AN ECONOMIC COMPARISON OF CONTROL
METHODS OF WYOMING BIG SAGEBRUSH IN
SOUTHWESTERN MONTANA
by
Jian Yi Du
A thesis submitted in partial fulfillment
of the requirements for the degree
of
Master of Science
in
Applied Economics
MONTANA STATE UNIVERSITY
Bozeman, Montana
December 1988
i i
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Jian Yi Du
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iv
ACKNOWLEDGEMENTS
I
would like to express my sincere appreciation to my
members,
Ronald
Dr.
N.
Myles J.
Watts,
chairman,
committee
Dr. R. Clyde Greer and Dr.
Johnson for their time and commitment during work on
this
thesis.
Appreciation is also expressed to Dr. Jeffrey T. LaFrance for his
help in preparation of this thesis.
Special
thanks
go
to my family
understanding over the years.
for
their
encouragement
and
v
TABLE OF CONTENTS
Page
APPROVAL .................................................. .
i i
STATEMENT OF PERMISSION TO USE ............................ .
i i i
ACKNOWLEDGEMENTS .......................................... .
iv
TABLE OF CONTENTS ......................................... .
v
LIST OF TABLES ............................................ .
Vii
LIST OF FIGURES ............................................ .
ix
ABSTRACT .................................................. .
X
CHAPTER
INTRODUCTION .................................... .
Statement of the Problem ..................... .
Objectives of the Project .................... .
Outline of the Thesis ........................ .
2
2
2
LITERATURE REVIEW ............................... .
3
3
METHODOLOGY ..................................... .
6
Study Site and Field Data ................... ..
Formulation of Forage Response Function
to Sagebrush Treatment .................... .
Development of DP Model ...................... .
Stages .................................... .
State Variables· ........................... .
Years since treatment ..........•........
Value of Forage yield per
animal-unit-month ................... .
Transition Probabilities .................. .
Calf price equation .................... .
Transition probabilities ............... .
Decision Alternatives ..................... .
Expected Additional Yield ...... ~ .......... .
Treatment Costs ........................... .
The Discount Rate ......................... .
7
8
10
11
11
11
12
13
14
15
16
18
19
19
vi
TABLE OF CONTENTS-Continued
Page
Recursive Equation ........................... .
Terminal Value ............................... .
20
22
RESULTS ......................................... .
23
Statistical Estimates of the
Response Function ..................... ~····
DP Solution .................................. .
28
CONCLUSIONS ..................................... .
34
BIBLIOGRAPHY .............................................. .
37
4
5
23
APPENDICES
A:
Original Data ..................................... .
43
B:
Derivation of the Signs of the Response
Function Parameters ............................... .
48
The Calculation of Equivalent Weight of Calves
and Average Cost of Raising a Calf ................ .
51
Statistical Test for Normal Distribution of
Calf Price Equation Residuals ..................... .
54
is Tests on Response Function ............. .
56
C:
D:
E:
Hypo~hes
vi ;
LIST OF TABLES
Page
Table
Probability distribution of calf price at
year t, given calf prices at year t-1
and year t-2 ...................................... .
17
2.
Summary of treatment costs ........................ .
19
3.
Response function statistics (1) .................. .
24
4.
Present value of the additional net returns
(dollars/hal and optimal retreatment
intervals (years) generated by sagebrush
treatment method, given specific treatment
costs and previous calf price condition ........... .
30
Present value of the additional net returns
(dollars/ha) and optimal retreatment
intervals (years) generated by burning,
given specific treatment costs, previous
calf price condition, and different k values ...... .
32
Present value of the additional net returns
(dollars/ha) and optimal retreatment
intervals (years) generated by rotocutting,
given specific treatment costs; previous
calf price condition, and different k values ...... .
33
Production level of perennial grasses
with spraying (kg/ha) ............................. .
44
Production level of perennial grasses
with burning (kg/ha) .............................. .
44
Production level of perennial grasses
with rotocutting (kg/ha). ......................... .
45
Production level of perennial grasses
with plowing and seeding (kg/ha) .................. .
45
Production level of perennial grasses
on control ( kg/ha) ................................ .
46
1.
5.
6.
7.
8.
9.
10.
11 .
vi i i
LIST OF TABLES-Continued
Page
Table
12.
Calf price (do11ars/100 1bs.) ..................... .
46
13.
Sagebrush canopy cover ............................ .
47
14.
Response function statistics (2) .................. .
57
ix
LIST OF FIGURES
Figure
Page
1.
Response function shape ............................ .
9
2.
Calf price intervals and midpoints ................. .
16
3.
Production response to treatment ................... .
25
4.
Production response to burning with
different k values ................................. .
26
Production response to rotocutting with
different k values ................................. .
27
5.
X
ABSTRACT
Big sagebrush (Artemisia tridentata Nutt.) is a very inefficient
user of water and competes strenuously for moisture with more
desirable forage plants such as grasses, forbs, and other shrubs.
Several control methods have been developed in hopes of increasing
range productivity for domestic livestock grazing. Spraying with 2,4D, bJrning, plowing and seeding, and rotocutting are the primary means
of controlling sagebrush.
The
objective of this thesis was to conduct an economic
comparison among these four Wyoming big sagebrush control meth~ds and
determine optimal retreatment intervals.
Production of perennial
grasses was measured on experimental plots in southwestern Montana 12
years during the period 1963-1986. The data were used to ~stimate the
treatment response function.
Sagebrush control is a stochastic
dynamic problem and as such the problem was formulated within a
stochastic dynamic programming framework.
The economic criterion was
the expected present value of additional net returns from sagebrush
treatment. Decision alternatives included i.n the DP model were keeping
or retreating the sagebrush. The state variables were years since
treatment and expected value of forage yield per AUM which was defined
as a function of calf price. Based upon a statistically estimated
second-order autoregressive difference equation,
the calf price
transition probability distribution was developed.
Given the data available to this study, Wyoming big sagebrush
treatment methods of spraying and burning were economically feasible,
with spraying the most profitable. Rotocutting was only marginally
feasible, and plowing and seeding was not feasible. From the study
results it was also concluded that in addition to treatment method.,
treatment cost and the quantity of sagebrush killed, the expected
present value of additional net return and optimal retreatment
interval also depend upon prev1ous calf price trend.
CHAPTER 1
INTRODUCTION
Statement of the Problem
Big
sagebrush
(Artemisia tridentata Nutt.) grows on
nearly
60
million hectares in the Western United States (Beetle,
1960).
Beetle
described
shrub.
It
big
sagebrush
characterized
by
an
as
an
erect,
aromatic smell and three
sagebrush may appear either dwarfed,
with other plant species for water,
desirable
Sneva
5.7
species
gray-green
toothed
shrubby or
leaves.
treelike.
is
Big
Competing
nutrients, and space, it is not a
for domestic livestock
grazing.
Rittenhouse
and
(1976) reported that perennial grass production declined 3.3 to
percent
economically
replacement
1 ivestock
for
every
feasible,
of
1
percent
domestic
increase
livestock
Therefore,
the subject of considerable research and
only
a
strategies.
sagebrush
efficiency
grazing.
It
deal
with
the
prefer
enhance
economic
application.
considerations
domestic
have
However,
of
these
is important to transform the biological response
treatment into monetary value for a comparison of
among
If
sagebrush management strategies
been
studies
brush· cover.
producers
sagebrush with species which would
grazing~
few
in
available treatment choices for domestic
to
econom,c
livestock
2
Objectives of the Project
The
among
purpose
Wyoming
four
wyomingensis,
D,
burning,
based
big
sagebrush
(Artemisia
comparison
tridentata
ssp.
Beetle and Young) treatment methods: spraying with 2,4plowing
upon
perennial
of this study is to conduct an economic
expected
grass
and seeding,
present
and rotocutting.
value of
Comparisons are
additional
net
returns
production realized after application
of
the
stuqy focuses
grazing,
of
four
treatments.
While
treatment
water
this
of
on domestic
sagebrush may impact soil
retention,
and wildlife habitpt.
considerations in the sagebrush
livestock
erosion
and
the
contamination,
Although these are important
management decision,
they are beyond
the scope of this study.
This
study
will
address
the
following
sagebrush treatments economically feasible?
they
be
imposed?
What
level
question:
If so,
of sagebrush
canopy
Are
how often
cover
these
should
warrants
imposing the treatment?
Outline of the Thesis
The next chapter reviews the literature on previous
of
for
sagebrush treatment methods.
choosing
a
Chapter 3 presents the
sagebrush management strategy.
contains empirical results and discussion.
topic of Chapter 5.
The
comparisons
methodology
fourth
chapter
Concluding remarks are the
3
CHAPTER 2
LITERATURE REVIEW
Sagebrush
management
publications,
treatment.
the
net
with
strategies
most
Previous
benefits
have been the
focus
concentrating on the biology
of
of
many
sagebrush
economic analyses computed the present value
from
a single
treatment of
of
over
sagebrush
a
prespecified period.
Among
the
destroying
Graham
big sagebrush were Elwe.ll. and Cox
(1951),
Hull et
of
first to demonstrate the effectiveness of
al.
2,4-D
and Hull and Vaughn (1951).
(1952)
in
usefulness in
(1950),
and
for
Cornelius
and
Working
ind~pendently,
Bohmont (1954) demonstrated
the usefulness
Wyoming,
and
eastern Oregon.
with 2,4-D became
2,4-D
widespread.
Hyder
Gradually,
(1953)
its
demonstrated
the brush control
program
The reported annual increase ih forage
yield after the treatment varied from as low as 85 percent in a threeyear observation (Cook, 1966) to as high as 295 percent in a five-year
observation
effective
(Alley
and
treatment
1958).
Bohmont,
Johnson
life
of
14-17 years on
not·
a
new
(1969)
areas
not
reported
grazed
by
livestock.
Burning
is
sagebrush-bunchgrass range.
Fire
tool
for
of
manipulation
was a .• commonly used
techniquet 50
~~~.
years
ago.
conducted
After
a
long-term ·ecological
in Idaho in 1936 and 1937,
study
with
two
burns
the usefulness of burning as
a
4
sagebrush
Ralphs,
management
al.
Because
1978).
increased
strategy was gradually
of its lower cost,
it
(8ritt6n
and
interest in burning has
From data presented by Pechanec
in the past 20 years.
(1954)
revealed
was concluded that within four years
after
et
burning,
forage production increased 90 percent; after 15 years it was still 33
percent
greater
Mueggler
and
production
than on an unburned
sagebrush
range.
Studies
Blaisdell (1958) show a 61 percent increase
of
three years following burning as compared to an
range.
The
effective
life
(1954)
lasted for at least 15 years.
forage
untreated
of burning reported by Pechanec
Whiteworth (1963)
by
et
al.
claimed
12
years of increased capacity following burning.
Plowing and seeding, and rotocutting are two mechanical treatment
methods.
·
A
study
average
increase
plowing
and seeding,
hand,
were
by
Kearl and
Brannan
showed
( 196 7)
an
following treatment of 133 and 369 percent for disk
it was thought
and rotocutting,
that
costs
respectively.
for
these
On the
treatment
other
methods
considerably higher· and results less dependable than burnihg and
spraying,
so
.. mechanical methods are not considered to be
a
viable
alternative" (USFS, 1973, p. 54).
Assuming
prespecified
conducted
an
that the response to the treatment was constant over an
effective
treatment
life,
Kearl
economic comparison among several
methods including rotary beating,
scraping with patrol or grader,
railing,
and
Brannan
sagebrush
(1967)
treatment
disk plowing and seeding,
and spraying by computing the present
net return from a single treatment.
From the
results
of
value
of
their
study it was concluded that patrol and disk plowing followed by
5
seeding were the most productive methods.
terrain
and
Spraying was desirable when
topography prohibited use of the patrol or disk
plowing
followed by seeding.
A similar
study by Nielsen and Hinckley (1975)
internal rate of return for spraying,
rotobeating, and chaining.
chaining
burning,
were economically feasible;
return
were
was obtained with burning.
similar
planning
to
period
but plowing
considered.
relatively
not
As
seeding,
and
seeding,
The shortcomings of the
prespecified and only a
the authors pointed out,
and
The highest internal rate
those of Kearl and Brannanls
was
plowing and
the
It was claimed that .burning, spraying, and
rotobeating were not economically feasible.
of
calculated
(1967)
in
single
study
that
the
treatment
was
.. Although this method is a
easy way to calculate the internal rate of return,
as accurate as one may desire" (Nielsen and
Hinckley,
it
is
1975,
p.
over
the
13) .
and Payne (1986) used data from a common site
Wambolt
1963 to 1981 time period to compare tour sagebrush treatment metHods-burning,
The
spraying with 2,4-D,
criterion
level.
The
to
objective,
then
and rotocutting.
used was sagebrush canopy cover and forage
conclusion
sagebrush
plowing and seeding,
increase
reached was that "if control of
understory
production
is
spraying with 2,4-D and burning are
than rotocutting or plowing and seeding" Cp. 319).
production
Wyoming
the
more
big
management
effective
6
CHAPTER 3
METHODOLOGY
For
maker
the
purpose of this study,
the objective of
the
decision
is to maximize the present value of the additional net
from
livestock
sagebrush
grazing.
The decision maker will then
treatment methods and levels.
Specifically,
returns
choose
the
among
decision
maker will maximize
00
PV
= L CRt
(3.01)
- Ct)/(1 + r)t
t=l
where
PV = present value of additional net returns
Rt
= additional
Ct
= treatment
r
= discount
To
maximize
returns in year t, which is the product of
the unit value of the forage and the additional yield
due to sagebrush treatment
costs in year t
rate
equation (3.01),
method
of treatment,
making
these
the decision
how much to treat,
decisions,
maker
chooses
and how often to treat.
the decision maker is
forage production response to the treatment,
concerned
with
the
In
the
the value of the forage,
and the costs of the treatment.
In
burning,
this
study four treatment methods were
plowing
and seeding,
and rotocutting.
analyzed under one intensity level,
analyzed:
spraying,
The treatments were
i.e., various amounts of spray or
7
fuel
quantities
were
intensities were not available.
the
experiment
Data
not considered.
to
analyze
different
Thus, the implicit assumption is that
from which the data were obtained was
designed
such
treatment
was
that the treatment intensities were optimal.
The
forage
production
response
to
sagebrush
estimated using data obtained in Montana.
Study Site And Field Data
The
big
study site from which forage production response to
sagebrush treatments was
measured is located approximately 27 km
west of Dillon in southwestern Montana.
wheatgrass habitat type,
Wyoming
The big
sagebrush--bluebunch
which receives about 310 mm of precipitation
annually--is typical of much rangeland in the region.
A ~ore detailed
description of the site is provided by Wambolt and Payne (1986).
Production
2,4-D,
plowing
and seeding,
sampled in 1964,
and
1986,
from
. responses
three
1965.
observations
11 years.
treatment was replicated four times resulting
(4x11)
40
for
spraying,
observations
plowing
(4x10)
for
and
year
44
and
The
four
of the study control (no brush treatment) were
as the proxy variable for environmental
in
seeding,
replications
perennial
were
its production response began
and
serve
1963
Because burning was applied one
rotocutting,
to
with
1965, 1966, 1967, 1970, 1976, 1977, 1978, 1981', 1985
totaling
Each
(spraying
and rotocutting) applied in
later (1964) than the other treatments,
in
treatments
burning.
averaged
conditions.
Total
grass production was used to represent desirable forage
as
8
no livestock browse was
encountered and
utilizable forbs were scarce
(Wambolt and Payne, 1986).
Formulation of Forage Response
Function to Sagebrush Treatment
The
value
subsequent
two
of
the treatment depends upon
to the treatment.
the
forage
response
In formulating a forage response model,
considerations are important,
the appropriate algebraic form and
the relevant variables.
The choice of a response function depends upon the biology of the
response
seems
process.
that
Whi1e such knowledge is
desirable
application,
reinvades,
reach
and
forage
a
would
peak,
after
increase
eventually
an
limited,
extended
after
decline
period
intuitively
of
as
it
the
treatment
the
sagebrush
time
approach
equilibrium which may or may not be the same as the equilibrium
an
prior
to treatment. Such a response function is shown in Figure 1.
The
choice
biology
of
the
treatment
type
influence
forage
forage
produced
for
may
Environmental
of variables should also depend upon the
response
and
time
production.
space,
be
the
water,
It
process.
period
and
also
hypothesized
since
treatment
Since sagebrush
nutrients,
influenced by the
conditions
is
quantity
may be
determining the forage response level.
competes
underlying
that
application
with
other
of
forage
the quantity
of
important
the
killed
sagebrush.
considerations
in
Little environmental data were
available (precipitation,
temperature, etc.), so the average level of
forage
control
response
on
the
was chosen
as
a
proxy
for
the
9
envirunmental influences.
Based
upon
the
considerations mentioned
above, the following response function was chosen:
(3.02)
where
Y(t) = level of forage production, defined as the ratio of
forage production with treatment to forage production
without treatment at time t (years since treatment
was applied)
k
=quantity of killed sagebrush, defined as sagebrush
canopy cover before treatment was applied (t=O) minus
sagebrush canopy cover after .treatment occurs (t=l)
e
= Euler's number (2. 71828· · · · · ·)
e
=long-term equilibrium level of forage produced by the
treatment relative to the control since limit Y(t) = e
t~
ut
=random disturbance term, ·assumed to be normally
distributed with mean zero
y
p
R
0
D
u
c
T
I
0
N
L
I
E
v
E
L
T
YEARS SINCE TREATMENT
Figure 1.
Response function shape.
10
a,
r,
~'
If a>O,
~>0,
e
B, and
are parameters to be estimated from the data.
o>O, o<O and B>O, the function has the desired form (for
the derivation of the .parameter
environmental
forage
influences
response
problem
of
ratio
signs,
see
Apendix
B).
That
were incorporated by expressing Y(t)
of the treatment to the
unavailability
of
enviromental
control
data
and
the
as
a
solves
the
avoids
the
complexity of estimation even if such data were available.
Development of DP Model
The
optimal
policy for sagebrush treatment involves choosing
sequence of decisions on whether and/or when to treat sagebrush,
that
the
expected
maximized.
present
A decision
value
of
additional
specifies one of the
net
possible
such
returns
is
alternatives,
given expected physical and economic conditions at a particular
in time.
a
point
A stochastic dynamic problem such as this can be efficiently
handled by dynamic programming.
Dynamic
programming
technique to solve
in
is
a
useful
mathematical
multistage decision problems.
The pioneering work
dynamic programming was done by Richard Bellman in
development
optimization
1957.
and applications have flourished since that time.
dynamic programming is a general type of approach to problem
and
the
individual
Further
particular
situation~
equations
used must be developed
to
Since
solving,
fit
each
a discussion of the development of the DP model
for the sagebrush management problem follows.
11
Stages
In dynamic programming, the problem is segmented into a number of
stages,
with
a
policy
decision
being
made
at
each
stage.
The
specification of stage and the total number of stages to be considered
depend
upop the particular problem being studied.
management
problem,
annually.
one
long
not
the
Therefore,
year.
Also,
decision
of
For the
whether to
sagebrush
retreat
is
made
the stage is a time interval and its length
the number of stages chosen should be
sufficiently
so that the decision rule and terminal value will be stable
be
affected by the number of stages.
horizon
is assumed,
Thus a
is
100-year
and
planning
with a decision concerning keeping or retreating
Accordingly,
sagebrush made annually in the late summer.
there
are
100 stages, each one year in length.
State Variables
The
condition
particular stage
The
or
state
of the problem
under
analysis
is defined by the magnitude of the state
at
a
variables.
state variables in the optimization problem at hand must describe
both the physical (additional yield potential) and economic conditions
that
which
are
will
be encountered at a given stage.
affects the additional forage yield,
two
such
variables
describing
the
Years
since
treatment,
and value of forage yield
physical
economic
and
conditions.
Years
since
deterministic
the
treatment
treatment.
Years
since
treatment,
t'
state variable which denotes the number of years
was imposed.
Choosing the range of t largely
is
a
since
depends
12
upon
the
effective life of the treatment.
Since the
lucus
of
the
forage production response to treatment was fairly flat after about 20
years (see Figure 4), the t range was assumed from 1 to 30.
Value
month
of
forage yield per
animal-unit-month.
An
animal-unit-
(AUM) is the amount of forage required to maintain a 1000 pound
cow for one month.
state
The value of forage yield per AUM is a
stochastic
variable in the model since the future value is not known
certainty.
example,
costs
The calculated value depends upon the technique used.
one
could
other
Another
with
than
way
For
take residual ranch income after payment of
forage
and compute the
average
of handling the problem would be to
use
value
all
per
market
AUM.
value,
usually expressed as a monthly lease rate.
Each
unique
this
analysis
of a sagebrush treatment project
may
characteristics that influence the appropriate AUM
study,
the
calculating
the
have
some
value.
monthly lease rate was not chosen as the basis
value
of forage yield per AUM
upon
the
In
for
following
considerations:
1.
In
Montana,
most
domestic
livestock producers
are
ranch
owners. Leasing of private land for grazing is not a dominant feature.
Leasing of public land for grazing is prevalent, but the lease rate is
not competitively determined.
2.
year,
Since
the
the
length of ranch lease is usually more
lease rate is somewhat vague
than
and does not reflect
one
annual
changes in product price.
Residual
per AUM :
income was used to compute the expected value of forage
13
(3.03)
E(VF) = (E(CP) x EW - AC)/LGS
where
E(VFl = expected value of forage per AUM
ECCP) =expected calf price per 100 lbs.
EW
=equivalent weight of a calf, which was 4.62 hundred
lbs. (for details, see Appendix C)
AC
=average costs to raise a calf, which was $262.41 (for
details, see Appendix C)
LGS
= length of grazing season, which was 9 months
Note
that
investment
value
of
if
imposing
sagebrush treatment was
by livestock producers,
forage
viewed
then the definition
of
as
an
expected
yield in equation (3.03) would make calf
price
an
important determinant in the optimal investment decision process.
The
results of previous empirical studies generally seem to indicate
that
livestock
The
price
studies
(1977)
by Freebairn and Rausser {1975),
claimed
investment
has significant influence on producerls
in
that
cattle price is a
breeding herd inventory.
investment.
and Martin
significant
Rucker et
and
Haack
determinant
al.
(1984)
of
also
showed a positive correlation between cattle price and inventory size.
Furthermore,
the authors pointed out, "Using pasture more intensively
to increase production during periods of high prices and letting
recover
during
periods of low prices through less intensive
grazing
could be quite rational behavior of ranchers in semiarid regions
as Montana" Cp.
likely
to
132).
expect
them
such
If a cattle cycle does exist and ranchers are
future
prices to continue to
follow
a
cyclical
pattern, then, to expand the operations in response to an upward price
trend,
they
may improve the productivity of the land by
investment.
-
....,...
'
14
"Various
sagebrush
control methods are important
ways
to
increase:
range productivity" (Kearl and Brannan, 1967, p. 9).
Transition Probabilities
As
defined earlier,
the
state
stochastic
variable--expected
value of forage per AUM--is a function of calf price.
continuous
Calf price is a
random variable for which transition probabilities must be
calculated.
Calf
price
equation.
equation
The
was
observations over 13 years (1974-1986) on calf
in Montana (Montana Agriculture Statistics,
was
expressed
Index (CPI).
stochastic
estimated
using
per 100
pounds
price
1983-1986).
Calf
price
in 1986 dollars after deflating by the Consumer
Price
The best fit equation was a second-order
difference equation.
autoregressive
Equation (3.04) gives the
estimated
coefficients with t-values in parentheses .
CPt
= 45.8162
(2.394)
+ 1.0841CPt-l - . 6388CPt_ 2 + et
(3.996)
(3.04)
(-2.328)
where
CPt
= calf
price in year t
CPt-1
= calf
price in year t-1
CPt-2 = calf price in year t-2
et
The
=
random disturbance term
standard
deviation of the estimate,
first-order autocorrelation are 14.3640,
The
hypothesis
that
.59,
the residual is normally
rejected (see Appendix D).
adjusted
R2
'
and
the
and . 06, respectively .
distributed
was
not
15
The
complex
estimated
root
calf
price difference equation has
a
cohjugate
which implies a convergent time path with
a
repeating
cycle every 8.43 years.
prevailed
of
for a century.
Savin,
1977;
the cycle,
numerous
price
The cyclical behavior of livestock prices has
distinct
economists
(McCulloch,
1975,
and Anderson, 1977) have questioned the existence
1977;
th~
While a few
economics literature of the last decades
contains
and reasonably well-defined explanations
cycle phenomena.
In his classical paper "The Cobweb
of
the
Theorem,"
Ezekiel (1938) showed that the cyclical nature stems from the response
lags
between
the deviations in past prices and in
current
outputs.
Nerlove (1958) introduced the concept of adaptive expectations to
modeling of agricultural markets.
demonstrated
maximizing
From
that
the
A
~tudy
by Long and Plosser
consumption-production
plans
individuals may be an explanation for price
the debates mentioned above,
the
(1983)
chosen
by
fluctuations.
it was inferred that the remaining
issue is which theorem may explain the price cycle better, rather than
whether this cycle does exist.
The debate is beyond the scope bf this
study.
Transition
probabilities.
The required conditional
probability
for calf price is specified below:
CPji
= PR(CPt =
CP J· 1·
=the probability of going to the ith calf price state
in year t, given the jth previous calf price state
in year t-1 and t-2
i
I CPt-l CPt_ 2 = j)
(3.05)
where
16
The
$97.5.
the
calf price range was from $60 to $135,
Fiure 2
which is centered at
shows the calf price midpoints and intervals used
in
model.
Since
was
not
the hypothesis that the residual is normally
rejected,
the
calf
price
transition
distributed
probabilities
were
computed using the standardized normal variate below:
Z
= (CPm
-
(3.06)
~)/~
where
Z
= the standardized normal variate
CPm =calf price midpoints
~
= the estimated mean
~
= the estimated standard deviation
The price transition probabilities are presented in Table 1.
- Midpoints 67.5
60
I
82.5
J
75
112.5
97.5
90
I
105
I
.I
127.5
120
1
135
-Intervals (dollars) Figure 2.
Calf price intervals and midpoints.
Decision Alternatives
A policy decision refers to a plan to make a decision based on
predetermined policy under each possible condition.
to
be
made
from
a set of available
alternatives
Decision alternatives considered in this model are:
0) Retreat the sagebrush.
The decision
at
each
a
has
stage.
17
1) Keep the sagebrush.
Note
that
variable;
i.e.,
quantity
of
years
since
on
a
deterministic
state
stage,
the
additional forage yield at the next stage is known
variable,
effect
is
once a decision is made at a particular
certainty given a decision.
state
treatment
the
However,
with
value of forage is a stochastic
and the decision made at a particular stage
value of forage state at the next
stage.
has
no
Value
of
forage state transition is a function of the previous calf prices.
Table 1.
Probability distribution of calf price at year t,
calf prices at year t-1 and year t-2.
Previous Calf
Price Condition
67.5
67.5
67.5
67.5
67.5
82.5
82.5
82.5
82.5
82.5
97.5
97.5
97.5
97.5
97.5
112. 5
112. 5
112. 5
112. 5
112. 5
127.5
127.5
127.5
12 7. 5
127.5
67.5
82.5
97.5
112. 5
127.5
67.5
82.5
97.5
112. 5
127.5
67.5
82.5
97.5
112. 5
12 7. 5
67.5
82.5
97.5
112. 5
127.5
67.5
82.5
97.5
112. 5
127.5
given
Calf Price Midpoints
CPt
67.5
82.5
97.5
112.5
127.5
.4761
.7291
.8980
.9738
.9955
. 1170
.2981
.5557
.7910
.9306
.0099
.0485
. 1611
.3745
.6331
.0000
.0026
.0170
.0721
.2148
.0000
.0000
.0000
.0048
.0274
.3604
.2214
.0918
.0248
.0045
.3240
.4004
.3273
. 1768
.0635
.0904
.2224
.3588
.3897
.2846
.0080.
.0375
. 12 31
.2688
.3878
.0000
.0020
.0136
.0570
. 1620
. 142 3
.0459
.0102
.0014
.0000
.3728
.2421
. 1041
.0303
.0059
.3049
.3955
.3444
. 1966
.0748
.0773
.2019
.3479
.3948
.3006
.0062
.0316
. 107 4
.2503
.3781
.0201
.0036
.0000
.0000
.0000
.15 79
.0549
.0129
.0019
.0000
.3858
. 2628
. 1195
.0381
.0075
.2892
.3911
.3558
.2178
.0872
.0659
. 1812
.3312
.3933
.3194
.0011
.0000
.0000
.0000
.0000
.0283
.0045
.0000
.0000
.0000
.2090
.0708
.0162
.0011
.0000
.6255
.3669
. 1562
.0465
.0096
.9279
.7852
.5478
.2946
. 1131
18
Expected Additional Yield
Expected
with
equal
additional yield was defined as the quantity of
treatment
(kg/ha)
minus that without treatment which
to the average quantity of forage produced on the
addition,
its
value
forage
was
set
control.
In
should be transformed into AUMs in order to
be
consistent with the measurement of expected forage value.
Formulation
of the expected additional forage yield equation
is
as follows:
Recall the treatment response function
(3.07)
where
= forage
Yr(t)
production level with treatment in the tth
year
YNT(t)
= forage
production level on the control in the tth
year
so
(3.08)
thus
(3.09)
which is the additional yield.
While
there is some difference of opinion on the forage required
per AUM, the Forest Service recommendation for the region of the study
site
Also,
of
YNT
353kg (USFS,
was
set
control, 180.21kg.
AUM terms is:
1983) with a 50% proper use rate
equal
to average level from
was
1964-1986
chosen.
on
the
Therefore, the expected additional forage yield in
19
EAY = [YT(t) - YNT(t)J/(2 x 353)
= 180.21
x Eak~t 0 e&t +
= .2552549
x Ea~t 0 e&t
ce+
ce-
1)J/(2 x 353)
l)J
(3.10)
Treatment Costs
Treatment
cost
is an important
consideration in
choosing
the
most desirable method of treatment and the optimal_ retreatment period.
The cost of the treatment varies depending upon individual situations.
Nielsen
and Hinckley's (1975) treatment costs were adjusted
to
1986
(Table 2).
Table 2.
Summary of treatment costs.
Treatment Costs
Nielsen and Hinckley's
(1975) Estimate ($/ac)
Adjusted to 1986a
($/ha)
Spraying
5.82
21.37
Rotocutting
7.37
37 .19
Burning
4.00
20.18
21.00
105.97
Treatment Method
Plowing and Seeding
aAdjusted for inflation from 1975 to 1986 by CPI (Economic Report
of the President, 1987). CPI is 161.2 and 328.4 for 1975 and 1986,
respectively, implying an inflation adjustment factor of 2.0372.
Transforming from acres to hectares as well as inflation results in a
total adjustment factor of 5.046.
The Discount Rate
The additional information necessary to compute the present value
of the benefits from the treatments is the discount rate,
~'
defined as 1/(1 + r), where r is the real interest rate (51.).
which was
20
Recursive Equation
A recursive equation identifies the optimal policy for stage
given
the
optimal
following
policy
for stage
three properties.
First,
(n-1).
It
must
n,
possess
the
a decision is to be made at
any
stage n. Second, a decision, together with the state of the process at
stage
n,
Third,
determines
the
state of the process at
the
next
stage.
for any stage n, the state and the decision determine expected
returns for that stage.
Bellmanls
formulation
principle
of
optimality provides the basis
of a recursive equation and for the
solution
This principle states that given the current state,
decision
for
decisions
defined
the
remaining
adopted
as
the
in
stages
is
An
the
technique.
an optimal policy
independent
the previous stages.
for
of the
optimal
sequence of decisions that optimizes
policy
policy
the
is
objective
function.
The
present
objective of the sagebrush treatment problem is to
value
Application
of
of
the
additional
net returns
principle
of
from
optimality
forage
gives
the
maximize
prod~ction.
following
recursive relationship:
5
K
L PR-1 ·
i =1
x [rr(t, EVF;) +
J
( 3 . 11 )
5
R
(L
i =1
where
PR··
x [rr(l, EVF;) +
1
J
21
PC·) = present value of additional net return at
J
stage n, given previous calf price condition PCj
n
=stage, 1, ... , 100
t
= years since treatment, t = 1, ... , 30
pc.
J
= previous calf price condition, there are 25
combinations· of such condition, so j = 1, ... ,
25 as presented in Table 1
K
= keep the sagebrush
R
= retreat the sagebrush
PR·.
1J
=probability of moving from the jth previous
calf price state to the ith calf price state
as presented in Table 1
n(t, E(VF;)) =immediate return with the ith calf price midpoint and t years since treatment, which is the
product of E(VF;) and expected additional yield
as defined in equation (3.10)
E(VF;)
= the ith expected value of forage per AUM as
defined in equation (3.03)
A
= discount rate
pc-:J
= previous calf price condition at stage n-1
where
1 <= j <= 5'
J = 1 ' 6, 11 ' 16' 21
6 <= j <= 10'
J = 2' 7' 12' 17' 22
i f
11 <= j <= 15' then J = 3' 8, 13' 18, 23
16 <= j <= 20,
J = 4, 9, 14' 19, 24
21 <= j <= 25'
J = 5' 10, 15' 20, 25
TC
= treatment cost as presented in Table 2
At each
stat~,
an optimal policy decision which yields a
present value between the two alternatives was chosen.
recursive. equation, the
maximum
By using
solution procedure moves backwards stage
stage, finding an optimal policy for each state at every stage.
this
by
22
Terminal
Valu~
The solution procedure, which begins by solving for PV 1 , requires
a value for PV 0 . Here PV 0 was set equal to zero.
23
CHAPTER 4
RESULTS
Statistical Estimates of the Response Function
The
using
response
the
function for each treatment method
was
estimated
same functional form with and without the restricting
with SAS/ETS SYSNLIN regression software.
was Marquardt-Levenbery.
were 2.5,
1.0,
rotocutting;
The estimation method
The starting values for a,
~'
o,
~=0
used
& and
e
.9, -.3 and 1.0 in the case of spraying, burning, and
and 2.7,
.1,
13.5, -4.5 and 1.0 in the case of plowing
and seeding, respectively.
The
the
Graphs
results of estimation are presented in Table 3.
production responses for each treatment are presented in
of
Figures
3, 4, and 5.
When the functions were estimated without the restriction
rotocutting
as
hypothesized,
parameters
it
influence.
The
equation
where
well
and
as burning,
from
all the parameter
the t values associated with
signs
the
was inferred that all the variables had a
exception
was
the parameter a
the t value was .76.
rotocutting and burning,
respectively.
in
the
~=0
for
were
as
estimated
significant
rotocutting
The R2 are .5548 and .6731
for
It is not surprising that the
influence of the quantity of killed sagebrush, k, is not significantly
different from zero
for spraying since the variance among k values is
24
small
to
from
for the treatment.
.17
for
burning
and
The k ranged from
rotocutting,
.08 to .21 and from
respectively,
but
.11
was only
.12 to .17 for spraying.
Table 3.
Response function statistics (1).a
e
Treatment
3.19503
-.09262
.60218
-.22325
.64334
(-.18)
(1.69)
(-2.03)
(1.15)
3.80688
.61121
-.22648
.65794
(5.48)
(1.73)
(-2.06)
(1.21)
1.35806 -.45720
.90482
( 1 . 92 ) ( -1 . 98)
(4.30)
1.64720
1.41106 -.44491
.89303
(3.26)
(1.81) (-1.84)
(3.80)
SSE
46.97062
.6643
47.01323
.6640
25.81935
.5548
29.05609
.4990
12.37374
.6731
13.92577
.6321
49.39890
.6482
49.39919
.6482
Sprayingb
(. 98)
Sprayingc
Rotocuttingb
22.47123 1.31507
( . 76)
Rotocuttinge
( 1 . 88)
3.96645
.37028
.77263
-.18009
.65967
(2.66)
(1.75)
(2.09)
(-2.19)
( 1 . 14)
1.97634
.74013
-.17257
.60645
(2.55)
(1.84)
(-1.91)
( . 89)
Burningb
Burningc
Plowing
and
Seedingb
2.65600
Plowing
and
Seedingc
2.67712
(1.10)
(1.72)
-.00386 13.59783 -4.79980 1.05726
(-.01)
(3.38)
(-3.34)
(5.28)
13.61179 -4.80470 1.05744
( 3. 4 3)
(-3.39)
(5.35)
aThe numbers in the parentheses are the t values.
bFunction was estimated without restriction ~=0.
cFunction was estimated with restriction ~=0.
In the case of the regressions with the restriction
~=0,
all the
parameter signs were as hypothesized, and from the respective t values
61I
I
PLOWING AND SEEDING
BURNING
5;
F
0
/
R
4
-
I
I
,I
'
'
3: i
I
A
T
I
I
i
""""""
I ~~- -----------~~
I
1
ROTOCUTTING
"
"""
\
"""
---~\ ---~ ~
,/
-'
-//
,.
:
\
I
I
J\
E
G
SPRAYING
)---\"'
,(
'\
I
I'
!p
1
0
--.__ --.__
"-
''
I 2 J:.!I I/
II//'I: III
0
I
"""----' -'" ""
----
I
---.._
I I
~---: :_-: o=------=-- - - - - - -- - . . - -. ______ ------- ---==---==
~ ----- -------'---,
--
-
c
I'\.)
(JI
-
-
-·
.
'r
~----------~--- --~~--.-- --T·----.-------r-·-.---,-------..---,----.--T·----r--T--,-
0
2
4
6
8
10
12
14
---r
16
YEJ\RS SINCE TREATMENT
Figure 3.
Production response to treatment.
18
'
20
r
22
1
24
4
-.._______
/
/
F
0 3
I'
II
I
''
'
'I'
R
G
~
----'-...
/
I/
/
/
~~ ~
~
_-- -
--------.
~"-
'-- '--
"'
"'"' ~ '
"-.
--- --,,
',,
K=.1 0
'-._ '-._
'-._ '-._
','
'-
' ',
/; /
K=.05
.
', ',,_
1ff,
1 /
~~/(.'/ ,1
'
.i I
E
R
A
T
tl
'
------
II!1/ / /
A
'-._
~
K=.15
'"-~~
----------'--'--~~
"-.
K=.20
"-..
"-,~
---
I
------
.
-------
y
I 1
---------- - ~: -:~-: --.: ~-: : .:~
"-....__'----.-....._
, , , - , , , ,....__ -....._ ..::::::- ::::---._
0
0
--·--r----·--1
0
•-:>
(..,J
r---
--,---T---.----r-----,---T-----r---~---.--
-1-
6
8
10
12
---r----,...r--r---~
1-1
16
18
20
YE ..L\.RS SINCE TRE ..\T1:1ENT
Figure 4.
Production response to burning with different K values.
22
24
N
(j\
4J
!~""\
I
I
I
I
F 3
0
R
A
~
!
i
I
/
I
I I
\
/~--.."-
I
I I
G
' I/
E 2 II /
K=.1 0
\\
""
i II
1
K=.OS
"'-"'-
\
"
"
/-----------
K=.15
\
\..
"" """'
-----._ '
/I ,
-----
AT
/1/I :
R I jj/I! /
I 1 ~~I
0
:
K=.20
~"
"----- .____ ______
"'"'"' "'------
""
.
~-:::-::~-7:. -:;:;:;. .~ - -
-.J
------- -- ---'::. ::-:-_-::::::::-
0 \-- -,---·r--,--·--y-·· ---,---,--.-,---.--r--r---,---,-~-----,-·-r-----r---r
0
rv
2
4
6
8
10
12
14
16
18
-r---.--T
20
YEARS SINCE TREATMENT
Figure 5.
Production response to rotocutting with different k values.
22
24~
28
it
was
except
inferred that all the vari·ables had a
e
parameter
for
burning, where
significant
influenc~
The R2
the t value was .89.
are .6640, .4990, .6321, and .6482 for spraying, rotocutting, burning,
and plowing and seeding, respectively.
The response function for each treatment was also estimated using
an alternative model specification k~(at 1 eBt +B).
allows
This specification
k to have an impact on long-term equilibrium,
but the fit was
not as good as for the previously specified function.
Some hypothesis tests show that:
1.
There is a significant structural difference among these four
treatments for the restricted model
2.
There
is
a
(~=0).
significant
structural
rotocutting and burning for the unrestricted model
3.
difference
between
(~~0).
Quantity of killed sagebrush does have significant
influence
on the unrestricted model in the case of burning and rotocutting.
For the details of these hypothesis tests, see Appendix E.
From
function
the
estimation results it was concluded that the
reflected
the
changes
in
forage
production
response
after
the
treatment was applied and explained a high percentage of the variation
in the observed values.
DP Solution
Solution of the recursive equation yields the optimal retreatment
interval
and expected present value of additional net returns for all
combinations of states and stages.
There are 750 states at each stage
of the sagebrush treatment decision model.
Before the results of
the
29
DP
model are discussed,
two important assumptions made in developing
the model should be stated:
1.
The
calf
price transition probabilities obtained
from
the
historical time-series analysis are valid for the future.
2.
the
The
subsequent responses to retreatments are identical
initial
response
treatment
was
interval,
then
identical
response
to
treatment.
For
example,
imposed at year t with a 10-year
after
if
an
optimal
a retreatment application
at
with
initial
retreatment
year
t+lO,
an
would begin to occur in year t+11 as the
initial
and
dynamic
response in year t+l.
5,
Tables 4,
model,
programming
~=0
6
summarize
the
results
of
the
given the response func.tion with the
for all four treatments,
without the restriction
at different k levels, and without the restriction
restriction
for burning
~=0
for rotocutting
~=0
at different k levels, respectively.
Ta.b 1e
4
shows the
results of
comparing
the
four
methods,
given the response function with the restriction
specific
treatment
plowing
and
seeding was not economically feasible;
only marginally feasible;
feasible.
was
~=0
.and the
concluded
that
rotocutting
was
and spraying and burning were
economically
The highest expected present value of additional net return
obtained with spraying.
increases
the
From the results it was
costs.
treatment
(i.e.,
The results also show that as calf price
calf price in year t-2 is lower than in year
expected present value of additional returns would
t-1),
increase
and
the optimal retreatment interval would decrease, since an upward price
trend
usually
implies
higher expected future
prices.
The
reverse
30
Table 4.
Present value of the additional net returns-(dollars/ha) and
optimal
retreatment
intervals (years)a
generated
bb
sagebrush treatment method, given specific treatment costs
and previous calf price condition.
Previous Calf
Price Condition
CPt-2
Treatment Method
Spraying
Rotocutting
Plowing and
Seeding
Burning
67.5
67.5
67.5
67.5
67.5
67.5
82.5
97.5
112.5
127.5
141.30
l37.35
135.36
134.68
134.36
( 8)
(10)
( 12 )
( 13 )
( 1 4)
11 . 78
11 . 12
11 . 06
11.04
11 . 03
( 1 3)
(30)c
(30)c
(30)c
(30)c
101.26 (10)
100.46 ( 9)
( 9)
99.24
(9)
98.75
( 9)
98.62
82.5
82.5
82.5
82.5
82.5
67.5
82.5
97.5
112.5
127.5
(7)
145.32
139.56 ( 8)
( 9)
135.51
132.28 ( 12 )
130.94 ( 1 5 )
13. 15
10.64
9.92
9.93
9.95
( 1 0)
( 14)
(30)c
(30)c
(30)c
102.22
100.04
98.33
97.24
96.71
( 11
( 11
( 11
( 11
( 11
)
)
)
)
)
---------------------------------------------------------------------97.5
97.5
97.5
97.5
97.5
67.5
82.5
97.5
112.5
127.5
148.96 ( 7 )
144.66 ( 7 )
( 8)
138.52
( 9)
134.22
130.95 ( 11 )
13.94
11 . 85
9. 78
8.92
9.09
( 1 0)
( 11 )
( 15 )
( 30) c .
(30)c
104.40
102.03
99.84
97.47
96.40
( 1f
( 11
( 11
( 12
( 12
112.5
112.5
112.5
112.5
112. 5
67.5
82.5
97.5
112.5
127.5
1 51 . 49
148.29
144.50
138.03
133.60
14.65
12.29
10.86
9.24
8.89
(10)
(11)
(12)
(16)
(29)
106.22
104.35
100.82
98.95
97.39
(11)
(11)
(12)
(12)
(12)
127.5
127.5
127.5
127.5
127.5
67.5
82.5
97.5
112 . 5
127.5
151.16
150.00
147.80
144.72
138.08
14.85
12.65
1 2 . 46
11 . 02
8.94
(10)
(11)
( 11 )
( 12 )
(27)
106.27
103.55
102.38
1 00. 7 6
99.05
(11)
(12)
(12)
( 12 )
(12)
(7 )
(7)
(7)
(8)
(9)
(7)
(7)
(7)
(7)
(8)
j
)
)
)
)
NFd
aoptimal retreatment intervals were in parentheses.
bTreatment costs were presented in Table 2.
cThe optimal retreatment interval might be longer than 30 years.
dNot economically feasible.
~
31
relationship
also holds with a downward price trend.
equal
calf
prices at years t-2 and t-1, a higher
would
result
in
a lower expected present value
For
the
price
of
combination
additional
returns and usually a longer optimal retreatment interval as
to
a
lower
given
net
compared
price combination, since successive higher prices
would
imply lower expected future prices.
For different previous calf price
conditions,
values
the
expected
present
and
optimal
retreatment
intervals
vary
from $130.94 to $151.49 and 15 years to 7
years
spraying,
and $96.40 to $106.27 and 12 years to 9 years for
for
burning,
respectively.
In
Table
5,
k varies from .05 to .25 in the case
given the specific treatment cost.
previous
calf
increase
and
increasing
price
condition,
of
The results suggest that given the
the expected
present
value
optimal retreatment interval would ·decrease
k
value.
Since
burning,
burning will still
be
an
would
with
the
economically
feasible treatment method with a .05 k value, it implies that imposing
burning
on
an
area
with
a sagebrush canopy
cover
:as
above
is
feasible.
The
given
results of varying k value from .05 to .25
the
specific
treatment
method,
on
are presented
rotocutting,
in
Table
Rotocutting
was
below
The results also show a positive correlation between
.15.
not an economically feasible method, with a k
value
and
the
expected
present value and the quantity of killed
negative
correlation between the optimal retreatment interval and the
quantity of killed sagebrush, other things equal.
sagebrush,
6.
a
.
.
32
Table 5.
Present value of the additional net returns (dollars/ha) and
optimal retreatment intervals (years)a generated by burning,
given specific treatment costsb,
previous
calf price
condition, and different k values,
Previous
Calf
Price
Condition
Burning
k=.05
k= .10
k= .15
k=.20
k=.25
67.5 67.5
67.5 82.5
67.5 97.5
67.5 112.5
67.5 127.5
47.50
46.30
45.90
45.74
45.54
(11)
(12)
(12)
(12)
(13)
81.46
79.95
79.14
78.82
78.73
(10)
(10)
(10)
(10)
(10)
105.50 (10) 126.29
104.59 (9) 124.06
103.34 (9) 124.21
102.85 (9) 123.55
102.71
(9) 123.37
(9)
(9)
(8)
(8)
(8)
82.5 67.5
82.5 82.5
82.5 97.5
82.5 112.5
82.5 127.5
47.75
46.65
45.49
44.83
44.49
(12)
(12)
(13)
(14)
(15)
82.10
80.26
78.83
77.62
77.23
(11)
(11)
(11}
(12)
(12)
107.89
104.29
102.54
101.82
101.19
(10)
(11)
(11)
(10)
(10)
127.80
124.81
122.49
121.04
120.33
(10)
(10)
(10)
(10)
(10)
144.83 (10)
141.57 (10)
139.04 (10)
138.03 (9)
137.08 (9)
97.5 67.~
97.5 82.5
97.5 97.5
97.5 112.5
97.5 127.5
49.76
47.59
45.93
45.17
44.43
(l2)
{12)
(13)
(13)
(14)
84.09
82.05
79.40
78.08
77.19
(11)
(11)
(12)
(12)
(12)
108.71
106.31
104.08
102.25
100.57
(11)
(11)
(11)
(11)
(12)
128.93
126.20
123.66
121.56
120.11
(11)
(11)
(11)
(11)
(11)
146.23
143.20
140.38
138.05
137.00
112.5 67.5
112.5 82.5
112.5 97.5
112.5 112.5
112.5 127.5
49.67
48.76
47.67
45.78
44.61
(12)
(12)
·(12)
(13)
(14)
85.68
82.48
80.86
79.30
78.00
(11)
(12)
(12)
(12)
(12)
110.54
108.64
106.36
103 17
101.58
(11)
(11)
(11)
(12)
(12)
130.85
128.72
126.17
123.71
120.81
(11)
(11)
(11)
(11)
(12)
148.30 (11)
145.96 (11)
143.~5 (11)
140.43 (11)
138.11 (11)
127.5 67.5
127.5 82.5
127.5 97.5
127.5 112.5
127.5 127.5
49.74
49.39
48.73
46.63
45.84
(12)
(12)
(12)
(13)
(13)
83.63
83.12
82.15
80.80
79.38
(12)
(12)
(12)
(12)
(12)
110.60
107.87
106.68
105.03
103.27
(11)
(12)
(12)
(12)
(12)
130.87
130.04
128.48
124.79
122.77
(11)
(11)
(11)
(12)
(12)
148.18
147.29
145.61
143.26
139.42
aoptimal retreatment intervals are in parentheses.
bTreatment costs are presented in Table 2.
143.38
140.71
140.57
139.85
139.65
( 9)
( 9)
( 8)
( 8)
( 8)
(11)
(11)
(11)
(11)
(10)
(11)
(11)
(11)
(11)
(12)
33
Table
6.
Present value of the additional. net returns (dollars/ha)
and optimal retreatment intervals (years)a generated by
rotocutting, given specific treatment costs ,
previous
calf price condition, and different k values.
Previous
Calf
Price
Condition
CPt-1 CPt-2
Rotocutting
k=.05
k= .10
k= .15
k=.20
k=.25
(7)
( 8)
( 9)
( 9)
( 9)
67.5 67.5
67.5 82.5
67.5 97.5
67.5 112. 5
67.5 127.5
13.41
12.73
12.61
12.57
12.56
( 13 )
(30)c
(30)c
(30)c
(30)c
63.00
60.62
59.24
59.04
59.02
82.5 67.5
82.5 82.5
82.5 97.5
82.5 112. 5
82.5 12 7. 5
14.36
12.16
11.36
11.30
11 ,• 29
( 10)
(14)
(30)c
(30)c
(30)c
(8)
63.76
62.04 (8)
58.46 ( 11 )
57.18 (30)c
57.14 (30)c
16.89
14.18
11.41
10.23
10.35
(9)
(10)
(14)
(30)c
(30)c
(7 )
69.47
(
8)
63.25
( 9)
59.87
57.25 ( 11 )
55.77 (30)c
125.04 ( 7 )
(7)
121.82
116.14 ( 8)
112.66 ( 1 0)
110.76 ( 11 )
112.5 67.5
112.5 82.5
112. 5 97.5
112.5 112.5
112. 5 127.5
17.78
14.80
12.93
10.82
10.27
( 9)
(10)
( 11 )
( 15 )
(29)
( 7)
71.84
( 8)
64.74
( 8)
63.27
( 9)
59.47
56.51 ( 11 )
127.39
124.96
122:.04
115.66
112.00
(7)
(7 )
(7)
( 8)
( 9)
127.5 67.5
127.5 82.5
127.5 97.5
127.5 112. 5
127.5 12 7. 5
15.39
15.24
14.96
12.06
10.50
( 10)
(10)
( 1 0)
( 12 )
( 16)
( 8)
66.31
.
( 8)
65.83
( 8)
64.93
( 9)
60.34
( 9)
59.51
127.81
126.88
125.10
117.54
115.69
( 7)
(7)
(7 )
(8)
(10)
(16)
(30)c
(30)c
120.53
117.38
115.79
115.48
115.40
122.29 ( 7 )
119.18 ( 7 )
114.39 ( 9)
112.85 (10)
111.67 ( 1 3)
---------------------------------------------------------------------97.5 67.5
97.5 82.5
97.5 97.5
97.5 112. 5
97.5 127.5
aoptimal
NFd
NFd
( 8)
( 8)
retreatment intervals are in parentheses.
0 Treatment costs are presented in Table 2.
cThe optimal retreatment interval might be longer than 30 years.
dNot economically feasible.
34
CHAPTER 5
CONCLUSIONS
management
Sagebrush
strategies
have
been
the
subject
considerable research and application for a long time.
a
However, only
few studies deal with economic considerations of these
The
of
strategies.
previous economic analysis completed simply computed the
present
value of the net benefits (Kearl and Brannan, 1967), or calculated the
internal
rate
treatment
of return (Nielson and Hinckley,
of sagebrush,
1975) from a
assuming that the response to the
single
treatment
was constant over a prespecified effective treatment life.
This
study conducted an economic comparison among
big sagebrush control methods--spraying with 2,4-D,
and
seeding,
and
rotocutting.
four
Wyoming
burning,
plowing
Production of perennial grasses
measured on experimental plots in southwestern Montana 12
the
period 1963-1986.
production
dynamic
criterion
from
model
were
AUM
response function.
problem
stochastic
The data were
and
dynamic
was
as
used to
programming
framework.
stochastic
economic
the expected present value of additional
sagebrush treatment.
a
formulated
The
net
Decision alternatives included in
were keeping or retreating the sagebrush.
during
nonlinear
estimate a
Control of sagebrush is
such the problem was
year~
was
The state
within
a
choice
returns
the
DP
variables
years since treatment and the expected value of forage yield per
which
was
defined as a function of calf
price.
Based
upon
a
35
statistically
estimated
equation,
calf
the
developed.
second-order
price transition
Optimal
retreatment
autoregressive
probability
difference
distribution
intervals as well as control
was
method
selection were model outputs.
Burning and spraying as control methods of Wyoming big
were
found economically feasible.
Spraying was the most
Rotocutting was only marginally feasible,
sagebrush
profitable._
and plowing and seeding was
not feasible.
From the study results it was also concluded that in addition
treatment
killed,
method,
the
retreatment
two
treatment
present
value
cost,
and
of additional
the_ quantity
net
of
return
sagebrush
and
optimal
interval also depend upon previous calf price trend.
For
an
upward
trend would result in a higher present value of additional net
return
and
combinations of previous calf prices with equal mean,
to
a shorter optimal retreatment interval as compared to a
downward
trend.
The
results
of this study are consistent
with
the
conclusion
reached by the USFS (1973) that .. mechanical methods are not considered
to be a viable alternative .. (p.
and
Hinckley's
(1975)
54), and also consistent with Neilsen
suggestion
to the extent
seeding was not economically feasible,
and
Brannan's
(1967)
that
plowing
but not consistent with
claim that plowing and seeding
was
and
Kearl
the
most
productive method.
Any
limitation
study
of
this type has limitations.
The
major
potential
of the study is that the decision rules are valid only
the extent that the subsequent responses to retreatment are
to
identical
36
to
the
initial
questionable
treatment
response.
This
assumption
because of the largely unknown nature of the
might
pe
biological
response process.
A second
limitation of this study is that the study plots
not actually grazed by livestock,
by
clipping,
Therefore,
a
and
prudent
thereby
concern
were
but forage production was estimated
AUMs
is
of
production
whether
this
accurate estimates under typical grazing conditions.
were
approach
estimated.
provides
37
BIBLIOGRAPHY
38
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42
APPENDICES
43
APPENDIX A
ORIGINAL DATA
44
Table 7.
Production level of perennial grasses with spraying (kg/hal.
Replication
Year
1
2
3
4
1964
1965
1966
1967
1970
1976
1977
1978
1981
1985
1986
660.7
566.7
118.6
811.2
280.3
169.5
128.0
358.4
517.8
123.2
103.8
721.5
560.3
329.9
833.6
546.2
182 .1
222.8
661.1
404.1
72.7
136.9
635.4
735.1
160.9
880.6
462.1
431.4
327.5
379.7
303.0
58.1
91.0
473.9
618.5
250.5
482.1
277.5
135.5
54.9
389.2
324.7
81.2
134.5
Table 8.
Production level of perennial grasses with burning (kg/ha).
Replication
Year
1
2
3
4
1965
1966
1967
1970
1976
1977
1978
1981
1985
1986
314.1
82.6
630.8
538.6
133.5
131.2
478.0
577.8
181.6
101.4
402.9
144.6
822.9
563.4
195.2
251.2
601.5
758.0
114.0
119.8
352.3
103.8
678.9
395.5
183.3
171.8
419.2
541.9
103.5
159.8
230.7
85.6
901.0
439.4
322'., 4
291.9
605.5
954.7
273.4
235.0
----------------------------------------------------------------------
45
Table
9.
Production
( kg/ha).
level of perennial
grasses
with
rotocutting
Replication
Year
1
2
3
4
---------------------------------------------------------------------1964
1965
1966
1967
1970
1976
1977
1978
1981
1985
1986
Table
570.6
330.5
218.0
430.1
215.7
63.6
46.2
206.4
338.9
82.8
214.3
10.
Production level
seeding (kg/ha).
294.9
259.4
134.1
780.6
463.3
185.3
180.1
238.8
559.3
195.1
182.0
223.0
278.0
55.7
405.7
287.9
118.9
"67.6
227.0
310.1
132.9
225.7
of perennial grasses with
495.0
526 .. 9
37.1
933.1
272.3
109.4
104.3
271.6
399.0
150.8
156.5
plowing
and
Replication
Year
1
2
69.1
262.8
101.8
544.4
212.5
142.2
71.9
245.3
217.8
126.7
121.8
143.9
554.7
1.50 .1
795.3
300.6
37.5
69.9
131.2
220.9
96.7
195.5
3
4
---------------------------------------------------------------------1964
1965
1966
1967
1970
1976
1977
1978
1981
1985
1986
50.5
546.6
264.4
592.2
216.9
364.2
354.8
658. 1
132.5
106.6
168.4
67.0
600.7
340.. 2
509.3
176.6
148.9
214.1
378.3
225.3
91.6
233.7
46
Tabfe 11.
Production level of perennial grasses on control
(l<g/ha).
Replication
Year
1
2
152.4
133.1
16.5
.145. 6
147.1
102.3
169.5
506.0
370.1
174.2
355.4
198.6
184.8
56.9
255.5
248.4
153.3
53.3
303.9
456.2
148.9
181 . 1
3
4
---------------------------------------------------------------------1964
1965
1966
1967
1970
1976
1977
1978
1981
1985
1986
Table 12.
243.6
187.7
48.3
174.5
145.9
129.2
98.0
216.7
365.8
82.0
235.0
131 . 3
84.2
30.4
248.7
88.9
83.4
48.6
223.0
236.6
113. 3
201.0
Calf price (dollars/100 lbs.)
Year
Current Dollars
1986 Dollarsa
. 1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
30.40
30.80
36.30
41.40
70.30
89.80
76.80
62.70
60.30
62.40
60.90
62.20
62.60
67.60
62.75
69.92
74.90
118.15
135.65
102. 19
75.59
68.50
68.67
64.27
63.40
62.-60
aAdjusted for inflation from 1974 to 1986 by CPI (Economic
of the President, 1987)
Report
47
Table 13.
Treatment
Spraying
Sagebrush canopy cover.
Replication
1
2
3
4
Burning
1963
1964
.1333333
.1430000
.1673333
.1870000
.0100000
.0100000
.0040000
.0124000
.1470000
.2120000
.0836666
.1366666
1
2
3
4
Rotocutting
Plowing
and
Seeding
1
2
3
4
.1773333
.1416666
.1253333
.1826666
.0083333
.0100000
.0153333
.0088888
1
.1863333
.0870000
.1203333
.1593333
.0266666
.0184000
.0065000
.0190000
2
3
4
1965
.0000000
.0000000
.0000000
.0000000
48
APPENDIX 8
DERIVATION OF THE SIGNS OF THE
RESPONSE FUNCTION PARAMETERS
50
For
the extremum to be a maximum at t
(8.04) should
be negative.
Hence,
positive under the conditions
5.
Since
e is the
~>0,
= -liB,
the sign
of
equation
the sign of parameter a should be
l>O and B<O.
long-term
equilibrium
level
of
forage
response, its sign should be positive.
Thus,
the
desirable
function are: a>O,
~>0,
signs of the parameters
l>O, B<O and B>O.
in
the
response
51
APPENDIX C
THE CACULATION OF EQUIVALENT WEIGHT OF CALVES
AND AVERAGE COST OF RAISING A CALF
52
The
calculation of equivalent weight of calves and
the
average
cost of raising a calf is presented in this appendix.
To simplify the model, weights of cows and bulls were adjusted to
equivalent weight of calves by the cattle to calf price ratio.
If we assume:
1.
There is a herd on 100 cows;
2.
The culling rate, replacement rate, and weaning rate are .16,
0.17, and .85 respectively;
The
3.
ratio
of bulls to cows is .04 and bulls are kept for
4
years; and
The
4.
average weights of cows,
bulls,
and
calv~s
are
1100,
1500, and 450 pounds, respectively.
Then, the equivalent weight of calves would be
=
EW
where
.8157
X
(1100
X
.16 + 1500
X
.01) + 450
X
(.85- .17)
.8157 is the average cattle to calf price ratio
from
= 462
1974
to
1986.
The
average
cost
of raising
a calf
was
obtained
usi~g
the
following formula:
CP x EW - AC
= GF
x LGS
(C.01)
where
CP
=
average ca 1 f price per 100 l bs. from 1974 to 1986 in
1986 dollars, which i s $79.65
EW
=
equivalent weight of a ca 1 f, which is 4.62 hundred
lbs.
AC
=
average cost of raising a calf
GF
=
average grazing fee from 1974 to 1986 in 1986 dollars,
wh i c h i s $ 11 . 73
53
LGS =length of grazing season, which·is 9 months
Thus, the calculated value of AC in equation (C.Ol) is $262.41.
54
APPENDIX D
STATISTICAL TEST FOR NORMAL DISTRIBUTION OF
CALF PRICE EQUATION RESIDUALS
55
The
the
statistical test for the hypothesis that the residuals
from
calf price equation are normally distributed is presented in this
appendix.
The test statistic, fully described in Kmenta (1986, pp. 265-267),
is
(0.01)
where n is the number of observation,
~
1 and
where
~
2 which are defined as
~
2
is the second moment,
skewness; and
~
b 1 and b2 are the estimates
variance;
~3
4 is the fourth moment, kurtosis.
is the
third
of
moment,
They may be obtained
using PROC MEANS statement in the SAS/ETS software package.
The
following
values were obtaineo for the calculation
of
the
value of the test statistic (0.01):
~2
= 206.324,
~3
= 1.63486,
b1
= .000000304306,
b2
= .00007916
~4
= 3.36981
The val.ue of the test statistic (0.01) in this case is 4.125,
the
tabulated value of X~ at 5i. level of significance is
~whereas
5.991.
hypothesis of normality at the 5i. level would not be rejected.
The
56
APPENDIX E
HYPOTHESIS TESTS ON RESPONSE FUNCTION
57
Some
hypothesis tests on the response function are presented
in
this appendix.
The test statistic used is
Fv, q, n-k
=
(SSEr - SSEur)/q
(E.01)
where SSEr and SSEur are the sum of square residuals in the restricted
and unrestricted models respectively, n is the number of observations,
q
is the number of restrictions,
k is the number of
estimated in the unrestricted model,
the
parameters
and v is the significance level.
All statistical tests are based upon a significance level of .05.
The
measurements
statistic
(E.01),
needed for calculating the value of
in addition to those presented in
the
Table
3,
test
are
presented in Table 14.
Table 14. Response function statistics (2).a
Treatment
e
a
Cross
Burning
and
Rotocuttingb
6.47422
Cross
A11 of
Four
Treatmentsc
SSE
1.06423
-.298103
.98950
(2.67)
(-2.83)
(4.28)
1.90881
2.74048
-.879551
(4.85)
(4.33)
(-4.31)
(1.72)
.73013
(2.29)
47.54433
1.21879 201.63429
.5272
.4856
(10.54)
---------------------------------------------------------------------aThe numbers in the parentheses are the t values.
bFunction was estimated without restriction ~=0.
cFunction was estimated with restriction ~=0.
1.
To
test
fo~ a significant structural change among the
forage responses with the restricted model (~=0),
four
the null hypothesis
··,;
58
i. s
Ho=
as
= ClB = ClR = ap
a-s
= 1s = a'R = 1p
<Ss = dB = <SR = dp
es
The
measurements
= eB = eR = Bp
for
calculating the value of
the
test
statistic
(E.Ol) are
SSEur
= 201.63429
= 47.01323 +
q
= 12
n
= 172
k
= 16
SSEr
The
= 139.39428
value
tabulated
there
29.05609 + 13.92577 + 49.39919
of the test statistic (E.01) is
value of F. 05 , 12 , 156 is 1.82.
then
5.80,
whereas
So the null hypothesis
the
that
is no structural change among these four forage responses would
be rejected.
2.
has
To
test the hypothesis that the quantity of killed sagebrush
a significant influence on the model for rotocutting,
the
null
hypothesis is
~R
Ho:
The
measurements
= 0
for
tE.Ol) are
SSEr
= 29.05609
SSEur
= 25.81935
q
=
calculating the value of
the
test
statistic
59
n
= 44
k
= 5
The value of test statistic (E.Ol) is then 4.89, whereas the tabulated
of F .os, 1 , 39 is 4.10.
value
killed
sagebrush
has
no
So the hypothesis that the
significant influence
on
quantity
of
model
for
the
rotocutting would be rejected.
3.
has
a
To test the hypothesis that the quantity of killed
influence
~ignificant
on the model
for
burning,
sagebrush
the
null
hypothesis is
~B = 0
Ho:
The
measurements
for
calculating the value of
the
test
statistic
(E.01) are
SSEr
= 13.92577
SSEur
= 12.37374
q
The
n
= 40
k
=5
value
of the test statistic (E.01) is
tabulated value of F.os,l, 35 is 4.12.
the
then
Therefore,
4.39,
whereas
the
the hypothesis that
quantity of killed sagebrush has no significant influence on
the
model for burning would be rejected.
4.
change
To test the hypothesis that there is a significant structural
between the two forage responses (burning and rotocutting) for
the unrestricted model
Ho:
a.B
= a.R
f3s
= f3R
({3~0),
the null hypothesis is
60
The
measurements
for
calculating the value of
the
test
statistic
CE.Ol) are
SSEr
= 47.54433
SSEur
= 25.81935
q
=5
n
= 84
k
= 10
The
= 38.19309
value of the test statistic (E.Ol) is then
tabulated value of
there
+ 12.37374
is
responses
no
F.os, 5 , 74
is 2.35.
Therefore,
3.6236,
whereas
the hypothesis that
significant structural change between
the
two
(burning and rotocutting) for the unrestricted model
would be rejected.
the
forage
(~~0)