be treated as con-
tinuous or discrete random variables. Again a uniform distribution,
be it discrete or continuous, should be used unless there are a priori
reasons for assigning higher probabilities of selection to certain
values or ranges of values. In this case acreage levels will be treated
as discrete variables with possible values being integer multiples of
ten lying on the interval between 50 acres and total available acreage,
240 + v . The probability of any particular admissible value being

1
selected is given by the expression:

p = 710(240:%;:50)¥T' 5.3
The denominator on the right hand side of 5.3 is a general expression
for the number of possible values of v2 or v3 given a particular value
of v].

Actual construction of a management strategy begins with the
generation of a sample observation from the distribution of v]. In
this instance let the value of that observation be 240, which implies
that 240 acres of land are to be rented under this strategy. Next,
values of v2 and v3, the crop acreage levels, must be determined.
Again using a discrete uniform process generator, one of these two
variables is selected for consideration. Let v2, the number of acres
planted in corn, be the variable selected. A sample observation from
the distribution of this variable is then generated. In this case let
its value be 220 acres. Clearly the constraint on total acreage is not

violated by this value, since v3 is considered to be equal to zero until

145
it is assigned another value. This is, then, a feasible value for v2.
The constraint on remaining available acreage is next updated to state
that:
240 + v -v 5,4

3 1 2
240 + 240 - 220 = 260

<
11

and a value for v3 is, in turn, established. It should be noted that
had the value of v2 been greater than 430, a value of v3 less than 50
would have been implied. Because values less than 50 are not permitted
for v2 or v3, v3 would have been set to zero and corn acreage would
have been expanded to 480. It should also be noted that the process of
control variable level determination is somewhat more complex when more
than two resource-using variables enter into a single constraint. The
procedures used in such cases are explained in Donaldson and Webster
(1969) and in Appendix C, and the computer program developed for the
implementation of the GREMP model is fully applicable to problems of
this sort.

Once values for the three controllable systems input variables have
been specified, all that remains in constructing a feasible management
strategy is the determination of a value for the control rule parameter,
v4. A sample observation from this variable's distribution is selected--
in this instance let its value be 2, which implies a date of May 26--

and the strategy is complete. It is:

240 acres of land rented

<
_.a
ll

v2 = 220 acres of corn to be planted

= 260 acres of soybeans to be planted

May 26, the date after which all unplanted acreage is shifted

to soybean production regardless of specified values of v2 and v3.

146

This simple example demonstrates some of the features which make
Monte Carlo programming such a flexible procedure. Choice variables can
be continuous or discrete, and they need not be assigned values which
correspond to strictly quantifiable entities. In addition, both upper
and lower limits can be established for a choice variable once activated
without necessarily forcing it into the management strategy at a non-
zero level. Other features of Monte Carlo programming are outlined in
Appendix C.

5.3.2 Determination of the Distribution of
System Output Levels

 

 

Once a strategy has been constructed, the associated distribution
of system output levels must be determined. This is done sequentially
as each strategy is generated using the Monte Carlo simulation tech-
niques described in Section 3.3 of Chapter III. System performance under
a given strategy is simulated for a large number of sample states of
nature, each defined by a sample vector from the joint probability
distribution of relevant stochastic system input variables. In this
way sample system output levels are determined, which can be used to
construct a cumulative distribution function for the underlying distri-
bution.

In the current example, net cash income is the system output
variable which serves as the basis for the evaluation of alternative
strategies. It will be recalled that crop prices, crop yields, and
time available for fieldwork were judged to be the stochastic exogenous
system inputs which have an important impact on system performance.

Subjective probability distributions for all of these factors were

147
specified in Section 3.4 of Chapter III. Using techniques described in
Appendix A, twenty sample vectors of levels for each of these stochas-
tic system inputs were generated and read into the computer program
which implements the GREMP model. A computer simulation model which
determines the net cash income level realized under a specific manage-
ment strategy in any given state of nature was also described in Section 3.4
of Chapter III. This model is incorporated into the larger GREMP model
as a subroutine which is called after the generation of each alternative
strategy. Using the stochastic system input data from the main program
and the control variable levels for the new strategy, it calculates net
cash income for each of the twenty states of nature. The twenty sample
income levels associated with the management strategy defined in the

preceding subsection are given in Figure 5.2.

5.3.3 The Evaluation of Alternative Strategies

Alternative management strategies are evaluated within the GREMP
model by applying the criterion of stochastic dominance with respect to
a function, with an interval measurement of decision maker preferences
defining the relevant levels and upper bound absolute risk aversion
functions. Evaluations are made sequentially as strategies are generated.
If a particular strategy is not dominated by any current member of the
efficient set, it, too, becomes a member of the efficient. In such
instances the control variable values which define the new strategy and
the associated set of sample system output levels are stored. If the
new strategy is dominated by any member of the current efficient set, on

the other hand, it is eliminated from further consideration. Similarly,

u = $10,707.58

148

-17271.70
-15571.83
- 8576.60
- 1328.38

1644.65

5658.63

7705.34
10057.10
11737.75
13146.29
13676.88
15136.41
19763.05
22865.28
26578.39
28998.96
29472.42
29695.14
33951.82

0 = $15,464.33

Figure 5.2 Sample Observations from the Distribution
of Outcomes Associated with Strategy One

149

members of the efficient set which are dominated by the new strategy
are removed from the efficient set. Because the criterion of stochastic
dominance with respect to a function is fully transitive, this proce-
dure ensures that no member of the efficient set for the set of strate-
gies already examined will be eliminated and that only information on
actual members of the current efficient set will be saved.

Returning to the example being discussed, let the decision maker
be the first of the three decision makers for whom actual interval
measurements were described in Section 4.8 of Chapter IV. Both the
lower and upper bounds of his risk aversion interval, it will be recalled,
decrease monotonically over the range of income levels for which pre-
ferences were measured. Let the first strategy generated by the GREMP
model be that defined above in Section 5.3.1. Being the first strategy
considered, it automatically becomes a member of the efficient set.

Let the second strategy generated be:

v1 = 0 acres rented

v2 = 50 acres of corn to be planted

v3 = 190 acres of soybeans to be planted

v4 = May 26, the date after which all unplanted acreage is to

be planted in soybeans
The set of net income levels associated with this strategy is given in
Figure 5.3. Given this decision maker's preferences, this distribution
dominates the distribution of income levels associated with the first
strategy. After two iterations, then, the efficient set is comprised
only of the second management strategy.
Let the third strategy generated by the GREMP model be defined by

the following control variable levels:

150

~11482.39

u = $2979.44

9829.
9463.
.16
3452.
606.
.79
2097.
3043.
3214.
3545.
3700.
5900.
6354.
7663.
10637.
.02
11360.
.27

5351

731

10921

14041

15963.

65
32

86
25

9O
55
38
55
89
3O
32
25
01

01

26

o = $7691.31

Figure 5.3 Sample Observations from the Distribution
of Outcomes Associated with Strategy Two

151

v1 = 160 acres of land rented

v2 = 120 acres of corn to be planted

v3 = 280 acres of soybeans to be planted

v4 = May 26, the date after which all unplanted acreage is to

be planted in soybeans

The distribution of net cash income levels associated with this strategy
is given in Figure 5.4. When compared to the distribution of income
levels associated with the second strategy--the only member of the
current efficient set-~this distribution neither dominates nor is
dominated. After three iterations, then, the efficient set is comprised
of the second and third strategies.

The process continues in this manner until a prespecified number
of iterations have been completed. As noted earlier, there is no
guarantee that a true optimum will be identified. If a sufficient
number of strategies are evaluated, however, it is almost certain that
a nearly optimal strategy will be included in the efficient set. The
number of iterations specified depends on the characteristic of the
problem being analyzed. Donaldson and Webster (1968) suggest that 2000
strategies be examined. Experience to date with the relatively small
problems considered in this study, however, indicates that 500 to 1000

iterations are often quite sufficient.

5.3.4 General Comments on the GREMP Model

The flexibility of the generalized procedure for the identification
of preferred choices described above is its greatest strength. It can
easily be adapted for use in the analysis of a diverse range of practical

decision problems without requiring that major simplifying assumptions

u = $10,152.21

152

-12468.33
-11453.37
- 8569.24
— 4272.46

740.
4556.
8818.
9095.

10008.
10424.
10543.
10785.
16644.
18476.
23776.
25010.
25050.
26685.
30877.

21
02
28
91
77
22
O7
49
11
43
9O
99
58
05
08

o = $12,516.70

Figure 5.4 Sample Observations from the Distribution
of Outcomes Associated with Strategy Three

153

be made. With regard to problem formulation, the use of a random

search procedure to generate strategies for consideration permits
considerable flexibility with respect to both the types of control
variables which can be specified in the definition of a management
strategy and the types of constraints which can be imposed on them.
Choice variables can be discrete or continuous. They can be controllable
system input levels, feedback control rule parameters, or even indicators
of the form of a feedback rule. Constraints on control variables can

be linear or non—linear and can take forms more complex than those per—
mitted in decision models based on mathematical programming.1 With
regard to the determination of system output distributions associated
with alternative strategies, the use of Monte Carlo simulation tech-
niques greatly facilitates the realistic representation of the stochastic
processes by which the outcomes of particular choices are determined.

In the simple example discussed above, price, yield, and time available
for fieldwork are all considered explicitly as random factors which
affect the outcome of any choice. Few restrictions are placed on the
form of exogenous system input distributions, and the relationships

among these factors, controllable system inputs, and system outputs can
be quite complex in nature. With regard to the representation of
decision maker preferences, the use of stochastic dominance with respect
to a function also contributes greatly to the flexibility of the approach

without sacrificing the logical power of decision theory based on the

 

1A discussion of some of the types of constraints in control
variables which are possible in the GREMP model is given in Appendix C.

154

expected utility hypothesis. The interval preference measurements used
with this criterion can be as precise or imprecise as is required for
any particular decision analysis.

The GREMP model is also relatively efficient computationally. In
the analysis of one test problem with thirty-five choice variables and
twelve linear constraints, for example, 1000 alternative strategies are
generated and evaluated using less than seventy seconds of CPU time in
a CDC6500. Furthermore, the core size of the computer program which
implements the GREMP model is relatively small, and the degree of compu-
tational accuracy required for interval calculations is not unusually
great. This suggests that it may be possible to design computer soft-
ware which will permit the use of the GREMP model on a moderately sized
personal computer.

Several criticisms of the GREMP model can be made. As was noted
earlier, there is no guarantee that the model will identify a true
optimum, since alternative strategies are generated in a random manner.
Furthermore, the efficient set of choices may be quite large if the
decision maker's preferences are not measured precisely enough. Neither
of these criticisms is particularly serious, however. With regard to
the first, in many instances it can be argued that a good solution to
a well-formulated problem is preferable to an optimal solution to a
problem which bears little resemblance to that actually facing the
decision maker. The flexibility of the GREMP model, then, compensates
for this weakness. With regard to the size of the efficient set, it can
always be reduced by making more precise interval measurements of pre-
ferences, but it should be remembered that such reduction may lead to the

exclusion of the decision maker's preferred choice from the efficient set.

155

A more valid criticism of the procedure is that it provides no
direct information on how the composition of the efficient set might
change if probability assessments were altered or if alternative values
were assigned to certain key system design parameters. Such informa-
tion can be obtained only by specifying the changes and repeating the
procedure. For complex problems a sensitivity analysis conducted in
this manner can be a costly and time-consuming process.

The GREMP model can also be criticized because it requires that
the evaluation of alternative strategies be based on the distribution
of a single system output variable, usually some measure of income or
wealth output variable. In reality, most decision makers are concerned
with more than one performance criterion. They have multiple objectives,
and they consider trade-offs among these objectives when making choices.
Decision theorists have focused considerable attention in recent years
on the construction of preference measures which depend on more than one
performance criterion and on the incorporation of such preference mea-
sures into a decision analysis. Unfortunately, however, the criterion
of stochastic dominance with respect to a function has not been extended
to the multivariate case, and there is no indication that such an exten-
sion will be made in the near future. Despite the acknowledged existence
of multiple goals which affect choices, it can be argued that in many
instances the consideration of only the most important objective may
be adequate. The added accuracy attained from a more complete analysis
may not justify the added cost of such an undertaking. When it is
judged that more than one performance variable must be considered in a

decision analysis, however, the GREMP model can be used if modified

156
slightly. All that is necessary is that a single-valued multiple cri-
terion utility function be incorporated into the model to replace the
evaluative component based on stochastic dominance with respect to a
function.

Finally, two criticisms of a practical nature can be made of the
model. First, it should be noted that, although the program which com-
plements the GREMP model can be easily adapted for use in the analysis
of a wide range of problems, the user is required to supply several
problem-specific subroutines and so must have some programming skills.
Though expertise in computer programming is certainly not required, this
may preclude the use of the model in some instances. Second, it can be
noted that in the analysis of complex decision problems with a large
number of control variables, a large number of strategies must be
examined before a nearly optimal one is identified. Computational costs
can be considerable, then, for large problems. It should be noted, how-
ever, that with careful problem formulation and with the use of feedback
control rules the size of the feasible set of strategies can be greatly

reduced.

5.4 An Application

 

In this section the GREMP model is applied to the analysis of the
sample problem discussed in the three preceding chapters. Efficient
sets of choices are identified from a set of 500 feasible strategies
for each of the three decision makers whose preference measures were
given in Section 4.8 of Chapter IV. Since this sample problem was the
basis for the discussion of the GREMP model in the preceding section,
the reader should be familiar with its essential features. Therefore,

the results of this application will be presented without further discussion.

157
The efficient set of choices for decision maker A is comprised of

1 Levels of land rented

the eight strategies defined in Table 5.1.
range from 0 to 160, with four of the eight strategies calling for land
rental levels of 80 acres. Soybeans are the predominant crop in each
plan, which is understandable given the cost-price relationships for
the two crops. At low corn acreage levels, the feedback control rule
parameter has little effect, so the switching date is of little impor-
tance in these strategies. Mean income levels realized under the
efficient strategies range from slightly less than $3000 to slightly
above $10,000. Minimum income levels vary little from one strategy to
another. Maximum income levels, however, are significantly affected by
land rental levels. It is also interesting to note that the efficient
set need not be a mean-variance efficient set. For example, strategy 2
dominates strategy 4 by the mean-variance criterion, but both are in the
decision maker's efficient set determined by stochastic dominance with
respect to a function.

The efficient set of decision maker B is comprised of the nine
strategies defined in Table 5.2. In this case, land rental levels
tend to be higher than those called for by the strategies in decision
maker A's efficient set. At higher acreage levels, the mix between
corn and soybeans becomes more even, though most of the available acreage
is planted to soybeans in each strategy. Mean net income levels tend
to be higher within this set of strategies, but the dispersion of

possible net income levels is also greater. Given the differences in the

 

1System output levels are not enumerated for each state of nature.
Rather, the mean, standard deviation, minimum value and maximum value
are given for each distribution.

 

158

 

 

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159

 

 

 

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160
preference measurements for decision makers A and B, the dissimilarity
between these two efficient sets is understandable. The interval measure-
ment of absolute risk aversion for decision maker A indicated a high
level of risk aversion at negative net income levels--i.e. he has an
apparently strong aversion to losses. Decision maker B, on the other
hand, has much lower levels of absolute risk aversion at low income
levels, and his efficient set contains strategies which, while providing
opportunities for the realization of high income levels, also can result
in substantial losses.

The efficient set of decision maker C is comprised of the strategies
defined in Table 5.3. Of interest in this case is the fact that each of
these eleven strategies is a member of the efficient set of either
decision maker A or B. In a sense, then, it can be said that this
decision maker's preference measurement lies between those of the other
two decision makers.

Several general comments can be made about these results. First,
they provide clear evidence of the discriminatory power of the preference
measures based on the interval approach. The largest of three efficient
sets contains only two percent of the total number of strategies examined.
Second, the results demonstrate once again that preferences do have an
important impact on the choices made by an individual. Finally, it
should be observed that a search of only 100 feasible strategies identi-
fied many of the strategies included in these three efficient sets.

This indicates that in some cases the evaluation of an extremely large

number of strategies may not be necessary.

161

 

 

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CHAPTER VI

COMBINED PRODUCTION AND MARKETING DECISIONS BY
CASH GRAIN FARMERS: AN EXTENDED APPLICATION

6.1 Introduction

 

The formulation of an integrated set of operational techniques for
the analysis of decisions made under uncertainty was the focus of the
preceding chapters of this study. A simple example related to land
rental and crop production decisions made by cash grain farmers has
been used to illustrate these techniques. In this chapter the useful-
ness of the methodological tools developed above is demonstrated further
by expanding the earlier example to include the consideration of alter-
native marketing strategies in conjunction with the selection of a
cropping plan. Two modes of marketing will be evaluated: the sale of
all production at harvest in the cash market and forward contracting.1
The objectives here are to examine how these two modes of marketing
can best be combined in the formulation of a marketing strategy which
is appropriate for a particular decision maker, to determine the degree
of interdependence between crop production and marketing strategies,
and to examine the impact of changes in preferences on combined produc-

tion and marketing strategies.

 

1To simplify the discussion, other marketing alternatives such as
hedging in the futures market and participation in government price
stabilization programs will not be considered, nor will the use of crop
storage as a marketing tool be examined. These alternatives could be
incorporated into the analysis presented below, however.

162

163

Cash grain farmers make major resource allocation decisions under
conditions characterized by uncertainty with respect to both prices and
output levels. With the increased dependence of the feed grain sector
on foreign markets in recent years, the impact of price uncertainty on
cash grain producers has been particularly strong. Many have come to
realize that marketing as well as production decisions have a major
impact on the level of income they realize. The common practice of
selling all production at harvest in the cash market often results in
the receipt of prices which are low relative to those which could be
realized under alternative modes of marketing. Furthermore, a strategy
comprised only of cash sales at harvest does nothing to diminish the
degree of price uncertainty faced by the producer over the period prior
to and during planting--the period when major allocative decisions must
be made. As a result, many producers find it desirable to consider for-
ward pricing some or all of their planned production. By contracting
to deliver a certain quantity of grain on a future date at a specified
price, the producer sells all or some portion of his crop in advance.

In doing so, he establishes with certainty a price for at least part of
his total production, thereby greatly reducing the degree of price
uncertainty he faces.

The reduction in price uncertainty achieved through forward con-
tracting can be of considerable value in some situations. Advance know-
ledge of the price to be received simplifies the planning process.
Furthermore, if the contract price is high enough, a producer who forward
contracts may almost ensure that he will realize an adequate level of

income. There are also costs associated with contracting, however.

164
The other party in the contract, the buyer of the grain, often has

access to better information than that available to the producer and it
is unlikely that he will offer a price higher than that he himself
expects to receive. Of equal importance is the fact that the seller,
though he protects himself against the effect of an unexpected downturn
in price, also foregoes the opportunity to benefit from unexpected price
increases. Finally, grain must be delivered according to the terms of
the contract even if production falls short of the amount which is
forward priced. If yields are unusually low or if poor weather condi-.
tions prevent the planting of some acreage altogether, the producer
faces the prospect of being forced to purchase grain on the cash market
to meet the terms of his contract. Given the advantages and disadvan-
tages of forward contracting, then, the producer must determine how
much, if any, of his anticipated production he wishes to contract.
Clearly preferences, financial position, price expectations and produc-
tion plans affect this decision.

The determination of optimal forward contracting levels for agri-
cultural producers facing yield and price uncertainty has been examined
in a mean-variance framework by McKinnan (1967). He shows that under
relatively simple conditions the optimal level of forward contracting
depends on five fundamental parameters: the standard deviations of
crop yield and product price at harvest, the expected crop yield, the
forward contract price (which is also the expected price at harvest),
and the correlation coefficient between crop yield and harvest price.
McKinnan's work has been extended by Ward and Fletcher (1971) and by
Heifner (1972), and Barry and Willmann (1976) provide an interesting

165

empirical application of contracting theory based on mean-variance
analysis. The applicability of the results of each of these studies

is limited by the somewhat restrictive assumptions associated with the

use of the mean-variance criterion--assumptions of normally distributed
net returns and risk averse behavior. The requirement that net returns

be normally distributed may be particularly unrealistic. More critical,
however, is the failure in each study to consider a factor which is of
primary importance in an applied context. All treat forward contracting
decisions as though they can be made at only one point in time. In
reality producers have forward pricing opportunities open to them over

an extended period of time. In such a context the decision of whgg to
contract may be as important as the decision of how much to contract.
Producers continually evaluate forward pricing opportunities, and it is
not unusual for an individual to enter into contracts at several different
times. Once a contract is made, however, it must be honored. Choices
made in the present, then, affect opportunities in the future. Therefore,
it is important to consider forward contracting decisions in a more
dynamic analytical framework.

The techniques developed in this study allow the relaxation of the
restrictions on probabilities and preferences imposed by the use of
mean-variance analysis. More important, however, they permit the evalua-
tion of both production and forward contracting decisions in a more
dynamic framework in which the roles of learning and adaptive behavior
are treated more explicitly than in previous studies. In the analysis
below, attention will again be focused on decisions made by a cash

grain farmer producing corn and soybeans. As in the example discussed

166

in earlier chapters, flexibility is introduced into the production
planning process through the incorporation of a simple stopping rule

for corn planting. Marketing strategies for both corn and soybeans are
defined by more complex feedback control rules which are applied
repeatedly over a seven month period extending from mid-January to
mid-August. Particular attention will be given to the examination of
both the interdependence between production and marketing strategies and
the impact of changes in decision maker preferences on the choice of a

combined production-marketing strategy.

6.2 Problem Formulation

 

The basic decision situation in this extended example is the same
as that described in Section 2.5 of Chapter II. The operator of a
relatively small southeastern Michigan cash grain farm needs to realize
a substantial level of income from his farming operation in order to
meet his debt repayment commitments of $35,000 annually and to cover
family living expenses. He wishes to choose a management strategy
which, given his risk preferences and the range of opportunities open
to him, will best satisfy this need. The time is January 1979, and,
because land rental decisions must be made, the farmer must formulate
at least a tentative management strategy now.

The system of concern in this example is comprised of a set of
production and marketing processes which constitute a farming operation.
The performance of this system is measured by a single system output
variable: annual net income for family living expenses and firm expan-
sion after all debt repayment commitments have been met. The level of
income realized is affected by the structure of the system and by exo-

genous and controllable system inputs.

167

The structure of the production and cash marketing processes
embodied in this system was described in Section 2.5 and remains
unchanged in this example. It will be recalled that standard crop
budgets for corn and soybeans were presented and that relevant planting
and harvest periods were specified for each crop, as were rules of thumb
which determine the priority of each crop during planting and harvesting.
Important state variables related to the production process included
production expenses incurred to date, acreage of each crop planted or
harvested to date, and bushels of each crop harvested to date. All
production was sold in the cash market at harvest in the original
example. The harvest price of each crop was multiplied by the number
of bushels harvested to determine the value of marketing receipts for
each crop, the key state variable used to describe the marketing process.

The incorporation of forward contracting into the analysis requires
that several structural features associated with this mode of marketing
be specified and that several new state variables be defined. Struc-
turally, it must be recognized that contracts, once made, are binding,
and that if production falls short of the amount contracted, enough
grain must be purchased in the cash market at harvest to cover the
deficiency. It must also be noted that, unlike a hedge in the futures
market, a hedge based on a forward contract cannot be lifted. The new
state variables which must be defined include the number of bushels of
each crop contracted to date and the current level of receipts forth-

coming at harvest from quantities of each crop contracted.1

 

1Several additional state variables will be defined during the
discussion of the feedback control rules which determine the marketing
strategies for corn and soybeans.

168

Exogenous system inputs in the original example included the
following stochastic environmental factors: the price at harvest of
each Crop, the number of days available for fieldwork in any particular
planting or harvest period, and the yield of each crop for each allow-
able planting-harvest combination. Consideration of forward contracting
requires the specification of an additional set of stochastic factors:
the harvest delivery forward contract prices for corn and soybeans

over the period when contracting decisions are to be made. Clearly this

‘ firth"

set of prices will affect the desirability of forward contracting and
the level of income realized under any management strategy which calls
for the forward contracting of either crop, and clearly these prices
cannot be known with certainty in mid-January when a management strategy
must be formulated.

Only four control variables were considered in the example discussed

in previous chapters. They were:

v1 = acres of land rented

v2 = acres of corn to be planted

v3 = acres of soybeans to be planted

v4 = the date after which all unplanted acreage is to be

planted in soybeans
The consideration of forward contracting decisions requires the specifi-
cation of several new control variables, which will determine the number
of bushels of each crop contracted at any particular time. Contracting
levels, pgr_§g, will not be selected directly. Rather, a feedback
control rule which determines the desired contracting level at any point

in time will be specified for each crop. The form of this rule is the

169

same for both corn and soybeans and is similar to that discussed in

Section 2.3.1 of Chapter II. It is:

where

DBC

DBC

EC

ZA

ZB

ZC

t

ECt(vaAt+vxZBt+vyZCt) 6.1
desired bushels contracted at time t

current expected size of harvest

the percentage difference between the current contract
price, CPt, and the current expected harvest price,

EPt; 1.e. ZAt = (CPt-EPt)/EPt

the current daily rate of change in the contract price:

1.e. ZBt = dCPt/dt

the difference between the desired percentage of the
expected crop contracted and the actual percentage
contracted. The desired percentage contracted is defined

by the express1on Apt/DAPt+vz’ where APt and DAPt are
current actual and desired acreage planted and v2 is a
parameter to be selected. The actual percentage con-
tracted is defined by the expression BCt/Ect, where BCt is
the number of bushels contracted to date and ECt is the
expected size of the harvest. It follows that ZCt = APt/DAPt

+Vz'Bct/Ect'

In addition to v , v , v , and vy are parameters to be selected.

Z W X

The inclusion of each term in this contracting rule can be justified

by appealing to commonly recognized rules of thumb regarding forward

contracting. The first term ZAt, reflects an assessment of the funda-

mental position of the market. If ZAt is positive, the current contract

170
price is above the current expected harvest price. As ZAt becomes
larger, the attractiveness of current pricing opportunities increases
and the decision maker is expected to want to contract more. Similarly,
if ZAt is negative, the current pricing opportunity is not an attractive
one and the desired level of contracting will be less. The parameter
which weights this factor, vw, is expected to be positive.

The second term, ZBt’ reflects, in a very simple way, a technical
assessment of market conditions. If ZBt is positive, the contract price
is increasing, and most technical analysts would recommend that contracting
be delayed. If ZBt is negative, and if fundamental analysis indicates
that the current pricing opportunity is a favorable one, on the other
hand, many technical analysts would recommend that forward contracts be

entered into. Following this reasoning, the parameter which weights

this factor, vx, is expected to be negative.

The third term, ZCt, reflects the degree to which current contracting
levels coincide with desired contracting levels, In this case desired
levels are a linear function of the percentage of desired acreage
actually planted. The parameter vz shifts the intercept of that function
and is expected to be negative, reflecting the fact that many decision
makers hesitate to contract before planting begins unless pricing
opportunities are particularly favorable. When ZCt is positive, the
decision maker is expected to desire to contract more unless other
factors indicate the current pricing opportunity is not a good one.

When ZCt is negative, the desire to contract is diminished. Therefore,
the parameter which weights the importance of this factor in the con-

tracting rule, vy, is expected to be positive.

171

As stated above, a control rule of this form is specified for
each crop. Since four parameters must be selected for each rule, a
total of eight new control variables are required in this example.
Control variables v5, v6, v7, and v8 correspond to Vw’ vx, vy, and v2
for the corn contracting rule, and v9, v10, v1], and v12 correspond to
the same parameters for the soybean contracting rule. Introduction of
the contracting rules into the analysis also requires the specification
of several new state variables. In addition to those mentioned earlier,
state variables indicating the daily rate of change in the contract
price, the expected size of the harvest, and the expected price at
harvest must be monitored for each crop. The daily rate of change in

the contract price is defined by the expression:

dCP

t - .-
‘dt— ‘ “Pt apt-1"dt 6'2

where CPt and CPt-l

and dt is the number of days between price observations. The expected

are successive observations of the contract price

size of harvest for a particular crop is defined by the expression:
ECt = (BY)(DAPt) 6.3

where DAPt is the desired acreage to be planted and BY is a base

1

yield. The expected harvest price for a particular crop is defined

by the following simple expectations model:
t t-l+'17CPt-2 6.4
This is simply a weighted moving average of the three most recent con-

EP = .SCPt+.33CP

tract price observations. Finally, it should be noted that in the

1BY is set at 100 bushels per acre for corn and 33 bushels per
acre for soybeans .

172
application of each rule, the number of bushels to be contracted at any
time is set at zero if the number of bushels already contracted is equal
to 150 percent of the expected crop.

As in the previous example, the planning and decision horizons are
both said to be a single crop year. It will be recalled that the stopping
rule for corn planting is checked at the end of the second, third, and
fourth planting periods. The two contracting rules are consulted a
total of twelve times--four times prior to the commencement of planting,
at the end of each of the six planting periods, and twice between
planting and harvest. The exact dates for each application of the two
rules are given in Figure 6.1.

The problem in this example, then, is to identify a management
strategy defined by a total of twelve control variables which maximizes
the expected utility of a particular decision maker. Three of these
choice variables specify controllable system input levels directly. The
other nine are parameters in feedback control rules which are consulted
periodically over the entire decision horizon.

6.3 The Determination of Subjective
Probability Distributions

 

 

Subjective probability distributions for crop yields, time avail-
able for fieldwork, and crop prices at harvest were specified in
Section 3.4 of Chapter III. They remain unchanged in this example. It
is also necessary to specify subjective distributions for the harvest
delivery contract price of each crop at each of the eleven dates after
January 10 listed in Figure 6.1. These prices cannot be known with
certainty at the time when a strategy must be selected, and their distri-
butions can have an important impact on the distribution of net income

levels realized under any particular management strategy being considered.

173

January 10
February 10
March 20
April 20
May 10

May 18

May 26
June 3
June 11
June 19
July 20
August 20

Figure 6.1 Dates for the Application of Forward
Contracting Rules

174

For either crop, elements in the series of contract prices are
expected to be correlated with each other and with the cash price at
harvest. Therefore, individual contract price distributions cannot be
specified independently. Even if this could be done, it is unlikely
that a decision maker would have clearly formulated expectations con-
cerning contract price levels at each of eleven dates over a seven
month period. To simplify the specification of these distributions,
then, the following procedure which reflects the author's own subjective
assessments was used. The harvest delivery contract price for each
crop on January 10, 1979, the date when a strategy is to be selected, is
known with certainty. That for corn is $2.08, and that for soybeans is
$6.31. For any particular state of nature, the cash price at harvest
can also be specified, this price representing a sample observation from
the underlying subjective price distribution. In addition, contract
margins charged by elevators are relatively constant and can be set at
$.10 per bushel for corn and $.25 per bushel for soybeans.1 Given this
information, it is assumed that the harvest delivery contract price
offered on any date between January 10 and the contract delivery date
can be adequately forecasted by linearly interpolating between the
January 10 price and the price at harvest less the contracting margin.2
In Figure 6.2, for example, the contract price of corn on January 10 is

$2.08. The cash price 295 days later November 1 is $2.32. With a

 

1Contract margins cover the elevator's operating expenses and
represent a premium paid to the elevator by the farmer for the reduction
in price uncertainty.

2Approximate delivery dates are October 15 for soybeans and
November 1 for corn.

175

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176

contract margin of $.lOper bushel, then, a hypothetical contract price
offered on that day would be $2.22. The line between the contract prices
of $2.08 on January 10 and $2.22 on November 1 is used to forecast the
contract price on intermediate dates. On May 10, 120 days after

January 10, for example, the forecasted contract price is $2.14.
Forecasts of the contract price of each crop at each of the dates
specified in Figure 6.1 can be made in this manner. By making such
forecasts for a number of sample harvest price observations, a crude
multivariate distribution of intermediate forward contract price levels
can be specified.

Clearly forecasts based on such a simple model are subject to con-
siderable error. To reflect this fact during the actual specification
of sample observations from the contract price distributions, the fore-
cast of each intermediate contract price was multiplied by a factor of
1+e, where e is a normally distributed random variable with mean 0.0 and
standard deviation .05. To further enhance the realism of the model,
the multiplicative error terms for successive dates were correlated to
reflect the fact that observed contract price levels are autocorrelated.
A plot of values based on this more complex model is also shown in
Figure 6.2. In this instance, relatively good pricing opportunities
occur in March and July, and contract prices during the period when
planting takes place cluster around the price forecast line. It is
interesting to note that the contract price in this example never exceeds
the actual cash price at harvest.

The distribution of net income levels realized under any particular

management strategy is determined by simulating system performance under

177

that strategy in each of a set of sample states of nature.1 One

possible management strategy is defined by the following control
variable levels: v1 = 240 acres rented; v2 = 190 acres corn; v3 = 290
acres soybeans; v4 = May 26, the date after which all unplanted acreage
is planted in soybeans: v5 = 12; v6 = 1; v7 = .90 and v8 = -.30 for the
corn contracting rule; and v9 = 17, v10 = -1, v1] = .50, and v12 = -.20
for the soybean contracting rule. The sample observations which define
the distribution of net cash income levels associated with this strategy
are given in Table 6.1 along with the other information pertaining to
system performance in each of the twenty states of nature considered.
The cropping plan in this case is identical to that which maximized
expected net returns in the earlier example which precluded forward
contracting, and the net income levels realized under the strategy with-
out contracting are also given in Table 6.1.

Several observations can be made about the differences in net
income levels realized under these two strategies. First, the average
net income level is higher under the strategy which precludes contracting.
This is to be expected, given the contract margins on corn and soybeans
of $.10 and $.25 respectively. Second, the variability of net income,
as measured by the standard deviation of the sample observations, is
reduced by forward contracting, again as would be expected. The reduc-
tion is not a sizeable one, however. It is also interesting to note

that in state 14 when yields for both corn and soybeans are apparently

 

1The computer program which implements the simulation model used
in this phase of the analysis is subroutine DISGEN in the listing of
the GREMP model at the end of Appendix C.

178

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ePeEem e cmec: mucescoeemm Emmem F.e m_nee

179
quite low, contracting costs the producer a considerable amount of
money and turns an already bad situation into a worse one. The problem
is particularly serious for corn. Pricing opportunities apparently
appear to be quite favorable, since more corn is contracted than would
normally be produced. At harvest, however, the cash price is $.09
above the average price for which corn was forward contracted, and a
total of 14,631 bushels must be purchased on the cash market at this
higher price to meet the terms of the contracts. Finally, it should be
noted that quantities of corn and soybeans contracted vary considerably
from one state of nature to another. This reflects the fact that there
are considerable differences in the attractiveness of the forward con-
tracting opportunities available in each state. Clearly the contracting

rule is sensitive to these differences.

6.4 (Decision Maker Preferences

 

Interval preference measures for the three decision makers dis-
cussed in the illustrations presented in Chapters IV and V will be used
to order the alternative strategies considered in this example.
Orderings will also be made for a risk neutral decision maker--a decision
maker whose absolute risk aversion function is always equal to zero--to
determine whether forward contracting rules can be identified which
lead to higher expected net income levels than those realized under
strategies which preclude forward contracting.

The net income level distributions associated with the two strate-
gies identified in the preceding section--one with contracting and one
without—-were ordered for each of the four decision makers considered

in this example. The strategy which precludes forward contracting is

180
preferred to that with contracting by the risk neutral decision maker
and by decision maker A. The two strategies cannot be ordered for
decision makers B and C. It is interesting to note that the non-contracting
strategy is preferred by decision maker A, whose preferences were most
conservative in the previous example. This may be attributable to the
aversion of this decision maker to the sizeable losses incurred under
the contracting strategy in state 14 when crop yields are unusually low

and contracting levels are high.

6.5 The Identification of Preferred Choices

 

By incorporating the simulation model described in Section 6.2 into
the Monte Carlo risk programming model described in Chapter V, an
efficient set of strategies can be identified for each of the four
decision makers considered in this example. Each strategy specifies a
flexible production plan and feedback control rules which direct forward
contracting decisions for both corn and soybeans.

Constraints placed on the values of crop production and land rental
control variables were discussed in Section 5.3 of Chapter V. They
remain unchanged in this example. Only upper and lower bound constraints
are placed on the parameters of the two contracting rules. It will be
recalled from Section 6.2 that each rule has four parameters. Experi-
mentation with the model indicates that the following ranges of admissible
values are reasonable and do not unduly constrain the set of feasible

choices:

181

OSVWSZO w=5, 9 6.5
~205vx50 x=6, 10
05vy51.0 y=7, 11
-.7Sv250 z=8, 12

Parameters vw and vx are restricted to integer values. Parameters vy

and v2 are restricted to values which are integer multiples of 0.1.

In this example 1000 randomly selected strategies were examined
and ordered for each decision maker's preferences. This is twice the
number of strategies considered in the earlier example. The fact that
each strategy is now defined by twelve control variables rather than
only four implies that many more feasible strategies exist, however,
and therefore more strategies should be evaluated. The efficient
sets identified range in size from a single element for the risk neutral
decision maker and three elements for decision maker A to thirteen and
seventeen elements for decision makers B and C respectively. It is not
possible to discuss all the efficient strategies for each decision
maker, therefore, a representative strategy from the efficient set of
each decision maker is defined in Table 6.2.

The representative strategy for the risk neutral decision maker
is that which maximizes expected net income under contracting. Detailed
information on system performance under this strategy in each of the
twenty states of nature considered is given in Table 6.3. The
expected net return is slightly less than that realized under
the expected net return maximizing strategy from the earlier example
which precludes contracting, but in many respects the two strategies

are remarkably similar. The two cropping plans are nearly identical,

182

 

 

 

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183

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184

and the contracting rules employed in the strategy which permits for-
ward pricing do not result in extensive forward contracting for either
crop in most states of nature. The paremeter weighting the first term
of each contracting rule is assigned a relatively high value, which
implies that contracting is strongly discouraged when the contract price
is below the expected harvest price and strongly encouraged when it is
above the expected harvest price. This is precisely the behavior one
would expect from an individual who seeks to maximize expected net
returns. The parameter weighting the second term of each contracting
rule is assigned a strongly negative value. This implies that contract
price movements also have an important impact on this decision maker's
choices. Even if the pricing opportunity is a favorable one, he will
delay his commitment to a forward contract if the contract price is
rising. Again this conforms to intuitive expectations about the
behavior of such a decision maker. Finally, the third parameter of
each contracting rule is relatively small, which implies that this
decision maker places little emphasis on achieving a desired level of
contracting under almost any market conditions.

Net income levels realized under an identical production strategy
which precludes contracting are also given in Table 6.3. Comparison of
the two net income distributions indicates that contracting can lead to
both higher expected net returns and less income variation, though
actual differences between the two strategies are minimal. It is also
interesting to note that the corn contracting rule is particularly
successful in this instance, the average corn contracting price being

well above the cash price at harvest in three of the four states of

185

nature when more than 1000 bushels are contracted. The soybean con-
tracting rule is somewhat less successful.

Information on system performance under the preferred strategy of
decision maker A is given in Table 6.4. This strategy is markedly
different from that of the risk neutral decision maker. Its production
component calls for no land rental and for the planting of all available
acreage in soybeans. This is identical to one of the efficient produc-
tion plans identified for this decision maker in the earlier example.
Because no corn is to be planted, only the soybean contracting rule
has any impact on system performance. As the information given in
Table 6.4 indicates, this rule results in substantial levels of forward
contracting in most states of nature. The parameter weighting the first
term of the soybean contracting rule, v9, is relatively large, indicating
that this decision maker also places considerable emphasis on fundamen-
tal analysis--on differences between the forward contract price and the
expected cash price at harvest. Unlike the risk neutral decision maker,
however, decision maker A assigns relatively little weight to technical
analysis--to the analysis of contract price movements--as is indicated
by the small absolute magnitude of parameter v10. Also in contrast to
the strategy of the risk neutral decision maker, the relatively large
value of parameter V11 indicates that much stronger emphasis is placed
on attaining some desired level of contracting under most conditions,
and the fact that v12 is assigned a value of 0.0 implies that this
desired level is 100 percent of the expected crop once all acreage has

been planted.

 

186

 

 

 

 

 

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187

Comparison of the net income levels realized under this strategy
with those associated with a management strategy which precludes con-
tracting but has an identical production plan is of considerable interest.
The information given in Table 6.4 indicates that both the probability
of negative net income levels and the magnitude of losses are reduced
by the contracting rule. The probability of realizing relatively high
income levels is also reduced, however, and expected net income is con-
siderably higher under the strategy without contracting. When these two
strategies are ordered using decision maker A's preference measurement,
the strategy with contracting dominates that without.

Information on system performance under the representative strategy
from the efficient set of decision maker B is given in Table 6.5.1 In
this instance 160 acres are rented, 140 acres are to be planted in corn,
260 acres are to be planted in soybeans, and May 18 is the date after
which all unplanted acreage is planted to soybeans. The two contracting
rules are quite different in this strategy. That for corn places a
strong emphasis on differences between the current contracted price and
the expected cash harvest price, the value of parameter v5 being 19.0.
Relatively little emphasis is placed on price movements, as evidenced
by the small value of v6, and a rather strong emphasis is placed on
contracting a specified percentage of the projected crop. This rule
results in moderate to high contracting levels in thirteen states and is

quite successful, with the average contract price being above the cash

 

1The preferred strategy of the risk neutral decision maker is also
in decision maker B's efficient set.

188

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189

harvest price in ten of the thirteen states in which there is contracting.
The soybean contracting rule, on the other hand, is characterized by low
values for each of its four parameters. It results in contracting levels
below 1000 bushels in all but three states of nature and so has little
effect on net income levels realized.

Comparison of the distribution of net income levels realized under
this strategy with that associated with an identical production plan
without contracting indicates that contracting results in a slightly
higher expected net return and in some reduction in income variability
Given decision maker B's preferences, the strategy with contracting
dominates that without.

Information on system performance under the representative strategy

1 In this case the produc-

for decision maker C is given in Table 6.6.
tion plan calls for the rental of 80 acres, for 50 acres of corn, and
for 270 acres of soybeans. May 25 is the date after which all unplanted
acreage is planted to soybeans, but the switching rule is inactive due
to the low level of desired corn acreage. The contracting rules for
both crops place considerable weight both on differences between the
current contract price and the expected cash harvest price and on con-
tract price movements. The corn contracting rule emphasizes the con-
tracting of a certain percentage of the crop in most states of nature,

however, while the soybean rule does not. Neither rule results in

substantial contracting levels in most states of nature, though the

 

1The representative strategy of decision maker B is also in
decision maker C's efficient set.

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191
corn rule does lend to contracting more than half the number of bushels
harvested in three states of nature. The average contract price for
corn exceeds the cash harvest price in twelve of the twenty states of
nature, while that for soybeans exceeds the cash harvest price in only
two of the six states in which there is contracting.

The distribution of net income levels realized under this strategy
is quite similar to that associated with an identical cropping plan and
no contracting. Expected returns are slightly higher under the strategy
with contracting, however, and, given decision maker C's preferences,

it is preferred to that without contracting.

6.6 Further Discussion of the Results

Several more general observations can be made regarding these
results. First, they demonstrate once again that decision maker pre-
ferences have an important impact on the choice of a management strategy.
Nowhere is this more evident than in the differences between the pre—
ferred strategy of the risk neutral decision maker and the representa-
tive strategy from the efficient set of decision maker A, whose level
of absolute risk aversion is high over the negative range of net income
values. Both cropping and marketing strategies are quite different, as
are the associated net income distributions. The preferred strategy
of the risk neutral decision maker calls for the rental of 240 acres
and forairelatively balanced mix of corn and soybean production; it
calls for the application of contracting rules which lead to an avoidance
of forward pricing except in instances when pricing opportunities appear
to be particularly favorable. The representative strategy for decision

maker A, on the other hand, calls for no land rental and specialization

192
in soybean production and for the application of a contracting rule
which results in the forward pricing of a substantial portion of the
expected crop under nearly all market conditions. The expected net
income level of $10,756.27 under the preferred strategy of the risk
neutral decision maker is more than $7,400 higher than that associated
with the representative strategy of decision maker A, but the latter
is much less likely to face substantial losses under his preferred
strategy.

The results also show that the introduction of forward contracting
has only a minor impact on crop production strategies. Levels of land
rental, relative acreages allotted to corn and soybeans, and stopping
dates for corn planting specified in the strategies included in the
efficient set of each decision maker show few significant changes. The
most notable change is the inclusion of several cropping plans which
devote more acreage to corn in the efficient set of decision maker C
when forward contracting is incorporated into the management strategy.

A third observation is that, though many of the strategies identified
appear to have some minor flaw, efforts to construct improved strategies
through a search of a larger number of strategies or by simply changing
parameter values which seem to cause problems rarely resulted in the
identification of strategies with substantially better performance. In
an effort to identify a better strategy for the risk neutral decision
maker, for example, 250 additional strategies for which land rental is
set at 240 acres were evaluated. The strategy defined in Figure 6.3 was
identified as that which maximizes expected net returns. It has an

expected net income level of $10,868.57. This is higher than that of

193

Crop Production Plan

v1 = 240 acres rented

v2 = 190 acres corn

v3 = 290 acres soybeans

v4 = May 26, the date after which

all unplanted acreage is
planted to soybeans

Corn Contracting Rule Parameters

v5 = 20.00
v6 = -12.00
v7 = .20

v82 = -.40

Soybean Contracting Rule Parameters

v9 = 7.00
v10 = -4.00
v1] = 0
v122 = -.30

Expected Net Return: $10,868.57

Standard Deviation of Net Return: $15,381.73
Lowest Sample Net Return: $-l7,108.70
Highest Sample Net Return: $33,857.36

Figure 6.3 The Expected Net Return Maximizing Strategy

194

the preferred strategy identified in the original searches of strategies
with and without contracting. The increase in the level of expected net
income is not a very significant one, however.

Finally, it is interesting to note the composition of the efficient
set for each decision maker when strategies with and without contracting
are evaluated simultaneously. For the risk neutral decision maker the
results based on the original search indicate that a strategy without
contracting--that defined in Table 6.l--is preferred to all others. The
strategy defined in Figure 6.3, which does involve contracting, has a
still higher expected net return, however, and so is preferred by this
decision maker.1 For decision maker A, it was found that each of the
eight efficient non-contracting strategies is dominated by at least one
of the efficient strategies with contracting. For decision maker B
seven of the nine efficient non-contracting strategies were dominated
by at least one efficient contracting strategy, and for decision maker C
all eleven efficient non-contracting strategies are dominated. In
general, then, strategies with contracting clearly outperform those

which preclude it under the conditions specified in this example.

6.7 Implications for Further Research
The problem of identifying combined production and marketing strate—
gies which are well suited for a particular decision maker's situation
is an important one. The analysis presented above is not intended to
be a source of definitive solutions to this problem. Rather, it demon-

strates an approach to the analysis of choices of this sort which employs

 

1A more intensive search of non-contracting strategies failed to
identify any with a higher expected net income level.

195

the techniques developed in this study to identify reasonable management
strategies which perform well under relatively realistic conditions.
More work needs to be done, however, to make the model described above
the basis for truly reliable prescriptions.

With regard to problem formulation, other marketing alternatives
need to be incorporated into the analysis. The impact on individual
management strategies of government price stabilization programs which
put an effective floor on crop prices and provide a form of disaster
insurance for participants may be particularly important, for example.
Similarly, the use of the futures market rather than forward contracting
may be an attractive alternative for larger producers. In addition to
the inclusion of other marketing alternatives, more careful specification
of the feedback control rules which direct marketing decisions may also
make the model a more reliable prescriptive tool.

Improved assessments of probability distributions for all the
stochastic factors considered in this example are also needed. Particular
attention should be given to the specification of more realistic distri-
butions for forward contract price offerings over the course of the
planning horizon. As was done in the example above, this can, perhaps,
be best achieved by modelling the relationships among current contract
price, cash price at harvest, and intermediate contract price offerings.
In addition to improved specifications of subjective probability distri-
butions, it may also be desirable to consider more than twenty sample
states of nature in a decision situation such as this one in which a

large number of random factors have an impact on net income levels.

196

Once modifications of this sort have been made, a more systematic
exploration of the effect of decision maker preferences, probability
assessments, and scale of operation on the nature of preferred forward
pricing rules would be of particular interest. Can shifts in the para-
meter values of a specific rule be related to differences in factors
such as these? Are contracting rule parameters relatively insensitive
to changes in some of these factors? Is the choice of an expectations
model dependent upon preferences? These are important questions which
should be considered in future research. The objective of such research
should not necessarily be to identify invariant behavioral rules which
can be applied in any situation. The results above indicate that such
rules are not likely to exist. Rather, the goal should be to better
understand both the contracting rules themselves and the complex inter-

actions among the variety of factors which affect decision maker choices.

CHAPTER VII
SUMMARY AND CONCLUSIONS

7.1 A Review of the Methodological Tools
Developed in this Study

 

This study has focused on the development of techniques designed
for use during four important phases of an applied decision analysis:
problem formulation, the determination of subjective probability distri-
butions, the measurement of decision maker preferences, and the identifi-
cation of preferred choices. When considered together, the procedures
described in this study represent an integrated set of operational
techniques which facilitate the application of powerful theoretical
tools based on the expected utility hypothesis in an applied setting.

In the discussion of problem formulation, two important considera-
tions were stressed: the need to structure the problem being analyzed
by identifying and classifying factors which will have an important
impact on the outcome of the decision being made, and the need to give
careful attention to the definition of what is to be decided. With
regard to the first of these considerations, the usefulness of systems
identification (Manetsch and Park, l977a) as an aid in structuring the
decision maker's perception of a particular practical problem was
demonstrated. With regard to the definition of what is to be decided,
the need to recognize the existence of future opportunities for learning

in many decision situations and the desirability of flexible management

197

198
strategies in such instances were emphasized. The incorporation of
feedback control rules into a management strategy was shown to be one
way in which considerations of flexibility can be introduced into a
decision analysis.

The value of combining direct probability assessments of underlying
stochastic factors with the modelling of more complex stochastic pro-
cesses was stressed in the discussion of the determination of the dis-
tribution of outcomes associated with any particular choice. Under this
approach, which is suggested by Spetzler and Stael von Holstein (1975),
direct encoding techniques are used to elicit information on a decision
maker's expectations about future levels of environmental variables
which cannot be controlled by the decision maker but have an important
impact on the outcome realized under any particular strategy. Monte
Carlo simulation techniques are then used to determine the combined
effect of these exogenous stochastic factors and a particular management
strategy in the distribution of system output levels. The use of both
direct encoding and modelling allows considerable flexibility in the
specification of subjective probability distributions of underlying
variables and in the representation of the stochastic system under
consideration--flexibility that is often lost when a more strictly
analytical approach is taken.

One important criticism of systems modelling and Monte Carlo
simulation is that correlations between random factors are often
ignored. This is due largely to the fact that procedures for the
generation of sample observations from multivariate distributions have

been developed for only a few special distributions. An important

199
contribution of this study is the development of a generalized multi-
variate process generator, which is described in detail in Appendix A.
This greatly enhances the value of Monte Carlo simulation as a tool in
the applied analysis of decisions made under uncertainty.

The need for preference measurement techniques which are more
flexible and more reliable than existing procedures is stressed in the
discussion of the measurement of decision maker preferences. In response
to this need, a new approach to preference measurement has been developed
as a part of this study. This new procedure permits the construction
of interval measurements of a decision maker's absolute risk aversion
function. Perhaps its most important feature is that it allows direct
specification of the degree of precision with which preferences are to
be measured. In contrast, single valued utility functions are exact
but often inaccurate preference measures, and commonly used stochastic
efficiency criteria are based on inexact, inflexible assumptions about
preferences.

Interval measurements of absolute risk aversion are used in con-
junction with the criterion of stochastic dominance with respect to a
function (Meyer, 1977a) to order alternative choices. This recently
developed stochastic efficiency criterion can be used to evaluate alter-
native choices for classes of decision makers defined by upper and lower
bound absolute risk aversion functions. As such, it is both more
flexible and potentially more discriminating than other efficiency
criteria based on stochastic dominance.

Results of an empirical test of the interval approach to the
measurement of decision maker preferences demonstrate its value. They

show that it leads to a lower probability of eliminating a decision

200

maker's preferred choice from the effiCient set of choices than
that realized with a single—vaerd utility functidn and that
permits the identification of efficient sets smaller than those
associated with the criteria of first and second degree stochastic
dominance. The results also show how the precision with which pre-
ferences are measured can be varied under this approach.

The identification of preferred choices is the primary objective
of any decision analysis. The final methodological contribution of
this study is the formulation of a generalized risk efficient Monte
Carlo programming model (GREMP), which combines random search proce-
dures, Monte Carlo simulation, and evaluation by the criterion of
stochastic dominance with respect to a function in a single analytical
framework for the identification of preferred choices. This model is
both flexible and computationally efficient, and it is well suited for
the analysis of a wide range of practical decision problems. The use
of Monte Carlo programming procedures to generate alternative strategies
for consideration facilitates the introduction of flexibility into the
definition of a management strategy, since strategies can be defined by
feedback control rules as well as by specific values of control variables
over the entire planning horizon. The incorporation of Monte Carlo
simulation techniques into the model facilitates the more realistic
representation of the complex processes by which the control variable
levels determined by a particular management strategy and a set of
stochastic environmental factors interact to determine the properties
of the distribution of outcomes associated with any choice. Finally,

evaluation by the criterion of stochastic dominance with respect to a

201

function permits the ordering of choices in a manner fully consistent
with the expected utility hypothesis without requiring that restrictive
assumptions be made about decision maker preferences or the form of
system output distributions.

The methodological tools developed in this study are of considerable
value. It should be noted, however, that they are not intended for use
in all decision situations. They allow considerable flexibility con-
cerning the degree of detail incorporated into any decision analysis,
but they are intended primarily for application on a computer and can
be expensive to use. They do not replace budgeting techniques or
mathematical programming but supplement these and other existing
methodological tools. As is true of any aid in the decision process,
the procedures developed here should be employed only when the benefits
associated with their use exceed the possible added costs. It should
also be noted that these procedures are not all that is required to
successfully resolve a practical decision problem. Rather, they repre-
sent a set of tools which facilitates some aspects of the decision

process-~a process which involves a wide range of activities.

7.2 Empirical Findings

 

This study focuses primarily on methodological rather than empirical
issues. Several empirical findings associated with the illustrative
applications of the procedures developed in this study are worthy of
note, however.

As a test of the interval approach to the measurement of decision
maker preferences, preference measurements were made for seventeen farmer

participants in a marketing extension workshop. As reported in Section 4.8

202

of Chapter IV, the resulting preference measurements provide rather
strong evidence that many decision makers exhibit both risk loving

and risk averse behavior. The interval measurements for thirteen of the
seventeen respondents included negative as well as positive values. The
results also showed that strictly decreasing absolute risk aversion
functions are not as common as is often suggested. Though certainly not
generalizable to a larger population, these findings do call to question
the wide acceptance of the proposition that absolute risk aversion
functions tend to be positive valued and strictly decreasing, and so
suggest that further research in this area may be warranted.

The procedures developed in this study are applied in the analysis
of two related examples concerned with choices made in the operation of
a cash grain farm. The first example focuses on land rental and produc-
tion planning decisions when prices, yields, and time available for
fieldwork are uncertain. In the second example, these same decisions
are considered in conjunction with the choice of a flexible marketing
strategy involving cash sales at harvest and forward contracting. The
results of both applications demonstrate the flexibility of the proce-
dures developed in this study. They also indicate that decision maker
preferences have an important impact on both the selection of a cropping
plan and on the choice of an appropriate marketing strategy. Finally,
they show that the degree of interdependence between production and
marketing strategies is not particularly great, which suggests that it
may be possible to analyze these two components of a management

strategy separately in some situations.

203

7.3 Implications for Future Research

 

The techniques developed in this study can be of use in the analysis
of a diverse range of practical decision problems. Clearly more work
can be done, for example, on the analysis of alternative production and
marketing strategies for agricultural firms. Of particular value would
be attempts to develop flexible marketing strategies which are suitable
for a broad range of decision makers and decision situations. Another
potentially important application of these techniques is in the analysis
of alternative pest management strategies. Such strategies are, in
effect, feedback control rules designed to direct actions in complex
decision situations characterized by a considerable degree of uncertainty,
and the GREMP model, with its combination of random search and simulation,
is well suited for the identification and evaluation rules of this type.

The methodological tools developed in this study can also be of
use in the analysis of major investment-disinvestment decisions, both
private and public. They permit the explicit consideration of flexibility
in the analysis of such choices--a factor which, as Masse (1962) notes,
is of critical importance when major resource commitments are to be
made in an uncertain environment. At a still higher level of aggrega-
tion, these tools can also be employed in the analysis of policy decisions
having outcomes which are strongly influenced by stochastic factors in
the environment in which they are implemented.

The techniques developed in this study greatly facilitate the
application of decision theory based on the expected utility hypothesis
in the analysis of practical decision problems. They have considerable

potential value, but they can be made more effective if they are refined

204

still further. There is a need, for example, for further testing of
the procedures used to implement the interval approach to the measure-
ment of decision maker preferences. Experiments should be conducted to
identify measurement scales which allow preferences to be adequately
represented in a wide range of decision situations, and alternative
modes of questioning should be evaluated. Will a measurement grid
which works well in the analysis of choices having outcomes which are
concentrated around a certain value be adequate when preferences are to
be measured over a much broader range of outcomes? Over how wide a
range of systems output levels can absolute risk aversion be assumed
to be constant? Does the range depend on the level of system output?
In the neighborhood of how many system output levels should preferences
be measured? Upon how many choices should each measurement be based?
These are but five of the many questions which need to be answered.
More research is also needed on the representation of preference
relationships which depend on more than one system output variable and
on the development of multivariate stochastic dominance criteria. Though
some work has been done in the latter area by Levy (1973), Levy and
Paroush (l974a, b) and Kihlstrom and Mirmon (1974), further research
is needed. Particularly valuable would be an extension of stochastic
dominance with respect to a function to the multivariate case.
Additional refinements are also needed in the GREMP model. It may
be possible, for example, to increase the efficiency of this procedure
by incorporating learning rules which lend at least partial direction
to the random search. Such rules could be applied at the end of each

100 iterations of the model and might have the effect of reducing the

205

range of values for control variables over which the search is to be
made in subsequent iterations. A second type of rule might cause the
search to be stopped if no new strategies have entered the efficient set
for a certain number of iterations.

Finally, it must be noted that methodological advances alone will
not eliminate all the difficulties encountered in the applied analysis
of decisions made under uncertainty. The two applications of the
techniques developed in this study point to the importance of and the
need for an improved information base in most decision situations. This
need is particularly strong with regard to decisions made by agricultural
producers who face so many different types of uncertainty. Efforts must
be made to supply producers with probabilistic price forecasts and to
teach them to use such information effectively. Frequently, all the
information needed to make forecasts in probabilistic terms is readily
available to the agencies or individuals who predict future price levels,
but in most cases price forecasts simply state a most likely value, or
at best, an interval within which the price is expected to fall.
Similarly, more complete information is also needed about how yields
are affected by timeliness and by stochastic factors in the environment.
More consideration needs to be given to such factors in the design of
agronomic experiments.

The information base for the analysis of decisions made under
uncertainty could also be improved by research designed to identify
systematic relationships between levels of absolute risk aversion and
selected decision maker characteristics. Such information could be of

use to policy analysts who wish to consider the impacts of uncertainty

206
on the choices made by representative firms. It could also be of use
in situations when the importance of a choice to be made does not
warrant the expenditure of sources required to construct an interval

measurement of the decision maker‘s preferences.

In conclusion, then, the methodological tools developed in this
study can, in their present form, be employed in the analysis of a wide
range of practical decision problems. They represent an important
improvement in the set of techniques available for the applied analysis
of decisions made under uncertainty. Further efforts are needed, however,

to make these tools both easier to use and more effective.

APPENDICES

APPENDIX A
A GENERALIZED MULTIVARIATE PROCESS GENERATOR

A.l Introduction

 

The analysis of decisions made under uncertainty requires that infor-
mation on the probability distributions of system output variables be
determined for each alternative strategy being considered. In most
cases the properties of such distributions depend on the controllable
system input levels which define any particular strategy and on the
probability distributions of stochastic environmental factors which
cannot be controlled by the decision maker. When there is only one
stochastic environmental factor or when the relationship between con-
trollable and exogenous system inputs and system outputs is relatively
simple, the properties of system output distributions can be derived
analytically.1 When several stochastic factors having probability
distributions from different families must be considered or when input-
output relationships are not of a convenient form, however, analytical
techniques cannot be used to determine the properties of system output
distributions. In such instances, Monte Carlo sampling procedures and
numerical simulation techniques are frequently used to obtain the

necessary information. Sample states of nature are defined by selecting

 

1See Anderson and Doran (1978) for an excellent review of the con-
ditions under which information on such probability distributions can
be determined analytically.

207

208

values for each exogenous system input in a pseudorandom _fashion.
Numerical simulation techniques are then used to determine the system
output levels associated with a particular strategy for each state of
nature. The resulting values constitute a random sample from the pro-
bability distribution of system outputs. They can be used to calculate
sample moments of the underlying distribution, or they can be used to
construct an approximate representation of the cumulative distribution
function of the system output variable.

A process generator is a procedure, usually programmed for imple-_
mentation on a computer, which generates pseudorandom sample observa-

1 As such, process

tions from a specified probability distribution.
generators are a basic building block in the procedure described above.
Process generators have been developed for a wide range of standard
univariate probability distributions including the discrete and con-
tinuous uniform, exponential, gamma, beta, chi—square, normal, lognormal,
geometric, binomial, hypergeometric, and Poisson (Naylor, et al., 1966;
Schmidt and Taylor, 1970; Newmann and Odell, 1971). In determining the
properties of a distribution of systems outputs, the use of univariate
process generators is appropriate if all underlying stochastic factors
can be assumed to be independently distributed. When this assumption
cannot be made, a multivariate process generator is required.

Process generators have been developed for several multivariate

distributions, most notably the multivariate normal and the Wishart

 

1The term "pseudorandom" refers to the fact that, although values
generated by a process generator have all the properties of a random
sample, they are actually generated in a deterministic fashion.

209
distributions (Naylor, et al., 1966; Newmann and Odell, 1971). In many
cases, however, the properties of the joint distribution of a particular
set of stochastic environmental factors may not be adequately approxi-
mated by either of these distributions. There is a need, then, for
multivariate process generators which are more flexible than those
currently available.

A major contribution of this study is the development of a workablel
procedure for the generation of random variates from multivariate pro-
bability distributions with non-normal marginal distributions. The
formulation and implementation of this procedure is the primary focus
of this appendix. Before introducing the generalized multivariate
process generator, however, basic concepts related to the generation of
random variates are first reviewed, and several commonly used univariate
process generators are presented. The generation of random variates
from the multivariate normal distribution is then discussed, since this
procedure is used in the more general process generator developed in
this study. Finally, the algorithm for the generation of random
variates from multivariate distributions with specified marginal dis-
tributions and correlation matrix is presented along with an explana-
tion and listing of the computer program used to implement it.1 An
empirical example is also presented to demonstrate the efficacy of

this procedure.

 

1Though developed independently, this procedure is similar in
many respects to methods described in an unpublished paper by Coleman
and Saipe (1976), which describes several procedures for the generation
of sample observations from bivariate distributions.

210

A.2 Basic Procedures for the Generation
of Random Variates

 

Several approaches have been developed for the generation of
random variates. They include the inverse transformation method, the
rejection method, and the composition method, all of which are described
in Naylor et al. (1966). Because it is the most commonly used and
because it serves as the basis for the algorithm to be discussed below,
only the inverse transformation method will be reviewed here.

The cumulative distribution function, F(x), of the continuous
random variable, x, is defined over the interval (0,1). Associated
with each value of x, then, is a value r lying on the interval (0,1)
such that

r = F(x) A.l
Similarly, if the inverse of F(x) can be determined, the following
relationship will hold:

x = F-1(r) A.2
In this case any particular value of r uniquely determines a value of
x if F'](r) is a continuous, monotonically increasing function.. By
generating a set of uniformly distributed random variables lying on
the interval (0,1) and calculating the corresponding valueSnyx deter-
mined by equation A.2, a set of sample observations from the probability
distribution of x is generated. This is the inverse transformation
method for generating random variates.

Following Manetsch and Park (1977b) the validity of the inverse
transformation method can be demonstrated in the following manner. Let

r = F(x) A.3

211

where r is a uniformly distributed random variable defined on the
interval (0,1). By the definition of a cumulative distribution function

F(xo) = Pixfxo} = r0 A.4
where x0 is a specific value of x and r0 is the corresponding value
of r. Since r is uniformly distributed

- = l

G(ro) - P{rSrO} rO A.5

Since F(xo) and r0 are equal, Equation A.5 implies that
< ..

P{r-F(x0)} - F(xo) A.6
and since F(x) is assumed to be a monotonically increasing function in
x, it also follows that

PiF"(r)Sx01 = F(x P{xfxo} A.7

0) =
Therefore, the distribution of the random variable generated by the
inverse transformation method is identified to that of the random

variable x.

Application of the inverse transformation method is shown graphically
in Figure A.1. A cumulative distribution function F(x) is drawn on the
left, and its inverse is drawn on the right. In the particular example
shown the randomly selected value of R is .5, and the associated value
of x is 6.3.

The inverse transformation method is the basis for commonly used

procedures for generating random variates from several standard probability

1For uniformly distributed random variables

g(ro) = rO'OSrosl

0 elsewhere
and

r0 r0

vogue: :oNNeENonceNN emce>cN
ezN Xe mmNeNNe> soecem No :oNNeNecme N.< ecsmNN

 

 

212

 

o v N 3 o o v N
11 d d d d. I q A u q a I d d d d
u . .
_ L N N N.
.
. .
_
. . z . z.
.
_ . .
— O
N 11111111 . o .. o
A ..
L O a O.
.. c— u o.—
Xlahv —.Im LIAXvu

 

213
distributions. Most notably, it is the basis for process generators
for exponential, gamma and beta random variables. Procedures for the
generation of variates from each of these distributions are presented

below with the computer programs which implement them.1

A.2.l An Exponential Process Generator

 

The density function of an exponential random variable is given

by the expression:

f(x) = oe"°‘x OSx A.8
.0 otherwise?.

This function can be integrated to obtain the cumulative distribution

function, F(x)

F(x) = Ix oe'ax dx 1-e'0‘x = r. A.9

0

The inverse of F(x) is given by the expression

F”1 (r) = x = --1;ln (l-r). 4'10
If r is a uniformly distributed random variable on the interval (0,1)
this expression reduces to

x = - a— 1n (r). A.11
By generating values of r at random and calculating the associated
values of x using Equation A.ll, sample observations from an exponential
distribution with parameter a can be generated.

A computer program which implements this procedure is listed in
Figure A.2. Three parameter values must be supplied by the user: ND,

BL, and XMEAN. ND is the number of variates to be generated, BL is

 

1These programs can easily be made into subroutines and incorporated
into larger programs.

2The mean and variance of this distribution are l/o and l/az
respectively.

214

CUTPUT)

=INPUTeTAPEE=

r

FULHWFE

TNT-

rlIL )

5‘
I C L o
(D pmxru
VN + 1)
F ))r......
P.) N1. 0‘02 0
Xr1.LN.\Il.rU 9N1.
N.......N! 91166ct.]
11.14. C 01.:
or! OH:..I\1L(C(I\
ASAIFAiTT
RTE “1555
nun-1...! PJEIPT pin.
OAVK D- N In! D. .i.-.t
EFL—U: :Fnun..1q
EDWARDLKXIWC NFC

heart.
”In.
1.1.:

Figure A.2 A Program for Generating Exponential

Random Variates

215
the lowest value the random variable can take,1 and XMEAN is the expected

value of the distribution.

A.2.2 A Gamma Process Generator

 

The density function of a gamma distribution is

 

0 otherwise?

An analytic expression for the cumulative distribution function of this
distribution does not exist. It can be shown, however, that the sum of
k exponentially distributed random variables, each having an expected
value of 1/a, is a gamma random variable with a density function identical
to that given in Equation A.12 (Schmidt and Taylor, 1970, p. 265).
Random variates from gamma distributionS‘flncwhich the parameter k is an
integer, then, can be generated by summing k variates drawn from an
appropriate exponential distribution.

A computer program which generates gamma random variates by this
method is given in Figure A.3. In this case four parameter values must
be supplied by the user: ND, BL, XMEAN, and K. ND, BL, and XMEAN are,
again, the number of sample observations to be generated, the lower

bound of the distribution, and the expected value of the distribution.

 

1In the deviation above, BL was assumed to be zero, but this need
not be the case.

2The mean and variance of this distribution are k/a and k/a2
respectively.

216

OUTPUT)

(u

E

D.

A

TI

9

T

U

p.

N.

I

N.

C»

E

P.

A

TI

9

T

U".

P 9

z. N.

Ur»

0E)

en. an I

Tvnl. c.“

U 9? T.

D LA. e.

IleanN... 5.!

I 'L 0

(DC; Vrrb

AN] 1)

V )h \I )F9..

My)l..\ F. N .5... 0

AOL... y)‘ r... 9......

rUNU .14 91oF... N..NU.1
1....» : 1N n. 971:

P. 9’1. :IL {Nile

ANSL lupin (NIT

D..(N.n..u nu er.r_£.r.

roenuvrchueL a. + T ”on:

Figure A.3 .A Program for Generating Gamma

Random Variates

oar. Q. D..)IN.LN.N\111
CHE-.0..0:: _NFIVCN.
hr Duluhuv fiLhN r...) erN F L.

Cufierwnie
1b....
In"

217

K is the second parameter of the gamma distribution.1 The parameter K
cannot always be treated as an integer. Though not reviewed here, a
procedure for generating variates from gamma distribution with non-
integer parameters is described elsewhere by Phillips and Beightler

(1972).

A.2.3 A Beta Process Generator

The density function of the beta distribution is

a-1 -1
.1.) 18.3er
2

O elsewhere.
Again, an analytic expression for the cumulative distribution function
cannot be derived. As Naylor et a1. (1966) note, however, the random
variable defined by the expression

x = xl/(x1+x2), A.14

where x1 and x2 are both gamma random variables with identical values
of a and values of k such that k1 = a and k2 = 8, has a beta distribu-
tion with parameters a and B. If a and B are integers, then, beta

random variates can be generated using an extension of the procedures

 

1The parameter k can be determined by solving the following
expression
k g (p-BL)2
2

a
where u and a2 are the mean and variance of the distribution being
modelled and BL is the lower bound of that distribution. The program
computes the value of the parameter a automatically.

2The mean and variance of this distribution are

u = a/(a+B)
and

 

o2= oB
(a+8+1)(51812

218
developed above. Two gamma variates from appropriate distributions are
generated and a beta variate is defined according to Equation A.14.1
A computer program for generating beta variates is given in
Figure A54. The user must supply values for five parameters: ND, BL,
BU, K1, and K2. ND is the number of variates to be generated. BL and
BU are the upper and lower bound values which the random variable can
take. Though set at zero and one in the derivation above, they can be
set at any values. K1 and K2 are the two shape parameters of the beta

distribution. Their values are determined by solving the following

two equations

 

3 2 2
K2 = Y :27 211+GJY-5 A.15
6
and K1 = 4%é- A.16

where y and 52 are the mean and variance of the distribution once it

has been normalized so that all values be on the interval (0,1).2

A.2.4 A Generalized Univariate Process Generator
Each of the process generators above is based on a simple applica-
tion of the inverse transformation method to the exponential distribu-

tion. For many distributions, however, the inverse of the cumulative

 

1When a beta distribution has non-integer parameters, the gamma
variates required in Equation A.14 can be generated using the procedure
developed by Phillips and Beightler (1972).

zMore formally,

Y = (u-BL)/Bu-8L)

and

52 = yz/(Bu-BL)2
1where u and y2 are the mean and variance of the actual distribution to
be modelled.

219

INPUT-TAPFE=CUTPUT)

(Nu

AN. no;

T3rh 1

{.0 C K),

L. .41 91F-
1: 1.1!

.1. 01. :(Gr.

AN... Enid.» O +

:LJePL

'

5'
75"
J’.

)

O

))*(5U-
X

10.79

)

pa A4)c r..—
K. 5.29. e
9,) 0 4. On.
11‘ 17:»..1
—— -( G 'YJLI
Jin.r.(£.((

FC+I¢TT

PIE.) 0 o FwLIO..A,L?1anNt

floa.1run.r.£AG
h. a. N. .N 9,. ..

:4

IN AFC f....1 u- p.

F...‘71rwrh-.
FF.L01J?U....10..:n/.:c PCP
pLDGCrUCV.th.PrLC.X.F—c F.

1.. LJrL.-.
1.. 11........
1a.

Figure A.4 A Program for Generating Beta

Random Variates

220

distribution function cannot be derived analytically, and simple
analytical links to other distributions for which such inverses can be
calculated may not exist. Most notably, this is true for the normal
distribution. In the case of many empirical probability distributions,
the exact functional form of the cumulative distribution function may
not even be known. Nevertheless, the inverse transformation method

can still be used in such instances. Values of the cumulative distri-
bution function F(x) at specified levels of the random variable x can
be used to construct a rough approximation of the entire function by

1 In Figure A.5, for

linearly interpolating between known points.
example, six points on a cumulative distribution function are specified:
the upper and lower bounds, and three intermediate points which divide
the cumulative into quartiles. Given such a diagramatic representation,
the value of the random variable x associated with any randomly selected
probability level is easily determined. As shown in Figure A.5, for
example, the value of x corresponding to a probability level of .2
is 44.

A table look-up function (Llewellyn, 1965) is the equivalent of
such a diagram on a digital computer. The table look-up function
TABEX, a listing of which is given in Figure A.6, has access to an

array of X values, ARG, and an array of corresponding probability

levels, VAL, each array having K elements. Given an argument value,

 

1Points in the cumulative distribution function can be determined
analytically, by numerical integration of a known probability density
function, or they can simply be sample observations from an empirical
distribution. As Schlaeffer (1959) notes sample observation arranged
in order are reasonable estimates of the fractiles of the underlying
distribution.

221

CNN

oNN

mucNoo czocz me co eemem
:oNNucou coNuoaNNNmNo e>NNeN352u
e No coNNeNcemmcoem eNeENxoeoa< c< m.< ecemNN

oo— om ow ON on om ow om ON ON

 

 

 

S.
3.

222

AV

t
1

O 1
Y L(J)-VfL(J-1))/

T
*(
d-

O)!‘
5’15
1.1+
\INUu
310+
.0!)
(P9)
591
RA.
DNJ
0"
TV?

1 VGWP
NQeUA
wCZYfi.
flLINNRla)

1L
ISJV2JIN

T"

U x (Phil

.1

1|.

(Elna-I erUvi

NM

“ICC. .nNaLNtNr... Z n1.e1\
FP.[IT(¢?(1Ur r.

1.

2

1

Llewellyn (1965, p. 4-22).

pr.

0
TO

K

Source:

Figure A.6 A Table Look-up Function Subprogram

223

DUMMY, this function subprogram computes a value of the function defined
by ARG and VAL. If x is a random variable and if R is a variable lying
on the interval (0,1) which corresponds to a value of F(x), then the
FORTRAN statement

R = TABEX(VAL, ARG, X, K) A.l7
calculates the probability level on the cumulative distribution function
corresponding to any specified value of X. One of the interesting pro-
perties of this particular table look-up function is that by simply
switching the positions of VAL and ARG in the calling statement, values
of the inverse function can be calculated. The FORTRAN statement

X = TABEX (ARG, VAL, R, K) A. l8
for example, calculates the value of X which corresponds to any
specified probability level, R.

A computer program which uses the inverse transformation method
to generate variates from a probability distribution defined by a
specified set of points on the cumulative distribution function is
given in Figure A.7. The user must supply values of ND, the number of
variates to be generated, and K, the number of points on the cumulative
distribution function to be specified explicitly. Paired values of the
random variable and the cumulative distribution function are then read
into arrays ARG and VAL. These values must be arranged by the user in
ascending order.

The procedures developed in this section are of interest in them-
selves, since they are all useful tools for Monte Carlo sampling. More
important in relation to the primary purpose of this appendix, however,
they are used extensively in the generalized procedure for the genera-

tion of variates from multivariate distribution.

224

OUTPUT)
FUNCTICL

,b N

F. 0

p. I o

A T R I
T. U pl. \,
' E h}. \I
.l I R 1
U R O .

P T In
.N D Cy F... Is
I E T; N L
= T. D .1 fr
c». A» o to ..\
F R F5 N ) .
P E VKJ E .KI \l
A K 1 rt 9 .u
T E TD 5 Y. K.
9 6 Art. A» r} 14L
.l .Lr: V a. \1
Us)?» UC x.” \I U CV1
P35 MX I I 5 1!.
7-..... US. (I Q) t 1U
CLT T. C. R R.( 5".
CA a4 D. A.» LLb \1 r: 1...».
TVS 0N. 9r. K 9P \a . V
U Orr. N D ,T. 9 LA» )nu+
F)T SA L 1A P :9 d!)
has‘ TC a.» (I 9 L) (r \a
If] N L? L (1 CF]
((RDIV L AA AX ) X‘ F9.
NGANC Ruhr VV V 3.) t.L b. . lb
rsLuV P e V Fu 0) or... n A.. oVul‘
Gin ) nl.) )urnw CC C 0 RV. TV?»
V For ECFV 10 Q’nhfi.)1flu T KC..F.-R
Fuhnuruovlruc QFrLllnuachVI n.” 9 ob!

C 1 F1 11...: .. .l\ 01...»..C: .. rm; Y r . v
HIPQRIQ$:OII(X€((( CI:w() b 2
LC..rtr.F_C5FT.5R F.F.(TTT- 7.1.5.0.. : luv...
PNP.(R.E(U :\ .LMFFEZA. TN HYI‘LI ru
Pat v PF.” Pfi.LL.NLLlA~LYlu/v n. p.r.1.l_. _rallevl T
ONUIUSbA PT FTZFREU Kn (?c1hv F
C..1..|lr.a. .11Vrtf...hr\ .. 2 r. graft.-.» U104? A5, «Lit. 7 (IV
hrh. R r?! h..D..r..n.:.p.an..Frv F3. r..r.._1aT (FF.IUr\r.

pl. Flu, r... F. 1...
A A An ts
E rqu r... SF. C31!» 2. «.
D. FT C C «1 .n
11.2
C CC C C

Figure A.7 A Generalized Univariate Process Generator

225

A.3 The Generation of Variates from the
Multivariate Normal Distribution

As stated above, the multivariate normal distribution is one of
the few multivariate probability distributions from which a workable
process generator has been developed. Each element of the vector of
random variables, x, of which a multivariate distribution is comprised
is normally distributed with specified mean and variance. The multi-
variate distribution is fully described by the vector of expected
values for each of its marginals, u, and by a positive definite,
symmetrical variance-covariance matrix, 9, which is defined by the
following expression.

9 = E[(x-u) ' (x-u)] A-l9
If the elements of x are not correlated, the off-diagonal elements of
this matrix will equal zero and each variate in x can be generated
independently using procedures such as those outlined in the preceding
section. When correlations are present, however, this approach is not
satisfactory.

Naylor, et al. (l966) describe a procedure for the generation of
variates from the multivariate normal distribution which is based on a
theorem proved by T. M. Anderson (l959). That theorem states that if
.z is a vector of independent standard normal variates, there exists a
LJnique lower triangular matrix, C, such that

x = C2+u ‘ _-, A.20
IFt follows directly that the variance-covariance matrix of (x-u)--which
1'3 also the variance-covariance matrix of x, since u is a vector of

Ctaristants—-is defined by the expression CC'. Therefore,

9 = CC' A.2]

226

The "square root method" can be used to derive a set of recursive for-
mulas for computing the elements of C from those of 0.

Once the elements of C have been calculated, a vector of independent
standard normal variates, 2, can be generated, and the vector x can be
found using Equation A.20. To generate a large number of sample vectors
from a given multivariate distribution, the elements of C need to be
calculated only once. The final two steps described above are then
repeated for the generation of each vector of variates.

A computer program which implements this procedure is listed in
Figure A.8. The user of this program must specify ND and MN, the num-
ber of sample vectors to be generated and the number of elements in
each vector. He must also specify the mean and variance of each of the
MN marginal distributions. Finally, the off-diagonal elements of the
correlation matrix which are non-zero must be specified.1 The variable
IND is set to a non-zero value when the last correlation coefficient is
read. This program can be used to model multivariate normal distribu-
tions with up to fifty elements. More information about its structure

is provided on comment cards included in the listing.

A.4 A Generalized Multivariate Process Generator
In this section a generalized procedure for the generation of

sample vectors from multivariate probability distributions is described.

1For any pair of random variables x and y, then correlation coefficient
[3 is defined by the expression

3<Y
p =35L
xy 0 o
X y
vaiere o is the covariance between x and y and Ox and 0y are the standard
(ieeviation of the two random variables.

227 .

PROCRA" HVNHR T INPUT: OUTPUT: TAPE5='INPUT0 TAP56=°UTPUT)
COHMON /BLUCKI/ C(50.50).COR(50; 5°)!PAR‘50: 50).V(50).HN
DIHENZ'TICN X150). XT‘IEATJL'SO): 2(50)
READ HUNTER 0F SANPLE VECTORS TO BE CENERATED AND NUHDER
OF ELEMENTS IN ENC“ VECTDR
- RE’«D(511('U) '11) "N
c READDSEgNIANDm VARIANCE OF EACH HARGINAL DISTRIBUTION
‘I.I"
5 READ(5. 201) XNFAN(I). V(I)
CONSTRUCT CQ'tnCl ATION HATRIX
FIRST SET Al-l_ LLI;NENTS EQUAL To ZERO.
DO 10 I~1.K?4
DO 10 J'- I "N
COR(I J): O O
IF(I E0 J) COR(IaJ)=I.O
IO CONTINUE
NEXT READ ALL NDN’ZFRO ELENCNTS IN LUUER TRIANCULAR PORTION
OF "ATRIX SET 1ND EQUAL TO A NON-ZERO VALUE ”HEN THE LAST
ELEMENT IS TO BE RC! D
15 PEACHJ-IO?) Ind COR(I: J):IND
THIS STATEV WEJT SFTS NON- ZERO VALUES IN THE UPPER TRIANGULAR
PORTION 0? TH!" NATHIX
COH(J. I)~“-CDH(I- J)
IF(IND EC 0) GO T015
CONSTRUET LOSER TRIAHGULAR "ATRIX C. - IDHN‘T- _...-.___ ‘7 -V -7

CAL COE
BECIgngSCENERATE SAMPLE VECTORS
'C GENERATE MN INDEPENDENTLY DISTRIBUTED STANDARD NORMAL VARIATES.
DO 20 J=1.RN
R1=RANF(-l)
R2=RANF'(~-l)
THIS TECHNIGUC FOR THE GENERATION OF NORMAL VARIATES 18 FROM
HANETSCH AND PARKIIQI /B)
20 ZtJ)=(-2 O‘ALOG(R1l)i§ SFCOSIG 28315R2)
C CENESATEOVARIATES FROM DESIRED MULTIVARIATE NORMAL DISTRIBUTION.

J H
X(J)=XHEAN(J)
25 X(J)=X(J)PC(JoK)QZ(K)
30 CONTINUE ' -.-_ _. T“- -.._. V '" "7 '
FOO) (X(J)oJ¢IoNN)

on

00

on (500

0 ti

on

100 FORMAT(215)

101 FORMAT<2¥IO 2)

102 FORMAT(?IS.FIO 2. 15) ‘ ’ "' ‘ ‘ _-
200 FORMAT(10FB.2)

SUBROUTINF CUFF ' *“IFT ~ .___. h "F'
C THIS SUBROUTINE L ONSTRUCTS THE LOWER TRIANCULAR HATRIX C.
C IT HAS WRITTEN "Y R. NEAL PEI ERSO
ggflTgN {INIUE'TA‘I/ (2‘50: 50). COR(500 50). PAR‘SOO 50,0 V‘SO): "N
= a n ..,

DO 20 I=I.HN
'20PAR(I.I)=COR(I:I) ' _- *“"‘”_-__ W"—
IF(HN.LT, 7’) CO TU 70
D0 60 J=¢_’t N
JH1=J'I
SUN-=0 T — -- F -.._. -TT- T ‘

3° SUH=SUN+PflR(Jo K)I§2
PAR(J: J)=JGRT( I. ’SUH)
IF‘JJEG.HN) GO TU 70 V—'_ ”——*"—".m-—_’—‘_“." —""’

o
IF(PAR(J. .n. so 0 I on T0 so ~--------—- - - --
so UH— 0

DO 40 K=1.JMI
40 SUM: SUPI+PAR( I. K) ePAR (J. K)
PAR(I.J)=(COR(I.J)—SUM)/PAR(J.J) “'“““-““*““"
50 CONTINUE
60 CONTINUE
70 DO 80 5:

. J)*SGRT(V(I))

Figure A.8 A Program for Generating Multivariate
Normal Random Variates

228

This procedure can be used to generate random variates from multivariate

distributions having up to fifty marginal distributions. Marginal dis-

 

 

tributions can be of any form, and they need not all belong to the same
family of distributions. All that is required is that enough informa-
tion be available to construct the cumulative distribution function of
each marginal and to specify the correlation coefficient between each
pair of random variablesirlthe distribution.1 The only condition on the
matrix of correlation coefficients is that it be positive definite and
symmetrical, as required for internal consistency.

The generalized multivariate process generator described below is
based on the premise that correlations between the elements of a marginal
distribution are, to a large extent, preserved as the elements undergo
successive inverse transformations. The degree to which correlations
are actually preserved under such transformations is difficult to
determine analytically in all but the simplest of cases. Levels of
correlation are not expected to be maintained exactly. It is expected,
however, that high levels of correlation between elements of the initial
distribution will lead to high levels of correlation between corresponding
elements of the transformed distribution. All experience to date, some
of which is reported below, strongly supports this basic premise.

The procedure for generating sample observations from a generalized

Inultivariate probability distribution is relatively simple. Once the

1As Coleman and Saipe (1976) observe, the higher order product
"Kaments of a multivariate probability distribution may have an impor-
tant impact on its character. These moments are difficult to estimate
reliably and are rarely considered explicitly in the description of
WnJltivariate processes. They cannot be considered in the process

developed here.

229

cumulative distribution function of each marginal and a correlation
matrix have been specified for the multivariate distribution to be
modelled, a simple vector, 2 is generated from a multivariate normal
_distribution having a correlation matrix identical to that specified.
For the sake of convenience all the marginals of the multivariate nor-
mal distribution are standard normal. Each element of the vector 2 is
then transformed to a uniformly distributed random variable defined on
the interval (0,l). This is done by simply finding the value on the
cumulative distribution of a standard normal random variable associated
with each element of z.1 The resultant vector,fy, is a sample vector
from a multivariate distribution, all marginals of which are uniform
distributions defined in the interval (0,1). The correlation matrix

of that distribution is not known exactly, but it is expected that it
can be reasonably well approximated by the correlation matrix originally
specified. Finally, each element of the vector u is transformed by the
inverse transformation method to a sample observation from the corre-
sponding marginal distribution of the multivariate distribution being
modelled. The resultant vector, x, can be viewed as a sample observa-
tion from a multivariate distribution having marginals identical to
that being modelled and a correlation matrix which, though not necessarily
identical to that originally specified, is expected to be a reasonably
good approximation of that matrix. For most purposes this procedure
Should provide an adequate representation of the distribution being

"Madelled. The sample properties for an example are discussed below.

1The fact that the transformed variable is uniformly distributed
f<>llows directly from the proof of the validity of the inverse trans-
fc>rmation method given in Section A.2.

230

A computer program which implements this procedure is listed in
Figure A.9. Several types of inputs must be supplied by the user of
MVGEN. ND and MN are the number of sample vectors to be generated and
the number of elements in each vector. MN cannot exceed fifty. K is
the number of data points to be specified for the construction of the
cumulative distribution function of each marginal distribution. It can
take a value as high as lOO. K paired values of the random variable
and cumulative distribution function of each marginal must be supplied
by the user. These values, which must be arranged in ascending order,
are read into the two dimensional arrays ARG and VAL. Finally, non-
zero off diagonal elements of the correlation matrix must be supplied
by the user. These can be read in any order. The variable IND is set
to a non-zero value when the last correlation coefficient is read.

The structure of the program closely follows the outline of the
procedure described above. After all necessary variables are initialized,
sample vectors are generated sequentially by first generating a sample
vector from a multivariate normal distribution and then transforming
the elements of that vector first to variates of a uniform distribution
and finally to variates from each marginal of the multivatiate distri-
bution being modelled. More information about particular aspects of
the program is given in comment cards.

Program MVGEN is designed to permit the user to specify marginal
(iistribution of any form. When all the marginals of a particular multi-
variate distribution can be adequately described by one or more standard

Flrobability distributions, it may be more convenient and more efficient

computationally to modify MVGEN so that only the parameters of each

231

PROGRAM HUGEDH INPUT. OUTPUT. TAPE 5=‘INPUT. TAPEb'OUTPUT)

COP‘MUN /81 GOP I/ C( 50. 50). COR(SO. SO) PARISO; 50). PIN

COMMON IBl OCFP/ A' ’C( '50. 100) VAL (50. 10

DIMENSION RV(;O).U(50). E(50) VALNCQI).ARON(4I)

DIMENEION Y(. )0)

If NFYI THO DATA STATFHFNTS ASSICN VALUES TO VARIABLES
SFD TO CONSRTUCT A TABLE LOOK UP REPRESENTATION OF THE
NVIRSE QT THE (UMU'ATIVE DISTRIBUTION FUNCTION OF A
IANJARD NORMAL RANDOM VARIABLE.

DATA kN/dl /

DATA ARON! 3 5.-I 96,-! 645.-1. 439.-I 281.-I.150.-I 0370-. 925.
1*.81.- 7'>b.- b74.~ 596. -.52 .-.454.- 386 312 253.-.189.
?-. 12b. - 0" IO: 0 Or ()an . 1260 .187: 253 .13 215860 . 4540 52‘. . 5980
3 674. 755 841. 9d‘1.1 03 7 I 150.1. 2811. 439 I b45.1.960o3.5/

C READ THE NUNBCR OF SAHPLE VECTORS TO BE GENERATED. ND. AND
c THE Nunucn o» clzr:cnrs IN EA(H VECTOR. MN.

RtiAIM 5. IOU) NI). MN
g READ THC hwhrCP n7 POINTS TO DEKDEFINED ON THE CUNULATIVE
C

mac-4

DISTRIDUTICN Cl EACH HARGINAL.
RFR.D(5.IOI)

READ DATA FOINTSK FOR EACH HARGINAL.
DO 5 I 1 MN

DO 5 J 1

READ ‘ I ARC(I.J) VALII:
F SOHAL ELEMENTS OF )THE CORRELATION MATRIX TO ZERO.

(
ALL 0
DO 10
O
I

0
Ru
4

)
A
MN
DO 1 MN
COR( 0. O
IF(I E0 J) COR(I J)=l O
CONT I IIK'E
AD ALL NOH- ZERO CORRELATION COEFFICIENTS SET IND EGUA L
N325 ZERO VALUE HHEN THE LAST CORRELATION COEFFICIENT IS
\D
FAD(5.103l I.J.COR(I.J).IND
COR (1
F(IND E0 0) GO JTD
'PUCT LOHER TRIANGULAR MATRIX C.
CALL COLF
”” VRLN(1)~O O
012? I= .KN
20 VA:N(I)=VA1N<J)+ OPS
c BEGISDTOOCENERAgg RANDOM VARIATES.
. K“: I
c CALL NORVE: HHICH RETURNs A VECTOR OF INDEPENDENTLY DISTRIBUTED
c STANDARD NTFMAE PANDOH VARIATE
cnuL NO‘V" (Tu
C CONS JTSJgg EA'EPLE VECTOR FROM A MULTIVARIATE NORMAL DISTRIBUTION.

. O?
F DI
I I-
J I.
J:

)

00
~u+4n~
a):

~ߤmm

O
E
O
O
5

O
3r
9r
H

O
O
.3
.4

LC

1:1 MN
RV(I)“O o
‘ “’“ DO 25 J=I ' "
25 RV(I)=RV(I)+C(I J)§E(J)
30 CONTINUC
C TRANSFORM THE VECTOR OF HULTIV RIATE NORMAL RANDOM VARIABLES
C INTODS VECTOR OZ UNIFORMLY DISTRIBUTED RANDOM VARIABLES.
35 I=I.M
35 UIIJ=TAIVEX(JAIN ARON RVII).KN)
C TRANSFORM T'6f IINIFCRHLY DIS RIBUTED RANDOM VARIABLES INTO
C RANDO m VARIABLFS FROM THE SPECIFIED BETA DISTRIBUTIONS.
DO 40 1:1.MN
’ 4O Y(I)=TABE¥I(I U(l). K)
50 NRITE(O.200) (Y(M) H31. MN)
100 FCRMATIZIC)
101 FORMAT(15)
102 FORMAthrlo 2)
103 FORMATt215.F10 2.15)
200 FORMAT<IOFO 2) ' "H ' '

SUDROUTINE NORVECCNN.E)
DIMENSION E(I)

‘ DO 5 I=I.HN " “" '“' ”
R1=RANF(-I)
R2=RANF(-I)
E(I)=(-2.0*ALOG(RI))G|.5*COS(6.2831§R2)
EEBURN

lo

FUNCTION TABEX(VAL.ARG DUMMY
c THIS FUNCTION SUBPROSRAM IS FROM LLEHELLYN(I965).
_"DIHENSION VALII). ARG(1)

DO 1 J22!“
IFIDUHNY. CT. fiRG‘J)’ GO TO I
TABEX=(DUHNY‘ARG(J— I))‘(VAL(J)’VAL(J-1))/
1(ARC(J)-ARC(J-I))+VAL(J’1
RETURN
I CONTINUE
J3K
'_ GO TO 2 I t I V k"
END

9
I

Figure A.9 A Generalized Multivariate Process Generator

232

Figure A.9 (cont'd.)

FUNCTION TABEXI(I. DUMMY.K
CONTON 48IOCK2/ ARG(50.IOO). VALCSO.IOO)

DO J-‘.K
IF(DUHNY CT VAL(I.J)) GO TO I
2 TADEXI=<DUMMY VAL(I.J —l))I(ARG(I J)-ARC(I J-I))/
I(VAL(I. J)*VAL(I. J-I))+ARC(I. J-I)
RETURN
I CONTINUE

“K
GO TO 2
END

SUBROUTINE COEF
C THIS SUJROUTIP'E CONSTRUCTS THE LOUER TRIANOULAR MATRIX C.
C IT HAS WRITTEN BY R NEAL PETERSON

COMMON IDLOCKI/ C(SOo50) CORISO-SO). PARIDO. 50). "N

DO IO I-l. HN

DO IO JtloNN

DO 20 I=I.HN

IF(NN.LT 2) CO TO 70

DO 60 J=2.HN

JMI=J-I

SUH=O.

DO 30 K= I.

30 SUM: SUM+PAR(J K)*&?

’ PAR(J.J)~SGRT(I 'SUH)
IF(J‘EO MN) GO TO 70
JP13\’+1
DO 50 I‘JPI HN
PAR(I. J)“ -O
IF!°AR(J.J). E0 0. ) GO TO 50
SUH=0.

K=I.JH 1K

DO 40
“ 40 SUI1=SLJM+PARf I PAR(J.
PAR(I.J)=(CO OR( SUH)/PAR(J. J)
50 CONTINUE
60 CONTINUE

70 DO 80 I=I.HN
DO 80 J=I.I
80 .J) -PAR(I.J)

C(I
RETURN
END

233

marginal distribution need to be specified rather than specific points
in the cumulative distribution function. In this study, for example,
it was often necessary to generate sample vectors from a multivariate
beta distribution. Accordingly, a special multivariate beta process
generator was developed. A listing of program MVBETA, which implements
this procedure, is given in Figure A.lO.

Program MVBETA is identical to MVGEN in most respects. It differs
only in that parameter values for each marginal beta distribution--BL,
BU, Kl, and K2--are supplied by the user instead of actual points on
the cumulative distribution function.1 The program then uses numerical
integration techniques to construct the cumulative distribution function
of each marginal. The other inputs to MVBETA are the same as those
required for MVGEN.

A.5 An Empirical Test of the Generalized

Multivariate Process Generator

The procedure described in the preceding section is based on the
premise that correlations between random variables are, to a large
extent, preserved as the variables undergo similar transformations. A
simple experiment was designed and conducted to test the validity of this
premise. Program MVBETA was used to generate 1000 sample vectors from
a multivariate beta distribution with marginal distribution parameters
and correlation matrix equal to those defined in Tables A.l and A.2.

It should be noted that each marginal distribution is skewed. The

first three are negatively skewed, while the fourth is positively

 

1These parameters are defined as in the preceding discussion of the
univariate beta process generator.

234

PROCRAH HVhCTA(INPUT.DUT?UT.TAPE5=INPUT.TAPEbiDUTPUT)

CD'V‘iDN /' I! OI'I'I/ C(SO DULCDRU‘ “O 50). PAR(50.50).HN

COHM'H /H1U(K.'/ ARC(SO.IOO).VAI-(DO.I 100)

DINENJIUH RV(.;O).U(DO). C(50).VALN(41) ARCN(41)

DINENHIUN Y(Hpo
THE NEXT Tun DATA STATFMENTS ASSICN VALUES TO VARIABLES
USED TO CONJRIUCT A TABLE LOOK UP REPRESENTATION OF THE
INVERSE OF Tut CUNULATIVF DISTRIBUTION FUNCTION OF A
STANDARD hfihhAL RANDOM VARIABLE

DATA VN/QI/

DATA (\HCN/“ J 5o'1.96.‘1.645n“1.439p-l.2810:1.1501--1.0370‘.3250

00W“,

1- 841.-. H::.- 674.— 598.- :24.-.454.—.386. 312 253.- .
2- 125 — 0215.0 0. OSS..I?6..IB9..253. 312. 386. 454..524..59B.
3 674. 7?“. UII..9:'5 I 0%).) 150.1 ’81.1 439.1 645.1.960.3. 5/
C READ THE NUOJEH OF SAMPLE VECTORS TO BE GENERATED. ND. AND
c THE Nwmurv 3r rIFNENrS IN EA(2H VECTOR NN
RLADIL.IHII NU. r31
C READ THE R¢h LR 0F POINTS TO BE DEFINED ON THE CUHULATIVE
C DISTRIBUTION 0; EACH MARCINAL. K.
RFfiUlS-IOI) K ,
DARCnI O/LIUA1(K-I) a ‘
DD- IOKDAU':
DO 51-: r“.
C READREAxArrTr“C FQPBEAC?V MARGINAL DISTRIBUTION.
It: .‘.'.] '1' I). V I
C USE EULER INII‘DPATION TU INTFGPP.TE THE DENSITV FUNCTION OF
C EACH N; :I ' ?.N TH' S N£.NNIR THF (UMJLATIVE DISTRIBUTION
C FUNCTgONIO! EA'H HARCINAL IS CCNSTRUCTED
A (I .1)
VAI<I II- n o "- ‘- * - - "~- ‘ ~
RANC‘. 'fi‘)‘ ‘L‘I
CA=CAMMAIKI)
GO=CANMA(RPI
- .-— KA= K1_1 ._.. _-. —. < _.-..-,--- -__-_ -7—— . _.
KB: K2— 1
KD=KI+K2
GD=GANMAIKHI '
- - - co =CD/(OAKGB) n ----- ‘ -n- -» - »-w- ~~
CON =CCNOIJU
Bzolo
P=0 0
DO 5 J‘? ” *“
DO I N=I.IO
P=P+CDN¢<I¢£KA>*((I.0-B)*9KB)
I B- D+DD
AR 0(I JI-ARG(I J- I)+DARC*RANGE ‘ ' ' “" '
VAL(I.J)-P
IF(VAL(I.J> CT 0 999999) VAL(I.J)=O.999999
IF(J LT K) CO TO 5
ARC(I.J)LUU
VALII.J)nI.0
D CONTINUE
C SET SOLISFI DIACONAL ELENENTS OF THE CORRELATION MATRIX TO ZERO.
DO 10 J1I.HN
COR(I-J)-U O
IF(I EQ J) c0R(I.J)=I.O .
IO CONTIr

c READ ALL R‘fi-*Il COPRELATIOII COEFFICIENTS SET IND EQUAL
g T8 CENENTE’ERQ VALUE NHEN THL LAST CORRELATION COEFFICIENT IS
15 REAH<3. 103) I. u COR(I J) IND
cei:lo.1>- 0h'(1.J)
IF (IND [U 0) GO TO 15
c CONSTRUCT L2H;R TRIANCULAR MATRIX c
, CALL curs
VALNIII=O 0
00120 1= LLHN
20 VALN<IIHVNIN JI+ 025 ‘*
atczgnto rcr-Lnnrr RANDOM VARIATES
IO bh‘l.PMI
CALI. NOC“CC. b.24'CH RETURNS A VECTOR OF INDEPENDENTLY DISTRIBUTED
STANDARD nowmn1_ p.\~oom UARIATES ,

COnggogrégqrfig vrQIDR FROM A HULTIVARIATE NORMAL DISTRIBUTION.

O (“5 O

25 RV(I)— PVIII-IC(I J)§E(J)
30 CONT INV

C TRANSFORM TH: VTCTDR OF HUI_TIVA “I ATE NORMAL RANDOM VARIABLES
C INTODS gECTCR LI UNIFOR.ILY DIST RID DUTED RANDOM VARIABLES.
5 -I.PH
35 U(I)=lr‘>x\“': N.ARGN. RV(I).KNI
C TRANSFORM THt HH’FOQHLY OIEWTPIDIIED RANDOM VARIABLES INTO
C RANDBB XSRIP’ Ll-rL FROM THE SPECIFIED BETA DISTRIBUTIONS.
3101!!“
40 Y(IJ—TA“EKIII UII) K)
50 NRITE'!..: :uv: (Y(M). M=I.HN)
zoo FuirflAIIDISI
IOI FUWMAT(I;')
1°? LQHMT I‘ '2 215‘
103 FORRAI<911 IIO 2.15)
200 FORfiATIIoru 2)

END

Figure A.lO A Multivariate Beta Process Generator

235
Figure A.10 (cont'd.)

UDPOUTINE NORVEC(NN E)
HEN‘ION E(

5 I“! MN

RANF(-I)

RIFi-l)

5
D
DO
R
R R
S 6=(-2. OOALOG(R1))09.:oCDS(b.2831*R2)
E

I
1:
P-
(I
ET
ND

FUNCTION TAHFYTVAE.ARC DUMMV.K
c THIS FUNCTION SUIPPD’RAM 15 FRon LLEHELLYNII965).
BIMFNCIQNK VAT(I).ARC(I)
l 1 J‘
IF(DU”“/ CT ARO(J)) 00 TO 1
2 TABrx=tDuHHY- ~AHGIJ- 1v)o<VAl(J)-VAL(J-1)Il
ItAEthI>-au.le 1))4VAL(J-1)
REIURN
I CONTINUE

J=
GOK T0 2
END

FUNCTION Tnntx1(I.DDNNV.K)
~-vggnTON éBkULKZI ARC<50.100>.VAL(50.100>
IFtDuNNT'c: VAL(I. 4)) GO TO
2 TABEXI=(DUHHY- VAl.(I J— I))I(ARG(I. J)-ARG(I J-IIII
1(VALééod)~VALII.J-I))+ARC(I J-I I

RETU

I CONTINUE
J=K
on T0 2 ~4~-- _”__._ufl-..-. ~»
END

‘FUNCTION CANNA<R)
KK=K~1

A=1.o '
IF(KK.LE.I) 00 TO 10
° DO 5 1:2.KK
5 A=A§FLOAT€II
10 GAMMA=A
RETURN
END

SUBROUTIr: (0?
C THIS SUBRO'JTINE CONSTRUCTS THE LONER TRIANCULAR HATRIX C.
C IT HAS NRITTtN HY R NEAL PETERS

ggmmgu /BLOCKI/ C(SO.30).COR&50.50).PAR(50.50)."N

I. 121- NM
' DO 10 J=1aNN
10 C(I.J)“O
DO 2 I=I.HN
20 PAR(I:1)=CCR(I.1)
IF(MN LT 3) GO T0 70
DO’bO J=20HN
JMIuJ‘I
UN=0

DO 30 K=I.JMI
30 SUM=SUM+PAR(J.K)I§P
PAR(J.J)=SUHT(I.-SUM)
IF(J.EG RN) GO TO 70
JPI=I+I
PI MN

I.

I O
éF(PAR(J.J) so 0 ) co TO so
DO 40' u I
40 SUH=SUN+ W?
50 CONTINUE
60 CONTINUE
70 DO 80 I=I.HN

DO 80 4:1

80 C(I-JI-PAR (i u)

RETURN
END

"1
. (I KIRPAR(J.K)
OR I J)°aUH)/PAR(J.J)

236

Tab1e A.1 Parameters of Margina1 Beta Distributionsa

 

 

 

Distribution K1 K2 BL BU p o
1 2 1 22.0 31.0 28.0 2.12
2 3 2 70.0 110.0 91.0 8.00
3 3 2 4.50’ 9.00 7.20 .90
4 1 4 1.75 3.20 2.04 .24

 

aDistributions 1 and 2 can be considered to be nationa1 average
yie1ds for soybean and corn respective1y, whi1e distributions 3 and 4
can be considered to be nationa1 average prices for the two respective
crops.

Table A.2 Specified Corre1ation Matrix

 

 

 

 

9‘! fig Distribution ,., 7.

Distribution 1 2 3 4
1 1 .75 -.30 --20
2 .75 1 ‘-20 "30
3 -.30 -.2o 1 -45
4 -.20 -.30 .45 1

 

237
skewed. Program MVNOR was then used to generate 1000 sample vectors

from a multivariate normal distribution having marginal distributions
with means and variances identical to those specified for the marginal
beta distributions in Table A.l and a correlation matrix identical to that
of the multivariate beta distribution. The sample correlation matrix
for each set of observations was then calculated and compared to that
which was specified. The procedure for generating sample observations
from the multivariate normal distribution was derived analytically, and,
in the limit, the sample correlation matrix for data generated with
MVNOR should converge to the correlation matrix actually specified.
Therefore, the sample properties of the observations generated with
MVNOR provide a standard against which the performance of MVBETA can
be judged.

The sample correlation matrix for the data generated with MVNOR
is shown in Table A.3. For the most part, it conforms closely to that
specified. The mean absolute deviation for off diagonal elements is
only .016. The sample correlation matrix for the data generated with
MVBETA is shown in Table A.4. It, too, conforms closely to that
specified, and the mean absolute deviation for off diagonal elements of
.018 is only slightly larger than that for the data generated with MVNOR.

These results indicate that, at least in this particular case,
correlations between random variables are preserved to a large extent
as the variables undergo the series of transformations required to make
their marginal distributions take a specified form. Clearly additional
tests should be performed, but all experience to date with the generalized

multivariate process generator has suggested the results presented here.

238

Table A.3 Sample Correlation Matrix for

Multivariate Normal Data

 

 

 

 

Distribution
Distribution 1 2 3 4
l l .765 -.320 -.195
2 .765 l -.180 -.295
3 -.320 -.180 l .417
4 -.195 -.295 .417 l

 

 

 

 

 

Table A.4 Sample Correlation Matrix for
Multivariate Beta Data
Distribution
Distribution 1 2 3 4
l l .753 -.313 -.192
2 .753 l -.l74 -.291
3 -.313 -.l74 l .404
4 -.192 -.291 .404 1

 

239
Given the degree of imprecision with which correlation coefficients
are generally specified, the inaccuracies associated with the use of

this approach can hardly be considered to be significant.

A.6 Conclusions

 

The multivariate process generator presented here is a reliable,
easily implemented procedure which greatly enhances the value of Monte
Carlo simulation techniques in the analysis of decisions made under
uncertainty. It facilitates the representation of stochastic processes
which cannot accurately be described by a multivariate norma1 distribu4
tion and, when used in conjunction with other simulation techniques,
permits the incorporation of a greater degree of realism into the analysis
of complex problems. As such, its contribution to the set of techniques

developed in this study is an important one.

APPENDIX B

IMPLEMENTATION OF THE INTERVAL APPROACH TO THE
MEASUREMENT OF DECISION MAKER PREFERENCES

8.1 Introduction

 

Implementation of the interval approach to the measurement of
decision maker preferences was discussed in Section 4.6 of Chapter IV.
In this appendix a more technical explanation of its implementation is
presented, with particular attention being given to the description of
several computer programs which greatly facilitate the process. More
specifically, six steps in the implementation of this preference
measurement technique are identified and discussed: the specification
of a measurement scale, the generation of sample distributions, the
identification of a boundary interval for each pair of distributions,

the construction of the questionnaire, administration of the question-
naire, and the use of interval preference measurements to order alter-

native choices.

B.2 Specification of a Measurement Scale

 

Implementation of the interval approach begins with the specifica-
tion of a measurement scale--a set of reference levels of absolute risk
aversion which serve as the basis for preference measurements. Because
this scale determines the degree of precision with which preference
measurements can be made, careful attention should be given to its

specification. Two related questions which must be considered at

240

241
this point are: how many reference levels to include in the measurement
scale and where to locate these levels on the scale.

The number of reference levels on the measurement scale depends to
a large extent on the number of choices the decision maker will be asked
to make in measuring absolute risk aversion in the neighborhood of any
particular system output level. There is a direct relationship between
the number of questions to be asked and the minimum number of reference
levels on the measurement scale. If the decision maker is required to
make only one choice, only one boundary interval need be specified, and
only two reference levels are required to define that interval. If a
second choice is to be required, two additional boundary intervals must
be specified, since the interval which is the focus of the second ques-
tion will depend on the decision maker's response to the first. Four
reference levels are required to define the total of three boundary inter-
vals called for in this case. Similarly, seven district boundary inter-
vals defined by eight reference levels are required for the construction
of a three question sequence, since a separate boundary interval is the
focus of the choice made at each mode of the hierarchy of questions
defined in Figure 8.1. It follows by similar reasoning that if N Choices
are to be made in measuring absolute risk aversion in the neighborhood
of a particular system output level, the measurement scale should be
comprised of at least 2N reference levels.

Once the number of reference levels to be specified has been deter-
mined, their location in risk aversion space must be established. Refer-
ence levels on the measurement scale need not be placed at regular
intervals. As was noted in Chapter IV, it is usually advisable to con-

centrate reference levels in that region of absolute risk aversion

242

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243
space where the decision maker's actual level is expected to fall or
in the regions where relatively small changes in absolute risk aversion
have the greatest impact on preference orderings. Experience to date
indicates that most of the detail on the measurement scale should be
concentrated in the risk aversion interval between -.0001 and .0010.
Actual measurements for a variety of decision makers have tended to
fall most frequently within this interval, and tests on several empirical
decision problems have indicated that choices are most strongly affected
by changes in absolute risk aversion within this range. A suggested set
of sixteen reference levels is given in Figure 8.2. These define fifteen
boundary intervals upon which choices in a four question sequence could
be focused. A measurement scale for a three question sequence could be
constructed by using every other reference level, and that for a two
question sequence could be constructed by using every fourth reference

level.

8.3 The Generation of Sample Distributions

Once a measurement scale has been specified, the sample probability
distributions which are the basis for the choices used to reveal the
decision maker's preferences must be generated. Program NORGEN, which
is listed in Figure 3.3, is used to construct these distributions, each
of which is actually a set of sample observations drawn from a normal

distribution with specified mean and standard deviation.1 These sample

 

l

A normal distribution is used because it is convenient. Any other
underlying distribution can also be the basis for the generation of
sample distributions.

244

.0100
.0050
.0025
.0015
.0010
.0008
.0006
.0004
.0003
.0002
.0001

0
.0001

.00025

.0005

.0010

Figure 8.2 A Suggested Absolute Risk
Aversion Measurement Scale

245

0.

FR
RFOO
E0 .1
8 NC
MNOA
UAIF
NET
.1 "AG
‘9 IN.
EE 9V1

Figure 8.3 A Listing of Program NORGEN

PNDED

A NDN

T 9 U

00 0.00

TENRR

UTCA

PAIGE

NRTMEH )

IEUAT 2

:NBT D R

5.5.7.80 N 1

[GR Mn U 1

P T. A 09 3

Apr—SN R9 8

TEIA. Q In; a...

v DEB 99 o

10 MT Do, 6

UTHYS T o o (

D. C S S S

13:» Q 9 9‘...“ fl.

UNENN N 0 C N.

00 00 A)...‘ t A

QINII EDT 5 E )

TTITT HNU . ”.1 5

UU UU YUP. t v- I

PBSBB 10.1 t + \I Q

NINII ORR ) ) 2 2

IRCER KIT. .1 «J I o

(TITT v(S lXN 9 0

NSTSS [TI .1 RQUY a 1..

EIAII ”Ann. ()0 Tsirr

GDVDLD 0 r. )0 run-LR) 30.0.

P. R )L.LNION))CTIIY I o 9

OFEGG OFL O 90 911L39 c.. 9 0.2.).

Nccukku rc/rylnell. .A14”.11 1.3.;
FIT. 9..an :C 9:((t(( c..JF.F..tF2

MRCYY. QTDIIQJFFOTT 41UU(((

AF. LL SSS ( NAN. obi‘ N..NTT.1

F.F...r._R.F ls: (7.5".LibiuntiTIlhéf

CMLEE CEF?NTIQ?.LLTR11V~V

0UPD..L oATT IT. : .. (r F 111...»..va RP:

RNV: NNDESAOR (\01? : : : 2 RCPOOOM.

P AUUNRIRDPHDRCXMYKPCCFFFF
D S U m:

A Fruo n\

EFHHP PP. a... J‘Lr.

POTTI r7 In...
«J11»...

CCCCC C

246

observations are considered to be possible system output levels, and
each is said to have an equal probability of occurrence.

The user of NORGEN must specify values for five parameters: NE,
N0, YMEAN, STD, and IROUND. NE is the number of sample distributions to
be generated, and ND is the number of sample observations defining each
distribution. Recommended values for NE and ND are 40 and 6 respectively.
The generation of forty sample distributions almost guarantees that at
least one pair of distributions will have its boundary interval at any
specified level. The use of six-element distributions is justified by
the ease of explaining the probability associated with each element and
by the fact that this number of elements is sufficient to allow for
considerable complexity in each distribution. YMEAN and STD are the
mean and standard deviation of the underlying normal distribution from
which the sample elements of each distribution are drawn. In practice,
YMEAN is usually set equal to 0.0, which implies that the expected
value of the mean of each sample distribution is also 0.0. That expected
value can be shifted to any level, y*, however, by simply adding y* to
each element of a distribution. The appropriate value for STD depends
on the characteristics of the decision situation being analyzed. If
STD is assigned too high a value, the dispersion of the sample distri-
butions will be great and the assumption of constant absolute risk
aversion over the range of system output levels on which they are defined
may be difficult to justify. If STD is assigned too low a value, on the
other hand, the points defining each distribution will be so highly
concentrated around a single system output level that choices between

distribution will be difficult to make. Experience to date indicates

247

that a value of STD between one and five percent of the entire relevant
range of system output levels is appropriate. Finally, it is often
desirable to round the system output levels defining each sample
observation to the nearest 10, 50, or 100 units. This can be accom-
plished by specifying a value for IROUND. If IROUND is set equal to

50, which is the recommended value, all system output levels are rounded
down to the nearest 50 units. A sample output from program NORGEN is

given in Figure 8.4.

8.4 Identification of Boundary Intervals

After a measurement scale has been specified and sample distribu-
tions have been generated, the boundary interval for each pair of dis-
tributions must be identified. The boundary interval for two distributions,
(A1, 12), is an interval in risk aversion space such that decision makers
whose absolute risk aversion functions lie everywhere below A] unanimously
prefer one distribution, while those whose absolute risk aversion functions
lie everywhere above A2 unanimously prefer the other. Clearly a boundary
interval is not unique. If (A1, A2) is a boundary interval for two
distributions, for example, and if A3 < A1 and A4 > 12’ then (A3, A4) is
also a boundary interval for these two distributions. In measuring pre-
ferences, however, it is desirable to specify boundary intervals which
are as narrow as possible.

The absolute risk aversion reference levels which define the
measurement scale constitute the set of potential endpoints for boundary
intervals. If a measurement scale is comprised of four reference levels,
-.0010, 0, .0005, and .0010, a total of six boundary intervals can be
constructed: (-.0010, 0), (0, .0005), (.0005, .0010), (-.0010, .0005),

248

000000 0.00000 0.00900 00090... n.30.u..-o...u 005000 003001.. 003073 .......a-..u:0 0.1.0000
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Figure 8.4 Sample Output from Program NORGEN

249

(D, .OOlO), and (-.ODlO, .OOlD). Because relatively narrow boundary
intervals are sought, however, only the first three-~those defined by
adjacent reference levels--are of interest. For any pair of sample
distributions it is necessary to determine which, if any, of these
three intervals can be said to be a boundary interval. Given the defini-
tion of a boundary interval, this requires the identification of the
highest reference level, A1, such that all decision makers less risk
averse than A] prefer one distribution and the lowest reference level,
A2, such that all decision makers more risk averse than A2 prefer the
other distribution.

Program INTID, which is listed in Figure 3.5, is used to accomplish
this task. Given a set of absolute risk aversion reference levels and
a set of sample distributions, it identifies the narrowest boundary
interval for each pair of distributions. It does this by applying
stochastic dominance criteria developed by Meyer (1977b) in "Second
Degree Stochastic Dominance with Respect to a Function." Subroutine
SDLB of INTID orders distributions for classes of decision makers whose
absolute risk aversion functions are bounded only from below by applying
the following criterion: cumulative distribution function F(y) is
unanimously preferred to cumulative distribution function G(y) by all

decision makers more risk averse than k(y) if and only if:

I!” [G(x)-F(x)]dk(x)20 By 1 B.l

This subroutine, then, is used to identify the upper bound of the

boundaryVinterval. Subroutine SDUB, on the other hand, orders distributions

 

1This criterion is based on Meyer's (l977b, p. 479) Definition 4.

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251

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253
for classes of decision makers defined only by an upper bound in absolute

risk aversion using the criterion: cumulative distribution function
G(yfi is preferred to cumulative distribution function F(y) by all
decision makers less risk averse than k(y) if and only if:

I; [G(x)-F(x)]dk(x)50 By 1 3.2

This subroutine is used to identify the lower bound of the boundary
interval.

In determining the boundary interval for a particular pair of
distributions, program INTID tests each interval on the measurement
scale until a boundary interval defined by two adjacent reference levels
is identified. Given the reference levels--.OOlO, 0, .0005, and .0010,
for example, the interval (-.0010, 0) is considered first. If sub—
routine SDUB indicates unanimous preference for one distribution at
absolute risk aversion levels below 0.00l0 and subroutine SDLB indicates
unanimous preference for the other at absolute risk aversion levels
above 0, (0.0010, 0) is a boundary interval. If this criterion is not
met, the interval (0, .0005) is evaluated in the same manner. The
program steps up the measurement scale in this way until a boundary
interval is identified or until all possible intervals have been
examined.

Several parameter values must be specified by the user of INTID.

NE and ND are defined exactly as in program NORGEN; they indicate the
number of distributions to be considered and the number of elements

defining each distribution. Maximum values of NE and ND are 40 and lO

 

1This criterion is based on Meyer's (1977b, p. 482) Theorem 5.

254

respectively. N6 is the number of reference levels on the measurement
grid. Its maximum value is 64, the number of reference levels required
for the specification of a six question sequence. Two other types of
inputs must be supplied by the user of INTID. First, the output of
program NORGEN--the NE distributions, each having ND elements--must be

read into INTID.1

These data are stored in two arrays: NAME, an array
of distribution names, and R, an array of sample points. Second, the
reference levels defining the measurement scale must be specified and
read into array RA. A sample output from program INTID identifying
boundary intervals for the distributions given in Figure 8.4 based on

the measurement scale defined in Figure 8.6 is given in Figure 8.7.

8.5 Construction of the Questionnaire

At least one pair of distributions for which the boundary interval
lies between any two adjacent reference levels on the measurement scale
should be identified by program INTID. Once this has been done, a
hierarchy of questions can be established, with each question focusing
on a different boundary interval. The hierarchy of questions associated
with the measurement scale defined in Figure 8.7 is given in Figure 8.8.

In general the first question of such a hierarchy should focus on
the boundary interval at the center of the measurement scale. That in
Figure 8.8, for example, focuses on the boundary interval (.0001, .0002),
which is defined by the fourth and fifth reference levels of the eight-

level measurement scale. The two questions at the second level focus

 

1Programs NORGEN and INTID are written so that the output of NORGEN
can be catalogued as a permanent file and read into INTID from that file.

255

.0050
.0010
.0006
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.0001

-.0001
-.0005

Figure 8.6 An Eight Element Measurement Scale

1256

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258

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259

on the intervals at the center of the two segments of the measurement
scale created by the first question, and the four questions at the
third level focus on the boundary intervals at the center of the four
segments created by the second set of questions.

Once the hierarchy of questions has been specified, the number of
system output levels at which direct measurements of absolute risk
aversion are to be made should be determined. Experience to date has
shown that direct measurements in the neighborhood of three to four
system output levels provide an adequate basis for the construction
of an absolute risk aversion function over even a broad range of system
output values. If, for example, annual income is the system output
variable for which preference information is to be elicited and the
relevant income range is from 0 to $20,000, direct measurements of
absolute risk aversion could be made in the neighborhood of $3,000;
$l0,000 and $l7,000.

In order to specify the choices used to elicit information on pre-
ferences in the neighborhood of a given system output level, the sample
distributions generated by NORGEN must be shifted to that level by
adding a constant to each element. The boundary intervals between dis-
tributions do not change when the expected value of their respective
means is shifted away from zero. This is true because the reference
levels on the measurement scale, A, represent constant levels of absolute
risk aversion, such as would be associated with a utility function of
the form
u(y) =-e-Ay 8.3
When the mean of any distribution is shifted by adding a constant value

to each of its elements, the associated expected utility is altered only

260

by a positive multiplicative factor for decision makers with this form
of utility function.1 If two distributions are shifted by the same
amount, then, their relative ranking by decision makers more or less
risk averse than any specified value remains unchanged.

A set of sample questions designed to elicit information on pre-
ferences for income in the neighborhood of $10,000 is given in Figure 8.9.
They are specified in the manner described in Chapter IV. It should be
noted that the choices actually presented to a decision maker are
dependent upon his responses to prior questions in the hierarchy.

8.6 Administration and Interpretation
of the Questionnaire

 

 

Before the questionnaire is administered, the decision maker should
have a clear understanding of its objective, which is to obtain an
accurate representation of his preferences. The system output for which
preferences are to be measured should already have been clearly defined
and should be recognized by the decision maker to be the primary indica-
tor of system performance he will consider when making a choice in the
situation being analyzed.

Administration of the questionnaire is straightforward. The

decision maker is presented with several series of choices such as those

 

1Let the mean of the distribution of a random variable y shift
from zero to y*--i.e., let the random variable w equal y + y*.

E[u(w)] fco - e'xw dw

In” _e'l(y+y*) dy

..*..-
= e 1y [_w-e 1y dy
a E [u(y)]

where a is a positive constant.

261

 

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cw hm_o m hm_o

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262

specified in Figure 8.8. Each series measures preferences in the
neighborhood of a particular system output level. Completion of a
questionnaire comprised of four three-question series takes approxi-
mately twenty minutes. Experience to date has shown that decision makers
find this preference elicitation procedure more interesting and more
informative than the interview process required to elicit a single-
valued utility function.

Interpretation of the results is also quite straightforward. Using
the series of questions in Figure 8.9 as an example, consider the case
in which the decision maker prefers DIST 5 in question (1), DIST 7 in
question (4) and DIST l7 in question (5). Referring to Figure 8.8,
preference for DIST 5 over DIST 20 indicates that the decision maker
is not less risk averse than .000l; i.e., that

r(y) > .0001 3.4
Similarly, preference for DIST 7 over DIST 4 indicates that his level
of absolute risk aversion is such that

r(y) < .0010 3.5
Finally, from his preference for DIST l7 over DIST 3 it can be inferred
that

r(y) < .0006 8.6
As noted in the lower line of Figure 8.8, then, these three responses
indicate that the decision maker's level of absolute risk aversion lies

on the interval [.000l, .0006] in the neighborhood of y = $l0,000.

263

8.7 The Use of Interval Measurements of
Preferences to Order Choices

 

 

The interval approach to the measurement of decision maker pre-
ferences was devised for use with the evaluative criterion of stochastic
dominance with respect to a function. It determines upper and lower
bounds on a decision maker's absolute risk aversion function, the basic
information on preferences required for the application of their cri-
terion. In order to actually implement stochastic dominance with respect
to a function in the ordering of choices, however, utility functions
having absolute risk aversion functions which correspond to these upper
and lower bounds must be constructed. The link between absolute risk
aversion functions and utility functions is straightforward, but analytical
relationships between the two can be found only in certain special cases.1
Because no particular functional form is specified for the upper and
lower absolute risk aversion functions constructed under the interval
approach, the determination of the associated utility functions by
analytical means is a difficult if not impossible task.

Program UFUNC, which is listed in Figure 8.10, resolves this pro-
blem. It employs numerical integration techniques to generate values
for the utility functions associated with the upper and lower bound
absolute risk aversion functions at regular intervals over the relevant

range of system output levels. In effect, these values define the two

 

1As Pratt (1964) notes,

u = Ie'fr
where u is a uti1ity function and r is an absolute risk aversion function.
The two constants of integration are arbitrary, corresponding to the
arbitrary scale and origin of the utility function.

264

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RUG 9 PH 9“ WC VT Tc _ _ HT 2 TO 2 35 ‘w

CCC ” CC C C CC. CC _ . . CC.

Figure 8.10 A Listing of Program UFUNC

265

utility functions, since values not calculated directly can be
determined by linear interpolation, as in Figure 8.11. Such an
approximate representation of a function is called a table look-up
function (Llewellyn, 1965).1

Several inputs must be specified by the user of UFUNC. First, the
range of system output levels over which the utility function is to be
constructed is defined by specifying the minimum and maximum values,
YMIN and YMAX. Values of the utility function over this range are
calculated by solving the following two differential equations recur-

sively with Euler integration:2

_Q My) = l 0 My) 37
dy My) -r(y) 0 U'U) '

The solution technique requires that initial values of u(y) and u'(y)

be specified at some level of y. Within the program, u(0) and u'(0) are
automatically set at 0 and 1.0 respectively, so this condition is met.

It is also necessary to specify a value of DY, the output increment.

The smaller the value of DY, the more accurate the numerical approximation
of the utility function value will be. In cases where system output has
been specified in dollars, values ranging up to 5.0 have proved to be

adequate.3 If the range of systems outputs is large, more values of the

 

1See Appendix A for a brief discussion of table look-up functions.

2Manetsch and Park (l977b) provide an excellent discussion of
numerical integration in general and Euler integration in particular.

3Stability conditions under Euler integration require in this case
that a value of DY be selected so that the absolute value of the following
expression be less than 1.0 where R* is the minimum value of the decision
maker's absolute risk aversion function:

(2-(ov) (8%)) (0;)2(R*)2 - 8(01) (R)

If the minimum value of r(y) for a decision maker is set at -.01, an
extremely low level, the stability conditions are met if DY = 5.0.

266

UUO

120

100‘

60

a
l 1 n l 1 1 1 L A L

-2000 2000 4000 6000 8000 10000 12000 14000 y

-20 .

-40 P

 

Figure 8.11 Representation of a Utility Function by
Interpolation Between Known Points

267

utility function will be calculated than are needed for the table look-
up representation. A final parameter to be specified, then, is YINT.
It defines the size of the interval between system output levels for
which values of the utility function are to be specified in the table
look-up function. In application to date, YINT has been set equal to
50 or 100. The minimum value for YINT is determined by the following
expression

YMAX-YMIN
399

The user of UFUNC must also supply information on the decision maker's

YINT > 8.7
absolute risk aversion function. The interval approach to the measure-
ment of preferences determines upper and lower bounds on a decision
maker's absolute risk aversion function in the neighborhood of several
system output levels. In the example shown in Figure 8.12, direct
interval measurements were made in the neighborhood of y = 3000,

y = 10,000, and y = 20,000. The upper and lower bound functions are
considered to be constant over the range of y values for which each
measurement applies:l Values for the two absolute risk aversion
functions at system output levels other than those where direct mea-
surements have been made are determined by linear interpolation

between known absolute risk aversion values or by linear

 

1This is a result of the assumption that the decision makers absolute
risk aversion function is constant in the neighborhood of any particular
system output level (see Section 4.5 of Chapter IV). The range of system
output levels over which a given measurement holds is dependent upon the
dispersion of the sample distributions used to elicit the preference
information. In the example in Figure 8.12, distributions generated by
NORGEN with STD set equal to 500 were used. As expected, nearly all
points in the sample distribution fall within two standard deviations
of the specified mean, y*. Therefore, the interval measurements are
said to be valid for system output levels in the range y* i 1000.

/

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3..

269

extrapolation for system output levels outside the range over which
direct measurements are made. The points marked with an X in Figure 8.12
convey all the information on the upper and lower bound absolute risk
aversion functions that is required by UFUNC. They occur at six system
output levels. Therefore, parameter KINT is program UFUNC, which indi-
cates the number of points for which specific information is required,
is set equal to 6. A series of six data cards is then read by the pro-
gram, each card setting values for three variables: ARG, which is the
level of y; VALL, which is equal to the lower value of r(y); and VALU,
which is equal to the upper value of r(y). The values for the example
in Figure 8.12 are given in Table 8.1. The utility functions generated
by UFUNC for their example are graphed in Figure 8.13.

Once values of the utility functions associated with the upper
and lower bound absolute risk aversion functions have been calculated,
they serve as inputs to program NSTDO, which orders distributions of
system outputs according to the criterion of stochastic dominance with

1 The

respect to a function. Program NSTDO is listed in Figure 8.14.
logical foundation of this procedure is explained in Section 4.4 of
Chapter IV and, more extensively, in Meyer (l977a).

Several parameter values must be specified by the user of NSTDO.
ND and NE again define the number of sample observations defining each

system output distribution and the number of distributions to be con-

sidered. Their maximum values are 40 and 50 respectively. Data defining

 

1This is a slightly modified version of the program written by Meyer
for the application of stochastic dominance with respect to a function
described in "Further Applications of Stochastic Dominance to Mutual
Fund Performance."

Table 8.1

270

An Example of Preference Data Input

for Program UFUNC

 

 

 

ARG VALL VALU

2000 -.0001 .0001
4000 -.0001 .0001
9000 0 .0003
11000 0 .0003
16000 -.0001 .0001
18000 -.0001 .0001

 

271

 

 

4000 ~ /
/
/
3600 e l
/
/
3200 I
/
/
/
2800 v I,
/
/
2400 " //
/
/
' /
/ Lower Bound
/ Utility Function
1600 " / ------- Upper Bound
/’ Utility Function
/
/
1200 " /
/
/
8m: . /
./
/
/’
400 ' l/’
/’
/
/

 

200 400 600 800 1000 1200 1400 1600 1800 2000

Figure 8.13 Utility Functions Associated with Upper and
Lower Bound Absolute Risk Aversion Functions

272

SUBR”UTINC RU' DCR
crmvr" w.) r.: N..C(iOi).CP(101) 1(50) R(40 50) RA410.4)

TUI‘WI. 1
DO 400 I=l.Nl)l
AH=T(1)
N“!
II=I+1
DO 401 J-II.ND
flFSAfi.LE .(JJ) G3 TO 401
|;:
AMTT(J)
401 CONTINUE
T(N)=T(I)
T(I)=AH _ - _
400 CONTINUE '
RETURN
EN
SUB°OUTIIN‘ CUMCAL(H1.H2)
fiOHdOH IJD NL.H3.C(IOI).CP(101).T(50).R(40.50).RA(10:4)
1:1
K231
C(1)=O O
CP(11=0 O
BNDtND
NDQ=ND¢2+I
DO 500 1'? N02
IF(K1 CT NU “JD ¥.2 GT ND) GO TO 550
IF(KI GT ND) (‘0 10
IF(KZ’ CT ND) (:0 TU 55]
IF(R<P‘.I.K )--R(M,".'.’..'?)) 551.552.5521
551 C(I)=C(I-l)*(1.0/B
CP(I)=R(HI.KI)
K1=K1+1
GO TO 500
55? C(I):C<I~l)
CP(I)=R(HI.K1)
K1=KI+1
KQKKQ'I
GO TO 500
553 C(I)=C(I~I -(l OIBND)
cr<1):n\rm 3K2)
K2=K2+1
GO TO 500
550 C(I)=0 O
CP<I)=0 O
500 CONTINUE
RETURN
END

€E1REUT5NE SD(NU.II)
""1 1' JD NL NCHC(LOI)CP(101 o D
EXTEPNI‘L U} U: ) T150): R(4O 50). RACIO 4)

PEA; NU

HU=0 0

N122fiND

NUE=2”ND

DU 600 I=I.Nl)'3_'

};(21L( LE)1) CO ID 650

1 NL) (551 6J21653

652 NL= NI- '1

"U: 0 0

GO TO 00
653 Hk'é?"‘_i+(U2(CP(NL+I). II)‘U2(CP(NL1: II1)V‘C(NL)
NLINL“1 ’

=1.NU.‘ °

DD 601
2kCP(P“‘lI.11);U2(CP(NL)II

TU—(' I))QC(NL)
654 622 6: 5

IF(NU:TU‘
655 NU MU+TU

601 ”VF
654 TT£HU3CINL1+UCICP'NL+11.II)

0:.

NU:

CP

GO TO

‘UIRCTT.II)

0 O
(NL+I)“G
{CO

651 RU=MU+§UI(CP(NL+I):II)-U1\CP(NL): II))'C(NL)

L-N
.--.. - DO e02. al.-41mm.
IF(Nl. LE 1) CO TO 550
TU=<UI<CP<H'_+I).Il)-UI(CP(NL).II))§C(NL)
IF(N-J+TU)obb.o:2.b 557
656 MU fiU+TU
NL=NL~I
60? CQNTINUE
65] T5 :HU"(NI)+UI(CF’NL+I) II)
0 'UIIkTSI] X)
Hu=0 O
CPIHL+11=G
600 CONE NWE
650 REIJRN
END

Figure 8.14 A Listing of Program NSTDOa

aThis is a modified version of a program written by Jack Meyer.

273

Figure 8.14 (cont'd.)

PV’UC"?K‘-M NSTDO( INhUT. OUTPUT. TAPE

5 PUT. TAPEb'OUTPUTa TAPEQ)
CL‘IPM'VI‘J ND. Niz. NC. C( 10'. ).- CP(101 ). T

(

E

IN

50).R(40.50).RA(10.4)
OO).VALU(400).KDIH.SHALLoolFF
(40.50).CE2(40.50).NANE(40)

Cf—N‘V’H‘H fnl ”CV"! :‘.F(‘(40'7).Vfi.l'
Dihflm lOrJ ICCDL(4U 3}.NSD(DO). C
PFI‘Y'Jffi- ‘7‘)0) N3. -.NL

950 F’C-U'in. (4"l .0)

' Na“ -1

LHJYTF .ncmn
300 f(iT2r({H1,7x SHNAHF. .21X.11HDATA P01NTS.23X.16HMEAN STD)
D M l N
' P"'-.1'('3. ‘000‘. NA 'F(I)
4mm rpm-uxzu)
[3 iwxn mmm 1. N1)
REKL';-LQOJ) R(1.NMM)
5000 CT"'IP
5001 F0 -?'FiO 7)

'(
4
1

 

 

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“RITE(Q:4\)l) (N8IO‘I)I\J-11320’OANEANISTD "

400 FO'TMJTHIH .A10.IO(IX.F8 2)) "-
401 FDéMAr(1H ,lox.10(1x.FB.2).2(4x.F8.2)) .
“ CALL ROFDER " .

DIFF. KDIM

1
Hr I).VAt).L(HMM) VALUUINM)
"901 FO""\T(1H .F 5X 2E213)
Boo CC.NT].’.‘¥5F
DO 104 J=1.N
3130 0
BB=0 n “‘ “
DO 10:. K‘HJ‘JI)
Bl=PL-‘<! (VL'NI'HK'JI
02=U?*xl O/UND)rUJ
”105 CONTIHJS

I‘D

(
(

CE1(JI)--UH(BI.I)
C u‘(' 1‘-U12(BZ.1)
104 CONTL
103 CCrTiru
' RI (Ffb.303)
303 ggflr1al<zw1 10x. 4HNAME.6X. 6HVERSUS.6X. 4HNAME.31X. bHGROUPS)
11. 1'!.N:
DO 106 J=l.Nl -
IF<J 20 I) G? TO 106
OD 107 K 1.NC
IF(CEl(I.K) CF CE1(J.K) AND.CE?(I.K).CE.CE2(J.K)) GO TO 60
IF(C?1(I.K) LE.CEI(J.K) AND CEZ(1.K).LE.CEZ(J.K)) GO TO 61
NSD(K)~=(I
GO TO 10’
60 CALL CL'“‘.A! (J I)
CRLL EszU 1 ’ ‘
IFtEU CL 0 O) CO TO 52
IF<CF1(!.K).EG CE1(J.K) AND.CEQ(X.K).EO.CE2(J.K)) GO TO 61
NSDH'PI"?
GO T I“’ ' “

CALL é: LU.K) -~-~-
IF<EH c: o 0) GO TO 63

CD TO 107
63 N"1"¥’)=-l

107 CONTINUE
C 304) NAME(I).NAHE(J).(NSDtK).K=1.NG)

304 FO'N« (EH 6X.A30.10X.A10.2X.1016)
105 cou*zuu: "‘*‘
:14 MPI7"’H.30AT
305 rGPHAT<.)
115 CQuIIHUC

STOP

END

274
Figure 8.14 (cont'd. )

FUNCTION UJI(XN.J1)

CU*“R a NO ‘ C(1C1) ).CP(101).T(50) R(40 50) RA A(10.4)
gUhVQN IDIOCKB/ ARC( (400).VALL(400) VALU(400). KDIH.SHALL.DIFF
V-KDIrw- l
U1=lABEXE<VALL.SMALLoDIFFokaDUHflY)
RquRN
END -- ~- .., --—-— 7 —2
FUNCTION U?(X.I!)
COVm£n .<D N uc.Cx10 ) CP(101).T(50).R<40.50)2RA(10 4)
> ggmuurV/bxorve/ 9(400). VALL(400).VALU(400).KDIN SHALL.DIFF
K: moxmu!
agglguEXE(VALU.SHALL DIFF. K.DUHHY)
END "- - ~ - ~--
UNCTION UI!(Y.J5)
SDKHON HD.NLH,.C(IC1).CP(101) T( 50) R( 40.50).RA(10.4)
commmu IBIOCKZ / ARC<400) VA ML (4 00) VAL LU(400).KDIH.SHALL-DIFF
K=VDIP
U11: TA3EX<ARG.VALI-.X K)
REIURN
END
FUICTIDN U12(X.JJ)
CUMW :N ND N: rJC C(101).CP<101L H50 R( (40.50).RA( (10.4)
comm u /!lu(MH/ ARG<400> VA 400) VA ALU(400) KD DIH.SHALL.DIFF
K=nuiM
urzv rALEX( RC VALU.X.K)
PEfxwuq
END

FUNCTION T\PF>F(VAL SMALL DIFF. KoDUNHY)

312:”3IDN VALII)

DUI“ .- f UMP: r ~3r: 1‘. L1

I?MINO\MA!I(1 U‘DUfl/Dl IF? I O) K)
TAEZXE1(VflL(1*I)—VAL(I)HH(DUM ~FlOAT¢I~ ~13ODIFF)/DIFF+VAL(I)

' RETURN
END
FUNCTION TAXEX(VAL. ARC DUMMY.K)
Dlrmr:wzltu Vhl_(1).ARG 1) '_ _
DO 1 J ?.¥
2: « AHU(J)) CO 10 1
2 'Ilggéx-ZTD’ 5.231%“ A" ;( Pl))G(VAL(J)~VAL(J~1))/
1(ARG(J)-AKG('~1))+VAL(J~1) _
RFTU=N
1 COmrIr HE
J:
00 TO 2

275

the NE distributions are read next by the program. This information is
stored in arrays NAME and R. Finally, data on the decision maker's
preferences must be read into the program. Values of SMALL, DIFF, and
KDIM--the smallest system output value for which utility values are to
be assigned, the difference between system output levels for which
utility values are to be assigned, and the number of system outputs for
which utility values are to be assigned--are read first. Then data on
the utility functions associated with the decision maker's upper and
lower bound absolute risk aversion functions are read into arrays ARG,
VALL, and VALU, which are defined as above in program UFUNC.1

A simple output from program NSTDO is shown in Figure 8.15. This
is an ordering of four distributions for the decision maker whose inter-
val preference measurement is graphed in Figure B.l2. The symbol 1_
indicates that the first distribution named is preferred to the second;
:l_indicates that the second distribution is preferred to the first;
and Q_indicates that the two distributions cannot be ordered by the cri-
terion of stochastic dominance with respect to a function for the class
of decision makers whose absolute risk aversion functions lie within
the specified bounds.

The combined power of interval measurements of decision maker pre-
ferences and the criterion of stochastic dominance with respect to a
function is demonstrated by the results presented in Sections 4.7 and 4.8
of Chapter IV. Clearly these are two related analytical tools which can

be of considerable value in the analysis of decision made under uncertainty.

 

1The program reads these data from a permanent file which is the
catalogued output of program UFUNC.

276

NAME

nUrU11

9.7g“

TTT
889.
7.11..

VERSUS

1....“

TTT
SS S
III
DFUD

2» L 2

17 T-
(urea;
11?:
PL— .PL

011

10..“

TTT
SSS
TIT.
firth;

739.15

TTT.
(.51.:
71111..
Pair:

111

15.2.3

TTT
883
III

Ftnlrl

0.414

TTT
SFUCV
17.71
a: PI.-

Figure 8.15 A Sample Output from Program NSTDO

277

In this appendix the relatively straightforward procedures used to

implement these techniques in a practical setting have been described.

APPENDIX C
IMPLEMENTATION OF THE GREMP MODEL

C.l Introduction

 

In this appendix the computer program used to implement the GREMP
model is described, and some of the special features of the model are
discussed. The objective is to acquaint the potential user with the
more technical aspects of this procedure and to suggest ways in which
the model can be adapted for use in the analysis of particular decision
problems.

The basic structure of the computer program which implements the
GREMP model is shown in the flow chart in Figure C.l. The program begins
with an initialization phase, during which parameter values are specified
and required data are read in. The program then goes through a specified
number of iterations during which strategies are generated at random,
the outcomes of each strategy are simulated for a number of states of
nature, and the efficient set is updated. Once the desired number of
alternative strategies has been generated and evaluated, information on
the elements in the efficient set is printed, and the program terminates.

As was noted in Chapter V, this procedure is not designed to identify
a truly optimal choice. Rather, it simply generates a large number of
strategies, and, on the basis of evaluative information supplied by the
user, it identifies an efficient set of choices from those considered.
The particular value of the GREMP model is that it can be used to analyze

problems for which an optimal solution cannot be determined analytically.

278

 

279

Start

 

 

‘Initialization
Phase

 

 

 

 

 

' Strategy
Generation

 

 

 

 

 

 

Simulation of
Outcomes

 

 

 

 

 

Update
Efficient Set

 

 

 

       
      
  

Desired
Number of
Strategies
Examined?

 

 

   

Print Information
on Efficient Set

 

Figure C.l General Flow Chart of Program GREMP

280

The discussion in subsequent sections is organized in a manner
similar to the computer program itself. Data requirements and suggested
program parameter values are first examined. Next, the procedure by
which strategies are generated is described. The simulation of the
outcomes associated with each strategy generated and the evaluation of
strategies by the criterion of stochastic dominance with respect to a
function are then briefly discussed, with references being made to the
more extensive descriptions of these procedures given in Chapters III
and IV and in Appendices A and B. A complete listing of the program is

included at the end of the Appendix.

C.2 The Initialization Phase

 

During the initialization phase of the program, run parameter
values are established, some or all of the constraints on control
‘variable levels are specified, and data defining alternative states of
nature and the decision maker's preferences are read in.

The run parameters define certain general characteristics of any
particular application of the model. They include: ND, ITNS, NV, NC,
NVC, MAXNO and NCONS. As in other programs developed in this study,
ND is the number of sample observations defining the distribution of
outcomes associated with each strategy being considered. As such, it
is also the number of states of nature to be defined. In the applications
of the GREMP model discussed in Chapters V and VI, a value of 20 was
specified for ND. In many practical instances a larger value of ND
would be desirable. The maximum value in this version of the program

is 20, but this can be augmented by simply changing the appropriate

281

array dimensions.1 ITNS is the number of iterations the model will
perform-~i.e. the number of sample strategies which will be generated
and evaluated. The value of ITNS depends entirely on the characteristics
of the problem being analyzed. ITNS was set at 500 and lOOO in the

two applications of the GREMP model discussed in this study. It may be
desirable to specify larger values of ITNS when problems with a large
number of choice variables are to be analyzed or when the identification
of a more nearly optimal strategy is desired. If the simulation model
used to generate sample observations from the distribution of outcomes
associated with each strategy is quite complex, however, the cost of
each iteration may be so high that a much lower value of ITNS must

be specified.

NV is the number of control variables used to define a management
strategy in the problem being considered. The set of control variables
can be divided into as many as ten categories, with NC being the number
of categories. In the applications discussed in Chapters V and V1, for
example, it was convenient to divide the control variables into three
categories: resource acquiring activity levels, resource using activity
levels, and control rule parameters.2 Once the number of categories has

been specified, the number of control variables in each category must be

 

1Each stochastic system input variable must be dimensional to ND;
e.g. in the listing at the end of this chapter, the second argument in
each array in common blocks 2 and 3 is set to ND. In addition, T and
R are dimensioned to T(ND) and R(21,ND), and C and CP are dimensioned to
C(2*ND+l) and CP(2*ND+l).

2See Eisgruber and Lee (1971) for an interesting discussion of why
choice variables need to be classified in such a manner when strategies
are to be constructed in a sequential manner.

282

read into the appropriate element of the array NVC—-i.e. NVC(l) is

set equal to the number of control variables in category one; NVC(Z)

is set equal to the number in category two; etc. Since the classifica-
tion of variables is intended to be mutually exclusive and exhaustive,

the following equation must hold:

NC
NV = 2 NVC
i=1

i C.l
In some instances it may be infeasible for all NV of the control
variables to be set at non-zero levels in the specification of a
strategy. When this is the case, it may be desirable to impose a
limit on the number of control variables considered in defining each
strategy. MAXNO is used to impose such a restriction; its value must
be less than or equal to NV.1
Finally, NCONS is the number of linear constraints to be imposed
on the control variables which define a management strategy. In
general, all these constraints must be of a "less than or equal to"
form.2 Up to 25 linear constraints can be imposed in the current
version of the model. This number can easily be expanded, however.
Once the run parameter values have been specified, the program
reads information on two sets of constraints. Members of the first set
limit the range of allowable values for the NV control variables by

establishing a minimum value, VMIN, a maximum value, VMAX, and the

magnitude of the interval between values, VINT. If, for a particular

 

1See Donaldson and Webster (1968) for a more complete discussion
of how such restrictions can be used.

2See Dent and Thompson (1968) for a discussion of the difficulties
caused by constraints of other forms and for an explanation of how these
difficulties can be overcome.

283
control variable, VMIN = O, VMAX = lOO, and VINT = 25, then possible
values for that variable are 0, 25, 50, 75, and 100. If VINT is set
equal to an integer, all values of the control variable will be integer.
If, on the other hand, VINT is set equal to a very small value, the
set of allowable values approaches that of a continuous variable
between VMIN and VMAX.

Linear constraints such as those used in the specification of a
linear programming model comprise the second set of constraints. The
program reads input-output coefficients and resource availability levels
for each of the NCONS constraints of this type. All input-output
coefficients are first initialized to equal zero. Non-zero values are
then read into the two dimensional array A by specifying the constraint
number, I, the control variable number, J, and the desired value of
A(I,J).1 The variable LAST is simply a flag which, when set to a non-
zero value, indicates that the last non-zero input-output coefficient
has been read. Next, the NCONS resource availability levels are read
into the array F, and the specification of linear constraints is completed.

The initialization phase continues with the program reading data
which define levels for each stochastic exogenous system input
variable in each of the ND states of nature used in the determination

2

of system output distributions. The number of exogenous system input

 

1Control variables are ordered in the following manner. The
first NVC(l) control variables are the elements in category one, the
next NVC(Z) control variables are the elements in category two, etc.

2These data are generated externally to the program using tech-
niques described in Appendix A.

284

variables depends on the characteristics of the system being considered
in a particular decision analysis. In general, then, the user will
supply his own READ statements here. The version of the program listed
at the end of this appendix is that used in the analysis of the problem
discussed in Chapter VI. Therefore, data on contract prices, cash
prices, crop yields, and days available for fieldwork are read.1
Finally, information on decision maker preferences required for
the application of stochastic dominance with respect to a function is
read by the program. More specifically, the data points generated by
program UFUNC, which define the utility functions associated with the
decision maker's upper and lower bound absolute risk aversion functions,
are read.2 First, however, values of YMIN, DY, and KDIM-aparameters
of the table look-up functions used to represent those utility functions--
are read. YMIN is the minimum value of the system output variable for
which a utility value is calculated, DY is the interval between system
output levels for which utility values are calculated, and KDIM is the
total number of data points. Once these values are established, KDIM
values of ARG, VALL, and VALU--a system output level, a lower bound

utility value, and an upper bound utility value--are read into the

appropriate arrays.3

 

1All these data are read from separate permanent files. This is
often more convenient than using data cards.

ZSee Appendix B for a listing and description of this program.

3These values are read from a permanent file which is the catalogued
output of UFUNC.

285

C.3 Strategy Generation

 

All user supplied inputs are read into the program during the
initialization phase. At the beginning of the first iteration and of
all subsequent iterations, all control variables are set to zero, all
constraints are reset to their original values, and several variables
used to monitor the strategy generation process are set to zero. The
generation of a feasible strategy then begins.

The segment of the main program which constructs each strategy is
listed in Figure C.2. The sequence of operations is such that all the
elements of one control variable category are assigned values before
those in the next category are considered. After updating values of
ILO and IHI, the lowest and highest variable numbers of the elements
in the variable category being considered, the program calls subroutine
SELECT.1 This subroutine is a discrete uniform process generator which
randomly selects a variable, V(J), from the set of variables in the
category under consideration. If that variable has been considered in
the construction of the current strategy, the value of IND(J) will be
non-zero and subroutine SELECT will be called again. If it has not
been considered, the value of NEX(I), the number of variables within
the category already examined, is augmented by one and IND(J), the
indicator for the variable is set equal to NEX(I). Subroutine LEVSET

is then called. Like SELECT, it is a discrete uniform process generator.

 

1When none of the control variables in a particular category imposes
constraints on any of the others, it is possible to bypass calls of sub-
routines SELECT and CHECK. In the version of the program listed at the
end of this appendix, this is done for variable category 3, the set of
control rule parameters.

50

65
70

286

:7-
I
Z
0
H

HZ a—H’JiQI-a
one
DI

CXAI‘O “C
*:¢~
Ola-40

mZXTZOI—VVUU)
AHH'TIA

I?“ II Ii “'ilHH

-r-
l
"UWH‘V"‘<ZZZ’.F' + I!

, .H
7K J,NCHNS-NIN)
T.hvt<1)) GO TO 50

‘—i

‘W'TTDDZFTHDFO
w: 1pqum‘lmHH

...I'tv-tvr-
‘l~l.a
A

fiOHHf'JOHZv-«OHOHH
Z""~.
a—al"

x:

1*

Z

L”:

‘
J

i
..-
I

Figure C.2 Statements in GREMP Hhich Generate

Feasible Strategies

287
Given values of VMIN, VMAX, and VINT for variable J, it sets a level for
that control variable.1

Once a value for a particular control variable has been set, the
feasibility of that value must be checked. This is done by calling
subroutine CHECK. If all the constraints imposed in the management
strategy are linear and of a "less than or equal to" form, and if all
values of the array A in column J are of the same sign, CHECK is as
listed in Figure C.3. The feasibility of the specified level V(J) is
checked against each of the NCONS constraint levels. If a constraint is
violated, the value of V(J) is adjusted to a level which is feasible.
Once all constraints have been met, each constraint level is updated
to reflect the resource requirements associated with the level of V(J)
and control is returned to the main program.

In many instances, there may be a need to impose non-linear con-
straints on some of the control variables. One common non-linear
constraint takes the form:

V(J)*V(K) = O, C.2
which implies tha V(J) and V(K) cannot both have non-zero values. Such
a constraint can easily be incorporated into subroutine CHECK by adding
statements such as the following:

IF(V(K).NE.0.0) V(J) = 0.0

IF(V(J).EQ.0.0) GO to 20

V(K) = 0.0 C.3

NEX(Z) = NEX(2)+l

IND(k) = NEX(Z)

 

1As was noted in the preceding section, if VINT is set to a very small
value, the variable V becomes, for all practical purposes, a continuous
variable. So LEVSET can approximate a continuous uniform process generator.

288

S‘JIIFILCJ TINI' CHECVMJ: NCONSININ)

COHMUI.‘ /BIUCK1/ V(25) VMIN(25).VHAX<25):VINT(25)
COMMON /BlDCK2/ A(25o25)oF(25)oIND(25)
IF(V(J).EG O. 0) GO TO 20 '
DU 10 1:1)NCOIJS

Il*(A(I-J).EG O. 0) GO TO 10

IF(F(I).GF AII.J)*V(J))7 901-010
LEV=TV/VINT (d)

K”. Jl-‘rl‘LUr’ll (l IiVHT-VINT(J)

I!’(~‘\J).L'l'.VHIN(d)) V(J)‘-'0.0
IF’(\'{J).GT.VHF‘.X( )) V(J)=VNAX(J)

"Fir '3' IN‘ F:

I?-(V~d).EG.0.0) GO TO 20

N] MININ“ 1

I“? 15 I'TIII'C NS

F(I)=—F(1) V(JHH'HIHJ)

Hi. TURN

EN];

Figure C.3 A Simple Version of Subroutine CHECK

r‘.

289
If V(K) already has a non-zero value, V(J) is set equal to zero and
control is returned to the main program. If V(J) is not equal to zero,
V(K) is set equal to zero. The number of variables in its category--in
this case variable category two-~is augmented by one and the indicator
for this variable, V(K), is set equal to NEX(Z). These final two state-
ments ensure that V(K) will not be considered again later in the strategy
generation process. Other types of non-linear constraints can be treated
in a similar manner. Examples of constraints which closely parallel that

given in Equation C.2 include:

V(J)*V(K)2F(I) C.4

V(J)/V(K)SF(I) C.5
and

if V(J)2F(I), then V(K) = O C.6

Subroutine CHECK can also be adapted to make the strategy genera-
tion process more efficient in some instances. For example, if the
only constraint on two variables V(J) and V(K) is

V(J)+V(K) = F(I). c.7
then a value of V(K) can be established automatically once a value of
V(J) has been specified and vice versa. Still another useful alteration
of subroutine CHECK in some situations involves the addition of statements
which alter levels of VMIN or VMAX for all or some variables in a
category to reflect the impact of the level selected for another
variable. This was done in the applications presented in this study
when maximum allowable acreages for corn and soybeans were adjusted
upward to reflect additions in available acreage due to land rental.

The strategy generation process continues until all the variables

in each category have been considered or until the maximum number of

290

variables have entered the control strategy at a non-zero level. At
this point, however, it is possible that the strategy constructed does
not use the available resources to the fullest extent. Therefore, it
may be desirable to expand the strategy out until one or more additional
constraints become binding. This is done by subroutine EXPAND.
Donaldson and Webster (l968) provide a good discussion both of the
rationale for expanding strategies and of the means by which this can
be done. It is difficult, however, to specify a general form for
subroutine EXPAND. In most instances its form is problem-specific and,
so this subroutine must be supplied by the user. In the listing pre-
sented at the end of this appendix, EXPAND is used to increase acreage

levels of corn or soybeans, depending on which was specified first, so

that all available acreage is used.

C.4 The Simulation of Strategy Outcome§_

 

When the strategy generation phase of each iteration has been com-
pleted, subroutine DISGEN is called. This user—supplied subroutine
simulates system performance under the newly specified strategy for
each of the ND states of nature. Subroutine DISGEN can be as simple or
as complex as the problem under analysis requires. It can be as simple
as the example listed in Figure C.4, in which the level of system
output realized is a simple linear function of five control variables
and of the stochastic price, yield, and cost levels associated with
each. Alternatively, DISGEN can be as complex or still more complex
than that in the version of the program listed at the end of this

chapter.

291

SUBRDUTINE DISGEN
COMMON /BIUCK1/ Pt5,20).Y(5.20).C(5.20)
Cowman IEIUCKE/ V(SJ
COEan /DIULK3/ ND.NP.R(21.20)
DD 5 J=I.ND
( 'I\’)::O.()
13‘]: t)

N
$5.J)=R(NP.J)+(P(I.J)*Y(I:J)-C(I.J))*V(I)
D

OUT ".3

URN

"1370030

U
(
E
N

Figure C.4 A Simple Version of Subroutine DISGEN

292

A brief comment should be made about the variables passed to and
from DISGEN. First, it should be noted that R, the array of system
output levels passed back to the main program, has two dimensions. The
first argument of R is always set equal to NP in subroutine DISGEN.
This parameter identifies the system output distribution during the
evaluative phase of each iteration. NP is passed to the subroutine
and should be passed back unchanged. Values of all control variables--
the array V--and all exogenous system input levels are also passed to
and from the subroutine unchanged.

Finally, it should also be noted that there is no reason why the
distribution for more than one system output variable cannot be deter-
mined with each call of DISGEN. The evaluative criterion used in the
GREMP model, stochastic dominance with respect to a function, can,
however, only evaluate alternative strategies by considering the distri-

butions of a single system output variable.

C.5 Updating the Efficient Set

 

During the final phase of each iteration the efficient set is
updated by comparing the system output distribution of the newly
generated strategy to the distribution of each strategy in the efficient
set. This is done by subroutine NSTDO, which is simply a modified
version of the program written by Meyer to implement the criterion of
stochastic dominance with respect to a function. This program and the
required inputs to it were described in the final section of Appendix B.
Those same inputs must be passed to subroutine NSTDO from the main pro-
gram. They include values for NO, the number of sample observations

defining each system output distribution and values in the array R, the

293

array of simple system output levels for each plan in the current
efficient set and for the newly generated plan. Information in decision
maker preferences is passed from the main program directly to function
subprograms U1, U2, U11, and U12, which are called by subroutine NSTDO.

As described in Section 5.3.3 of Chapter V the evaluative process
involves the pair-wise comparison of the distribution of system output
levels associated with the newly generated strategy with that associated
with each strategy in the efficient set. Such comparisons are made until
the new strategy is dominated or until all possible comparisons have
been made. If the new strategy is dominated, control is returned
immediately to the main program. If it dominates strategies in the
efficient set, those which are dominated are eliminated and the arrays
describing the efficient set--R, AMEAN, STD, and VLEV--are rearranged.
The program limits the size of the efficient set to twenty strategies.
This is done because it reduces storage requirements and because
efficient sets having more than twenty elements are difficult to work
with. When the size of the efficient set reaches the maximum allowable
level, newly generated strategies are still compared to each element in
the efficient set, but they enter the efficient set only if they dominate

one or more of its existing elements.

C.6 Output of the GREMP Model

Information on selected run parameter values, on the linear con-
straints specified, and in the range of allowable values for each control
variable is printed before the first iteration is begun. This information
can be of value for diagnostic purposes. When the final iteration has

been completed, information on each element in the efficient set is

294

printed. First, the mean and standard deviation of the system output
distribution associated with a particular strategy are printed. then the
control variable levels defining the strategy are printed, and, finally,
the set of sample observations defining the system output distribution
is printed. It should be noted that performance indicators other than
those upon which the evaluation of alternatives is based could also be

printed if this were desired.

295

c.7 A Listing of the GREMP Model

0
8
F. I
D. 0 S D
‘ 2 T r... N.
T O P. L l
O 2’ 9 B
T In. N or a. _
f. (2 I T .
D. a 9 Mn 9. 0 .
I up... v- I T .
T PLA‘ . v
0 Ole H L a...
F \1 IE 91 “L ..t.
f. 5 9 Q D. RD L
P 2 ...I K TR 0 P
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BIBLIOGRAPHY

BIBLIOGRAPHY

‘* Anderson, J. R., l974a. "Simulation: Methodology and Application in

Agricultural Economics." Review of Marketing and Agricultural
Economics 42:3-55.

l974b. "Sparse Date, Estimational Reliability, and
Risk-Efficient Decisions." American Journal of Agricultural
Economics 56:564-72.

1975. "Programming for Efficient Planning Against
Non-Normal Risk." Australian Journal of Agricultural Economics
19:94-107.

1976. “Risk Efficient Monte Carlo Programming. A
Manual." Miscellaneous Publication No. 6, Department of Agricul-
tural Economics and Business Management, University of New
England, Armidale, Australia.

Anderson, J. R., J. L. Dillon, and J. B. Hardaker, 1977. Agricultural
Decision Analysis. Ames, Iowa: The Iowa State University Press.

 

Anderson, J. R. and H. E. Doran, 1978. "Some Properties of Simple
Functions of Random Variables." Review of Marketing and Agricul—
tural Economics 46:37-47.

Anderson, T. N., 1958. An Introduction to Multivariate Statistical
Analysis. New York: John Wiley and Sons, Inc.

Aoki, Masanao, 1967. Optimization of Stochastic Systems. New York:
Academic Press, Inc.

. 1975. "Output Decisions by a Firm: An Example of a
Dual Control Problem with Information Externa1ities," in Adaptive
Economic Models, Day, R. H. and T. Groves, eds. New York:
Academic Press, pp. 157-194.

Arrow, K. J., 1971. Essays in the Theory of Risk-Bearing. Chicago:
Markham Publishing Company.

Arrow, K. J. and M. Kurz, 1970. Public Investment, the Rate of Return,
and thimal Fiscal Policy. Baltimore: Johns Hopkins Press.

302

303

Barnett, V., 1975. "Probability Plotting Methods and Order Statistics."
Applied Statistics 24:95-108.

 

Barry, P. J. and D. R. Nillman, 1976. "A Risk Programming Analysis of
Forward Contracting with Credit Constraints." American Journal of
Agricultural Economics 58:62-70.

 

Black, J. R., N. A. Tinsley, E. Fuller, and C. Cuykendall, no date.
"Corn, Soybean Farm Planning Guide.“ Telplan Program 18, Form 1,
Department of Agricultural Economics, Michigan State University,
East Lansing, Michigan.

Boussard, J. M. and M. Petit, 1967. "Representation of Farmers'
Behavior under Uncertainty with a Focus-Loss Constraint." Journal
of Farm Economics 49:869-80.

 

Bradford, L. A. and G. L. Johnson, 1953. Farm Management Analysis.
New York: John Wiley and Sons, Inc.

Chapman, L. J., 1967. "Illusory Correlation in Observational Report."
Journal of Verbal Learning and Verbal Behavior 6:151-155.

Charnes, A. and w. w. Cooper, 1959. "Chance Constrained Programming."
Management Science 6:73-79.

 

Chow, G. C., 1973. "Effect of Uncertainty on Optimal Control Problems."
International Economic Review 14:632-645.

 

Churchman, C. N., 1968. The Systems Approach. New York: Dell
Publishing, Inc.

 

Cocks, K. D., 1968. "Discrete Stochastic Programming." Management
Science 15:72-79.

Coleman, D. R. and A. L. Saipe. 1976. "Modeling Bivariate Relationships
for Simulation," unpublished manuscript.

Coleman, D. R. and A. L. Saipe, 1977. "Simulating a Lognormal Time
Series with Prescribed Serial Correlation." Management Science
23:1363-1364.

Conklin, F. S. A. E. Baquet, and A. N. Halter, 1977. "A Bayesian
Simulation Approach for Estimating Value of Information: An
Application to Frost Forecasting." Technical Bulletin 136,
Agricultural Experiment Station, Oregon State University,
Corvallis, Oregon.

Cyert, R. M. and J. G. March, 1963. A Behavioral Theory of the Firm.
Englewood Cliffs, New Jersey: Prentice-Hall, Inc.

304

Day, R. H., 1963. Recursive Programming and Production Response.
Amsterdam: North Holland Publishing Company.

 

. 1964. "A Behavioral Theory of the Firm by Cyert and March.“
Econometrica 32:461-464.

 

 

. 1971. "Rational Choice and Economic Behavior." Theory and
Uec1sion 1:229-251. ‘*

. 1975. "Adaptive Processes in Economic Theory," in Adaptive
Economic Models, Day, R. H. and T. Groves, eds. New York: Academic
Press, pp. 1-38.

 

Dent, J. B. and J. R. Anderson, eds., 1971. Systems Analysis in Agri-
cultural Management. Sydney, Australia: John Wiley and Sons, Inc.

 

Dent, J. B. and S. C. Thompson, 1968. "The Application of Monte Carlo
Techniques to the Feedmix Problem." Farm Economist 11:230-248.

Donaldson, G F. and J. P. G. Webster, 1968. "An Operating Procedure
for Simulation Farm P1anning--Monte Carlo Method." Department
of Agricultural Economics, Wye College, University of London.

Dreyfus, S. E., 1968. "Introduction to Stochastic Optimization and
Control," in Stochastic Optimization and Control, Karreman, H. F.,
ed. New York: John Wiley and Sons, Inc., 99. 3-24.

 

Ehrlich, I. and G. 5. Becker, 1972. "Market Insurance, Self-Insurance,
and Self-Protection." Journal of Political Economy 80:627-648.

 

‘riEisgruber, L. M. and G. E. Lee, 1971. ”A System Approach to Studying
the Growth of the Farm Firm," in Systems Analysis in Agricultural
Management, Dent, J. B. and J. R. Anderson, eds. Sydney, Australia:
Wiley and Sons, Inc.

 

Fishburn, P. C., 1970. Utility Theory for Decision Making. New York:
John Wiley and Sons, Inc.

Forrester, J. W., 1961. Industrial Dynamics. New York: John Wiley
and Sons, Inc.

 

Freund, R. J., 1956. "The Introduction of Risk into a Programming
Model." Econometrica 24:253—63.

 

Friedman, M. and L. J. Savage, 1948. "Utility Analysis of Choices
Involving Risk." Journal of Political Economy 56:279—304.

Grayson, C. J., 1960. Decisions Under Uncertainty: DrillinggDecisions
by Oil and Gas Operators. Boston: Graduate School of Business
Administration, Harvard University.

 

305

Hadar, J. and W. R. Russell, 1969. "Rules for Ordering Uncertain
Prospects." American Economic Review 59:25-34.

 

Hanoch, G. and H. Levy, 1969. "The Efficiency Analysis of Choices
Involving Risk." Review of Economic Studies 36:335-46.

 

Hazell, P. B. R., 1970. "Game Theory--An Extension of its Application
to Farm Planning under Uncertainty." Journal of Agricultural
Economics 21:239-52.

1971. "A Linear Alternative to Quadratic and Semi-

variance Programming for Farm Planning under Uncertainty."
American Journal of Agricultural Economics 53:53-62.

 

Heidhues, T., 1966. "A Recursive Programming Model of Farm Growth in
Northern Germany.“ Jgurnal of Farm Economics 48:668-684.

Heifner, R. L., 1972. "Optimal Hedging Levels and Hedging Effectiveness
in Cattle Feeding." Agricultural Economics Research 24:25-36.

Hicks, J. R., 1946. Value and Capital, 2nd edition. London: Oxford
University Press.

 

Hirshleifer, J., 1970. Investment, Interest and Capital. Englewood
Cliffs, New Jersey: Prentice-Hall, Inc.

 

Hogarth, R. M., 1975. "Cognitive Processes and the Assessment of
Subjective Probability Distributions." Journal of the American
Statistical Association 70:271-91.

 

Holt, C. C., F. Modigliani, J. F. Muth, and H. A. Simon, 1960. Planning
Production, Inventories and Work Force, Englewood Cliffs, New
Jersey: Prentice-Hall, Inc.

 

Howard, R. A., J. E. Matheson, and D. W. North, 1972. "The Decision
to Seed Hurricanes." Science 176:1191-202.

Johnson, G. L., 1961a. "Some Reflections on the Nature of Managerial
Problems," in Managerial Processes of Midwestern Farmers.
Johnson, G. L., A. N. Halter, H. R. Jensen, and D. W. Thomas,
eds. Ames, Iowa: The Iowa State University Press, pp. 142-149.

1961b. "Implications of the IMS for Study of Responses
to Price," in Managerial Processes of Midwestern Farmers,
Johnson, G. L., A. N. Halter, H. R. Jensen, and D. W. Thomas,
eds. Ames, Iowa: The Iowa State University Press, pp. 150-169.

1976. "Philosophic Foundations: Problems, Knowledge,
and Solutions." European Review of Agricultural Economics 3:207-234.

Jo

306

Johnson, G. L. and C. F. Lard, 1961. "Knowledge Situations,“ in
Managerial Processes of Midwestern Farmers, Johnson, G. L.,
A. N. Halter, H. R. Jensen, and D. W. Thomas, eds. Ames, Iowa:
The Iowa State University Press, pp. 41-54.

 

Johnson, G. L. and C. L. Quance, eds., 1972. The Overproduction Trap
in U.S. Agriculture. Baltimore, Johns Hopkins Press.

 

Johnson, G. L. and L. K. Zerby, 1973. What Economists Do About Values.
East Lansing, Michigan: Department of Agricultural Economics,
Center for Manpower and Public Affairs, Michigan State University.

Kahneman, D. and A. Tversky, 1973. "On the Psychology of Prediction."
Psychological Review 80:237-251.

 

Kahneman, D. and A. Tversky, 1972. "Subjective Probability: A
Judgement of Representativeness." Cognitive Psychology 3:430-454.

Karreman, H. F., ed., 1968. Stochastic Optimization and Control.
New York: John Wiley and Sons, Inc.

Keeney, R. L., 1973. "A Decision Analysis with Multiple Objectives:
The Mexico City Airport." Bell Journal of Economics and Management
Science 4:101-17.

Keeney, R. L. and H. Raiffa, 1976. Decisions with Multiple Opjectives:
Preferences and Tradeoffs. New York: Wiley and Sons, Inc.

 

Kihlstrom, R. E. and L. J. Mirman, 1974. "Risk Aversion with Many
Commodities." Journal of Economic Theory 8:361-388.

Kirk, o. 5., 1970. Optimal Control Theory. Englewood Cliffs, New
Jersey: Prentice-Hall, Inc.

Kleindorfer, P. and H. Kunreuther, 1978. "Stochastic Horizons for the
Aggregate Planning Problem." Management Science 24:485-497.

Knight, F. H., 1921. Risk, Uncertainty and Profit. Boston: Houghton-
Mifflin.

Levy, H., 1973. "Stochastic Dominance, Efficiency Criteria and Efficient
Portfolios: The Multi-Period Case." American Economic Review
63:986-994.

Levy, H. and J. Paroush, 1974a. "Toward Multivariate Efficiency
Criteria." Journal of Economic Theory 7:129-42.

Levy, H. and J. Paroush, 1974b. "Multi-Period Stochastic Dominance."
Management Science 24:428-441.

 

307 -

Lewis, C. I., 1955. The Ground and Nature of the Right. New York:
Columbia University Press.

Lin, W. W. and H. 5. Chang, 1978. "Specification of Bernoullian
Utility Function in Decision Analysis." Agricultural Economics
Research 30:30-36. —'

Llewellyn, R. W., 1965. "FORDYN--An Industrial Dynamic Simulator."
Department of Industrial Engineering, North Carolina State
University, Raleigh, North Carolina.

Low, A. R. C., 1974. "Decision Taking under Uncertainty: A Linear
Programming Model of Peasant Farmer Behavior." Journal of
Agricultural Economics 25:311-22.

 

McInerney, J. P., 1969. "Linear Programming and Game Theory Models."
Journal of Agricultural Economics 20:269-278.

 

McKinnon, R. I., 1967. "Futures Markets, Buffer Stocks, and Income
Stability for Primary Producers." Journal of Political Economy
75:844-861.

MacRae, E. C., 1975. "An Adaptive Learning Rule for Multiperiod
Decision Problems.” Econometrica 43:893-906.

 

Manetsch, T. J., 1978. "Theory and Practice of Model Building and
Simulation," in Agricultural Sector P1annipg, Rossmiller, G. E.,
ed. Department of Agricultural Economics, Michigan State
University, East Lansing, Michigan.

 

Manetsch, T. J. and G. L. Park, 1977a. System Analysis and Simulation
with Applications to Economic and Social Systems, Part I, 3rd edition.
Department of Electrical Engineering and System Science, Michigan
State University, East Lansing, Michigan.

 

Manetsch, T. J. and G. L. Park, 1977b. System Anglysis and Simulation
with Applications to Economic and Social Systems, Part II, 3rd
edition. Department of Electrical Engineering and System Science,
Michigan State University, East Lansing, Michigan.

 

Markowitz, H., I959. Portfolio Selection--Efficient Diversification
of Investments. New York: Wiley and Sons.

 

Massé, P., 1962. Optimal Investment Decisions. Englewood Cliffs, New
Jersey: Prentice, Inc.

 

Meyer, J., 1977a. "Choice Among Distributions." Journal of Economic
Theory 14:326-36.

308

Meyer, J., 1977b. "Second Degree Stochastic Dominance with Respect to
a Function.“ International Economic Review 18:477-87.

. 1977c. "Further Applications of Stochastic Dominance to
Mutual Fund Performance." Journal of Financial and Quantitative
Analysis 12:235-242.

 

Modigliani, F. and K. J. Cohen, 1961. The Role of Anticipations and
Plans in Economic Behavior and Their Use in Economic Analysis and
Forecasting, Studies in Business Expectations and Planning, No. 4,
Bureau of Economic and Business Research, University of Illinois,
Urbana, Illinois.

 

 

 

Mood, A. M. and F. A. Graybill, 1963. Introduction to the Theory of
Statistics. New York: McGraw-Hill, Inc.

 

 

Naylor, T. H., J. L. Balintty, D. S. Burdick, K. Chu, 1966. Computer
Simulation Techniques. New York: John Wiley and Sons, Inc.

 

Newman, T. G. and P. L. Odell, 1971. The Generation of Random Variates.
New York: Hafner Publishing Company.-

Nott, S., G. Schwab, S. Harsh, J. Black, M. Kelsey, 1977. "Enterprise
Budgets Michigan, 1977." Agricultural Economics Report No. 314,
Department of Agricultural Economics, Michigan State University,
East Lansing, Michigan.

Officer, R. R. and A. N. Halter, 1968. "Utility Analysis in a Practical
Setting." American Journal of Agricultural Economics 50:257-77.

 

Phillips, 0. T. and C. S. Beightler, 1972. "Procedure for Generating
Gamma Variates with Non-Integer Parameter Sets." Journal of
Statistical Computation and Simulation 1:203.

 

Pratt, J. W., 1964. "Risk Aversion in the Small and in the Large."
Econometrica 32:122-36.

 

Pratt, J. W., H. Raiffa, and R. Schlaifer, 1965. Introduction to
Statistical Decision Theory. ‘New York: McGraw-Hill, Inc.

 

Rae, A. N., 1971. "Stochastic Programming, Utility, and Sequential
Decision Problems in Farm Management." American Journal of
Agricultural Economics 53:448-60.

 

Ramsey, F. P., 1931. "Truth and Probability,” in The Foundations of
Mathematics and Other Logical Essays, Braithwaite, R. 8., ed.
London: Routledge and Kegan Paul.

 

Robison, L. J. and R. P. King, 1978. "Specification of Micro Risk
Models for Farm Management and Policy Research." Agricultural
Economics Report No. 349, Department of Agricultural Economics,
Michigan State University, East Lansing, Michigan.

309

Sage, A. P., 1968. Optimum Systems Control. Englewood Cliffs. New
Jersey: Prentice-Hall, Inc.

 

Samuelson, P. A., 1967. "General Proof That Diversification Pays."
Journal of Financial and Quantitative Analysis 2:1-13.

Savage, L. J;, 1971. "Elicitation of Personal Probabilities and
Expectations." Journal of the American Statistical Association
66:783-801.

 

Schlaifer, R., 1959. Probability and Statistics for Business Decisions.
New York: McGraw-Hill, Inc.

 

Schmidt, J. W. and R. E. Taylor, 1970. Simulation and Analysis of
Industrial Systems. Homewood, Illinois: Richard D. Irwin, Inc.

 

 

Spetzler, C. S. and C-A. S. Stael von Holstein, 1975. "Probability
Encoding in Decision Analysis." Management Science 22:340-58.

 

Stael von Holstein, C-A. S., 1970. Assessment and Evaluation of Sub-
jective Probability Distributions. Stockholm: The Economic
Research Institute at the Stockholm School of Economics.

 

Tobin, J., 1958. "Liquidity Preference as Behavior Towards Risk."
Review of Economic Studies 25:65-78.

 

Tsiang, S. C., 1972. "The Rationale of the Mean-Variance Analysis,
Skewness Preference, and the Demand for Money." American Economic
Review 62:354-372.

 

Tversky, A. and D. Kahneman, 1973. "Availability: A Heuristic for
Judging Frequency and Probability." Cognitive Psychology 5:207-232.

von Neumann, J. and O. Morgenstern, 1944. Theory of Games in Economic
Behavior. Princeton, New Jersey: Princeton University Press.

Wald, A., 1947. Sequential Analysis. New York: John Wiley and Sons, Inc.

 

Ward, R. W. and L. B. Fletcher, 1971. "From Hedging to Pure Speculation:
A Micro Model of Optimal Futures and Cash Market Positions."
American Journal of Agricultural Economics 53:71-78.

Watson, S. R., J. R. Weiss, and M. L. Donnell, 1979. "Fuzzy Decision
Analysis." IEEE Transactions on Systems, Man, and Cybernetics
SMC-9:1-9.

Whitmore, B. A., 1970. "Third-Degree Stochastic Dominance." American
Egonomic Review 60:457-459.

 

w;

-a:"*

310

Winkler, R. L., 1967. "The Assessment of Prior Distributions in
Bayesian Analysis." ooyrnal of the American Statistical Association
62:776-800.

1969. "Scoring Rules and the Evaluation of Probability
Assessors." Journal of the American Statistical Association
64:1073-1078.

Zadeh, L. A., 1973. "Outline of a New Approach to the Analysis of
‘ Complex Systems and Decision Processes." IEEE Transactions on
Systems, Man, and Cybernetics SMC-3z28-44.

 

Zellner, A., 1971. An Introduction to Bayesian Inference in Econometrics.
New York: Wiley and Sons. Inc.