II. Concepts and Principles Involved in the Economics of Fertilizer Use
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135
ECONOMICS OF FERTILIZER USE
were present. However, formal statement and elaboration of the law of
the minimum has been attributed to Liebig, the famous German chemist.
Initially, Liebig, describing the relationship between crop yields and
applied plant nutrients, stated that “The crops on a field diminish or
increase in exact proportion to the diminution or increase of the mineral
substances conveyed to it in manure,” and also, that “by the deficiency
or absence of one necessary constituent, all others being present, the
soil is rendered barren for all those crops to the life of which that one
constituent is indispensable” (Russell, 1950). These two quotations provide the basis of the law of the minimum as propounded by Liebig, and
later by P. Wagner of Germany. Few would take issue with the latter
quotation, but many have disputed the former statement.
Examination of the disputed statement shows that if only one nutrient were limiting crop yields the resulting yield equation for a given
soil would be
P = a!+ px,
(1)
where P is the estimated crop yield, a! reflects the combined effects of
all soil nutrients present in the soil and other growth factors, and p indicates the yield increase produced by additions of the deficient nutrient
XI. The relationship described by this equation is shown in Fig. 1.
01
0
I
I
I
I
I
40
80
120
160
200
POUNDS OF NUTRIENT APPLIED PER ACRE
FIG.1. Theoretical yield curve if crop responses were increased in direct proportion to the amount of applied nutrient.
Under these conditions, the yield increases produced by applications
of the nutrient, XI, would be constant regardless of the total quantity
added. This situation, of course, is unrealistic, implying that unlimited
production can be obtained from a plot of land. If a linear relation is
136
ROBERT D. MUNSON AND JOHN P. DOLL
found to exist in experimental results, it probably implies that the treatment combinations and rates used have not “bracketed” the response
range, thus limiting the usefulness of the information. Further references to the “law of the minimum” are presented in “Liebig and After
Liebig,” edited by Moulton ( 1942).
2. The Law of Diminishing Soil Yield or the Law
of Diminishing Increments
The basis of this concept conforms to the theory of diminishing returns proposed by Ricardo about 1817 (Ricardo, 1951). Liebig, in some
of his writings on soil tillage and manuring operations, subscribed to
this theory. However, formal development of the concept concerning
soilplantfertilizer relationships fell to men such as Mitscherlich, Baule,
and Spillman. A description of the mathematical expressions proposed
by these workers is discussed below.
a. MitscherZich. In 1909, Mitscherlich ( 1909) mathematically expressed the law of physiological relationship or the law of diminishing
soil yield concerning cropyield response to applied plant nutrients. He
stated that this law “was known before Liebig’s time.” In developing his
equation, Mitscherlich discarded Liebig’s assumption that the yield increase produced by fertilization was directly proportional to the quantity
of nutrient or nutrients applied. Noting that yield increases from fertilization usually increased at a decreasing rate, Mitscherlich made two
assumptions: ( 1 ) A maximum yield, A, exists for a crop grown on a given
soil. (2) The yield increment from fertilizer additions is proportional to
the decrement from the maximum yield, A. From these assumptions he
developed the following equation:
&!
= (A  y)k
dx
where dy/dx is the rate of increase in yield, Y, produced by additions
of a deficient nutrient, x; k is a proportionality constant which Mitscherlich later proposed as being constant, different, and independent for each
growth factor. By integrating equation (2)and converting to logarithms
to the base 10, the expression becomes
log (A
 Y)
= log A
 cx
where c = 0.4343k. This equation may also be written
Y
=
A(l
 10cs)
or, to the base e
Y = A(1  ek*)
(3)
ECONOMICS OF FERTILIZER USE
137
when none of the deficient nutrient is contained in the growth medium.
However, the expression becomes
y = A(1  ~ O   C ( L . + ~ ) )
(5)
where b is the quantity of the nutrient supplied by the soil and seed.
In actuality, b has a varying value if estimated in different years. Dean
and Fried (1953) discussed the use of b in predicting the amount of a
nutrient (phosphorus) in the soil and indicated that it may sometimes
give erroneous estimates of the amount of nutrient in the soil.
This discussion will not include the controversial issue of the constancies of the “effect factors,” c, for the various nutrients (Mitscherlich,
1947; Van Der Pauuw, 19521953; Willcox, 1955), but will deal only
with its agronomic and economic utility in analyzing experimental data.
Mitscherlich recognized the importance of the economics of crop fertilization and included sections in his original paper showing how his
equation could be used to determine the most profitable rate of fertilization. He discussed the following costs involved in crop production:
1. “Fixed costs” or those costs that remain constant regardless of the
yield level, e.g., taxes and interest on investment.
2. “Variable costs” or those costs that vary with the magnitude of the
total yield, e.g., the quantity of fertilizer applied and the harvesting
and handling cost.
The method outlined for determining the most profitable rate of fertilizer application is easily applied once the parameters of the equation
have been estimated. These parameters can be estimated algebraically,
graphically, or by the method of least squares. The method of least
squares involves an iterative process which can be time consuming if
electronic computers are not used. Mitscherlich ( 1909), Rauterberg
(1939), PimentelComes (1953), and Eid et al. (1954) have described
methods for determining these parameters.
Black (1955), writing on the evaluation of nutrient availability and
predicted yield response from fertilization, described Mitscherlich‘s
method for determining the most profitable rate of fertilizer application,
assuming unlimited amounts of capital are available. This rate can be
determined by the equation
x,= log [2.3cA(M  C,)/Cz]
 cb
C
where X, is the number of units of fertilizer that produce the highest
profit, Czis the cost per unit of fertilizer, M is the value per yield unit
of crop, C, is the cost per yield unit, and the “constants,” A, b, and c
138
IlOBEnT D. MUNSON AND JOHN P. W L L
are the determined values described above. Net profit, P, for any ratc
of application can be found by solving the equation
+
P = A ( M  CS)(l  lo”* )  (Cl C2x)
(7)
where C1is the fixed cost per unit area of land associated with production and the meaning of the other symbols remains the same.
b. Buule. Baule (1918),a German mathematician, became interested
in Mitscherlich’s law of physiological relationship and proposed a modification of it. According to Baule, it is not the absolute increase in yield
that reflects the effect of fertilizer, but the percentage increase in yield
or the percentage of the maximum possible yield produced by fertilization. He showed that when one uses this concept the percentage increase
in yield is independent of the quantity of nutrient present in the soil.
Baule used an “effect quantity,” h, inversely proportional to Mitscherlich‘s c value. This quantity, the “Baule unit,” is defined as the amount
of fertilizer nutrient required to increase the yield onehalf of the maximum possible. In other words, each additional quantity, h, will increase
yield by half of the decrement from the maximum yield. One Baule unit
will increase yield within 50 per cent of the maximum; 2 Baules, 75 per
cent of the maximum, etc.
Baule also suggested that all nutrients required for plant growth
should be included in the yield equation. This leads to the following
generalization of the Mitscherlich equation
Y = A(l
...
 lOcl~)(l  1Oc~~). * . (1  lo&)
(8)
..
where c1, cz,
cn are the “effect factors” for the nutrients xl, xz,
,
Xn. Y and A are the values previously defined. This equation provides
for yields of zero when a given essential nutrient is not present in a soil
nor added in a fertilizer application. For example, considering an equabl = 0, and q and bz are some positive
tion for two nutrients, if XI=
quantities
y
=
A(1
becomes
y = A(1
 10~(n+bl))(l ~ O  C ( Z Z + ~ Z ) )
 lo~CO))(l ~ O  C ( ~ S + ~ Z ) )
and
Y=O
because 10c@) = 1.
This function does not allow for yield decreases above the maximum
yield. However, as discussed later, it can be used to determine most
profitable nutrient combinations.
Willcox (1947), who has long been a proponent of the Mitscherlich
139
Bade method in the analysis of yield response to fertilization, described
a method for estimating yield equations using a standard yield diagram.
The diagram is used to estimate the parameters of the equation. After
the equation is obtained, the same techniques of economic analysis outlined previously apply in determining the most profitable rates of fertilizer application. One of the major drawbacks of this method is in the
analysis of yeartoyear yield variations and the difficulty of estimating
equations for more than one nutrient.
c. Spillman. In 1924 Spillman edited a book dealing with the law of
diminishing returns and its relation to biological production relationships.
In that publication ( Spillman, 1924), and a subsequent research bulletin
( Spillman, 1933), Spillman developed a mathematical expression of crop
yield response to plant nutrients. This equation is
ECONOMICS OF FERTILIZER USE
Y=MARo
(9)
where Y is the yield expected from x amount of applied nutrient, M is
the theoretical maximum yield attainable, A is the theoretical maximum
increase in yield obtainable by increasing x , and R is the ratio of successive yield increments. R is defined as
R = AYz
  = AYs
AY,  ... A Yn
AY1 AY2 AYa
A Yn1
where AY1 equals the yield increase produced by the first increment of
fertilizer, A Y is
~ the yield increase produced by the second increment, etc.
Another expression of Spillman's equation is
Y = M(1 Rz)
(10)
Spillman's equation is similar to Mitscherlich's with the primary exception that the ratio ( R ) of successive yield increments is not assumed
to be constant for a given nutrient but may vary with soil and climatic
conditions. The M of the Spillman equation corresponds to Mitscherlich's
A value, while Spillman's A and Rm correspond to Mitscherlich's
and
respectively.
Spillman (1933) outlined algebraic and graphic methods for evaluating the parameters of the equation for a given set of yield data.
Magistad et al. (1932) presented the leastsquares solution for determining the parameters of the equation from experimental data. Farden
and Magistad (1932) applied Spillman's equation in an economic analysis of the most profitable rates of fertilizer for pineapple cultures.
For three nutrient variables Spillman (1933) proposed the equation
Y = Ma(1  R"*)(l  RP+*)(l  RHc)
(11)
where the exponential terms n + a, p b, k + c are quantities of ni
+
140
ROBERT D. MUNSON AND JOHN P. DOLL
trogen, P205,and KzO,respectively, available from the soil, plus that
applied in the fertilizer. The other symbols are as previously defined
except that Ma specifies the maximum yield for three nutrients.
The above discussion has outlined some of the important early developments concerning production relationships and the profitability of
fertilizer rates. Recently, more emphasis has been placed on the economic
analysis of yieldresponse data. The following section outlines the economic analyses now being applied to cropyield data.
B. THEAPPLICATIONOF THE PRINCIPLES
PRODUCTION
ECONOMICS
TO FERTILIZER
USE
The fundamental purpose of production economics is the maximization of net farm income, It provides criteria that can lead to the most
profitable and efficient use of agricultural resources. In general, a farmer
has certain resources available for his use in production operations. Usually, these are owned, but they may also be rented or purchased on
credit. Production economic principles consider the costs and returns of
each production alternative and provide criteria that aid the farmer in
using his resources in the most profitable manner.
Yield curves or production functions, such as those developed by
Mitscherlich and Spillman, provide a basis for production economics
analyses. The purpose of these curves is to establish a continuous cause
and effect relationship among variables such as fertilizers and crop yields.
Although Spillman and Mitscherlich specified a certain mathematical
form for the production function, other types of equations may also be
used. Regardless of the form of equation used, the relationship must
display diminishing returns if it is to be consistent with known biological
phenomena.
For plant nutrientyield relationships, the theory of diminishing ret u r n s can be stated as follows: As the quantity of one variable input or
factor of production necessary for crop growth, in which the soil is deficient, is applied in increasing amounts while other inputs or factors
of production are held constant, the added product or yield increase
caused by the variable factor will eventually decrease. The economic
analysis which follows requires that the production function display
diminishing returns.
OF
1. SingleNutrient Analysis
The purpose of the production function or yield curve is to predict
yield response to fertilizer applications. The predicted curve, along with
crop and fertilizer prices, is used to determine profits that can result
from using different rates of fertilizer. This is done by considering the
ECONOMICS O F FERTILIZER USE
141
value of the added yield and the cost of the fertilizer required to produce that added yield. As an example, suppose an increment of nitrogen
( A N ) produces an increase in corn yield ( AY) . Then, if P, is the price
of nitrogen per pound and P, is the price of corn per bushel, (P, x AN)
is the cost of the added nitrogen and ( P , x AY) is the value of the
added yield. As long as the cost of the added nitrogen is less than the
value of the added corn yield, profits can be increased by additional
nitrogen. There will be a particular nitrogen application and resultant
yield at which the profit from nitrogen fertilization will be a maximum.
Using either more or less nitrogen will reduce profits. This optimum
exists when
P,(AN) = P,(AY)
(124
or,
A Y P,
AN  P,
With a yield equation, the expression on the left side of (12b) becomes an approximation of the slope of the yield curve. When the increments of nitrogen (AN) become infinitely small, the approximate slope
(AY/AN) is replaced by an exact expression for the slopethe first
derivative of the production function (dY/dN). Therefore, (12b) can
be replaced by
dY  P
_.,
dN  P,
The above analysis is called a “marginal” analysis. The yield increase
or additional yield caused by an additional nutrient input is called the
marginal physical product (MPP) of that nutrient and expresses the
rate at which the yield curve is increasing or decreasing.
An example of an economic analysis of a singlenutrient yield curve
has been selected from Paschal and French (1956). The experiment used
in this analysis, experiment 10 in their bulletin, deals with the response
of irrigated corn to nitrogen on Greenleaf silt loam. It was conducted
at Ontario, Oregon, in 1952. The experimental site had received a 20pound application of nitrogen the year before the experiment was conducted, but had not been fertilized for seven previous years. Fifty pounds
of Pz05were applied to all plots in 1952. The experiment was designed
to estimate optimum or most profitable fertilizer recommendations.
Twelve rates of nitrogen, ranging from 0 to 320 pounds per acre, were
used.
An arithmetic marginal analysis of the observed corn yields is presented in Table I. A corn price of $1.40 per bushel and a nitrogen cost
of $0.15 per pound were used to calculate the costs’and returns pre
142
ROBERT D. MUNSON AND JOHN P. DOLL
TABLE I
An Arithmetic Marginal Analysis of the Response of Irrigated Corn
to Nitrogen, Greenleaf Silt Loam, Ontario, Oregon, 1952“
~~
~
~~
Pounds of
nitrogen
applied
per acre
Bushels
of corn
per acre
(Y)
Added
yield
( W
Added
nitrogen
(AN)
0
40
64.6
90.4
118.2
132.4
140.7
141.0
146.8
141.2
147.1
145.8
147.4
143.6
25.8
27.8
14.2
8.3
0.3
5.8
5.6
5.9
1.3
1.6
3.8
40
40
20
20
20
20
20
20
40
40
40
80
100
120
140
160
180
200
240
280
320
Cost of
Value of
added
yield
nitrogen
(AY X PY), (AN X pn)c
$36.12
38.92
19.88
11.62
0.42
8.12
7.84
8.26
1.82
2.24
5.32
$6.00
6.00
3.00
3.00
3.00
3.00
3.00
3.00
6.00
6.00
6.00
” Source: Paschal and French (1956).
c
Mean of four replications.
Price of corn ( P J is $1.40 per bushel and cost of nitrogen (P,) is $0.15 per pound.
sented in this table. Other costs could have been included if relevant.
For the prices used, the data indicate that it would be profitable to
apply at least 120 pounds of nitrogen per acre. Although yield responses
resulting from higher nitrogen rates are too erratic to establish a trend,
it appears that net profit might still be increasing at 160 pounds per acre.
A Spillman yield equation was fitted to the experimental data by an
iterative leastsquares procedure which requires an initial estimate of R,
the ratio previously defined. The estimated equation is
P, = 150.17  89.40 (0.75)cf
(13)
where Y, is the mean corn yield of the jth fertilizer treatment and xj is
the quantity of fertilizer measured in 20pound units of nitrogen.
In analysis of variance presented in Table 11, the treatment sums of
squares is divided into two components, that due to regression and that
due to deviations from regression. Because the treatment sums of squares
represent the yield variation which the regression attempts to “explain,”
it is appropriate to fit the equation to the treatment means or totals rather
than the individual observations. The authors (Paschal and French)
emphasized the fact that the regression analysis is a supplement to the
analysis of variance rather than a substitute for it.
143
ECONOMICS OF FERTILIZER USE
TABLE I1
Analysis of Variance for Corn Yields on Irrigated Greenleaf Silt Loam,
Ontario, Oregon, 1952"
Source of variation
Degrees of
freedom
Replications
Fertilizers
Due to regression
Deviations from regression
Error
Sums of squares
3
11
315.10
2,783.85
945.29
30,622.22
29,567.41
1,063.53
2
9
T o hl
Mean squares
33
2,191.95
47
33.759.46
14,783.71
118.17
(36.66
Source: Paschal and French (1956).
The yieldresponse curve calculated from equation (13) is shown in
Fig. 2. Little or no yield response to nitrogen applications larger than
160 pounds per acre were observed. However, 320 pounds did not appear
to cause a marked decrease in total yields.
The most profitable rates of nitrogen can be found by equating the
derivative of (13) to the nitrogen:com price ratios and solving for the
amount of nitrogen. If nitrogen costs $0.15 per pound, the most profitable
rate would be 157 pounds if corn were $1.12 per bushel, 172 pounds if
corn were $1.40 per bushel, and 185 pounds if corn were $1.68 per bushel.
The dashed curves in Fig. 2 represent the 67 per cent confidence
limits derived for equation (13). The circles on the dashed curves denote the most profitable amounts of nitrogen for these limits. When nitrogen costs $0.15 per pound and the price of corn is $1.40 per bushel,
the amounts of nitrogen are 155 and 192 pounds, respectively, or a range
of 37 pounds between the upper and lower limits.
BU. PER ACRE
1
160
120
a0
0
0
40
I
0
80
1.40
1.68
I
I
160
240
16. N APPLIED PER ACRE
320
FIG.2. Yield curve, 67 per cent confidence limit curves, and most profitable rates
of nitrogen application (at $0.15 per pound) in irrigated corn experiment on green
leaf silt loam, Ontario, Oregon, 1952 (Paschal and French, 1956).
TABLE I11
Predicted Total Costs and Returns, Marginal (Additional) Costs and Returns, and Returns per Dollar for Nitrogen Applications on
Irrigated Corn, Greenleaf S i t Loam, Ontario, Oregon, 1952"
Pounds
of
Bushels
nitrogen
of
applied
yield
per acre increase
20
40
60
80
100
120
140
160
180
220
240
260
22.35
16.76
12.57
9.43
7.07
5.30
3.98
2.98
2.24
1.68
1.26
0.94
0.71
280
0.53
200
o
b
Value
of
Costof
yield
added
increaseb nitrogen
$31.29
23.46
17.60
13.20
9.90
7.42
5.57
4.17
3.14
2.35
1.76
1.32
0.99
0.74
$3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
Returns
Returns per dollar
above
onlast
nitrogen 20 lb. of
costsb
nitrogen
$28.29
20.46
14.60
10.20
6.90
4.42
2.57
1.17
0.14
0.65
1.24
1.68
2.01
2.26
$10.43
7.82
5.87
4.40
3.30
2.47
1.86
1.39
1.05
0.78
0.59
0.44
0.33
0.25
Total nitrogen
application
Total yield
increaaeb
Pounds
Cost
Bushels
Value
20
40
60
80
100
120
140
160
180
$ 3.00
22.35
39.11
51.68
61.11
68.18
73.48
77.46
80.44
82.68
84.36
85.62
86.56
87.27
87.80
S 31.29
54.75
72.35
85.55
95.45
102.87
108.44
112.62
115.75
118.10
119.87
121.18
122.18
122.92
200
220
240
260
280
Source: Paschal and French (1956).
Corn price is $1.40per bushel and nitrogen price is $0.15per pound.
6.00
9.00
12.00
15.00
18.00
21.00
24.00
27.00
30.00
33.00
36.00
39.00
42.00
Average
Total return per
return
dollar
to
spenton
nitrogenb nitrogen
$28.29
48.75
63.35
73.55
80.45
84.87
87.44
88.62
88.75
88.10
86.87
85.18
83.18
80.92
$10.43
9.12
8.04
7.13
6.36
5.72
5.16
4.69
4.29
3.94
3.63
3.37
3.13
2.93
id
0
kl
p
5
2
Z
8
8
L(
'd
g
145
ECONOMICS OF FERTILIZER USE
Total costs and returns, additional or marginal costs and returns, and
returns per dollar spent are presented in Table I11 and Fig. 3. These data
were calculated using a corn price of $1.40 per bushel and a nitrogen
price of $0.15 per pound. The curves in Fig. 3 demonstrate economic
principles which should be considered when making fertilizer recommendations. The optimum rate of nitrogen application is the same, 172
pounds, for either the criterion of total return above total cost (value
of total yield increase minus total cost of nitrogen application) or the
marginal criterion (additional or marginal costs should equal the value
80
40
[
o.
Total return above cost of
I___+
N

Cost a1 total application
ADDITIONAL RETURNS

^^I.
..'
Value of additional
16. N APPLIED PER ACRE
FIG.3. Costs and returns from rates of nitrogen on irrigated corn Greenleaf silt
loam, Ontario, Oregon, 1952 (nitrogen at $0.15 per pound, corn at $1.40 per bushel)
(Paschal and French, 1956).
of the additional or marginal yield). The returns per dollar spent, however, do not determine the most profitable rate of fertilization. Because
the yield curve exhibits diminishing returns, returns per dollar spent
decrease as nitrogen applications increase. Therefore, returns per dollar
spent is not a satisfactory criterion for determining the most profitable
rate of fertilizer.
However, returns per dollar spent or net returns per dollar can be
used to determine the minimum amount of fertilizer that is profitable.
The optimum rates of fertilization discussed above were calculated assuming that all costs other than the cost of fertilizer are insignificant