Title Page
 Table of Contents
 Fertilizer efficiency from the...
 Discussion, William G. Brown
 The fertilizer problem: Resource-enterprise...
 Discussion, H. B. Cheney
 Review of fertilizer research in...
 A critical evaluation of fertilization...
 Problems in design of soils experiments,...
 Statistical problems in designing...
 Discussion, James S. Plaxico
 Use of experimental data in making...
 Some economic considerations of...
 Discussion, H. B. Peterson
 Roster of attendance

Group Title: Farm management in the West : problems in resource use report ;, no. 1
Title: The economics of fertilizer application
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00055261/00001
 Material Information
Title: The economics of fertilizer application conference proceedings
Series Title: Farm management in the West problems in resource use report
Physical Description: 81 p. : ill. ; 27 cm.
Language: English
Creator: Western Agricultural Economics Research Council -- Farm Management Research Committee
Conference on the economics of fertilizer application, (1956
Publisher: The Council
Place of Publication: Corvallis Ore
Publication Date: [1956?]
Subject: Fertilizers -- West (U.S.)   ( lcsh )
Fertilizers -- Research   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
conference publication   ( marcgt )
non-fiction   ( marcgt )
Bibliography: Bibliographical footnotes.
General Note: Cover title.
General Note: Conference held January 16 and 17, 1956.
Funding: Electronic resources created as part of a prototype UF Institutional Repository and Faculty Papers project by the University of Florida.
 Record Information
Bibliographic ID: UF00055261
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: oclc - 08094882

Table of Contents
    Title Page
        Title Page
    Table of Contents
        Table of Contents
    Fertilizer efficiency from the agronomic point of view, D. G. Aldrich
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
    Discussion, William G. Brown
        Page 7
        Page 8
    The fertilizer problem: Resource-enterprise and tenure relationships and criteria for optima, H. Russell Shaw
        Page 9
        Page 10
        Page 11
        Page 12
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        Page 21
        Page 22
    Discussion, H. B. Cheney
        Page 23
        Page 24
        Page 25
        Page 26
    Review of fertilizer research in the West, Bert A. Krantz
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
    A critical evaluation of fertilization research, Glenn Johnson
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
    Problems in design of soils experiments, F. G. Viets, Jr.
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
    Statistical problems in designing experiments to study the economics of fertilizer application, Bernard Ostle
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
    Discussion, James S. Plaxico
        Page 53
        Page 54
        Page 55
        Page 56
    Use of experimental data in making statistical inferences, Burton L. French
        Page 57
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    Some economic considerations of fertilizer use in American agriculture, E. L. Baum
        Page 69
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        Page 78
    Discussion, H. B. Peterson
        Page 79
        Page 80
    Roster of attendance
        Page 81
Full Text
C/ /...o .o
Peter E. Hildebrand
Agricultural Economics

Farm Management in the West

Problems in Resource Use

Report No. 1

The Economics of

Fertilizer Application

Conference Proceedings
Farm Management Research Committee
of the
Western Agricultural Economics Research Council
Corvallis, Oregon
January 16 and 17, 1956


The Western Agricultural Economics Research Council consists of the
Heads of Departments of Agricultural Economics of the Land-Grant Colleges
in the Eleven Western States, and Hawaii, and the Heads of the Agricultural
Marketing Service and Agricultural Research Service of the United States De-
partment of Agriculture. The Council was formed in 1947 to strengthen, inte-
grate and coordinate state and regional research in Agricultural Economics in
the Western Region.

Intensive study is given to special areas through committees of the Council.
At the present time, three council committees are in existence. They are: the
Committee on the Economics of Water Resource Development, the Committee
on the Economics of Range Resource Development, and the Western Regional
Farm Management Research Committee.

Plans for a Farm Management Research Committee were made in 1953, and
the Committee was officially established late in 1954. Its purpose is to provide
opportunities for professional growth through mutual exchange of ideas and the
stimulation resulting from work conferences. The conferences would investigate
new areas, new methods, and new approaches that may prove fruitful in farm
management research. A grant-in-aid from the Farm Foundation makes it pos-
sible for the Council to sponsor these work conferences.

The first meeting of the Committee was held in:Fort Collins, Colorado,
June 16-17, 1955. At this meeting, the research problems common to the West
were discussed and classified into research areas. From these areas, the
" Economics of Fertilizer Application" was selected for intensive study at a
meeting in Corvallis, Oregon, January 16-17, 1956. This report presents the
formal papers delivered at that meeting.

The Chairman of the Committee wishes to express his gratitude to the par-
ticipants. They made it possible to achieve the objective of providing opportuni-
ties for professional growth by their contributions to this conference.

Rex D. Rehnberg



D. G. Aldrich.................................... .. .... 1

William G. Brown, ........................................ 7

H. Russell Shaw.......................... .... ......... .. 9

H. B. Cheney. .. o ........... ..................*. .23

Bert A. Krantz ................. ........ ........ ... .... 27

Glenn Johnson .......... ....... .. o.......... ........ .. 33

F. G. Viets, Jr......... .......................... ..... 41

Bernard Ostle.......... ..... ... .....0............ ... ..... 47

James S. Plaxico. ........ ....... .......... .. ............0 53

Burton L. French .....0............. ..... ..... ........... o 57

E. L. Baum .. ...0... ...... .... .............. .. ... .... 69

H. Bo Peterson ................ .......... .... ..........0 79

ROSTER OF ATTENDANCE.................................... 81


D. G. Aldrich1

Literally thousands of papers have been published, and in all probability
thousands more will be published by agronomists on the relationship between
crop yield and fertilizer use. Two objectives characterize the majority of
these papers. First, they record for a particular soil and crop the kind and
amount of fertilizer required to produce maximum yield. Second, from this
data they attempt to determine the most efficient fertilizer application rate.
The two most common criteria for efficiency are the smallest amount of ferti-
lizer associated with the largest increase in yield per unit of fertilizer and
the quantity of fertilizer associated with maximum financial returns from the
crop allowing for the costs of the fertilizer and any additional production costs
resulting from increased yield.

Using either or both of these criterion as a basis for evaluation, agronomists,
after may years of careful field experimentation, have learned that fertilizer
efficiency is influenced by a number of factors. Some of these involve soils and
their management, crops and their sequence, climate (including soil moisture
supply), and systems of farming. These factors may vary from one locality to
another, from farm to farm, and even from field to field. The manner in which
each of them influences the efficientuse of fertilizers is discussed briefly in this

Soils and Their Management

Although crops differ in their needs for nutrient elements, their fertilizer
requirements depend largely on the soil. The native nutrient-supplying power
of the soil in turn is dependent upon the chemical composition of the parent
rock material from which the soil is derived and the conditions of climate and
vegetation under which it is formed. For example, the highly weathered soils
of the eastern United States presently receive about 3 times as much phospho-
rus and 10 times as much potassium per acre annually as do the less weathered
soils of the Pacific Southwest. On the other hand, soils of each of these areas
receive from 5 to 6 times as much nitrogen fertilizer per acre as do the rich
prairie soils of the Middle West.

Studies on the phosphorus content of soils in southern California, rela-
tively unaffected by variations in climate, revealed that wide variations in the
phosphorus supply of adjacent fields could be attributed to differences both in
source of parent material and in chemical composition of the same parent mate-

A recent review of nitrogen fertilizer practices in southern California
citrus orchards indicated sandy soils required 2 to 3 times as much nitrogen per
acre for maximum production as did finer-textured, hardpan soils. Thus, ferti-
lizer requirements vary from region to region and within regions, from soil
type to soil type and within soil types.

1Chairman, Department of Soils and Plant Nutrition, University of Cali-
fornia, Davis and Berkeley.

A number of methods can be used to assess the fertilizer requirements of
different soils--soil tests, tissue tests, quick tests, pot tests, and biological
tests; but the most reliable method for establishing the efficient use of ferti-
lizers is field experimentation. The knowledge gained by field trials on par-
ticular types of soil can usually be extended to individual farms and fields on
similar soils by the use of soil maps, soil analyses, or other fertility apprais-
al techniques, and by general experience with local soils. Although fertilizer
requirements vary with different soil types, past management practices of
soils also have a great influence on these requirements.

Soil management includes not only fertilizing, manuring, and crop rotation,
but also cultural practices, drainage, irrigation, and water conservation.
Management of the soil may alter its properties to such an extent that two fields
with the same type of soil, but under different management, may require very
different fertilizer treatment.

For instance, one field has been used to grow cereal crops for many years,
without manure or fertilizer. An adjacent field with the same acid -soil type
has been used over the same period of years under a system of management
which included a rotation of legumes and cereal crops and the use of manure
and fertilizer. When the time comes to devise an identical cropping system
for the two fields with a suitable fertilizer program, it is obvious that treat-
ments for the first field will differ from those of the second. The first field
will in all probability need more lime and fertilizers thanthe second.

Again, let us consider two fields of the same soil type, with similar slope
and drainage. One has been well managed to prevent loss of soil by erosion;
the other through mismanagement has lost a good portion of its original sur-
face soil. A much more rigorous program of soil management and fertilizer
dressings is required on the eroded field to restore it to usefulness than on
the field which has been consistently well handled.

Crop Differences

Experience and research show that different crops differ in the amounts
of several plant nutrients that they need and in their sensitivity to deficiencies.
Some crops are better foragerss" for the plant nutrients already in the soil
and respond less to these nutrients in fertilizers than other crops on the same
soil; some crops, also, are shallow rooted while others take nutrients from a
considerable depth. Recognition of these specific requirements is important
to the efficient use of fertilizers. The brief statements that conclude this sec-
tion indicate a few of the relative differences among some common crops when
grown in soils moderately deficient in available nitrogen, phosphorus, and po-
tassium, but under favorable conditions of climate, including rainfall.

In general, the grasses and cereals show greater response to nitrogen
than to the other plant nutrients. Such crops as corn and sugar cane produce
high yeilds per acre and require large amounts of nitrogen. Common appli-
cations to these crops are 50 to 100 pounds of nitrogen per acre, but both may
receive much more when all the growth factors are favorable. Grass meadows
and pasture respond to heavy applications of nitrogen if sufficient soil moisture
is available for optimum growth. Under most conditions, however, grasses
should be grown in mixtures with legumes; this practice improves the quality
of the forage, reduces the need for nitrogen fertilizer, and if the deeply rooted
species are used aids in:maintenance of soil structure. The successful grow-
ing of these legumes often reduces the need for nitrogen but increases the need
for minerals, especially phosphorus, potassium, and calcium. In someareas some

of the nutrients needed by the plants in small quantities must also be supplied.
For instance, applications of zinc and boron are required for legumes in some
parts of the U.S.

Excessive applications of nitrogen in fertilizer or manure to cereal crops
such as wheat, barley, rice, and oats often cause lodging, which leads to de-
creased yields. These crops ordinarily receive not more than 20 to 40 pounds
of nitrogen per acre. Heavy nitrogen application to cereal crop serving as a
nurse crop for grass and legume seedings often increases the shading and re-
duces the stand and growth of the seedlings.

In contrast to the grasses and cereals, legume hay crops need little if any
nitrogen from fertilizers, except perhaps a little when sown, but require pot-
ash, phosphate, and often calcium. Under favorable soil conditions they obtain
their nitrogen from the abundant supplies in the air if the seed is inoculated
with the proper bacteria.

Ingeneral, root crops and legumes respond more to phosphorus and potassi-
um than to nitrogen fertilizers, although this depends a great deal on soil type;
for instance, on muck or sandy soils potassium is often needed in substantial
amounts. Truck crops--tomatoes, and lettuce in particular--also respond well
to phosphate fertilizers. Ordinarily the amount of phosphate required by most
legumes and root crops ranges between 40 and 80 pounds of available P205 per
acre. Potatoes and vegetable crops under intensive production may respond
well to twice these amounts where other factors are favorable.

Crops that are especially responsive to potassium fertilizers are white
and sweet potatoes, sugar beets, mangolds, tobacco, tomatoes, cotton, corn,
and legumes. Grass crops and cereals other than corn are generally less re-
sponsive to potassium than to nitrogen and phosphorus fertilizer. But again
this varies with the soil type. In certain sections of the U.S., potatoes often
receive as much as 150 to 200 pounds of K20 per acre. In contrast, corn and
cotton ordinarily receive only 30 to 60 pounds per acre. Tomatoes and tobacco
receive intermediate amounts.

Crop responses to animal manures. Unfortunately there is not enough
manure available on most farms to permit an adequate application to each field
each year. In a rotation the emphasis should be given to the use of manure on
the more valuable crops which respond well to.applications of plant nutrients
and to improved soil structure and also on those with a long growing season.
The plant-nutrient content of manure naturally depends on the kind and amount
of feeding stuffs and litter and on the way the manure is made and used. Manure
usually supplies useful quantities of N, P, and K. For root crops, experiments
show that smaller dressings of P and K fertilizers are needed on manured land
than on similar land without manure. With ordinary applications, however,
there appears to be no need to cut down the nitrogen fertilizer on the manured
land for crops requiring large amounts of nitrogen, because the chemical and
physical effects of the manure enable the crops to respond profitably to more
nitrogen than is afforded by manure.

Sugar beets, potatoes, fodder beets, turnips, cabbage, corn, and oilseed
respond particularly well to manure. Legumes and cereals generally respond
less. The legumes can utilize atmospheric nitrogen, and the cereals have a
lower potassium and phosphorus requirement than many other crops. Liquid
manure is used well on such crops as fodder beets, potatoes, turnips, cabbage,
rutabaga, rape, maize, and spring cereals. It is a poor fertilizer for legumes.

The principle of diminishing returns holds with manure as with fertilizers;
generally the most efficient use of manures is obtained with medium rates of
application. This is particularly true of the first crop grown after manuring.
Applications of 5 to 10 tons per acre are most commonly used, but much higher
rates are sometimes used in intensive vegetable cropping. In established stands
of legume seedings, a rate of 2 to 5 tons per acre is sufficient. The opportuni-
ties for increasing agricultural efficiency through better care and use of ma-
nure can scarcely be overemphasized.

Crop rotations and rates of application. Where systematic crop rotations
are followed, the place of the crop in the rotation often determines whether
nitrogen fertilizer is required. If corn or small grains follow legumes, for
example, they may obtain all or most of their nitrogen needs from the decom-
posing roots and residues of the legume crops; on the other hand, if they are
grown the second or third year after the legume crop, they may require a rela-
tively large quantity of nitrogen fertilizer. However, if all other factors are
favorable, corn usually responds very markedly to side dressings of nitrogen
in midseason, even following a legume.

If a crop in the rotation is heavily fertilized, there may be some phospho-
rus and potassium not used by that crop, and this residual material may help
to meet the fertilizer requirements of the following crop. Residual effects are
uncommon with chemical nitrogen fertilizer, however, because this form of
nitrogen, if not used by the crop, is readily leached out of the soil.

The use of manures affects the rates of fertilizer application. When ma-
nure is applied, less quantities of fertilizer may be needed.

The amount of fertilizer used for various crops is also influenced to a
large extent by the acre value of the crop. The crop of highest acre value gen-
erally receives the bulk of the fertilizer. During World War II and succeeding
years of food shortage, some countries took steps to encourage the use of fer-
tilizer on food crops at the expense of other crops. While the practice of giving
preferential treatment to high-value crops is sound, there is sometimes a ten-
dency to use such large amounts of fertilizer on these crops as to be uneconom-
ic, while other crops on the farm are slighted. This is especially true on
farms where potatoes, tobacco, vegetables, or similar intensive crops are
grown in the same field year after year. On a few farms in various parts of
the United States, for example, large amounts of phosphorus and potassium
have accumulated in the soil from high annual applications through the years.
On these farms the amount of fertilizer now used on special crops is often
higher than necessary. Uneconomic use of fertilizers results from too large
applications, because the response of crops per unit of fertilizer applied di-
minishes as the quantity of fertilizer is increased beyond that needed for a
good crop. But too small amounts are also often uneconomic. In soils quite
deficient in some nutrient, enough must be applied to provide normal growth,
smaller amounts may be essentially wasted. The farmers' problem is to find
the rate of fertilization that will provide the greatest net return rather than
the greatest yield, considering the whole rotation of crops and the long-time
productivity of the soil.

Climate and Soil Moisture

Differences in soil moisture greatly modify fertilizer requirements. Gen-
erally, soils in dry regions are not so leached as those in humid regions, and
contain more nutrients. Further, moisture supply in these regions is usually


a greater limiting factor than the nutrient supply. But irrigation may change
the fertilizer need completely. Under dry farming, fertilizer application may
not increase crop yield, but crops may respond wonderfully to nitrogen and
phosphorus on the same soil under irrigation. In one experiment conducted at
Yuma, Arizona, an application of 375 pounds of nitrogen per acre increased
the acre yield of seed cotton by 1280 pounds when irrigation water was applied
to maintain a soil moisture tension of 0.2 atmospheres in the root zone, but by
only 693 pounds when irrigation water was applied to maintain a soil moisture
tension of 9 atmospheres. To utilize high fertility levels effectively in the soil,
irrigated crops may respond to additional irrigation over those growing on soils
of low or medium fertility. Also abundant fertility reduces water requirements
of plants. Of course, only factors that inhibit normal growth, nutrient defi-
ciency, drought, or disease raise the water requirements of plants,

Similarly, in humid regions, control of runoff water through terraces may
so improve soil-moisture conditions that fertilizers will give good responses
where they did not before. Other things being equal, the more favorable the
environmental conditions for producing high yields, the greater is the quantity
of fertilizer that can be used to advantage.

Phosphates help to promote root development and encourage early maturity
in crops. In some areas with a short growing season and especially those with
a moist climate, phosphate fertilizers are used to hasten the maturity of the
crop. On the other hand, heavy applications of nitrogen encourage a heavy top
growth and tend to delay ripening of crops, make the crop more susceptible to
disease, and reduce the mineral content of the plant. For these reasons it may
be necessary in a cool, moist climate to limit the rate of nitrogen application.
Certainly high amounts must not be used when the mineral elements are defi-

Systems of Farming

The system of farming determines to a considerable extent the kind of
fertilizers that should be used and the rate of application. For one thing, fer-
tilizer treatment of crops depends upon whether manure is used. It is evident
that fertilizer practices on livestock farms should be somewhat different from
those on farms where little or no livestock is kept and no manure is used.
The fertilizer practice used by a market gardener growing his crops intensively
must differ from those of a farmer growing similar crops extensively. The dry-
land farmer.~in'allprobability, uses fertilizers at quite a different rate of appli-
cation than does his neighbor on an irrigated farm, even though soil types of the
farms are nearly identical.

For each system of farming which depends upon crop, soil, climate, kinds
and rates of fertilizer applied and equipment available, there are many ways of
applying fertilizer which affect fertilizer efficiency significantly. Some of the
common methods of applying fertilizer are banding along the row, drilling with
the seed, broadcasting before or after plowing, deep drilling after plowing,
plow-sole or deep furrow application, foliar application, side dressing, bedding,
top dressing, starter solutions and applications in the irrigation waters. The
choice of the proper method of application is as important in determining ferti-
lizer efficiency as applying the right kind and amount of fertilizer. No single
fertilizer-placement pattern has been found that is superior for all crops and
under all conditions.

Broadcasting on the soil surface is usually less desirable than localized
placement of the fertilizer in relation to the seed or plant. Phosphate and pqt-
ash move downward only very slowly in many soils and may remain near the
surface out of reach of plant roots when the surface layer of the soil becomes
dry or is cultivated. A number of investigators have shown that nitrogen may
be lost by broadcasting ammonia-containing fertilizers on the surface of alka-
line soils or applying these same fertilizers in the irrigation water. The
losses may approach 50 per cent of the total ammonia nitrogen applied. Such
losses are effectively prevented by putting the fertilizers an inch or two into
the soil.

Banding fertilizers to the side of the rows, in furrow bottoms or beds,
and drilling the fertilizer with the seed gives good returns from limited quan-
tities of fertilizers, Direct contact between fertilizer and seed should be a-
voided because many crops are seriously injured by such a practice.

Many experiments have shown that deep placement of fertilizers increases
yield. This is particularly true in arid regions where soils dry out to a con-
siderable depth or where deep-rooted crops are grown. The efficiency of deep
fertilizer placement, however, is dependent upon the textural, drainage, and
aeration characteristics of the soil. Heavy soils having poor aeration and drain-
age show little benefit from deep fertilizer applications.

Foliar applications of fertilizers circumvent soil interactions which may
render the applied fertilizer unavailable to the crop and enables the farmer to
supply his crops directly at critical stages of growth with a variety of essen-
tial plant nutrients.

From the foregoing agronomic considerations, it is apparent that the effi-
cient use of fertilizers is influenced by many factors. As agronomists strive
for greater fertilizer efficiency, it will be necessary for them to give more and
more attention to those factors which supplement the soil' s inherent ability to
give up nutrients to crops. As we move in this direction, it is becoming evi-
dent that the soil becomes more and more a medium through which water, ox-
ygen, and nutrients are conveyed to plant roots, and less and less the direct
source of plant nutrients.


William G. Brown

Dr. Aldrich's paper is interesting from the economic viewpoint. The two
criteria for efficiency listed are familiar to economists. The first criterion
is "the amount of fertilizer associated with the largest increase in yield per
unit of fertilizer." This is the point of application where the greatest average
increase in yield occurs. It would be the point where returns per dollar of
fertilizer are greatest. According to economic theory, this application would
be the minimum amount which would ever be applied. That is, the farmer
would put on at least this much or else none. It would be irrational" for a far-
mer to apply an amount short of this point where the average marginal physical
product is greatest because more money could be made by putting the same
amount of fertilizer on fewer acres.

Dr. Aldrich's second criteria, "the quantity of fertilizer associated with
maximum financial returns from the crop allowing for the costs of the fertilizer
and any additional production costs resulting from increased yield," is even
more familiar to economists. In economic jargon, this point would occur where
marginal revenue was just equal to marginal cost. This can readily be shown
mathematically. However, the important point in regard to either the first or
second criterion is that in order to apply either criterion, the functional re-
lationship between yield and inputs of fertilizer must be estimated. That is,
an assumption of continuity within the experiment must be made. If an estimate
of the continuous, functional relationship is not made, then the agronomist can
pick only one of the treatments in the original experiment as the optimum"
rate of application. For example, if 0, 40, 80, and 120 lbs. of nitrogen were
applied to corn, the agronomist could pick only one of these treatments unless
he turns to regression analysis. And there seems to be no reason why one of
the 0, 40, 80, and 120 lb. treatments will always be the optimum rather than,
say, the 50, 60, 70, or 90 Ib. treatments. The chief advantages of regression
analysis in such a case is (1) that it allows estimation of yields in between the
actual treatments in the experiment and (2) is at the same time statistically
more efficient. All the observations can be used to estimate a particular treat-
ment mean. Thus, confidence limits for any relevant treatment can be "nar-
rower" and tests of significance more sensitive than by the older methods such
as finding the least significant difference between treatment means.

Once the continuous, functional relationship between yield and fertilizer
inputs has been estimated, then the "exact" optimum input for any price situa-
tion can be determined. (Even the tenure and capital position of the farmer can
be taken into account.) However, the exactness" of the solution through the
use of regression and calculus seems to cause many agronomic economists to
shy away from the method. There is too much variability in the experimental
data to justify finding solutions exact to the pound!" exclaim many agronomists.
It is indeed true that there is considerable variability in the basic yield data.
But the variability is still there no matter what method we use to make fertili-
zer recommendations. For example, if the standard error of the most exact
and unbiased recommendation is 20 pounds of nitrogen, the problem is not
solved by making an inexact or rule-of-thumb recommendation; the rule-of-
thumb recommendation only increases the margin of error. Furthermore,
the more refined method of regression analysis gives a better evaluation of the
variability that is involved.

department of Agricultural Economics, Oregon State College, Corvallis

A second objection often raised to the proposed use of regression is that
"there are too many factors affecting the response to fertilizer which make
yield impossible to predict." However, does anyone believe that the use of
less precise methods solves this problem? Precise analysis can be qualified
just as easily in terms of the basic experiment as can recommendations made
by less formal methods. Furthermore, more progress towards measuring the
effect of these variable factors can be made. But first the problem of response
estimation must be considered in relation to economic theory and statistical

Dr. Aldrich's paper lists many of these variable factors which must be
considered in estimating yield responses. For example, soil types often differ
so much that response estimates must be made separately for each type. Sim-
ilarly, past management and cropping practices alter the response to fertilizer.
Of. course, different crops vary in their response to fertilizer. Dr. Aldrich
also lists water as another factor that can change the response of a crop to fer-
tilizer applications.

The problems that these variable factors present is a big one. However,
basic economic considerations can be helpful in solving these problems. It may
be necessary to estimate separate response functions for each major crop and
soil type. Recent advances in soil nutrient tests may make it possible to take
some of the effects of varying soil fertility into account in the yield-estimating
equation. Perhaps water can also be included in the estimating equation. Cer-
tainly, there is excellent opportunity for progress in these directions. Dr.
Aldrich's paper gives an excellent discussion of factors which must be consi-
dered when estimating yield response to fertilizer.


H. Russell Shawl

The main objective of this paper is to present the problem and concepts in
the optimum use of fertilizer in the economic sense. We will be concerned
primarily with indicating the solution but not in the research techniques nec-
essary for generating the information essential to the solution. In presenting
the problem, however, many of, the general techniques applicable for finding
an approximation of needed data are suggested.

Under a system of private ownership of resources and perfect markets,
the optimum allocation of resources can come about only if (1) resource own-
ers maximize incomes, (2) consumers maximize individual satisfactions from
their incomes, and (3) firms maximize profits. Under such a system consumer
preferences are reflected through the firms back to the resource owners by
means of prices. That is, relative prices indicate to resource administrators
(owners and firms) how valuable the respective resources and services are to
consumers. When the physical productivity of resources in different uses are
related to prices, then the relative values of resources become known. If re-
sources are allocated always first to those products which yield the highest re-
turn and finally so that each resource earns the same at the margin in every use,
then the optimum allocation has been reached.

At the farm level decisions in resource allocation center about: (1) how
best to employ a given technique, (2) which techniques and types of resources
should be used in producing a single commodity, (3) which products or com-
binations of products should be produced, (4) how many resources in aggre-
gate should be used on each, (5) how much of each resource should be used in
producing each product, and (6) how should resource use be spread over time.
The basic technical criteria for making choices of these types are three in
number. They are: (1) factors must be combined so that the marginal rate of
substitution between any pair used in the production of a commodity is equal
to the inverse of their price ratio; (2) factors must be used in quantities so
that the marginal rate of transformation of factor into product is inversely
equal to the price ratio between factor and product; and (3) products must be
produced in proportions so that their marginal rate of substitution is equal to
the inverse of their price ratio. In order to satisfy any of these three, de-
tailed knowledge of the production function is absolutely essential.

What do these conditions mean when applied to the fertilizer problem? It
is important to note that the economics of the problem enters only after the
physical input-output relationships have been determined. In addition, with-
out specific orientation of fertilizer trials toward eventual economic analysis,
much physical data are practically useless so far as the economist is concerned.

1Department of Agricultural Economics, University of California, Davis,

The Nature of the Relationship of Fertilizer Use to Yield
of a Single Crop

We shall first consider the nature of the relationships between fertilizer
and yield and later apply the conditions which must be satisfied in order to
maximize profits It is well known that the addition of increasing quantities
of a fertilizer element to a given area of l1nd results in increases in output
but at a decreasing rate as more is added It is also known that, if enough is
added, the output will reach a maximum and eventually decline. Such a rela-
tionship is illustrated in figure 1.




Figure 1. Yield response to a single fertilizer element.

Such a curve could be derived for any element holding all other elements
constant at some given level. This immediately raises a question whether the
relationship remains the same if other elements are held constant at other
levels. To consider this problem we must go to a three-dimensional diagram
illustrating yield responses from various combinations of two nutrient elements.
Consider for example a situation where two elements are perfect substitutes.
The yield surface would then appear as in figure 2.

There are two features to note: (1) the shape of the indivual isoyield curves
and (2) the successive increments of output as more of either element or a given
ratio of the two is added. In this example only the part of the curve in figure
1 from A to B is shown. Any vertical slice of the surface in figure 2 would be
of the shape indicated in figure 1, and substitution of element two for element
one is at a constant rate as indicated by the curve EF. If we return to a two-
dimensional diagram, the isoyield contours appear on the xy plane as shown
in figure 3.

1 Maximize the value of the product from a given value of resources or
conversely minimize the cost of a given output.
2 Exceptions to this general rule will be discussed later in the paper.
3In all subsequent diagrams the added yield only will be illustrated, i.e.,
the yield without fertilizer (OA) being omitted.
4 Perhaps the only situation where this would occur is where a nutrient is
supplied from two different sources.

Figure 2.

Figure 3.

LU _
< -J

(3 >

A production surface indicating perfect substitution
between two nutrients.

Production surface with perfect substitution and
diminishing marginal productivity projected in
two dimensions.

Since the isoyield or isoproduct contours are straight lines, the two fertilizer
elements substitute for each other at constant rates. Diminishing returns are
expressed also in that the isoproduct curves representing equal increments
move farther apart as distance is increased from the origin. This indicates
that it requires more of either element (or any combination) to produce equal
successive increments of output.

The Nature of Substitution Relationships

For purposes of exposition factor substitution relationships may be con-
sidered as belonging to one of seven basic types or combinations of them. The
number may be reduced or expanded since each may be considered to shade into
the next closest related type. Certain basic characteristics make these seven
types unique either as limiting cases or as intermediate between limiting cases.
These are presented graphically in figure 4A to 4G.

Figure 4A. Factor substitution at a con-
stant rate (already discussed) throughout
all factor ratios. A special case is
where factors substitute at a 1:1 ratio.


Figure 4A.

Figure 4B. Factor substitution at a di-
minishing rate throughout all factor ra-
tios and intersecting each axis. This
implies that neither factor is absolutely
essential for this level of output.


Figure 4B.

Figure 4C. Factor substitution at a di-
minishing rate throughout all factor ra-
tios but with the output contours becoming
asymptotic to each axis. This implies
that each factor is absolutely essential
to the process. A special case is sub-
stitution at a constant elasticity.

Figure 4C.


Figure 4D. Factor substitution at a di-
minishing rate throughout a central range
of factor ratios only but becoming asymp-
totic to or identical with a line parallel to
each axis at the extreme factor ratios.



Figure 4D.

Figure 4E. Factor substitution at a di-
minishing rate throughout a central range
of factor ratios and after becoming par-
allel to each axis at a point diverging
from the axes. A special case of this
ftypemay be designated as becomingas-
w ymptotic to or identical with a pair of fac-
tor ratio lines.
I --


Figure 4E.

Figure 4F. No factor substitution (" sub-
stitution" at a point) with each extreme
parallel to one axis. This is the case of
technical complements.



Figure 4F.


Figure 4G. No substitution with no pos-
sibility of changing the factor ratio with-
out reducing the product to zero.



Figure 4G.

By combining these types of isoproduct contours many other types of sub-
stitution may be specified. For example, a combination of type B and type E
may be very common.

The Nature of the Production Surface

The production surface (Y f (Fi) specifying the relationship of yield to all
combinations of fertilizer) may be considered to be composed of an infinite num-
ber of one or more of the type of contours illustrated in the previous section
with each contour representing successively larger values of output. We now
turn to consider the nature of the production surface and shape of successive
contours. It is essential to recognize that the yield will eventually decrease
for each individual nutrient or ratio of nutrients if sufficient is added In
the following example we consider the case where the individual effects of the
fertilizer are additive, i. e., there is no interaction. This implies that only
the level but not the shape of the production surface changes above a single con-
stant nutrient line regardless of the amount of input of the other nutrient. In
figure 5, for example, the yield at M is the sum of AJ and DN and similarly the
yield at R is the sum of AJ and ES. The input OE of nutrient 2 gives the maximum
yield increase for this fertilizer element. Similarly the input OB of nutrient 1
gives the maximum yield increase for this element The combined effect of the
two nutrients gives a maximum for both together at Q. When this set of response
relationships is reduced to two dimensions, product contours as illustrated in
figure 6 result. Here again Q represents the maximum output. The line EQ defines
the maximum level to which nutrient 2 may be added and still result in increas-
ing yield while holding nutrient 1 constant at any level between O and B. Simi-
larly BQ defines the limit for adding nutrient 1 while holding nutrient 2 constant.
BQ and EQ define the limits within which the rate of substitution between the
factors is negative (i.e., between -toand zero). These lines are defined as
ridge lines. There are an infinite number of isoclines all radiating from the

This becomes obvious, since, if this were not true, it would be possible
to produce unlimited quantities of product on a single acre or square foot of
land except in the unlikely case where the surface became asymptotic to some
maximum yield.
2Note that OE and OB are not necessarily equal.
3An isocline is defined as a line connecting all isoproduct contours at points
with the same rate of substitution.

Figure 5.

Production surface with no interaction but
diminishing marginal product for both nutrients.


Figure 6.

Product contours with no interaction between
nutrients and a dinimishing marginal product for
both nutrients.

point Q toward the axes and lying within these limits .

The more general type of fertilizer production function would allow inter-
action. Such a situation is illustrated in figure 7. As in figure 6, the isopro-
duct contours will converge on a single maximum point. The ridge lines, how-
ever, will not be parallel to the axes, and they are not necessarily straight lines.
It is assumed that the ridge lines are positively sloped since most yield-respon-
se data indicate that a negative effect of one fertilizer element is likely to occur
at lower levels of input of that element when the second element is at a low rath-
er than a high level. There is no reason to believe that the isoclines (and ridge
lines) converge on the origin since this implies isoproduct curves at low levels
of input are of the typegiven infigure 4C, and this requires that no increase in
yield can be obtained from one element without at least a small quantity of the
other being present, Infigure7, S denotes the marginal rate of substitution

S = 0


S= 0.0


Figure 7.

Product contours with interaction between nutrients and a
diminishing marginal product for both nutrients.

1The significance of the isoclines will become apparent in cost minimization
discussed later.

of Nutrient 2 for Nutrient 11

Economic Optima

Earlier it was indicated that the optimum condition for combining factors
was where the marginal rate of substitution of one factor for the other is equal
to the inverse of their price ratios. To illustrate, suppose that the price of
element 1 is twice that of element 2. If all combinations of the two inputs
which could be purchased by a given expenditure are plotted as in figure 8, the
locus of such points would be a straight line with a slope of 0. 5 (the inverse of
the price ratios). This line would be tangent to some output contour at the point
where it is intersected by the isocline joining all rates of substitution equal to
0. 5. It may be noted that at no other point will the same expenditure result in
a larger product, or conversely for the same output on other combination will
result in a smaller expenditure. Cost, therefore, has been minimized. For



Figure 8. Cost minimization.

1The rate of substitution of Nutrient 2 for Nutrient 1 is defined as the quan-
tity of Nutrient 1 which is exactly replaced (not changing output) by an increment
of Nutrient 2. Since substitution can take place only "at the margin," the rate
of substitution and the marginal rate of substitution amount to the same thing.
In moving along any product contour from a high ratio of Nutrient 1 to a lower
ratio, Nutrient 2 is being substituted for Nutrient 1. As Nutrient 2 is increased
by equal increments, Nutrient 2 replaces successively smaller quantities of
Nutrient 1. The rate of substitution of Nutrient 2 for Nutrient 1 is decreasing.
Similarly increasing Nutrient 1 results in a diminishing rate of substitution of
Nutrient 1 for Nutrient 2. A curve convex to the origin represents a diminishing
rate of substitution between nutrients.' The slope of the isoproduct curve at any
point is the marginal rate of substitution of Nutrient 2 for Nutrient 1.

any price situation, the optimum combination of nutrients falls on some speci-
fic isocline.

The optimum total quantity of nutrient elements remains to be specified.
If the production surface is sliced vertically parallel to the x axis (holding the
other element constant), the resulting configuration is given in figure 9. The
slope of the curve AB at any point denotes the marginal product of element 2.
With the price of the product and factor given, a line may be drawn from the
origin connecting quantities of product that are equal in value to corresponding
quantities Nutrient 2 .

E 8

ol C


Figure 9. Optimum level of use of a single nutrient.

This line has a slope equal to the ratio of the price of the nutrient to the price
of the product. If a line is drawn parallel to the price ratio line and tangent
to the output curve (i. e. DE) then no other level of input can be selected which
will increase the margin between cost and revenue. Profit has been maximized.
This point must coincide with the optimum factor combination indicated earlier.
These conditions must in fact be satisfied simultaneously. The required equa-
tions to be solved simultaneously for the quantities of the two nutrients and the
level of output are: (1) the equation specifying the production function, (2) set-
ting the M.R.S. equal to the inverse price ratio of the nutrients, and2(3) setting
one marginal product equal to the ratio of the factor to product price .

The above specifies the optimum for a single crop in a single year. In many
instances there may be a considerable residual effect of fertilizer either direct-
ly or indirectly through crop residues. If the same crop follows, the added yield
from the residual (discounted for time) may be added to that of the first year and
the resulting total response function calculated as before. This introduces the
problem of distribution of fertilizer over time. The problems of both residual

1The cost of the factor would include any variable cost of application. Any
added cost of growing or harvesting the crop due to the increasing yield would
be included by subtraction from the product price.
The equation setting the second marginal product equal to its appropriate
price ratio may be derived from (2) and (3). It may be noted also that the ratio
of the two marginal products is identical with the marginal rate of substitution.

effect and distribution over time may be solved by finding the yield surface re-
sulting from continuous application of the same set of treatments after the yield
response pattern had become constant. This still does not solve completely the
problem of year-to-year adjustments to changing prices of nutrients and pro-
duct when residual effects are present.

The problem of variation in quality of product with changes in fertilizer
application probably has not received sufficient attention in the past. This of
course is of no consequence if the product is sold and no recognition of quality
differences are made in prices received. In many cases, however, quality ef-
fects may be exceedingly important. Notable examples might be cotton and for-
age. Perhaps the most convenient means of accommodating variation is to con-
vert the physical production function to a value function and solve the set of
equations in the same way as before except that now the price of the product is
automatically equal to 1.0.

The price of the nutrient elements used should include a premium depending
upon the earning capacity of expenditures on other factors elsewhere in the busi-
ness since otherwise the marginal earnings on fertilizer will be driven down to
1.0 leaving the returns to the farm as a whole lower as the result of failing to
employ the last units of expenditure in neglected higher earning uses.

Multiple Products--the Rotation

Fertilizer use may be influenced indirectly through shifting the effects of
other resource limitations of the farm as a whole such as seasonal labor, ma-
chinery or water requirements. These adjustments, however, express them-
selves through product selection. In addition multiple products (in effect rota-
tions) may have some direct influences in themselves. These may arise from
intereffects of crops on nutrient requirements or from concentrations of crop
preditors. In any case, it is advantageous for the farm operator to choose a
sequence of different crops rather than concentrating solely on a single one.
Within any one year all crops are highly competitive. That is, any increase in
one can come about only as a result of decreasing another. Only over several
years can the full affects of the rotation be realized. In any one year the line
AB in figure 10 might represent the alternative outputs of product from given
land with a given fertilizer application. Over a period of years, however, the
resulting outputs are likely to drift as in the manner indicated by CD and finally
become stable as for EF.

The optimum condition for the selection of enterprises is given by the tan-
gency of GI to EF at H where GI represents the largest total revenue possible
from the given resources used in deriving EF. The slope of EF (the marginal
rate of substitution between the two products) at H is equal to the slope of GI
(the inverse of the price ratio between the products). With an increase in fer-
tilizer elements applied, the output quaiitities would increase (initially at least)
so that a new curve representing a higher fertilizer application could be drawn,
perhaps, beyond the points A, H, and B. The interaction between crops implies

This will not only shift the relative elevation of the various constant pro-
duct curves which formerly corresponded to given values but will change their
shape if the various nutrient elements have differential effects on quality.

that product responses to fertilizer must be developed for whole sets of rotations
rather than for single crops. In addition to the adjustments required as before
for capital limitations and residual effects, there may be further adjustments
in rotations which are optimum for individual farms due to the fact that the pro-
portions of types of soil differ among farms. This is particularly true for live-
stock farms where most of the feed is homegrown.

a. G
a: A H

I -

0 F/ B
Figure 10. Economic optimum in crop selection.

Leasing and Economic Optima

Leasing is essentially a means by which a tenant buys the services of land,
buildings, and sometimes other resources from the landlord. With a sharing
arrangement leasing might also be viewed as a means by which the landlord buys
the services of labor, power, machinery, and other resources from the tenant.
The economic function of the lease is twofold: (1) it provides a basis for com-
bining resources of two owners into a single firm and (2) it distributes income
to the resource owners within the firm. Leasing systems are efficient only if
they will allow channelling successive quantities of resources into their rtost
profitable uses and in so doing maximize net farm income in the long run If
leasing systems prevent or discourage resources from being invested in ways
which will maximize income to the farm as a unit, the leasing system mist be
defined as inefficient.

Hurlburt gives four conditions necessary within the leasing arrangement to
encourage operation of the firm at maximum profit from the combined resources
of landlord and tenant These are:

(1) The share of the owner factor of variable input must be the same
as the share of output of product obtained from it.

An ideal leasing system may not result in the optimum allocation of re-
sources through other faults such as insufficient knowledge of production or
price relationships, or other aspects of managerial quality.

2Hurlburt, Virgil L., Farm Rental Practices and Problems in the Midwest,
North Central Regional PublicationNo. 50, Iowa State College, Ames, Iowa,
October 1954. These are incentive conditions in that they prevent conditions
which would encourage a deviation from the optima given earlier and do not in
themselves encourage deviation.

(2) The shares of all products must be the same.
(3) Each resource owner must receive the full share of the
product earned by each unit of resource he contributes.
(4) Each resource owner must have opportunity to receive
return on investment made in one production period but
not forthcoming until a subsequent period.

These conditions are necessary not to discourage optimum use of resources
unless all resource and output quantities are specified in accordance with the
optimum conditions and agreed to in total at the signing of the contract. How-
ever, there is no allowance for changing conditions unless the whole contract
is completely negotiated as changes occur in prices, etc. This in itself would
discourage optimum use of resources in an "all or nothing" contract as just
indicated as an alternative. Under a cash lease the first two conditions are
satisfied since the tenant pays the full cost of the fertilizer and receives the
full benefit from the revenue. The deviation caused by an unduly high or low
rent on land (condition 3) may not be serious except in a case of extreme cap-
ital limitation on the part of the tenant. In a share lease if the proportion of
fertilizer cost carried by the tenant is larger than his share of the product,
this will tend to restrict fertilizer input below the maximum profit level. If,
for example, he paid the full cost while obtaining onehalf the crop and the de-
cision regarding quantity of fertilizer were his, he would restrict input of fer-
tilizer to that level where the return from his share was just equal to his cost.
For the farm as a whole the last dollar of expenditure on fertilizer would still
be yielding $2.00. This would restrict fertilizer input very considerably below
the optimum level. It is not likely that the direct effect of differences in shares
of the product would have much influence on fertilizer use except as it would
change the crop combination. With a share rent, if the first condition is sat-
isfied, the third will be also so far as fertilizer is concerned. The conditions
apply equally to livestock share leases.

The fourth condition is probably at least as important with regard to fer-
tilizer use as is the first. In order to satisfy the latter condition, three pri-
mary provisions should be provided in the lease. These are provisions for
renewal, and compensation and length of termination notice. These are all pe-
culiar to leasing and tend to increase uncertainty and so (1) shift the cropping
pattern to as short production period as possible (cash crops) and (2) reduce
the quantity of inputs including fertilizer. Without a long-term lease (of any
type) or provision for compensation for residual effect of fertilizer, use of
fertilizer nutrients will be restricted below the optimum level. This combined
with a shift away from rotations (which require considerable time to yield their
benefits) may induce a severe deviation from optimum resource use. Any com-
pensation for unused fertilizer should contain a premium including the expected
rate of return discounted to the time of termination of the lease.

In view of the foregoing, certain conclusions seem warranted. It is im-
possible to allocate fertilizers at their optimum without knowledge of their
prices, the product prices, and complete yield data within the relevent range.
Optimum fertilizer mixes change as relative prices of the nutrients change.
Recommendations based on a given fertilizer ratio can rarely be correct. Wth
such related factors as type of crop, soil, variety, rotation, water, and climate
(as well as several others probably of lesser importance) causing variation in
yield response to fertilizer and the variety of possibilities within these items,
there is a considerable challenge and rather urgent need to construct a priority
system for determining which problems to consider first and how much effort is


justified on each. In addition, some system of extrapolation useful for sim-
ilar" soils, varieties, climates, etc. would seem essential.


H. B. Cheney1

Let me make it clear at the start that basically I am in general agreement
with the concepts and approach that were made in outlining the problem of op-
timum use of fertilizer in the economic sense. Unfortunately, as will be true
of many agronomists, I am not sure that I understand the full implications of
many of them.

I would like to think that once we tear away the veil of specialized termino-
logy of the agronomist, horticulturalist, or soil scientist and that of the econo-
mist, basically we will have much in common in our concepts and approach to
the fertilizer problem. However, we may be a little like the blind man and the
elephant in that we tackle the problem from different viewpoints and come up
with different concepts or apparently different approaches to the problem.

The soil scientist, I am sure, often sees primarily certain specialized seg-
ments of the problem from the physical scientist viewpoint whereas the econo-
mist often becomes impatient when the physical scientist doesn't, supply the
kind of data he needs to work out his theoretical concepts and functions.

Another observation seems appropriate relative to physical data required
for sound economic decisions and fertilizer practice within the framework of
the total farm production plan. Due to the complexities of the interacting effects
of a large number of variables on crop production I would suggest right at the
start that economists need to be satisfied with less than perfect data--even from
the best designed experiments. In other words, viewpoints, backgrounds, and
concepts as well as facts are going to be important in weighing evidence and ar-
riving at decisions on fertilizer use. We probably never will have all of the
facts that are essential for arriving at completely logical answers for all situa-
tions. However, that should not prevent us from striving to improve both our
facts and our concepts or our hypotheses in the approach to the fertilizer problem.

Several points that seem to me to have a bearing on this general problem
are as follows:

1. As indicated in the generalized growth curve shown below the actual
growth curve is probably of a sigmoid type if the full range, particularly at
low levels, is included. In most fertilizer experiments we are dealing with
the B portion of the curve. Many fertilizer experiments, of course, are not
adequate to fully define even the B portion of the curve. To be fully certain
that the B portion cf the curve is adequately defined at the upper levels, it is
usually necessary to have some points in the C portion. Moreover, numerous
factors aside from the variables under consideration affect the exact shape and
position of the growth curve,

2. The term substitution of one nutrient element for another is often
used by economists and also by soil and plant scientists but with different
meanings implied. This may lead to misunderstanding unless we keep clearly
in mind the meaning intended. I assume that the economist is talking about
economic substitution whereas the soil and plant scientist means physio-
logical substitution.

IHlead, Department of Soils, Oregon State College, Corvallis.





3. Considerable emphasis has been placed upon fitting response curves
particularly near the maximum-yield level. The economist is particularly con-
cerned with the nature of the function that describes the yield curve. In con-
trast the soil and plant scientists who have to a considerable extent used analy-
sis of variance to aid in interpreting their data have been somewhat less con-
cerned. However, an increasing number are recognizing the utility of obtain-
ing information that can be used to develop a production surface rather than only
an occasional point on it.

4. I must confess that many of us have often looked at the returns per dol-
lar spent over the entire amount of fertilizer applied rather than at the marginal
return for successive increments of fertilizer. I am convinced, however, that
this is less true now than it was a few years ago.

5. We should recognize that the specialist has many prejudices as he search-
es for the optima. For example, many fertilizer experiments have been con-
ducted with a two-element factorial even including a wide range of rates, and
the results have been misleading. If such an experiment is conducted on a soil
that proves to be somewhatdeficient in a third nutrient, or if another factor such
as stand enters in to influence the results, we have only a partial answer. Con-
sequently, we need to be humble as we approach this goal of the optima.

Several years ago, I had the opportunity of working with representatives
from agricultural economics and all of the production departments in making
the Iowa production capacity study. Each small subcommittee had the job of
determining the effects of the selected practices on the potential yield of corn

and other crops. Our goal was to estimate the average yields and production
obtainable by farmers in the top 2 or 3 per cent of managerial ability. When
the increases estimated by the individual specialized groups, including rota-
tions, fertilizers, lime, insect control, weed control, varieties, etc., were
all added together we arrived at an average yield for the poorest soil area of
the state of over 115 bushels per acre. This was approximately 25 bushels per
acre more than the highest average farmer yield known on the most productive
soil in the state. It became obvious that each of us was confounding certain in-
teraction effects of a combination of practices and letting our biased viewpoint
overestimate the real effects. It seems to me that we must continually guard
against letting our biases and prejudices sway us as we search for the optima.

6. Continual recognition must be give to the large number of variables
that ultimately affect the nature of the production function or production sur-
face. No matter how you group them, there is a large number of such factors.
Most research attempts to hold constant all but a select few and vary those.
few. In the past several years factorial experiments have been used to facili-
tate the study of two or more variables. However, the utility of the factorial
design in studying several variables over a considerable range has limitations.
The new rotatable designs offer possibilities that need more study to further
extend our knowledge of production surfaces obtained where several variables
are studied simultaneously over a rather wide range.

7. In developing concepts of optimum use of fertilizers we need to give
some consideration to how the results will be extended from one or several ex-
periments to the innumberable situations that are encountered under practical
farm situations We can draw conclusions and develop specific recommenda-
tions for the optimum use of fertilizer for a given experimental site but still
not solve the larger problem. It seems to me that we should keep clearly in
mind that we are interested in solutions that have predictive value that can be
applied to the farm situation. This may affect the entire approach and the type
of accuracy needed or desired in the interpretation of individual experiments.

If the results of fertilizer experiments are to be most useful to farmers,
it seems to me that we must have one or more methods of comparing the far-
mer fields with the experimental results obtained on specific fertilizer exper-
iments or groups of experiments. Most states are attempting to use a com-
bination of soil tests, soil type, crops grown, crop yields, and management
practices to compare the results on experimental sites with farm fields in pre-
dicting anticipated response to fertilizers. Others are using various types of
plant-tissue analyses as a similar tool for comparing the unknown with a known
situation. Personally, I don't expect our predictive value ever to be perfect.
I like to think of it in terms of an estimation of the odds that a farmer has of
achieving certain results under a given situation.

8. In our discussions of this problem all of us tend to assume that the far-
mer makes his decisions on a completely rational, logical basis strictly with-
in an economic framework. This assumption probably is satisfactory for this
particular conference. However, I am sure we all appreciate that decisions
on actual change of practices such as fertilizer use involve emotions, social
values, personal values and status as well as economic considerations.


Bert A. Krantz1

I welcome this opportunity to meet with your group as I am convinced that
much greater progress can be accomplished by closer cooperation between
workers in the fields of soils and agricultural economics. To be the most ef-
fective, this cooperation should start in the planning stage. Too often, we in
soils have been guilty of waiting until we have a particular project finished and
then we get together with the economist to see what economic analysis can be
worked out. I think a meeting like this is a good start in the right direction.
In fact, it is a great satisfaction to me to see that economists are becoming in-
terested in fertilizer research. Less than a decade ago, when I was stationed
at the North Carolina Experiment Station doing fertilizer and soil management
research with corn, this interest was found to be practically nonexistent in the

The subject assigned me is so broad that the best I can hope to do is to hit
a few of the highlights. In an effort to cover the fertilizer research in all areas
of the West, I wrote to all the heads of soils or agronomy departments in the
eleven Western states, and their response was excellent. Some states sent me
enough material for a whole conference on their material alone, so my job was
to try to condense it.

Fertilizer use is relatively new in much of the West. There was a time
when it was thought that the application of water was all that was needed to grow
bumper crops. Experience has taught us otherwise, and fertilizer consumption
is increasing much faster in the West than in the country as a whole.

Crop yields have been greatly improved by fertilizer application when used
with other sound soil and crop management practices. Nitrogen is the most
universally deficient fertilizer element in the irrigated West. This is not sur-
prising when one considers that much of the irrigation takes place on desert
soils where vegetation is sparce and organic matter content low. Under irriga-
tion nitrogen fertilizers will usually give economic responses on nonleguminous
crops that are not planted immediately following a legume such as alfalfa. Phos-
phorus is the second most universally deficient element. Response to phospho-
rus, however, is by no means consistent. In some fields, crops will respond
markedly to phosphorus application while not at all in others. This variability
is often related to past fertilization history.

Deficiency of zinc and other trace elements is found in certain soil areas
and is often accentuated where deep-cut leveling is done on irrigated lands.
Where deficiencies occur, the application of these trace elements is recom-
mended in adequate amounts, since the cost of treatment is usually rather low.
Deficiencies of sulphur and potassium are also found in limited areas. How-
ever, since these deficiencies are of local and not general importance in the
West, I will confine most of my discussion to nitrogen and phosphorus fertili-
zation work.

Soil Scientist, Western Soil and Water Management Section, Soil and
Water Conservation Research Branch, Agricultural Research Service, U,S.
D.A., Billings, Montana.

Response Curves Influenced by Many Factors

Numerous experiments have been conducted in the West to try to estab-
lish the response curves of various crops to fertilizer elements. This is not
a simple task when dealing with dynamic biological systems. The response
curves of a crop to given nutrients vary with soils, climate, management, and
many other factors. Since Dan Aldrich gave a thorough treatment of this sub-
ject this morning, I will merely list a few of the factors which influence the
shape of response curves such as:

1. The nutrient supply power of the soil.
2. The inherited capability of the variety used.
3. The yield potential of the area, length of growing season, etc.
4. The relationship to soil-moisture level.
5. The quality factor such as sugar tonnage compared with total plant
tonnage or crude protein compared with forage yields.
6. Climatic factors and other environmental factors.

In spite of this, soil and plant scientists need this information so they at-
tempt to cont rol or characterize some of these variables while studying others.

Examples of fertilizer-rate studies on the major crops of the area were
found in reviewing the work of each Western state. No attempt will be made to
summarize these experiments in this discussion; however, they would in most
cases afford good opportunities for cooperative studies with economists.

In many cases other physical or management factors must be studied and
corrected before response to fertilizer is obtained even though a nutrient de-
ficiency exists. A good example of this is a recently expanded research pro-
gram on mountain meadows which was started about five years ago. Since
many of these meadows remain wet by either high water table or continuous
"wild flooding" irrigation methods, it was found that water control was the
No. 1 problem. After water-control methods were established, it was then
possible to get greatly increased yields by fertilization. However, Willhite
(4) found that in order to reap the maximum benefit from fertilization, he
had to change from the standard one-cut harvest system to a two-cut hay har-
vest system. After these other production factors were adjusted the 1952 hay
yields were increased from 3.3 tons without nitrogen to 5.4 tons in plots re-
ceiviig 480 pounds of nitrogen. .The per acre crude protein yields were pro-
gressively increased from 910 pounds with no nitrogen up to 2300 pounds with
the 960-pound nitrogen application.

In recent years J. L. Paschal of Production Economic Branch, USDA, has
made economic analysis of data from fertilizer rate experiments and has calcu-
lated the most profitable rate of application under these conditions. This work
has been done on several crops, and the results on alfalfa (1), corn (3), and
-sorghum (2) have recently been published. I will not discuss the details of
these studies since several other papers are concerned with this type of inves-

The most profitable rate-of-application concept has added to the usefulness
of these results and this type of analysis should become more pertinent in fu-
ture experiments which are planned cooperatively by economists and agrono-

Phosphorus Fertilization in the West

As mentioned above, response to phosphorus application is rather erratic.
When soils were first placed under cultivation, their phosphorus content was
related to the parent material and soil formation processes, and the expected
response could often be related to a specific soil type. However, since soils
have been put under cultivation, the phosphorus status has changed consider-
ably due to depletion by cropping and additions by manures and fertilizers.
The rate of phosphorus application has varied greatly by crops. In the case of
certain high-value crops such as vegetables and potatoes, the rate of applica-
tion has been far in excess of needs, while other crops have been inadequately
supplied. Thus it becomes apparent that the expected response to phosphorus
will vary greatly depending upon previous management which varies from field
to field.

In an experiment at Tucumcari, New Mexico, R. W. Leamer obtained
marked increases in alfalfa yields from phosphorus applications up to 480
pounds P205 per acre. In this experiment all of the phosphorus was applied
broadcast in 1951 at planting time. During the next four years, the phospho-
rus uptake in the increased hay yields and higher phosphorus content was equi-
valent to all of the 60- and 120-pound P205 app locations. In the case of the 240-
and 480-pound applications, the phosphorus recovery was 86 and 63%, respec-
tively. These extremely high recovery figures indicate excellent efficiency of
applied phosphorus and little, if any, phosphorus fixation under these soil con-

TheTucumcari experiment is a part of a regional phosphorus study that
was started in 1951. At the other three locations, the responses were less
striking, but relatively high residual effects were noted.

The residual value of applied phosphorus has been observed in other ex-
periments and in farmers' fields. These scattered observations indicate that
phosphorus fixation in certain Western soils is not the problem we once thought
it to be. However, further work on this phase is needed to clarify the whole
picture on the major Western soils.

In soils where high residual effects exist, the problem of phosphorus fer-
tilization can be considered from the standpoint of fertilizing the rotation with
large applications to the most responsive crops rather than small applications
to each crop in the rotation. Because of the carry-over effect, it seems logical
to operate at a relatively high level of phosphorus fertility in the soil rather
than to strive for maximum efficiency of small annual applications of phospho-
rus. A known exception to this situation is found in the acidic soils of the Pa-
cific Coast where phosphorus fixation is an important problem.

The development of the sodium bicarbonate method by Sterling Olson of
ARS shows considerable promise for estimating available phosphorus content in
Western soils. Further experimentation is necessary to adequately calibrate
this test on inherently different soils. However, as this type of information is
accumulated, it should be extremely helpful in determining the economic phos-
phorus-fertilization program for various field conditions.

Nitrogen Fertilization in the West

The situation with nitrogen fertilization is quite different. There is less
residual effect of applied nitrogen since in addition to crop removal it is

subject to tie up by plant residues and to losses from leaching, erosion, deni-
trification, and volatilization. The last two of these losses may occur evenunder
the best-known soil-management conditions, and the fundamental principles in-
volved in these losses are not completely understood. Also, we do not have ade-
quate analytical methods for assessing the available soil nitrogen or the nitrogen-
supplying power of a soil.
As I looked over the material which I received from the various states, I
was impressed with the large number of annual nitrogen-rate experiments which
are being conducted often with no investigation of the nitrogen level or nitrogen-
supplying power of the soil. We know that even within any one climatic region,
soils differ greatly in their native nitrogen-supplying power, and this is further
complicated by induced differences due to previous fertilization and cropping his-
tory. To add to this dilemma, the nitrogen-feeding power of plants varies con-
siderably and is altered by soil and environmental factors. With all of this vari-
ability one can rightly question the advisability of the large number of the annual
nitrogen-rate experiments. As I mentioned earlier, fertilizer use in the West
is relatively new, and in many areas there is a need for building up a background
of the general fertilizer needs of the various crops under local conditions. The
most convenient way to do this is by annual rate studies.
It seems to me that in many areas we have obtained this needed background,
and it is time that we shift our emphasis to a more basic approach. One means
of accomplishing this would be by studying some of these problems on a more
regional basis and in this manner try to understand some of the wide differences
which do exist in the response from field to field and area to area. This would
require a rather complete study of the available nitrogen supply, the nitrogen-
supplying power of the soils and a study of behavior of both the indigenous and
applied nitrogen. Some progress has been made along these lines--a good ex-
ample is the work of Dr. A. H. Hunter and his associates in Oregon, and Dr.
H. M. Reisenauer and others in Washington who are conducting extensive studies
on wheat to establish, on a more quantative basis, the relationship between soil
nitrogen, soil moisture, and wheat yield.
As mentioned earlier, nitrogen is the most universally deficient fertilizer
element in the West. However, the response differs greatly when management
factors such as soil moisture, variety, and plant population are varied. The
soil moisture by fertilizer interaction is extremely important in the West under
both the dryland and irrigated conditions. These interactions should receive
more attention by both the economist and the soil scientist in multivariable ex-
As we get a better understanding of more of these factors and their inter-
actions, we need to study the integration of all production factors for maximum
economic crop production and sustained soil productivity. To be most effective,
this should be done on a large-scale or pilot-plant" basis. Here, again the
economists and soil scientists should coordinate their efforts.
I recently attended an SCS Irrigation Guides Conference and was very much
impressed with all of the factors that the irrigation engineers are using to deter-
mine the total and peak water needs to design the best irrigation system for any
particular farm. Admittedly, many of the factors which they use such as con-
sumptive use, depth of rooting, and water-holding capacities of various soils
are based on assumptions; however, by putting all of these factors into a for-
mula, they are able to arrive at something much better than a guess. In other
words, they are making use of the best known information. It appears to me
that we in soils have fallen far short of this in providing information for the


planning of the best fertilization or soil-management program for specific farm

Last Monday, I heard a newspaperman give his predictions for 1956. In
this he mentioned the trend toward large farming enterprises and predicted that
this trend would increase and our approach to farming would be more like that
of a businessman. With this trend, I can visualize that there will be a greater
call for more detailed information on the economics of fertilizer application as
these large farmers will approach their problems with sharp pencils in an ana-
lytical manner.

In this short time I have merely been able to give a sketchy review of the
fertilization research in the West. I realize that I have left many important
phases untouched but hope that I have brought out a few points which will be help-
ful for discussion purposes and help give you an appreciation of the importance
of proper fertilization to Western agriculture.


1. Paschal, J. L. Economic Analysis of Alfalfa Yield Response to
Phosphate Fertilizer at Three Locations in the West.

U.S. Bur. Agr. Econ. F. M. 104. 1953.

2. Paschal, J. L. and Evans, C. E. Economic Interpretation of
Yield Data from a Nitrogen Rate Experiment with Irrigated Grain

Soil Sci. Soc. Amer. Proc. 18: 454-458. 1954.

3. Paschal, J. L. and French, B. L. A Method of Economic Analy-
sis Applied to Nitrogen Fertilizer Rate Experiments on Irrigated

USDA Tech. Bul. 1141, 1956.

4. Willhite, F. M., Rouse, H. K. and Miller, D. E. High Altitude
Meadows in Colorado: III. The Effect of Nitrogen Fertilization on
Crude Protein Production.

Agron. Jour. 47: 117-121. 1955.


Glenn Johnson1

Since World War II, cooperation between agronomists and economists in
the design and interpretation of fertilization experiments has grown rapidly.
Such studies are under way at a number of experiment stations including Iowa,
Kentucky, Michigan, North Carolina, and Virginia. In Canada, rather exten-
sive experiments are under way at Guelph. These evidences of cooperation on
the part of agronomists makes it inappropriate to continue the protestations
long made by economists, including myself, that the design of agronomic ex-
periments does not permit economic interpretation of experimental results.
The plain truth is that, in many states, economists no longer have anything to
complain about. I, for one, greatly appreciate the cooperation I have received
from agronomists. The discussions of Drs. Aldrich and Cheney here indicate
that much cooperation can be expected here in the West. However, in case any
of you feel the need for references in neg otiating cooperative research with
agronomists, I would recommend the Heady-Schrader and Redman-Allen ar-
ticles which have appeared recently in the journals Another fine reference
is Robert Hutton' s TVA pamphlet entitled An Appraisal c: Research on the
Economics of Fertilization. ..The forthcoming proceedings of last spring's TVA
symposium on the economics of fertilization should contain some useful materi-
al, especially with respect to design, interdisciplinary negotiations, and the
considered points of view of economists, statisticians, and agronomists some-
what accustomed to working together. All of these are rather open minded"
(yet meaningful), mature expositions which discuss the need and basis for co-
operative work between economists and agronomists.

For the remainder of this paper, I am going to assume that members of
the Western Agricultural Economics Research Committee are more interested
in an evaluation of the results of the cooperative work now getting under way
than they are in another harangue about the neglect of economics in the design
of fertilization experiments.

Criterion for Evaluation

The hope in initiating these cooperative experiments was that they would
better help farmers find the most profitable combinations and amounts of fer-
tilizer to apply in producing different crops under varying price conditions.

Department of Agricultural Economics, Michigan State University, East
Lansing, Michigan.

Earl O. Heady and W. D. Schrader, The Interrelationships of Agronomy
and Economics in Research and Recommendations to Farmers,"" The Agronomy
Journal, October 1953; John C. Redman and Stephen. Allen, Some Interre-
lationships of Economic and Agronomic Concepts," Journal of Farm Economics,
August. 1954, p. 453.
Robert F. Hutton, An Appraisal of Research on the Economics of Fer-
tilizer Use, Report No. T 55-1, Tennessee Valley Authority, Division of Agri-
cultural Relations, Agricultural Economics Branch.

Some of the current investigations also have methodological objectives which,
of course, are instrumental or intermediate ends in attaining the objective of
helping farmers.

A "critical evaluation" such as I have been asked to make requires a cri-
terion or standard. My'criterion for evaluating this research will be useful-
ness to farmers either directly or ultimately through further research appli-
cation. In making fertilization decisions, a farmer faces a set of conditions
which he cannot change such as soil type, past treatment of his soil, water
supply, topography, etc. He also knows that even after he has done his best
to control those elements in the situation which are not fixed, that many chang-
es and events will occur to cause fertilizer responses to be higher or lower than
he anticipates. What the farmer in such a situation could use most advantage-
ously is information on how to adjust those things he can control which applies
in (1) a fixed situation exactly similar to the one he faces and (2) a situation
subject to the same, or perhaps to. more, disturbing influences which cause
yields to vary from expectations.

In more specific terms, the farmer faces the problem of adjusting X,
....X in the following production function so as to maximize the profits he
makesgfrom producing Y.

Y= f(X1, ..... Xg/Xg + .... Xn) +u

Y = yield of a large land area, perhaps seldom less than 10 acres in size.
X, ...., X are the controllable elements in the situation he faces.
X ...,X are the fixed elements in his situation, and
u stands for variations in yield caused by variations in elements or con-
ditions in the situation other than X .... ,X The causes
of u may include variations of certain of the X .... from
the levels at which then are supposedly fixed

Obviously, if the fixed elements in an experimental situation vary substan-
tially from those on a farmer's farm, the results of the experiment are not ap-
plicable. Functions estimated from experimental data produced with certain of
the X ,....., X fixed at different levels than at which they are fixed on the
farm o+f Uarmers attempting to use the estimated functions can be viewed as
either (1) a different production function than exists on these farms, or, (2) more
generally, as a different subfunction from the subfunction on the farms, both of
which are from a more general function which treats the troublesome "fixed"
elements as variable. Regardless of which way the situation is viewed, the ex-
perimental production function is not useful to a farmer, if based on different
fixed conditions than he faces.

The source of variance in experimental yields should at least be as great
as those on the farm. Why this is true is not obvious at first thought. The
reason is found in the decision-making process. A farmer likes to know how
much reliance he can place in experimental information applied on his farm.
If causes of variation are controlled in the experiment which he cannot control,
he will not know how much reliance he can place in the results. He will say,
"Sure, it works that way at the college, but it won't work that way for me on my
farm." And, he is right. Though, ideally, a farmer would like the experimen-
tal results subjected to the same number and amounts of disturbing elements

encountered on his farm, he would not object strenuously if the experimental
results were subject to more disturbing elements than he faces provided they
are still acceptably accurate.

The typical experiment designed jointly by economists and agronomists.
Now that we have looked very briefly at what a farmer logically wants or needs,
let us look at our experiments. The typical experiment, recently designed by
agronomists and economists, investigates a production function about as follows:

Y = f(N, P, K, /X, .... Xn) +u


Y is yield expressed on a per acre bsis but measured from 1/50th
1/125 or perhaps only 1/250 of an acre.
N, P20 and K20 are the common independent variables. In some in-
stances only two of these three have been varied, the other being held
fixed. In other instances, subportions of the experiment investigate
fertilizer placement, side-dressing, or some other cultural practice
as a third or fourth independent variable.
X, .... ,X commonly include the details by which soils are classed and
typed; the recommended practices with respect to cultivation, ferti-
lizer placement, insect controls, rotations, p. h. level, disease con-
trols, and varieties; uniform drainage; minimum slope; near maximum
soil uniformity; experimental harvesting methods, etc. Field and plot
sizes are typically small, and the degree of control over elements which
vary locationwise is typically much tighter than a. farmer could maintain.
As most data available from such experiments are for one year only,
weather is, for practical purposes, also a controlled variable.
u is the variation in yield not associated with the experimental variations in
N, P, and K. This variation is caused by uncontrolled elements in the
situation which vary from plot to plot such as slope, past soil history,
mistakes of researchers, drainage, depth of topsoil, insect and disease
infections, etc. Undoubtedly, many of these elements are actually de-
viationsof the X ....,X from levels at which they are supposedly
controlled. Over a period of years, weather is a major source of un-
controlled variation.

Evaluation of Typical Experiments

In evaluating these experiments, each of the following will be examined in
order: (1) the independent variables which are studied, (2) the fixed elements
in the experiments which a farmer could control, (3) the fixed elements in the
situation which a farmer cahnot control, (4) the unexplained deviation about these
functions, and (5) some miscellaneous problems,

The independent variables in the experiments. Generally speaking, the list
of independent variables in these experiments is not long enough to permit solu-
tion of a high proportion of the fertilization problems pointed out by Dr. Aldrich
this morning. Pesek's and Heady's Iowa study treated N and PZ0 as variable.
At Michigan we have treated N, P205, K20, and, perhaps, one cultural practice
such as fertilizer placement as variable. Thus, there are many important fer-
tilizer problems we are not studying. This, however, is not a serious criticism.
Science must limit itself to investigations which can be handled. A little bit is
done at a time--later the results are put together. All we need do at this point
is recognize that there are important problems (involving variables we have

fixed) which we are not answering. These are often technical problems involv-
ing very little economic analysis which are nonetheless of first importance to
farm managers. They can often be handled advantageously by purely agronomic

Fixed elements which the farmer can control. A large proportion of the
fixed elements in an experimental situation are ordinarily controllable by the
farmer. Many of these elements are somewhat complementary while still others
tend to represent levels of technology more than inputs. The complementary
elements are easily fixed in" proper proportions" and generally are. In most
instances, experiment designers appear to have done a rather good job of se-
lecting levels at which to fix such elements; i. e., recommended practices have
been followed which tend to be both desirable (from economic and agronomic
viewpoints) and attainable by the typical commercial farmer.

Fixed factors which the farmer cannot control. Experimental work today
appears somewhat less commendable when one compares the levels at which
elements the farmer cannot control exist in experiments with the levels at which
these same elements exist on the farms of our clients.

Almost any experimental site has certain unique characteristics which limit
the number of farms to which results secured from it are applicable. This is
partially a result of the small experimental fields commonly used and partially
a result of the attempts made in selecting such sites to minimize within-field
variability. If variability were minimized around levels near the average for
large numbers of farms, minimization of variability would be a desirable thing.
This, however, is not always the case. For example, most sites are more level
than the fields of farmers who might use the results. Also, when a very pro-
ductive soil is being studied, experimental sites are likely to "average" some-
what higher" with respect to soil type than are the less uniform large fields
of farmers which often have areas of less productive soil types intermingled
with the dominant productive soils. When soils of lower productivity are studied,
the reverse is likely to be true, i. e., a farmer' s field of predominately the
same soil is likely to average" more productive than the experimental plot.
The use of larger experimental fields which encompass within themselves
more of the variability assumed under a soil type, it appears, would tend to in-
sure that average levels of the supposedly fixed elements correspond in experi-
mental and applied situation. However, a number of agronomists including Pesek
from Iowa State College argue quite the opposite. Large fields, they feel, are
more likely to include variations in soil or past treatment of the soil thus intro-
ducing additional variance in the resulting yield data. The correctness of Pesek's
facts is attested by tests of significance applied to yields from different sectors
of certain Michigan experimental fields. In several instances, plots in third-
and quarter-sections of experimental fields had average yields significantly dif-
ferent than in other third- or quarter-sections despite the random distribution
of treatments over the entire field We also found significantly different aver-
ages on two fields supposedly identical enough to be in the same rotation experi-
ment. The correctness of Pesek's facts about variance does not necessarily
lead to the conclusion that small experimental fields are desirable. Confining
experimental observations to small areas may cause the resultant experimental
functions to be so unique as to unduly limit their applicability. From the stand-
point of a scientist wanting results for rigorously defined situations, Pesek' s
position is correct, but from the farmer's standpoint his position appears to un-
duly limit the applicability of results. I believe, for instance, that there are
relatively few 10-, 20-, or 40-acre fields in Michigan which have the "average"
soil type which exists on any of our six current or eight contemplated experi-
ments. If, instead, our experiments could "average out" a range of differences

within soil types more comparable with the ranges which exist in the fields of
farmers, our results would have a wider range of applicability.

This is not a fatal criticism of recent cooperative work between agronomists
and economists. It is, however, important enough to consider in designing fu-
ture research. As it is not a problem on which there is agreement, consider-
able discussion of it needs to take place among agronomists, statisticians, econ-
omists,. and farm managers. I include "farm managers" here because the pro-
blem of accuracy is present. Accuracy is, in a sense, a noneconomic considera-
tion in decision-making which is more likely to be understood by an interdisci-
plinary than by a purely economic approach to management.

Evaluation with respect to variance. Very large between-plot variances,
not associated with the independent variables, are characteristic of all agro-
nomic experiments. The recent cooperatively designed experiments are no
exceptions. Even the best of the functions secured to date, the Pesek-Heady
production surface for corn has large unexplained deviations. An examination
of their scatter diagrams published in the Journal of Farm Economics would
convince most entrepreneurs about to invest money that less variation would be
required in on-the-farm results before fine, year-to-year marginal adjustments
are worth worrying about. From the scientific viewpoint the unexplained devia-
tions from the Pesek-Heady function are so large that rather serious questions
exist concerning (1) the appropriate mathematical function to fit, and, hence,
(2) the accuracy of optima located on the selected functions.

Data produced by pasture fertilization experiments when I was at Kentucky
exhibited such wide variances that no reliable functions could be estimated from
them. Mylastconversation with Harold Jensen concerning subsequent corn ex-
periments at Kentucky indicated that the variances being encountered are such
thatit is difficultto.secure significant coefficients. A similar situation exists in
the potato fertilization data being secured in Ontario by Phil Wright.

P. R. Johnson' s North Carolina corn data, while capable of yielding statis-
tically significant coefficients for several functions, contain so much between-
plot variance that clear-cut choices between alternative functions could not be
made. In this case, extensive replication of points yielded significant coeffi-
cients but, of course, did not reduce the average unexplained deviation. This
left the number of different functions yielding statistically significant results
much too numerous to attach economic significance to fine marginal adjustments
on any one of them. Cliff Hildreth's technique of estimating points on a concave
production surface avoided but did not solve or eliminate the problem of select-
ing an appropriate function.

This last year, a three-variable, red clover experiment with replication
somewhat similar to that carried out by Pesek and Heady was conducted in
Michigan. Rebsulti:- great variance with no evident response.to a single nutrient
or combination of nutrients. The same was true of two similar experiments
with oats. In the case of corn data, from a rotation Michigan experiment, sur-
faces are evident but choices among alternative functions are extremely diffi-
cult. Bad weather and timing spoiled a more extensive N, P, K experiment
designed to investigate the feasibility of continuous corn on one of Michigan's
richest corn soils. This year' s potato experiment at Michigan produced data
showing some evidence of a P, K potato production function. Last year' s corres-
ponding potato data were characterized mainly by variance--in fact, little else
was evident in the data.

Before examining the consequences of these large variances, it seems ad-
vantageous, from the standpoint of fairly evaluating this new type of research,
to note again that such variances are characteristic of yield data from plots of
sizes commonly employed in agronomic work. The standard deviations of yields
from replicated plots at one point on a production surface seem to compare rath-
er closely with the standard errors of 1, 2, and 3 variable production functions.

The customary desire of agronomists to replicate extensively attests their
long acquaintanceship with these large variances. As an aside, I would like to
point out that a large proportion of my own professional work has dealt with
the kind of data one encounters in price, in agricultural policy, and, more re-
cently, in utility analysis. Difficult as the measurement and quantification pro-
blems are in these areas of analysis, I was hardly prepared for the variances
I encountered on my first excursions into the "no man's" land between econo-
mics and agronomy. The social sciences seem to have no monopoly on supply
of in-exactness."

After this digression, let us investigate the consequences of large variances.

What are the consequences for the farmer? Earlier in this paper we saw
that farmers should object more to over- than to under-control in experiments
of the causes of unexplained variance, provided accurate estimates of the func-
tions can be secured. The undesirable consequence is that variance often pre-
vents, as we have already seen, construction of reliable estimates of the rele-
vant production functions.

Small experimental plots appear to be an important cause of between-plot
variance not associated with fertilizer treatment. For example, the average
per acre yield of 1/5 acre plots has an expected standard deviation only
/ T- as large as the expected standard deviation of theper acre yields of
the ten 1/50 acre plots in the same area.

As farmers probably make more fertilizer decisions with respect to areas
ten acres or larger rather than smaller in size, the standard deviation of their
expected yield from a given treatment is, very fortunately, much lower than
the standard errors of estimate of our experimental production functions which
are based on plots only a small fraction of an acre in size. The fact that far-
mers should not expect to experience the standard deviations we experience in
experimental work is comforting, indeed. It means that our work may be use-
ful, after all--provided we can secure reliable estimates of sufficiently general
functions to serve an important number of farmers.

The above discussion suggests that we should use larger plots in our ex-
perimental work. Why, for instance, should we not harvest plots nine times as
big a, we are now harvesting thereby reducing our between-plot variance by
] V7 or 1/3 r The benefits would be very substantial, i.e. it would be
easier for researchers to select and fit appropriate functions. The farmer,
while not objecting to more causes of variance in experimental results than he
himself experiences, could not logically object to "averaging out" causes of
variance which he (the farmer) is in even better position to average out. Nor
should he object to losing some information about between-plot variance unless,
of course, he is fertilizing a mixed garden

Some researchers may point out that consolidating, say nine adjacent plots
into one plot, loses the advantages secured from scattering these nine plots
randomly over the entire experimental area. This is a valid point if the causes

of variance are not uniformly distributed over the field. If this situation exists,
as it probably does, it suggests that both between-plot variance and representa-i
tiveness could be increased by increasing plot size and field size, i.e., by sam-
pling larger plots over a wider geographic area. In fact, it would probably be
desirable to sample, not a field, but the entire geographic area to which the ex-
perimental results are going to be applied. In carrying out this suggestion, the
total number of plots could probably be reduced somewhat thereby offsetting the
increased costs of a larger plot. As labor, not land, is the expensive item of
experimental costs, total costs should not be increased greatly by adopting this

One way of handling the causes of unexplained variances is to measure them
and include them in the study as independent variables. Most studies contain
provisions for running P205 and K20 tests for ;each plot with this procedure in
mind. Three difficulties encountered to date in this connection are probably
solvable but not well handled as yet. They are: (1) the difficulty of measuring
N at all, (2) the apparent unwillingness of agronomists to attach very much sci-
entific significance to any soil test, and (3) the problem of specifying functional
relationships to take into account both applied and residual nutrients in the soil.

Miscellaneous problems. Under this heading I would like to discuss, among
other things, a tendency which has developed in the evaluation of production
function research. In two fairly recent articles, differences in net profits were
compared at different optima resulting from greatly different price relation-
ships. As net revenue was not greatly different at these points, the investi-
gators implied that the research necessary to estimate the production function
was not worthwhile.

On production functions exhibiting fairly constant returns to scale and/or
a high degree of substitutability among inputs, wide price variations may call
for great changes in scale and/or input combinations but bring about only small
changes in net profit. To agree that the discovery of areas of near-constant
returns to scale or of high degrees of substitutability if of no economic conse-
quence is absurd. Yet, critics observe theconsequences of such conditions and
draw the same conclusions.

Also under the miscellaneous category, we should look at the problem of
built-in, year -to-year changes in the conditions which are fixed for any one
year. Fertility residuals build up from year to year. Thus, the three problems
previously mentioned in connection with soil tests must be solved in order to
combine data from different years in order to average out the influence of

Another problem which results from building up fertility levels is that plots
with unreasonably high residual fertility levels lose their usefulness--the fields
are, in a sense, self-destructive.

R. E. Hutton, and D. W. Thorne, "Review Notes on the Heady-Pesek
Fertilizer Production Surface," Journal of Farm Economics, Vol. 37, Febru-
ary 1955.. p. 118. Emil Ranchenstein, Forage-Grain Substitution; Its im-
portance in the Economics of Milk Production," Journal of Farm Economics,
Vol. 35, p. 562.



1. It is time to quit complaining about lack of cooperation between agronomists
and economists.
2. Instead, it is time to start evaluating the cooperative work now under way.
3. Cooperative experiments now under way appear to be based on suitable fixed
levels of those factors of production which farmers can control.
4. However, current experiments appear to be based on inappropriate levels
of some factors of production which the farmers cannot control. These
factors are mainly those which vary within soil types and classes such as
slope, cropping history, drainage, and mixtures of different types of soils.
5. The failure (noted in 4) of fixed conditions on experimental sites to corres-
pond to fixed conditions on a greater number of farm fields is probably due,
in part, to small experimental fields.
6. Data produced by current experiments are characterized by excessively
large variances not associated with variations in fertilizer treatments.
7. The excessive variances (noted in 6) are probably due, in part, to the small
size of plots commonly harvested.
8. The range of farm situations to which estimated production functions apply
and the accuracy with which such functions can be estimated could probably
be increased by:
a. Increasing the size of plots now harvested, and
b. Extending the area sampled so as to average out" more of the condi-
tions which vary from one small area to another to something more
nearly approximating the conditions found in a farmer's field. It may
even be advantageous to move away from experiment farms and fields
to samples of the entire geographic area to which the results will be
9. Production function research cannot be fairly evaluated in terms of the pro-
fitability of adjustments to optima--it is of as much economic significance
to find that wide changes in combinations and amounts of inputs can be made
without affecting profits as it is to find that such changes would affect pro-
10. Work on methods of measuring and analyzing the production of initial and re-
sidual fertility nutrients is urgently needed.


F. G. Viets, Jr. 1

I. What Is Our Goal?

It is assumed from the disciplines represented at this conference that the
soil scientist and agronomist have a responsibility toward providing suitable
physical data for an economic analysis of the costs and returns resulting from
various fertilization practices. It is the purpose of this paper to discuss from
the standpoint of the soil scientist some of the factors which must be kept in
mind in establishing the objectives of fertilizer experiments. In general, my
remarks will be confined to areas of inference, some aspects of the problems
of variability in physical data, factors which may cause biases in experiments
which appear to be well designed, size and shape of plots, and toa consideration
of properties of individual fertilizer elements and their reactions with soils in
so far as they affect experimental objectives. I will leave to Dr. Ostle, who
succeeds me on the program, the specific problems of experimental design once
the general objectives have been set.

In asking the question: "What is our goal?" I have in mind how closely do
we expect to pin down this problem of fertilizer-application rates. I do not
know the answer, but I believe that it can be more closely pinpointed by asking
a series of rather simple questions some of which are not easy to answer.

The first question: Is it sufficient to be able to make a fertilizer recom-
mendation for a crop irrespective of where or how it is grown? The answer
to this, of course, is "No." It is a,simple fact that about 300 lbs. of nitrogen
are required in the tops of corn producing 200 bushels per acre, whereas when
the yield potential is only 20 bushels per acre only about 30 lbs. of nitrogen are
required. Since yield potentials differ vastly because of factors in the physical
environment, and because the nutrient-supplying capacity of soils differ, there
can be no single answer.

The next question which we might ask in narrowing our problem further is:
Is a single fertilizer recommendation for a crop sufficient when that crop is
grown on soils of similar morphological conditions? The soil surveyor classi-
fies soils on the basis of what he can detect with two senses--sight and touch.
From this information he classifies soils into series and types. He can go fur-
ther and group soil types into families of similar individuals such that they
might behave similarly under prescribed soil-management conditions. Regard-
less of how good such delineations and groupings may be for some purposes,
they are of only limited value in forecasting fertilizer requirements. Most
soils with respect to most fertilizer elements show considerable hysteresis;
i. e., they do not return to some standard state the following season. The

Soil and Water Conservation Research Branch, Agricultural Research
Service, USDA, Fort Collins, Colorado.

amount of crop residues, the amount of fertilizer applied, the presence or ab-
sence of nitrogen-fixing legumes in the rotation--all affect the response which
may be expected to the fertilizer applications of the next season. Thus it is
not sufficient to merely say that if you are going to grow corn, and if the soil
is Podunk silt loam, you should use 80 lbs. of nitrogen.

However, at the present time in the West we have not proceeded very far
beyond this point. For example, in the fertilizer recommendations for non-
legumes on irrigated soils in most states of the West at present we do not go
beyond recognizing two management conditions of that soil. One is where leg-
umes were not grown in the proceeding season and application of barnyard ma-
ure was not made. The other is where the crop was preceded by heavy applica-
tions of manure or luxuriant growth of a nitrogen-fixing legume.

The two proceeding questions have not narrowed our problem or our area
of inference sufficiently so that we can arrive at an answer which is even close.
Is our goal a fertilizer recommendation based on some test that will adequately
characterize the ability of the soil to supply the nutrient in question? I believe
that herein lies our hope for the future, but it is a hope based on ignorance of
all of the difficulties which may be encountered in adequately standardizing or
calibrating soil tests. In the West we probably have not made the progress on
this score that has been made in the Central states on phosphate and potash.
In the West we are further along in having good calibrations of phosphate tests
than we are for any other element, but progress is slow. With respect to ni-
trogen, our most important element from the standpoint of need and quantity
required, we are still in the dark ages. I believe that our goal should be to
base our fertilizer recommendations on adequate site characterization through
well-calibrated soil tests. I believe I can understand how miffed a farmer must
be when he asks about a fertilizer recommendation and gets the answer: "Use
50 to 100 lbs. depending upon conditions." What conditions? If you ask that
same farmer how much his truck weighs, or how much gas its tank will hold,
he can probably give you a rather accurate answer.

Thus far in our discussion we have presumed that every year had the same
kind of climate. We know that this is not true. We have wet years, dry years,
hail, cool seasons, hot seasons, etc. In the Plains states we know that we can
get wheat to respond to nitrogen during years of favorable moisture. The same
amount of nitrogen applied in a year which turns out to be dry may actually cut
yields. How can we get a very precise answer to the question of fertilizer rates
under these conditions ?

In the wheat-growing areas of the Pacific Northwest gearing the rate of
nitrogen application to the stored and anticipated soil moisture and to available
nitrogen in the soil appears to be a very promising attack on the problem of
nitrogen rate. This approach is not very promising in the Great Plains where
the rain comes in the spring and summer when it is too late to fertilize wheat.

In the Mountain Meadow areas of Wyoming and Colorado we have clover
years" and "'nonclover years." This is controlled by some factor in the en-
vironment that we do not understand. Clover requires much more phosphate
than grass. Grass is benefitted by fertilizer nitrogen, clover is not. What
ratio of phosphate to nitrogen do we use when we do not know the type of year it
is going to be ?

What place does plant tissue analysis hold in the economics of fertilizer
application? As you know, so-called critical levels for some of the plant

nutrients have been worked out for some of the crops. The critical level is
that level in some part of the plant which is adequate to guarantee the maximum
yield or perhaps 95% of maximum. If the nutrient in question falls below this
critical level, a yield reduction will be suffered. These critical levels have
been established generally with little or no regard for economics. This ap-
proach has been particularly fruitful with sugar cane, which requires two sea-
sons for its growth, thus enabling the planter to correct a nutrient deficiency
in the incipient stages. This approach is also being used on sugar beets, par-
ticularly in California. At the present time, and at moderate cost for samp-
ling and analysis, the nutrition of the beet plant can be much more accurately
regulated for yield and sugar production than can be done by soil tests and
nitrogen-response curves.

II. The Problem of Variability and the Need for Control
and Mathematical Estimates of it

Variability is the bugaboo of the scientist making quantitative measure-
ments of natural phenomena. It is particularly serious in fertilizer-response
experiments where frequently a series of plots, treated alike, will show a range
in yields which, converted to dollars, would be sufficient to pay for the cost of
the fertilizer. In the last 20 years rather elaborate designs have been developed
for control and estimate of experimental error. This has very frequently led to
very precise investigations of treatment effects and estimates of plot error that
are low. The question of whether the results would be applicable across the
fence, where the land might have had a previous history which was different,
has largely been ignored. Ignored also has been the question of whether the re-
sults could be applied elsewhere in the supposed area of inference. Insufficient
attention has been paid to calibration of tests for site characterization, and pro-
bably too much attention has been paid to measurement of small differences on
the experimental site.

An investigator replicates his treatments so that he will have more reli-
able treatment means and estimates of the error involved in those means. He
randomizes his treatments within the replicates in order to get an unbiased
treatment mean and error.

Most agronomists and soil scientists have used their estimates of error to
determine whether two treatments differed significantly. A frequent error is to
conclude that since two treatments have not been shown to be different that,
therefore, they are ihe same, This is not the case at all. One thing which I
have noted is that economists in fitting various functions to fertilizer-response
data have been concerned only with the treatment means with the goal of reduc-
ing the residual sums of squares to a minimum. In many cases they have ig-
nored the error terms which are available for evaluating whether one function
is a better fit of the data than some other function. I hope that this point will
be thoroughly discussed at this conference because it has a bearing on the type
of designs and number of replications that should be used in obtaining future
physical data.

I would like now to discuss some consequences of the "null" hypothesis
which is generally used by agronomists and soil scientists in evaluating treat-
ment differences. This hypothesis is almost universally used without much
thought being given to the consequences of its adoption. In effect, the null
hypothesis states that two treatments are drawn from the same population,
and that their difference is zero. If the ratio of the difference to the standard
error of the difference is found to be above a certain value, the investigator
concludes that the treatments are different and are from different populations.

This places the burden of proof on anything that is new.

In the Columbia Basin of Washington, zinc-deficiency symptoms-on:bean's
occurred over a widespread area. Our investigations showed that an applica-
tion of about 10 lbs. of zinc per acre, costing about $4.00, would correct this
deficiency for a period of probably ten years. Even if we ignored the carry-
over value of zinc from year to year, and assumed that it must be paid for in
one year by additional produce,, the $4.00 cost could be offset by yield in-
crease of about 50 pounds of beans. Now in practice, a least significant differ-
ence between treatments of 300 to 400 lbs. per acre is usually obtained. That
means that a farmer could have lost annually about 300 lbs. of beans due to
zinc deficiency, which he could have corrected at a cost of only 50 lbs. of beans
and we would not have advised him to do anything about it, if we used the null

A similar situation can apply to phosphate fertilization. In effect, we fre-
quently tell the farmer to let the deficiency or yield reduction get severe enough
so that we can detect it with our experimental methods before we recommend
that he correct the deficiency. Oftentimes the cost of the corrective measure
is so small compared to the least significant difference, converted to dollars,
that the use of the null hypothesis is ridiculous. There is no reason why we
cannot adopt some other hypothesis, which takes the cost of the treatment into
account, as an alternative. This is not a matter of statistics; it is a matter of

III. The Problem of Biases in Plot Work

Control of plant competition. You are all familiar with the fact that the
growth of a plant affects the growth of its neighbor. Plant roots extend lat-
erally a considerable distance from the plant and affect the amount of nutrients
and water available to its neighbor. It is also affected by its neighbor. For
this reason, the section of the plot harvested for the determination of yield is
always surrounded by a border area of the same crop grown in the same way as
the portion to be harvested. Comparatively narrow border areas are sufficient
for light and factors in the soil environment that affect plant growth, but we do
not know how to provide adequate borders for aerial environmental factors like
humidity, air movement, and temperature.

Control of runoff. The method of treatment of a particular plot may affect
the infiltration rate of the soil. One treatment may improve the intake rate,
and another may decrease it. Hence, if water is allowed to run downslope
from a plot with a low intake rate and runs across a plot with a high intake rate,
then the plot with the high intake rate receives more runoff than would be the
case under field conditions. This is a factor in biasing the yield of plots which
is not commonly recognized. There are many experiments set up to study the
effect of treatments on erosion and runoff in which plots are not adequately pro-
tected from the runoff of other plots.

Control of erosion and deposition of soil. In long-term experiments it is
frequently noted that soil erodes from certain plots and is trapped in the plant
residues of other plots. This can happen through erosion by either wind or
water. At Big Spring, Texas, there is a good example of such differential
erosion and deposition. A profile of these plots indicated differences in eleva-
tion from plot to plot of almost two feet in a 40-year period due to differential
deposition and erosion. Obviously, the plots receiving the deposition, because
of their proximity to erodible areas, did not truly represent the condition which

would exist in a large field.

Control of snow drifting. Snow drifts into areas wherever a barrier is im-
posed and wind velocity is reduced. The result can be that a plot on the lee
side of a plot with considerable standing residue will receive much more snow
drift than would occur in an open field. How to protect plots with adequate bor-
ders against deposition of soil by snow or wind is a problem for which we do
not have an adequate answer.

IV. Size and Shape of Plots

Much can be said on this subject, and it is closely related to the problem
of biases previously discussed. In general, in experiments with a duration of
only one year the plots can be much narrower than in long-term experiments.
The reason for this is that crop residues and soil move laterally on the surface
due to tillage operations. In general, we feel that a long-term experiment with
comparatively short crops, like small grains, should have a minimumwidth
of 12 feet. For crops which grow tall, like corn, the minimum width should
not be less than 15 feet.

V. Consideration of Properties of Individual Fertilizer Elements
and Their Reactions with Soil

There are a few things which we should know about the properties of nitro-
gen, phosphorus, potash, and other essential nutrients in setting the objectives
of our experimentation. These considerations may be summarized as questions
as follows:

Does it leach?
Is it volatile ?
Is it" fixed" or does it have high residual value to succeeding crops?
Are super-optimum amounts injurious to yield or quality of crop?
What is the cost of effective quantities of the nutrient in relation to the ex-
pected returns?

The answers for these questions are not unique for a fertilizer element,
but depend on the interactions of the nutrient with other soil constituents, rain-
fall, and sometimes the kind of crop.

If the nutrient is readily leached from the soil, fixed in a form unavailable
to plants, or lost by volatilization or erosion, then our approach to the fertili-
zer problem is one essentially of squeezing the last dime of efficiency out of a
particular application. If the nutrient is not leachable, and remains available
over a long period of time, then we can turn our attention to fertilizing the soil
rather than the crop.

If large amounts of fertilizer are detrimental to yield or to the quality of
the crop, then we have a different problem than if such detriments did not occur.


Bernard Ostlel

I. Introduction

The planning of experimental programs to investigate the effects of ferti-
lizer application on crop yields is the concern of both the agronomist and the
economist. Unfortunately, the criteria on which these two base decisions are
not always the same. The agronomist, for example, tends to plan his experi-
ment with a view to assessing the differential effects of varieties and/or ferti-
lizers on yield. The economist, however, usually wishes to know everything
about the effect of fertilizer on yield from both an agronomic and economic
point of view. That is, he wishes to know all that the agronomist wants to know
and more. He wishes to determine the conditions under which a farmer should
operate to attain a maximum profit. He is also interested in the social impli-
cations. It would be desirable, therefore, if the agronomist planned more ex-
periments to produce data of use to the economist as well as to himself.

II. Types of Experiments

Experiments to locate significant differences. Most experiments are de-
signed to locate significant differences among treatments, or to estimate mean
yields or components of variance. Agronomic experiments are practically all
of this type.

Experiments to describe response surfaces. Some experiments are plan-
ned with the one thought in mind: find out all possible about how one variable
(the dependent variable) acts over a preassigned range of values of the other
variables (the independent variables). That is, it is desired to map the behav-
ior of a function over a specified range of values of the independent variables.
Because the range of values of the independent variables is already determined,
it is possible to specify (in advance of any experimentation) a complete set of
conditions under which the behavior of the function will be investigated.

Experiments seeking maxima (or minima). In certain instances, the main
objective of an experiment may be the seeking out of a set of conditions that will
permit a function to achieve its maximum (or minimum) value. This is a diffi-
cult task unless some previous work has been undertaken to partially map the
behavior of the function. The problem of finding a relative, or local, maximum
(or minimum) is not so difficult, however, since it does not require the pre-
liminary mapping of the function. Further problems exist when a process con-
sists of several stages and the maximum of product No. 1 (stage No. 1) does
not necessarily lead to the maximum of product No. 2 (stage No. 2) and so on.

IStatistician, Montana Agricultural Experiment Station, Bozeman, Montana.

2Some preliminary experimentation may have been performed to aid in set-
ting the range over which the independent variables are to vary.

Experiments seeking an economic maximum (or minimum). Experiments
designed to provide data amenable to economic analysis are often more involved
than either of the types discussed above. This is so because it is necessary
to know the answers which experiments of types B and C were planned to pro-
vide as well as the economics of producing and marketing a particular product.
Not only is this true, but it must be realized that the conditions which maximize
a function ( in the sense of preceding paragraph) may not be the same conditions
which produce an economic maximum.
III. Methods of Planning Experiments
As might be expected, more than one method of planning experiments is
available. For the sake of definiteness, these have been classified into three
groups: (1) the practical approach, (2) the classical approach, and (3) the
statistical approach.
The practical approach. The practical approach is to select those partic-
ular combinations of levels of the factors involved which the researcher pre-
dicts will produce satisfactory yields and/or profits, and then perform experi-
ments under just those sets of conditions. If a fortunate selection of conditions
has been made, this approach may be of great help in locating an optimum value
of a function. Historically, however, the selection of operating conditions has
been affected by conservatism, and this has led to the response surface being
described at conditions far removed from the optimum. Also, it will usually
be inadequate to describe a process since many gaps will exist in the available
data. There is another possible end towards which this type of experiment may
be directed which should not be overlooked. That is, by selecting sets of con-
ditions using the practical approach, we may check on the operability of the pro-
cess (viz., whether it can be done) and on the economic feasibility of the pro-
cess (assuming it to be operable) at what are thought to be either limiting or
desirable conditions. It goes without saying that some of these selected condi-
tions should be chosen to verify any laboratory experiments that have already
been undertaken.
The classical approach. By the classical method of experimentation is
meant that all factors except one are fixed (held constant) at some level while
this one factor is permitted to vary over some predetermined range. Next,
change the level of one of the factors previously held constant and then let the
original varying factor again take on all values in which we are interested.
This procedure is continued until the experimenter has tried all combinations
of factor levels that are of interest. This approach is satisfactory if all one
wishes to do is describe a function. Probability statements will, of course, be
difficult to obtain due to lack of proper randomization.
A slightly different version of the classical approach, which might be named
the sequential classical" method, is as follows. First, the procedure is the
same as in the classical method; that is, the levels of all factors except one are
fixed, and this one is permitted to vary over a predetermined range. This, of
course, enables the experimenter to determine the effect of this one factor (i. e.,
plot a curve showing how it affects the characteristic being measured) at the par-
ticular levels of the other factors selected by the experimenter. There is no
guarantee that the curve representing the effect of the factor being examined will
be unaffected (in addition to a possible translation) if one or more of the other
factors are allowed to act at different levels. The next step in the sequential
classical approach is to fix the first factor (which was permitted to vary), at the
level which produced an "optimum" effect and allow a second factor to vary over
some predetermined range, all other factor being held constant. This then leads
to an" optimum" level of the second factors.

Such a procedure is carried out, permitting a different factor to vary at each
stage, until such time as the experimenter feels he has pinned down the best
possible set of conditions under which to operate. It is quite possible, of
course, that several cycles (progressions through the factors in a definite order)
may be necessary before a suitable stopping point is reached. A word of warn-
ing is appropriate, however. This method may never lead to a maximum (or
minimum) if the response surface is of a certain nature (1). Then, too, the
lack of randomization makes probability statements difficult.

The statistical approach. Before attempting to describe the statistical
approach to the planning of experimental programs, it is well to point out
why the preceding methods may not be considered statistical. The practical
approach is not, in general, amenable to statistical analysis for three reasons:

(i) Data are not always available on a complete grid (i.e., in all cells of
an n-way table, there being n factors). In other language, data are not
available at all combinations-of all levels of the factors which the ex-
perimenter desires to vary. This makes any possible statistical ana-
lysis extremely difficult, especially if there is no symmetry in the dis-
position of entries in the n-way table.

(ii) The trials are generally performed in a systematic fashion, which does
not permit the easy calculation of measures of confidence in any infer-
ences made.

(iii) The systematic consideration of the various conditions may permit fac-
tors other than those being controlled to exert an influence on com-
parisons of interest, thus making unconfounded (unambiguous) results
impossible to attain.

Classical experimentation suffers from these same restrictions, although per-
haps a full set of data may be available for examination.

What, then, is meant by the statistical approach or by the statistical plan-
ning of experiments ? It is the planning of the complete sequence of steps in the
experimental program so that:

(i) Data will be obtained which are relevant to the problem.

(ii) The data will be amenable to statistical analysis.

(iii) Within a certain preassigned margin of error (in a probability sense)
one or more questions may be answered. That is, the results of the
experiment may be evaluated objectively within limits dictated by the
inherent variability of the process.

What are the salient points to be considered when planning an experiment?

(i) One must know exactly what the problem is and any hypotheses to be
tested should be clearly stated. A mathematical formulation is to be

(ii) The factors to be included in the experiment must be decided upon; that
is, the fundamental variables must be defined. This, effectively, clas-
sifies all variables into two groups: those included as controlled factors

and those which are not controlled but over which we randomize in an
attempt to even out the effects.
(iii) The range and number of levels of each factor to be investigated must
be stipulated.
(iv) The number of replications of the experiment under each set of con-
ditions must be fixed. It is desirable, of course, that the number of
replications be the same for each set of conditions.
(v) A method of assigning treatments to the experimental units must be
established in accordance with some scheme of randomization so that
the "experimental error" may be assumed in a state of statistical con-
(vi) Some functional form should be assumed to represent the distribution
of random errors. The usual assumption is a normal distribution.
(vii) All possible outcomes should be considered and provision made to be
certain that these are sufficient to answer the problem. If not, a new
plan is called for.
(viii) Assuming that the performance of the experiment has been carried out
as stipulated in the preceding steps and that any assumptions made
seemed justified in view of such performance, the plan should specify
the type of statistical analysis to be carried out. This.is, it will be
seen, closely tied in with point (vii).
(ix) The plan should also indicate the manner in which the conclusions are
to be presented so that they may be understood readily by those per-
sons who are not well trained in statistical theory and methods.
IV. Fertilizer Experiments
It will not be unrewarding if I spend a few moments on fertilizer experi-
ments as they are commonly planned. Seldom, in recent years, do we find the
agronomists concentrating on only one nutrient at several levels or on several
nutrients at one level each. The statistician may, I believe, take credit for
some of this improvement in experimental procedure because of his constant
reminder of the need to consider interactions anrd response curves. Conse-
quently, we now find experiments being conducted which involve, say, N, P,
and K, and each of these at several levels, or rates, of applications.
If a complete factorial scheme of experimentation is used, one may study
the main effects of the various nutrients as well as the interactions between
them. Since several levels of each nutrient are employed, one may examine
the response (regression) curves which tell how yield is affected by changing
the rate of application of the fertilizer. A word of advice may be helpful at
this point: try to use a balanced design for simplicity of the analysis. Like-
wise, when deciding on rates of application, keep them equally spaced (i.e.,
in arithmetic progression) and use as many levels as possible. You should
remember that economizing as to the number of rates (or levels) can soon
eliminate the possibility of an adequate study of yield response to rate of appli-
V. Economic Aspects
One must not, of course, rely wholly on experiments such as those outlined
in the preceding section. Marginal productivities, costs and so forth, must be

considered if one is to reach sound economic conclusions. This has been done
in some instances, and the results have been most illuminating. I trust the
economists, in cooperation with the agronomists, will pursue these points fur-

VI. Suggestions for Future Experiments

We have suggested that the researcher should not be too restrictive when
selecting rates of application. However, if he heeds this advice, he may well
find himself faced with another problem, that is, the difficulty in obtaining
enough homogeneous plots on which to conduct his experiment. If the plots are
not homogeneous (that is, fairly similar in all respects), the resulting analyses
may be suspect. Recourse in such instances should be made to incomplete
block designs and various schemes of fractional replication.

To aid in the exploration and description of response surfaces, as well as
the location of maxima, one should consider the methods advocated by Box (1)
and by Box and Youle (2). I believe the approach to the problem suggested by
these authors is of prime importance to all of you concerned with the problem
we are discussing here. It certainly represents some of the better thinking on
such problems which has been formalized and published in recent times.

VII. Summary

The problem of designing agronomic experiments to study the economics
of fertilizer application was discussed. Various types of experiments were re-
viewed and several alternative approaches were examined.

VIII. References

1. Box, G. E. P. The exploration and exploitation of response surfaces:
some general considerations and examples. Biometrics, 10:16-60, 1954.

2. Box, G. E. P. and Youle, P. V. The exploration and exploitation of re-
sponse surfaces: an example of the link between the fitted surface and the
basic mechanism of the system. Biometrics, 11:287-323, 1955.

3. Ostle, Bernard. Statistics in Research. The Iowa State College Press,
Ames, Iowa, 1954.


James S. Plaxicol

I am sure that the excellent papers presented by Drs. Viets and Ostle have
made it painfully clear to each of us that designing experiments which provide
the data desired by agronomists and economists is a complex task. In discus-
sing these papers I have elected to make some rather general introductory re-
marks to be followed by a consideration of certain specific points developed in
the two major papers.

The fact that this is a conference of agronomists, economists, and statis-
ticians implies that the design of experiments directed at economic evaluation
is of necessity a joint effort of the three disciplines represented. Furthermore,
it is essential that each member of the" team" understand and appreciate the
viewpoints and problems of his co-workers. At the same time it seems to me
that each member of the team has a rather definite area of responsibility in for-
mulating the specific problem to be investigated and in designing empirical pro-

Selecting the Problem

In selecting a problem from the almost infinite array of relevant and im-
portant fertilizer economics problems, and in designing research procedures,
we should recognize that an army of well-coordinated research workers could
profitably devote many years to relevant problems relating to fertilizer eco-
nomics. Yet it is likely that resources devoted to this area during the foresee-
able future will be modest relative to the magnitude of the problem. Thus we
as researchers are faced with a real world" allocative problem. Our pro-
blem is simply how can we allocate research resources available to us so as to
maximize returns to the resources in terms of gains by individual farmers and
society. If we are to allocate our resources in an optimum manner we must of
course face the question posed by Dr. Viets, namely what is our goal, i.e. what
do we wish to accomplish. Only after we have ranked existing questions rela-
tive to their importance and have formulated a specific research problem can
we proceed to select or develop the experimental design which promises to pro-
vide the data desired at a minimum cost.

I consider the selection of the problem to be investigated, i. e., the speci-
fication of the model to be estimated and the determination of the level of signi-
ficance required, to be the problem of current paramount importance in fertili-
zer economics research. The difficulty of selection is intensified by the fact
that fertilizer responses are not independent of other management practices.
For a given crop we may write:
Y = f(xl,2 ---Xn) + (x 1 ------x n) + e
y = yield
x xn = variables which can be controlled by the experimenter

Department of Agricultural Economics, Oklahoma A. & M. College, Still-
water, Oklahoma.

x x = variable which cannot be controlled by the experimenter
1 n
e = random variations

The variable x, x would include such items as the levels of nitrogen,
phosphorous, potassium, irrigation, soil fertility, rotation, plant spacing, etc.
In like manner the x' x' factors would include such effects as rainfall, tem-
perature, wind velocity, epc.

I believe that most agronomists and economists would expect some degree
of interaction (complementarity) between the different factors which can be con-
trolled by the experimenter and the farmer. If we wish to estimate the com-
plete surface involved it would be necessary to vary each of the "controllable"
factors simultaneously. Yet if there are seven such "controllable" effects and
we select a complete factoral design 2, 187 plots would be required per repli-

Obviously then it is necessary to reduce the number of variables to man-
ageable size by determining a priori the parts of the surface which are of most
interest. (This would correspond to steps II and III as outlined by Dr. Ostle.)
In practice most cooperative experiments with which I am familiar take N, P,
and K as variables with the other factors fixed at some level. It is my opinion
that this choice is not necessarily an optimum solution under all conditions.
Perhaps a given number of plots would yield more information if the optimum
iatio of N, P, K were determined (i.e., estimated) a priori and the nature of
the response to different levels investigated by experimentation at several dif-
ferent locations or for different crops. Another alternative would be to vary
one or more nutrients simultaneously with plant spacing or level of irrigation.
Again more information on one portion of the surface is gained at the expense of
giving up information relative to some other segment of the surface.

The question of necessary replications, i.e., what level of statistical sig-
nificance is required, needs careful consideration. Again, with limited re-
search resources, the estimation of one segment of a surface with a high de-
gree of accuracy may involve a very large opportunity cost in terms of data on
a larger segment of a surface estimated at a lower level of significance. Since
a surface estimated for a given year has a rather fleeting usefulness due to year
-to-year weather variability and shifts in the function due to technical advances,
it is my opinion that this question merits considerable attention on the part of
agronomists, statisticians and economists.

Soil and Tissue Tests and Yield Variability

I agree with Dr. Viets that the development of soil tests which would char-
acterize the ability of a soil to supply a nutrient in question would be of tremen-
dous value. At the same time I wonder if we cannot visualize an eventual fur-
ther development in the direction of a chemical-physical soil test which would
predict the response surface and the range of year-to-year variability. Cer-
tainly the development of better-calibrated predictive tests could greatly re-
duce the number of experiments required to produce the types of decision mak-
ing information needed by farmers. Thus, it is clear that attempts to calibrate
soil tests should be an important consideration in experiments designed to char-
acterize surfaces.

There is, as Dr. Viets indicates, a need for the development of a family
of pure (i.e., residual effect removed) response surfaces associated with varied
rainfall-soil moisture relationships. Such data would enable one to ascertain

the manner in which optimum inputs of fertilizer would be influenced by weather
variability. There are at least two possibilities: (1) Lboth slope and intercept
are to a substantial degree a function of moisture, (2) intercept is a function of
weather but slope is independent of weather. Obviously these different possi-
bilities would lead to entirely different optimum economic decisions.

An analysis of eight years of cotton response data at Stillwater, Oklahoma,
suggests that weather conditions may have relatively little effect on the optimum
inputs of fertilizer under the conditions of the experiment. However, it is clear
that these results should be tested at different locations and with different crops.
There is, of course, a good possibility that in many areas the sequence of wet
and dry years may have an important effect on optimum fertilization practices.
That is, fertilizer applied during a dry year may remain as a residual in the
soil so that little response would be secured from applied fertilizer the follow-
ing year regardless of conditions. In areas such as the great plains I consider
work dealing with the nature of response surface variability to be of prime im-

It is important to recognize that if we wish to solve the risk and uncertainty
aspects of the fertilizer-allocation problem we must design and conduct the re-
search with this objective in mind. I believe the most hopeful approach is to at-
tempt to establish functional relationship between response surfaces and observ-
able factors that cannot be controlled but that can be measured and probability
distribution estimated. The alternative is to conduct experiments over a time
period sufficiently long to directly provide probability estimates. This latter
approach would appear to be prohibitively expensive in terms of time and funds.

Plant-tissue tests offer an interesting possibility as a tool in fertilizer eco-
nomics research. Such tests are a part of a corn rotation fertility experiment at
the Virginia Station. During the first year of the tests no yield response was
secured due to an extremely dry season. Yet there was an excellent relation-
ship between the levels of the different nutirients in the plant tissues and applied
fertilizer. This would seem to suggest that if tissue tests can be correlated
with yield, they may be utilized to aid in the estimation of the relationship be-
tween yield and effects not controlled in the experiment. In any event this ap-
proach may offer a fruitful area of exploration.

Selection of the Design

As indicated previously, the appropriate design can be selected only after
the problem has been properly formulated and the level of significance desired
has been determined. However, I am convinced that if we are to make a signi-
ficant dent in the fertilizer economics problem we must adopt designs which
allow us to estimate surfaces with a minimum of observations. The work of
Box, Youle, Hader, Hunter, Mason, et al. would, as Dr. Ostle suggests, seem
to offer much in this area. One of the Box designs, for example, provides es-
timates of all second-order effects of athiee-factor surface with only 15 obser-
vations per replication. These estimates are, in certain respects, superior to
those derived from the 27 observations in a complete factorial. However, if
third-order effects are expected this particular design is inadequate. The Box
designs are being used in fertilizer tests at the North Carolina and Virginia
Stations. These designs have the added advantage of being very easy to ana-

It is well to note that a priori knowledge of the nature of the response sur-
face can greatly increase the efficiency of any design, and the Box Composite
designs in particular. Perhaps this raises questions relative to the applicability

of the Box designs to biological research where variability is the rule rather
than the exception. More important, however, this characteristic suggests
that a thorough evaluation of existing data is an important prerequisite to fur-
ther experimentation.

In many areas there is considerable fertility-rate data which have not been
subjected to economic analysis. There is good reason to believe that an exami-
nation of these data by agronomists and economists can provide a basis for cur-
rent decision making and for future planning of experiments.

Cooperation Important in All Stages

The basis for cooperation between agronomists and economists lies in the
fact that when the two work together then efforts complement and supplement
each other. Thus minor additions to or variations in a design developed to test
a series of purely agronomic hypotheses can provide the basis for economic
estimation and analysis which may make the agronomic interpretation more
meaningful to decision makers. This relationship is true in the problem-for-
mulation, research-design, and data evaluation and analysis stages.

Dr. Viets points out (1) economists have largely ignored available error
terms for evaluating the goodness of fit of alternative functions, (2) the test
of hypothesis procedures commonly followed by agronomists are not adequate
guides for decision making. These two related observations are a prime ex-
ample of the need for closer working relationships between agronomists, eco-
nomists, and statisticians. A recent paper by Dr. Mason of North Carolina
suggests that error terms offer a method of evaluating the fit of a function which
may be superior to methods commonly used by economists. Likewise the use
of production functions along with choice guides familiar to economists would
seem to be the appropriate tools of economic analysis. It is to be hoped that
this conference will help us see how we may work together as a team so as to
exploit such opportunities.


Burton L. French

The need of economists and agronomists for experimental evidence of plant
response to fertilizer has been presented in earlier papers. It has been gen-
erally accepted that some functional relationship exists between plant nutrients
and the quantity of a crop produced. Whether this is a continuous or a discrete
function and the precise mathematical form of the function are problems for which
no answer has been developed that generally can be considered the "true" law
of plant response. How are the statistical inferences to be drawn from experi-
mental evidence so that the economic solutions can be presented and interpreted
in terms of a farm operate? s needs? What mathematical function represents
the relationship between the yield of the plant and the quantity of nutrients ap-

Logic of Plant Response

The optimum application of a plant nutrient is seldom equal to any level
applied in the physical experiment. Also, the process of analysis based on the
concept of equating the values of the additional product to the value of the re-
lated unit of input assumes the product and the factor to be minutely divisible.
Some estimate of the expected yield of plants for all possible values of the fac-
tor should be available. Therefore, the response function should be a continu-
ous stochastic function that can be fitted by statistical procedures which will
provide probability estimates about the predictions.

In determining the mathematical form of the function by which the data
from fertilizer experiments are to be analyzed, the logic based on the physio-
logy of biological growth is of primary importance. Under the assumption
that all growth factors except those under study are fixed at a given level, the
amount of product will increase as the quantity of the nutrient factor is increased.
As the quantity of the factor applied is increased at a constant rate, the product
will increase at a decreasing rate--the law of diminishing increments. At par-
ticularly low levels of one or more of the fixed growth factors, the quantity of
the product may be decreased as extremely large quantities of the growth fac-
tor under study are applied. This is apparently2true of nitrogen fertilizer ap-
plied to corn when the level of moisture is low.

Econometrician, Production Economics Research Branch, Agricultural
Research Service, Washington, D. C.

2This is well illustrated by the yields of corn at increasing inputs of nitro-
gen applied at different levels of phosphorous in the corn-nitrogen-phosphate
experiment conducted in Iowa. Here the fixed level of moisture was probably
too low. See Heady, E. O., Pesek, John, and Brown, William. Crop Response
Surfaces and Economic Optima in Fertilizer Use. Iowa Agr. Expt. Sta. Res.
Bul. 424, Table A-14, p. 330.

When the other fixed growth factors are held near the physical optimum,
the quantity of product increases at a decreasing rate as the variable factor is
increased until a theoretical maximum is approached. This condition was ap-
proximated in irrigated corn-nitrogen experiments conducted in Nebraska, Or-
egon, and Washington.

Besides the controlled factors included in an experiment, uncontrolled
sources of variation influence the results. In general the analysis of variance
is the usual technique to test significant differences between the different appli-
cations of each factor. Estimating a functional relationship between treatment
mean yields and the quantifiable factors of production is supplemental to the
analysis of variance rather than a substitute for it. Use of a functional rela-
tionship is comparable to that used in tests of the linear, quadratic, or cubic:ef-
fects in the treatments for factorial experimental designs without formally es-
timating the variable constants of a predicting equation.

In addition, the experimental observations may provide information about
the specific form of the mathematical function that best represents the rela-
tionship between the quantity of product and the amount of factor input.

Mathematical Functions Used

Mathematical techniques used to describe the relationships between quanti-
fiable input factors and output are:

1. Exponential
2. Gompertz
3. Polynomials
a. Quadratic
b.. Quadratic: square root
4. Cobb-Douglas power function.

In analyzing experimental data on fertilizer applied to tobacco, Spillman ob-
served that for each added unit of fertilizer applied the ratio of successive in-
creases in yield approximated a constant. From this he suggested the form
(1) Y.=M AR of the exponential function where the ratio is represented by
the variable constant R. Y. is the yield, M is the maximum yield it is theoreti-
cally possible to attain, aTid A is the increase in yield.between the yield at no
application of fertilizer and the theoretical maximum.

The exponential function does not provide satisfactory estimates of the seg-
ment where the experimental data exhibit diminishing total returns (negative
marginal product). However, if the yield of the crop is diminishing for allfer-
tilizer levels beyond the zero level of application, the exponential might appro-
ximate the response very well with the constant A having a positive sign and

These experiments are discussed in detail as experiments 10, 32, and
51 in Paschal, J. L., and French, B. L. A Method of Economic Analysis Ap-
plied to Nitrogen-Fertilizer Rate Experiments on Irrigated Corn. U.S. Dept.
of Agr. Tech. Bul. 1141 (in press).

Spillman, W. J. Use of the Exponential Yield Curve in Fertilizer Experi-
ments. U.S. Dept. of Agr. Tech. Bul. 348. 1933.

the yield becoming asymptotic to the value M, which is a minimum instead of
a maximum.

A satisfactory method of estimating the constants of the exponential func-
tion for a single input variable by least-squares methods of estimation has been
developed. However, this method requires the estimation of the ratio between
incremental yields, R, and the computation of a correction for that value by
successive iterations. At present, Mendum and Burrows are developing a fea-
sible method of approximation to least-squares estimates of the constants of
the exponential function for two or more variables.

Some fertilizer experiments show a relationship in which yield increases
at an increasing rate at low levels of fertilizer application (generally called
a sigmoid curve). That is, some of the fertilizer must be present in the soil
before the nutrient is available for the plant' s use. In the experiments conducted
by Glover, this was particularly evident in the high apparent consumption of
phosphorus. This was attributed to the fixation of phosphorus by traces of iron
in the sand. A mathematical function that is consistent with this logic is the
Gompertz function, which is written as (2) y (M-ARXj) in which e is the base
of the naperian logarithms. This function isicomplex, however, and the econom-
ic estimates need to be made by approximation.

Another general class of functions that has been widely used are the poly-
nomial functions. The quadratic member of this family (3) 2
Yj = a+blXj+b2Xj
j 1 lj 2 lj
is the function generally used. The quadratic function makes no assumption
about the incremental yields or elasticities of production. It will reflect a nega-
tive total product if the application of the fertilizer is made at such low levels
of other nutrient factors as to make the nutrient under test restrict plant growth.

Transformations of the quadratic function have been made. They appear
to provide a more reasonable estimate of the relationship than the quadratic
function. The independent variable, X., has been transformed to the square
root of the independent variable,-; X. This function is similar to the quadratic,
but the maximum yield is reached al much higher inputs of fertilizer. Within
the range of most of the inputs tested, the "quadratic square root" equation
approximates the exponential if the yields do not reflect a segment of negative
total product.

A function that has been employed by economists in making estimates of
production relationships is the Cobb-Douglas power function, a function linear
in the logarithms of the dependent and of the independent variables. The major
assumption of this function is that the elasticity of production is constant for
the entire range of the data. A second assumption is that there is not a range

lStevens, W. L. Asymptotic Regression, Biometrics 7:247-267, 1951.

2Glover, J. The Nutrition of Maize in Sand Culture, The Journal of Agri-
cultural Science 43 (2):154-165. 1953.

3Heady, Pesek, and Brown, op. cit.

of decreasing total product. This function is difficult to fit to fertilizer-re-
sponse data because thereis not an admissible value for the logarithm of an in-
put of zero. Thus, it must be fitted to yield observations corrected for the
yeild of the check plot, which assumes that there is no experimental error in
the check-plot data.

These five functions were fitted to results from an experiment designed
to study the effect of nitrogen applied to corn in southwestern Nebraska in 1952.
They are illustrated in figure 1. At the lower levels of nitrogen input, the"
"quadratic square root" increases at a faster rate than the others. The quad-
ratic function increases at the slowest rate.

As the level of input is increased, the Gompertz, the exponential, and the
"quadratic square root" functions are similar in prediction levels. The quad-
ratic reaches a specific maximum at approximately 220 pounds of nitrogen and
136 bushels of corn and then suggests a diminishing total product. This is in-
consistent with the experimental observations, which, in general, appear to
fluctuate about a horizontal line beyond 200 pounds of nitrogen applied. The
Cobb-Douglas function increases at an approximate constant rate from 60 pounds
of nitrogen applied. Above 200 pounds of nitrogen applied, it departs from the
experimental levels of yields. Predictions based on each of these functions
would differ widely, as shown by the estimated most profitable rates of nitro-
gen application. Just which of these should be used for prediction?

The "best" fitting function, in terms of the sums of squares of deviations
is the Gompertz, but the improvement over the fit of the exponential function
is relatively small. The fit of the "quadratic square root" is almost as "good"
as the exponential (see table 1). However, only one function had a mean square
of deviations less than the mean square of the experimental error. The econom-
ic estimates and the predicted yields from these two functions are shown in
tables 2 and 3. All these measures differ by little more than the measures of
variability of the equations.

Figure 2 illustrates the estimates of the exponential, quadratic, and quad-
ratic square root functions, using the data of the application of nitrogen fertili-
zer to corn at a 320-pound level of added phosphorous from the Iowa experi-

The experimental observations appear to reflect a diminishing yield at the
280- and 320-pound levels of nitrogen application. As in the Nebraska example,
the quadratic function reached a maximum at a yield above the observations,
then declined rapidly. Here, predicted yields from the exponential and quad-
ratic square root functions were comparable up to the highest rate, 320 pounds
of nitrogen. At this level, the quadratic square root had passed the maximum
yield and entered the diminishing total product phase. However, in these ex-
periments the levels of nitrogen application were not high enough to establish
definitely the fact of a diminishing total product.

llbid., p. 330, table A-14.

Corn Response to Nitrogen, Hardy, Nebraska, 1952
BU./ACRE ....E





120 160 200 240 280 320

Figure 1

Corn Response to Nitrogen, Iowa, 1952




120 160 200 240 280 320

Figure 2


Table 1. Sums of squares and mean squares of deviations about regression,
five mathematical functions and experimental error in experiment,
experiment of nitrogen applied to corn, Hardy, Nebr., 1952

Function d. f. Sums of Mean
: squares square

Y. = M-ARXj 9 808.70 89.86
Y. = a+b Xj+b2 X 9 3,715.50 413.83

Y = a+b1Vj +bZXj 9 1,925.00 213.89

Y. = aX 10 4,424.90 442.49
. = e(M-AR j) 9 525.35 58.37
Experimental error 44 3,846.28 87.4

Table 2. Estimated most profitable rates of nitrogen application to corn, es-
timated yield at that rate, and maximum yield, two mathematical
functions, Hardy, Nebr., 1952

Functo : Most profitable Yield at Maximum
Functonrate of nitrogen MPR yield
: Pounds Bushels Bushels
Y. = M-ARXj 164.0 121.6 + 1.22 127.8

Y. = a+b1~. +bZXj 180.0 121.9 +2.14 127.7

Hildreth has developed procedures for estimating points on a production
surface of unspecified algebraic form from data on outputs produced by vari-
ous combinations of inputs when the outputs are subject to diminishing returns.
This procedure was illustrated in greater detail in the paper presented at the
TVA Symposium. In this, paper, he extended the procedure to the economic

Hildreth, Clifford Point Estimates of Ordinates of Concave Functions,
Jour. Amer. Statis. Assoc., 49 (267) September 1954.

Hildreth, Clifford Discrete Models with Qualitative Restrictions, TVA
Symposium on the Methodological Procedures in the Economic Analysis of Fer-
tilizer Use Data. Iowa State College Press, Ames., 1956 (in press).

Table 3. Yields, costs, and returns at the most profitable rate of nitrogen applied, two mathematical func-
tions, Hardy, Nebr., 1952 a

Predicted Total returns Increase in Nitrogen Cost Net re-
yield at at the most returns applied of turns
Function zero appli- profitable from at nitro- from
cation of rate nitrogen MPR gen nitrogen

:Bushels Bushels Dollars Bushels Dollars Pounds Dollars Dollars
Y.=M-ARj 21.0 121.6 170.24 100.6 140.8 164.0 24.60 116.24

Y.=a+b l-J+bZX 18.6 121.9 170.66 103.3 144.6 180.0 27.00 117.62

Corn prices at $1.40 per bushel in the field and nitrogen priced at 150 a pound applied to the corn.

implications of the results and presented them in a corn-nitrogen price map
which specifies the corn-nitrogen price relation corresponding to the number
of pounds of nitrogen to apply to maximize the returns from nitrogen.

Analysis of Nonquantifiable Factors

Several factors in plant growth are difficult or impossible to quantify in a
functional relationship. These include spacing, application of moisture, level
of moisture, placement of fertilizer, and type of material carrying fertilizer
nutrients. For example, the quantity of water made available by irrigation and
as rainfall can be measured, but other conditions, such as time and method of
application, may have as much effect on yield as the total quantity. Two treat-
ments in which the same quantity of water is applied in the same number of
applications, but at different dates, may produce different yields. Two treat-
ments in which twice the quantity of water is applied in one as in the other may
produce nearly the same yield, if the smaller quantity is applied at critical

Comparisons of such nonquantifiable factors may be made by studying the
effect of these factors o0 the response to fertilizer as represented by the differ-
ent response functions. The curves may approach the same maximum, but
one factor may cause the yield to approach the maximum at a lower rate of fer-
tilizer input than another.

In some experiments, tests of hypotheses outlined by the experimental de-
sign result in no significant differences between treatments other than fertili-
zer. The test of significance determines whether or not the differences between
treatments are greater thanthe differences within treatments. Even though the
test reveals no greater differences between treatments than within treatments,
it cannot be concluded that there are, in fact, no differences between treatments.
The analysis of variance test of hypothesis minimizes the possibility of indicat-
ing a difference between treatments when no real difference exists. However,
this test may fail to measure differences that do exist. Therefore, merely not
rejecting a hypothesis of no difference between levels of other treatments does
not permit pooling of data for all levels when estimating the response curve for
a quantifiable factor. It is known, a priori, that there is a difference, for ex-
ample, between plots that receive no phosphate and plots that receive 100 pounds
of phosphate. Therefore, the response curve for each treatment of nonquan-
tifiable factors needs to be analyzed independently and the comparisons made
between the maximum returns to nitrogen for each response curve.

1For the discussion of the price map, see Hildreth, Clifford, Economic
Implications of Some Cotton Fertilizer Experiments Econometrica. Volume
23 (Z3); 88-98. January 1955.

2Rhoades, H. F., Howe, O. W., Bondurant, J. A., and Hamilton, F. B.
Fertilization and Irrigation Practices for Corn Production on Newly Irrigated
Land in the Republican Valley. Nebr. Agr. Expt. Sta. Bul. 424, 1954.

3Finney, D. J. Response Curves and the Planning of Experiments Indian
Jour. Agr. Sci. 23:167-186. 1953.

Previous discussions have presented a possible means of evaluating non-
quantifiable factors or nonmeasurable factors. In the analysis, economic con-
siderations dictate the use of certain variable factors such as plant nutrients or
water under irrigation. There are also the variables which influence the yield
of a crop independently or in conjunction with others that are a "free" good,
soil structure, available plant nutrients, and soil moisture for example. This
class of variable is usually held fixed in the experiment by choice of site.

Determination of the optimum applications of nutrients is important for the
first class of variables but only in joint determination with the second class of
variables. Yield is a function of all of these--
(4) Y= f (X1, X2, ...., X, X ., X ). Where Y is the yield,
X1 ..... X are economic variale factors, and X, .. ., X are free
variable factors. The optimum application of nutrients lis then determined by
the general method of solution, but the simultaneous solution of the partial deri-
vations set equal to the ratio of the price of the product to the price of each
nutrient will yield a function--
(5) X. = f (P Pi, P ...., P X+, ...., X ). Where the P.'s are the
prices of theeconomic factors and + ... the free variable factors.
k+l n
Let us assume that Xk+, .. .., X can be represented by a measure such
as available nitrogen, phosphorous, ornpotash in the soil, the quantity of soil
moisture at the time of applying fertilizer when the major part of the total sup-
ply is obtained prior to that time, or an index of one, or combinations, of
these, and indexes of other free but not easily quantifiable factors such as
structure. We then have a predictive function by which each different farm
situation can be integrated into the nutrient-application recommendations. This
depends, therefore, on the value of quantitative estimates that are feasible at
the farm level.

Effect of Fewer Experimental Rates

The functional relationships previously discussed present different statis-
tical and economic inferences when applied to different experiments. To illus-
trate, estimates of the functional relationships for the exponential, quadratic,
and quadratic square root function were made, using only subsamples of the 12-
rate experiment mentioned earlier. The four subgroups studied were: (1) The
first 4 rates; (2) the first 5 rates; (3) 5 rates equally spaced over the entire
range of the experiment; and (4) 6 rates distributed over the range of the experi-
ment. In all groups the first rate--zero pounds of nitrogen applied--was in-
cluded. In those groups distributed over the range of the experiment, the 12th
rate--16 20-pound units of nitrogen applied--was included. The respective esti-
mated curves for the first 4 rates and the 5 rates equally spaced over the en-
tire range of the experiment are shown for all three functions in figure 3, com-
pared with the estimated functions that utilized all 12 treatments of the experi-

These illustrations provide an indication of how estimates of limited ex-
periments deviate from estimates of more complete experiments. When they
are distributed over the entire range of the experiment, these estimates indi-
cate that the number of treatments might be reduced from those applied in this
experiment, and that meaningful estimates for recommendations to farmers can
still be obtained. However, care must be taken in designing the experiment to
include, if possible, the optimal level of the nutrient. A test of the efficiency

1See the discussion by D. J. Finney, op. cit.

Hardy, Nebraska, 1952

1st. 4 RATES


Y-a+b, X+bgX2
-- 200 1



0 80 160 240 320 0 80 160 240 320

----- Subsample 12 rates
SIncluded in subsample o In experiment

Figure 3

of a geometric progression of rates attractive to many soils persons has not
been made, but it is expected to be less favorable than 5 rates over the range
of the experiment analyzed.

In general, statistical summary procedures for analyzing data from experi-
ments designed with consideration of a functional relationship between a factor,
or factors, and the resulting product have been discussed. There has been little
evidence that any one function will be specified as the "law of plant response,"
and it is likely that different functions should be applied to different sets of ex-
perimental data.

Little consideration has been given to date to determining a probability dis-
tribution of the estimates for one function between different years in which sim-
ilar experiments are conducted or between different locations where similar ex-
periments are conducted. The Oregon Agricultural Experiment Station, in co-
operation with the Soils and Water Conservation Branch, ARS, is now assem.-
bling such a collection of data in the five-year wheat experiment conducted on
47 different farm sites. Analysis of these series of experiments may provide
more information as to the use of a particular function or functions and the abil-
ity to make estimates of yields and economic recommendations over a broad
area and between different years. Prior experiments have tended to be single,
one-year experiments which from the standpoint of statistical inference appear
to have limited validity for broad recommendations to farmers.



E. L. Baum

I. Introduction

During the last two decades American agriculture has experienced a major
technological revolution. This has enabled fewer farmers to produce larger
quantities of food and fiber for an increasing population. New and improved
machines and improved crops and livestock have contributed significantly to
increased farm output during these last twenty years.

Fertilizer, too, has made an important contribution to increased farm pro-
ductivity. Some idea of the size of its contribution can be gained from the knowl-
edge that fertilizer consumption has increased from 6 million to 23 million
tons in the United States from 1934 to 1954. Along with the increased fertilizer
use there has been a rapid rise in the average plant food content of fertilizers
from 18 percent in 1934 to 27 percent in 1954. This recent improvement in
average analysis is largely a result of improved fertilizer production technology
and lower costs per unit of plant food.

The Tennessee Valley Authority through its national fertilizer development
program has given much attention in recent years to areas of fundamental study
related to the economics of fertilizer use on farms. TVA carries on a program
of agronomic and agronomic-economic research mainly through cooperative
arrangements with land-grant colleges and by sponsoring meetings on this sub-
ject. In addition, TVA has its own greenhouse facilities for making preliminary
tests on new fertilizers developed at Muscle Shoals, Alabama.

Only recently have economists recognized fertilizer use as an area for em-
pirical research. Elementary economic principles texts usually use a fertili-
zer example to illustrate the principle of diminishing returns. Generally, how-
ever, these examples are hypothetical and do not represent factual data from ex-
perimental plots. It is sufficient at this point to indicate that agricultural re-
searchers have not thoroughly explored the physical and economic production
relationships between fertilizer use and crop response.

II. Belated Development of Fertilizer Economics Research

Several reasons can be cited that may explain the big gap in" economics
of fertilizer use" research between the early work of Mitscherlich and Spillman
and the present time. Some of the reasons are as follows:

Head Agricultural Economist, Agricultural Economics Branch, Division
of Agricultural Relations, TVA. Many of the ideas presented in this paper were
discussed at a TVA sponsored fertilizer economics symposium held in Knoxville,
Tennessee, June 14-16, 1955. The Proceedings of the symposium will appear
in a book titled Methodological Procedures in the Economic Analysis of Ferti-
lizer Use Data edited by E. L. Baum, Earl 0. Heady, and John Blackmore,
Iowa State College Press, 1956.

1. Before the last decade, few agricultural economists received ade-
quate training in mathematical economics and in relevant tech-
niques of statistical analysis. The recent emphasis upon econome-
trics in graduate schools has provided an increasing number of
agricultural economists with the necessary training to conduct
meaningful agronomic-economic research.

2. Agricultural subject-matter disciplines have encouraged such ex-
treme degrees of specialization that unnecessary barriers have
existed between agricultural economists and agronomists. For
example, economists sometimes indicate that they cannot make
meaningful fertilizer recommendations from a one-variable ferti-
lizer experiment, having only three rates of application, frequent-
ly at very low levels. The economist may further indicate that
meaningful economic recommendations for fertilizer use would
have to depend upon a yield surface, or a multivariable yield equa-
tion that accounts for the simultaneous effect of several nutrients
on yield. Furthermore, the application rates of all nutrients
should include rates at high enough levels to warrant economizing
considerations. By this means, the economist may determine
least-cost and/or maximum revenue fertilizer-use situations.
To make matters even more difficult, the concept of nutrient sub-
stitution as recognized by economists has sometimes been denied
by agronomists as a biological impossibility. Many hours can be
and have been consumed in the process of untangling the economic
and biological concepts of substitution.

Economists tend to consider the fertilizer-use problem as a pro-
duction problem in which there is a functional relationship between
inputs and outputs, with the relationship generally nonlinear in
nature. This approach yields desirable production coefficients
for farm production programming techniques. In addition, such
information presents the nature of physical response of plant
growth to fertilizer application. These data furnish the informa-
tion for practical fertilizer recommendations which may be made
for particular crops or rotations on soils of a given level of fertil-
ity. These recommendations may, with such information, be
varied as price-cost relationships change.

Agronomists sometimes design their agronomic research so that
it appears they have conceptualized their problems as a compari-
son of discrete phenomena. Although this concept is not neces-
sarily different from that employed by economists, there are es-
sential differences which create problems where interdisciplinary
cooperation needs to be improved. Many agricultural economists
who develop research in the field of fertilizer use find it difficult
to understand why agronomists have not run their application-rate
trials at higher levels. They also wonder why agronomists are so
insistent on numerous replications, and why they have seemingly
avoided multivariable experiments.

On the other hand, agronomists like other production scientists,
have been dismayed by the confusing jargon developed by the econ-
omists to describe what appear to be essentially simple ideas.
Many soil-fertility researchers are inclined to believe that the
rate experiments of such great interest to economists are less im-
portant than research on more fundamental aspects of plant growth.

3. The statistical techniques necessary to handle multivariable fer-
tilizer experiments had not been developed until recently. Now
researchers can plan sound experiments from which economists
may make economic analyses suitable for meaningful fertilizer
recommendations. In addition to these new techniques, there have
been new developments in economic analyses that have enabled
economists to make sound interpretations of the agronomic data.

4. In many areas of the United States, fertilizer has become an im-
portant factor of production during the last decade. Prior to this
time relatively little fertilizer was used in these areas. Hence,
the problem in the economics of fertilizer use did not seem to be
important enough for agricultural production economists. Instead,
they allocated their limited time and energy to the then relatively
more important factors of production.

The above discussion, in my opinion, explains the dearth of economic re-
search dealing with fertilizer use before the last decade. Although methodolo-
gical procedures used in the agronomic and economic analyses of fertilizer use
data have been greatly improved, we have by no means achieved the desired
types of data and the techniques of securing and analyzing these data. In regard
to this matter, TVA is currently giving much attention to research designed to
test and evaluate new and improved statistical and economic methodologies in
the economic analysis of fertilizer response data under different situations with-
in the United States.

There is a new and growing pressure for interdisciplinary studies between
the agricultural production scientists and agricultural economists. This new
development has grown out of the present concept of the farm production unit
in terms of the economists' model of a"farm." Much credit can be given to
new analytical techniques available to agricultural production economists in
demonstrating2to farmers how they can plan their farm as a unit for maximum
net revenues. Recommendations on all production matters must be synchro-
nized if the farm is to maximize returns and the farm operator--not the pro-
duction specialist--must choose among the kinds and combinations of production
factors to be employed and the kinds and amounts of products to be produced.
This implies that the production specialist cannot answer the farmers' problems
by merely supplying a single "best answer" to a particular production problem.
The farmers must receive information on a range of applicable alternatives.
In helping to organize farm businesses for maximum returns, farmers should
be made aware of the interrelations between their own contributions and those
made by professional agricultural specialists.

Cf., Glenn L. Johnson's Interdisciplinary Considerations in Designing
Experiments to Study the Profitability of Fertilizer Use in E. L. Baum, et
al., op. cit., chapter 2.

2Cf., Earl R. Swanson's Selecting Fertilizer Programs by Activity Anal-
ysis in E. L. Baum, et al., op. cit. chapter 12.

Agricultural economists are now finding it highly desirable to work as col-
laborators with physical production scientists inmanykinds of agricultural pro-
duction experiments. This is made possible through the newly developed tech-
ques which enable them to treat the economic problem of the farm as a whole.
With the general breaking down of barriers between the physical scientists and
the economists, we are beginning to see the economists working with the agron-
omists and the statistician in the initial design phases of fertilizer experiments,
as well as in the course of the analysis itself.

This changing role of the agricultural production economist is no doubt the
basis of the theme of this meeting here in Corvallis. If economists are to con-
tribute effectivelyto such research, it would be desirable that they secure a
broad understanding of agronomic research and of statistical methodologies as
well as the economic principles and methods of economic analyses applicable
to such problems.

III. Economic Research Considerations for Practical
Application of Fertilizer Use Data

As I have indicated, the field of fertilizer economics is relatively new;
there is much fundamental research yet to be done in the near future. However,
the agricultural production economist ought to be concerned with practical or
applied aspects as well as the fundamental considerations. The manner inwhich
crop yields vary with a whole range of varying applications of nitrogen, phos-
phorus, and potassium in combinations is essential information for sound farm
planning. These variations in application combinations need to be tested for
many conditions, such as (1) different crops, (2) different soils, (3) different
climatic conditions, (4) for tenants and owners operating under different capital
situations, and (5) varying price-cost relationships. Such knowledge enables
the farmer to plan for an economical and in many cases an expanded use of fer-
tilizer. Although most farmers cannot determine precisely the equivalence be-
tween the marginal yield and price ratio of the fertilizer nutrient ingredient and
the product, they may be able to determine their ability to fertilize their land if
suitable nomographs can be developed from detailed response surfaces. Of
course, the determination of this particular coefficient is only one of the many
and important coefficients necessary for farm planning.

Detailed coefficients secured from a response surface of constructed yields
based on varying amounts of nitrogen, phosphorus, and potassium applications
also enable the farm manager to select the proper technical coefficients. How-
ever, these should be consistent with the operator's notions on expected price
variability and crop variability. In the final analysis, the technical coefficient
of production and the price-cost relationships will be based on an estimate of
the most probable occurrence that may exist over a wide range of possible oc-
currences. The results of a study recently completed in Iowa revealed that the2
conditions of yield and price uncertainty tended to restrict the use of fertilizer.

Cf., William G. Brown' s Practical Applications of Fertilizer Production
Functions in E. L. Baum, et al., op. cit., chapter 10; Roger C. Woodworth's
Organizing Fertilizer Input-Output Data in Farm Planning in E. L. Baum, et
al., op. cit., chapter 11.

M. A. Anderson, L. E. Cairns, E. O. Heady, and E. L. Baum, Use and
Acceptance of Fertilizer in Iowa, 1953, Iowa State College Agricultural Experi-
ment Station Special Report, In Process, 1956.

We need more knowledge about the effects of uncertainty factors on fertilizer
use. This knowledge for farm planning and linear programming can be im-
proved so that better predictions may be made of the economic effects of fer-
tilizer in the whole farm program.

Each suggested level of fertilization--based on the relevant points on the
response surface--can be treated as a separate activity or investment oppor-
tunity. If these data are properly treated, the optimum level of fertilization
relative to other alternative investment opportunities may be estimated, and
the complete farm organization can then be predicted. Information provided
in this form will generally encourage economical and probably expanded use
of fertilizer because of the knowledge of the marginal returns for increments
of fertilizer inputs. Thus, the use of this particular resource by farmers op-
erating under varying capital situations can be specified. Another important
research consideration for practical application of results is that of securing
crop response functions based on many input variables as categories. This
approach is essential because the production coefficient for and the return from
any one input category is a function of the amount and kind of many other input
categories with which it is combined. Meaningful research in the economics
of fertilizer use may include the different fertilizer nutrients, seeding rates,
seed varieties, variable soil-moisture levels, soil fertility levels, and other
technologies involving crop production. It is entirely possible that a fertilizer-
rate study involving variable rates of only one nutrient may result in a signifi-
cantly lower response curve than if that particular nutrient were varied along
with one or two other nutrients as well as perhaps seeding rates. Let us con-
sider, for example, the problem of appraising the productivity of a new crop
variety primarily developed for increased yields. Before its use is widespread,
multivariable response studies should be conducted which would include rates
of the important fertilizer nutrients at sufficiently high levels. Such an experi-
ment would provide a basis for adequately demonstrating the yield-boosting
effect of this particular innovation.

The development of workable solutions to practical problems is the main
goal of fundamental research and methodological considerations. It is apparent
that sound fundamental research can result in a more efficient use of fertilizer
if it provides the necessary refinements for getting more practical recommen-
dations for the individual farmers. My idea of practical" is that the recom-
mendations recognize the important variables governing the products on each
particular farm; these variables include price-cost relationships, level of soil
fertility, tenure, capital position of the farmer, the farmer's personal aims,
his risk and uncertainties, the particular crops and varieties to be grown, and
the like.

1i., Earl Oo Heady's Methodological Procedures in Fertilizer Use in E.
L. Baum, et al., op. cit., chapter 1; Earl R. Swanson, Ibid., chapter 12.

C2 L, W. L. Park's Methodological Problems in Agronomic Research
Involving Fertilizer and Moisture Variables in E. L. Baum, et al., op. cit.,
chapter 7.

3CL, Earl W. Kehrberg's Some Problems Involved in Fitting Production
Functions to Data Recorded by Soil Testing Laboratories in E. L. Baum, et
al., op. cit., chapter 8.

4An expansion of these points is presented by Earl O. Heady in E. L. Baum,
et al., op. cit., chapter 1.

IV. Production Function Considerations

In a discussion of the economic interpretation and the use of fertilizer re-
sponse data, some attention should be given to the methodologies used. I do
not plan to spend much time on this subject, since other papers are mainly con-
cerned with this problem area.

It is important that we are aware that our investigation of the techniques
proceed on the assumption that smooth curves and surface's are representative
of the yield--plant nutrient relationships. It implies a certain orderliness in
the underlying relationships which keeps the observations from being complete-
ly haphazard. The variance of the observations from such a smooth surface
permits estimation of error in most cases. The use of conventional factorial
designs permits such experimentation from which agricultural production econo-
mists can make satisfactory recommendations concerning fertilizer use. Some
fertilizer-response research is now being conducted that uses the "cubic com-
posite" or Box design in estimating the response surfaces. It is claimed that
this alternative technique would yield estimates of linear and quadratic primary
effects and linear interactions between yield and nitrogen, phosphorus, and
potassium. In addition, it is further claimed that this "cubic composite" de-
sign will give these estimates with much greater efficiency per unit of observa-
tion than would a factorial design in nitrogen, phosphorus, and potassium.

Another approach is that of experimentation at discrete points in the two-
or multidimensional field to secure the most economical means of developing
the data needed for sound economical fertilizer-response coefficients.

It appears that the crux of the relative efficiency problem involves the abil-
ity of the researcher to select the treatment combinations specifically for the
expression to be fitted. This places a great burden on the researcher, as it is
a subjective selection based on past experience.

The methodological problems involved merit our attention in the immediate
future, since every fertilizer recommendation made to farmers implies knowl-
edge of the mathematical nature of the response function. Without question,
greater knowledge of the nature of response is needed for the most efficient
development of experimental designs that would yield data in such form that can
be best used by economists. If we knew that the mathematical form is of a
quadratic nature, a "cubic composite" design might be the most efficient. How-
ever, another design may be more efficient if the mathematical form proves to
be logarithmic or exponential. More emphasis should be given to the utiliza-
tion and testing of the relative applicability and efficiency of alternative experi-
mental design procedures so that economists would have a foundation to deter-
mine the design applicable to their particular experiments.

Cf., Richard C. Anderson's Comparison of Discrete and Continuous
Models in Agricultural Production Analysis in E. L. Baum, et al., op. cit.,
cFapter 3; David D. Mason's Functional Models and Experimental Designs for
Characterizing Response Curves and Surfaces in E. L. Baum, et al., op. cit.,
chapter 5.

Cf., Clifford Hildreth's Discrete Models with Qualitative Restrictions in
E. L. Baum, et al., op. cit., chapter 4.

An important point for economists to recognize is the great danger of mak-
ing recommendations to farmers based upon a nonop ium economic" point on
the surface, or upon an incorrectly defined surface. If the economic loss from
being at a nonoptimum point on the response surface is small relative to the loss
from being at an incorrectly defined surface, then efforts should be devoted to
experiments to define more precisely certain points on the n-dimensional field
rather than attempt to specify the entire surface. At this stage in the develop-
ment of fertilizer production functions relevant in the economic analysis of
plant nutrient response data, it is important that all pertinent experimental
approaches should be subjected to rigorous study to determine their applicabili-

The soundness of fertilizer recommendations based on plant nutrient re-
sponse curves or surfaces depends to a large extent on the economic research-
er's ability to interpret the experimental error in the results of the analysis.
Another problem with which the economist is faced is that of treating the resi-
dual or latent effect of fertilizer materials. It is necessary to do this in order
to make an adequate economic appraisal of the agronomic experimental results.
The economist is concerned with the distribution of production functions with 2
respect ,to the carryover effects from one crop year to the next in the rotation.
The optimum level of fertilization will undoubtedly require alternative fertili-
zation rates from year to year.

The amount of carryover of the important nutrients will depend upon the
climate, physical and chemical properties of the soil, crops grown, soil man-
agement practices, and the like. If there are carryover effects, the optimum
level of fertilization can be determined by equating the discounted value of mar-
ginal responses with the marginal cost of each fertilizer increment.plus the
compounded interest on the investment in the residual fertilizer.

Knowledge of the residual effects can reduce the importance of uncertainty
considerations if the farm operator knows that although the weather during the
first year may be unfavorable, probabilities are high for getting a large resi-
dual effect in the following years. If this were the case the farmers would not
be so timid about using fertilizers. In addition, knowledge of residual response
functions allows farmers to discount fertilizer returns to fit their own particu-
lar capital and uncertainty situations. On a practical basis the size of the dis-
counted rate would differ with each farmer. If we can develop our research to
supply information on time sequences of yield responses we can aid the farmer
in using fertilizer to fit his own unique circumstances.

V. Importance of Fertilizers in Conservation Farming

I have previously stated that in the past there was a considerable tendency
to treat the use of fertilizers as a resource practice apart from other resource
practices. If we consider fertilizer in isolation, the broader role and economic
significance of fertilizer is overlooked. In addition, this approach bypasses
many of the broader methodological problems in the economics of fertilizer use.

W. W. McPherson cited this possible danger in a paper titled Mathemati-
cal Functions Used in Estimating Production Functions presented to the South-
ern Farm Management Research Committee, Memphis, Tennessee, January
5, 1954.

2Continued cropping of the same crop is considered a rotation here.

Productive research in this particular field should consider the interaction
of crop combinations or crop and fertilizer applications. The whole farm ap-
proach enables recommendations to be made on optimum combinations of en-
terprises and fertilizer plans. These determinations, of course, are made
jointly. Fertilizer is one factor of production which can quickly give returns.
Hence, the use of fertilizer can belconsidered an income complement" in
major farm adjustment problems. It is well known that most of the major
farm adjustment problems require several years in planning and in operation
before increases in the income flow are apparent. Some examples are: (1)
the adjustment from exploitive to conservation farming systems, and (2) ad-
justment from dryland to irrigation farming.

The requirements of farm adjustments cause these major shifts to be dis-
counted in the farmers' decision-making process because of capital limitations,
opportunity costs of using funds, and uncertainty relating to the outcomes in
the more distant future. A resource such as fertilizer--one that has a relative-
ly short period for transforming investment into profit--can help overcome
some of these effects and speed up adjustments.

Importance of fertilizer in shifting towards a conservation system of farm-
ing is that it closes the income gap soon after adopting a conservation plan.
This effect deters farmers from shifting to a system of farming that features
few or no conservation practices. Fertilizer--if properly programmed into
the over-all farm plan--can play a vital role in conserving the nation' s land
resources. Of course, the ability to properly begin applying fertilizer depends
upon our having adequate knowledge of crop response to varying rates of appli-
cation of the major nutrients and important related variables governing crop

Fertilizer may be considered as a capital investment in developing a farm
plan. We may determine that fertilization of rotation crops would be more
profitable than investment in any livestock practice. This may be true with
limited capital. However, as capital becomes more available, it becomes
more profitable to invest in more livestock and, finally, to use capital in pas-
ture improvement or renovation. Pasture renovation requires fertilization.

VI. The Changing Fertilizer Technology

Economists working on problems in fertilizer-use economics should keep
abreast with the rapidly changing technology in the production of chemical fer-
tilizers. During the past decade the use of nitrogen has tripled, potash use has
doubled, and phosphate use has increased by 60 percent. The ratio of plant
nutrients--nitrogen, phosphorus, and potassium--was about 1:2:1 before 1945;
currently this ratio is approaching 1:1:1. Prices of fertilizers have decreased

1Cf. Earl O. Heady's Fertilization in Relation to Conservation Farming
and Alfocation of Resources Within the Farnm in E. L. Baum, et al., op. cit.,
chapter 13.

2Cf., T. P. Hignett's Our Changing Fertilizer Technology in E. L. Baum,
et al., op. cit., chapter 14.

greatly in recent years, especially prices for nitrogen. Before 1950, nitrate
of soda was the main source of nitrogen; now most of the nitrogen fertilizer
is derived from synthetic ammonia. The use of ammonia and nitrogen solutions
for direct application and the use of solid ammonium nitrate have grown spectac-
ularly since 1945. The production of urea currently is increasing sharply, and
there is an increase in the processes that produce compound fertilizers such
as ammonium phosphates and nitric phosphates.

There are other .major advances in technology, such as mining and benefi-
ciating phosphate rock. Recent reductions in freight rates on fertilizer-grade
acid should encourage the trend to wet-process phosphoric acid in the manu-
facture of concentrated superphosphate and ammonium phosphates. The in-
creased availability of wet-process acid may encourage its use in mixed fer-
tilizers in conjunction with ammoniation, in enriched superphosphate, and in
liquid fertilizers.

In addition to the above-mentioned technological advances, there are others
such as mining, beneficiation, and refining of potash salts. These advances
have been reflected in increased use of high-grade salts and decreased cost.
There has also been the development of combination fertilizer processes and
liquid fertilizers.

The above-mentioned technological advances in fertilizer production--as
well as advances in methods of fertilizer application--have profound economic
implications. This is true not only in the planning of the production of a single
crop in whole farm production planning, but also for the general economic wel-
fare of agriculture. It is therefore important that agricultural economists con-
duct studies in the macro-economic analysis of fertilizer use so that they can
evaluate farm fertilization programs of expanded use of lower cost and improved
fertilizer materials within the context of a national agricultural policy. In
addition, of course, more and better fertilizer response surfaces on all types
of major soil associations under varying climatic conditions and cropping and
soil-management practices are urgently needed. It is these micro-economic
studies, however, that provide the relevant technical coefficients of production
that are necessary for the proper development of the macro-studies more con-
cerned with policy implications.

VII. Acquainting Farmers with Fertilizer Response

We may be able to find worthwhile notions about crop response from ferti-
lizer and related variables from personal experience, observation, and dis-
cussion with other farmers in a particular area. These sources may not give
us complete information, but it is a practical recognition that many plans for
fertilizer use are determined in this manner. It is important that we improve
farmers' ideas on fertilizer response through other educational means, such
as magazines, newspapers, radio, commercial organizations, and college bul-

In the final analysis, the farmer will decide on the amount to be invested
in fertilizer. Can recommendations be based solely on experimental plot ex-
periments? Can management practices be treated as a constant? The answers
to these questions are "no."' However, experimental research conducted by
experiment station personnel can serve as guides and indicate the nature of
the response surface, yield isoquants, and fertilizer isoclines, Fertilizer-use
recommendations require a great deal of judgment to fit response data to soil
tests and different levels of management. The success in conveying the results

of experimental research to farmers in the form which will enable them to plan
economical fertilizer use depends upon the agricultural workers' ingenuity.

VIII. Summary Statement

To summarize this paper, the economic analysis of fertilizer-response
data and the incorporation of fertilizer-use data in whole farm plans are rela-
tively new fields of agricultural economic research. In order to develop more
meaningful fertilizer-response coefficients--as well as other production co-
efficients necessary for sound economic recommendations and whole farm plans--
agricultural economists must work more closely with agricultural production
scientists and statisticians. In this manner, economists may take part in the
original planning and play an active role in the experimental work so as to make
sure data are in a form suitable for economic analysis. More emphasis should
be placed on correlating crop response to fertilizers with soil tests and other
factors influencing crop production.,

The economist is concerned with the mathematical form of the fertilizer
production function, alternative experimental designs, and estimating pro-
cedures. He should be concerned also with the considerations of sufficient rep-
lications, as well as predicting the response curve and its standard error.

Of paramount importance is the development of experimental designs which
allow both statistical and economic efficiency in the estimation of response sur-
face s.

Research data from many experiments--along with soil tests, chemical
quality of the crop, and other data on relevant variables--must be accumulated
before economists can develop general estimating functions. If the proper
technical coefficients are developed, a sound economic fertilizer program may
be selected in the complete context of the farm business--fertilizer being con-
sidered as an investment opportunity. It is important that agricultural produc-
tion economists participate in research which provides estimations of the effects
of the application of different nutrient rates and ratios for different crops and
rotations. These relationships should be developed for various soil associa-
tions, under variable climatic and soil moisture conditions, variable manage-
ment practices, and other variables important in agricultural production. Ques-
tions about soil fertility depletion and fertility buildup--as well as questions per-
taining to timing of fertilizer applications in crop rotations--may be answered
through the above types of research.

Finally, economists should be aware of the rapidly changing fertilizer tech-
nology, trends in use, and the over-all economic policy implications from the
expanded use of this increasingly important factor of production.


H. B. Peterson

The material contained in these papers appears to be largely of a non-
controversial nature. Mr. French has made a contribution by treating some
actual field-trial data with mathematical techniques which have been employed
to study input factors. His suggestion that it is unlikely that any one function
will be specified as the "law of plant growth"should not be too discouraging.
It suggests to me that population with which we want to work must be defined.
The best function for that population must then be determined and used. This
population should be defined in terms of the fixed variables that influence the
yield of the crop in question.

It is my opinion that, when defined, the population must be sampled at
random and that this be done by taking a large number of samples of a simple
design. The larger and more variable the population included as a recommen-
dation unit, the greater should be the range of recommended rate of fertilizer.
In order to recommend a "best" rate for any farm an individual case study
should be made.

With an increase in the use of irrigation over the entire nation we should
be aware that there is a strong interaction between soil moisture and response
to nitrogen fertilizer. Where irrigation is used, water should be removed
from the group of..fixed variables, or the use of water should be considered in
describing the population.

It appears to me the paper of Dr. Baum comes as a fitting climax to our
meeting. His is an excellent summary of what has been presented during the
last two days. Certainly the information presented by French and Baum enable
us to take from these meetings a better understanding of the problems we face.

1Head, Department of Agronomy, Utah State Agricultural College, Logan,


Members of the Farm Management Research Committee

Joseph Ackerman
Warren Bailey

Chester Baker
Roland Bevan
Howard Diesslan
Joseph T. Keeler
Rex Rehnberg

Gordon Sitton
Douglas Strong
(Alternate for
Ernest Morrison)
Jo P. Swanson
John W. Thomas
Andrew Vanvig
Tim Wallace
(Replacing G. P.
Garland P. Wood

Others in Attendance

D. G, Aldrich

Owen L. Brough, Jr.
William Brown
Douglas Caton
Burton L. French
Glenn Johnson
Bert A. Krantz
Bernard Ostle
J. L. Paschal
Ho B. Peterson
J. S. Plaxico
Henry Stippler
Russell Shaw
Frank Yiets
G. B. Wood

Farm Foundation, Chicago, Illinois
Prod. Econ. Res. Br., ARS, Washington, D.C. (Secre-
Montana State College, Bozeman, Montana
University of Idaho, Moscow, Idaho
Farm Foundation, Chicago, Illinois
University of Hawaii, Honolulu, Hawaii
Colorado A & M College, Fort Collins, Colorado (Chair-
Oregon State College, Corvallis, Oregon
Utah State College, Logan, Utah

Washington State College, Pullman, Washington
New Mexico State College, Las Cruces, New Mexico
University of Arizona, Tucson, Arizona
University of Nevada, Reno, Nevada

University of Nevada, Reno, Nevada

University of California, Berkeley, and Davis, Califor-
Washington State College, Pullman, Washington
Oregon State College, Corvallis, Oregon
Prod. Econ. Res. Br., ARS, Berkeley, California
Prod. Econ. Res. Br., ARS, Washington, D.C.
Michigan State University, Lansing, Michigan
ARS, U.S.D.A., Billings, Montana
Montana State College, Bozeman, Montana
Prod. Econ. Res. Br., ARS, Fort Collins, Colorado
Utah State Agricultural College, Logan, Utah
Oklahoma A & M College, Stillwater, Oklahoma
Prod. Econ. Res. Br., ARS, Corvallis, Oregon
University of California, Davis, California
ARS, U.S.D.A., Fort Collins, Colorado
Oregon State College, Corvallis, Oregon

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