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II. Concepts and Principles Involved in the Economics of Fertilizer Use

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

soil-plant-fertilizer 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 crop-yield 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



- 10-cs)



or, to the base e

Y = A(1 - e--k*)



(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, 1952-1953; 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), Pimentel-Comes (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 one-half 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



...



- lO-cl~)(l - 1O-c~~). * . (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 10-c@) = 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 year-to-year 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=M-ARo

(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 Yn-1

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 least-squares 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 yield-response data. The following section outlines the economic analyses now being applied to crop-yield 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 nutrient-yield 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. Single-Nutrient 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 slope-the 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 single-nutrient 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 least-squares 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 20-pound 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 yield-response 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



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