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This is to certify that the

dissertation entitled

OPTIMAL POST-HARVEST GRAIN MARKETING STRATEGIES
IN A RISKY MARKET ENVIRONMENT

presented by
Jing—yi Lai
has been accepted towards fulfillment

of the requirements for

PhOD.

 

degree in Agricultural Economics

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MAY BE RECALLED with earlier due date if requested.

 

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1/” WWW“

OPTIMAL POST-HARVEST GRAIN MARKETING STRATEGIES
IN A RISKY MARKET ENVIRONMENT

By

Jing-yi Lai

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Agricultural Economics

1999

 

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ABSTRACT

OPTIMAL POST-HARVEST GRAIN MARKETING STRATEGIES
IN A RISKY MARKET ENVIRONMENT

By

Jing-yi Lai

A stochastic dynamic programming model is used to solve the post-harvest grain
marketing-storage problem, allowing for a risk averse decision maker. Both theoretical and
numerical evidence are provided to support the hypothesis that selling storage partially
throughout the marketing period can be an optimal marketing strategy for risk averse farmers.

Simple marketing decision rules that are easy to understand and implement are
illustrated and applied to Michigan corn producers. The results show that risk averse farmers
generally spread their sales out over the marketing year in order to avoid risk. In fact, risk
averse farmers prefer to gain a certain income earlier by selling a portion of the production
at harvest, and then use the rest to speculate on an increase in future prices. The main
difference between the risk neutral and risk averse cases is that the risk neutral farmers use
the entire stock to speculate on a higher price, while risk averse farmers use only a portion
of it. The findings also indicate that low storage costs and low interest rates result in holding

the grain longer for all degrees of risk aversion.

 

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The economic value of the optimal marketing rules is measured by the farmer’s
willingness-to-pay (WT P) for the right to apply the optimal marketing strategy rather than sell
the production immediately at harvest. It is found that W77D is economically significant and
the optimal storage rule is especially valuable for risk neutral farmers. The optimal marketing
strategy under risk aversion results in an increase in the level of producers’ expected final
wealth and a decrease in the variance of returns. The increase in the level of expected final
wealth is significant at low degrees of risk aversion while the reduction in variance of returns

is significant at high degrees of risk aversion.

To my parents

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ACKNOWLEDGMENTS

I would like to thank all who have served on my thesis committee for providing keen
insights in their respective areas of expertise. I extend my deepest gratitude and greatest
respect to my major advisor, Dr. Robert Myers. I have benefitted tremendously fi'om his
invaluable instruction and been inspired by his academic discipline, which have stimulated my
potential and enabled me to complete the degree.

I am also extremely grateful to Dr. Steven Hanson, who has always been a source of
motivation and guidance for me ever since I joined the department. I am indebted to Dr.
James Hilker and Dr. John Ferris for their input at various stages of this research. Their
comments have strengthened this dissertation profoundly.

My sincere thanks go to all my graduate colleagues in the department who have
allowed me to extend my horizon and made my graduate study fruitful and enjoyable. A
special mention goes to Agustin Arcenas (Oggie), Nazmul Chaudhury (Rana), Kei Kajisa,
Sharon Lerner, Thomas Moen, Janet Owens, Julie Stepanek, and Lorie Srivastava, who
constantly provided me with warmth, encouragement, and precious friendship.

I would also like to express my admiration and appreciation to KeShin T swei, for his
love and care in walking me through my most difficult moments during the overseas study.
I am blessed to have him as part of my life.

The biggest thanks go to my family for their unlimited love and enduring support in

the long process of completing my degree. I owe them the most.

 

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TABLE OF CONTENTS

LIST OF TABLES ............................................ viii
LIST OF FIGURES ........................................... ix
CHAPTER I INTRODUCTION ................................ 1
1.1 Problem Statement ................................ 1
1.2 Objectives of the Study ............................. 4
1.3 Outlines of the Dissertation .......................... 5
CHAPTER H LITERATURE REVIEW ........................... 7
2. 1 Decision Criteria ................................. 7
2.2 Decision-Making for Post-Harvest Grain Marketing ............ 10
2.3 Dynamic Programming in Farm Management ................ l 5
2.4 Summary ...................................... 20
CHAPTER III THE DYNAMIC DECISION MAKING MODEL ........... 22
3. 1 The Theoretical Model ............................... 22
3 .2 Formulation of the Optimal Marketing Strategies ............. 26
3.3 Summary ....................................... 39
CHAPTER IV THE PROBABILITY-BASED PRICE FORECAST MODEL. . . 41
4.1 DataAnalysis ..................................... 43
4. 1. 1 Unit-Root Tests ............................. 45

4.1.2 Graphical and Descriptive Analysis of the Dependent
Variable .................................. 51
4.2 The EconometricModel .............................. 56
4.2. 1 Model Specification .......................... 58
4.2.2 EstimatedResults ............................ 60

 

 

 

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4. 3 Time-Dependent Transition Probabilities .................. 62
4.3.1 Procedure to Generate the Price Transition Probability

Space .................................... 63

4.3.2 Simulation Results ........................... 69

4.3.3 Model Validation ............................ 76

4.4 Summary ....................................... 85

CHAPTER V OPTIMAL STORAGE RULES ......................... 86

5. 1 The Empirical Problem ............................... 87
5.2 Optimal Marketing Policy for Nearly Risk Neutral Producers ...... 93
5.3 Optimal Marketing Policy for Mildly Risk Averse Producers ...... 96
5.4 Optimal Marketing Policy for Highly Risk Averse Producers ...... 108
5.5 EvaluatinganOptimal Marketing Policy .................... 114
5 . 5. 1 Expected Optimal Marketing Volumes .............. 1 15

5.5.2 Probability Distribution ofFinal Wealth ............. 1 19

5.6 Summary ........................................ 123

CHAPTER VI ECONOMIC VALUE OF

THEOPTIMAL MARKETING STRATEGY ............... 125

6.1 Definition ofWillingness to Pay ......................... 126

6.2 Results ......................................... 129

6.3 Summary ........................................ 136
CHAPTER VII CONCLUSION ................................... 13 7
APPENDD(A STATISTICAL HYPOTHESIS TESTS ................... 142
APPENDIXB PROPERTIES OF UTILITY FUNCTIONS ................. 147
APPENDIXC THEBISECTION METHOD .......................... 149
REFERENCES .............................................. 151

vii

 

 

 

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Table 4-1
Table 4-2
Table 4-3
Table 4-4
Table 4—5(a)
Table 4-5(b)
Table 4-5(c)
Table 4-5(d)
Table 4-5(e)
Table 4-6
Table 4-7
Table 5-1

Table 6-1
Table 6-2
Table A-1

LIST OF TABLES

Results fiom Phillips-Perron Unit Root Tests .................. 47
Results fi'om Augmented Dick-Fuller Unit Root Tests ............ 49
Descriptive Analysis of the Rate of Price Change for Corn .......... 54
Econometric Models .................................. 61
Price Transition Probability from November to January ............ 70
Price Transition Probability from January to March .............. 71
Price Transition Probability from March to May ................ 72
Price Transition Probability from May to July .................. 73
Price Transition Probability from July to September .............. 74
Average Seasonal Price Index for Simulated and Historical Prices ..... 79
Average Price Change Rates for Simulated and Historical Prices ..... 79
Cutofl‘ Prices for Different Rate of Storage Costs and Interest Rates

for Nearly Risk Neutral Decision-Makers ..................... 95
Expected Final Wealth Results ........................... 130
Willingness-to-Pay Results .............................. 134

The Log-Likelihood Functions under Various Orders of Seasonal
Effects in Mean (J) and Variance (l) ......................... 146

viii

Figure 4-1

Figure 4-2

Figure 4-3

Figure 4-4

Figure 4-5(a)
Figure 4-5(b)
Figure 4-5(c)
Figure S-1(a)
Figure S-1(b)
Figure 5-1(c)
Figure 5-1(d)
Figure S-1(e)
Figure 5-2(a)
Figure 5-2(b)
Figure 5-3(a)
Figure 5-3(b)
Figure S—3(c)
Figure 5-3(d)
Figure 5-3(e)

Table 54

LIST OF FIGURES

Logarithm of Prices for Corn ............................ 44
Rate of Price Changes for Corn ........................... 52
Squared Rate of Price Changes for Corn ..................... 53

Comparison of Simulated Average and Historical Average Price
Movements .......................................
Price Distribution for each Periods of Marketing Season Conditional
on the Average Price in November ......................... 82
Price Distribution for each Periods of Marketing Season Conditional

78

on a Low Price in November ............................. 83
Price Distribution for each Periods of Marketing Season Conditional

on a High Price in November ............................. 84
Optimal Marketing Policy for Mildly Risk Averse Producers for the

Month of November .................................. 97
Optimal Marketing Policy for Mildly Risk Averse Producers for the

Month of January .................................... 99
Optimal Marketing Policy for Mildly Risk Averse Producers for the

Month of March ..................................... 103
Optimal Marketing Policy for Mildly Risk Averse Producers for the

Month of May ...................................... 104
Optimal Marketing Policy for Mildly Risk Averse Producers for the

Month of July ...................................... 105
Optimal Marketing Policy for the Month of November at Different

Rates of Storage Costs .................................. 107
Optimal Marketing Policy for the Month of November at Different

Interest Rates ...................................... 107
Optimal Marketing Policy for Highly Risk Averse Producers for the

Month of November .................................. 109
Optimal Marketing Policy for Highly Risk Averse Producers for the

Month of January .................................... 110
Optimal Marketing Policy for Highly Risk Averse Producers for the

Month of March .................................... 1 11
Optimal Marketing Policy for Highly Risk Averse Producers for the

Month of May ...................................... 112
Optimal Marketing Policy for Highly Risk Averse Producers for the

Month of July ..................................... 113
Expected Marketing Volumes for Various Degrees ofRisk Aversion . . . . 1 18

ix

 

 

 

Tablc 5-

Table 5-

Table 5-

Table 5-

 

Table 5-5

Table 5-6

Table 5-7

Table 5-8

Expected Optimal Marketing Volumes for Mild Risk Aversion under

Various Storage Costs and Interest Rates .................... 120
Expected Optimal Marketing Volumes for High Risk Aversion under
Various Storage Costs and Interest Rates .................... 121
Expected Optimal Marketing Volumes for Low Risk Aversion under
Various Storage Costs and Interest Rates .................... 122
Probability Distribution of Final Wealth ..................... 124

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CHAPTER I

INTRODUCTION

1.1 Problem Statement

The Agricultural Marketing Transition Act of 1996 eliminated much of the price and
income protection that had been provided to U. S. farmers for decades. While there continues
to be debate over whether this policy change is going to lead to more commodity price
volatility, it is clear that the new legislation has shifted the responsibility for managing
agricultural price and income risk squarely on the shoulders of individual farmers. One area
in which price risk management is of primary concern is in determining post-harvest grain
marketing strategies. Most grain producers have enough storage capacity to store at least
some of their grain between harvests. They then must choose when and how much to sell
over the storage season. Selling at the right time (at a relatively high price) versus the wrong
time (at a relatively low price) can have a big impact on farm returns. Thus, farmers face
considerable risk in their post-harvest grain marketing decisions.

To improve farmers’ risk management performance in post-harvest grain marketing,
better storage-marketing strategies are necessary. Farmers currently are able to obtain market
information, including realized and forecasted prices in the spot and futures markets, to guide

their marketing decision-making. However, it has been observed that farmers often are

 

  
 

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overwhelmed by the unstructured nature of information flows and the complex decision-
making process (Anderson and Mapp 1996). Simple marketing decision rules that are easy
to understand, have a proven track record, and are based on market information readily
available at the time decisions are made, could be of considerable value and assistance in
managing price and income risk in post-harvest grain marketing. Such decision rules would
provide decision aids directly through reducing the transaction costs of collecting data and
updating market information, as well as simplifying the complexity of the decision process.
This may also lead to an enhancement in farm returns and decline of income variation.
Although several private marketing advisory services presently are available, these services
typically are designed for commodity contract traders.l Most marketing strategies
recommended by such services usually ignore the individual farm’s business situation and
farmer risk attitudes making them problematic for farm use.

Decision theoryunder uncertainty has been well developed and widely applied in
research on agricultural commodity marketing. However, most of the focus has been on joint
production and marketing decision problems during the pre-harvest season. Less attention
is given to post-harvest storage-marketing management, which may be especially important
today with the expansion of on-farm storage capacity allowing producers to store their
harvest to profit from expected increases in cash prices over the storage season. Moreover,
in previous studies little effort has been devoted to explaining how information derived from

theoretical models can be transferred into applicable decision rules decision-makers can

 

‘ The limited research that has been conducted to evaluate the information fiom these

private advisory services has found little evidence to support either the statistical or
economical emciency of their usefulness in farm management (e. g., Martines-Filho and Irwin
1995; O’Brien and Wisner 1995; Zulauf and Irwin 1998).

directly

 

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directly utilizez. The results therefore are less often incorporated into extension programs.
The missing link between theory and practice is largely because theoretical models are
perceived to be restricted and unrealistic, and therefore of limited usefulness in the real world.
For instance, some research uses static models (e.g., Mapp et al. 1973; Turvy and Baker
1989; Lapan and Moschini 1994) that assume decision-makers are concerned only about two
points in time (the present and some future terminal date) implicitly imposing the restriction
that all decisions are irreversible. Other research suggests an overly simplified marketing
strategy in which, at any period, decision-makers either market or retain the entire stock,
implying partial sales are not optimal (e. g, Fackler and Livingston 1996). Decision-makers
in real world situations, however, face a far more complex scenario and grain producers with
storage capacity often sell their production gradually between harvests. Since price
uncertainty is continuously resolved as the marketing period progresses, they will respond to
the changing conditions by altering their expectations of filture prices and adjust the timing
of sales based on the new information. Selling storage partially throughout the marketing
period may spread the price risk over time for farmers who experience risk aversion.
However, this proposition has not been adequately shown or explored.

This study is expected to make a contribution to modeling the post-harvest grain
marketing decision problem based on realistic assumptions about post-harvest marketing
scenarios and incorporating the goal of risk management. Theoretical support for partial sales
as an optimal marketing strategy when risk aversion is incorporated in the model will be

provided. Furthermore, decision rules easily understood and implemented by users will be

 

2 For exceptions see King et al. (1988); Alderfer et al. 1992; Berg (1987), and Fackler
and Livingston (1996).

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developed to facilitate storage marketing advisory and extension activities. The research will
be particularly usefirl to analysts who are designing post-harvest marketing advice for crop
producers or extension researchers who are making storage-marketing recommendations to

farmers.

1.2 Objectives of the Study

The first research objective is to specify a multistage decision-making model using
methods of stochastic dynamic programming in a discrete-time framework to describe crop
producers’ post-harvest storage-marketing problem. A generalized marketing decision rule
will be developed from the model. The model extends Fackler and Livingston’s (1996) risk-
neutral model by considering the producer’s risk attitudes. Rather than the sell-all-or-none
marketing strategy derived by Fackler and Livingston, an optimal partial selling rule is
developed from the model in this study.

The second objective is to apply the theoretical model to the storage-marketing
problem faced by Michigan corn producers. Empirical support for the optimal partial sales
rule will be provided. Optimal marketing decision rules will be presented in an easily
accessible and understandable form that is can be used by farmers and extension agents. A
sensitivity analysis will be conducted to investigate the impacts of changes in parameters in
the dynamic programming model on the optimal timing of sales.

The third research objective is to evaluate the economic value of the optimal

marketing strategies and provide quantitative support for the usefulness of marketing decision

models.
no stora

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models. Farmers’ willingness to pay for the right to adopt the optimal sales rule, using the
no storage case as a benchmark will be computed. The resultant expected final wealth and

the associated variances (risks) will also be examined.

1.3 Outline of the Dissertation

The dissertation is organized into seven chapters. Chapter 11 gives background
information on decision theories under uncertainty, stochastic dynamic programming models,
and their applications in farm management. A review of previous studies on post-harvest
grain marketing is summarized. This chapter concludes with discussion of a preferred model
to represent farmers’ post-harvest storage-marketing management problems.

A normative stochastic dynamic programming decision model is developed in Chapter
III. A generalized optimal marketing policy is derived mathematically to provide theoretical
support for a optimal partial sales rule, as opposed to the sell-all-or-none marketing strategy
arising from previous work.

Chapter IV estimates the stochastic structure of the risky market environment for
Michigan corn producers. The stochastic element in a dynamic programming model is
featured by transition probabilities mapping stochastic states across stages. The details of the
procedure used to convert the econometric results into Markovian transition probability
matrices is illustrated. The validation of the transition probability matrices is also
investigated.

Chapter V contains the estimation of the optimal storage-marketing policy for various

 

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degrees of risk averse corn producers in Saginaw, Michigan. This chapter also provides
empirical evidence to support the theoretical result that proportional sales may be optimal
marketing strategies when decision-makers exhibit risk aversion. The optimal marketing
policy for each decision period throughout the storage season will be presented. Sensitivity
analysis is conducted to investigate the impacts of various parameters in the decision model
on the optimal timing of sales. An application of the optimal storage rules is given and
discussed.

Chapter XI studies the economic value of the optimal marketing strategies by
computing farmers’ willingness to pay for the right to apply the proposed marketing decision
rules rather than sell production immediately at harvest. The resultant expected final wealth
and the associated variances (risks) from the application of the optimal storage rules are
examined.

Chapter VII concludes the dissertation by summarizing the major results of the study.
The strengths and weakness of the study are also discussed, along with suggestions for future

research.

 

 

 

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CHAPTERII

LITERATURE REVIEW

This chapter will provide background information on decision theories under
uncertainty, stochastic dynamic programming models, and their applications in farm
management studies. A review of previous research on post-harvest grain marketing is

summarized followed by discussion of a preferred model for application in this study.

2.1 Decision Criteria under Uncertainty

Many farm management issues can be represented as decision-making problems in a
stochastic environment, where decisions are made to maximize a certain objective when
outcomes are not known with certainty. Agricultural economists have developed a variety
of mathematical programming techniques to model farm-level decision problems in a risky
environment. The predominant decision criterion among all these techniques is expected
utility maximization which assumes monotonic and concave utility functions implying a degree
of risk aversion (Lambert and McCarl 1985). Direct use of expected-utility-maximization
models in earlier research, however, was limited due to the computational complexity of

nonlinear objective functions. Some of the more popular methods used in earlier empirical

 

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applications are mean-variance models developed by Markowitz (1959) and minimization-of-
total-absolute-deviations models (MOTAD) by Hazell (1971).

The mean-variance model assumes that the solution to expected-utility-maximization
problems is equivalent to maximization of a function of the first two moments of the income
distribution, based on the assumption that a decision-maker chooses among alternatives solely
on the basis of his or her expected income, the associated variance, and the degree of risk
aversion he or she holds (see Scott and Baker 1972; Barry and Willmann 1976; Turvey and
Baker 1989; Garcia et al. 1994). This criterion is consistent with expected utility theory and
is second-degree stochastic dominantl under the condition that income is normally distributed
or the decision-maker’s utility firnction is quadratic (Levy and Hanoch 1970). MOTAD is an
approximation technique to quadratic programming using a linear programming algorithm.
It assumes that the agent makes decisions by minimizing the total absolute deviations, both
positive and negative, around a decision variable (see Mapp et al. 1979; Gebremeskel and
Shumway 1979). MOTAD is modified to Target-MOTAD by Tauer (1983) where the agent
minimizes only negative deviations from a pre-specified target income, based on the idea that
decision makers are concerned with avoiding only unfavorable outcomes (see Kaiser and
Apland 1989; Bauer 1991).

It appears that a primary purpose for developing the mean-variance and MOTAD
models is to approximate optimal solution under expected-utility-maximization but with more

tractable methods that are easier to implement and are less data intensive. In spite of the

 

‘ The second-degree stochastic dominance technique is one other decision criterion in

selecting preferred activity by comparing the ordering of distributions of associated returns
proposed by Meyer (1977). A more detailed discussion of this approach will be discussed
later in this section.

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computational advantages, however, solutions derived fiom these approximation models are
suboptimal unless certain constraints are met, which can be very restrictive and perhaps costly
in practice. For example, non-normal probability distributions often provide better
representations of yields or income in agricultural production than normal distributions (Day
1969), which implies using farm income as an objective in mean-variance models may violate
the normality condition. The increasing power and availability of mathematical programrrring
software are making the numerical solution of expected utility maximization easier to achieve.
Better and more sophisticated approaches to directly maximize expected utility are adopted
in more recent literatures (see Ho 1984; Karp 1988; Lambert and McCarl 1989; Martineq and
Zering 1992; Adam and Anderson 1995). Lambert and McCarl (1985), Kaylen et al. (1987),
and Preckel and Vuyst (1992) have detailed discussions on how to obtain numerical solutions
of expected-utility-maximization problems directly using nonlinear programming techniques
with a focus on pursuing efficient ways to incorporate the probability information of random
events.

An alternative technique in the theoretical framework for modeling decision making
under uncertainty is based on the distribution of returns. Two of the most popular approaches
include safety-first models proposed by Roy (1952) and the method of stochastic dominance
by Meyer (1977). The safety-first model assumes that the decision-maker places constraints
on the probability that returns fail to achieve a certain goal (see Hatch et a1. 1989; Dorward
1994). Musser et al. (1981) conclude that the safety-first model is appealing in extension
applications since it is simple to implement and is easy to explain in the limited time available,
and the results are straightforward and easily understood. The stochastic dominance

technique is a risk efficient criterion, under which preferred alternatives are selected by

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coefficient
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comparing the cumulative distributions of the associated returns (see King and Lybercker
1983; Ristor et al. 1984; McClintock 1991). This approach is theoretically attractive because
it is based on the expected utility hypothesis but puts few restrictions on utility functions--
positive marginal utility of income and risk aversionurather than a specific functional form.
In empirical applications, both approaches demand sizable data to give a complete ordering
of distributions from alternative decisions. The requirements can be additionally difficult to

meet in farm-related research since historical data on farm income are often not available.

2.2 Stochastic Dynamic Programming in Farm Management

Many farm activities are dynamic and stochastic in the sense that events develop over
time as uncertainty regarding future events is resolved and that there decisions made today
influence opportunity that will be available in the fiiture. They often are formulated in a
multistage decision process, which is characterized by the task of selecting a sequence of
decisions to meet a certain decision criterion, where current actions affect future
opportunities.

Two classes of mathematical programming techniques have been applied to multistage
decision processes in farm management. Both of these techniques allow for random
coeficients in the objective function and the constraint set. The first approach is usually
called the discrete stochastic sequential model. This approach explicitly constructs a
“decision tree” to describe all the possible decision alternatives, states of nature, the

consequences of the actions given the various states of nature, and their corresponding

10

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probabilities. Based on the decision tree, a set of matrices can be arranged, with the
replication of rows of activities for each state of nature at each stage and the introduction of
transfer columns linking the multi-stage decision process into a single linear programing
problem. The discrete stochastic sequential model is developed by Cocks (1968) and applied
by Rae (1971a and b) in farm management studies. A more recent application includes
Lambert and McCarl (1989) and Kaiser and Apland (1989). Lambert and McCarl use this
approach to solve post-harvest grain marketing problems under price uncertainty. Kaiser and
Apland solve the joint production and marketing problems of crop farms in a multi-stage
framework under yield, output price, field rate and field time risks. Discrete stochastic
sequential models also have been used to determine optimal forward contracting levels for a
representative Texas farm (e. g., Barry and Willmann 1976), to analyze grth of farm firms
(e. g., Johnson et al. 1967), to assess alternative production, marketing and financial strategies
for an Iowa crop and Livestock farm (e.g., Johnson and Boehlje 1986), and to evaluate
farmland investment (e.g., Schnitkey et al. 1989).

A major impediment to the application of this approach is its computational burden.
The programming matrices become considerably large with even modest increase in the
number of stages and states. This inherent computational problem has been accommodated
in the literature by modeling only a few states of nature or restricting the number of decision
variables. For example, random variables which are not critical to the problem or add only
little risk are modeled detenninistically; a set of actions are pre-selected as potential solutions.

The method of stochastic dynamic programming is another popular multi-stage
mathematical technique developed by Bellman and others in the 19505. It has become a

standard tool in economic analysis and operations research. This mathematical technique

11

formula

which C:

Where {
I if. I.’
state Veg

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formulates the dynamicdecision problem by application of Bellman’s Principle of Optimality2

which can be represented as:

V;(x;) = max {ft(x,, ”2) + EtVM(xM)} (2.1)

at x“l = g,(x;,u;

x0 rs given

where { V; },T=0 denote the value functions defined as the optimized values of the state at
t; { f, },T=0 are return functions ofien characterized by utility or monetary returns; x, is the
state vector which describes the condition of the decision process; 11, is the control vector
refers to all the potential activities being selected to reach the objective function;
and { g, },T=0 are state transition equations mapping state vectors between two consecutive
stages. The first term on the right-hand side of equation (2.1) is the immediate reward or
profit, the second term constitutes the continuation value, and the optimum action at the
current period is the one that maximizes the sum of these two components.

In practice, the dynamic programming algorithm is implemented by iterating on the
equation starting at the terminal period and moving backwards recursively through time.
Without plotting out all the possible actions and outcomes (as in the discrete stochastic

sequential models), the computational efforts are reduced and the algorithm is generally more

 

2 Bellman’s definition of this principle: “An optimal policy has the property that
whatever the initial state and initial decision are, the remaining decisions must constitute an
optimal policy with regard to the state resulting from the first decision.” (Dixit and Pindyck
1994)

12

efficient

   
 

framewc
is more '
between
world si
continuo
therefore

I
control rr
diSCTCllza
' Particu‘.
Stale inter

Pregram

result in n

 

 

 

eficient. Dynamic programming problems can be solved under a continuous- or discrete-time
fiamework, depending on the research problem at hand. The continuous-time model usually
is more usefirl to yield qualitative insights (Karp 1988) such as in the case of a binary choice
between continuation and stopping of holding stocks (F ackler and Livingston 1996). In real
world situations, however, instant response to the environment that is implied in the
continuous-time setting is less likely to occur. A great number of applications in the literature
therefore use discrete-time models 3.

To implement dynamic programming numerically in a discrete framework, states and
control matrices with a finite number of elements have to be specified, which usually involves
discretization and truncation of a continuous variable. The discretization involves assigning
a particular value to represent all actual values of each variable which fall within a specified
state interval. Usually, the assigned values are nridpoints of the intervals. With some dynamic
programming formulations, the state transition are such that midpoints of the current state do
not necessarily match with the predefined state space variables in the next stage which can
result in numerical error in solutions, particularly if a coarse grid is used (Taylor and Novak
1992). Such error can be substantially reduced, if not completely eliminated, by using a value
obtained by interpolation of the states that bracket the actual value of the state variable. Since
linear interpolation is computationally inexpensive, it has been widely used to approximate
numerical solutions to dynamic optimization problems. Approximation errors are commonly

seen in this application while the accuracy of the approximation depends on the number of

 

3 For exceptions see Ho (1984) and Karp (1988). Ho examines dynamic joint

consumption and hedging problem and Karp considers joint production and hedging
problems.

13

$13185 ar.

approxir
executio'
|

T
manager _
plan that
approach
Maser.
Ihflunfic
under he
POaqu\
I996). to

Zering 1

 

states and the distance between them. State spaces with small intervals give more reliable
approximations. However, this also expands the size of the space and therefore makes the
execution more time consuming.

The method of dynamic programming was originally introduced into farm
management by Burt and Allison (1963) to consider the problem of choosing a crop rotation
plan that will maximize the present value of returns over a given planning horizon. The
approach was not widely used until more recently when computing capabilities significantly
increased, making numerical solutions to dynamic optimization problems less costly.
Dynamic programming has been adopted to solve joint production and marketing problems
under yield and price uncertainties (e. g., Taylor and Novak 1992; Tronstad 1991), to make
post-harvest grain storage and marketing decisions (e.g., Berg 1987; Fackler and Livingston
1996), to design optimal hedging strategies (e. g., Tronstad and Taylor 1991; Martinez and
Zering 1992; Lence et al. 1992 and 1994), and to derive optimal replacement and
management policies for animal feeding problems (e.g., Lambert 1989; Frasier and Pfeiffer
1994; Tronstad and Gum 1994).

Dynamic models allowing the decision-maker to revise his or her initial decision have
been shown to outperform static models. The dynamic models provide potential for high
returns and a more realistic portrayal of sales patterns in post-harvest marketing (e. g.,
Lambert and McCarl 1989). These models also generate a lower variance in after-tax income
and accumulate more wealth (e.g., Tronstad and Taylor 1991), and increase profits with little

change in uncertainty over fixed hedging strategy (e. g., Martinez and Zering 1991).

14

F
into pro
discusset
farm mar.
and prod
Strategieg
“Omalls
PTOductit
PTOductit
WCles an
pmdUCtit
lil'eSlock
"rider pre
alid mark
Wei-51'0” S

'5 Possib}

 

2.3 Decision-Making for Post-Harvest Grain Marketing

Farm management problems encountered by producers can generally be categorized
into production and marketing decisions. The mathematical programming techniques
discussed above have been widely used in analysis of these two major farm problems. Most
farm management literature related to animal production often takes both marketing decisions
and production decisions into account at the same time; for example, feeding and marketing
strategies for cull beef cows. This is because for animal production the timing of marketing
normally implies adjustments with respect to the duration of the production process. Hence,
production and marketing decisions essentially are simultaneously determined. The
production process for a storable commodity such as a crop is heavily locked into seasonal
cycles and major marketing activities usually start after the planting season is over. Crop
production and marketing decisions are therefore more separably than is the case for
livestock. Nevertheless, Taylor and Novak’s study (1992) concerning storable commodities
under pre-harvest or intra-year time frames, argues for the nonseparability of grain planting
and marketing decisions. They find that under a progressive income tax schedule, or risk
aversion scenarios, the optimal cropping strategy is substantially influenced by whether storge
is possible or not. This shows that production and marketing decisions for storable
commodities are still interdependent, mainly through output expectations during pre-harvest
periods and the cash flows of farm income. Karp (1988) also points out that knowledge that
the producer will be able to revise his positions in a futures market at subsequent stages
affects both his initial production and hedging decisions in the planting season.

A few farm marketing studies for storable commodities have focused exclusively on

15

marketini

Lambert
stated as
fumers u
that espe

Similari-

Utility fun

 

prOdUCers

marketing problems during post-harvest period (King and Lybecker 1983; Berg 1987;
Lambert and McCal 1989; Fackler and Livingston 1996). Post-harvest marketing can be
stated as a problem of storage-marketing management which is especially important for
farmers with on-farm storage capacity. Although some household studies have demonstrated
that especially with small farms, the optimal storage decision cannot be fully explained by
speculative behavior alone but also household food-security motives (e. g., Saha and Stroud
1994), post-harvest marketing strategies are typically designed to increase revenues through
storage until a time when prices have reached a more favorable level.

Lambert and McCarl (1989) proposed a sequential expected-utility-maximization
model for post-harvest marketing of wheat using four types of utility functions, including
linear, quadratic, negative-exponential and power firnctions. The data show that alternative
utility functional forms have little influence on optimal marketing plans. Their model is
informally tested in approximating actual sales patterns observed in a small group of Oregon
producers. A major weakness of this model is that the marketing strategies are pre-
determined and only allow the farmer to either sell or retain the entire stock in storage.

In order to reduce the size of the programming matrix, pre-selecting marketing
strategies has often been used (eg., Bailey and Richardson 1985). These pre-selected
strategies usually are those that have actually been used on the field and thus are more likely
to be accepted by farmers, although the solution may not be optimal. King and Lybecker
(1983) presented a model to select optimal post-harvest marketing strategies for Colorado
pinto bean farmers. Stochastic dominance is used to evaluate the alternative strategies. A
noteworthy feature of their approach is that farmers are allowed to choose partial sales out

of the storage and the marketing strategy is continually reevaluated to respond to the

16

 

incoming

I
E

Stain ma"
ll harvel
neutralitjl
l
costs. 8
SOlV'ed b.
the same

Other “to

”Wither

Scenario

 

 

incoming marketing information.

Berg (1987) developed a dynamic programming model to investigate post-harvest
grain marketing. Marketing decisions are assumed to be made at monthly intervals, beginning
at harvest time and ending before the subsequent harvest. The paper first assumes risk
neutrality with an objective of maximizing the expected value of the margin over storage
costs. By imposing a density function on market price at each stage, the decision problem is
solved by backward recursion. A break-even price for each corresponding stage that yields
the same objective function value for either selling or retaining the entire stock is derived. In
other words, the optimal strategy is to sell the entire stock if current market price is higher
than the respective break-even price, or to retain the entire stock otherwise. This risk neutral
scenario results in only two possible marketing actions.

Berg also expands his model to include risk aversion. The objective firnction is
specified to maximize a sequence of certainty equivalents of net revenue. This implies that
the decision maker has identical utility functions for each stage and the entire marketing
period’s utility is an additive firnction of a sequence of utility over the time interval. The
optimal marketing policy is represented by the monthly break-even prices, which indicate the
lowest market prices that would call for selling a certain proportion of the stock. For
example, results from a Monte-Carlo simulation show that, in August (initial stage), a market
price of 56. 18 DM/dt‘ is the lowest price at which it is optimal to sell 60 percent of the initial
stock, 56.21 DM/dt is the lowest price to sell 70 percent of his initial. Therefore, if the

current market price falls within 56.18 and 56.21 DM/dt, the optimal marketing volume is 60

 

‘ “DM” is deutsche mark and “dt” is hundred kilograms.

l7

 

 

Nes’erths

han'esti:

theoretic
’1
Optimal ;
increase
that an ir
and the \-
alllllldes
TUWey a
"’03! SIUI
Optima] n
considete

A
prObabih't

 

percent of the initial stock. This type of marketing policy is directly usable by producers.
Nevertheless, the numerical results of the optimal marketing policy is presented for only the
harvesting time (initial stage). No further discussion on strategies for the subsequent periods
is given. Additionally, while Berg succeeds in providing numerical evidence to support partial
sales as optimal marketing strategies for risk averse decision-makers, there is no formal
theoretical model to support these results

Berg also analyzes the impacts of interest rates and the degrees of risk aversion on the
optimal decision nrles. The results indicate that a higher interest rate, which implies an
increase in opportunity costs of inventory, will shorten the storage period. It is also found
that an increase in risk aversion leads to a decline in both the expected profits from storing,
and the variation of the returns, due to a more cautious marketing policy. The impacts of risk
attitudes on optimal marketing policies are also discussed in other research (e.g., Karp 1988;
Turvey and Baker 1989; Trostad 1991). With exceptions (e.g., Lambert and McCarl 1989),
most studies conclude that the decision maker’s risk preference has a significant effect on
optimal marketing behavior. This suggests decision-makers’ attitudes toward risk should be
considered in developing decision-making models.

Another limitation of Berg’s model is the assumption of statistically independent
probability density functions for the monthly market prices. Studies in time series of
commodity prices haveprovided a large body of evidence that prices are significantly positive
autocorrelated (e.g., Tomek and Robinson 1990; Williams and Wright 1991). This
information should be utilized to better forecast the price movements in post-harvest grain
markets.

Fackler and Livingston (1996) developed a stochastic dynamic model to solve a post-

18

han'est :

 

of stora.

and lliin

 

With the
mar ketin,
While 0p.
SequenCe
adillSt WI
Cutoff p r.
Prices are
Weekly 0‘
and dealt.
to “$an

\

p0llcjes E

harvest marketing-storage problem for a risk neutral producer. They view current sales out
of storage as irreversible due to a realistic assumption that the transactions costs are high
enough to preclude any replenishing of stocks once they have been sold. The dynamic
programming problem is shown to be a standard optimal stopping problem where a binary
choice between continuing and stopping holding the grain is derived. That is, the optimal
marketing rule reduces to a simple procedure: sell everything when the current price is higher
than a cutoff price; otherwise store everything into the next period. This decision rule is
consistent with that of Berg’s study in the risk neutral case, but may be overly simplified in
practice because grain producers with storage capacity may plan on spreading the price risk
over time by selling their storage partially throughout the marketing season.

Fackler and Livingston applied their simple sales rule to the cases of North Carolina
and Illinois soybeans in a continuous time frame. The optimal cutoff prices are compared
with the cutoff prices derived fi'om myopic and open-loop marketing strategies. The myopic
marketing policy considers only the prospects for sales in the current and the next period
while open-loop marketing strategy does not utilize incoming information and describes a
sequence of marketing actions as if the expected optimal controls in firture periods will not
adjust when more information becomes available. Empirical results show that the optimal
cutofl’ prices are always larger than the open-loop cutoff prices and the open-loop cutoff
prices are always larger than the myopic cutoff prices. The maximum difference between the
weekly optimal and open-loop cutoff prices occurs at the beginning of the marketing season
and declines monotonically as the end of the season approaches. This implies that the ability
to recognize that future optional controls can change makes a practical difference in optimal

policies especially over the early part of the marketing season. This is because at the

19

beginnlr.

 

consider
I
largest.

of storag

evaluate.

explicitl}
making.
decision

Tronstad

2.4 S

 

which is
ObleCtive
nor be kt
foundallo
mean‘Var
Stochlsllc
resmcllve

Mastic

beginning of the marketing season there are simply more future time periods in which to
consider selling, the time value of holding stocks beyond the next time period therefore is
largest. Results also indicate that the myopic cutoff prices greatly understate the unit value
of storage stocks since the option value to sell the stock in subsequent periods is not correctly
evaluated. The myopic strategy call for sales to be made too early.

The Berg, and Fackler and Livingston studies are two of the very few studies that
explicitly illustrate applicable commodity storage “decision rules” in practical decision-
making. A lot of farm management literatures suggest that their findings can be used to aid
decision makers but do not explain how to actualize the models in practice (e.g., Bauer 1991;

Tronstad 1991).

2.4 Summary

Many farm management problems are formulated in a multistage decision process
which is characterizediby the task of selecting a sequence of decisions to meet a certain
objective. In this situation, current actions affect firture opportunities and the outcomes will
not be known with certainty. Maximizing expected utility is the predominant theoretical
foundation for risk analysis. Alternatives include approximations of expected utility (e.g.,
mean-variance models and MOTAD) and methods based on distributions of returns (e.g.,
stochastic dominance analysis and safety-first models). These alternatives suffer from
restrictive constrains and intensive data requirements, respectively. Discrete sequential

stochastic models and dynamic programming are two classes of mathematical programming

20

 

 

 

techniqu-
manager

compare

 

computa

to explic:
decision
marketin
PTDgT arm
T
Storageq
The Simp
selling 51

for fan-”e

 

techniques which have been adopted to solve many multistage decision processes in farm
management. Dynamic programming is based on Bellman’s Principle of Optimality which,
compared to discrete sequential stochastic models, has an advantage of a large saving in
computational effort and computer storage.

Direct expected-utility-maximization models in a dynamic framework, which are able
to explicitly incorporate decision-makers’ risk attitudes and capture the nature of sequential
decision process under uncertainty, seem to best represent the producer’s post-harvest
marketing decision problem. The increasing power and availability of mathematical
programming software are making the numerical solution of the model easier to achieve.

The previous discussion also highlights the need for decision rules for post-harvest
storage-marketing management, which can be directly used in decision support programs.
The simple sales rule to retain or market the entire stock may be overly simplified in practice.
Selling storage partially throughout the marketing period may spread the price risk over time

for farmers who exhibit risk aversion.

21

marketin

 

theoretic;

marketin

3.1 '1

A};

CHAPTER III

THE DYNAMIC DECISION MODEL

This chapter studies a normative model of the grain producer’s optimal post-harvest
marketing strategy. A generalized optimal marketing decision rule will be derived from the
theoretical model to provide the theoretical support that a partial sale may be an optimal

marketing strategy when risk aversion is incorporated in the decision model.

3.1 Theoretical Model

The producer with on-farm storage facility is intending to store the harvest to profit
fiom an expected increase in the cash price during the marketing year. Periodically through
the storage season, the marketing decisions are re-evaluated based on updated market
information. We model the producer’s marketing problem as a stochastic, dynamic,
optimization problem in a discrete-time framework. The following assumptions formalize the

structure of the problem:

A1: A risk-averse producer seeks to maximize his expected utility of terminal wealth.

Terminal wealth is defined as the future value of initial wealth plus the compounded

22

 

rV—fi—

 

 

 

 

A4:

cash flows arising up to the terminal date. The initial stock of harvested production,
so, and initial wealth, w0 , are given. The cash flows are subject to only one source
of uncertainty: cash price at which the grain will be sold.

The marketing season begins at the harvest and ends before next harvest. The
planning horizon is divided into T decision periods at identical intervals. At the
beginning of each decision period, the producer knows his wealth and stock levels,
as well as the market price, and chooses an amount of grain to sell in the spot market.
Producers pay storage costs for carrying stocks over into the next period.

The producer receives a constant interest rate on his wealth. For simplicity, we
assume that the producer does not engage in other marketing activities nor invest in
any other risky assets.

High transaction costs prevent the producer from replenishing storage once it has
been sold, and short sales are not allowed. Thus, sales can never be negative or larger
than the available storage.

Stochastic state variables follow a Markov process. That is, the probability
distribution for all future prices depends only on the current value of the state

variables.

A1 is a standard decision criterion used in the multi-period decision models.

Maximization of additive expected utility of cash flows (incomes or profits) is another

alternative which is also popular in the literature (e. g. Berg 1987 ; Martinez and Zering 1992).

Maximization of expected utility of terminal wealth implicitly assumes the existence of

efi‘ective and accessible capital markets which allow the decision maker to borrow and lend

23

 

COGS
lITt‘i'c
Pdeu

and Fa

Valiablc

”bleeds

 

 

with negligible transaction costs. Thus the stability of cash flows from the marketing
activities is not a great concern to the producer but the risk surrounding his terminal wealth
is of concern. This specification of the producer’s decision objective, therefore, applies to
farmers in developed countries better where capital markets are more complete than those in
developing countries. A3 allows us to abstract from interest rate risk in order to focus on
cash sale marketing strategies only. It is not unreasonable to expect interest rates to be fairly
constant over a marketing season. Under A4 sales out of storage can be viewed as
irreversible investments: stocks can not be recovered once they have been sold. For many
producers it is a realistic assumption which has been adOpted by King and Lybecker (1983)
and Fackler and Livingston (1996).

The structure of the problem is summarized by the following model specification and

variable definitions:
max E0 U(wT)
“1.1.7.3
subjective to
Wm = (“OM + 11.61, — cl(S,-€1,)] (3.1)
SM = s' - ql (3.2)
0 s q, s s, (3.3)
13... ~f.(p,.,lp,,z,..) (3.4)
s0 and wo are given
where
U(-) = atwice differentiable utility function characterized by U ’ >0 and U ” < 0;

24

E

etitration

that S!

r
mggeSts

(3'1). We

 

where p \

\

 

 

P:

‘1:

monetary wealth at the beginning of period t, a deterministic state variable; wT is
the terminal wealth;

the cany-in stock level at period t; a deterministic state variable;

the spot price at period t, a stochastic state variable;

quantity sold at period t, the control variable;

a constant per unit storage rate;

a constant interest rate;

a vector of deterministic variables, representing trends and seasonality;

a probability density function for p,+1 conditional on p, and 2“,.

Equations (3.1) and (3.2), respectively, are wealth and storage state transition

equations allowing no depreciation of stocks in storage. From (3.2) it is clear

thats

and q, are simultaneously determined when either of them is chosen, which

suggests that the optimal solutions will not change if the control variable, q,, is replaced by

the carry-over stock, SM Equations (3.3) and (3.4) follow A4 and A5 respectively. Using

(3.1), we can express terminal wealth corresponding to each decision period as

“77 = (l+r)[wT_1+pT_qu-l- “(ST-r “qr-1)]

T-l
= (1+r)“rw,+p.q.-a(s,-q.)1 + 2 (1+r)T”[P',q,-a(s,-q.)1 (3.5)

r=r+1

where t= 0,1,....T-2.

25

3.21

optimal
over sto
termina.‘

canbec"

subled t

for Perio

Where .8:

 

3.2 Formulation of Optimal Marketing Strategies

This section studies the derivation of the optimal marketing decision rule. A general

optimal decision rule will be proposed at the end of this section. The series of optimal cany-

over stocks are obtained by using the dynamic programming algorithm and starting at the

terminal period and working backwards to the initial period. The solutions to the problem

can be derived by solving recursively the Lagrangian functions

2.9,(wpsppt) = max E,U(w7.) + il,(5,'S,.1)

’rol

t= T-l, T-2,....,21, 0

subject to s; 2 0 and

T-i
WT = (1+r)T"[W,+P,(S,-S,.i)'08,.11 + Z (1+r)T"[fi,(S,-S,.,)-CIS,.1]

r=r+1

for period t and u; is the multiplier for s; — SM 2 0.

With one period to go in the marketing year, we solve the problem

9T-1(WT-..ST_1.PT_1) = max ET—1U(WT) + pT-I(ST-l—ST)
‘r

(3.6)

(3.7)

(3.8)

where ST 2 O and WT = (1+r)[w1H +pT_;(sT_l-sT)—asr]. The first-order necessary

conditions at period T -1 are

26

 

where

wealth

 

underl}

WSllil'e
CORdltio

lefi in SII

 

This 0m

prodUCer

 

 

 

59“

 

 

= -U’(wT';T-1)(l+r)(pr_l+u) - “m s 0, (3.9.1)
as,
s; 2 0. s; t—U’(w;;T—1)(1+r)<p7-.+a) — 111-1] = 0; (3.9.2)
7' = Sr-r - ST 2 0, “r-r 2 0, uT_1(sT_l - ST) = 0; (3.9.3)
duT_l

where U ’(w; ;T -1) represents the first derivative of utility function with respect to terminal
wealth evaluated at the optimum of period T-l, and * denotes the optimal value of the
underlying variable.

Note that (3.9.1) is always non-positive since U ’('), 1+r, a, and Pr-r are all
positive, and pm is non-negative. Therefore, to avoid violating the Kuhn-Tucker
conditions 3; must equal zero. Consequently, decision at T -1 is to sell everything that is

left in storage and the Optimal temrinal wealth reduces to

w; = (1+r)[wT_l + Pr—rsr-rl- (3.10)

This occurs because the planning horizon ends at the beginning of period T, so that carry-
over stock after period T-l has no value left. The producer hence will sell all his storage
available at period T -1 regardless of price outcomes. The result is reasonable when the
producer has a capacity constraint and hence has to sell all his storage before the next harvest.

Moving back to the previous period at period T-2, we solve the problem

E97220974,Sr-szT-z) = max Er-2U(‘;’7:;T'l) + “r-2(Sr-3’Sr-2) (3'11)

SH

27

where

terminal

The firs

all 5‘

where l
optimun

"1&th r

AceOrdi r

 

where U09; ; T - l) is the utility function evaluated at the optimum of period T -1; optimal

terminal wealth revised from (3. 10) becomes:

M7; = (l +r)2 [wT_2 +pT_2(sT_2-ST_1) -asT_l] + (1 ”)fir-iSr-r (3.12)

The first-order necessary conditions for (3.1 1) are

agT-Z

(9.3'T_l

 

= ET_2U’(WT';T-2)[-(l+r)2(pT_2+a)+(l+r)p'T_1] - “r-2 5 0, (3.13.1)

s;_, 2 o, s;_,{E,_,U’(w,t;T-2)[-(i+r)2(pT_,+a)+(1+r)p'T_;] - p,_,}=o; (3.13.2)

 

= ST 2 - 31:1 2 0, “r-2 2 0, rrT_2(sT_2 - 3311) = 0; (3.13.3)

where U ’05; ;T -2) represents the first derivative of utility function evaluated at the
optimum of period T-2. Note that the solution to the first-order conditions is a unique
maximum because the objective function is strictly concave and the constraint set is convex.

To derive the optimal solution, let

K = E,_,U’(w;;T-2) [ —(1+r)2(p,.,2 +a) + (r +r),3,._l ]. (3.14)

According to the first-order Kuhn-Tucker conditions, there are three possible optimal

outcomes:

28

With th

optimal

then s."

then Sr-

011m;

 

reSpecs,

 

(l) K < 0 5 117-2 and 37:, = 0;
(2) K = ‘11-: > 0 and 57:] = 37-2;

(3) K = l‘r-z = 0 and 31:: equals the amount that solves K =0.

With the introduction of the discount factor, B, which is equal to the inverse of (Hz), the

optimal marketing decision rule at period T -2 can be described in the following fashion‘:

 

 

If

ET_2U’(w”T'; T—1)pT_2 2 ET_2U’(M7T'; T-l) [31‘5“ -a] ls,,=o , (3.15.1)
then 37:, = 0, which says to sell everything;
If

Er-z U’(M7T';T- 1)pT_2 5 151-2 U’(WT';T- l) [B1314 -a] l’r-f‘r-z , (3.15.2)
then s;., = 3122’ which says to store everything.
Otherwise, 3T1, equals the amount that makes

ET_2 U’(w"T';T-1)pT_2 = Er2 U’(WT';T- 1) [3131—1 —a] , (3.15.3)
which is greater than zero but less than 3”, implying a proportional sale;
' Notice that I: :0 and S :5 denote “evaluated at ST-l = 0 and 57-1 = ST_2

. T-l T-l T-2

respectively.”

29

price, ,
today!
Thisbe
decides
which is
future I
altemat
is 5.0}.
the futu:

Current 1

imntedia

leVels‘ 1h

 

 

ir we examine (3. 15. 1)-(3. 15.3), we see that, at period T -2, the current market
price, Pr-r represents the benefit of immediate sale of a unit of storage. Rather than sell it
today, the producer always has an option to wait for one more period and sell it at period T -1 .
This benefit of sale in the firture dates captures the option value of waiting. Ifthe producer
decides to sell the given unit of grain today, the forgone option value is an opportunity cost,
which is measured by the highest expected market value of the stocks when it is sold at some
future time, subtracting storage costs and discounting to the present. Here the only
alternative to the immediate sale is to sell at period T -1, so the option value of waiting
is [3’7“ -u dollars per unit. The opportunity cost is highly sensitive to the uncertainty over
the future value of the stocks so that the expectation on future prices has a large impact on
current decision-making.

The marginal utility in (3.15) can be viewed as the “weights” on the benefit of
immediate sale and the option value of waiting. The weights are determined by the state
levels, the magnitude of the imbedded parameters such as the storage cost, and the functional
form of utility which represents the producer’s risk attitude. Note that the weights are not
constant but varying with the control levels. The optimal marketing decision rule hence
suggests that if the weighted benefit of immediate sale is high enough to compensate the loss
of weighted option value, it is optimal to sell the underlying unit of grain. Moreover, the
producer should keep selling grain until the next unit of sale has equal weighted benefit of
immediate sale and weighted option value of waiting (equation 3.15.3), or until the whole
stock is sold out (equation 3. 15. 1). In other words, if the current market price is high enough
or the expected future market price is low enough, then the weighted benefit of immediate

sale is always greater than the weighted option value of waiting for all levels of storage

30

 

availab'. :

   

saleisn.

risen Li.

weighte
“hole 5
expects
always 2

the pro:

Constant
n'Sh-neu
Storage
disco“.
consider
and F35
Mfifls

oppom,‘

salts. )

 

 

 

available. Thus, the producer should sell everything. Conversely, if the weighted benefit of
sale is not high enough to offset the weighted loss of option value, it is optimal to store the
given unit of grain until next period. He should further keep storing the grain until the
weighted benefit of immediate sale equals the weighted option value of waiting, or until the
whole stock is stored. In other words, if the current market price is low enough or the
expected future market price is high enough, then the weighted benefit of immediate sale is
always less than the weighted option value of waiting for all levels of storage available. Thus,
the producer should store everything (equation 3.15.2).

It appears that if the producer is risk-neutral, the “weights” in (3. 15) will become
constant, and thus the optimal solution will always occur at a boundary. This means that the
risk-neutral producer’s optimal marketing strategy at period T-2 is to either sell the entire
storage or sell nothing, depending on the performances of current cash price and the
discounted expected future cash price (minus storage costs). Only two possible outcomes are
considered in this case, which is consistent to the results of Berg’s risk-neutral model (1987)
and Fackler and Livingston’s profit-maximization model (1996). We also see in (3.15) that
higher storage cost reduces the benefit of storage and higher interest rate implies a higher
opportunity cost of waiting. Intuitively, both are expected to encourage more immediate
sales. Nevertheless, the changes in these two parameters may influence the producer’s
decision in an opposite way through the changes of weights. The impacts will be more
complicated and can only be illustrated by numerical simulation.

Substituting 57:, into equation (3. 12) yields optimal terminal wealth at period T -2

"7; = (1 +r)2 [WT—2 +p1'-2 (ST—2-57:1) - (13711] T (1+r)fiT—|ST.-I° (3- I6)

31

Note th

If 5.7.4

functior
0f peric

illustrat:

Casel

where L

 

Note that if s;_, = 0, terminal wealth in (3. 16) reduces to

W; = (“02097—2 +171-237-2)-

If s;_, = 37-2, terminal wealth in (3.16) reduces to

117; = (1+r)2wT_2 + [ -(1+r)2a + (l+r)p~T_l]sT_2.

Now we move one period further backward to period T -3 and solve the corresponding

function in (3.6). Since the optimal solution at period T -3 is derived from the optimal result

of period T-2 which is discrete, the first-order conditions at period T -3 should also be

illustrated in each of the possible separate situations. There are three cases to consider.

Case 1: If s;_, = 0 at period T -2, the first-order conditions at period T —3 are

391-3

331—2

 

s;_2 2 0, SH {Er-3 U’ 117;; T-3)[-(1 +r)3 (pT_3+ a) + (1 +r)2p'T_2] - 111-3} = 0;

 

= ST'J‘ '- 57:2 2 O, “T'3 2 O, ”T‘3(ST‘3 — ST__2) = 0;

~ It _ 3 a a 2 ~ ' v
where WT - (1+r) [wT_3 +PT-3(ST_3‘Sr-2)'asr-2l + (1+r) P14574-

32

= ET_3U’(15;;T—3)[-(1+r)3(pT_3+a) +(1+r)2p"T_2] - 111.3 s 0, (3.19.1)

there ‘

are sim.

Case}

 

 

 

 

 

Case 2: If 5,1, = 3m at period T -2, the first-order condition at period T -3 is

as - . ~
T 3 = Em U’(1TIT;T-3)[‘(l+r)3(PT-3+<1)'(1+020 +(1+r)pml

- it“, s 0, (3.20)

 

as”

Where W; = (1 Tr)3[wT—3 +pT-3(ST—3—ST.-2) - (15712] + [—(l ”920' +(1+r)fiT—1]S;-2'
Note that only the very first part of the first-order conditions is displayed. The rest

are similar to those in case 1 and thus are not elaborated here or in the following discussion.

Case 3: If 0 < 37:, < 57—: at period T-2, the first-order condition at period T -3 is

 

 

 

691-3 = E U/(W"T-3)[-(l+r)3(P +11) +(l+r)2p~ ]-p S 0 (3 2l)2
ST-2 72.3 T s T-3 T-Z 7'3 ’ '
2 as
a 7‘3 = ET-3U/(Wfif'3)l‘(1”NWT-3+“) + (1”)2fiT—2]

57-2

/ ~* 2 ~ ‘ 657:1

+ ET_,U(wT;T-3)[(1+r) (pT_2+a)’(1+r)PT_1laT“ - 11m
r-2
= E,_3U’w;;T-3)[-(1+r)3(p,._,+a) +(1+r)’157--2] - up.

where

EH, (1’05; ;T-3) [(1 ")2 (fin-<1) - (1 "H7741

(1+r)2ET—3 U/(W;;T-3)[fi7._2 - (B5721 '11)]
= 0
according to (3. 15.3) by envelop theory.

33

produce

Optimal

Oss.

then s:

the" s.’

 

 

 

where w; = (1+r)’[wT-3 +p.-.<s.-. -s;..)- «min + (1 +r)2 [13730112 -s;.i) - asrin

"’ (1 +’)fir-15;-r-

Note that although (3.19.1) and (3.21) have the same form, the it); inside the
marginal utility function are different. Summarized fi'om the first-order conditions, the
optimal marketing decision rule at period T -3 has two different forms depending on the
producer’s expectations on the optimal marketing strategy in the next period.

First, if, given the information available at period T -3, the producer expects his
optimal marketing strategy at period T-Z to be to sell at least some storage, i.e.

0 5 37:, < ST_2, then the optimal marketing decision rule at period T -3 is as follows.

If

ET-3U/(W~;;T'2)PT-3 >- 157-3 U/(W;;T'2) [315mm] l,T_2.o. (33-2-1)
then s;_2 = 0, implying sell everything.
If

1v:,_;t/’(u7;;T—2)p,_3 s ET_3U’(117T';T—2)[B[3T_2-a] hm”, (3.22.2)
then 3,3,2 = 37-3, implying store everything.
Otherwise, 37:2 equals the amount that makes

EH U’(113;;T-2)p,_3 = 14:H U’(w;;T-2) [BfiT_2-a] , (3.22.3)

34

attic

hand.

then .1

the“ s.

Olhem

 

 

 

Whjch

 

 

lhe pT0.'

 

which is greater than zero but less than ST_3, implying a partial sale.
Secondly, given the information available at period T -3, if the producer, on the other

hand, expects his optimal marketing strategy at period T -2 to be to store everything,

i.e. 57:, = 51-2, then the optimal marketing decision rule at period T -3 becomes as follows.

ET-3U/(w;r T—2)pT-3 2 ET-3 U,(W7ta T-Z) [BzfiT-l — Ba — a] '37-}:0 3 (3231)

then 3712 = 0, implying sell everything.

If

19:,_3U’(1t3;,T—2)p,_3 s E,_;U’(w;,T—2) [1321574431141] (Ham, (3.23.2)

then s;_2 = ST_3, implying store everything.

Otherwise, 37:2 equals the amount that makes

ET_3U’(1i'IT',T—2)pT_3 = Er-3 U’(117;,T--2)[sz5'7_l - Ba - a] , (3.23.3)
which is greater than zero but less than ST_3, implying a partial sale.

Equations (3 .22) and (3.23) say that, for every unit of storage available at period T -3,

the producer can choose to sell it today and earn an instant value equal to the current price,

35

or sell it at future dates and capture the option value of waiting. Here the option value of
waiting is an opportunity cost of immediate sale measured by the highest present value of the
stocks being sold at period T -2 or period T - 1. This, in turn, is determined by the expectation
of the optimal marketing strategies in the coming decision periods before the terminal date.
That is, given the information available at period T -3, if the producer expects his optimal
marketing strategy at period T -2 to be to sell some storage, as opposed to store everything,
then the option value of waiting is the present value of cash price at period T -2 minus the
storage rate, ( 5171-2 -u dollars per unit). On the other hand, if the producer expects his
optimal marketing strategy at period T-2 to be to store everything and sell it later at period
T -1, then the option value of waiting for per unit of storage is the expected present value of
cash price at period T -1 minus the discounted storage cost over two periods,
( 013'“ - Ba - 11 dollars per unit).

The cash price at period T -3 is the value of immediate sale for per unit of grain. The
marginal utility, again, is a “weight” which is varying with the control levels. Similarly, the
optimal marketing decision rule suggests that the producer should sell until the weighted
benefit of immediate sale is equal to the weighted option value of waiting for the next unit of
grain. However, if the current cash price is high enough, or the expected future market price
is low enough then the benefit of immediate sale is always greater than the option value of
waiting for all levels of available storage. In this case the producer should sell everything
since it is not worthwhile to wait. On the other hand, if the current cash price is low enough
or the expected firture prices are high enough that the benefits of immediate sale is always less
than the option value of waiting for all levels of available storage, then the producer should

store everything.

36

Comparing (3.22) with (3.23), we can see that, in addition to other factors such as
interest rate and storage rate, the producer’s major concern in decision-making has been
placed on the current market price and the very next cash price at which period he anticipates
to sell his stocks. For example, in the case that at period T -3 the producer expects his next
sale to occur at period T -1, it seems that the corresponding optimal decision rule in (3.13) is
based on the two prices: pT_3 and fir-r- Notice, however, that this does not imply
that Pr-z plays no role in optimal decision-making. Instead, the information has been
incorporated to form the expectation of the optimal marketing strategies at future decision
periods, which in turn determines the option value of waiting.

From the previous discussion, a generalized optimal marketing decision rule can be
derived. Given the information available at period t, if the producer expects the very next
period at which he will sell his storage be period j (i.e. he will store everything available after

current period until period j—l), the optimal marketing decision rule at t is:

If

;_.
E,U’vi'r;;t)p; 2 E,U’(1iz;;t)[B"/p"j-a:[3"'] lsmw, (3.24.1)
i=1

then s,:1 = 0, implying sell everything.

If

,_.
E,U’w;;1)p, s E,U’w;;1)[pr-I,3j-a§p“](W (3.24.2)

then 3,:1 = 3,, implying store everything.

37

Otherwise, 3;, equals the amount that makes

;_.
E,U’(ri’r;;t)p; = E,U’W;;t)[B"/13j-u:[3’], (3.24.3)
i=1

which is greater than zero but less than 5,, implying a partial sale.

A general form of the optimal final wealth becomes:

'-1
w; = (1+r)T"rw.+p,(s.-s.:.)-as.:.1 - us.:. it (1 v)“

i=t+l

“I

1' (1+r)T_'-lfi)(sri-Sr:r)-mr:r] - (3.25)

n

t:

k

The generalized optimal marketing decision rule suggests a two-step decision-making
process. At the beginning of period 1, using the available information, the producer first
makes an expectation on the optimal marketing strategies of future periods before the terminal
date. According to the expected future optimal marketing strategies, he anticipates his next
sale occurring at period j. That is, he expects to store all the stocks he will have after the
current period until period j-l, and sell either all or only a portion of the stocks at period j.

The producer knows that for every unit of storage currently available, the benefit of
immediate sale is the current market price, [2,. If he decides to store the given unit of grain,
he will earn an option value of waiting which is the discounted cash price subtracting the
discounted storage costs between period t and period j, 5'71)“; — off [3’ " . The option value

i=1
of waiting is an opportunity cost of immediate sales and is greatly sensitive to the future
prices. The producer then calculates the expected weighted benefit of immediate sale:

E, U'(1i'r;;t)pt and the expected weighted option value of waiting:

38

E, U’ W; ;t) [fijB‘ '1 - (1% [3“] for a given unit of sale. He should choose the sale level that
i=1

has equal weighted benefit of immediate sales and weighted option value of waiting.
However, if the weighted benefit of immediate sales is always larger than the weighted option
value of waiting for all levels of stocks available, he should sell everything in storage. If, on
the other hand, the weighted benefit of immediate sale is always less than the weighted option
value of waiting for all levels of stocks available, he then should store everything.

From the decision rule it is clear that current market price, the expectations on future
cash prices, interest rates and storage costs, as well as the individual’s wealth, stock level,
and risk attitudes, all may influence the timing of optimal marketing in different ways. These

impacts will be illustrated systematically in a later chapter.

3.3 Summary

A stochastic dynamic programming decision model in discrete-time is developed to
maximize the expected utility of the decision-maker’s final wealth from the post-harvest
marketing-storage management problem, under the assumption that high transaction costs
preclude any replenishing of stocks once they have been sold. Under this assumption, sales
out of storage can be viewed as an irreversible investment which makes this storage
management problem the equivalent of exercising a financial option (F ackler and Livingston
1996). The concept is rooted in real options theory (Dixit and Pindyck 1994). The essential
contribution of this study is to recognize the opportunity cost of the option to invest is a

significant component of the marketing decision.

39

A generalized marketing decision policy is derived from the normative model. The
optimal marketing level is to sell until the weighted marginal return from an immediate sale
is equal to the weighted option value of the given unit stock, whereby the marginal option
value is the discounted conditional expected unit value of the stocks if it is marketed in a
future period, while taking into account that the optimal control can change in future periods.
This provides the theoretical support that a partial sale of storage, as opposed to sell-
everything-or-nothing, may be an optimal marketing strategy when risk aversion is

incorporated into the decision model.

40

CHAPTER IV

THE PROBABILITY-BASED PRICE FORECAST MODEL

This chapter describes econometric estimation of the stochastic structure ofMichigan
corn prices for use as an input into modeling optimal storage decisions. In post-harvest grain
marketing, the source of uncertainty is mostly from price changes. Potential benefits from
post-harvest marketing, therefore, highly depend on the ability to understand price movements
in the commodity being stored. Because post-harvest storage exposes farmers to price risk,
and we assume farmers are risk averse, single point price forecasts are not sufficient and we
need to characterize the entire probability distribution of cash price movement.

In stochastic dynamic programming models, the stochastic state of the process is
usually controlled by a transition probability density function. To simplify the analysis, a great
deal of work assumes a finite number of stochastic states that follow a first-order Markovian

relationship in a general form as:

y: = 0' +Byp-1 + 78p] + p: (41)

where y; is the stochastic state variable; 51-1 denotes other explanatory variable(s) that are
deterministic or known at time H; u, is a disturbance term usually assumed to be normally
distributed with zero mean and constant variance (e. g., Weersink and Stauber 1988; Tronstad

and Taylor 1991; Taylor and Novak 1992), or have heteroskedastic variance (e. g., Martinez

41

 

and Zering 1992; Fackler and Livingston 1996); and a, B and y are parameters. The
conditional distribution of y, , given y“l , can be directly generated from (4.1). In the
literature, 51-1 has often been defined as seasonal component, time trend, and/or time to
maturity for prices of futures contracts and the basis (e.g., Tronstad 1991; Yager et a1. 1980;
Tronstad and Gum 1994). Other economic indicators, such as a binary dummy variable to
reflect export market changes or period of harvesting (e. g., Tronstad and Taylor 1991 ; Taylor
and Novak 1992) are also sometimes used. Since 51-1 is time-dependent, the resultant
transition probabilities, in turn, change across stages. If 51-1 were not included in the model
then the transition probabilities would be constant across stages (e. g., Karp 1987; Kaiser and
Apland 1989; Schroeder and Featherstone 1990). Parameters o. and [3 in some studies are
allowed to change over time (e.g., Yager et al. 1980) indicative of time-varying transition
probabilities as well.

Conceptually, price is a continuous variable but it will be discretized into finite states
for mathematical convenience in the dynamic programming model used here. The stochastic
element in the discrete-time model is featured by a transition probability space, using
conditional probabilities to describe the risky environment of concern. In the next section of
this chapter the data are described and preliminary analysis undertaken. Then the econometric
results are discussed, followed by details of the procedure used to turn the econometric result
into Markovian transition probability matrices. The chapter will conclude with an evaluation
of this probability-based price forecast model and the resultant transition probabilities, by

comparing price movements simulated by the model to the historical pattern.

42

i1 llat

Ca
closing p1
Odober l
10%,

11
before a
translon
outliers

trend in

exam:
Spike in
HOWey
Charm
“Onstat
general

0fmetl

4.1 Data Analysis

Cash prices for corn at Saginaw, Michigan, are used in this study. The data are daily
closing prices sampled on every Wednesday. The time period covers the first week in
October of 1975 to the last week in September of 1996. Total number of observations is
1096.

In many asset pricing studies it is common to take the natural logarithm of the data
before attempting to model price movements. An advantage of the logarithmic
transformation is that it tends to eliminate trends in variance, and reduces the impacts of
outliers. Figure 4-1 illustrates the behavior of the historical logarithmic prices. No systematic
trend in either levels or variances are obvious from the graph.

The sample autocorrelation functions and partial autocorrelation functions are
examined for the log prices. The decaying pattern of autocorrelations and the single large
spike in the partial autocorrelations at lag 1 suggest a first order autoregressive model, AR( 1).
However, the sample autocorrelations die out very slowly, which is indicative of a large
characteristic root in the AR(l) model. This implies that the underlying series is likely to be
nonstationary and, therefore, conventional econometric methods and statistical inferences
generally are not applicable. This unit root hypothesis can be formally tested using a variety

of methods. The details of the unit root tests will be discussed in the next section.

43

 

39.: dam - :2 .80 £83285 68 5c coca 8 assumes 2. 25mm

30>

coe— maz veg nee N22 30— coo. owz nag nag one. 32 vwfi $0- Nwa $0— owa one. 2.2 :b— 3.2 2.2
od

 

.1 v.9

ad

 

 

 

 

 

 

m.—

44

Dir

4.1.1 Unit Root Tests

Among statistics for testing the existence of unit roots in a time series, the augmented
Dickey-Fuller and the Phillips-Perron tests are perhaps the most popular. Both methods are
intended to deal with the potential presence of serially correlated residuals. While the
augmented Dickey-Fuller method attempts to account for serial correlation explicitly in the
regression models, the Phillips-Perron method adjusts the test statistics but allows the
disturbances of the regression models in the procedure to be weakly dependent and
heterogeneously distributed. The simplicity and broad applicability have led to widespread
use of the Phillips-Perron method. However, it has been well documented in the literature
that Phillips-Perron suffers from severe size distortions when there are negative moving-
average errors (e.g., Phillips and Perron 1987; DeJong et al. 1992). In this case, the
augmented Dickey-Fuller tests are likely to be more useful in practice (e. g., DeJong et al.
1992; Perron and Ng 1996). To increase the robustness of the tests, this study adopts both

approaches for testing the hypothesis of a unit root.

Phillingerrgn Unit-Root Tests

According to the testing strategy proposed by Perron (1988), we first utilize the

statistics Z(r'i) and Z(t&) derived from the regression model

P = p + [3(1-T/2) + (1PM +17

1

(4.2)

l

to assess whether there is evidence for rejecting the null hypothesis of a unit root. Here T is

the sample size and P, is the logarithm of price level (p,). Formula for the test statistics are

45

available in Perron (1988). The test statistics may be sensitive to the value of the truncated
lag parameter I in the Newey-West estimator, and sometimes they are critical to the test
results. Choosing a specific value of I, however, is an empirical matter. Perron (1988)
suggested that, when the residuals in (4.2) are negatively autocorrelated (e. g., MA(1) with
a negative parameter), a large value of I will minimize the bias and mean square error. The
Newey-West adjustment with 10 and 20 lags are considered here.

The upper portion of Table 4-1 presents the test statistics and their null hypothesis for
the cash corn data. It shows that none of Z(&) and Z(t&) are significant at any conventional
significant levels, implying no evidence being found to reject the hypothesis of a unit root,
except 2(6) at the 10% level for both levels of I.

Proceeding further we need to verify that regression (4.2) is an adequate
representation of the data generating process. We use the statistics, Z(t;.,) and Z05), to test
the existence of a drift ( H0: f1 = 0 ) and the time trend ( Ho: [3: O ), respectively, and the
statistics Z(<1>,) and 2(4),) to test the joint hypothesis of (a, (‘3, .11 ) = ( 1,0,0) and (51,6) =
(l, 0). All statistics for various levels of I are insignificant at standard significance levels.
These results suggests the absence of a time trend for the series and further verifies a more

powerful set of test statistics can be obtained fi'om the regression model:

P, = 11* + a’PH + u,' (4.3)

The test is then carried out via the statistics, Z(a*), Z(t,,.), Z(t,,.), and Z(<I>,) with the results
presented in the second portion of Table 4-1. Z(a*) and Z(t,,.) are used to test the null
hypothesis of a." = 1. Both are insignificant at the 1% but not at the 5% and 10% levels. To

firrther verify the results are reliable, in the sense that the specification in regression (4.3) is

46

Table 4-1 Results from Phillips-Perron Unit Root Tests

 

 

 

Statistics Estimates Critical Values ' Null Hypothesis
Cash Prices
[=10 I=20 10% 5% 1%

 

 

01:09882 8=2><10'6 p =o.0102

 

 

 

2(&) -18.630 .20.550 -l8.3 -21.8 -295 6:1

20,-.) -3059 -3212 -312 -341 -3.96 (i=1

20,-.) 1.184 0.871 2.72 3.08 3.71 11:0

20,) 0.739 0.741 2.38 2.78 3.46 B=0

2(4),) 4.753 5.178 5.34 6.25 8.27 a=1andp=0

Z(<I>2) 3.170 3.453 4.03 4.68 6.09 a=1andii=ri=0
01* = 0.9886 B“ = 9.854x10-3

Z(a*) -18.115 -20.065 -113 -141 -207 a*=l

20,.) -2995 -3135 -257 -2.86 -343 a*=1

20,.) 1.127 0.776 2.16 2.52 3.18 tr*=O

2(o,) 4.491 4.978 3.78 4.59 6.43 0t*=1and u*=0
8:0.9993

2(a) -1.106 -1.198 -5.7 -8.1 -13.8 azl

5’11) -0710 -0741 -1.62 -195 -2.58 azl

 

' Critical Values are collected fiom Hamilton (1994) and Enders (1995).

47

appropriate, we test for the hypothesis of 11* = O and the joint hypothesis of 01* = 1 and 11*
= O by calculating statistics Z(t,,.) and Z(<I>,), respectively. From Table 4-1, Z(t,,.) are
insignificant for both values of I at all significant levels. Z(<I>,) for 1= 10 is insignificant at the
1% and 5% levels but not the 10% level, while for 1= 20 it is insignificant at the 1% level but
not the 5% and 10% levels. The evidence tends to suggest a regression without random drift

as in equation (4.4) is a more appropriate representation of the data generating process:

P, = 61PM + a, (4.4)
2(6.) and Z(t&) are calculated to test hypothesis of f1 = l and the results are

reported in the lower portion of Table 4-1. From Table 4-1, no evidence has been found to

reject the null hypothesis of unit root. Overall, these results strongly support the hypothesis

that the cash corn price series has a unit root.

A Di k -F ller Tests

The statistics of the augmented Dickey-Fuller tests, and the associated hypothesis and
their critical values are summarized in Table 4-2. The null hypothesis of a unit root can be

tested using the 1:, statistic from the regression:

AP, = a0 + bot + 'yP,_l + i: a, APH. + d, cos(21tw,/ 52) + dzsin(21rw,/52) + a, (4.5)

i=1

where the cosine and sine firnctions are used to accommodate seasonal effects. The w, variable

denotes the number of weeks after October in a given crop year. Formula for the 1, statistic

48

Table 4-2 Results fi'om Augmented Dick-Fuller Unit Root Tests

 

Statistics Estimates Critical Values ' Hypothesis

 

Cash Prices 10% 5% 1%

 

 

y = .-1.241><10'2 a, = 9.681><10'3 b, = 2x 10*5

 

1:, -2.549 -3.12 -3.41 -3.96 y = O

to, 2.201 2.72 3.08 3.71 610 = 0 given 7 = 0
1,, 0.625 2.38 2.78 3.46 a1 = 0 given 7 = 0
(I), 0.196 5.34 6.25 8.27 y = a1 = 0

(1)2 2.254 4.03 4.38 6.09 a0 = y = a1 = 0

7 = -1208x10'2 a0 = l.054><10'2

 

 

 

1,, -2.603 -2.57 -2.86 -3.43 y = 0
1,,“ 2.525 2.16 2.52 3.18 ao=0giveny=0
(I), 3.193 3.78 4.59 6.43 a0 = y = 0
y = -5><10“
1: -0.418 -1.62 -1.95 -2.58 y = 0
' Use asymptotic critical values from Davidson and Mackinnon (1993).

49

can be found in Dickey and Fuller (1979). To appropriately select the lag length, q, in
equation (4. 5), we start with a relatively long lag length and test down the model by the usual
t-test. Once a tentative lag length has been determined, the Ljung-Box Q-tests are conducted
to ensure that no significant autocorrelations are in the residuals (see Appendix A). Here, q
= 5 was selected for the given cash price series.

From the top portion of Table 4-2, '1:t is larger than the critical values at the standard
significance levels, implying no evidence being found to reject the null hypothesis of unit root
(1 = 0 ). To further verify the regression specification is appropriate, we use statistics Ta, and
1,, to test the existence of drift ( a0 = O ) and time trend ( a, = 0 ) , respectively, and (D3 and
(D2 to test the joint hypothesis of y = b0 = 0 and a0 = y = b0 = 0, respectively. From Table
4-2, all four statistics are insignificant even at the 10% level. These results suggests the
absence of a time trend for the series and also suggests a more powerful set of test statistics

can be obtained fi'om regression models:

AP = a0 + 7P,_1 + f: a, APH. + d,cos(21tw,./52) + dzsin(21rw;./52) + e, (4.6)

i=1

AP = '11PH + f: a, APH. + d,cos(27rw,/52) + dzsin(27rwl./52) + a, (4.7)
i=1

The statistic 1,, calculated from equation (4.6) is insignificant at the 1% and 5% levels
but significant at the 10% level. The test for the presence of the drift in (4.6) is given by rm,
which is insignificant at the 1% level but significant at the 5% and 10% levels. The statistic

(I), for testing the joint hypothesis of a0 = y = O is insignificant, even at the 10% level. The

50

test for the null of unit root in the regression without drift in (4.7) is given by statistic t which
is insignificant at standard significant levels. These results also tend to support the hypothesis
that the given cash corn price series has a unit root, which is consistent with the conclusion

fi'om Phillip-Porron tests.

4.1.2 Graphical and Descriptive Analysis of the Dependent Variable

The common practice to handle a series with a unit root is to take a first difference
of the series in order to obtain a stationary representation. Define AP, as the first difi‘erence
of the variable, i.e., AP, = P, - P, ,, . As P, denotes the logarithm of price, AP,, in turn, is
approximately the rate of change in price levels (or price growth)‘. Figure 4-2 illustrates the
path of the difi‘erenced series. It appears that the series fluctuates around a constant level.
The amplitudes of fluctuation vary over time, however. The squares of the difi’erenced series
are plotted in Figure 4-3 which helps to identify the volatility of the series. The squared price
growth rates exhibit periods of unusually large volatility and periods of relative tranquility.
Hence, it is reasonable to believe that the variance of the series is not constant and may be

serially dependent. The conventional assumption of homoskedasticity apparently is

 

inappropriate.

1 AP, = 111p; " lnpt-l
= 1" (pr /pr-1)
= 1n {1 + [ (pr 'pr-l)/pr-1]}
1:: (pr “pr-l) /pr-l

where we have used the fact that for x close to zero, In (1+x) z x.

51

662 dam - :2 .89 58 8.. amaze 8:36 29m N4 2.85

80>

cam. gm. #02 nag mag 32 8a. 3a— :3 $3 32 32 32 32 32 $2 owo. 3.2 2.3 2.2 052 2.2
and-

 

I
T

mmd-

1

r omd-

1
1

26.

.. 2.?

 

 

 

. 3o-

 

 

 

1. mod

 

 

.. 26

 

 

 

cud

52

382 am - 32 .89 58 é .855 SE co 82 388m 3. 2:?

30>

.32 3.... 32 82 32 ...,2 82 $2 :2 52 £2 22 33. $2 32 $2 82 22 at: 2.2 22 22
2:2,. .1 ...: ._._ ._. .__. ... .12 .:.£_._._. ... ._ .... -3, ...,. f. _ 8...: .2. 28¢

. e :3

 

.f 8.0

 

 

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1. v0.0

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nod

53

Table 4-3 summarizes the descriptive statistics for the rate of change in price levels.
The sample coeflicients of skewness and kurtosis of the variable are -O.802 and 9.375,
respectively.2 Both are significantly difi‘erent from the population values for the normal
distribution, 0 and 3, at the 5% confidence level. They indicate that the unconditional
distribution of the series is skewed and has fat tails.

Atest for normality is further conducted using the Kolmogorov-Smimov goodness-of-
fit test. The idea of the test is to investigate the significance of the difference between an
observed distribution and a specified population distribution. The maximum difference
between the two cumulative distributions provides the test statistic, Dm (see Appendix A).
The null hypothesis of a normal distribution for price growth for cash corn is rejected at the

5% level.

Table 4-3 Descriptive Analysis of the Rate of Price Change for Corn '

 

I Sample Mean I Sample Variance Skewness Kurtosis" I Dm ° I

[7.713 810'3 I 12.5538 -0.8022 9.3752 I 0.0776 I

 

 

 

 

‘ Units are (AP, I"100).

b Calculated by standardized data. If normal, the kurtosis is 3.

° Kohnogorove-Smironov statistic. The null of normality is rejected at the 5% significant level
whereby the critical value is 0.0412.

In large samples

S ~ AN(0, 6/ 7)

K ~ AN(3, 24/7)
where AN represents asymptotic normal distribution, and T is the sample size. The formula
of the parameters can be found in Appendix A.

54

Sample autocorrelations and partial autocorrelations of the rate of price changes are
calculated. The magnitude of autocorrelations and partial autocorrelations is very small. In
terms of absolute values, the largest of autocorrelations and partial autocorrelations up to 200
lags is about 0.08. It is worth of pointing out that there is no evidence that the magnitude of
autocorrelations and partial autocorrelations become smaller as the time lag becomes larger.
Some autocorrelations and partial autocorrelations at long time lags are as large as those of
the first several time lags. This indicates the dependence between distant observations are not
necessarily weaker than that between nearby observations. In practice, it suggests that older
price movements may be as useful as recent price movements in prediction of future price
changes.

Also, the seasonal patterns observed in the correlogram may support the existence of
a seasonal cycle in the data. The seasonal pattern of changes is the most common regularity
observed in agricultural prices. For crops, seasonality arises from climatic factors and the
biological grth process of plants, and from marketing patterns where many crops are
harvested once a year and, depending on perishability, may be stored for sale through one or
several marketing seasons. These agricultural products characterized by seasonality in
production and marketing patterns may cause the prices and price changes to show seasonal
patterns in either their levels and/or volatilities. Normally, price levels of storable
commodities are lowest at harvest time, rise as the season progresses, and reach a peak prior
to the next harvest. Seasonal effects are also present in a cyclic pattern of price volatilities,
where higher volatility is found during the planting and growing months and low volatility is

found in the winter months.

55

4.2 The Econometric Model

An autoregressive process is adopted to describe the conditional mean of the
logarithmic price changes and the conditional variance of the innovation is described by a
generalized autoregressive conditional heteroskedastic model (GARCH). The GARCH was
extended by Bollerslev (1987) from the autoregressive conditional heteroskedastic model
(ARCH) first developed by Engle (1982). This model is able to directly describe the common
heteroskdedastic features found in high fi'equency commodity price data. The GARCH(p, q)

is specified in the form:
0,2 = a) + fate; + $310127. (4.8)
- 1

where a, is the disturbance term of the conditional mean equation following a specified
distribution with zero mean and heteroskedastic variance of .

Seasonality can be captured by seasonal dummy variables which implicitly assume the
variable “jumps” each period. Fackler (1986) suggests the use of sine and cosine functions
as a smooth function of time to model the seasonality in the form of:

’ w, _ w,
21 w“ cos( 21cm] + tum. 3142137] (4.9)

i: I

where I is the degree of frequency in seasonality variables; M/i is the length of each cycle, for
example, if the data is sampled daily, M is equal to 365 (or 366 depending on the given

calender year) so the denominators equal 365 (or 366) for one-year cycle and 365/2 (or

56

 

 

366/2) for half-year cycle and so forth; w, is the number of periods from the beginning of
each one-year cycle, for example, if the data is sampled daily, w, is equal to {1, 2, , 365
(366)}; and \v1 , and V” are the coefiicients of the associated variables. The seasonal effect
in (4.9) is cyclical and can be combined in different fi'equencies to provide different forms and
lengths of cycles. Likelihood ratio tests and joint F ¢test have been used to select an
appropriate level of I (e. g., Fackler and Livingston 1996; Wang 1996).

When the trigonometric seasonal effects are specified in the variance equation, it
implies the volatility of the series being concerned is time-varying. This specification of time-
dependent volatility is deterministic which is significantly different from the stochastic setting
in ARCH or GARCH models. The volatility prediction of the seasonality model as in (4.9)
is conditional on the time period for which the forecast is made, while the volatility prediction
of the ARCH or GARCH models converges to a constant unconditional variance that does
not depend on the time period. Therefore, the GARCH specification is best suited to model
volatility when volatile and tranquil periods emerge randomly, while the deterministic
seasonality model is best suited to model volatility that appears cyclically (Wang 1996). To
accommodate the two firndamentally different effects, Fackler (1986) developed a process
represented by both a deterministic seasonal component and a GARCH component, which

does a good job of characterizing the variance process for the Saginaw corn prices used here.

57

4.2.1 Model Specification

The general form of the econometric model in the study is specified as:

 

w
AP = p+yAP_ 2+:£l[dl cos(21r ")+d21.sin(21r;;.)] +8,

W
52’! 1 (4.10)
2
[Int-l ~, f(026()v)

8

I
2 2 2 W, . W,

o =m+ae_ + o_+ coser— + .sm27r— 4.11
r .. B” gm. < 5sz w, ( 52”” ( )

where (2,, is an information set available at t-1;f(.) is a student 1 density function with zero
mean, variance 0’, and v degrees of freedom; w, in the sine and cosine functions is the number
of weeks since the first week of October in the given crop year; denominators in the sine and
cosine functions are the specified cycle length in weeks, so 52 (r' or j = 1) indicates a one-year
cycle, 26 (i or j = 2) indicates a half-year cycle, etc. We also rescale the dependent variable
by multiplying by 100, so it becomes a percentage change.

In our preceding data analysis, the null of normality for the price growth data has been
rejected and the series seems to have more observations in the tails than does a normal
distribution. This leptokurtosis characteristic has been found in the literature on asset prices
and it is usually dealt with by assuming that the disturbances follow the student t distribution
(e.g., Bollerslev 1987; Yang and Brorsen 1992). In the study, f (.) is assumed to be a
Student’s t density firnction with v degrees of freedom where v is a parameter being
estimated. In the literature, the degrees of freedom are usually reported in the form of UV.
This is because when 1/v .. 0 the t-distribution approaches a normal distribution. Hence,

58

when the estimation of UV fails the standard significant test, it implies that a normal
distribution is insufiicient to represent the given data series. Likelihood Ratio (LR) tests are
also used in the literature for testing the null of UV = 0 against NV > 0 (see Appendix A).
However, when testing against the null hypothesis of conditionally normally errors, i.e., l/v
= 0, UV is on the boundary of the admissible parameter space, and the usual test statistics will
likely to more concentrated toward the origin than a Chi-squared distribution of degree 1,
implying a conservative test for conditional normality (Bollerslev 1987). The hypothesis of
a well-calibrated distribution will be formally tested by the Kolmogorov-Smimov goodness-
of-fit tests on the standardized residuals3 of the models. The null hypothesis of a well-
calibrated distribution for the econometric model with conditionally t-distributed errors
requires the standardized residuals to follow a standard t-distribution.

Likelihood ratio tests will be undertaken to examine the order of seasonal firnctions
in mean (J) and variance (1). The sequential tests begin by specifying high-order seasonal
functions for both mean and variance and test down the model by reducing the order
sequentially (see Appendix A for details).

The estimation of the parameters in the econometric model will be conducted using

the maximum likelihood estimation (MLE).

 

3 A standardized residual is the residual from the conditional mean equation, divided
by the squared root of the conditional variance, is /6 .
l I

59

4.2.2 Estimated Results

Estimates of the model are reported in Table 4-4. Coefficients in the conditional mean
equation are all significantly difierent form zero at the 5% level, except the second degree of
cosine function in the seasonal term. Coefficients in the conditional variance equation are also
all significant at the 5% level, except for the cosine function, which is significant at the 10%
level.

Ljung-Box Q-statistic of the standardized residuals, and the squared standardized
residuals, are calculated (see Appendix A). They are used to test for the hypothesis that there
does not exist residual autocorrelation and conditional heteroskasticity, respectively, not
captured by the model. From Table 4-4, the Ljung-Box statistics show that the standardized
residuals do not exhibit substantial autocorrelations in either mean or variance at the 5%
significance level.

The unconditional sample kurtosis, é‘l (c‘r’)2 , differs significantly from the normal
value of three. The estimate of UV is highly significant. The Likelihood Ratio test statistic
LR1,,,0 for the GARCH(1, 1) model with conditionally normal errors is much higher than the
critical value of x: at 1% significance level. Both provide strong evidence to reject the
hypothesis that normal distribution is suflicient to represent the conditional distribution of this
sample data. The Kolmogorov-Smirnov statistic D“m used to test the calibration of the
models is reported. Ifthe model is appropriately specified, the standardized residuals should
follow a standard t distribution with 9 degrees of freedom. The Kolmogorov-Smirnov test
reject the hypothesis at even the 10% level, implying the model is not well calibrated. This

is also shown in the implied estimate of the conditional kurtosis, 3(r3 — 2)(r3 — 4)‘1 = 7.888,

60

The Model: GARCH(1, l )-t

W!

52/1

 

2 w
AP, = y AP,_2 + 2 [d,,.cos(21r ) + d2}.sin(21r—')] + a,
1'1 52/1
s,|Q,_, ~ f(0, 01v)

0,2 = 0.) + nail + pof_,+ w,cos(2nw,/52) + w,sin(2rrw,/52)

Table 4-4 Econometric Models

 

 

MEAN Estimates VARIANCE Estimates
9 0.0506 (0.0488) a) 1.6255 (0.0078)
a?” -0.5776 (0.0000) ii 0.0934 (0.0013)
3,, -0.3893 (0.0007) [3 0.7658 (0.0000)
“n -00021 (0.4929)* lb. 0.6222 (0.0614)*
3,, 0.3211 (0.0029) \IIZ -l.1059 (0.0000)
Degrees of Freedom
0“ 0.1913 (0.0000)
STATISTICS
Sample Skewness - 0.5824 LR,,,,=0" 74.9968
Sample Kurtosis 6.6342 Dm ° 00555"
Autocorrelations: Ljung-Box "
Q(5) = 3.6719(0.2991) Q2(5) = 1.1303(0.7698)
Q(10)= 9.4968(03021) Q2(10)=10.8081(0.2128)
Q(15) = 13.1055(0.4397) Q2(15) = 12.8787(0.4572)
Q(20) = l7.9l30(0.4614) Q2(20) = l6.3879(0.5655)
Q(30) = 26.8897(O.5243) Q2(30) = 22.7677(0.7446)

 

' Asterisks denote the coefficient is significant at the 5% significant level. The values in
parentheses are p-values.

” Likelihood ratio test statistic for GARCH(1,1) with conditional normal errors (l/v = 0)
against conditional t-distributed errors (l/v > 0).

° The critical value of K-S tests is 0.041 1 at the 5% significant level and 0.0493 at the 1%
significant level.

‘ The values in parentheses are the degrees of order in serial correlation. Q-statistic is a
test for serial correlation in the residuals; Qz-statistic is a test for residual GARCH
effects. The values in parentheses are p-values.

61

which is not close to the unconditional sample analogue.

Second-order and first-order seasonal functions in the conditional mean and the
conditional variance, respectively, are found to best represent the cash price movements in
the study, i.e., (J‘, 1*) = (2, 1). These are consistent with other research in which the
seasonal efi‘ects in means or variances have also been found to have lower order (e. g., Yang
and Brorsen 1992; Fackler and Liverston 1996).

In the model, the ARCH and GARCH terms are highly significant which suggest
variances of Saginaw corn price changes are not constant but change over time. Seasonal
components in both conditional mean and variance are also significant, implying these price
changes are cyclical in level and variability. Both confirm the findings of our preliminary
graphical examination. Summing up, the model succeeds in capturing the general features of
the price movements of storable commodities and removing serial dependence. Although the
result of Kolmogorov-Smirnov tests indicates that the model is not well-calibrated, it is clear
that a “fat-tailed” conditional t-distribution is superior to the conditional normal to account

for the observed leptokurtosis in Saginaw corn cash prices.

4.3 Time-Dependent fiansition Probabilities

The objective of the section is to generate the transition probability space of stochastic
price states for each decision period in the dynamic programming model. The stochastic
structure will be created based on the GARCH( l , l)-t model proposed in the previous section.

For convenience we rewrite the model as:

62

Alnp, = yAlnp,_2 + 5,., + e, (4.12)

om. ~ f(0, of, v)

0,2 = 00+ (18:, +130?» + <p,_, (4.13)

where 5,_, and (p,_, are deterministic seasonal components; and fl.) is a student 1
distribution with v degrees of freedom. Since seasonal effects and conditional

heteroscedasticity are included, the resultant transition probabilities are time-dependent.

4.3.1 Procedure to Generate the Price Transition Probability Space

The procedure to generate the time-dependent transition probability spaces of the
price states will be illustrated in this section. For illustration purposes, assume two states of
nature, {51 , p3} , and two decision periods, 1 and 1+1. The transition probability matrix
mapping the states at 1 to the states at 1+1, and the states at 1+1 to the states at 1+2 will be

generated, which are denoted, respectively, as:

l 2
, 1tll 1r.“ 2 1:ll 7t2r

II = 1 l and II = 2 2 (4.14)
1:12 7‘22 7‘12 7'22

where it; gives the probability that stateiattwill be followed by state j at 1+1 forr= 1, and
the probability that state 1' at 1+1 will be followed by state j at 1+2 for r = 2.

2
Therefore, 21:; = l, fori= 1, 2andr= 1,2.
j=1

63

Note that the marketing decision in the empirical application that follows is assumed
to be at two-month intervals whereas the econometric model is estimated based on weekly
sampled price data. The stages characterized by transition probabilities are therefore on ‘a
difi‘erent time frame from the econometric model. The way to accommodate the
inconsistency is to simulate the week-by-week prices from a given stage (first decision period)
toward the next stage (second decision period) using the estimated econometric model. The
price at the beginning week of the interval (between two decision periods) is then considered
the initial price (first decision period) and the simulated price at the last week of the interval
is the price state of the next stage (second decision period) which is conditional on the given
initial price. Those weekly prices between the two decision periods are only simulated in
order to arrive at the price of real interest. We assume T sub-periods between two decision
periods and let III and 1'12 denote the transition probabilities from 1 to 1+T and from 1+Tto

1+2T, respectively. The elements in 1'11 and II2 as defined in (4.14) then become:

Fl
ll

1, prob(p,.r=17, I 12517,.) i,j= 1, 2

:1
ll

i, prob(p,.2T=17, I p,.T=17,-) 1;] = 1, 2

where “prob (- | °)” indicates the conditional probability. The details of the procedure to

compute these probabilities are presented in the rest of the section:

Procedure:

Step 1 Choose the first price state p, = [7, .

Step 2 Simulate a value for the random shock é,” as follows:

2a Compute 0,2,, using

64

2b

63.. = at + a116,”) +01%?) +1». (4.15)
where E? and 6,2 in the variance equation of (4. 13) for one-period forward
are set to the value of the unconditional variance E (0,2) . The unconditional
variance can be obtained by taking unconditional expectations on (4.13) and
then solving for E(o,’) where E(of_,) =E(o,’) , and E(ef.,) =E(of_,) =

E (0,2) since E (a, )= 0. The unconditional variance then becomes:

(0 + ‘Pr-l

l—o-B

E(o,2) = (4.16)
The unconditional variance is conditional on the deterministic seasonal
component (pH .

The reason why the unconditional variance values are used here is
because the conditional variance in general is unobservable. It seems
reasonable to use the unconditional values to represent their average impact.
In the present study, transition probability matrices in the dynamic
programming algorithm are restricted to one dimension (a single state
variable, price). Volatility of price changes therefore is not explicitly specified
as a stochastic state although it maintains its influence on price movements

through disturbance term in the conditional mean equation.

Simulate a random draw, 11,, from the random numbers of a student 1

65

distribution with v degrees of freedom“ .

2c Construct a,” = u, (5,,l

Step 3 Compute a random draw on pm using

1115,” = mg, + ?E(Alnp,_,) + 8, + am (4.17)

where E(Alnp,_,) is the unconditional mean of the price changes which can be
obtained by taking unconditional expectations on (4.12) for one-period ahead and
then solving for E(Alnp,_,) where E(Alnp,_3) = E(A1np,_,) by definition. The
unconditional mean of price changes at 1-1 is computed as:

8

"2.. (4.13)
1 - v

 

E(Alnp,_,) =

which is determined by the constant coefficients in the conditional mean equation
and the deterministic seasonal components. Unconditional values are used for
stochastic variables before 1 because the optimal marketing policy in the current
study is not designed for particular crop year so the price simulation is designed to

be applicable to any crop year so the price transition probabilities are not conditional

 

‘ Random numbers of the student 1 distributions with v degrees of freedom can be

created fi'om a random variate (z) from the normal distribution and a random variate (x) fi'om
an independent chi-squared distribution, which is a special case of the Gamma distribution,
in the form of z/ W . Random numbers oft distributions therefore can be generated from
random numbers of normal distribution and Gamma distribution.

66

on a particular crop year..

Step4 Simulate a value for the random shock E“, as follows

2 C 0
4a. Compute 6,,2 recursrvely usmg

.2 . . . A2 .
where am 1s avallable ln 2c and (IM ln 2a.

(4.19)

4h. Simulate another random draw p2 from the Student’s 1 distribution with l9

degrees of fi'eedom.

A _ 52
4c. Construct 3,,2 — u2‘/o,,2 .

Step 5 Compute a random draw on p“.2 recursively using

lnfif+2 : lnljltrl + ?E(Alnpt) + 81+] + €142

where unconditional price change is used for lag price change.

(420)

Step6 Continue iterating forward in time until we have one Simulated realization of

pr+T|pr =51 '

Note here that once we move beyond two time periods we will be able to use

simulated values for ApM , where s 2 2, in the conditional mean equation, rather

than the unconditional expectations.

67

 

 

 

{rat

I(/)

Stefl Go back to step 1 and repeat N times to get N realizations of pm | p, = [72 .

_$_t_ep_§ AllocateallN realizations of p,,T| p, =17, into categories p_, or 52 and compute

relative frequencies. These relative frequencies are then used as:

pf0b[]),,T : [71 I I), = [71]

pr0b[p,.T = 172 l p, = 171] .

Step 9 Repeat starting fi'om choosing p, =17, in step 1 to get

pr0b[p;+r : [71 I P, : I72]

pr0b[p1+7' : [32 lpr : [72] '

We now have the complete transition probability matrix for many price states

between 1 and HT.

In the study, we assume the marketing season begins in early November and ends in
early September of the next year. Storage decisions are allowed to be adjusted every two
months, so a total of six stages (decision periods) are considered. The only optimal marketing
policy at the final stage is to sell all of any grain left in storage, which indicates five transition
probability matrices will be needed. Hence, we assume the producer re—evaluates the
marketing decision in November (1), January (1+7), March (1+2T), May (1+3 7), and July
(1+47), and then sells all of any grain left in Storage in September (1+5 T). To construct each

transition probability matrix, we start from specifying the price at the given stage as in step

68

one of the procedure.

The sample size used for the simulations (N) is 10,000. There are m (m > 2) price

 

levels considered, {51,172, p-m} , where [3, < [5, <....< ,5", , so the price states are
defined as:
State it = {pl 1) < 1325,} ;
State [5, = {pl 5'42th s < 54:213-3} , i=2, 3m-1; (4.21)
State 17... = {pl 12 2 17,..-12+E,,,} -

The state space for cash prices in the study is based on 15 (m = 15) price states, from
$1.6/bu to $4.4/bu in 20-cent increments. As defined in (4.21), each value is viewed as the
mid point of the underlying continuous price interval. For example, $2.0/bu represents prices
that are higher than $1.9/bu and less than or equal to $2.1/bu, and therefore any price that
falls into the interval is counted as the state of $2.0/bu. The upper bound price state, $4.4/bu,
represents prices that are higher than $4.3/bu and the lower bound price state, $1.6/bu,

signalizes prices that are less than $1.7/bu.

4.3.2 Simulation Results

The time-dependent price transition probabilities for the five decision periods are

given in Table 4-5(a) to (e). It appears that the price movements are generally different

69

 

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74

across time. In the early stages of the marketing season, prices of a given state at one period,
in general, are more likely to remain around the original state for the next period. Towards
the end of the marketing season, in general, the tails of the conditional distributions are fatter.
This means that the volatilities of price movements increase as the marketing season proceeds.
The results present a typical cyclic pattern of crop price volatilities where higher volatility is
usually found during the planting and growing months (March to September) and low
volatility is found in the winter months (November and January). Using $3 .0/bu, the mid level
of the price space, as an example, assume a January price of $3.0/bu. Then there is around
a 71% probability that price will remain in the price range from $2.8/bu to $3 .2/bu in March,
implying there is only a close to 30% probability that price will be 20 cents higher or lower
in March. Similarly, assuming a July price of $3.0/bu, there is about a 42% probability that
price will remain in the price range from $2.8/bu to $3.2/bu in September, implying there is
around a 60% probability that price will be 20 cents higher or lower in September, about two
times as high as March prices.

Prices in earlier stages also have a stronger tendency to increase than in the later
stages. Prices of November, January, March, May and July at $3.0/bu are likely to switch to
higher price states with 45%, 57%, 56%, 42% and 28% probability, respectively. Although
January and March prices, in general, have equal tendency to increase, March prices have
more chance to reach a higher price state than January prices. For example, the probability
that the March price of $3.0/bu will become $3.4/bu or higher in May is 10% higher than the
same case from January to March. This implies that prices are likely to move upward from
November to May, while the magnitudes of the increases from March to May, on average,

are larger than those from November to January and from January to March.

75

From Table 4-5(d) and (e), it is found that a given May price has roughly equal chance
of increasing or decreasing in July, while July prices are much more likely to decrease in the
following period. For example, a May price of $3.0/bu has about a 42% probability of being
higher in July and has about a 39% probability of being lower; while a July price of $3.0/bu
has around a 28% probability of increasing and a 58% probability of decreasing. This
suggests the rate of price growth begins decelerating from May to July, and the price path
general becomes downward sloping after July. The findings are quite consistent with the
usual seasonal pattern of storable commodity prices, which follows an earlier harvest low and
high peak in late spring or summer prior to the next harvest. In general, the transition

probabilities have captured the usual price pattern of storable agricultural commodities.

4.3.3 Model Validation

In this section, we proceed to examine the credibility of the price forecasting model
and the resultant transition probabilities. We first simulate price movements over the post-
harvest marketing period using our estimated model and compare the simulated price paths
with the actual historical path. Numerical evidence is provided by computing the average
seasonal index and average price change rates using simulated and historical prices. We
fiirther examine the credibility of the transition probability matrices by conducting a
descriptive analysis on the price probability distributions constructed from the transition

matrices.

76

Simulated Price Movements

To simulate the price movements over the marketing season, we use essentially the
same procedure used to estimate the transition probability matrices in section 4.3. 1 . The only

differences are:

1. Instead of setting p, equal to one of the pre-determined price states, we draw the
initial p, randomly fiom one of the historical prices for the first week of November.
We denote this initial price as 1')", ;

2. Instead of simulating the price series until the next decision period, the simulation
continues until the end of marketing season. Thus we obtain a set of simulated

price, { phljhzynp‘fl,...fihnrufifl I pt: [5,} , for each simulation, where T

denotes the final period of the marketing season.

We repeat the procedure N times and take the average of the N set of prices for each
week. Then we plot the simulated average prices on the same chart as the historical average
prices on Figure 4-4. The graph shows that the simulated prices are quite close to the
historical average pattern.

In addition to the graphical analysis, using the simulated prices, we compute the
average seasonal price index of each decision period over the post-harvest marketing season,
and the average price change rates between two adjacent decision periods, to compare with
the actual prices. The results are summarized in Table 4-6 and Table 4-7, respectively. The
base for the seasonal price index is an average of weekly prices of the given crop year. An

index of 93 .42 for November, therefore, indicates that the first week of November prices, on

77

 

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Table 4-6 Average Seasonal Price Index for Simulated and Historical Prices

 

November January March May July September
Simulated 93.42 95.51 98.70 103.91 105.42 98.46
Historical 91.94 96.38 99.33 105.74 107.96 99.68

 

Note: The base of the seasonal index is the average of the given crop year.

Table 4-7 Average Price Change Rates for Simulated and Historical Prices (%)

 

 

November January March May - July -

- January - March - May July September
Simulated 2.67 3.54 5.17 1.09 ~8.14
Historical 4.83 3.24 6.24 1.52 -8.63

 

Note: Price change rate is computed as (p, - p,_.) / [(p, + p,_,)/2].

average, are 6.58 percent below the average of the year. The results share a quite similar

pattern with the historical average.

The average of price change rates between two adjacent decision periods for

simulated prices are also compared to the historical average. Price change rates are the

difference between two prices of interested periods, divided by their average. In general, the

simulated price change rates are quite consistent to the historical average, especially from

January to March and From July to September. The major increases in prices occur from

March to May. Price changes start to decrease afier May, and become negative afier July.

The main inconsistency between simulated and historical prices is from November to January.

For simulated prices, the rate of price changes fi'om November to January is lower than that

79

fi'om January to March, while for historical prices, the rate of price changes from November

to January is lower.

Probability Distribution Conditional on the Initial Price State

The credibility of the estimated transition probability matrices in Table 4-5(a) to (e)
will be examined here by a descriptive analysis on the price probability distributions,
constructed from the matrices, for each decision period over the post-harvest marketing
season. These price distributions are conditional on a given price in the initial stage, which
gives an insight of a harvest price’s impacts on the subsequent price distributions throughout
the crop year. The harvest price is of interest because it is an indicator of the new crop
production, while the size of the new crop is a principal factor in yearly price variability and
may substantially influence the seasonal price pattern.

Again, for illustration purposes, consider the simplified case of two stochastic
states, p.l and 172 , and two decision periods, t and t+T , where the transition probability
matrices have been defined in (4.14). The probability distribution of each period conditional

on the initial state, [71 , are calculated by:

 

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80

Conditional distributions are computed for three difi‘erent initial price levels. They are
(1) average price: the historical average price of the first week of November ($2.23/bu), (2)
low price: average price minus one sample standard deviation ($1 .73/bu); and (3) high price:
average price plus one sample standard deviation ($2.72/bu). The price probability
distributions at each decision period conditional on each of the three initial prices are given
in Figure 4-5(a) to (c).

Results show that price levels at the initial stage have strong implication for the levels
of future price in any given crop year (as expected). That is, crop years with high harvest
prices, relative to low harvest prices, implies a high probability that prices will stay relatively
high over the marketing season. The weights of the probabilities tend to transfer to the higher
price states in earlier stages, implying that prices during early periods have a general tendency
to increase. The weights shifi back to the lower price states when the season approaches to
the end, indicating that prices prior to the next harvest have a higher chance of decreasing,
relative to the early season. These findings are consistent with the average historical price
change rate in Table 4-7 where the rate of price changes grows from January to May and
becomes negative after July.

In addition, the price distributions at any stage, in general, are quite symmetric except
for the cases when the initial price level is low. The asymmetry in the low price state is a
result of truncation effects due to the assumption of a finite stochastic state space. The
probability of a price below the lower boundary must be added to the probability of price
falling within the lowest price interval; likewise, for very high prices and the higher boundary.
The single pikes of the probability at the ends of the price space as in Figures a5, b4, b5, c3,

c4 and c5 are caused by the same effect.

81

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Furthermore, tails of the distribution get fatter as the marketing season proceeds. This
increasing dispersion of the distribution could simply be the result of a time effect because,
in later periods, there is more time for prices at the later stages to change. However, it may
also imply that, but not necessarily, the volatility of price movements are larger in the later
stages of the marketing season than in the early stages as shown in the transition probability
matrices. This is consistent with the usual seasonal price volatility pattern whereby low
volatility is found in the winter season and high volatility is found in planting and growing

seasons.

4.4 Summary

Time-dependent transition probability matrices are specified from a GARCH-t price
forecasting model, which is successful in capturing the seasonal patterns in mean and variance
of corn prices. The transition matrices are used to describe the stochastic structure imbedded
in price movements between stages in a dynamic programming decision model. The price
movement simulated fiom the price forecast model, and a descriptive analysis of the transition

probabilities, are found to imitate the historical price pattern well.

85

CHAPTER V

OPTIMAL STORAGE RULES

In this chapter, we will estimate the optimal decision rules for post-harvest grain-
storage marketing for Saginaw county in Michigan. The stochastic dynamic programming
model developed in Chapter III, and the time-dependent transition probabilities for corn prices
estimated in Chapter IV, are used to compute the optimal marketing policies for Saginaw,
Michigan, corn producers. A sensitivity analysis will also be conducted to investigate the
impacts of various changes in parameters in the decision model on the optimal timing of sales.
These parameters include storage costs, interest rates, and the degree of farmers’ risk
aversion. One of the contributions of this chapter is to provide empirical evidence to support
the theoretical result that proportional sales may be optimal marketing strategies when the
farmer’s risk aversion is incorporated into the decision model.

A description of the empirical problem is presented in the first section, followed by
an illustration of optimal marketing policies and the associated numerical analysis for nearly
risk neutral farmers, mildly risk averse farmers and highly risk averse farmers. The optimal
storage rules will be applied and the resultant by—monthly expected optimal marketing
volumes over the marketing season and the probability distributions of expected final wealth

will be calculated in the last section.

86

5.1 The Empirical Problem

We assume that a corn producer with an on-farm storage facility is faced with the
decision of how much and how long to store grain during the post-harvest season, given the
uncertainty of fiJture corn prices. The marketing year begins at harvest and ends before the
next harvest, due to limited storage capacity. High transaction costs prevent the producer
from replenishing storage once it has been sold.

To implement the simulation of the optimal decision rules and the associated
numerical analysis, several critical features of the problem have to be defined. These include
the number of stages, definition of state and control spaces, the specification of the utility
function, and the values of the parameters in the dynamic programming decision model

including interest rates, storage costs, and risk coefficients.

Stages

It is assumed that marketing decisions are made at the beginning of each of six stages
in a marketing year, with each stage corresponding to two months in length, beginning right
after the most recent harvesting time is over. In Saginaw, Michigan, the harvesting season
usually starts in early October. The first decision period in the study therefore is set at the
first week of November (initial stage, i.e. stage 1). The marketing decision is re-evaluated
in the first week of the months of January (stage 2), March (stage 3), May (stage 4), and July
(stage 5). In order to be prepared for the next harvest, the total stock must be sold by the

first week of September (terminal stage, i.e. stage 6) of the next year.

87

State and Control Spaces

State variables describe the environment of the decision process. They constitute the
vector of attributes that determine current and future stock/inventory values. According to
the dynamic programming model specified in Chapter III, the three state variables considered
in the study are (1) the cash price for corn, (2) the wealth level earned from the current year’s
marketing activities for com, and (3) the storage level for corn. The control variable is the
decision made in one stage which determines the state of the process in the following stage.
The control is selected to maximize the objective fiJnction of the dynamic programming model
which is the expected utility of final wealth. The amount of sales is the only control variable
in the study.

These state and control variables might best be viewed as continuous variables but,
for computational purpose, they are divided into discrete units. As previously discussed in
the section on constructing transition probability matrices (section 4.3.1), the state space for
cash prices is based on 15 price states, from $1 .6/bu to $4.4/bu in 20-cent increments. Each
value is viewed as the mid point of the underlying continuous price interval. For example,
$2.0/bu represents prices that are higher than $1 .9/bu and less than or equal to $2.1/bu, and
therefore any price that falls into the interval is counted as the state of $2.0/bu. The upper
bound price state, $4.4/bu, represents prices that are higher than $4. 3/bu and the lower bound
price state, $1 .6/bu, signalizes prices that are less than $1 .7/bu.

The possible sale levels of grain (control) are represented by the proportion of the
initial storage level (i.e. the amount harvested). It is assumed that the amount of initial

storage can be marketed in up to 11 equally sized portions. Hence the control space is

88

defined from O to 100 percent in 10 percent increments. Since the carry-over storage is the
balance of the beginning stock at a given stage, afier subtracting out the current periods sales,
the storage levels are able to achieve 11 different states, including an empty stock.

The wealth space at a given stage is specified at all feasible wealth levels, allowing for
every possible combination of price outcomes and sales in preceding stages. Storage costs
and interest income are taken account. The number of wealth states is different across stages
because the range of marketing opportunities expands with the time horizon. The initial
wealth at the harvest period is assumed to be zero. Furthermore, the values of the computed
wealth are rounded to the nearest 3-digit decimal numbers. In cases where the same wealth
levels arise from different current storage levels, these are considered separate states.

In dynamic programming with discrete states and control spaces, the method of
interpolation is usually adopted to next period’s value function when the exact state value is
not an element of the state space. Thus, the accuracy of the approximation depends on the
number of states and the distance between them. State spaces with small intervals give more
reliable approximations. However, this also expands the size of the space and therefore
makes the execution more time consuming. The benefit from specifying the state space along
the feasible state levels as we construct the wealth states in this study is that it minimizes the
approximation errors of the interpolation method. Moreover, since it excludes the infeasible
states, the time demanded for the empirical execution is shortened.

As an illustration, consider the case of 10% annual interest rate and a 1 cent/bu
monthly storage cost. The numbers of states in three dimensions, (price >< storage X wealth),

are 15 at the initial stage, 2265 at the second stage, 95460 at the third stage, 178170 at the

89

fourth stage, and 195180 at the fifth stage.‘ It has been found that the approximation errors
have been substantially reduced and the hours needed for computation are also reduced,
compared to the fixed-increment state space.

Note that since the marketing units are a proportion of the initial storage, wealth
levels, in tum, are scaled by the initial storage as well. A wealth level of 1.351, for example,
means 1.351 dollars per bushel of initial storage. In dollar terms, the wealth level is “1.351
X the amount of initial storage in bushels” dollars. This transformation of the argument in the
objective function will not effect the selections of the optimal solution in that the objective
fimction defined in the following section is homogeneous (the proof can be found in Appendix

B).

Utility Function

A constant Relative Risk Aversion (CRRA) utility function of the form

l-b
W

b>0 5.1
H) ( )

 

U(wT) =

is used to represent the decision-maker’s preferences. The parameter b denotes the
coefficient of relative risk aversion. A larger value of b suggests the decision-maker is more
relatively risk averse. w, is the terminal wealth.

A utility fimction in the form of equation (5.1) implies decreasing absolute risk

 

‘ The wealth space at the terminal stage need not be constructed. This is because

according to the results derived fi'om the theoretical model, the only optimal marketing policy
is to sell all of the grain in storage at the final stage.

90

aversion (see Appendix B). Namely, an individual whose wealth increases fears a given risk
less or at least not more. The idea comes from a reasonable assumption that, given a same
risky situation when the decision-maker becomes wealthier the same amount of expected gain
or loss from the gamble becomes a smaller portion to his or her total wealth so the agent may
behave as being less risk averse. The risk coefficient is set to characterize three different
degrees of risk aversion. These are b = 0.0001 for near risk neutrality, b = 5 for mild risk

aversion, and b = 10 for high risk aversion.

The Interest Rates and Storage C osts

Annual interests rate of 5%, 10%, and 15% are considered in the study to capture the
different levels of opportunity costs to holding grain.

Three monthly rates of storage costs of 0, l, and 2 cents per bushel are considered.
Storing and conditioning grain on the farm involves both fixed and variable costs. Once the
farmer invests in a bin and the related equipment, the farm business incurs the fixed costs each
year until all of the depreciation is recovered. Variable costs refer to certain costs that are
incurred only when holding grain. The major variable costs are associated with the foregone
interest income which has been accounted for through the discount factor. In addition,
variable costs involve extra labor to keep the grain in good condition, to cover the physical
loss, and to pay ofl‘ the insurance expenses.

The main purpose of the study is to look into the optimal timing of grain-storage
marketing given the scenario that the on-farm storage facility is in place already, as opposed

to the investment decision on the construction of the storage facility.

91

Transition Probabilities

The post-harvest grain marketing problem has been formulated as a stochastic
dynamic decision process. The state transition for storage condition is deterministic. The
state transition for wealth, instead, is random but the randomness is derived from prices.
Therefore, all uncertainty of the dynamic problem is accounted for by random prices. The
state of the stochastic marketing decision process is controlled, at any stage, by a set of
transition probability matrices in the discrete-time model. The time-dependent transition
probabilities have been estimated and are given in Table 4-5(a) to (e), based on which the
simulation of optimal marketing rules in the current chapter will be conducted.

There is a problem of truncation in using a discrete valued variable to approximate a
continuous variable. It is necessary to have upper and lower bounds on price, which requires
that the probability of price exceeding the upper boundary must be added to the probability
of price falling within the highest price interval; likewise, for very low prices and the lower
boundary. The immediate impact is the value function will be over estimated at the lower
bound on price and be under estimated at the upper bound on price. In some literature, the
problem is handled by providing an arbitrarily selected buffer zone on each end of the price
interval of primary concern (e. g., Yager et al. 1980), that is, by defining extra states on each
end of the space. The bias, however, will never be completely eliminated as long as a finite
state of the stochastic structure is assumed. A common practice to reduce the bias is to
increase the number of states of the stochastic variables, which, however, will also increase

the time of empirical execution.

92

Terminal Value

According to the theoretical results, all of the grain in storage has to be marketed by
the end of the marketing year. The terminal value function for any state, therefore, is the
utility of final wealth with the carry-over at the beginning of the terminal stage sold at the

given cash price.

5.2 Optimal Marketing Policy for Nearly Risk Neutral Producers

In this section, we will illustrate the optimal marketing policy for nearly risk neutral
producers (b = 0.0001) and investigate how changes in other parameters of the dynamic
model effect the optimal timing of sales.

The numerical results show that only two possible actions are optimal for nearly risk
neutral producers. These are selling or storing the entire stock level. This result is consistent
with the findings of previous studies of Fackler and Livingston (1996) and Berg (1987) in
which the dynamic programming models are specified under the assumption of risk neutrality.
The post-harvest marketing problem in this case becomes a standard optimal stopping
problem by selecting an optimal time during the decision horizon at which to stop holding the
grain. In this case, the optimal marketing rule is defined on a “cutoff price”. The cutoff price
stands for the marginal option value of the given stocks, which is the discounted conditional
expected unit value of the stocks if it is marketed in a fixture period while taking into account
that the optimal control can change in future periods. In other words, the cutoff price is the

marginal “option value” of the given stocks. Therefore, if current cash price is higher than

93

the cutofl‘ price, implying the marginal benefit of selling today is higher than that of holding
until a future period, the optimal marketing strategy is to sell everything. On the other hand,
if current cash price is less than or equal to the cutoff price, the optimal marketing strategy
is to store everything. Using the base case with a monthly storage cost rate of 1 cent per
bushel and an annual interest rate of 10% as an example, for the month of November the
cutoff price is $4.1/bu. Hence, if the cash price in November is higher than $4.1/bu, the
optimal cash grain sale is everything in storage; otherwise, the optimal marketing strategy is
to retain the entire stock.

The cutoff prices throughout the marketing years of the base case are presented in the
first column of Table 5-1. The numerical results of this study show that Optimal cutoff price
starts with a relatively high value at the beginning of the marketing season and decreases as
the end of the marketing season approaches. This is because at the earlier stages of the
marketing season, there are more future time periods in which to consider selling, with a
concept that is similar to the “time values” of financial options trading which suggests the
time value of a given contract diminish when its expiration date approaches. Nevertheless,
the expectation of firture prices has equivalent influence on the option values of given
inventory as well as the time value. When future prices are expected to be high, which
suggests a high option value of the stocks, a high current cash price is required to compensate
the high forgone benefits if the stock is sold today, which in turn bids up the cutoff price.

From Table 5-1, the highest cutoff price of the base case has been found in January
instead of the earliest stage of the decision horizon. This may be caused by the fact that
January prices are expected to have a stronger tendency to increase than November prices,

as shown in the underlying transition probabilities in the proceeding chapter. This result is

94

Table 5-1

Cutoff Prices for Different Rate of Storage Costs (Sk/bu) and Interest
Rates (r) for Nearly Risk Neutral Decision-Makers (b = 0.0001)
unit: dollars/bushel

 

 

 

Base Case Changes in Changes in
storage costs interest rates
Month r=10%k=1 k=2 k=0 r=15% r=5%
November 4.1 3.9 4.1 3.9 4.1
January 4.3 4.1 4.3 4.1 4.3
March 4.1 4.1 4.1 4.1 4.1
May 1.9 1.7 2.9 1.9 3.1
July 1.7 1.7 1.7 1.7 1.7
September ‘ - - - - -

 

According to the results derived from the theoretical model, the only optimal marketing
policy is to sell all of the grain in storage at the final stage.

horizon.

The cutoff prices of each decision period for different values of storage costs and

similar to the Fackler and Livingston’s study (1996) where the optimal cutoff price steadily
increases in the early marketing season and reaches its peak in March before gradually
declining toward the end of the marketing season which ends in the month of J une. In Berg’s

study (1987), however, the cutoff price is decreasing from the very beginning of the decision

interest rates are also presented in Table 5- 1. The two columns next to the base case illustrate
the cutoff prices for high and low levels of unit storage costs. The last two columns display
the cutoff prices for high and low levels of interest rates. It shows that cutoff prices decline
as the per-unit storage cost increases. This is an intuitive result since storage cost is a

negative income incurred when holding the grain. An increase in storage costs, therefore,

95

suggests a decrease in option values of the stocks which makes storage less desirable and thus
demands a lower current cash price to compensate for the decreased foregone benefits of
immediate sales. Similarly, as showed in the third column of Table 5-1, an increase in interest
rates triggers a lower cutoff price since the potential increase of interest income represents

the opportunity costs of holding the grain, which makes storage less attractive.

5.3 Optimal Marketing Policy for Mildly Risk Averse Producers

This section focuses on optimal marketing policy for mildly risk averse producers (b
= 5). The optimal marketing rules for each of the decision months will be illustrated
graphically. The impacts of changes in parameters in the dynamic model are also shown.

We, again, use the base case with a monthly storage cost of 1 cent/bu and an annual
interest rate of 10% to illustrate the numerical results. The optimal marketing policy at the
initial stage (the month of November) is described graphically in Figure 5-1 (a). Since this
is the first stage of the decision horizon, the unique feasible initial storage state is 100 percent
of the harvest and the single feasible initial wealth state is zero. Therefore, the optimal
marketing policy is dependent only on the price state.

The optimal marketing policy in Figure 5-1(a) is interpreted as follows. If current
cash price is less than $1.7/bu, optimal marketing strategy is to store the entire stock; if
current cash price falls between $1 .7/bu and $1 .9/bu, optimal marketing volume is 50 percent
of the storage; and if current cash price falls within $1.9/bu and $3.7/bu, optimal grain for

sales is 60 percent of the grain in storage. Similarly, if current cash price is between $3.7/bu

96

 

marketing volume as a proportion of
the initial stock

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4
price(S/bu)

Figure 5-1(a) Optimal Marketing Policy for Mildly Risk Averse Producers
for the Month of November

and $3 .9/bu, optimal marketing strategy is to sell 70 percent of the storage; if the current cash
price falls within $3.9/bu and $4.1/bu, optimal marketing volume is 90 percent of the grain
in storage; while if current cash price is higher than or equal to $4.1/bu, optimal marketing
strategy is to sell all of the grain.

Here we have two essential points worthy of notice. First, when the risk neutrality
assumption is replaced with risk-aversion, the numerical outcomes support the theoretical
results derived in Chapter III that partial sales, as opposed to selling everything or nothing,
become possible optimal marketing strategies. From Figure 5-1(a), partial sales are
recommended when current price is between $1.9/bu and $4.1/bu. When current price is
higher than this range, the optimal marketing strategy is to sell everything whereas when the
price is lower than this range, optimal marketing niles suggest the entire stock be retained.
Secondly, when current price increases, benefits from immediate sales become larger which
leads to a consistent increase in optimal marketing volumes.

For each unit of stock, the marketing decision to be made at any point in time involves

97

the choice between the action “sell at the current price”, which has a certain outcome, and the
uncertain prospect “retain the stock and sell at a future price”. For decision-makers who
exhibit risk aversion, at least in some price ranges, may prefer a “combination” of the certain
act and the risky prospect, instead of taking one way or the other as risk neutral decision-
makers do. Through this way, they may gain an early return by immediate sales and use the
rest of the stock to speculate on higher prices. In other words, risk averse decision-makers
are willing to use only a portion of the stock to speculate on higher prices, while risk neutral
decision-makers prefer to use the total stock.

Optimal marketing policy for the month of January is presented in Figure 5- l (b). To
demonstrate a dynamic problem with three states of nature on a two-dimensional graph, one
of the state variables has to be specified in separate charts. The three panels of Figure 5-1 (b)
from top to bottom display the optimal marketing rules for high, middle, and low wealth
states, respectively. Since a given current storage level is the result of previous corn sales,
it can lead to many wealth levels depending on the price of which the corn was sold.

The high-, middle- and low-level of wealth states are defined based on the previously
determined wealth space. The feasible wealth levels of a given storage state are split equally
into high, middle, and low ranges. The mid point value of each range is used to represent
each of the three wealth states, as shown in the legend of each panel. It appears that within
any of the three wealth-level states, a higher current storage state always comes with a smaller
representative wealth level and vise versa. This is because higher current storage means less
corn has already been sold last period so that realized wealth remains low. A high wealth
state arises because a gain from selling a large portion of the initial storage before the current

stage and leads to a low current carry-over. Note that if the entire stock is retained until the

98

marketingvolumeuaproportionof

muketingvolumeasaproportionof

marketingvolumeasapmponionof

 

 

 

 

 

 

 

 

 

—O—s=l.0

—-—s=0.9 w=0.382
+s=0.8 w==0.783
+s=0.7 w= 1.185
+s=0.6 w= 1.586
—6—s=0.5 w= 1.988
—-—s=0.4 w=2.390
—+—s=0.3 w=2.79l
+s=0.2 w=3.l93
-B-—s=0.l w=3.594

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price(S/bu)

Panel 1 High Wealth Level

 

 

theinitialstock

 

 

 

 

+s=l.0

—--—s=0.9 w=0.287
+s=0.8 w=0.594
+s=0.7 w=0.900
+s=0.6 w= 1.207
-9—s=0.5 w= 1.514
—-—s=0.4 w= 1.821
—+—s=0.3 w=2.128
+s=0.2 w=2.434
—B—s=0.l w=2.74l

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price($/bu)

Panel 2 Middle Wealth Level

 

 

 

theim'tialstock

 

 

 

 

 

 

 

 

+5: 1.0

—-—s=0.9 w=0.l92
+s=0.8 w= 0.404
+s=0.7 w=0.615
+s=0.6 w= 0.828
—e—s=0.5 w= 1.040
—-—s=0.4 w= 1.252
—+—s=0.3 w= 1.464
+s=0.2 w= 1.675
—B—s=0.1 w= 1.888

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price(S/bu)

Figure 5-l(b)

Panel 3 Low Wealth Level

Optimal Marketing Policy for Mildly Risk Averse Producers

for the Month of January

99

 

 

 

current stage, meaning no sales have been made in the previous stage, only one wealth state
is feasible. This wealth state is the negative compounded storage costs covering the storage
period since harvest. The negative wealth value does not violate the positive argument
assumption in the utility function because the decision criterion in the current study is based
on the utility of the final wealth, instead of the cash flows or the cumulative wealth at each
stage.

Empirical results on the optimal marketing policy for the month of January in Figure
5-l(b) can be interpreted as follows. From the second panel of Figure S-l(b), optimal
marketing strategy for producers who have 90 percent of the initial storage left (i.e. 10
percent sold in November) at middle wealth state (i.e. average sales price in November) is to
retain the entire stock if the current cash price is less than $1 .9/bu or falls within $3 .3/bu and
$3.9/bu. If the current cash price is between $1.9/bu and $2.1/bu, or is higher than $2.3/bu
but less than $3.3/bu, or falls within $3.9/bu and $4.1/bu, the optimal marketing strategy is
to sell 10 percent of the initial storage. If the current cash price falls between $2.1/bu and
$2.3/bu, optimal marketing volume is 20 percent of the initial stock. If current cash price
goes higher than $4. l/bu, the optimal marketing strategy is to empty the storage at the current
stage.

This empirical result does not seem as intuitive as that at the initial stage, in the sense
that the optimal marketing volumes do not always increase as current price increases. A
possible explanation is that the current price state not only represents the benefits of
immediate sales, which encourages marketing volumes to increase as current price increases,
but it also determines future price distributions which, in turn, determines the expected option

value of current stocks. Hence, if the increased expected option value is higher than the

100

increased benefits of immediate sales, the optimal marketing policy will suggest holding a
larger portion of the stocks and lead to selling less at a higher price. From Table 5-4(b),
January prices of $2.0/bu, $2.2/bu, $2.4/bu, $2.6/bu, $2.8/bu, $3.0/bu, $3.2/bu, $3.4/bu,
$3.6/bu, and $3.8/bu have around 47%, 49.5%, 51.9%, 53.8%, 55.5%, 56.9%, 58.5%,
59.4%, 60.3%, and 61% probabilities, respectively, to go up in March. The fact that there
is a high probability for a January price to go up when the current price level is high may
support the above argument.

However, simply looking at the probabilities of the possible future prices may not be
sufficient and could be misleading since risk averse agents’ decision-making process usually
is far more complicated than comparing the current price and the option value of the
inventory. Risk averse decision-makers put different weights on the current price and the
option value of the stock. Intuitively, highly risk averse agents put more weight on the
current price than the option value, relative to less risk averse agents, which tends to
encourage an immediate sale and, in turn, gain a certain income earlier. Risk neutral agents,
instead, have an equal weight on the current price and the marginal option value of the stock.
The magnitudes of the weights for risk averse agents depend on the marginal expected utility
of final wealth as shown in Chapter III. Namely, the decision-maker’s current wealth and
storage states, and his degrees of risk reversion, as well as the expectation on the future price
distributions, all have an influence on the values of the weights. These factors usually are
interdependent, which makes it difficult to predict the direction of changes in optimal
marketing volumes when the current price increases.

It appears that the optimal storage is generally much higher in November than

January. A reasonable explanation is that a risk averse decision-maker prefers to earn a

101

certain income earlier by selling a portion of stocks in the early stage, then use the rest of the
stocks to speculate the higher future prices. The findings are consistent with the results of
Berg’s risk averse models. The fact that January prices have a higher tendency to increase
than November prices may help to encourage waiting in January.

We continue the analysis by comparing the three panels in Figure 5- 1 (b). It is found
that, given a storage level and price outcome, the optimal marketing strategy at high wealth
states tend to suggest a smaller current marketing volume than at lower wealth states. Thus,
the numerical results suggest that wealthier agents are less willing to sell now and more likely
to store. This is because of the CRRA utility function, whereby a decision-maker becomes
less risk averse when his or her wealth increases, and therefore becomes more willing to take
the risk of storing.

Optimal marketing rules for the months of March, May and July are presented in
Figure 5-l(c), (d) and (e), respectively. Each figure consists of three panels to capture the
high, middle and low wealth states. Wealth states at each stage have to be re-constructed
based on the feasible wealth space of the given stage. Again, partial sales are optimal
marketing strategies under certain price and wealth states.

We proceed to compare the optimal marketing policies across stages. This shows
that, given the same scenario of current price, storage, and wealth states, optimal marketing
policies in the earlier stages generally suggest smaller volumes of sales than the later stages.
This is because time value of the stocks is diminishing and the future prices have less chance
to increase as we approach the next harvest. Corn prices are even expected to decrease in
late spring, as discussed in section 4.3.3. Both factors suggest a lower option value of the

given stocks which implies less benefit to holding longer. Another possible explanation is that

102

marketigvohrmeuaproportionof muketingvohmeuaproportionof

mket'ngvolumeofaproporfiouof

 

 

 

 

+s=1.0

—-—s=0.9 w=0.368
+s=0.8 w=0.778
+s=0.7 w= 1.187
+s=0.6 w= 1.596
—-6—-s=0.5 W'2.006
—o—s=0.4 w=2.415
—+—s=0.3 w=2.824
+s=0.2 w=3.234
—B—s=0.1w=3.643

 

1.6 1.8 2.0 22 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price($/bu)

Panel 1 High Wealth Level

 

 

 

 

+s= 1.0

———s=0.9 w=0.271
+s=0.8 w“0.582
+s=0.7 w-0.894
+s=0.6 w= 1.205
-e—s=0.5 w= 1.517
—o—s=0.4 w= 1.828
—+—s=0.3 w=2.139
+s=0.2 w=2.451
-B—s=0.l w=2.762

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price(S/bu)

Panel 2 Middle Wealth Level

 

 

 

+s=1.0

---s=0.9 w=0.173
+s=0.8 w=0.386
+s=0.7 w=0.600
+S=0.6 w=0.813
—O—s=0.5 w= 1.027
—-—S=0.4 w=1.241
—-1—S=0.3 w=1.454
+S=0.2 W‘B 1.668
+S=0.1 w= 1.881

 

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price(S/bu)

Panel 3 Low Wealth Level

Figure 5-l(c) Optimal Marketing Policy for Mildly Risk Averse Producers

for the Month of March

103

 

 

marketingvohmeuaproponionof marketigvohmeuaproportiouof

mu'ketirrgvohtmeuaproponionof

m.m] I

Figure 5-1(d)

1.0 ~
0.9 1
0.8 -<
0.7 <

.......

0,5 .

0.5 4

0.4

034--
024-

 

0.1
0.0

0.9 4

 

1.0 -
0.9 <
0.8 <
0.7 1

0.6 l----
0.5 1».
0.4
0.3
02 «-

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8

 

0.1
0.0

 

 

 

 

l 1
T f T f r fi T _‘

Panel 1 High Wealth Level

 

 

 

 

4L
1

A 1 A A A
V Y 1' Vi T Y T

Panel 2 Middle Wealth Level

 

 

 

 

 

 

Y Y T f U

Panel 3 Low Wealth Level

 

 

—O—s= 1.0

—-—s=0.9 w=0.356
+s=0.8 w=0.772
+s=0.7 w= 1.190
+s=0.6 w= 1.607
+s=0.5 W‘2.024
—o—s*0.4 W'2.441
—+—s=0.3 w‘2.858
+s=0.2 «H 3.275
+s=0.1 W'3.692

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price($/bu)

 

 

+s= 1.0

———s=0.9 w=0.255
+s=0.8 w=0.570
+s=0.7 w= 0.887
+s=0.6 w- 1.203
-O—s=0.5 w- 1.519
—o—s=-0.4 w= 1.835
—+—s=0.3 w=2.151
+s=0.2 w= 2.467
+s=0.l w=2.783

 

4.0 4.2 4.4 price (slbu)

 

 

+s= 1.0

—-—s=0.9 w=0.154
+s=0.8 w-0.368
+s=0.7 w=0.584
+s=0.6 w=0.798
—O—s=0.5 w= 1.014
--—s=0.4 w= 1.229
—lv—s=0.3 w= 1.444
+s=0.2 «P 1.659
+s=0.1w= 1.874

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price (S/bu)

Optimal Marketing Policy for Mildly Risk Averse Producers

for the Month of May

104

 

 

 

marketingvolumeasapmportionof marketingvohimeuaproportionof
theim'tialstock theilitillstock

marketing volume as a proportion of
the initial stock

1.0 -
0.9

 

 

0.8 «~-
0.7 1"

0.6 .

 

0.5
0.4
0.3
0.2
0.1

0.0 : .
1.6 1.8 2.0 2.2 2.4

1.0 -
0.9

 

 

T

A
r T

Panel 1

f ‘r

 

 

’—‘

 

+s=l.0

—-—s=0.9 w=0.404
+s=0.8 w=0.823
+s=0.7 w= 1.241
-)(—s=0.6 w= 1.658
—O—s=0.5 w= 2.077
-—-—s=0.4 w=2.495
—l-—s=0.3 w=2.913
+s=0.2 w=3.331
—B—s=0.1w=3.750

 

2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price(S/bu)
High Wealth Level

 

 

0.8
0.7 y.-

0.6 4

 

0.5
0.4
0.3

0.2
0.1
0.0

 

 

 

 

 

 

 

0.0

f I T

A
17

A
Y

-
f - - A - - A Q ‘ - - . - I
.

 

+s= 1.0

—-—s=0.9 w=0.275
+s=0.8 w=0.592
+s=0.7 w=0.909
+s=0.6 w= 1.225
—6—s=0.5 w= 1.542
—-—s=0.4 w= 1.859
—l~—s=0.3 w=2.176
+s=0.2 w=2.492
—B—s=0.1w=2.809

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price($/bu)
Panel 2 Middle Wealth Level

 

 

 

     

db
‘h
db
{I
d.
I.

 

 

1

A
r T Y I

A

A
T

j

 

+s=1.0

—-—s=0.9w=0.l46
+s=0.8w=0.361
+s=0.7 w=0.577
+8 =0.6 w=0.792
+s=0.5 w= 1.007
—-—s=0.4 w= 1.222
-1—s=0.3 w= 1.438
+s=0.2 w= 1.653
—B—s=0.l w= 1.868

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price($/bu)

 

 

 

Figure 5-l(e)

Panel 3 Low Wealth Level

Optimal Marketing Policy for Mildly Risk Averse Producers
for the Month of July

105

the weights a risk-averse decision-maker places on the current price and the option value for
the given stock also change over time. A risk averse agent may weight a dollar of certain
income more at the later stages when the selling opportunity is getting smaller, and therefore
is more willing to sell at any price in the later marketing periods.

In the months of March, May and July, it seems that, as the current price increases,
optimal marketing volumes increase consistently, as opposed to the counter intuitive storage
rules in the month of January. As a matter of fact, an inconsistent storage rule exists also in
the month of March at some specific wealth states. For the months of May and July, as the
current price increases, not only do the probabilities for price decreases rise in smaller
increments than January and March prices, but the probabilities for prices to go down also
increase. Hence, increases in the current price may not lead to an equivalent increases in
option values, as in the value of immediate sale, through the changes of future price
distributions. Couple this with the higher weights on current prices in later marketing periods
(May and July), and increases in current price may simply dominate the optimal storage rules.

Two selected comparative static results between various model parameters are
demonstrated in Figure 5-2. Figure 5-2(a) illustrates the optimal marketing rules for the
month ofNovember when the monthly storage costs increases from the base case of l cent/bu
to 2 cents/bu and decreases to O cent/bu, keeping the annual interest rates at the base case of
10%. Intuitively, a higher rate of storage cost makes storage less desirable because the
anticipated income fi’om storage decreases, which calls for a larger optimal volume of sales
for given price states at a given stage. For example, when the current price falls in the state
of $3.0/bu, the optimal marketing volumes for storage costs of 2 cents/bu, l cent/bu and 0

cent/bu are 90 percent, 60 percent and 50 percent of initial storage, respectively.

106

ofthe initial stock

marketing volume as a proportion

marketing volume as a proportion
of the initial stock

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4
priee($/bu)

 

[I storage cost = 0 storage cost = 1 cent/bu [:1 storage cost = 2 cents/bu

 

Figure 5-2 (a) Optimal Marketing Policy for the Month of November
at Different Rates of Storage Costs

 

 

 

 

 

 

 

 

 

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4
price(S/bu)

 

[I interest rate = 5% E interest rate = 10% Dinterest rate = 15%

 

Figure 5-2 (b) Optimal Marketing Policy for the Month of November
at Different Annual Interest Rates

107

Figure 5-2(b) gives impacts of interest rates on the optimal marketing rules for the
month of November. Annual interest rates increasing from the base case of 10% to 15% and
decreasing to 5% are considered. Similarly, an increase in interest rates makes storage less
attractive as the opportunity costs of storing become higher, which also entails a larger
marketing volume for given price states at a given stage. For example, if the current price is
$3.0/bu, the optimal marketing strategies for annual interest rates of 15%, 10% and 5% are
to sell 80 percent, 60 percent and 50 percent of initial storage, respectively.

The changes in storage costs and interest rates for the rest of the marketing periods
have a similar impact on the optimal marketing policies as in the initial stage. That is,
increases in either of the parameters suggest higher optimal marketing volumes for given

storage, wealth and price states at given stage.

5.4 Optimal Marketing Policy for Highly Risk Averse Producers

In this section, we change the risk coefficient from the base case of mildly risk averse
(b = 5) to highly risk averse (b = 10). The optimal marketing policy for the base case will be
demonstrated graphically for each decision period throughout the marketing year. A
comparison of the optimal marketing rules between cases of various degrees of risk aversion
will also be conducted.

Figure 5-3(a) displays the optimal marketing policy for the month of November. The
three panels in each of Figure 5-3 (b) to (e), again, represent the high, middle and low wealth

states for the months of January, March, May and July. The numerical results show that

108

 

 

the initial stock

 

 

marketingvolumeasaproportionof

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4
price(S/bu)

Figure 5—3 (a) Optimal Marketing Policy for Highly Risk Averse Producers
for the Month of November

partial sales are again optimal marketing strategies at certain price states.

When compared with the case of mild risk aversion, optimal marketing strategies for
highly risk averse producers, for given storage, wealth and price states at a given stage,
suggest a larger volume of sales at the beginning of the marketing year. For example, the
optimal cash grain sales recommended for highly risk averse producers is 80 percent of initial
storage at the initial stage when current price falls between $1 .9/bu and $3.7/bu whereas
under the same state scenario the optimal marketing volume for mildly risk averse producers
is 60 percent of initial storage. It is intuitively reasonable for highly risk averse producers to
price the stocks earlier in order to reduce risks of unfavorable price movement later. This
implies that portions of the stock may be sold at lower prices, compared to the case of mild
risk aversion, and the average storage time may decrease accordingly due to a more cautious
marketing policy.

The optimal marketing rules in January, at certain storage and wealth levels, during

certain price ranges, again suggest storing more as the current price increases. This also

109

marketingvohrmeuaproportionof
the"'l l

theilitial stock

MWuaproportionof

marketingvohrmeasaproportionof

 

 

 

 

 

-O—s=1.0

—--s-0.9 W'0.382
+s=0.8 w=0.783
+s=0.7 w- 1.185
—)(-—s=0.6 w= 1.586
+s-O.5 w' 1.988
—-—s=0.4 w= 2.390
—+—s=0.3 w=2.791
+s=0.2 w= 3.193
+s=0.1w=3.594

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price($/bu)

Panel 1 High Wealth Level

 

 

 
 

0.4 T .....
0.3 1"
0.2 -
0.1
0.0

 

 

.‘,

 

 

+s= 1.0

—-—s=0.9 w= 0.287
+s=0.8 w= 0.594
+s=0.7 w= 0.900
+s=0.6 w= 1.207
+s=0.5 w= 1.514
-—-—s=0.4 w= 1.821
—l—s=0.3 w=2.128
+s=0.2 w= 2.434
—B—s=0.1w=2.741

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price($/bu)

Panel 2 Middle Wealth Level

 

1.0
09 . ...................................................................................................................
0.3 . ................................................................................................................
0.7
06 'l .......

0.5 - — - A
““7“ ‘
0.3 1.. ' .
0.. /

0.1
0.0

 

  
  
 
     
 

 

 

 

'/

‘

 

 

+s= 1.0

—-—-—s=0.9 w=0.192
+s=0.8 w=0.404
+s=0.7 w*0.615
-)(--s=0.6 w=0.828
—9-—s=0.5 w= 1.040
—-—s=0.4 w=1.252
—l-—s=0.3 w= 1.464
+s=0.2 w= 1.675
—B-—s=0.1w=1.888

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 pricc($/bu)

Panel 3 Low Wealth Level

Figure 5-3 (h) Optimal Marketing Policy for Highly Risk Averse Producers

for the Month of January

110

 

 

 

marketingvohrmeasaproportionof marketingvolumeaaaproportionof

marketingvohrmeasapropoflionof

1.0

 

0.9

 

0.8 -
0.7 4
0.6 1

theinitialatoek

02«

 

 

0.5
0.4
0.3

 

0.1
0.0

1.0

 

 

1.6 1.8 2.0 22 2.4 2.6 2.8 3.0 32 3.4 3.6 3.8 4.0 4.2 4.4

Panel 1 High Wealth Level

 

0.9

 

0.8 w
0.7 <
0.6 ~

 

0.5 +----
0.4 +--
0.3 +-

 

02
0.1
0.0

1.0

1.6 1.8 2.0 22 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4

Panel 2 Middle Wealth Level

 

0.9

 

0.8 ~
0.7 «
0.6 ~
0.5 4

initial stock

021

0.4

3 0.3 +-

 

 

0.1
0.0

 

 

+s=1.0

—--8=0.9 w=0.368
+s=0.8 w=0.778
+s=0.7 w= 1.187
+s=0.6 w=1.596
—9—s=0.5 w=2.006
—-—s=0.4 w=2.415
-—+-—s=0.3 w= 2.824
+s=02 w= 3.234
+s=0.l w=3.643

 

 

 

price ($/bu)

 

 

 

+s=1.0

—-—s=0.9 w=0.271
+s=0.8 w=0.582
+s=0.7 w= 0.894
+s=0.6 w= 1.205
+s=0.5 w=1.517
—-—s=0.4 w= 1.828
—1-—s=0.3 w=2.139
+s=02 w=2.451
-B—s==0.1w-2.762

 

 

 

 

 

price (9130)

 

+s= 1.0

-—-——s=0.9 w=0.173
+s=0.8 w= 0.386
+s=0.7 w=0.600
+s=0.6 w=0.813
—6—S=0.5 w= 1.027
—o—s=0.4 W=1.241
—l-—s=0.3 w= 1.454
+s=02 w= 1.668
—B—s=0.1 w= 1.881

 

 

A
j

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price($/bu)

Panel 3 Low Wealth Level

 

 

Figure 5-3(c) Optimal Marketing Policy for Highly Risk Averse Producers

for the Month of March

111

1.0 -
0.9

 

 

0.8 1

0.1 .
0.6 4 ..

 

0.5
0.4
0.3
0.2
0.1
0.0

the'mitinlatoek

markefirgvohimeaaaproportionof

 

 

 

 

 

 

market'ngvohrneasaproportionof

0.0

theinitinlstock

marketingvolumeasaproportionof

0.0

A
Y

A A A A
T fir f Y

 

+s= 1.0

—-—s=0.9 w=0.356
+s=0.8 w=0.772
+s=0.7 w=1.190
+s=0.6 w=1.607
—G—s=0.5 w=2.024
—-—s=0.4 w=2.441
—+—s=0.3 w= 2.858
+s=02 w= 3.275
—B—s=0.l w=3.692

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price($/bu)
Panell High Wealth Level

 

 

 

 

 

 

 

A
Y

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price($/bu)
Panel 2 Middle Wealth Level

A A A A
f f f T

 

—O—s= 1.0

—-—s=0.9 w=0.255
+s=0.8 w=0.570
-—O—s=0.7 w=0.887
+s=0.6 w=1.203
—0-—s=0.5 w=1.519
—-—s=0.4 w=1.835
—t—s=0.3 w=2.151
+s=02 w=2.467
—8—s=0.1w=2.783

 

 

 

 

 

 

 

A A
‘F r T 1‘

Panel 3 Low Wealth Level

j

 

+5: 1.0

—-—s=0.9 w=0.154
+s=0.8 w= 0.368
+s=0.7 w=0.584
+s=0.6 w= 0.798
—O—s=0.5 w=1.014
—-—s=0.4 w=1.229
—+-—s=0.3 w=1.444
+s=02 w=1.659
-B—s=0.1w=1.874

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price($/bu)

Figure 5-3(d) Optimal Marketing Policy for Highly Risk Averse Producers

for the Month of May

112

 

 

 

 

 

 

theinitialatoek

 

 

 

 

marketingvolurneesaproportionof

0.0 . :

A
Av,

2.0 2.2 2.4

T

Panel 1

A
T

*7

A A
T Y

 

+s=1.0

—-—-s=0.9 w=0.404
+s=0.8 w=0.823
+s=0.7 w= 1.241
+s=0.6 w=1.658
-6—s=0.5 w=2.077
—-—s=0.4 w=2.495
—l—s=0.3 w=2.913
+s=02 w=3.331
—8—s=0.1 w= 3.750

 

2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price($/bu)
High Wealth Level

 

 

 

 

0.0

4»

marketing volume as a proportion of
theinitial stock

1.6 1.8

1.0

0.8
0.7
0.6
0.5

02
0.1

marketingvolumeasaproportionof

0.4 - -
0.3 - .

0.0 4 ¢

2.0 2.2 2.4

A
r

2.6 2.8 3

4)-

4r-
1-

A l
I j

.0 3.2 3.4 3.6 3.8 4.0 4

Panel 2 Middle Wealth Level

A
T

L

A
1

A r
f fir T r

 

 

V

 

+s= 1.0

—-—s=0.9 w=0.275
+s=0.8 w=0.592
+s=0.7 w=0.909
—)(—s=0.6 w=1.225
—9—s=0.5 w= 1.542
—o—s=0.4 w=1.859
—~l-—s=0.3 w=2.176
+s=02 w=2.492
—B—s=0.1w=2.809

 

.2 4.4 price (S/bu)

A
fi

 

 

 

 

+s= 1.0

—-—s=0.9 w=0.l46
+s=0.8 w=0.361
+s=0.7 w=0.577
—-)(-—s=0.6 w= 0.792
—9—s=0.5 w=1.007
—-—s=0.4 w=1.222
—l—s==0.3 w=1.438
+s=02 w=1.653
—B—s=0.1w=l.868

 

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 price($/bu)

Figure 5-3(e)

Panel 3 Low Wealth Level

113

Optimal Marketing Policy for Highly Risk Averse Producers
for the Month of July

 

 

 

occurs in the month of March but is under different wealth states rather than the three
representative wealth levels. Similar to the optimal storage rules for mildly risk averse
farmers, the optimal sales in higher wealth level states are generally lower than the lower
wealth level for a given storage level due to the CARA utility function. In general, the
optimal marketing strategies for the highly risk averse farmers share a very similar pattern as
those for the mildly risk averse farmers, and substantially different from those for nearly risk

neutral farmers.

5.5 Evaluating an Optimal Marketing Policy

In this section, the optimal marketing policies presented in preceding sections for
various degrees of risk averse producers will be applied to a set of simulated market
scenarios. For this purpose, a risky market environment first has to be specified. This will
be undertaken by constructing the price outcomes and the associated probabilities from the
transition probability matrices in Table 4-5. By applying the optimal marketing rules to each
of the possible price outcomes, while assuming that future prices are not known at the time
marketing decisions are made, the resulting expected marketing volumes of each decision
period will be examined, followed by the probability distributions of the expected final wealth

under application of the optimal marketing policy.

114

5.5.1 Expected Optimal Marketing Volumes

To obtain the expected optimal marketing volumes, a risky market environment
characterized by all possible price outcomes throughout the marketing period, and their
associated probabilities, has to be constructed. The stochastic structure of prices can be
computed fiom the transition probabilities specified in the preceding chapter. For illustration
purposes, a simplified case with two stochastic states, 17, and [72 , and two decision
periods, I and (+1 is considered 2. For convenience, we rewrite the transition probabilities

defined in (4.14) here:

1 “it "'21 2 7‘11 7‘21
II = r r and II = 2 2
7:12 7'22 7‘12 7‘22

where n; gives the probability that state i at I will be followed by state j at 1+1 for r = l, and
the probability that state i at (+1 will be followed by statej at 1+2 for r = 2.
Given initial price P..- , the conditional probability that price at (+1 is ,3} followed

by p—k at (+2 is affix; ,where i,j,k= 1,2:

pr°b(P¢+1=p—pp¢+2=fik 1 p55,) (5-2)
= prob(p...=i,|p.=13,) X prob(p..2=13.lp...=17,)

_ r 2 .. __
— 7:”. X 191. 1,},k— 1,2

 

2 Two decision nodes imply three periods in the entire decision horizon.

115

2 2
r 2 _ ._
where 21% nyxrtjk - l ,forr—lor2.
j: :

Now, we assume the probabilities for the initial price to be at state 1 and state 2
are K, and K, ,respectively, that is, prob(p,= pi) = K1 and prob(p,=172) = K2 . The

probabilities for all possible price outcomes across the decision horizon can be expressed as

— — — 1 2
pr0b(p::ppp(+|:pjrpuzzpk) = K; X “1] X njk (53)

where Z2: :2: 22: KIXnkrr; = l . There are 23 possible combination of price outcomes
in this case.

Furthermore, when the numbers of price states and decision periods expand, the
number of potential price outcomes increases exponentially . Assuming m price states and
T decision periods, the total number of possible price outcomes becomes mT+1 . In practice,
some of the price outcomes have negligibly low possibility of occurrences. Accordingly, the
associated probabilities are zero or so close to zero that can be ignored. Let 0" denote the
probability of the n-th price outcome as computed in (5-3), where n = 1, 2, , N, and N s
m’+1 . In this study, all the numbers needed to compute 9" are available from the conditional
probabilities in Table 4-5(a) to (e), except the probability of price occurrence at the initial
stage (re). An unconditional frequency tabulation of the historical prices at the corresponding
period is used to approximate the possibilities of price occurrence at the initial stage. The end
result is a set of N price outcomes.

Optimal marketing policies computed in preceding sections are now applied to each
of the N price outcomes. Given each price scenario, we obtain the optimal marketing

volumes over the marketing year, denoted as { q,", q,':l,...., qrzrin=r.2....,rv . Expected

116

marketing volumes at t then are calculated by:

Eq, = Z q,"xe" 1: 1,2, 7‘ (5.4)

where E is the expectation operator.

Expected marketing volumes over the marketing horizon for various degrees of risk
aversion are presented in Figure 5-4(a)-(c). It shows that if the optimal marketing rules for
nearly risk neutral producers are adopted, about 83% of the time the entire harvest is
expected to be sold in May, about the time the seasonal price growth usually reaches its peak.
Almost no transactions are expected to occur before March in this case. Around 8% of the
time the harvest is sold in July and 7% of the time in September.

Expected optimal marketing volumes during the marketing season for mildly risk
averse producers are substantially different from the case of near risk neutrality. The way to
interpret the results is also different in that only two actions are considered in the risk neutral
case while the marketing strategies are more sophisticated in the risk averse case. The
simulation results for mildly risk averse producers suggest that about 52 percent of the
harvest is expected to be sold right in November and around 39 percent is expected to be
marketed in May, while less than 3 percent is expected to be held until September.

At high risk aversion, the general pattern remain largely the same as in the case of mild
risk aversion. Over 70 percent of initial storage is expected to be marketed in November,
which is higher than the case of mild risk aversion. About 20 percent of the harvest is
expected to be marketed in May, and only about 3 percent is expected to be held on the farm

after May. It appears that November and May are the two most active marketing periods for

117

 

Percentage of Initial Stocks
es
:3

Optimal Marketing Volumes as

1 00%
80%
60%
40%
20%

0%

Percentage of Initial Stocks

Optimal Marketing Volumes as

l 00%
80%
60%
40%
20%

Percentage of Initial Stocks

0%

Optimal Marketing Volumes as

 

 

52 0%

 

 

 

 

 

 

 

 

 

NOV JAN MAR MAY JUL SEP
(a) Mild Risk Aversion
73.2%

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

NOV JAN MAR MAY JUL SEP
(b) High Risk Aversion

83.2%
0.0% 0.2% 1.3%

 

 

 

 

NOV

JAN MAR MAY JUL SEP

(c) Near Risk Neutrality

Figure 5-4 Expected Marketing Volumes for Various Degrees of Risk Aversion

118

 

risk averse farmers. These patterns are being caused by the high probability of price increases
implied in transition probabilities of J anuary and March, which suggest a high potential benefit
from waiting until May to sell. The weights of sales between these two periods, however, are
more balanced in the case of mild risk aversion than high risk aversion. It is also found that
an increasing degree of risk aversion results in a growing amount to be marketed at the
beginning of the marketing season. It is expected that the extremely risk averse producer will
sell everything at harvest. This also illustrates that the main effects of different attitudes
toward risk show up in the first decision month.

The impacts of storage costs and interest rates on the optimal timing of marketing are
given in Figure 5-5 to Figure 5-7 for various degrees of risk aversion. It appears that low
storage costs and low interest rates result in holding the grain longer while high storage costs
and high interest rates suggest call an earlier sale for all degrees of risk aversion. Especially
in the case of near risk neutrality, the major marketing period shifts from May to July when

there are low storage costs and low interest rates.

5.5.2 Probability Distribution of Final Wealth

For the n-th possible price outcome, producers’ final wealth can be computed as:
r
w, = Z (1+r)T"*'(p, q, — as, ) (5.5)
1:1

where r is the interest rate; s," is the storage at t given the n-th possible price outcome; or

is the unit storage cost.

119

 

 

 

 

52.0%

 

 

 

 

 

 

 

 

(a) base case (k= 1 cent/bu, r = 10%)

 

 

 

 

 

 

 

 

   

 

  

 

  

 

 

 

 

«a la)“ a 100°/
g a a .3 ° 72.4%
§ 80% E g 80%
g m 50.9% '5
> a > '3' 60% 4
a 5’ 5
5° «.5 g “3' 40% l
3’ . .
E g 2 g 20°” r.4%2.2% rs%o.7%
'21“- 2;“ 0% J
> >- e-
g 3. E g i a:

k = 2 cents/bu

 

(b) Changes in Storage Costs

 

 

 

 

48.3%

 

 

 

 

   

 

 

2.2%2.9% 1.7% 1.1%

   
 

 

 

Percentage of Initial Stocks
h
o
g

Optimal Marketing Volumes as a

 

 

 

Optimal Marketing Volumes as a
Percentage of Initial Stocks

r=15%

P

> 5’:

O
2

$522

r=5°

e\

(C) Changes in Interest Rates

Figure 56 Expected Optimal Marketing Volumes for Mild Risk Aversion under
Various Storage Costs and Interest Rates

120

 

 

73.2%

 

 

 

  

 

1.47.2.5“ 1.7% 1.1%

 

 

     

OptinnlMarket'mgVohtmesasa
PeroentageoflnitialStocka
N
s s

gaggsa

(a) base case (k: l cent/bu, r = 10%)

 

 

 

 

 

87.6%

 

 

 

 

    

 

 

9.9%
0.5%0.8%

k = 2 cents/bu

 

 

 

0.8%0.4°/

 

    

 

OptimalMarketingVolumesasa
Pereentageoflnitial Stocks
N
s a s

 

(b) Changes in Storage Costs

100%
80% -
60% «
40% ~
20% -

0% .

 

 

85.2%

...
.\.§

 

 

 

 

  

 

 

10.8%
r.3%r.2% 1.1%o.4*y

0..
5m
m

 

 

 

      

 

 

Optimal Marketing Volumes as a
Percentage of Initial Stocks

OptimalMarketingVolumesasa
PereentageoflnitialStoeks

MAY

 

(C) Changes in Interest Rates

Figure 5-6 Expected Optimal Marketing Volumes for High Risk Aversion under
Various Storage Costs and Interest Rates

121

 

 

83.2%

 

M?

 

8
x

 

8.2967. 1%

 

N
c
3

Percentage of Initial Stocks

 

0.0% 0.2%1-3%

ES

.3

OptimalMarketingVolumeeasa

>..

g5

>
0
Z

--i

 

 

 

0-
u}
U)

 

 

(a) base case (k= 1 cent/bu, r = 10%)

 

 

 

 

   

 

 

 

  

' 100% a 100%
a a a

g 80% 2 § 80%
m 60.6% .3 w

> 3 60% > 75' 60%
g5 g3

0 40‘/e lg b5 40%

E 20% 2 3’ 20%
0.0%0.2%l.3% g g

- a. 0% - go. 0%

>
2 i S 3 fig
k=0centlbu

(b) Changes in Storage Costs

 

 

 

 

   

 

  

 

 

; “KP/e a: 100%
fig 80'. 67.7% 2:5 80%
g m

3% 60% >°§ 60%
EE 40% 21.4% 9.4tyom g; 40%
3’ 20% 0.0%o.2%l.3% g g 20%
EE 0% ‘ > 0.. “go. 0%

>

g .5. g g 3 a
r=5%

(C) Changes in Interest Rates

92 2%

 

 

 

 

  

 

2.2%4. 1%

  
 

D.0%0.5%1.0%

 

 

_> > n.
g E 3 § 5 a
k=2centslbu

83.1%

 

 

 

 

 

8.2% 7.1°/

  

 

  

0.0%0.5% 1.0%

 

 

 

Figure 5-7 Expected Optimal Marketing Volumes for Near Risk Neutrality under

Various Storage Costs and Interest Rates

122

 

The probability distributions of final wealth for various degrees of risk aversion are
given in Figure 5-8. The distributions of final wealth seem to skew to the higher price level.
The final wealth distribution becomes more centralized as the relative risk aversion coefficient
increases. Namely, the possible outcomes of final wealth for higher risk averse producers is
likely to be in a confined range of levels while final wealth for lower risk averse producers has
more chances to drop to very low levels or jump to a relatively high level. The probability
distribution of final wealth can be usefirl for producers to identify final wealth distributions

obtainable from different marketing decision rules.

5.6 Summary

Numerical results show that when risk aversion is allowed in the decision model,
partial sales become possible optimal marketing strategies. This marketing rule may be better
suited for practical use than the overly simplified sell-all-or-none decision rules in the
literature.

It is also found that risk preference, storage costs and interest rates have substantial
impacts on the optimal timing of sales. Higher degrees of risk aversion call for an earlier sale.
Higher storage costs and higher interest rates suggest holding the grain longer. This implies
that analysts who will provide decision-making support to farmer§ marketing corn need to
take many difi‘erent factors into account to come up with effective and useful

recommendations.

123

 

 

20%

 

 

15%

 

 

10%

 

5%~

  

i-

 

 

0/0— T I I
1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8

Final Wealth (dollars per bushel of initial stocks)
(a) Mild Risk Aversion

 

20%

 

 

15%

 

10% J

 

5%-

 

 

 

0%-
1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8

Final Wealth (dollars per bushel of initial stocks)

(b) High Risk Aversion

 

20%

 

15%

 

 

10%

    

 

5%J

 

 

 

.... IIII III

1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8
Final Wealth (dollars per bushel of initial stocks)

(c) Near Risk Neutrality

Figure 5-8 Probability Distribution of Final Wealth

124

CHAPTER X1

ECONONIIC VALUE OF THE OPTIMAL MARKETING STRATEGY

The objective of this chapter is to measure the economic value of the optimal
marketing strategy by computing producers’ willingness to pay for the right to apply the
proposed marketing decision rules, rather than sell the production immediately at harvest.
This also provides quantitative support for the usefirlness of marketing decision models.

Economic value of the optimal marketing strategy is considered in two parts. First,
applying optimal marketing rules may lead to an increase in the average level of final wealth,
compared to the benchmark case when storage is not allowed. Secondly, the optimal strategy
may also reduce the associated variance of final wealth. Each of these potential outcomes
alone has positive impacts on the economic value of the optimal marketing strategy, and in
most situations the occurrence of the outcomes are interdependent. It appears that decision-
makers with different risk preferences weight these impacts differently. For example, facing
the same risky situation, highly risk averse decision-makers are less willing to give up a
certain amount of return in exchange for a high expected return than risk neutral decision-
makers. Willingness to pay is an assessment of the total impact. Accordingly, it is reasonable
to expect substantial differences in willingness to pay for the optimal marketing strategy
depending on different attitudes toward risk.

In addition, as concluded in Chapter IV, the level of initial price has a strong influence

125

on the price distributions of subsequent periods in the marketing season. This means that
price at harvest signals the stochastic structure of the price series for the given marketing year
which indicates the potential of profits from storing. Intuitively, the economic value of the
optimal marketing strategy is expected to be related to the initial price state. This relationship
will also be investigated in this chapter.

The definition of the producers’ willingness to pay for optimal marketing strategy is
illustrated in the first section of the chapter, followed by a discussion of numerical results and

economic implications in the second section.

6.1 Definition of Willingness to Pay

The willingness to pay for the optimal marketing policy is defined as the maximum
certain net retum, in monetary terms, that the producer is willing to pay for the right to apply
the marketing strategies rather than sell the production right at harvest. In other words, it is
the minimum monetary compensation to make a producer who sells all of his grain at harvest
able to reach the same level of expected utility as if he adopts the optimal marketing

strategies. Using mathematical notation, the willingness-to-pay (WT P) is determined by:

Ear/(w, + WTPX(1 +r)T) = E0 r/(wT') (6.1)

where E is the expectation operator with O indicative of the harvest time before the first

decision period; U(°) is the utility firnction; r is the interest rate; WT is the producer’s final

126

wealth when storage is not allowed, calculated by:

WT = pl (1 +r)T (6-2)

and w; is the producer’s final wealth when the optimal marketing strategy is applied,

computed by:

T
I:

where q, and sM are optimal marketing volumes and the carry—over at 1, respectively.

 

To solve for WTP in equation (6.1), given simulated price realizations and associated L

probabilities with U(-) replaced with the CRRA functional form, (6. 1) is rewritten as:

i [‘m + WYI'P;(1+T)T]l-bx9(n) = $137 (WT 1(n)2——:_l—_ b x901) (6.4)
"=1 ‘ n=1

 

where 0‘") is the probability for the n-th possible price outcome constructed from the

transition probability system as discussed in the preceding chapter by the following formula:

pf0b(p‘=§i, pzzp—w P3=]7k, » p1_1:[31. pr=l75) (6-5)
= Kt. X it:j X 791. X....X it; i,_/', k, l, S: l, 2, m
= 0‘") n=1,2,....,N

where 1t; gives the probability that state 17: at I will be followed by state p3, at r+1;x

= i, j, , I; y = j, k, , s; K,- is the probability for state if, to occur at the initial period

127

based on historical price outcomes of the corresponding period, therefore in, = 1 with
121 possible price states. 0‘") represents the probability for the n-th possible price outcome,
hence :0“) = l where N is the total number of possible outcomes. Given m possible
price states and T decision periods, the total number of potential price outcomes is mm.
However, some of these price scenarios are likely to occur only at a negligible probability, so,
in practice, N is less than Mr". WTP can be numerically approximated by the bisection
method (see appendix C).

Similarly, the willingness-to-pay conditional on the initial price (WTP(i)) is determined

by:
(4107(1) + WTP(i)x(r+r)T) = E, (10.40)) (6.6)

where (i) denotes that the associated variable is dependent on the i-th initial price state. To
solve for W7P(i), all possible conditional price outcomes and their respective probabilities
must first be defined. Following the discussion in section 5.5, the conditional probability can

be computed from the transition probabilities system, by the following formula:

prob<p.=5,, p.=5.. . p.-.=17.. p.=i5. lp1=17,-) (6.7)

_ r 2 r .. ._
— 1t,j >< 1:1,, X....>< n“ r,y,k,....,....,l,.s—l,2,m

= 0’) h=1,2,....,H

1

Allow pf”) to represent the conditional probability for the h-th possible price outcome given
I!

initial price equal to 17, , hence X pf”) = 1 where H is the total number of possible
h=l

outcomes. Again, given m possible price states and T decision periods, total number of

128

 

 

potential price outcomes is m7 when initial price is given. Since some of these price
scenarios are likely to occur only at a negligible probability, H is less than m T and varies

depending on the initial price state. Equation (6.6) can be rewritten as:

L670) + WTP(i)x(1+r)T]“”_ _)H: _(_____w (1')":ij< p“)

6.8
H) M _ , ( )

 

Since the left-hand side of (6.8) no longer has stochastic elements, WI‘PU) can be solved
directly from (6.8).

Note that WPT and WTPU) are scaled by the initial storage in bushels. They can be
interpreted as a constant marginal value per bushel that the producers are willing to pay for
the optimal marketing strategy if the alternative is to sell the production immediately at
harvest. Likewise, expected final wealth and conditional expected final wealth, calculated as

N H
z: w;(")>< 0‘") and Z w;(i)"" >< pf”) , respectively, also have been scaled per bushel.
- 1,4

6.2 Results

The expected final wealth from applying the optimal marketing strategies for varying
degrees of risk aversion are presented in the top of Table 6-1. They have been discounted to
their present value at the initial (harvest) stage so they can be compared with initial (harvest)
prices. The difference between discounted final wealth and the initial price are displayed in
italics next to the corresponding final wealth number. Standard deviations of final wealth are

also reported. The results indicate that the optimal marketing policy brings an increase in the

129

 

Table 6-1

Expected Final Wealth Results

 

 

 

 

 

 

Initial Price Ri_sk Preference
Mild High Near
Risk Aversion Risk Aversion Risk Neutrality
dollars/bu dollars/bu cents/bu dollars/bu cents/bu dollars/bu cents/bu
Expected Final Wealth
2.237‘ 2.301c 6. 346d 2.271 3. 419 2.375 13.820
(0.494)b (0.193) (0.101) (0.454)
Conditional Expected Final Wealth
1.6 1.743 14.316 1.690 8. 950 1.771 1 7.063
(0.232) (0.142) (0.336)
1.8 1.857 5. 693 1.832 3.193 1.920 12.005
(0.169) (0.095) (0.373)
2.0 2.045 4.542 2.023 2.280 2.115 11.495
(0.164) (0.082) (0.417)
2.2 2.250 4. 967 2.225 2.486 2.325 12. 470
(0.182) (0.091) (0.459)
2.4 2.456 5.554 2.428 2. 780 2.539 13. 914
(0.198) (0.099) (0.497)
2.6 2.658 5.815 2.631 3.111 2.756 15.573
(0.210) (0.106) (0.532)
2.8 2.866 6.604 2.833 3.302 2.966 16.546
(0.223) (0.112) (0.559)
3.0 3.069 6. 950 3.035 3. 475 3.175 17.450
(0.228) (0.114) (0.573)
3.2 3.269 6.857 3.234 3. 429 3.373 17. 284
(0.226) (0.113) (0.571)
3.4 3.463 6.311 3.152 3.156 3.560 16.009
(0.217) (0.109) (0.555)
3.6 3.653 5. 344 3.627 2. 672 3.737 13.666
(0.200) (0.100) (0.518)
3.8 3.826 2.529 3.817 1.669 3.887 8. 683
(0.135) (0.089) (0.468)
4.0 4.003 0.252 4.003 0.252 4.026 2. 620
(0.039) (0.039) (0.405)
4.2 4.200 - 4.200 - 4.200 -
4.4 4.200 - 4.200 - 4.200 -
' Unconditional mean of historical prices over the harvest period.
" Numbers in the parentheses are the standard deviation of the final wealth.
2 Expected final wealth has been discounted to the initial stage (harvest period).

Numbers in italics are the differences between the discounted expected final wealth (to
initial stage) when the optimal marketing rule is adopted and the initial (harvest) price at
which the production is sold if storage is not allowed.

130

 

 

 

expected return and a decrease in its variation regardless of the degree of risk aversion.
However, highly risk averse producers will have lower expected returns with smaller
deviations than lower risk averse producers. This implies that the conservative marketing
strategy for highly risk averse producers, which tends to call for an earlier and more
diversified (across the storage season) sale will lead to a relatively lower expected gain from
storage, but will reduce risk.

Conditional expected final wealth given specific initial prices are presented in the rest
of Table 6-1. Results show that the optimal marketing strategy results in an increase in the
levels of expected final wealth for all possible initial prices and various attitudes toward risk.
The gain is higher at lower risk aversion levels. However, variance (risk) is also higher in this
case. Along the same risk attitude, expected final- wealth increases consistently as the initial
price grows.

The patterns of the differences between discounted conditional expected final wealth
and their respective initial prices are similar at various degrees of risk aversion. They are
relatively high at the mid levels of initial price states, decrease sharply when initial price is
higher than $3.8/bu and become zero toward the end of the highest price states. This pattern
results from the benchmark case we use (no storage). When the harvest price is high, the
optimal marketing rule suggests selling a large percentage of production right at harvest,
meaning that only a small portion is stored to speculate on favorable future prices. The
potential gains from storage are therefore limited. When the harvest price is higher than or
equal to $4.2/bu, the optimal marketing rule for various risk attitudes suggest to sell the entire
stock at harvest, so no gains from the optimal storage rules is expected.

However, an exception occurs at the lowest price state. The margins are particularly

131

 

 

high when initial price drops to $1.6/bu. This is caused by the truncation effects from the
finite number of price states. Since price outcomes below $1.6/bu are considered $1.6/bu,
the price distributions over the marketing season are distorted as showed in Figure 4-5 (b4)
and (b5). The expected returns, in turn, are overestimated. This truncation effect has
particularly large impacts on the two price states that are closest to the lower boundary, in
which cases filture prices have the highest probability of dropping below the lowest price state
and thus the resulting expected final wealth is much likely to be overestimated.

In practice, given the fact that the 1996 farm program continues provisions for
commodity loans, the lower boundary price state can be considered as the level of loan rate
which is farmers’ lowest received price for the crop if participating the program. Therefore,
if we consider $1.6/bu is the loan rate, when harvest price drops to the lowest price state,
essentially there is no unfavorable outcome from storing. Accordingly, the expected final
wealth is always higher or never less than selling at harvest. Thus, the difference between the
discounted expected final wealth and harvest price is relatively high, compared with what
occurs at higher harvest prices.

Since the degree of risk aversion is part of the optimal marketing rules, optimal use
of the marketing strategy requires farmers to identify their real risk preferences. In practice,
however, it is impractical to specify a specific degree of risk aversion for many different
individual farmers. Table 6-1 can be especially useful in providing farmers with more relevant
information to select their preferred alternative considering the tradeoff between levels of the
expected final wealth and the associated variance.

Producers’ W at different attitudes toward risk are displayed in the top portion of

Table 6-2. W7? is measured in cents per bushel of initial storage. It appears that the higher

132

 

 

the degree of risk aversion, the lower the WTP, meaning that the economic value of the
optimal marketing strategy is higher for the group of nearly risk neutral producers than for
those that are highly risk averse. Highly risk averse producers have the lowest WT P, 2.925
cents/bu among the three risk preferences considered. It is slightly lower than mildly risk
averse producers’ 4.166 cents/bu but much lower than nearly risk neutral producers’ 13.82
cents/bu. As a percentage of expected discounted final wealth, WTP for mildly risk averse
producers’ is about 1.81%. In other words, for each dollar of expected return, highly risk
averse producers are willing to pay 1.81 cents for the right to apply the optimal marketing
strategy. It is higher than highly risk averse producers’ 1.3% but much lower than nearly risk
neutral producers’ 5.82%.

Producers’ WTP(i) conditional on initial prices are given in the rest of Table 6-2 with
the first column denoting the initial price level used for the calculation. Given any initial price,
W7P(i) are higher for less risk averse producers. Along the same risk preference, it is found
that the levels of W TP(i) are related to initial price states, though not with a strongly
systematic pattern. W7P(i) is relatively high at the mid levels of the initial price states and
decreases slightly as initial price grows. WTP(l') becomes zero when initial price reaches
$4.2/bu. This is reasonable because when initial price is already high, there is limited potential
for filrther increases. The optimal marketing policy apparently is not especially valuable in
this case.

Again, an exception occurs when the initial price drops to its lowest price state where
W7P(i) is significantly higher than other states of concern. This implies that the optimal
marketing strategy is particularly valuable to producers when the price at harvest is very low,

which is likely to happen in a large crop year. However, the particularly high W TP(i) at low

133

Table 6-2 Willingness-to-Pay Results

 

Initial Price Risk Preference

 

 

 

 

Mild High Near
Risk Aversion Risk Aversion Risk Neutrality
dollars/bu cents/bu (%) cents/bu (%) cents/bu (%)
Willingness-to-Pay: WTP
2.237‘| 4.166 (1.81)b 2.952 (1.30) 13.820 (5.82)
Conditional Willingness-to-Pay: WT P(i)

1.6 8.432 (4.84) 4.466 (2.64) 17.063 (9.63)
1.8 2.708 (1.46) 1.285 (0.70) 12.004 (6.25)
2.0 2.017 (0.99) 0.994 (0.49) 11.495 (5.43)
2.2 2.087 (0.93) 1.032 (0.46) 12.470 (5.36)
2.4 2.402 (0.98) 1.192 (0.49) 13.913 (5.48)
2.6 2.491 (0.94) 1.413 (0.54) 15.572 (5.65)
2.8 3.077 (1.07) 1.531 (0.54) 16.546 (5.58)
3.0 3.424 (1.12) 1.706 (0.56) 17.450 (5.50)
3.2 3.518 (1.08) 1.758 (0.54) 17.284 (5.12)
3.4 3.291 (0.95) 1.649 (0.48) 16.009 (4.50)
3.6 2.831 (0.78) 1.423 (0.39) 13.666 (3.66)
3.8 1.452 (0.39) 0.707 (0.19) 8.683 (2.23)
4.0 0.170 (0.04) 0.083 (0.02) 2.619 (1.02)
4.2 0.000 (0.00) 0.000 (0.00) 0.000 (0.00)
4.4 0.000 (0.00) 0.000 (0.00) 0.000 (0.00)

 

h

Unconditional mean of the historical prices over the harvest period.
Numbers in the parentheses are willingness-to-pay as a percentage of average
final wealth, discounted to the initial (harvest) stage.

134

 

harvest prices can be the result of the truncation effect due to the finite number of price states.
Since W7P(l') is computed relative to sell at harvest, if the harvest price is already at its low
within the given range of price states, after adjusting the forgone interest income and storage
costs, farmers simply have nothing to lose by storing and thus any storage policy is valuable.

It is worth noticing that the pattern of WTP(i) is consistent with the pattern of the
margins as a function of initial price. This may suggest that WT P for the optimal marketing
strategy is highly dependent on the resultant increases in the level of final wealth, induced by
the optimal marketing rules. Moreover, nearly risk neutral producers’ WT P5 are essentially
equal to the margins they gain from the optimal marketing activity. Risk averse producers
instead demand a portion of the margins to compensate for the additional risk of storage they
will be experiencing.

For a filrther understanding of the economic value of the optimal storage rules, the
unit value of WTP will be transformed into it’s aggregate value. For a typical Michigan
farmer who owns a 1000-acres of crop land, with half of the acres planted for corn, assuming
mild risk aversion, WTP of 3 to 8 cents, depending on the harvest prices, is about $1,875 to
$5,000 in aggregate value. The values would be about 5 to 15 percent of the average net
farm cash income. This percentage could be considered economically significant, highlighting

the importance of sound storage/marketing strategies.

135

 

6.3 Summary

Compared to no storage, the optimal marketing strategy results in an increase in the
level of producers’ expected final wealth and a decrease in the variance of returns. The
increase in the level of expected final wealth is significant at low degrees of risk aversion
while the reduction in variance of returns is significant at high degrees of risk aversion.

W7? for the optimal marketing strategy is higher for less risk averse producers. The
value is also dependent on the initial price state. The optimal marketing strategy is more
valuable when the price at harvest is lower than $1.6/bu, while is not useful when the price
at harvest is over $4.2/bu. The values of WT P for nearly risk neutral producers are equal to
the difference between their discounted expected final wealth and initial price, while risk
averse producers require a portion of the margin as compensation for the risk they have to
endure from undertaking storage.

The economic values of the optimal storage rules are around 5 to 15 percent of the
net farm cash income of a typical Michigan crop farm, which is considered economically

significant.

136

 

 

CHAPTER VII

CONCLUSION

The goal of this study was to derive an optimal post-harvest storage/marketing rule
which can be implemented fairly easily by the farmers and extension agents. A main
contribution of the study was to provide theoretical and empirical support for an optimal
partial sale, as opposed to the overly simplified sell-everything-or-nothing marketing strategy
discussed in much of the earlier research. The cause of these partial sales is risk aversion.
This optimal marketing strategy is more consistent with the real world situation where farmers
usually do not use a sell-every-or-nothing strategy.

The simulation results of the optimal marketing strategy for risk neutral farmers is
consistent with previous studies where the optimal rules are defined on a time—varying cutoff
price, which represents the option value of the given stock while taking into account the
possibility that the optimal control may change in the filture. This optimal marketing policy
is to sell any of the grain in storage if the current price is higher than the cutoff price;
otherwise retain the entire stock. The highest cutoff price occurs in January, when the time
value of the stock and the expectation on the stored commodity’s future prices are both high.
They have positive influences on the option value of the given stock and thus a higher current
price is required to compensate the foregone benefits from storing. The cutoff price then

decreases as the marketing season proceeds.

137

 

Partial sales become possible optimal marketing strategies when risk aversion is
incorporated in the decision model. The optimal marketing policy is to sell the grain until the
weighted marginal option value of the stock is equal to the weighted current price. This
implies that a combination of a certain income, from an immediate sale of a portion of the
storage, and an uncertain future income, from storing and selling the rest of the storage at
some future price, can be a preferred strategy for risk averse farmers. The weights are
determined by the individual farmer’s attitude toward risk, his current storage and wealth
levels, the expectation on future price distributions, and the opportunity costs of storage. Due
to their low tolerance for risk, highly risk averse farmers weight the current price more,
relative to low risk averse farmers, which leads to an optimal marketing strategy that allows
sales at lower prices. In general, the optimal marketing rules under risk aversion tend to
suggest a larger marketing volume in the later stages. This is because the time value of the
stock is diminishing as there is simply less chance for price to increase. Furthermore, the
price is expected to drop as next harvest approaches, implying the potential benefits from
storage are decreasing when the marketing season approaches an end.

An application of the optimal marketing rules for nearly risk neutral farmers indicates
an 83% probability that the entire harvest will be sold in May, which is the time the seasonal
price growth usually reaches its peak. However, the expected optimal marketing volumes for
risk averse farmers are substantially different from the risk neutral case, in that a large portion
of production is expected to be marketed right at harvest. For mildly risk averse farmers,
about 52% of harvest is expected to be sold right in November and around 39% is expected
to be marketed in May; and for highly risk averse farmers, about 70% of harvest is expected

to be marketed in November and about 20% of harvest is expected to be sold in May. It

138

 

appears that marketing activities are most active in November and May in risk averse cases.
These imply risk averse farmers prefer to gain a certain income earlier by selling a portion of
the production at harvest and use the rest to speculate an increase in future prices. The main
difference between risk neutral and risk averse cases is that the risk neutral farmers use the
entire stock to speculate on higher prices, while risk averse farmers use only a portion of it.

Furthermore, the sensitive analysis illustrated that as the degree of risk aversion
increases, a larger portion of harvest is expected to be sold in the early stages. An increase
in interest rates, which implies a higher opportunity cost of storage, makes storage less
desirable and leads to early sales. An increase in storage costs, which is a negative return
from storage, has a similar effect as higher interest rates on the optimal timing of sale. These
results suggest that optimal timing of sales are sensitive to the individual farmer’s degree of
risk aversion, and the values of storage costs and interest rates. For implementing optimal
storage decisions, it is essential to get good data on the values of these parameters.

It is also found that the optimal marketing policy brings an increase in the expected
return and a decrease in its variation. Highly risk averse producers will have lower expected
returns with smaller deviations than lower risk averse producers. This implies that the
conservative marketing strategy for highly risk averse producers, which tends to call for an
earlier and more diversified (across the storage season) sale will lead to a relatively lower
expected gain fi'om storage, but will reduce risk. In practice, the tradeoffs between the levels
of expected final wealth and the associated variance at various degrees of risk aversion
provide valuable information for farmers to select their preferred marketing strategy.

The economic value of the optimal marketing rules proposed in the study is measured

by the farmer’ s willingness-to-pay (WT P) for the right to apply the optimal marketing strategy

139

 

rather than sell the production immediately at harvest. The quantitative results indicate that
WTP is 13.82 cents per bushel for risk neutral farmers, 4.17 cents and 2.95 cents per bushel,
respectively, for mildly and highly risk averse farmers. For each dollar of expected return,
mildly risk averse producers are willing to pay 1.81 cents for the right to apply the optimal
marketing strategy. This is higher than highly risk averse producers’ 1.3% but much lower
than nearly risk neutral producers’ 5.82%. It appears that the optimal storage rule is
especially valuable for risk neutral farmers. The values are about 5 to 15 percent of the
average net farm cash income, which are considered economically significant. This means a
need for improved post-harvest grain marketing strategies is evident.

Within the scope of the current study, farmers are assumed to only look at cash prices.
The use of forward pricing instruments, such as filtures contracts and options, are not
considered. Optimal use of these risk management tools may reduce the income risk
confi'onting the farmers and may also enhance farm income and storage. However, the
availability of these additional risk management alternatives will increase the complexity of
farmers’ marketing decisions, and, therefore, applicable optimal marketing decision rules will
be particularly valuable for hedging strategies. Extending the current study in this direction
may be particularly interesting.

In the study, it is assumed that farmers maximize their utility of final wealth, implying
an efficient capital market accessible to farmers with negligible transaction costs to smooth
their farm income and to accommodate their production expenses during the planting season.
An alternative decision objective with credit constraints and consistent farm consumption

needs may also lead to an interesting extension of this work.

140

 

APPENDICES

141

 

APPENDIX A

STATISTICAL HYPOTHESIS TESTS

Ljung-ng (2 Test

The Q-statistic for testing autocorrelation is calculated as

000 = Tami": ri/(T—k) (A—I)
k=1

where r, is the k-th order sample autocorrelation function of the residuals from the variable
being tested. Q is asymptotically Chi-squared distributed with n degrees of freedom. If the
value of Q exceeds the value in a {(n) table, we can reject the null of no significant
autocorrelation. For the of Q2 statistic the formula is the same except rk is the k-th order
sample autocorrelation function of the “squared standardized” residuals. The standardized
residual is the squared residual divided by it’s conditional variance. Q is used to test for the
existence of residual autocorrelation not captured by the model, while Q2 is used to test for
the existence conditional heteroskasticity not captured by the model.

Q(n)-statistics in Table 4-4 are for n-th order serial correlation in the residuals; Q2(n)-
statistics are for n-th order serial correlation in the squared standardized residuals. Note that
we form the n correlations for an estimated ARMA(p,q) model, the degrees of freedom are
reduced by the number of estimated coeflicients. That is, Q follows a Chi-squared

distribution with n-p-q degrees of fi'eedom.

142

 

Qggfiig'gt 9f Skewness gnd Kurtosis

Coefficient of skewness (S)

Coefficient of kurtosis (K)

 

 

r
:1 3x2 (A'Z)
_1__ - 2
(“go a]
l T
HE 9"?"
:1 2 (A‘3)
1 _-2
(fi§ 0’: y) ]

where y, is the standardized value of the series considered; )7 is the sample mean of y, .

The standardized value of a variable is a sample value subtracted by it’s sample mean and

divided by it’s ample standard deviation.

Kglmgggrgv-Smimov Goodness-of-Fit Tests (K-S tests)

The objective of the K-S test is to investigate the significance of any difference

between an observed distribution and a specified population distribution. If the model is

appropriately specified, the estimated residuals should follow the pre-specified distribution.

The idea is to determine the cumulative distribution, S(x), from the sample, then plot it on the

same diagram as the cumulative distribution of the assumed population, F (x). The maximum

 

difference between the two distributions provides the test statistic, Dmam = |F (x) - S(x)| ,

and this is compared with the value D(0l) which are equal to -1—23 , ll— , or 1—6—3 at

v/T r/T 47”
the 10%, 5%, and 1% significant levels, where a is the significance level, and T is the sample
size. IfDm > 0(a), the null hypothesis that the sample came from the assumed population
is rejected.
In practice, to calculate the statistic D“, we first sort the estimated standardized
residualsinascendingordertoget {fr,,flz,....ji,,....,l‘rr} ,where l1,< {rm fori= 1,2,....T-
1. Then calculate the absolute value of the difference between F011.) and S(fl,) , where F (.)

is the cumulative density function of the null hypothesis and S( 1.1,.) = i/ T. Dm, is equal to

the maximum of |F(l'i,.) - S(fl,)| .

Likelihggd Rgig Test

Denote 22(0) and 29(0) as the values of the maximized log likelihood firnctions at
the unrestricted and restricted estimates, respectively, where 0 is the parameter vector. The
null hypothesis implies a set of m different restrictions on the parameter vector. The
likelihood ratio statistic is calculated as 2 [22 (0) - 2? (0)] and has an asymptotic chi-squared
distribution with m degree of freedom. If the restriction is valid, imposing it should not lead
to a large reduction in the log likelihood function.

The log-likelihood filnction for a sample of T observation is, assuming the conditional

distribution of the errors follow a Student’s 1 distribution:

144

 

l!

T
22(6) = no glwrv -2) “2} - (1/2)2Iog(o.’)
[=1

“2 (V / 2) (A-4)

2

-[(v+l)/2]Zlo 1+—
6 2(v- 2)

where v is the degree of freedom for the distribution and F(.) is the gamma function.
Likelihood ratio tests are conducted to select the appropriate order of seasonal
functions in the mean (J) and variance (1). The sequential tests begin by specifying high-order

seasonal functions for mean and variance and pare down the model by reducing the order

 

sequentially. Given 1 = l, the log likelihood filnctions under different value of J are calculated
and reported in the first two columns of Table 4—6. The regression with the smaller number
of S is taken as the restricted model in the null hypothesis. The selection of the degree of

seasonal impact by likelihood ratio test is conducted as followed:

1. Ho: J=3and1=l
H1: J=4and1=l

2x[-2738.0389 - (-2738.4415)] = 0.8052 < 12(2): 3.99, accept H0

2. Ho: J=2andl=l
H1: J=3and1=l

2x[-2738.4415 -(-2739.4701)] =2.0572 < 36(2): 3.99, accept H0

3. Ho: J=land1=1
H,: J=Zandl=l

2x[-2739.4701 - (-2743.2782)] = 7.6162 > 26(2) = 3.99, reject H0

145

Table A-1 The Log-Likelihood Functions Under Various Orders of
Seasonal Effects in Mean (J) and Variance (I):

 

 

 

Given Log-Likelihood Given Log-Likelihood
1 Functions J = 2 Functions
J -2743.2782 I = l -2739.4701
J -2739.4701 I = 2 -2739.2080
J -2738.4415 1 = 3 -2738.2286
J -2738.0389 I = 4 -2735.2011
Ho: 1 = 3 and J = 2
H1: 1 = 4 and J = 2

2x[-2735.2011 - (-2738.2286)] = 6.0550 > 76(2) = 3.99, reject HO

HO:
H1:

I=2andJ=2
I=3andJ=2

2x[-2738.2286 - (-2739.2080)] = 1.9588 < 36(2) = 3.99, accept HO

HO:
H1:

I=2andJ=2
I=4andJ=2

2X[-2735.2011 - (-2739.2080)] = 8.0138 < 76(4) = 9.49, accept HO

HO:
H1:

I=landJ=2
I=ZandJ=2

2x[-2739.2080 -(-2739.4701)] = 0.5242 < {(2) = 3.99, accept H0

146

 

APPENDIX B

PROPERTIES OF UTILITY FUNCTION

flgmggenegus Utility Functign

A function fit) is homogeneous of degree k if f(tx) = t"f(x) for all t > 0. An

important implication of this fact is that the slopes of the level surfaces of a homogeneous

function are constant along rays through the origin (Varian):

531(1) 8f(tx)

 

 

6x. 8x.
’ = ' (B- 1 )
610:) 6I (1x)
6x]. 8x}.

which implies if the objective is to maximize fix), the same optimization solution will be

reached by maximizing f(tx). In the study, the utility function in the form of
l-b
l

w
U(wT) = 1- is homogeneous of degreel-b. Allowing t = S— ,wheres0 isthe initial
0

 

storage, will not effect the optimal selections.

147

 

CARA mg CRRA Utility Functions

The relationship between absolute risk aversion (A a) and relative risk aversion (A ,) can

be described by equation (B-2):

Ar(wT) = wTAa(wT) (B-Z)

By differentiating this expression with respective to WT we obtain:

dAr/dwT = A“ + wT(dAa/dwT) ' (B-3)

 

Since A, is assumed to be constant in this study, the left-hand side of (B-3) is equal to zero.
Since A, and W7. are non-negative, dAa/dwT therefore must be less than or equal to zero,
implying the degree of absolute risk aversion decreases when the wealth level increase. This

means that an individual whose wealth increases fears a given risk less or at least not more.

148

APPENDIX C

THE BISECTION METHOD

The bisection method is based on the Intermediate Value Theory. According to the
theorem, if a continuous function defined on an interval assumes two distinct values, then it
must assume all values in between. In particular, if f is continuous, and fla) and flb) have
different signs then f must have at least one root x in [a, b].

The bisection method is an iterative procedure. Each iteration begins with in interval
known to contain or to “bracket” a root of f The interval is bisected into two subintervals
of equal length. One of two sub-intervals must contain a root off. This sub-interval is taken
as the new interval with which to begin the subsequent iteration. In this manner, a sequence
of intervals is generated, each half the width of the preceding one and each known to contain
a root f. The process continues until the width of the interval known to contain a root off

shrinks below an acceptable convergence tolerance. (Miranda and Fackler, 1997)

149

 

REFERENCES

 

 

 

150

 

REFERENCES

Adam, B. D., and K. Anderson. “Robust Risk Management Strategies for U. Hard Red

Winter Wheat Producers.” NCR I34, Commodity Price Analysis, Forecasting and .

Risk Management, 1995: 306-22.

Aldefer, R., S. Harsh, J. Hilker, and J. R. Black. “Farm Income Risk Management (FIRM):
A Decision Support System Approach.” NCR 134, Commodity Price Analysis,
Forecasting and Risk Management, 1992: 177-91.

Anderson, K. B. and H. P. Mapp. “Risk Management Programs in Extension.” .1. Agr.
Resour. Econ. 21(July 1996): 31-38.

Bailey, D. V., and J. W. Richardson. “Analysis of Selected Marketing Strategies: A Whole
Farm Simulation Approach.” Amer. J. Agr. Econ. 67(November, 1985): 813-20.

Barry, P. J ., and D. R. Willmann. “A Risk-Programming Analysis of Forward Contracting
with Credit Constraints.” Amer. J. Agr. Econ. 58(February, 1976): 62-70.

Bauer, L. L. “A Target MOTAD Analysis of Sweet Potato Marketing.” S. J. Agr. Econ,
23(December 1991): 123-29.

Berg, E. “A Sequential Decision Model to Determine Optimal Farm-Level Grain Marketing .

Policies.” Euro. Rev. Agr. Econ. 14(1987): 91-116.

Bollerslev, T. “A Conditionally Heteroskedastic Time Series Model for Speculative Prices
and Rates of Return.” Rev. Econ. and Stat. 69(August 1987): 542-47.

and J. R. Allison. “Farm Management Decisions with Dynamic Progranuning.”
Amer. J. Agr. Econ. 45(February 1963): 121-36.

 

Cocks, K. “Discrete Stochastic Programming.” Management Sci. 15(September 1968): 72-
9.

Davidson, R., and J. G. Mackinnon. “Estimation and Inference in Econometrics.” Oxford,
New York: Oxford University Press, 1993.

Day, R. H. “Exact Aggregation with Linear Programming Model-A Note on the Sufficient

Conditions Proposed by R. H. Day: Reply.” Amer. J. Agr. Econ. 51(August 1969):
686-88.

151

 

 

Dickey, D., and W. A Fuller. “Distribution of the Estimates for Autoregressive Time Series
with a Unit Root.” J. Amer. Stat. Assoc. 74(June 1979): 427-31.

Dixit, A K., and R. S. Pindyck. Investment Under Uncertainty. Princeton, NJ: Princeton
University Press, 1994.

DeJong, D. N., J. C. Nankervis, N. E. Savin, and C. H. Whiteman. “The Power Problems of
Unite Root Tests in Time Series with Autoregressive Errors.” J. of Econometrics,
53(July-September 1992): 323-43.

Dorward, A “Farm Planning with Resource Uncertainty: A Semi-Sequential Approach.”
Euro. Rev. Agr. Econ. 21(1994): 309-24.

Enders, Walter. Applied Econometric Time Series. Iowa State University. Wiley Press.
1995.

Engle, R. F. “Autoregressive Conditional Heteroscedaticity with Estimates of Variance of ‘
U.K. Inflaton.” Econometrica 50(July 1982): 987-1008.

Fackler, P. L. Futures Price Volatility: Modeling Non-Constant Variance. Paper presented
at AAEA Annual Meeting, Reno, Nevada, July 1986.

. and M. J. Livingston. “Optimal On-Farm Storage.” NCR 134, Commodity
Price Analysis, Forecasting and Risk Management (April 1996): 176-90.

Frasier, W. M., and G. H. Pfeiffer. “Optimal Replacement and Management Policies for Beef
Cows.” Amer. J. Agr. Econ. 76(November 1994): 847-58.

Garcia, R, B. D. Adam; and R. J. Hauser. “The Use of Mean-Variance for Commodity
Futures and Options Hedging Decisions.” Amer. J. Agr. Econ. 19(July 1994): 32-45.

Gebremeskel, T., and C. R. Shumway. “Farm Planning and Calf Marketing Strategies for
Risk Management: An Application of Linear Programming and Statistical Decision
Theory.” Amer. J. Agr. Econ. 61(May 1979): 363-70.

Hamilton, J. D. Time Series Analysis. Princeton, New Jersey : Princeton University Press, .
1994.

Hatch, U., J. A, and J. Segar. “An Application of Safety-First Probability Limits in a
Discrete Stochastic Farm Management Programming Model.” S. J. Agr. Econ.
29(July 1989): 65-72.

Ho, T. Y. “Intertemporal Commodity Futures Hedging and the Production Decision.” J
Finance 39(June 1984): 351-76.

Johnson, D. and M. D. Boehlije. “Managing Risk by Coordinating Investment, Marketing

152

 

and Production Strategies.” W. J. Agr. Econ. 1 1(December 1986): 235-36.

Johnson, S. R., K. R. Tefertiller, and D. S. Moore. “Stochastic Linear Programming and
Feasibility Problems in Farm Growth Analysis.” J. Farm Econ. 49(November 1967): ‘
908-19.

Kaiser, H. M., and J. Apland. “DSSP: A Model of Production and Marketing Decisions on
a Midwestern Crop Farm.” N. Cen. J. Agr. Econ. 11( July 1989): 157-69.

Karp, L.S. “Methods for Selecting the Optimal Dynamic Hedge When Production Is
Stochastic.” Amer. J. Agr. Econ. 69(August 1987): 647-57.

. “Dynamic Hedging with Uncertain Production.” Inter. Econ. Rev. 29(November
1988): 621-37.

Kaylen, M. S., P. V. Preckel, and E. T. Loehman. “Risk Modeling via Direct Utility
Maximization Using Numerical Quadrature.” Amer. J. Agr. Econ. 69(August 1987):
701-06.

King, R. P., J. R. Black, F. J. Benson, and P. A. Pavkov. “The Agricultural Risk
Management Simulator Microcomputer Program.” S. J. Agr. Econ. 20(December
1988): 165-71.

and D. Lybecker. “Flexible, Risk-Oriented Marketing Strategies for Pinto Bean
Producers.” W. J. Agr. Econ. 8(December 1983): 124-33.

 

Lambert, D. K. “Calf Retention and Production Decisions over Time.” W. J. Agr. Econ.
14(July 1989): 9-19.

. and B. A. McCarl. “Risk Modeling Using Direct Solution of Nonlinear
Approximations of the Utility Function.” Amer. J. Agr. Econ. 67(November 1985):
846-52.

. and . “Sequential Modeling of White Wheat Marketing
Strategies.” N. Cen. J. Agr. Econ. 11( July 1989): 105-15.

Lapan, H, and G. Moschini. “Futures Hedging Under Price, Basis, and Production Risk.”
Amer. J. Agr. Econ. 76(August 1994): 465-77.

Lence, S. H., K. L. Kimle, and M. L. Hayenga. “A Dynamic Minimum Variance Hedge.”
NCR 134, Commodity Price Analysis, Forecasting and Risk Management, 1992: .
129-43.

. Y. Sakong, and D. J. Hayes. “Multiperiod Production with Forward and
Option Markets.” Amer. J. Agr. Econ. 76(May 1994): 286-95.

153

 

Levy, H., and G. Hanoch. “Relative Effectiveness of Emciency Criteria for Portfolio
Selection.” J. Finance and Quantitative Analysis S(March 1970): 63-7.

Mapp, H. P., Jr., M L. Hardin, O. L. Walker, and T. Persaud. “Analysis of Risk Management
Strategies for Agricultural Producers.” Amer. J. Agr. Econ. 61(December 1979):
1071-77.

Miranda, M. J ., and P. R. F ackler. Lecture Notes in Computational Economic Dynamics,
The Ohio State University and North Carolina State University, 1997.

Martines-Filho, J, and S. H. Irwin. “Pre-Harvest Hedging Behavior and Market Timing
Performance of Private Market Advisory Services.” NCR 134, Commodity Price
Analysis, Forecasting and Risk Management, 1995: 275-300.

Martinez, S. W., and K. D. Zering. “Optimal Dynamic Hedging Decisions for Grain
Producers.” Amer. J. Agr. Econ. 74(November 1992): 879-88.

McClintock, D. N., F. Z. Ahmadi-Esfahani, and J. H. Johnston. “A Simulation of the Grain
Producers’ Decision Problem at Harvest.” Rev. Marketing and Agr. Econ.
59(December 1991): 271-82.

Meyer, J. “Two-Moment Decision Models and Expected Utility.” Amer. Econ. Rev. 77(June
1987): 421-30.

Musser, W. N., J. Ohannesian, and F. J. Benson. “A Safety-First Model of Risk Management
for Use in Extension Program.” N. Cen. J. Agr. Econ. 3(January 1981): 41-6.

O’Brien, D, and R. Wisner. “The Performances of Probability-Based Grain Marketing .

Strategies.” NCR 134, Commodity Price Analysis, Forecasting and Risk
Management, 1995: 323-30.

Perron, P. “Trends and Random Walks in Macroeconomic Time Series.” J. Econ. Dynamic
and Control. 12(June/Sept. 1988): 297-332.

and S. Ng. “Useful Modifications to some Unit Root Tests with Dependent
Errors and Their Local Asymptotic Properties.” Rev. Econ. Stud. 63 (July 1996): 43 5-
63.

 

Phillips, P. C. B., and P. Perron. “Testing for a Unit Root in Time Series Regression.”
Biometrika 75(September 1987): 335-46.

Preckel, P. V., and E. DeVuyst. “Efficient Handling of Probability Information for Decision
Analysis under Risk.” Amer. J. Agr. Econ. 74(August 1992): 655-61.

Rae, A. N. “Stochastic Programming, Utility, and Sequential Decision Problem in Farm
Management.” Amer. J. Agr. Econ. 53(August 1971): 448-60.

154

 

 

. “An Empirical Application and Evaluation of Discrete Stochastic Programming
in Farm Management.” Amer. J. Agr. Econ. 53(November 1971): 625-38.

Rister, M. E., J. R. Skees, and J. R. Black. “Evaluating Use of Outlook Information in Grain
Sorghum Storage Decisions.” S. J. Agr. Econ. 16(July 1984): 151-58.

Roy, A. D. “Safety-First and the Holding of Assets.” Econometrica 20(Ju1y 1952): 431-49

Saha, A, and J. Stroud. “A Household Model of on-Farm Storage under Price Risk.” Amer.
J. Agr. Econ. 76(Auguest 1994): 522-34.

Schnitkey, G. D., C. R. Taylor, and P. J. Barry. “Evaluating Farmland Investment
Considering Dynamic Stochastic Returns and Framland Prices.” W. J. Agr. Econ.
14(July 1989): 143-56.

Schroeder, T. C., and A. M. Featherstone. “Dynamic Marketing and Retention Decision for ‘
Cow-Calf Producers.” Amer. J. Agr. Econ. 72(November 1990):]028-40.

Scott, J. T., and C. B. Baker. “A Practical Way to Select an Optimum Farm Plan under
Risk.” Amer. J. Agr. Econ. 54(November 1972): 657-60.

Tauer, L. W. “Target MOTAD.” Amer. J. Agr. Econ. 65(August 1983):607-10.

Taylor, C. R., and F. S. Novak. “Optimal Flex Cropping and Storage of Spring Wheat under
a Progressive Income Tax.” Can. J. Agr. Econ. 40(November 1992):369-84.

Tomek, W. G., and K. L. Robinson. Agricultural Product Prices. Ithaca, New York:
Cornell University Press, 1990.

Tronstad, R. “The Efl'ects of Firm Size and Production Cost Levels on Dynamically Optimal
After-Tax Cotton Storage an Hedging Decision.” S. J. Agr. Econ. 23(November
1991):165-79.

and R. Gum. “Cow Culling Decisions Adapted for Management with CART.” .
Amer. J. Agr. Econ. 76(May 1994):237-49.

 

and C. R. Taylor. “Dynamically Optimal After-Tax Grain Storage, Cash Grain
Sale, and Hedging Strategies.” Amer. J. Agr. Econ. 73(February 1991): 75-88.

 

Turvey, C. G., and T. G. Baker. “Optimal Hedging Under Alternative Capital Structures and
Risk Aversion.” Can. J. Agr. Econ. 37(March 1989): 135-43.

Varian, H. R. Microeconomic Analysis. New York, New York: W. W. Norton & Company,
1992.

Wang, H. “Farmer Risk Management Behavior and Welfare under Alternative Portfolios of

155

Risk Managements.” Ph.D. thesis. Department of Agricultural Economics, Michigan
State University, 1996.

Weersink, S., and S. Stauber. “Optimal Replacement Interval and Depreciation Method for
a Grain Combine. W. J. Agr. Econ. 13(July 1988): 18-28.

Williams, J. C., and B. D. Wright. Storage and Commodity Markets. Cambridge, New
York: Cambridge University Press, 1991.

Yager, W. A, R. C. Greer, and O. R. Burt. “Optimal Policies for Marketing Cull Beef
Cows.” Amer. J. Agr. Econ. 62(August 1980): 456-67.

Yang, S. R. and B. W. Brorsen. “Nonlinear Dynamics of Daily Cash Prices.” Amer. J. Agr.
Econ. 74 (August 1992): 706-15.

Zulauf, C. R., and S. H. Irwin. “Market Efficiency and Marketing to Enhance Income for
Crop Producers.” Rev. Agr. Econ. 20(Fall-Winter, 1998): 308-31.

156

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