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III. Current Research on Economics of Fertilizer Use

III. Current Research on Economics of Fertilizer Use

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nl. (1956) conducted a 5 x 5 x 5 x 2 complete factorial with corn.

For estimation of the yield curve, it is very desirable to have the yield

estimates that these experiments produce, but the work and cost involved in conducting them are often excessive.

This caused research workers to attempt to reduce the number of

treatment combinations and yet obtain reliable estimates of the regression coefficients. The composite designs proposed by Box (1954) appear

to be applicable in this respect. These designs require fewer plots per

replication, and the results are readily analyzed by regression techniques.

Composite designs were initially developed for evaluating multifactorresponse surfaces in industrial research.

Cochran and Cox (1957)have presented a discussion of the application and analysis of composite designs for studying response surfaces.

The basis of these designs is a 2” factorial treatment arrangement, with

additional treatments included. A composite design requires a minimum

of 15 treatment combinations per replication for a three-factor experiment, while a S3 complete factorial requires 125 treatment combinations.

Hader d al. (1957)have discussed the use and analysis of these designs

in agronomic research. Baird and Fitts (1957)discussed their application in field experiments. Also, Tramel (195%) developed the triplecube design, a modifkation of the composite design, which requires a

minimum of 31 treatment combinations for a three-variable experiment.

Another design being used is the interlaced factorial developed by

C. G. Hildreth of Michigan State University. To the authors’ knowledge,

a description of this design has not been presented in the literature.

Composite designs may not be as readily adaptable to exploration

of response surfaces in agricultural research as they are in industrial

research. The use of these designs in agronomic research is still in the

trial phase. Whether or not they will receive widespread acceptance

remains to be seen, but indications are that their use is increasing. Where

multifactor experimentation is necessary, composite designs seem to hold

definite promise. However, complete factorials are needed to compare

estimates found when all of the treatment combinations are used with

those found when only the treatment combinations of the composite

design are used.





Currently, the fitting of yield curves and calculation of fertilizer

recommendations is being done primarily by agricultural economists.

As with earlier work, much of the current research is concerned with

the selection of equations that approximate the “true” yield function.

Although the selection of an adequate equation is undoubtedly impor-



tant, other factors such as cropping sequence, weather, residual fertility

should also be considered, This section will review recent research on

these phases of the economics of fertilizer use and present some references for the differing approaches.

1. The Spillman Equation

This type of equation is widely used, both in the form developed by

Mitscherlich and the form developed by Spillman. Paschal (1953) presented a graphic method of approximating the Spillman equation and

applied an analysis to alfalfa yields resulting from P2Oa applications.

Ibach and Mendum (1953) described an extension of the graphic method

permitting the estimation of equations containing more than one nutrient

variable and applied it to pasture-yields data from a 4 X 3 X 3 factorial

experiment with nitrogen, P20a, and K20. Ibach (1953) compared the

use of a Spillman equation, a quadratic equation, and a power equation

( Y = axb),sometimes called the Cobb-Douglas function, for predicting

fertilizer recommendations. Paschal and Evans (1954) presented further

applications of the Spillman equation determining the profitability of

nitrogen fertilization of irrigated sorghum.

Paschal and French (1956) used the Spillman equation in the economic evaluation of nitrogen fertilization in several irrigated-corn experiments. The Oregon experiment described in Section 11, B, 1 was

among these. They used an iterative method of least squares developed

by Stevens (1951), which requires an initial estimate of R, to estimate

parameters of the equations for experiments containing enough application rates to characterize the response curve. French (1956), in a

further analysis of data presented by Paschal and French ( 1956), compared the Spillman equation, the quadratic equation, the “square root”

equation ( Y = a bx1I2 c x ) , and the Gompertz curve ( Y = e(M-ARO)) .

He also compared equations estimated from fewer than the twelve fertilizer rates included in some of the experiments and concluded that at

least five or six adequately spaced fertilizer applications should be used.

Ibach (1958) presented further information on graphic estimation of

the Spillman equation, and Ibach and Mendum (1958) further developed graphic and least-squares analyses to estimate parameters for this


Having estimated the yield curve, the above workers calculated fertilizer recommendations in accordance with the procedures described

in Sections 11, B, 1 and 11, B, 2. It should be noted, however, that most

of the analyses using the Spillman equation have dealt with yield response to only one nutrient.





2. Other Types of Equations

Many workers have used equations other than the Spillman or

Mitscherlich function. The quadratic and square-root equations are

frequently used. These equations are linear in the coefficients and may

be readily estimated by least-squares regression procedures.

Heady and Pesek (1954) and Heady et al. (1955) presented an application of production economics techniques to yield data from experiments conducted in Iowa. These experiments were specscally designed

for economic interpretation. Isoquants, isoclines, and optimum recommendations were derived using quadratic and square-root yield curves

for corn, red clover, and alfalfa. The Iowa corn experiment discussed

in Section 11, B, 2 is an example of this research, An important finding

of these workers was that the ratio of nutrients in least-cost nutrient

combinations varied with yield levels. This suggests that different nutrient ratios should be used, depending upon the yield desired.

Pesek et al. (1959) applied an economic analysis of corn yields resulting from varying rates of nitrogen and different plant populations.

They found that, in order to make high rates of nitrogen profitable, the

plant population had to be high. The optimum number of plants per

acre was dependent upon the price of nitrogen and corn rather than

the cost of seed corn required to increase plant population.

Knetsch (1956) and Knetsch et al. (1956) analyzed corn-yield response to nitrogen on a Kalamazoo sandy loam soil in Michigan. These

workers utilized the equation (Y = aXbc'") which has been described

by Halter et al. (1957). Sundquist (1957) and Sundquist and Robertson

(1959) presented the results of an economic analysis of yield curves for

oats, wheat, and beans. In this work, the equation used by Knetsch

(1956) was compared with square-root and quadratic equations. It was

found that, although the former equation characterized the observed

yields equally as well as the latter equations, it required lengthy calculations to derive fertilizer recommendations. Other research using

quadratic and square-root equations has been reported by Brown et al.

(1956), Woodworth et al. (1957), Orazem and Smith (1958), and Doll

et al. (1958).

3. A Discrete Analysis

To avoid the problem of selecting an appropriate equation to represent the yield-response curve, Hildreth (1954) developed a method of

estimating discrete points along the yield curve or surface. This method

estimates the yields to be expected from each fertilizer application used

in the experiment. By using the discrete analysis, an investigator 'need



not select a continuous equation and may avoid possible biases that

could result from selecting an inappropriate equation. The estimates are

obtained by the method of maximum likelihood subject to the restriction of diminishing returns. Therefore, the yield increase caused by any

fertilizer increment must be equal to or smaller than the yield increase

caused by the preceding fertilizer increment. A curve drawn through the

yields estimated by this method would be concave to the abscissa. Estimates of yields resulting from nutrient inputs falling between any two

application rates included in the experiment must be obtained by interpolation.

Stemberger (1956, 1957) applied Hildreth's discrete analysis and a

continuous production function analysis to the same set of experimental

data. A study of these data suggests that differences among fertilizer

recommendations derived by the two methods are slight. Also, Stemberger indicated that computations involved in the discrete analysis were

long and tedious.

4. A Rotation Analysis

Stritzel (1958) presented an analysis of a corn, oats, meadow, meadow

rotation. The experiment was a 5 x 4 x 3 NPK factorial, in a randomized

block design replicated twice. Some plots were top-dressed with nitrogen

and PzOa during the rotation to compare initial and residual responses.

Using quadratic, square-root, and cubic equations, Stritzel analyzed yield

response for each year of the rotation. An economic analysis of the

corn yields was conducted. Nitrogen, Pz05,and KzO yield of nutrient

curves were used to evaluate nutrient uptake for some of the crops in

individual years.

The results for the complete rotation were analyzed by converting

crop yields into total digestible nutrients and also by converting them into

total value products, a measure of gross income. For both methods, crop

yields in the second, third, and fourth years were discounted or reduced

to account for uncertainty and time considerations. Equations were then

estimated from the four-year totals and an economic analysis was conducted. For relevant price combinations, yield isoclines calculated from

the four-year total digestible nutrient surface were almost perpendicular

to the Pa06 axis of the isoquant map. This differs from the isoclines estimated for individual years and indicates that P206should be applied at

approximately the same rate regardless of nitrogen applications, or that

minimum cost combinations would include the same amount of P205

regardless of yield levels. This seems consistent with the concept that

PzOa, due to its role in plant nutrition, should be maintained at a high

level regardless of application rates of other fertilizers ( E. C.Doll, 1958).



5. Weather Variability

One important cause of year-to-year variations in yield response to

fertilizer applications is weather, Brown and Oveson (1957, 1958) have

discussed variations in the response of spring wheat to nitrogen applications over a ten-year period. They demonstrated the use of a yield curve

representing the average response for the ten years. Climate and rainfall

variations are undoubtedly major causes of the differences in the yearly

response functions they present.

Orazem and Herring (1958) have analyzed the effects of soil moisture

at seeding time, rainfall during the growing season, and nitrogen on

grain sorghum yields in southwestern Kansas. Six years’ data were reported. Soil moisture at seeding time was found to have the largest effect

on yields, and the effectiveness of rainfall during the year increased as

initial soil moisture increased.

Knetsch and Smallshaw (1958), using a drought criterion developed

by Van Bavel (1953) which considers the daily precipitation, the daily

moisture loss, and the water-holding capacity of the soil, computed the

drought incidence in the Tennessee Valley for a thirty-year period.

Using the same drought criterion, Knetsch and Parks (1958) analyzed

an experiment with varying rates of nitrogen and irrigation water. A

continuous yield curve, expressing Starr Millet yield as a function of

nitrogen and drought, was calculated. Using the yield-response data and

weather records for the area, they predicted the probability of the occurrence of different drought intensities and the anticipated response to

nitrogen. An analysis which enabled probability estimates to be placed

upon response to fertilizers was applied to the results. The nitrogen

application that returned the most profit when averaged over all drought

conditions was considered to be the most profitable application in the

long run. Optimum rates of nitrogen and irrigation were also derived.

Although Knetsch and Parks state that their data are not conclusive, their

results indicate that average profits over time varied only slightly for

nitrogen rates between 120 and 210 pounds per acre. The analysis, however, did not adequately account for nutrient accumulation or depletion

over time.

6. Discussion

The mathematical approximation of the yield curve should be as

representative as possible of the “true” yield relationship. Also, these

curves should be, within limits, easily estimated and analyzed. Quadratic,

square-root, or other equations linear in the coefficients seem most nearly

to meet these objectives. These equations can be easily estimated by



least-squares regression procedures and readily lend themselves to economic analysis. J. P. Doll (1958) has described various types of equations,

linear in the coefficients, which might be used. Kenneth W. Meinken of

Rutgers University, in an unpublished manuscript reviewed by the

authors, has used a family of curves of the form (Y = a bx'" &).

In the analysis of twenty-eight sets of experimental data, t h i s type of

function appeared to be better than those previously used. Meinken also

proposed a different approach to the concept of interaction.

At the present time, very little of the reported research on the

economics of fertilizer use has dealt with the variations of yield curves

over time. One problem in this regard is that completely adequate research methods are not available for analyzing this type of data, even

when it is available. And, as suggested by Kempthorne ( 1957), problems

in the statistical analysis of such data are not small.

Recommendations derived from yield curves for a single year are

point estimates of unknown reliability. One of the goals of research in

the economics of fertilizer use should be to estimate the probable returns

from a given fertilizer application for an individual year and to estimate

the returns from that application rate if it were used over time. This is not

to say that problems related to depicting and analyzing yield responses

within individual years are not important, because the whole can only

be as accurate as its individual parts.





Soil tests, along with past-management information, are generally

being used to provide guidance in recommending rates of fertilization

farmers should use. These tests are effective for making such recommendations only after ample experiments have been conducted in the field and

the resulting yields of different crops to rates of fertilization have been related to soil-test values. In other words, as Bray (1948) has stated, ''. , .

the test must measure the total amount of the available nutrient . , the

crop must be allowed to indicate the significance of that amount in terms

of growth and response to the added nutrient."

Various methods have been used to relate soil-test values to crop-yield

response to fertilization. Few, however, have considered economic levels

of fertilization in conjunction with soil tests or the simultaneous effect of

more than one soil-test nutrient in analyzing experimental results. Both

of these are important considerations if recommendations are going to

be made to farmers on the basis of soil tests and supplemental information.

With regard to relating soil-test information to economic levels of

fertilization, Hanway and Dumenil (1955) used nitrogen soil-test results

and corn response to nitrogen fertilization for predicting the most profit-




able rate of nitrogen. These workers used a modification of the Mitscherlich equation to relate yield increases from nitrogen applications to the

soil-test values. Utilizing corn:nitrogen price ratios, they devised a graph

to estimate the most profitable rate of nitrogen application, given the

soil-test value. Other functional models could be used in analyses of t h i s

kind and, if adequate data were available, extended to include several

nutrients simultaneously. If the latter were done, perhaps variations in

results now attributed to error could be attributed to the appropriate


Pesek (195s) has proposed a functional analysis that considers both

the soil and fertilizer nutrient components. He suggests that the soil-test

values be used for the soil components in the yield equation. Hildreth

(1957) has proposed a somewhat similar approach, If a generalized

function were found to exist, these proposed methods would predict the

most profitable rates of nutrient application, if the soil-test values and

price relationships were known. It is possible that an experimental model

such as this could be further improved if weather data were incorporated

into the analysis.

The possible existence of a generalized production function, a continuous response curve for different initial fertility levels of a soil type, is

currently being investigated in greenhouse research at Iowa State College. The theoretical considerations and results of preliminary research

have been presented by Jensen and Pesek (1959a, b). Among the findings thus far, indications are that for certain fertility levels on a given

soil type, a generalized function may exist. Also, as might be expected,

the results indicate that interactions probably occur between the soil and

fertilizer nutrients.

By regression analysis, crop yields from unfertilized soils, as well as

fertilized, may be analyzed as dependent variables, with the different

soil-test values as independent variables. Such analysis can aid in determining nutrients that should be evaluated by soil tests. They can also

be used to reflect the contribution nutrients make toward yield, and they

indicate possible improvements in soil tests (Eid et al., 1954; Black, 1955).

D. D. Mason and others at North Carolina in unpublished results

have used four regression equations to relate corn yields from fertilized

plots to soil-test values. Different combinations of' soil-test values were

included as independent variables. The soil-test variables initially included were pH, Ca, Mg, P, K, and organic matter. Of the four regression

models studied, little difference existed in the amount of variation that

could be accounted for by regression analysis. Care should be exercised in

the interpretation of such analyses because, in some cases, soil-test values

of certain nutrients are highly correlated. Unless this is taken into con-



sideration the results of such analyses can be biased and lead to erroneous


IV. Conclusions

The authors believe that certain aspects of crop production research

need greater emphasis. Among these are:

1. Additional variables, such as climatic factors and soil properties,

should be measured and included in the analyses. The purpose of their

inclusion should be to estimate average returns over time, as well as

probable returns in a given year, resulting from varying amounts of


2. Analyses should be conducted on the physical, chemical, and

biological changes that occur in soils under continuous and rotational

systems of cropping. The introduction and widespread use of commercial

fertilizers has apparently affected current thinking on rotations; that is,

crops that add nutrients to the soil are no longer as important as they

once were. Therefore, a quantification of the nutrient additions and

depletions caused by different cropping sequences under various levels

of fertilization appears to be useful.

3. The “fertilize the soil” and “fertilize the crop” concept should be

compared and analyzed. The conditions under which each of these concepts is the most useful should be explored further.

4. Soil-test values should be utilized along with other measurable

variables in an attempt to determine generalized yield equations for soils.

5. The economics of various recommended practices should be studied

in conjunction with fertilization rates. Many of these practices would increase yield response to fertilizers and would usually involve little risk

to farmers.

Consideration of the above would add a time dimension to the

economics of fertilizer use and base the yield curve on a dynamic foundation rather than a static, single-year foundation. Because yield curves

constitute one of the basic elements of farm planning, casting them in a

dynamic setting would, in turn, more adequately permit the consideration

of time in farm-planning studies.

The use of yield curves in determining the profitability of fertilizer

applications has been outlined and discussed. Needless to say, all facets

of this subject have not been considered. A general review of the economics of fertilizer use has been presented by Nelson and Ibach (1957).

Also, the readers are referred to North Central Regional Publication No.

54 (1954),“Profitable Use of Fertilizer in the Midwest,” and to two books

based on seminars sponsored by TVA, “Methodological Procedures in the



Analysis of Fertilizer Use Data” (Baum et aZ., 1956) and “Economic and

Technical Analysis of Fertilizer Innovations and Resource Use” ( Baum

et al., 1957) for more extensive coverage of the subject.

Finally, a comprehensive bibliography is scheduled to be prepared

soon by the Agricultural Research Institute’s Committee on the Economics of Fertilizer Use for the Agricultural Board of the National Research Council. The outstanding membership of this Committee insures

the preparation of an excellent publication concerning the various aspects

of the economics of fertilizer use.


The major portion of this paper was prepared while the authors were on the

staff of the Division of Agricultural Relations, Tennessee Valley Authority, Knoxville,

Tennessee. We wish to thank Dr. L. G. Allbaugh, Director of the Division, Dr.

E. L. Baum, Chief of the Agricultiual Economics Branch, and Dr. George Stanford,

Chief of the Soils and Fertilizer Research Branch, for their aid and encouragement during the time this paper was being prepared.


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