WEED CONTROL DECISION RULES UNDER UNCERTAINTY
William Deen, Alfons Weersink,Calum G. Turvey,and Susan Weaver
A model of weed control, which took
into account the stochastic nature of crop
price, yield, and weed density, was developed
to assistfarmersin determiningweed densities
that justify herbicide application and the
optimal rate of application. In an application
of cockleburcontrol in soybeans,it was found
that the value of following the "if-then-else"
treatment strategy versus a fixed application
rate regardless of weed density was approximately $25 per acre at low weed numbers.
Profits of the marginaltreatmentstrategyare
strategy,but may
higherthan the "if-then-else"
not be sufficient to cover the additional
informational costs. Under both strategies,
the total amount of herbicide applied
decreaseswith increases in uncertaintyunder
the assumption of risk neutrality. The result
is due to the convex relationship between
weed density and yield loss. Under the
assumption of maximizing expected utility,
there are instances in which herbicide use
increases with risk aversion as per
conventional wisdom.
Introduction
Consumers, as well as farmers, are
concerned about environmental and human
health impacts associated with pesticides, food
contamination from herbicide residues,groundwater contamination resulting from leaching
of herbicides, and surface water contamination
through surface run-off from herbicide treated
fields. In addition, farmers are concerned with
potential operator exposure to herbicides,
increasingpest resistance,changingpest dominance, removal of existing herbicidesdue to political pressure exerted by consumers,and finally,
overall returns to their farming enterprise.
The authors are a Research Leader, the George
Morris Center, Guelph, Ontario, Assistant Professors,
Departmentof AgriculturalEconomicsand Business,University of Guelph,and a Research Scientist,AgricultureCanada,
Harrow, Ontario. Financial Assistance for the project was
provided in part by the Ontario Ministryof Agriculture and
Food's Food System 2002 program and the Agriculture
Canada/NSERCresearchpartnershipprogram.The authors
wish to acknowledge the helpful comments of Glenn Fox,
Scott Swinton, and two anonymous journal reviewers.
Total amount of pesticide applied is a
function of: (1) the number of applications;and
(2) the rate of application.A reductionin either
would result in reduced overall pesticide usage
and would also assist in alleviatingthe concerns
consumersand farmershave regardingthe safety
and profitability of pesticide usage. Reducing
total applied herbicidewhile maintainingprofitability would represent a Pareto improvement;
consumers would be better off since potential
environmentaland human health concernsassociated with the use of a pesticide would
potentially be reduced.
Economic threshold concepts are
management tools designed to promote more
efficient pesticide use in pest control. Biologists
tend to focus on reducing the number of pesticide applications through the use of discrete
choice threshold concepts. The suggested treatment strategiesinvolveapplyinga fixedpesticide
dosage if pest population density exceeds a
minimum: otherwise, the recommendation is
not to treat (Norton). The concept, often called
the "if-then-else"approach, has been applied
to control weeds in rice (Smith), barley
(Blackshaw),winter wheat (Doyle et al.), white
potatoes (Marraet al.), corn (Zanin et al.), and
soybeans (Weaver).In contrast,economists have
tended to focus on continuous choice threshold
concepts in determiningthe optimal application
rate for a given pest density. The resulting
marginal treatment strategy ensures profit
maximization, but considerable information is
required for the strategy to be implemented.
Studies using the marginal approach have
examined such aspects as increasing pest
resistance (Hueth and Regev), multiple-species
in a dynamic setting (Wetzstein et al.) and risk
(Tisdell).
Reichelderfer and Wetzstein both claim
that risk reduction is the major reason for
producers to apply pesticides. However, in a
recent review of risk in decision-making for
agricultural pest control, Pannell (1991)
concluded that risk may not necessarily lead
40
REVIEW OF AGRICULTURAL ECONOMICS, Vol. 15, No. 1, Januay 1993
to increased pesticide use. It may be true under
risk aversion if the source of risk is pest density
as shown by Olson and Eidman, but pesticide
application may decrease if the uncertainty
surrounds output price and pest-free yield.
Therefore,whether pesticides are risk-reducing
inputs depends on the relative balance of the
risk sources and whether the decision-maker
is risk neutral or risk averse. Recent empirical
examples that have incorporated uncertainty
such as Moffit et al., Osteen et al., and
Gillmeister et al. have tended to deal with the
uncertainty surrounding pest density only and
have assumed a normal probabilitydistribution
which is generally not the case (Brain and
Cousens). In addition, the relationshipbetween
crop damage and pest densityhas been assumed
to be linear which does not accurately reflect
the underlying biological interaction between
the species.
The purpose of this articleis to investigate
the relationship between price, yield, and weed
density uncertaintyon optimal herbicide application for control of cocklebur in Ontario
soybeans. The stochastic model developed can
be used to compare the "if-then-else"treatment
strategy of the biologist to the marginal treatment strategyof the economist.The contribution
this article makes is in the incorporationof crop
price and yield uncertainty, an empirically
determined probability distribution for weed
density,and the inclusion of a damage function,
consistent with the biology of the crop-weed
interference, into the weed threshold problem.
Theoretical Model
Biologists have assumed a fixed dosage
rate and determined the economic injury level
as the density at which treatment and no
treatment are equally profitable. Economists
have tended to assume fixed density levels and
determine the optimal pesticide applicationrate
as the dosage where the marginalreturnis equal
to the marginal cost. The following model
derived from Moffit (1988) of a profitmaximizing firm managing a pest population
(Do) can be used to show the differences
between the two approaches.
It =
PyY
- xP - C
(1)
Y = f(Ywf, D)
(2)
D = d(p, Do)
(3)
Py-P(Py)
Ywf -Y(YWf)
Do -d(Do)
(4)
(5)
(6)
where n is profit, Py is the price per bushel of
output Y (bushels per acre), T is the per unit
price of herbicide applied at a rate P, C is the
herbicide application cost per acre, Ywf is the
weed-freeyield (bushels per acre), D is the weed
population after treatment (number of plants
per square meter), and Do is the initial weed
population (numberof plantsper squaremeter).
Herbicideprice is assumedconstant in the profit
equation (1) so that the marginal per acre cost
of herbicide treatment does not varywith weed
numbers. Equation (2) is the damage function
that relates the impact of weed density on yield.
It is assumed that yields decrease with weed
density (aY/aD<0) and increasewith the weedfree yield (aY/aYwf>0). Equation (3) is the
control function that expressesthe existingweed
population as a functionof the herbicideapplied
and the initial infestation. It is assumed that
weed density decreases with the amount of
herbicideapplied (aD/IP<0) and increaseswith
the uncontrolled population (aD/aD0>0).
Equations (4), (5), and (6) represent the
probability density functions for the three
stochastic variables in the model; crop price,
weed-free yield, and weed density.
WEED CONTROL DECISION RULES UNDER UNCERTAINTY
Substitutingthe damage equation (2) and
the control equation (3) into equation (1) and
incorporating the probability distributions of
the stochastic variables results in the following
expected profit function:
E(n) = E{pF(YWfd(P, D)) -P-c}.
(7)
Under the biologist's definition of the
economic injury level, herbicide dosage is
constrainedto be at the specified label rate (pL).
This dosage is applied if the profit obtained
from herbicide application exceeds the profit
derived from applying no herbicide. The
economic injurylevel (DIL) or action threshold
(Moffitt et al.) is the population at which the
profits of treating with herbicide at the dosage
rate pL are equal to the profits from not
treating. The resulting simple decision rule for
the case of a single application is then to apply
herbicide at the specified label rate if the weed
density is greater than DEILand not to apply
if the population is less than DEIL.
Under the economist's definition of the
economic injurylevel, the herbicide dosage can
take on any non-negative value up to the
specified label rate (pL) and the focus of the
decision rule changes from the weed density
at which control should be initiated to the
optimal level of herbicide application. This
dosage is determined for various levels of weed
density by maximizingequation (7) with respect
to P and setting it equal to zero:
Ep
aDL
'
x
(8)
The dosage rate that solves equation (8) is the
point at which the marginal return of herbicide
application is equal to its marginal cost. The
pest population level prior to the application
of the profit-maximizing dosage rate is often
termed the economic threshold (Moffitt et al.).
The marginal product of herbicide is assumed
to be diminishingso that the dosage determined
by equation (8) is a maximum. Equations (1)
through(8) summarizethe informationrequired
to solve for either the economic injury level
with a discrete herbicide application choice or
the optimal level of herbicide dosage for
alternative weed densities.
Deen, Weersink, Turvey, Weaver
41
Empirical Specification
In this section, optimal strategies for
pesticide application under uncertainty are
developed for control of cocklebur in Ontario
soybean fields. After corn, soybeans are the
major field crop in Ontario representing
approximately 15 percent of the total field crop
area under cultivationand of total value of field
crop production. Cocklebur is studied because
it is one of the major broadleaf weeds in
Ontario affectingsoybean production. Both the
biologist's "if-then-else" approach and the
economist's marginal approach are measured.
Stochastic dominance is then used to evaluate
which approach is most appropriate depending
on risk preferences.
Cost and Efficacy of Herbicide Control
In Ontario, soybean weed control has
consistedprimarilyof applicationsof premergent
or preplant incorporated herbicides followed
by a postemergentherbicideto control any grass
or broadleaf weeds that may have escaped the
soil applied herbicidetreatments.Basagranplus
the surfactant Assist, until recently, has been
the only herbicide registered for the post
emergencecontrol of broadleafweeds. Basagran
plus Assist has provided the farmer with the
flexibility of choosing a postemergent "treatas
necessary"herbicide program.At the label rate
of application (2.25 L/ha plus 2 L/ha), the cost
of Basagranplus Assist is $22.50/acre.The fixed
application cost (tractor, sprayer, labor) is
$2.48/acre (Publication 60,1990 Crop Budgets,
Ontario Ministry of Agriculture and Food).
The most common functional form used
by researchers for the control function in
equation (3) is the exponential (Moffit et al.;
Moffit 1986; Osteen et al.; Gillmeister et al.),
which can be expressed formally as:
(9)
D = DoeUnder the current regulatory system, the
manufacturer of the herbicide conducts rate
efficacy tests and recommends a rate to be
registered under the Pest Control Products Act
based on these tests. Unfortunately, data
generated in these tests are not made available
to the public. But even if the data were
42
REVIEW OF AGRICULTURAL ECONOMICS, Vol. 15, No. 1, January 1993
available, it is unlikely that the rates tested
would be extensive enough to accurately
estimate the control function. As a result, the
parameter value of the control function must
be assumed using the guideline that label rates
of the herbicide must provide weed control
approaching 100 percent; a requirement of the
current regulatorysystem. At a 96 percent level
of weed control, the p value of the exponential
control function for Basagran is 1.4.
Damage Function
The damage function given by equation
(2) quantifies the degree by which a given weed
density is likely to produce damage if left
uncontrolled.The majorimpactof weed infestation in soybeans is generally through yield loss.
Cousens demonstratedthat crop-weedcompetition could be sufficiently explained by two
parametermodels in whichyield loss was related
to weed density. Cousens compared 17 models
and found the rectangularhyperbolamodel best
explainedthe data and was most consistent with
the underlying biology of the crop-weed interference. Expressed in terms of yield, the rectangular hyperbolic damage function model is:
ID
100(1 +
A)
(
loss
where I is the percentage yield
per unit
and
zero
weed density as density approaches
as
weed
A is the percentage yield loss
density
(D) approaches infinity. Data to estimate the
damage function were generated through field
experiments conducted at the Agriculture
Canada research station in Harrow, Ontario
from 1986 to 1988. In each year, four replicate
plots contained various weed densities established by overseeding weed seed and thinning
where necessary (Weaver). The parameters I
and A were estimated to be 17.4 and 66.6,
respectively, for cocklebur. These parameter
estimates were significant and explained over
97 percent of the variation in the pooled data
(Weersink et al.).
Y
Ywf 1-
Probability Distributions
The probability density function for
soybean price given by equation (4) is assumed
to be normal. Using historical Ontario soybean
prices over the time period from 1981 to 1988,
the nominal price per bushel was $7.45 with
a standarddeviation of $1.02/bushel. Deflating
these with the Ontario Index of Prices Paid in
Crop Production as a deflator, the average
soybean price per bushel in real 1988 dollars
was $8.07 with a standard deviation of
$1.10/bushel.
The estimation of probabilitydistributions
for weed-free yield is very difficult since yield
is often site specific and the influence of technological trends and management are difficult
to determine. As well, yield in the presence of
weeds is usually reported since few fields are
weed free. For this study, average yields for the
counties of Kent and Essex from 1980 to 1988
were used. The mean yield over this time period
was 37.35 bushels per acre with a standard
deviation of 5.44 bushels per acre. The measure
of dispersion for yield may be under-estimated
due to the aggregation effects associated with
the use of county data. A regression was run
on the data to detetmine if there were any
significanttrendsin the data over time, but none
could be determined. The level of yield loss due
to weeds must be chosen somewhat arbitrarily.
It will be assumed that maximumyield loss due
to weed infestation is approximately 13 percent
of the long-run average. The result is an
estimated weed-free yield of 42.95 bushels per
acre with a standard deviation of 6.26 bushels
per acre. It will be assumed that yield is
normally distributed for tractability,but yields
may follow a gamma density (Day).
Previousthresholdwork has assumed that
weeds are distributed uniformly over the field
and have a normal distribution for the probability of a given weed density (Osteen et al.).
However, most weeds are not distributed
uniformly(Cousens) and accuratedetermination
of weed density is difficult if extensive sampling
is not undertaken. Cussans et al. noted that
accurate sampling may not be possible due to
the prohibitive costs associated with sampling.
Brain and Cousens argued that the distribution
of weed populations follows the negative
binomial distribution. Such a distribution
indicatesthat there is an element of aggregation
or clumping in natural weed populations and
that when sampling, there is a high probability
WEED CONTROL DECISION RULES UNDER UNCERTAINTY
of obtaining a low weed density and a low
probability of obtaining a high weed density.
Weed distributiondatafor cockleburin soybeans
was obtained from a weed survey conducted in
1989 by Frick et al. and it was found to
approximatethe negative binomial distribution.
Incorporation of Probabilities into Model
The stochastic threshold model was
evaluated using 1000 Monte Carlo simulations
assuming that prices and yields are jointly
(negativelycorrelated)distributed,and thatweed
densities approximate a negative binomial
distribution. Prices and yields were assumed to
be independent of weed density. The mean of
the results provides an expected weed density
threshold under uncertainty. Optimal applications rate thresholds were determined for a
series of weed density ranges. For every weed
density range, the optimal application rate was
determined for each crop price and weed-freeyield random deviate set. If the benefit curve
at all points is below the cost curve, the optimal
application rate will be determined to be zero
and profit would be equal to profit times
resulting yield. Multiplying the mean optimal
application rate for each weed density by the
probability of that density occurring results in
an optimal application rate for that price, yield,
and weed density outcome.
Results
Risk Neutral
The "if-then-else"weed control strategy
resulting from the biologist's definition of the
economic injury level involves applying the
herbicide at the specified label rate if weed
density is greater than the economic injurylevel
(DEIL) and not to apply if the population is
less than DEIL. Assuming the producer has
perfect knowledge and expects soybean price
to be $8.07/bu and yield to be 42.95 bu/acre,
the economic injury level for cocklebur was
determined to be 0.63 plants/m2 with no
uncertaintysurroundingweed density.However,
uncertainty about weed density raised the DElL
or action threshold to 0.73 plants/m2.Changing
the probability distribution for crop price or
weed-freeyield also changes the economic injury
Deen, Weersink, Turvey, Weaver
43
level. Assuming standard deviations of $1.10
bu/acre for price and 6.25 bu/acre for weed-free
yield with uncertainty about weed density, the
cocklebur density at which treatment should
be initiated increased to 0.78 plants/m2. The
economic injurylevel increasedto 0.91 plants/m2
for an additional increase in uncertainty as
representedby probabilitydistributionsfor price
and yields with standard deviations of $1.75/bu
and 9.0 bu/acre, respectively.
The results are consistent with those of
Hall and Moffit along with Pannell (1990b) and
reflect the structure of the control function,
damage function, and the subsequent benefits
of herbicide treatment. The objective function
in this part of the analysis is to maximize
expected profits. Even with such an objective
function, risk can still affect optimal decisions.
Auld and Tisdell argued that because of the
convex relationshipassumed between crop yield
and weed density, increases in the variance of
initial weed density reduces expected yield loss.1
The intuition behind this statement is that the
same number of weeds causes less damage if
they are evenly distributed than if they are
concentrated in sections of the field. Pannell
(1990b) proved that the uncertainty in weed
density increases the economic injury level, as
was found here, and subsequently reduces the
overall level of pesticide use. Pannell (1990b)
also examined the impact of uncertainty about
weed competitiveness,weed kill, herbicide rate,
and weed-free yield and found that an increase
in uncertainty in these variables also increased
the economic injury level. Consistent results
were obtained in this studywith regardto weedfree yield. In addition, this study also found that
increases in price uncertainty under expected
profitmaximizationincreasethe economic injury
level, and therefore, reduce pesticide use.
In contrast to the "if-then-else"treatment
strategy which assumes a fixed herbicide application rate, the marginal treatment strategy
involves determining the optimal application
rate that will change continuously with changes
in weed density. Optimal Basagran application
1Swintonnotedthatthe relationshipbetweenweed
densityand yield may be sigmoidalrather than strictly
convex.Thiswouldlimitthe generalityof the riskeffects
on pesticideuse.
44
REVIEWOF AGRICULTURALECONOMICS,Vol. 15, No. 1, Januay 1993
rates for control of cocklebur in soybeans are
given in Table 1 for various levels of producer
uncertainty. Within each level of uncertainty,
the amount of herbicide applied increases with
weed density since the marginal benefit of
treatment increases (equation (8)). Expected
profits fall due to the yield loss resulting from
weed damage and the cost of the herbicide
treatment. As with the "if-then-else"treatment
strategy, the total amount of herbicide applied
decreases with increases in the uncertainty
surroundingsoybean price and weed-free yield.
The optimal amount of Basagranrecommended
at each level of weed density is approximately
5 percent less for the producer with high
uncertainty as compared to one with perfect
knowledge. The direction and size of the effect
from uncertainty surrounding weed-free yield
is consistent with the results of Pannell (1990b).
He noted, however,that while individualsources
of risk may have a small impact on the dosage
rate, multiple sources may have a rather
significant impact.
Averageapplicationrateswere determined
to be 0.37 l1/hawith no uncertaintyand 0.35 1/ha
with high uncertaintyregardingprice and yield.
These average rates were calculated using
proportions based on the total numberof fields
sampled and are quite low due to the high
proportion of quadrants sampled that had a
weed density of zero. Cocklebur density varied
considerably from field to field. Determining
the optimal applicationrates for each individual
field would have resulted in fields with higher
mean weed densities requiring higher rates of
Basagran for optimal control.
Figure 1 illustratesthe differencebetween
the alternative weed control strategies and the
informational value obtained from following
the optimal approach. At low weed densities,
expected profitability is greater for the no
treatment strategythan applying a fixed dosage
rate. The difference in profitability narrows as
weed density increases until eventually relative
profitability is reversed. The point of equal
profitability is the economic injury level that
was calculated previously to be 0.78 cocklebur
plants/m2under uncertainty.Thus, the "if-thenelse" treatment strategy involves not treating
at all until cocklebur density reaches 0.78 and
then applying the fixed rate of Basagran. The
value of this information to a producer who
applies herbicide on a schedule spray basis
decreasesfrom approximately$25 per acre with
no weeds to zero at the economic injury level.
The marginal approach involves determining the optimal herbicide application rate
for each weed density. The recommended rate
at very low weed densities may generate returns
that are less than the no treatment strategydue
to fixed application costs. The density at which
profitability is equated between the marginal
and no treatment approaches is where the optimal dosage should begin to be applied. Termed
the optimal economic threshold (Pannell), it
was calculated to be approximately0.15 plants/
m2. The amount of herbicide applied is greater
for the marginalthan the "if-then-else"
treatment
between
the
economic
threshold
strategy
optimal
and
the
economic
level
(0.15)
injury
(0.78).
Expected profit for the marginal approach is
always at least as great as the "if-then-else"
treatmentstrategy.Below the optimal economic
threshold, profitability of the two approaches
will be equal since no herbicide is applied. The
profit curves will again converge at high weed
densities where the optimal dosage rate is equal
to the fixed application rate which is assumed
to be the maximum rate that can be applied.
Risk Aversion
Under the assumption of risk neutrality,
a decision-makerwho is attemptingto maximize
expected profits will reduce the amount of
herbicideused with an increasein the uncertainty of weed density,crop price,or weed-freeyield.
An increase in risk associated with these
variables increases the weed density at which
herbicide application is justified under the "ifthen-else" treatment strategy and reduces the
optimal application rate under the marginal
treatmentstrategy.The reason for the reduction
in herbicide use is that risk has a negative
impact on the marginal productivity of the
herbicidedue to the convexrelationshipbetween
weed density and yield loss. The result that
herbicide use decreases with an increase in risk
runs counter to general perceptions and may
be due to the assumption that thus far, risk's
only influence is on expected profits. Feder,
WEED CONTROL DECISION RULES UNDER UNCERTAINTY
Deen, Weersink, Turvey, Weaver
45
Table 1. OptimalBasagranApplicationRates and ExpectedProfit for Control
of Cockleburin SoybeansUnder VariousLevels of Uncertainty
Weed
Density
(#/m2)
No Uncertainty
(aO
7y=0()
Profit
Rate
($/acre)
(1/ha)
Proportion
High Uncertainty
(a=1.75 ay=9.0)
Rate
Profit
(1/ha)
($/acre)
Uncertainty
(p=1.1 ay=6.25)
Rate
Profit
($/acre)
(1/ha)
0
.667
0.000
346.61
0.000
346.37
0.000
346.50
1
.177
0.740
306.04
0.721
306.55
0.695
307.73
2
.077
1.240
293.66
1.216
294.17
1.184
295.33
3
.040
1.530
286.42
1.506
286.93
1.472
288.08
4-6
.030
1.890
277.30
1.841
277.81
1.836
270.96
7-11
.007
2.310
266.80
2.291
267.31
2.249
268.47
12-18
.003
2.680
257.68
2.055
258.18
2.589
259.32
0.368
Expected Application Rate
0.351
0.361
op = standard deviation of soybean price (mean = $8.07/bu)
Cy= standarddeviationof weed-freesoybeanyield (mean = 42.95 bu/acre)
Figure 1. ExpectedProfit Under AlternativeWeed ControlStrategies
TreatmentStrategies
-
350
>
- .-
No. Treatment
OptimalRate
-.------
325
.. ... ...
[@-.
Li
Fixed Rate
.
~~~~~~~~~~-^--*-*-*-*-*-.---.
i
0
E
|
300
S.
x
275
a:
11X
250
[!!i
0.0
0.15
0.15
0
0.5
0.78
CockleburDensity(#/sq m)
1.0
46
REVIEW OF AGRICULTURAL ECONOMICS, Vol. 15, No. 1, January 1993
along with Robison and Barry and Olson and
Eidman, theoretically showed that under risk
aversion, uncertainty about weed densities
increases herbicide use. This was demonstrated
empirically by Osteen et al. However, these
studies assumed a linear damage function and
the convexityof this biological relationship was
shown to drive the results under risk neutrality.
To illustrate the effect of risk aversion
on optimal application rates, the weed control
problem is reformulated so that the decisionmaker is assumed to maximize expected utility
(U) which is a function of profits (n). It is
assumed that at each level of weed density, the
producer can choose among eight different
herbicide dosages similar to the optimum rates
given in Table 1. Profitability for each application rate is evaluated at the seven weed
densities is incorporated, dosage rate has a
negative impact on average net returns. The
minimum returns are obtained at the highest
weed density (15/m2) and the maximumvalues
are obtained with no weeds present. Relative
differences in maximumnet returns among the
alternative strategies solely reflects the cost in
applying more herbicide since there is no damage effect from weeds. Under the assumption
of extremeriskaversion,a decision-makerwould
prefer the highest level of herbicide application
on the basis of the maximin principle.
The risk efficiency of these alternative
application rate strategies was evaluated using
generalized stochastic dominance (GSD)
developed originally by Meyer. The optimal
solution procedure for GSD is:
11
Maximize
densities. These net returns per acre are
combined with the probabilities of weed
populations given in Table 1 to generate a
cumulative density function of net returns for
the seven application strategies.
Summary statistics for the alternative
herbicide rates are given in Table 2. Average
net returns per hectare increase with dosage
rate. However, if the probabilities associated
with net returns at each of the seven weed
J1
[F(n)-G(n)]U (n)dx
11U
subject to r (n)
'
? r2()Vn
U (i)
where F(i) and G(i) are the cumulative
distribution functions of net returns from the
associatedstrategiesFand G, respectively,Ul(n)
and U1l(ir) are the first and second derivatives
of a monotonically increasing von Neumann
Morgenstern utility function U((n), and rl(7)
?
Table 2. Summary Statistics on Net Returns ($/ha) Generated by Alternative
Herbicide Application Rates
Rate (1/ha)
Average (*)
Weighted Avg. (*)
Minimum
Maximum
0.00
245.88
322.73
162.70
346.61
0.20
252.42
321.75
169.03
341.61
0.26
257.55
320.64
175.36
337.58
0.40
258.75
320.33
177.04
336.61
0.60
264.63
318.45
186.52
326.61
0.80
269.82
316.10
197.06
326.61
0.90
272.10
314.76
202.58
324.11
1.00
274.16
313.91
208.15
321.61
The weights determined by probabilities associated with weed densities are given in Table 1.
WEED CONTROL DECISION RULES UNDER UNCERTAINTY
and r2(W)are the lower and upper absolute risk
aversion coefficients, respectively. If the
difference between F(r)-G(r) is negative, then
F dominates G for all risk aversion coefficients
between rl(x) and r2(n). Unlike other stochastic
dominancecriteria,GSD imposes no restrictions
on the absolute risk aversion coefficients, and
thereby allows for greater discriminatorypower
in evaluating risky prospects by narrowing the
range of r1(ir) and r2(r).
The computer program MEYEROOT
(McCarl) was used to rank the net return
distributions for each of the eight application
strategies at the three different levels of price
and yield uncertainty.At each level of uncertainty, a given application rate tended to dominate
strategies involving higher herbicide doses as
shown in Table 3. Thus, the impact of risk on
expected profit through the damage function
may outweigh the effects of risk aversion and
result in less herbicide use with an increase in
risk. However, there are instances involving
applying no herbicide or 0.2 1/hain which risk
averse individuals would prefer applying more
herbicide rather than less. The level of risk
aversion at which dominance changes from the
low application rate to a higher one increases
with the dosage rate. The values for the risk
Deen, Weersink, Turvey, Weaver
aversion coefficients are within values obtained
from previous studies, but direct comparison
involves appropriate scaling of the outcome
variable (Raskin and Cochran). The result for
the low applicationrate strategiesin comparison
to one another that herbicide use increaseswith
risk aversion is consistent with the findings of
Osteen et al.
Summary
In this article,a model was developed that
generatedinformationto assist farmersin determining weed densities that justify herbicide
application and the optimal rate of application.
Unlike many previous threshold models, this
model took into account the stochastic nature
of variables involved in the herbicide control
decision: weed-free-yield, crop price, and weed
density uncertainty.In addition, the differences
in the alternative weed control strategies were
explicitly considered.
The "if-then-else"weed control strategy
resulting from the biologist's definition of the
economic injury level involves applying the
herbicide at the specified label rate only if the
returns from application are greater than the
returnsfrom not treating.The value of following
Table 3. Risk Efficient Herbicide Application Rates (1/ha)
Application
Rates
0.00
0.20
0.36
47
Application Rates
0.40
0.60
0.80
0.90
1.00
0.00
0.20
.004
-
0.36
.005
.006
0.40
.006
.007
D
0.60
.007
.010
D
D
0.80
.009
D
D
D
D
0.90
D
D
D
D
D
D
1.00
D
D
D
D
D
D
-
D
D = row strategy dominates column strategy by GSD.
-D = column strategy dominates row strategy by GSD.
= risk aversion coefficient at which row domination of column is changed to column domination of row strategy.
48
REVIEWOF AGRICULTURALECONOMICS,Vol. 15, No. 1, January1993
the "if-then-else"treatment strategy versus a
fixed application rate regardlessof weed density
was found to be approximately $25 per acre for
cocklebur in soybeans at low weed numbers.
The marginal weed control strategy resulting
from the economist's definition of the economic
injury level involves determining the optimal
application rate that will varycontinuouslywith
weed density. The informational value of the
marginal versus the "if-then-else" treatment
strategy is greatest at the economic injurylevel
and then converges with increases in weed
density. In this study, it was estimated to be
approximately $2 per acre which may not be
sufficient to cover the additional costs and
complexity of the marginal approach.
Risk associated with density, crop price,
and weed-free yield was incorporated into both
the "if-then-else"and the marginalweed control
strategies. Risk can influence the optimal
decision under either strategy by affecting
expected profit or because of risk aversion.
Under the assumption of risk neutrality, an
increase in the uncertainty of the three
stochastic variables was found to lead to an
increase in the economic injury level or action
threshold and a decrease in the optimal application rate. Thus, an increase in risk was found
to lead to a decrease in herbicide usage
consistent with the findings of Pannell (1990b)
and Hall and Moffitt. The result can be
explained by the convex relationship between
weed density and yield loss so that an increase
in risk reduces the marginalproductivityof the
herbicide. It is, however, counter to the general
perception that herbicideuse increaseswith risk
(Pingali and Carlson). Under the assumption
of expected utilitymaximizationand uncertainty
only in weed density, it was found that the
herbicide use does increase with uncertainty
for extreme risk aversion.
The economicinjurylevel model developed
in this study has attempted to account for
several complex aspects of the herbicidecontrol
decision by incorporating a control function,
which allows for variable levels of control; a
rectangularhyperbolic damage function, which
describes the relationship between yield and
weed density; and risk associated with uncertaintyof crop price,weed-free-yield,and weed
density.It is apparentthat the abilityto estimate
accurate probability distributions for random
variables such as weed-free yield poses limitations to the usefulness of the model. Furthermore, the model in this study has simplified
several complex aspects of the biological system
underlying a decision regarding weed control.
This simplification is often argued to be
necessaryin order to avoid making the problem
so complex and involved that costs as well as
abilities of extension personnel or researchers
to determine economic injury levels would be
prohibitive. It appears that there exists a tradeoff between practical usage and complexity of
a model. This trade-off needs to be examined
furtherto determinewhetheror not complexities
of the weed control decision such as weed
dynamics,multiple species, crop quality effects,
optimal control options, and soil applied
herbicides can be incorporated into the model
without making the model too difficult to
practically apply.
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