Determining optimal fertilization rates under variable weather conditions


Talpaz, H.T.ylor, C.

Western Journal of Agricultural Economics 77(2): 45-51

1977


This paper presents a theoretical framework for incorporating the following sources of risk into the determination of optimal fertilization rates: (a) the influence of weather and other stochastic factors on the marginal product of fertilizer, and (b) uncertainty about the coefficients of the response function. the decision criterion considered is the maximization of profit subject to a risk constraint on the probability of not recovering cost of the fertilizer. the theoretical framework is applied to the fertilization of dryland grain sorghum in the Texas Blacklands. results indicate that the risk averse producer should substantially lower his fertilization rate if soil moisture at fertilization time is low.

DETERMINING
OPTIMAL
FERTILIZATION
RATES
UNDER
VARIABLE
WEATHER
CONDITIONS
Hovav
Talpaz
and
C.
Robert
Taylor-
This
paper
presents
a
theoretical
framework
for
incorporating
the
following
sources
of
risk
into
the
determination
of
optimal
fertilization
rates:
(a)
the
influence
of
weather
and
other
stochastic
factors
on
the
marginal
product
of
fertilizer,
and
(b)
uncertainty
about
the
coefficients
of
the
response
function.
The
decision
criterion
considered
is
the
maximization
of
profit
subject
to
a
risk
constraint
on
the
probability
of
not
recovering
the
cost
of
the
fertilizer.
The
theoretical
framework
is
applied
to
the
fertilization
of
dryland
grain
sorghum
in
the
Texas
Blacklands.
Results
indicate
that
the
risk
averse
producer
should
substantially
lower
his
fertilization
rate
if
soil
moisture
at
fertilization
time
is
low.
The
decision
criterion
commonly
used
in
making
fertilizer
recommendations
is
ex-
pected
profit
maximization.
However,
the
risk
averse
producer
who
bases
his
fertiliza-
tion
program
on
this
criterion
may
experi-
ence
a
serious
misallocation
of
resources
if
he
is
uncertain
about
the
influence
of
weather
on
the
marginal
productivity
of
fertilizer
and
about
the
response
function.
In
a
pioneering
article,
de
Janvry
pre-
sented
a
model
that
accounted
for
risk
due
to
weather
variability.
However,
he
implicitly
assumed
that
the
response
function
was
known
with
certainty.
This
article
extends
the
de
Janvry
framework
to
include
uncer-
tainty
in
the
response
function.
This
ex-
tended
model
is
applied
to
the
fertilization
of
dryland
grain
sorghum
in
the
Texas
Black-
lands.
Weather
risk
is
appraised
with
histori-
cal
records,
while
the
response
function
is
appraised
with
experimental
data
on
the
yield
response
to
different
fertilizer
rates.
Hoval
Talpaz
and
C.
Robert
Taylor
are
Associate
Profes-
sor
and
Assistant
Professor
of
Agricultural
Economics,
respectively,
Texas
A&M
University.
Seniority
of
au-
thorship
is
equally
shared.
Texas
Agricultural
Experiment
Station
Technical
Article
No.
12711.
The
authors
are
indebted
to
the
Editor
and
the
two
reviewers
for
their
valuable
and
constructive
com-
ments.
This
study
was
sponsored
by
NSF
and
by
the
Texas
Agricultural
Experiment
Station.
The
Decision
Model
In
a
recent
article,
R.
H.
Day,
et.
al.,
provide
an
excellent
discussion
about
firm
behavior
under
risk.
In
particular,
they
ex-
plore
variations
of
the
"safety-first
principle"
originally
developed
by
Roy.
One
variation
is
the
"strict
safety-first
principle"
advocated
by
Shackle
and
applied
by
Telser.
This
criterion
assumes
that
the
decision
maker
will
apply
his
resources
to
maximize
expected
profits
subject
to
a
constraint
on
the
probability
(8)
of
experiencing
a
loss.
Day,
et.
al.,
show
that
this
criterion
involves
a
minimum
acceptable
safety
margin.
If
the
safety
margin
is
less
than
the
decision
maker's
subjectively
specified
8,
resource
use
is
constrained
at
a
level
just
se-
curing
the
acceptable
safety
margin.
The
ap-
peal
of
this
criterion
becomes
apparent
by
recognizing
that
it
"
.
represents
a
com-
promise
between
expected
profit
maximiza-
tion
and
safety
margin
maximization"
[Day,
et.
al.,
p.
1296].
Robinson
and
Day
have
shown
that
this
principle
reflects
a
utility
function
with
a
lexicographic
ordering
in
the
expected
value-risk
space.
Therefore,
this
principle
can
be
rationalized
by
a
set
of
con-
sistent
axioms
of
behavior.
This
strict
safety-first
decision
criterion
is
used
to
evaluate
optimal
fertilization
rates
under
variable
weather
conditions.
Formally,
the
objective
is
to
maximize
the
profit
to
fer-
45
December
1977
Western
Journal
of
Agricultural
Economics
tilizer
for
a
crop
producer
who
operates
in
competitive
markets,
subject
to
a
risk
con-
straint
defined
as
the
probability
of
not
re-
covering
the
cost
of
fertilizer.
Stated
mathematically,
the
decision
model
is:
1)
MAX
E
[P•Y(N,W)]
mN
subject
to
2)
Pr
[P•D(N,W)
>
mN]
>
5
where:
E
=
expected
value
operator;
P
=
unit
price
of
the
product;
Y
=
yield
response
function;
N
=
fertilization
rate;
W
=
weather
variable;
m
=
fertilizer
price;
Pr
=
probabil-
ity;
D
=
Y(N,W)
Y(N=0,W)
=
increment
in
yield
attributed
to
fertilizer;
and
8
=
sub-
jective
loss
probability
threshold
(maximum
risk).
To
find
the
fertilization
rate
that
satisfies
this
decision
criterion,
we
must
first
find
the
probability
distribution
given
by
expression
(2).
The
stochastic
variables
in
the
above
model
are
weather
(W)
and
the
increment
in
yield
attributed
to
fertilizer
(D)
which
are
ex-
pressed
by
the
following
conditional
proba-
bility
density
functions:
3)
f
i
(DIN,W)
4)
f
2
IA)
where
W
is
specified
as
the
number
of
stress
days
after
planting
and
fertilization,
rinfl
A
is
available
soil
moisture
at
fertilization
time.'
From
the
definition
of
conditional
proba-
bility,
it
is
known
that
the
joint
conditional
probability
distribution
of
D
and
W
is:
5)
f
3
(D,WIN,A)
=
f
i
(DIN,W)
f
2
IA)
1
The
amount
of
water
deficit
experienced
by
the
crop
is
described
by
the
number
of
"stress
days"
during
the
growing
season.
Formally,
the
number
of
stress
days,
W,
was
calculated
by
Kissel,
Ritchie,
and
Richardson
as
W=(1—
n
),
n=1
Eo
where
n*
is
the
number
of
days
from
crop
emergence
to
harvest,
E
n
is
daily
evapotranspiration,
and
E
0
is
daily
Now
note
that
integrating
this
distribution
over
all
values
of
the
weather
variable
gives:
6)
f
4
(DIN,A)
=
f
3
(D,WIN,A)dW
=
.C
ee
f
i
(DIN,W)
f
2
(WIA)dW
which
is
the
conditional
probability
distribu-
tion
for
the
increment
in
yield
attributable
to
fertilizer.
Since
the
total
fertilizer
cost,
mN,
is
known
with
certainty,
the
problem
reduces
to
finding
the
probability
distribution
of
the
net
revenue
(R),
where
R
=
P•
D.
For
this
application
of
the
model,
it
is
assumed
that
price
(P)
is
known
with
certainty.
This
as-
sumption
is
approximately
valid
for
a
farmer
who
has
a
forward
market
contract
for
the
product
or
copes
with
price
risks
by
other
means.
With
price
known,
the
probability
distribution
of
R
is
obtained
by
transforming
the
probability
distribution
(6).
Applying
a
theorem
from
mathematical
statistics
for
ob-
taining
the
probability
distribution
of
a
func-
tion
of
a
random
variable
[see,
for
example,
Meyer,
p.
88]
gives:
7)
fs
(RIN,A)
=
1
P
1
.
4
(DIN,A)
By
integrating
this
probability
distribution
from
an
infinite
loss
(R
=
—00)
to
the
cost
of
the
fertilizer
(R
=
mN),
the
probability
of
not
recovering
the
cost
of
the
fertilizer
as
ex-
pressed
in
equation
(2)
is
obtained
8)
Pr(R
<
mNIN,A)
=
f
mN
f
5
(RIN,A)dR
1
mN
f
(DIN,A)dD
Referring
back
to
(6),
it
can
be
seen
that
an
alternative
expression
is:
9)
Pr(R
<
mNIN,A)
=
i
p
f
l
_
n
oo
N/P
[f;
f
i
(DIN,W)
f2
(WIA)dW]
dD.
As
de
Janvry
has
shown,
the
solution
to
this
type
of
decision
model
is
characterized
evapotranspiration,
and
E
0
is
daily
poten-
tial
evaporation
rate
above
the
plant
canopy.
For
further
discussion
of
the
stress
day
concept,
see
Kissel,
et.
al.
46
Optimal
Fertilization
Rates
Talpaz
and
Taylor
by
two
regions.
The
characteristic
of
one
re-
gion
is
that
Pr(mN
e
>-
8,
where
N
e
=
the
fertilization
rate
(N)
that
maximizes
ex-
pected
profit.
In
this
region,
the
"strict
safety-
first"
level
of
fertilization
(N*)
is
N
;
that
is,
N*
=
N
e
.
The
characteristic
of
the
other
region
is
that
Pr(mNe
I
N,,A)
<
8.
Here
the
minimum
acceptable
safety
margin is
not
met
by
apply-
ing
the
expected
profit
maximizing
rate,
N
e
.
For
this
region,
the
strict
safety-first
level
of
fertilization
(N*)
is
below
N
e
,
and
N*
is
found
by
setting
expression
(9)
equal
to
8
and
solving
for
the
N
that
gives
the
highest
expected
profit.
An
Application
This
section
presents
the
results
of
apply-
ing
the
model
to
evaluating
fertilization
rates
for
dryland
grain
sorghum
in
the
Texas
Black-
lands.
Both
weather
and
the
response
func-
tion
are
assumed
to
be
random
variables.
In
the
sections
that
follow,
response
uncertainty
is
considered
first,
then
the
weather
uncer-
tainty,
and
finally
the
two
types
of
probability
information
are
combined
with
the
use
of
equation
(9)
for
joint
evaluation.
Response
Uncertainty
Using
experimental
data
presented
by
Kis-
sel,
Ritchie,
and
Richardson,
the
following
response
function
for
dryland
grain
sorghum
is
estimated
with
ordinary
least
squares
re-
gression:
10)
Y
=
2674.46
+
27.88N
.323WN
.0804N
2
(16.99)
(5.51)
(5.73)
(3.07)
k
=
32
R'
=
.71
where
Y
=
grain
sorghum
yield
in
pounds
per
acre;
N
=
nitrogen
rate
in
pounds
per
acre;
W
=
number
of
stress
days
in
the
grow-
ing
season;
k
=
degrees
of
freedom;
and
the
values
in
.
parentheses
are
the
t-statistics.
Under
the
standard
regression
assump-
tions
made
in
estimating
a
response
function
of
the
form
(Y
=
g
o
+
A
I
N
+
f3
2
WN
+
0
3
N2),
it
can
be
shown
that
for
a
finite
sample,
f
l
(D
I
N,
W)
is
distributed
as
Student's-t:
r(k/2)
\
F
-
rk
k
--00<t<00
where:
(D—µ).‘/F
c
t
=
a
k
=
degrees
of
freedom
=
13;
+
13
2
+
ti3N
2
[
1/2
a
=
N'i'
We
i
)
+
W
2
V(13;
)
+
N
2
VO
3
)
+
2W
CVO;
4
)
+
2N
Cqi
'1
3
3
J
+
2WN
CV(ff2,f3;
)
with
=
standard
error
of
the
estimate
V(.
)
=
variance
CV(•
•)
=
covariance
.
This
gives
the
probability
distribution
re-
lating
to
the
uncertainty
about
yield
re-
sponse.
Weather
Uncertainty
Weather
uncertainty
in
the
Texas
Black-
lands
is
appraised
with
estimates
of
the
number
of
stress
days
for
three
ranges
of
soil
moisture
(Kissel,
Ritchie,
and
Richardson).
While
it
would
be
desirable
to
have
more
than
three
ranges
of
soil
moisture,
it
was
im-
possible
to
obtain
the
necessary
data
for
the
Texas
Blacklands.
To
allow
for
more
precise
probability
estimates,
a
continuous
probabil-
ity
function
was
fitted
to
the
stress-day
data
for
each
range
of
soil
moisture.
Climatic
and
biological
factors
suggest
that
the
probability
density
function
be
continu-
ous,
nonsymmetric
in
general,
and
concave
47
December
1977
Western
Journal
of
Agricultural
Economics
or
convex
based
on
the
value
of
its
param-
eters.
Nonsymmetry
is
needed
because
dif-
ferent
biological
effects
are
naturally
linked
to
low
W's
compared
to
those
linked
to
large
W's.
Photosynthesis,
respiration
availability
of
soil
nutrients,
probability
of
diseases,
and
pests
are
likely
to
be
related
to
W
in
a
non-
symmetrical
way.
Howell,
et.
al.,
computed
empirical
probability
distribulions
of
grain
crop
yields
as
a
function
of
soil
moisture.
They
showed
that
under
different
levels
of
soil
moisture,
the
distribution
function
is
generally
nonsymmetric.
concave,
or
convex.
The
Gamma
density
function
possesses
these
characteristics,
given
by
1
(a
-
-W
/
b
for
W
0
12)
f(W)
=
W
e
Ra)ba
0
forWCO
with
E(W)
=
ab
and
Var(W)
=
al3s,
where
a
and
b
are
parameters
to
be
estimated,
and
F(a)
is
the
Gamma
function
(note
that
the
Gamma
density
function
is
concave
for
a>1
and
convex
for
a<
1).
2
The
data
in
Kissel,
et.
al.,
are
given
in
terms
of
cumulative
distributions;
hence,
equation
(11)
must
be
integrated
in
order
to
estimate
its
parameters.
This
integration
is
carried
out
numerically
by
using
an
adaptive
Romberg
extrapolation
discussed
by
de
Boor.
Parameters
a
and
b
were
estimated
by
minimizing
the
sum
of
squared
deviations
using
a
non-linear
optimization
algorithm
based
on
the
Levenberg-Marquandt
algo-
rithm
discussed
by
Brown.
Since
the
number
of
stress
days
(W)
is
conditional
on
the
avail-
able
soil
moisture
(A),
it
was
possible
to
esti-
2
Additional
plausible
justification
for
the
application
of
the
Gamma
is
two-fold.
First,
it
can
be
shown
that
as
a
special
case
the
Gamma
distribution
is
symmetrical;
hence,
it
is
more
general
than
the
normal
distribution.
Second,
a
special
form
of
the
Gamma
distribution
(the
Erlang
distribution)
is
simply
a
summation
of
indepen-
dent
negative
exponential
random
variates
with
param-
eter
(1/b).
The
negative
exponential
was
extensively
used
to
describe
random
events
corresponding
to
dur-
ations,
which
is
what
W
essentially
is
[Phillips,
et.
al.,
pps.
218-220].
mate
the
following
three
conditional
proba-
bility
functions:
13)
f
2
(WI0<A<3.9)
=
Gamma
(2=2.45;b=14.52)
with
E(W)=35.6,
and
mean
squared
error
=
0.00089
14)
f
2
(WI3.9<A<6.5)
=
Gamma
("a'=3.38;13=6.54)
with
E(W)=22.14,
and
mean
squared
error
=
0.00215
15)
f,
(WIA>6.5)
=
Gamma
(a=0.539;b=18.71)
with
E(W)=10.09,
and
mean
squared
error
=
0.00451.
If
a
no
risk
situation
is
assumed,
equation
(10)
is
known
with
certainty.
This
means
that
the
maximization
of
equation
(1)
is
no
longer
subject
to
equation
(2),
and
the
expected
value
of
W
could
be
used
with
certainty.
Then,
differentiating
equation
(1)
with
re-
spect
to
N,
and
equating
the
derivative
to
zero
provides
the
optimal
value
for
N.
These
values
are:
for
E(W)
=
35.6,
the
no
risk
N*
=
72.26
lbs/acre;
for
E(W)
=
22.14,
N*
=
99.30
lbs/acre;
and
for
E(W)
=
10.09,
N*
=
123.50
To
obtain
the
loss
probability
for
this
em-
pirical
problem,
the
response
probability
dis-
tribution
(11)
is
combined
with
each
of
the
weather
probability
distributions
(13),
(14),
or
(15)
using
rule
(9).
With
the
general
weather
distributor
(12)
we
get:
1
mN/Pe-
r[(k+1)/21
16)
Pr(R
<
mN(N,A)
=
-
13
f_.
f
o
r(o)
rrk
(D-µ)k
2
(
0
-
'
y(k+0/2.\\/
a
(a
1)
(1
+
e
-w
i
b
dWdD.
k
r(a)b
The
complex
mathematical
form
of
this
dis-
tribution
requires
that
the
integrations
be
done
numerically
using
a
procedure
devel-
oped
by
Greville.
The
optimal
fertilization
rates
as
related
to
soil
moisture
and
the
loss
probability
are
shown
in
Figure
1.
By
specifying
the
value
of
8
acceptable
to
a
farmer
and
by
determining
the
level
of
soil
moisture,
one
can
use
this
figure
to
find
the
optimal
fertilization
rate.
For
example,
suppose
that
soil
moisture
is
between
3.9"
and
6.5"
and
that
8
is
specified
to
be
.50.
From
Figure
1,
it
can
be
seen
that
the
"strict
safety-first"
level
of
fertilization
(N*)
is
about
99
lbs/acre,
which
is
also
the
48
soil
moisture
less
than
3.9"
soil
moisture
between
3
:
9"
and
6.5"
soil
moisture
greater
than
6.5
-
0.0
.
10
.
20
In
O
4.
0
O
1111
.30
1.1P
.o
O
M.
.40
.
50
Talpaz
and
Taylor
Optimal
Fertilization
Rates
0.0
.
10
soil
moisture
less
than
31_
soil
moisture
between
31
and
6.5""
soil
moisture
greeter
than
6.5"
.
28_
O
z
JO
O
.
40
.
50
20
4'0
6
'
0
80
1
00
12
0
Optimal
N
Rate
[lbs./acre]
(N1
Figure
1.
Optimal
N
rates
where
there
is
uncertainty
about
weather
and
yield
response.
10
20
30
40
50
Expected
Return
Attributable
to
Fertilizer
(6/acrel
Figure
2.
The
relationship
between
expected
profit
and
risk
aversion
for
the
case
where
there
is
un-
certainty
about
weather
and
yield
response
49
December
1977
Western
Journal
of
Agricultural
Economics
expected
profit
maximizing
rate.
Note
that
for
this
level
of
soil
moisture,
the
expected
profit
maximizing
rate
should
be
used
as
long
as
the
specified
loss
probability
(8)
is
greater
than
.27;
that
is,
the
loss
constraint
is
not
binding
unless
8
is
less
than
.27.
As
another
example,
suppose
that
8
is
specified
to
be
.10
and
soil
moisture
is
between
3.9"
and
6.5".
In
this
case,
the
"strict
safety-first"
level
of
fer-
tilizer
is
about
75
lbs/acre.
Figure
1
can
also
be
used
to
find
the
loss
probability
associated
with
a
specific
fertiliza-
tion
rate.
For
example,
if
soil
moisture
is
be-
tween
3.9"
and
6.5"
and
60
lbs/acre
of
fer-
tilizer
is
applied,
the
associated
loss
probabil-
ity
is
0.05.
Figure
2
depicts
the
relationship
between
the
expected
return
to
fertilizer
and
the
probability
of
a
net
loss
for
the
three
soil
moisture
conditions
when
"strict
safety-first"
fertilizer
rates
are
applied.
As
an
example,
suppose
that
soil
moisture
is
between
3.9"
and
6.5"
and
that
the
acceptable
loss
proba-
bility
(8)
is
specified
by
the
farmer
to
be
.30.
Under
these
conditions,
the
expected
return
to
fertilizer
is
about
$33.00
per
acre.
And
if
the
loss
probability
is
.05,
the
expected
re-
turn
is
about
$28.00
per
acre.
So,
for
soil
moisture
between
3.9"
and
6.5",
the
farmer
who
wants
to
recover
the
cost
of
fertilizer
95
percent
of
the
time
rather
than
70
percent
of
the
time
will
give
up
an
expected
return
of
about
$5.00
per
acre
per
year.
Trade-offs
for
other
loss
probabilities
and
soil
moisture
levels
can
be
obtained
from
Figure
2.
Summary
and
Discussion
In
this
paper,
the
de
Janvry
model
for
find-
ing
the
optimal
fertilization
level
under
risk
is
extended
to
include
uncertainty
about
the
re=
sponse
function.
The
critical
assumptions
for
applying
the
model
were
complete
knowl-
edge
on
(a)
the
probability
distributions
for
weather,
(b)
the
mathematical
form
of
the
re-
sponse
function,
and
(c)
prices
of
products
and
resources.
If
a
long
time-series
of
data
were
used
to
estimate
the
weather
distribu-
tion
(as
in
this
paper),
the
first
assumption
is
likely
inconsequential.
One
way
to
overcome
the
weakness
implied
by
the
second
assump-
tion
is
to
estimate
various
functional
forms
with
a
subjective
probability
assigned
to
each.
If
a
fairly
general
form
is
used,
the
as-
sumption
would
not
appear
to
pose
a
serious
problem.
The
extension
to
two
sources
of
risk
may
pave
the
way
for
dealing
with
the
more
generalized
case
of
multiple
sources
of
risk.
Price
uncertainty
can
be
incorporated
by
ex-
tending
the
model.
However,
using
probabil-
ity
distributions
that
are
not
easily
integrated
by
analytical
means
will
significantly
increase
the
computational
burden.
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