ECONOMiC OPTIMA FROM AN
EXPERIMENTAL CORN-FER‘HUZER
PRGDUCTEON FUNCYEON: CAUCA VALLEY;
COLOMBEA‘ S. A... 1958

Thu]: for {he Dogma of M. 5.
MICHIGAN STATE UNIVERSITY

Enrique Delgado C.
1962

 

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Michigan State
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ABSTRACT

ECONOMIC OPTIMA FROM AN EXPERIMENTAL
CORN-FERTILIZER PRODUCTION FUNCTION,
CAUCA VALLEY, COLOMBIA, S.A.. 1958

by Enrique Delgado C.,

Latin America has a large number of farms where
fertilizers are needed. For those farms where use of ferti-
lizers is a common practice, little attention has been paid
to economic optima.

Michigan State University has initiated fertilizer
experiments in the Cauca Valley in Colombia, South America.
These have been multiple purpose experiments. This thesis
reports an analysis of data produced by an experiment
designed to permit determination of the most profitable
amounts of fertilizers to use.

The crop studied was corn and the variable nutrients
were nitrogen, phosphorus and potassium. In addition, three
different plant populations were studied and half of the
120 plot observed were irrigated.

A production function was fitted to the data
obtained from the field experiment. The function used was

as an incomplete second degree polynomial of the form:

Enrique Delgado C.,

2 2
Y = + b + + b K + + b + NK + +
a lN b2P 3 b4N 5P b6 b7I

2
+ + + .
has bgs blONs bllPs

This function was found adequate by inspection of the
distribution of unexplained residuals for the analysis of
the data according to standard statistical tests. The unex-
plained residuals for the experiment and function fitted
were graphically studied.

High profit points were computed under different
assumptions of price for both inputs and output. It was
observed that when the price of corn was low (between $1.00
and $1.30 Bu.) and the price of nitrogen was fixed at
$ .068 KG., the use of nitrogen was not profitable. When
the price of corn was $1.60 Bu., however, the use of 42.7
lbs. of nitrogen per acre became the most profitable quantity
to use. The increase in the price of corn showed that the
optimum quantity of phosphorus to use is also increased
although slightly (from 47.2 lbs/acre up to 51.6 lbs/acre
when the price of corn is changed from $1.00 to $1.60).

This study showed that use of estimated HPP quantities of
fertilizers increased profits about $24.75 per acre over
the use of no fertilizer.

The experiment was performed at only two levels of

irrigation; this limited inferences about changing marginal

Enrique Delgado C..
productivity of irrigation. No irrigation cost data were
available. Yields averaged 18.340 bushels per acre higher
on those plots which received irrigation.

The data here analyzed represent observations from
one year only. The promising results obtained under economic
analysis should encourage continuation of this kind of
research. Valuable experience was also gained.

The extension of results from this kind of economic
analysis to the farm level may provide positive assistance
in the general effort to increase productivity and standard
of living in agricultural sectors of several Latin American

countries.

ECONOMIC OPTIMA FROM AN EXPERIMENTAL
CORN—FERTILIZER PRODUCTION FUNCTION.

CAUCA VALLEY, COLOMBIA, S. A., 1958

BY

Enrique Delgado C..

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Agricultural Economics

1962

ACKNOWLEDGMENTS

The author is indebted to several persons and
institutions who helped, directly and indirectly, my graduate
work at Michigan State University.

The Ministry of Agriculture and the University of
Chile have given the necessary leave of absence to make use
of a fellowship granted by the Food and Agriculture Organization,
FAO. of the Uhited Nations whose financial help I sincerely
appreciate. I want to thank the former Minister of
Agriculture, sefior Jorge B. Salezer, for his personal interest
in obtaining the Presidential Decrete of permission to leave
the country.

Dr. Glenn L. Johnson, my major professor, with his
guidance, comprehension and high University spirit. has made
possible the achievement of this thesis and encouraged all
my graduate work.

Mr. Dennis Oldenstadt. from the Agricultural Economics
Department gave me invaluable help with his numerous
suggestions, supervision of the computations and reading
and correction of the drafts of this thesis.

Drs. Kirk Lawton from the Soil Science Department

and Leonard Kyle from the Agricultural Economics Department

ii

both gave me additional and useful information about the
data and conditions of the experiment whose results are
analyzed in this thesis; my thanks to both of them.

I am grateful to the Department of Agricultural
Economics which provided an assistantship during a month of
financial uncertainty.

The personnel working with MISTIC. and in the statistical
pool of the Agricultural Economics Department were always
willing to work out my computational requirements.

Finally. I wish to present my special appreciation
to my wife and daughter for having accepted sharing with me
the peculiar family life of a graduate student in this

country.

iii

TABLE OF CONTENTS
Chapter
I 0 INTRODUCTION 0 O O O O O O O O O O O O O O O

A. General background

B. Fertilizers in Latin America

C. The source of data for this thesis
D. Approach of this thesis

II. THEORETICAL BASIS OF THE PRODUCTION FUNCTION

A. Definition
B. A brief historical review
C. Mathematical forms

1) Exponential functions
2) Polynomial functions

i) Quadratic polynomial functions
ii) Square-root polynomial

functions

3) Cobb—Douglas or power function
4) Carter-Halter—Hodking function

D. Actual use
E. Selecting mathematical forms

III. THE HYPOTHESIZED MODEL . . . . . . . . . . .

A. Agronomic model
B. Empirical work

i) Statistical planning of
experiments

ii) The straight line function
iii) The curvilinear functions

IV. CHARACTERISTICS OF THE DATA HERE USED . . .

The Colombian Experiment

;iv

Page

|._.n

\lxlU'lw

10
15

18
20

20

20

21
22

23
24

27
27
31

31
34
39

45

45

Chapter

1) Irrigation

2) Plant populations
3) Fertilizer used
4) Corn yield

V. FITTING THE DATA TO THE HYPOTHESIZED MODEL

A. Seeking the estimated production
function

B. Some statistical remarks

C. Analysis of the Residuals

VI. ECONOMIC ANALYSIS OF THE DATA AND
STATISTICAL ESTIMATES . . . . . . . .

A. High profit point analysis under
different price assumptions

1) The effect of various prices

of nitrogen. PN, and of

corn. P“ on optimum

Y0
quantities of N to use
2) The effect of various prices

of nitrogen. PN'

quantities of N to use
3) The effect of various prices
of corn. PY, on optimum

quantities of P to use
4) The effect of various prices
of corn, PY' on optimum

quantities of seed. S. to
use
5) The effect of various prices

of seed, PS. on optimum

quantities of N. P and S
to use

on optimum

Page

47
47
48
50
52

52
55
57

61

61

67

69

69

71

73

Chapter Page

B. Comparison of some predicted
yields. E. with and without
irrigation under varying prices
of corn and fixed prices of N.
P, S, and a fixed quantity of

K 73

VII. EVALUATION. RESULTS AND IMPLICATIONS . . . . 78

A. Evaluation 78

B. Results 80

C. Implications 82

APPENDH A o o o o o o o o o o o o o o o o o o o o o 86
APPENDIX B O O O O O O O O O O O O O O O O O O O O O 91
BIBLIOGMPHY O O O O O O O O O O O O O O O O O O O O 92

vi

Table

LIST OF TABLES
Page

Production. consumption and net balance of
various fertilizers by regions of the
world, 1958 . . . . . . . . . . . . . . 15

Fertilizer consumption in selected South
American countries, 1958-59 . . . . . . l6

Hypothetical data demonstrating the
computational procedures of least square
regression . . . . . . . . . . . . . . . 37

Use of N. P. and K on irrigated and not
irrigated experimental plots with three
levels of plant population in the Cauca
Valley of Florida, Colombia. 1958 . . . 48

Different price levels of corn, N, P, and
S; the amount of K being constant . . . 62

HPP quantities of N, P and S. at various
prices of corn. P and PN. Values for

P

Y!

P and K are held at fixed amounts

P' S . 66

High profit number of plants per acre at
various prices of corn and with P

P

N!
P' PS and amount of K held at
specified levels . . . . . . . . . . . . 7l

vii

LIST OF FIGURES

Baule units of growth factors . . . . . .

Graph of

Graph of
with

Graph of
c X2

Graph of

Graph of
when

the function E a + b X . . . .

the function Y = a + b X + c X2
given values for a, b and X . . .

the function log Y a + b X +
whena=0.b>0,c<0 ....

the function Y = a + b log X . .

the function log Y = a + b log X
b<1..............

Effects of various PY on optimum levels of

N 0

Effects of various PN on optimum levels of

N 0

Effects of various PY on optimum levels of

P .

viii

Page
14

35

40

42

42

43

68

70

72

CHAPTER I

INTRODUCTION

The main objective of this thesis was to estimate the
quantities of fertilizer inputs which would have maximized
net returns in the production of corn on a fertilizer exper-
iment in Colombia, South America. HOwever, the author has
in mind additional objectives such as summarization and
translation of this work into Spanish to be published in
the near future.

It is well known that in general in Latin America the
research on and the teaching of agriculture at the University
level has been carried out for many years under unsatisfactory
conditions. Lack of funds has been one of the most serious
handicaps. This scarcity has been reflected in a shortage
of buildings, equipment, laboratories and other facilities.
On other occasions when requirements for physical facilities
were fulfilled, a shortage of instructors became a serious
problem. There were also cases where expensive facilities
and instructors were available but where there existed a
shortage of students.

Almost all the twenty Latin American countries have

at least one school of agriculture. Some of these schools are

1

2
as old as in the United States of America. For example,
Chapingo in Mexico, is more than one hundred-eighty years
old having been founded around 1776. There are several
countries with schools of agriculture founded at the end
of the last century. European influence is great in a large
number of our colleges. The bachelor degree in agriculture
is called ”INGENIERO AGRONOMO" in resemblance of the title
given by the Institute Agronomique du Paris, France. Colleges
in Chile reflect a strong French influence. In fact, by 1875
several agronomic experiments were carried out for French
professors specially contracted to promote agricultural
teaching. Other Latin American countries had similar
experiences.

For at least eighty years, experiments on fertilizer
use have been carried out in different Latin American countries
under the control of some College of Agriculture.. In
addition, private organizations have developed their own
experimental stations. The results of these experiments
have been, in many cases, poorly extended to the farmers.

The main reason for this limitation may be found in the
organization of schools of agriculture; in many instances,
extension programs were forgotten. Mbst of these experiments

were done by able professional people, some even at personal

3
sacrifice, but unfortunately the real value of the work
remained almost unknown in many instances.

Since 1930, important rearrangements in the experi-
mental stations were made in many Latin American countries.
During the decade of the 40's, the experiments were generally
worked out using statistical analysis. Of course the
statistical approach was used much earlier in some experiment
stations.

very useful agronomic results were obtained from
experiments on fertilizer use. However, the economic analysis
was often neglected, badly used or not used at all.

This thesis is written with the intention of showing
systematically an approach to the economic analysis of a
fertilizer experiment. It is hoped that this work will
eventually be printed in Spanish. As the explanation is in
a simple form, it is hoped that it will serve as a basic
reference in those places of Latin America where such a

reference is needed.

A. General background.

Although the results obtained from an experiment in
fertilizer use can be significant from an agronomic standpoint,

the results may have additional economic significance.

4
For the agriculturist and economist working in a team,
higher yields do not always mean higher profits. The additional
use of fertilizer becomes optimum (most profitable) when the
cost of the last unit of fertilizer used is equal to the
value of the additional output or, in the vocabulary of the
economist, when marginal factor cost equals marginal value
product.

In order to find the economic optimum in the use of
fertilizer, a number of approaches can be used. Some of
these approaches will be reviewed in Chapter II. With these
approaches, the analysis becomes straightforward for meeting
the minimum necessary conditions. In some of the well-
established experiment stations of today, fertilizer
experiments are being controlled by a group of people with
several different interests in the results. In many instances,
experiments are designed to provide the necessary data for
a variety of interests. The soil specialist, for example,
tries to find the best kind of fertilizer for a given type
of soil while the agricultural economist may be interested
in the high profit amount of that fertilizer to use.

It seems pertinent to say here that the economic
analysis that is going to be presented in this thesis can be

applied to fields other than the use of fertilizers, as for

instance, livestock feeding or any experiment where a group

of inputs are combined in the production of a product.

B. Fertilizers in Latin America.

In order to compare fertilizer production and con-
sumption in South America, Central and North America with
Europe, the following figures are given from a F.A.O. report:

Table 1. Production, consumption and net balance of various
fertilizers by regions of the world, 1958.

 

 

Production Consumption Balance

(Thousand metric tons)

 

South America

Nitrogen 291 132 + 159
P205 85 159 — 74
K20 16 91 - 75

North and C. America

Nitrogen 2,358 2,272 + 86
P205 2,398 2,259 + 139
K20 1,797 1,764 + 33
Europe

Nitrogen 4,712 3,293 + 1,419
P205 3,833 4,175 - 342
K20 5,074 4,221 + 853

 

1Annual Review of World Production and Consumption of
Fertilizer, Food and Agriculture Organization of the United
Nations, FAO, Nov.,l958.

 

6

Two conclusions are suggested for South America:
(a) with the exception of nitrogen, the other two fertilizers
are in a serious deficit production position, and (b) considering
that, in general, the intensity of fertilizer use is still
low, it can be concluded that South America must find additional
sources of phosphorous and potassium fertilizer in order to
increase crop productivity.

On taking into consideration the 1958—59 fertilizer
consumption in five South American countires, the following
is found, according to the same F.A.O. information:

Table 2. Fertilizer consumption in selected South American
countries, 1958-59.

 

 

Country N 1 P205 K20

(Thousand metric tons)

 

Colombia 7,000 17,500 5,400
Argentina 6,000 3,400 3,400
Brasil 33,000 73,000' 59,000
Chile 36,000 33,000 9,000
Peru . 36,000 14.300 4.000

 

Perhaps the most surprising facts shown by the figures
above is that Argentina, one of the world's top wheat and

corn producers, is the lowest consumer of fertilizers.

7

C. The source of data for this thesis.

The data here analysed were provided by an experiment
carried out at Florida, Cauca Valley, Colombia, South America.
Personnel from Michigan State University have been working
cooperatively with the Faculty of Palmira at the Cauca River
Valley. Full details on these data will be given in Chapter

IV of this work.

D. Approach of this thesis.

This thesis will first present the theoretical basis
for using production functions. This will be developed in
Chapter II, the main sub-heading of which will deal with a
short historical review of the use of production functions
from earlier days until now. Then, the mathematical meaning
will be explained briefly. At the same time, actual uses of
production functions will be summarized. The final part of
Chapter II will deal with the problem of selecting appropriate
mathematical expressions for production function analysis.

In Chapter III, the hypothesized model will be
presented. At the same time a brief review will be made of
the general agronomic conditions under which fertilizer
experiments are performed. Certain uses of mathematical

functions will be illustrated with an example.

8

In Chapter IV, Characteristics of the data used in
this thesis will be presented including the available
information on the Colombian experiment, the quality of the
data collected and the relevant characteristic of the
experimental design.

In Chapter V, data from the field experiment will be
fitted to the hypothesized model.

In Chapter VI, an analysis of the results obtained
from the previous chapter will be made. High profit points
(HPP) and predicted yields will be found for optimum quantities
of fertilizers at various price levels of inputs and output.

Finally, Chapter VII will be devoted to an evaluation

of the approach used and the results obtained.

CHAPTER II

THEORETICAL BASIS OF THE PRODUCTION FUNCTION

Use of input-output analysis has increased greatly
during the last several decades. This approach is of para-
mount importance to the economic analyst dealing with fertilizer
experiments. Mathematical functions have been employed to
explain and predict input-output relationships.

This chapter discusses the underlying concepts and
use of mathematical functions to describe relationships

between plant nutrients and crop yields.

A. Definition.

 

A function can be defined as the relationship between
two variables. One of the variables is dependent on the
others. For the usual production function, a dependent
variable Y (which can be translated as output per unit of

time) and X ,..., Xh, which are inputs per unit of time or

1

independent variables. In this case the production function
can be written as follows:

Y=f(xl,...,xn)

which means that output (Y) is a function of or depends on

the inputs X ,..., Xn. The inputs involved in the production

1

10
function can be classified into two groups, variable and
fixed. The usual notation for this is Xl'X2 where the
bar (|) means ”given X2 fixed." Relationships between

variables can be shown by word description, tabulation or

graph.1

B. A brief historical review.

The famous German scientist Justus von Liebig may
be considered the first person to devise the production
function concept. In his famous "law of the minimum" he
held that the yield of any crop is determined by changes in
the quantity of that factor which appears in lowest amount:
this factor is called the minimum factor. If this minimum
factor is increased, the yield of the product is increased
in proportion to that factor until another nutrient becomes
limiting. If another factor—-not at the minimume-is
increased, the yield of the product does not change. In
other words, Liebig intended to show that the yield of any
product is a linear function of the minimum factor.

Liebig's law was formulated about 1860. It has had

 

1 O I I
The function can be continuous or discrete. In

developing a function in production economics, cardinal
numbers (1, 2, 3, etc.) are used in contrast with the ordinal
numbers (lst, 2nd, 3rd, etc.) used in consumption economics.

11
a tremendous influence on agricultural scientists for almost
a hundred years, especially on agronomists and on farm
economists.

The well known graphic illustration of a water barrel
with the lowest stave representing the minimum factor that,
in turn, shows the limit of water (or profits in the case
.of farm business) was widely used in presenting Liebig's
concept. Under this circumstance, Liebig recognized constant
returns to the limiting factor but denied the presence of
factor substitution. He did not consider substitution of
resources and that farmers, for example, on considering the
factor-product price ratio can add the minimum factor profit-
ably as long as the marginal factor cost (MFC) is greater
than or equal to the marginal value product (MVP).

Researchers have rejected Liebig's formulation for
two main reasons: (a) factors of production are seldom
perfect complements and a given crop yield can be produced
with different quantities and combinations of nutrients

(such as P 0

2 5, N and K20), moisture, heat, etc., providing the

necessary minimum amount of each is present, and (b) successive
additions of factors limiting crop yields do not necessarily
result in linear additions to crop yields but result, instead,
in diminishing additions to crop yields and eventually

decreases in total yield.

12

Lawes and Gilbert at the Rothamsted Experiment
Station, England, demonstrated that the law of diminishing
returns operates in fertilizer use.1 Ewald Wollny at the end
of the last century (1897—98) conducted an experiment to test
Liebig's law of the minimum. He concluded that additional
amounts of nutrients cause a rise in production of a plant
which first is progressive and eventually becomes smaller
and smaller arriving at a limit where further additions of
nutrients prokae yield reduction.2

Jethro Tull thought that yield was an increasing
function of inputs which varied with manure or tillage.
He claimed that as more tillage was used on a crop, it
became less expensive and that crop yield increased. Most
of the people have rejected Mr. Tull's conclusion, in part,
' because he was an inventor and manufacturer of tillage
machinery3 and more importantly because it does not always

meet the test of experience.

 

1E. J. Russell, Plant Nutrition and Crop Production

(Berkeley: University of California Press, 1926).

2‘W. J. Spillman and E. Lang, The Law of Diminishing
Returns and the Law of the Soil (New YOrk: WOrld Bock Co.,
1924), p. 100. (Quoted from Ewald‘Wollny's ”Untersunchungen
uber den Einfluss der Wachstumsfaktoren auf das Produktronsver—
mogen der Kulturpflantzen,” Forschungen auf dem Gebiete der
Agrikulturophysik 1897-98, p. 105-06).

3Jethro Tull, Horse HOeing Husbandry (London:
William Cobbett, 1829).

13

Hellriegel thoroughly studied the application of
nutrients to a plant in 1880. He did his experiment using
nitrogen (N) on a barley crop. In the beginning, additional
amounts of N gave increments in yield followed, as the N was
increased in amount, by greater increments in yield--greater
even than those shown by Liebig's law. Finally, when still
more N was applied, a declining incremental yield response
was observed and the well known sigmoid yield curve became
evident. This experiment was repeated several times and the
yield curves were always similar.

E. A. Mitscherlich at the Experiment Station of
Koenigsberg, Germany, observed that the sigmoid yield curve
could be represented by a mathematical equation in order to
quantify the manner in which crop yields were related to
plant nutrients. His principal assumption was that maximum
yields would be obtained under ideal conditions unless any
essential growth factor were shown to be limiting.2

Baule, on enlarging Mitscherlich's ideas, suggested

'that the final yield is the result of all factors working

together. The ideas of Baule can be summarized as follows:

 

1E. J. Russell, Soil Conditions and Plant Growth

(New YOrk: Longmans, Green and Company, 1950).

2J. Redman and S. 0. Allen, Journal of Farm Economics,
m1 (August,1954). p. 457.

14

yield responses will be determined by the level of use of
the fixed factors. If only one factor is variable and the
others fixed, the response to this unique factor is lower

I 1
than when two or more factors are used as variables.

X.‘, X2.) X3

X.,X;/X3,... Xn

)(JX..,... X.

 

 

Figure l. Baule units of growth factors.

In Figure 1 9a Baule unit” effect is illustrated. A Baule
unit is the amount of a yield influencing factor necessary
to produce one half the maximum yield when other factors are
at their optima.
W. J. Spillman in the USA, a contemporary of Mitscherlich

in Germany, developed an equation dealing with the growth

 

lSpillman and Lang, op. cit., p. 147.

2Redman and Allen, op. cit., p. 458 fn.

15
factor's effect on plant yields.

From the early 30's, when Mitscherlich and Spillman
developed their equations, until now, a number of other
equations and mathematical functions have been used including
the Cobb-Douglas, Carter-Halter-HOCKing and a large number
of less specialized polynomial equations. Most of these
will be explained later on in this thesis.

Heady and othersl have used two typesof polynomials
in a prediction equation for corn. Johnson, when working
with a production function, has placed particular emphasis
on the distribution of "residuals? (u‘s) generated by
uncontrolled factors2 in selecting mathematical functions

to represent production responses.

C. Mathematical forms.

When trying to find the mathematical form of the most
suitable function for the analysis of available fertilizer

data, physiological and biological growth have to be

 

1
E. O. Heady, J. Pesek, and W} Brown, 9Crop Response

Surfaces and Economic Optima in Fertilizer Use," Research
Bulletin 424, Iowa Agricultural Experiment Station (1955).

2G. L. Johnson, FInterdisciplinary Considerations in
Designing Experiments to Study the Profitability of Fertilizer
Use,9 Economic Analysis of Fertilizer Use Data, edited by
Baum and others (Ames, Iowa, 1956), P. 26.

16

considered. When all the growth factors except one are
fixed, the expected increase in yield is influenced by the
varying proportions between the variable and the fixed
factors. Here the law of diminishing returns comes into the
picture. When the level of the fixed nutrients is extremely
low, the yield of the product may decrease as successive
additional amounts of the variable factor are applied. The
action of N under low moisture is a good example of the
preceding statement.1 On the other hand, when the fixed
nutrient factors are near the physical optimum, the total
physical product (TPP) should be expected to increase first
at an increasing rate and then at a decreasing rate until a
maximum (theoretical) is reached beyond which output shall
be expected to decrease. Experiments done in Nebraska,
Oregon and washington confirm this assertion.

The results of an input-output experiment using
fertilizers as inputs are not only subject to variations
because of controlled forces but also because of uncontrolled

forces. MOre details on this topic will be given later on.

 

lHeady, et al., op. cit. In Iowa, N was applied at
different levels of P O in a corn-N-P experiment; the fixed
level of moisture was found very low, p. 330.
2J. L. Paschal and B. L. French, "A Method of Economic
Analysis Applied to Nitrogen Fertilizer Rate Experiments on
Irrigated Corn," USDA Bulletin 1141 (1956).

17
Attempts to describe input-output relationships have
included use of the following functions:

1) Exponentials

2) Other polynomials:
i) quadratic and
ii) square root

3) Cobb-Douglas or power function1 (a special case of
the exponential Carter-Halter-Hbcking function) and

4) the exponential Carter-Halter-Hocking function.

An early mathematical formulation was due to
.Mitscherlich. His function that he called Alaw of diminishing
soil yield? intended to show that a maximum yield is obtained
when one of the essential growth factors is limited.

This can be expressed as

Y = A(1 - e—cx)
where,
A = maximum possible yield
e = a constant
c = a constant
x = input

Mitscherlich believed that additional yields brought about

 

1E. O. Heady and J. Dillon, Agricultural Production

Functions (Iowa State Press, 1961).

 

18
by one factor had no effect on the productivity of the other
nutrient factor. This belief was modified by Baule who used
the production function:

G X
Y=A(1-1011)(l-10

c X c X

22)...(1-10“)

where,
A = maximum yield
cn= effect factors

Xn= variable growth factors not considered by Mitscherlich.

l) Exponential functions.

 

An exponential function was developed by Spillman
in analyzing fertilizer data of a tobacco experiment.1 His
production function was:

Yj=M-ARx

where,

Yj = is the yield

M = maximum yield possible to obtain theoretically

A = the increase in yield between the yield with no
application of fertilizer and the theoretical maximum

R = the constant ratio of successive increments in yield

X = the variable.

 

1W; J. Spillman, “Use of the Exponential Yield Curve

in Fertilizer Exponents,9 USDA Technical Bulletin 348 (1933).

l9

Spillman found in the Baule and, consequently,
Mitscherlich's formulations a good deal of inspiration for
his own formula. The main difference between Mitscherlich
and Spillman is that the former claimed that the ratio of
successive increments in yield, given a unit increase of a
given growth factor, is the same for all crops and all soils,
providing no other factors are limiting.

Spillman's exponential production function has been
found useful for a number of input-output relationships but,
at the same time, it has been pointed out that it is unable
to give estimates of those segments where the data under
analysis show negative marginal products (or diminishing
total returns). On the other hand, it is said that the
exponential function gives a good answer providing that A
(the constant) has a positive sign and the crop yield becomes
asymptotic to M. In this case M is a minimum, not the
maximum, as originally assumed in the function.

The Spillman function, expressed for general cases,

can be written as follows:

_ x x x
Y-M(l-Rll) (l-R22) (l-Rnn).

20

2) Polynomial functions.

It is possible to conceive of an infinite number of
polynomial functions. HOwever, those equations involving a
third or higher degree have not been used much because they

have been considered unnecessary to describe data used.

i) .guadratic polynomial functions.

It is usual to speak about the family of polynomial
functions. One widely used member of this family is the
quadratic, being represented as follows:

Y = a + bl X + b2 X2
The quadratic function can be fitted using the method of
least squares and has the favorable Characteristic of permitting
terms to be added or subtracted giving a new pattern to the
function. When using this quadratic function, a negative
marginal product can be shown when, for example, fertilizer

application produces a restriction on the growth of the crop

under experiment.

ii).§guare-root polynomial functions.
A type of square root polynomial function was used

by Heady, Pesek and Brown1 as a prediction equation for corn:

 

1Heady, et a1. op. cit.

21

= + +
Y a+leN bZJP+b3N b4P+b5JNP

The square root seemed to provide better estimates

of relationship than the quadratic function.

3) Cobb-Douglas orgpower function.1

This function has been widely used by researchers
for the analysis of the input—output relationships. This
function has the form:

b1 b2 bn

Y = a X1 X2 ... Xn

This function becomes a linear function when the dependent
and the independent variables are transformed to logarithms.
Estimation of parameters by least squares is easy because of
the linearity characteristic.

Two of the chief restricting assumptions of this
function is continuously increasing yields if 0<bi and a
constant factor elasticity of production. ‘With this function,
yield can increase at an increasing, constant or diminishing
rate. However, the response curve can only show one of these
stages, not a combination. This is one of the limitations of
the Cobb—Douglas function. Another characteristic is that

it takes on a zero value where any input is zero.

 

lHeady and Dillon, op. cit.

22
4) Carter-Halter-Hocking function.1

This is an exponential function of the form

_ b1 N b2 P b3 K
Y — a N cl P c2 K c3

Before being fitted by least squares, this function also
requires a logarithmic transformation. Carter and Halter
(formerly from M.S.U.) and Hbcking worked out this function
as a more general form of the Cobb-Douglas.

The chief advantage of this function is that it is
more flexible than the Cobb-Douglas. HOwever, it requires
two parameters for each variable (i.e., for variable N,
parameters b1 and C1 are needed). Greater complexity in
locating various economic optima is another disadvantage.

In the Carter—Halter-Hocking function, the bi's and
ci's have to be positive and less than one in order for it
to reflect the law of diminishing returns. This function,
under this condition, will show a total product increasing
at a decreasing rate and eventually decreasing. Preliminary
results in fitting this function have shown that it is able
to describe the three stages of production satisfactorily.

1Journal of Farm Economics, VOl., XXXIX, No. 4
(Nov., 1957).

23
For a long time, it was the general belief that one
functional form could be found to describe the relationship
between plant nutrient and yields. No such single function
has been found yet. Researchers have to select, from among
the vast number of functions conforming to all or part of the
law of diminishing returns, one which does a reasonably good

job of describing their data.

D. Actual use.

During the last twenty years, increasingly wide use
of production functions has been made in the analysis of
data coming from different controlled fertilizer experiments.

A number of researchers making economic analyses of
fertilizer data have been using the production function
approach. A large number of studies could be mentioned but
perhaps the most illuminating are those performed at
Universities like Iowa, Michigan State, Oklahoma, California

and those made by technical departments in the TVA and USA.1

 

lH. Bertolotto, “Economic Analysis of Fertilizer
Input-Output Data from the Cauca Valley, Colombia” (unpublished
Masters Thesis, Department of Agricultural Economics, Michigan
State University, East Lansing, Michigan, 1959); G. I. Trant,
”Implications of Calculated Economic Optima in the Cauca
valley, Colombia, S. A.," Journal of Farm Economics, XL
(Feb., 1958).

 

24
Other workers dealing with dairy problems have also
made use of production function in their analytical approach
to the study of available research data.1 Production functions
have been as widely used for whole farms as for individual

enterprises and the results have been at least as reliable.

E. Selecting mathematical forms.

There is no definitive criteria for selecting a
production function to describe an input-output relationship.
However, Johnson has discussed several problems which should
be taken into consideration in designing an experiment and
in selecting functions.3 In discussing selection of
appropriate functions on the basis of objective statistical
tests, he states, "Perhaps a fruitful approach would be to
test the degree to which the alternative functions individually

meet the usual assumptions with respect to the distribution

 

E. Jensen, et al., "Input—Output Relationships in
Milk Production," USDA, Technical Bulletin 815 (1942).

2G. L. Johnson, ”Sources of Incomes on Upland Marshall
County Farms," Progress Report 1, Kentucky Agricultural
Experiment Station (Lexington, Kentudky, 1952); G. Tintner,
O. H. Brownlee, ”Production Functions Derived From Farm Records,”
JOurnal of Farm Economics, V01. 26 (1944): E. O. Heady,
FProduction Functions from a Random Sample of Farms,“
Journal of Farm Economics, V61. 28 (1946).
3G. L. Johnson, "Discussion: Economic Implications
of Agricultural Experiments,9 Journal of Farm Economics,
Vol. XXXIX. No. 2 (May. 1957). p. 391.

25
of unexplained residuals. If this were done, all functions
failing to meet these assumptions could be rejected on an
objective basis without the use of subjective judgment." The
usual assumptions regarding the residuals are: that the Ui
residuals are independently and normally distributed with
zero mean and constant variance. If these assumptions are
met then the usual tests for statistical significance can
be applied in choosing between alternative functions.

These tests, however, may fail to reveal statistically
significant differences between alternative functions.
Johnson recognized this possibility and concluded that in
such cases,90ne would be forced to turn to 'experimenters
judgment', expert opinion and independent information as a
basis for judgment.? The advantage of this approach rests
on the fact that objective statistical tests are first used
in arriving at a decision regarding the appropriate function.
Failing this, subjective concepts and judgments are used in
arriving at a decision.

In general, the function selected must provide a
good fit of the data in the region where: 1) yield is
increasing at decreasing rate and, 2) total yield is
decreasing. The Vgoodness of fit” can be finally determined

with the help of statistical measures. The most important

CHAPTER III
THE HYPOTHESIZED MODEL

The agricultural economist who designs a model for
use in producing and analyzing experimental fertilizer data,
should have in mind not only correlations and mathematical
manipulation of the data, but also the agronomic aspects
of what he is doing.

Early in this chapter, a brief review is made of
the general agronomic conditions under which a fertilizer
experiment is performed. After this, experimental design,
mathematical manipulation and fitting of functions are
discussed and illustrated in some instances with elementary

examples.

A. Agronomic model.

A general mathematical model for a production function
dealing with input-output relationships in using fertilizers
could be written as follows:

YC = f(N,P,K,I,S IC,M,L,W,R,FS) + u
The above formula can be interpreted as follows: the yield
of corn, Yd, is a function of the controlled variables

nitrogen, phosphorus, potassium, irrigation (I), and number

27

26
statistical tests are:

a) the coefficient of multiple correlation and multiple
determination which show and compare variance explained by
regression with the total variance of the yield data; and

b) the standard error of the parameters in the prediction
equation.

Another measure which can be determined is the
magnitude of residual variance not associated with the
regression. The plotting of alternative functions using at
the same time a scatter diagram of the experimental obser-
vations may also be useful. Or, by following logical expec-
tations, the technical relationships of the variables
represented in the function can be checked.

As Sundquist and Robertson1 have pointed out, the
pragmatic test of whether or not a particular production
function formulation is found in its ability to predict over
time. This kind of test "can be applied only by prediction:

further observation and further prediction."

1W. B. Sundquist and L. S. Robertson, ”An Economic

Analysis of Some Controlled Fertilizer Input-Output Experiments
in Michigan,” Technical Bulletin 269, Agricultural Experiment
Station, Department of Soil Science, Michigan State University.

28

of plants per unit of area (S) and of the fixed conditions
such as soil condition (C), management of the soil (M),
animal manure (L), climate and soil moisture (W), and systems
of farming (FS), plus a number of unexplained variations
(u's) which generate the unexplained variations (u) in Y.

It is advantageous to review the agronomic conditions
which are often fixed but which affect productivity of

fertilizers and, consequently, the final yield of the crops.

1) Soil conditions.

 

By considering the chemical composition, texture,
topography, slope and natural drainage of soil, we have a
starting point to assess fertilizer use. Of course, the
most definite way to establish fertilizer responses is field

experimentation.

2) Management of the soil.
Previous management practices (including crop rotation,
manuring, fertilizing, cultural practices, irrigation and

water conservation) are capable of producing significant

alterations in soil properties.1 Examples of this kind are

 

1G. R. Anderson, ”An Economic Evaluation of Three

Soil Nitrogen Tests," (unpublished Masters Thesis, Department
of Agricultural Economics, Michigan State University, East
Lansing, 1958).

29
found all over the world where two fields with the same type
of soil are receiving substantially different fertilizer
treatment. Greater response to nitrogen than to other
nutrients is generally shown by pasture grasses and cereals.
Sugar cane and corn, for instance, need large quantities of
nitrogen. Heavy application of nitrogen on pasture grasses
usually result in a greater response with adequate soil
moisture. waever, when grasses are being grown in mixture
with legumes, the required amount of nitrogen fertilizer is
reduced. On the other hand, the need for minerals such as
phosphorus, potassium or calcium may be increased.

Sometimes small amounts of "minor" mineral nutrients
are enough to produce crop response. Zinc, boron, sulphur
and iodine are good examples of the so-called Fminor elements."
.Potatoes, beets, corn and legumes are eSpecially responsive

to potassium fertilizers.

3) Animal manure.

Natural organic fertilizer is usually applied on the
more valuable crops with good responses. Manure is able to
improve both soil structure and biological activity in the
soil. Sugar beets, potatoes, turnips, corn and oilseeds

respond well to manure. Cereals and legumes usually respond

30
less well because legumes can utilize atmospheric nitrogen
and cereals have lower potassium and phosphorus requirements
than other crops. Diminishing returns hold for manure as for

other inputs.

4) Climate and soil moisture.

These factors can seriously affect the optimum amount
of fertilizers to use. In dry climates, fertilizer application
without provision for adequate moisture is often almost use-
less while under irrigation there may be a profitable
response- In rainy regions, water control with soil conser—
vation practices can improve soil moisture conditions making

possible better crop responses from the use of fertilizer.

5) Systems of farming.

It is important to consider the effect of farm
enterprises on fertilizer requirements. On livestock farms,
fertilizer use can be quite different from that on farms
where livestock is not kept and no manure is used. Further—
more, the kind of livestock and its management is another
point to be considered.

Usually, farming systems are closely related to crop,
soil, climate, etc. Fertilizer use has no unique pattern for

all crops and for all conditions. For instance, potash and

31
some phosphates have slow solubility and in many soils
remain near the surface out of the upward reach of plant
roots. On the other hand, nitrogen, especially "Salitre de

Chile? (Na NO and K N03), is extremely soluble under good

3
and even relatively poor soil moisture conditions. For arid
regions, experiments have shown that deep placement of
fertilizer may increase yield. In these cases, deep rooted
crops are usually grown. The advantage of deep fertilizer
placement depends on textural, drainage and aeration

conditions. This method of fertilizer placement is less

useful in heavy soils.

B. Empirical work.

Use of a mathematical model permits a more definite
statement of an unspecified agronomic model and is helpful
in understanding the scope of the latter.

But before discussing mathematical functions further,
it seems necessary to establish here those statistical

aspects essential in planning experiments.

i) Statistical planning of experiments.

In the first place, the data to be obtained from the

 

1"The Economics of Fertilizer Application,” Conference
Proceedings, Farm Management Research Committee of the western

Agricultural Economics Research Council, Corvallis, Oregon, 1956.

32
experiment must be relevant to the existing problem and be
suitable for statistical analysis. ‘Within a certain margin
of error, one (or more) questions must be answerable.
In planning controlled experiments, the chief points
to be considered are as follows:

1. The researcher should know the problem. If any
hypotheses is going to be tested, it should be clearly
stated. Usually a mathematical formulation is necessary to
express a production function.

2. All the factors or variables to be studied in the
experiment must be defined. As we have already seen in the
previous section, the independent variables are classified
in two groups: controlled and not controlled. The uncontrolled
factors are randomized to permit averaging out their effects
(see 5 below).

3. The range of variation and number of levels of each
controlled variable to be investigated must be stipulated.
Of course the figgg_controlled variables have no range and,
hence, only one level of use.

4. The number of replications of the experiment under
each set of conditions should be determined. Though the
number of replications should be the same for each level of
input to simplify statistical computations, other considerations

may offset this advantage.

33
5. A method of assigning treatments to the experimental
units must be established in accordance with some scheme of
randomization so that the "experimental error" due to
uncontrolled variables may be independent of the independent
variables.
6. The correlations between the independent variables

should be minimized. Ideally the rx X should equal zero
iJ'

where rX.X. is the correlation coefficient of the Xi and xi
1 J
independent variables.

7. Some functional form should be used to represent the
distribution of experimental errors. The usual form used is
the normal distribution though other assumptions are
frequently made and with justification.

8. The possible outcomes should be considered. Provision
should be made to be certain that a sufficient proportion of
these will answer the problem. If not, a new plan is required.

9. Assuming that the experiment has been carried out as
stipulated in the preceding steps and that the assumptions
made were justified, the type of statistical analysis should
be specified in the plan. This point has to be closely

related to item 8 above and section B of the last chapter.

10. The plan should also indicate the manner in which the

conclusion are to be presented so that they may be understood

34
readily by those persons who are not well trained in statis-

tical theory and methods.

ii) The straight line function.

 

Let us consider a simple linear function in two
variables
Y = a + UK + u

We can describe this function as follows:

Y = a variable dependent on X

E = (Y - u) = the predicted Y

a =-a constant

b = a constant

X = the independent variable

u = an unexplained residual generated by a set of uncontrolled

variables which are independent of X.

The previous equation is called the equation of the
function. It describes a straight line between two variables.
In order to determine such a mathematical relationship between
X and Y in a set of data, the constants a and b must be

estimated.

 

1 . . . .

This and the follow1ng section is based on M. Ezekiel's
Method of Correlation Analysis (New YOrk: John Wiley and
Son, 1953; 2nd edition). '

35

The symbol Y in the equation simply represents the
number of units of the variable designated as Y. These can
be acres, pounds, dollars, etc.; similarly X represents the
number of units designated as X.

The meaning of the constants a and b of the formula
will be explained first in a simple graphic way and then with
the aid of mathematics.

In Figure 2 we have the graphical representation of

the function Y = a + bX.

VALUE OF
Y'S

 

 

 

p.

VALUE OF X '3

Figure 2. Graph of the function E = a + bX

When the value of X is O, b times X is O (b.O = O) and

therefore E = a + 0 = a. The constant a gives, then, the
height of the line in terms of Y or vertical units at the
point where X is zero. we can see this in the lower left

part of Figure 2.

36

. l . .
In the same equation, every time X increases one

A ‘ A

unit, Y increases b times one unit, since Y is computed as a
plus b times X. Therefore, the difference of the height of
the vertical line measured in Y units between two adjacent
values of X, say from 1 to 2, is b units of Y as indicated
in the graph. This is true for every unit change of X Whether
0 to l, or from 25 to 26 or 99 to 100.

The difference between a point in the above diagram,
Yi' and the regression line is the residual variation, ui =

Y. - §.o
1 J.

Pipping the line by ”least squares.? A mathematically
determined straight line can be obtained from experimental
data by the method of least squares. In this process for
determining values of the constants a and b, all the
observations are considered and given equal weight.

The pattern of computation from the total number of
X and Y observations for determination of the straight line

by ?least squares? is as follows:

 

1The discussion of properties of a linear function

presented in this section is of course, well known by production
economists and is fully discussed in elementary algebra and
statistical text books. It is reviewed here primarily because
it serves as a basis for further discussion of curvilinear
functions of the type used in this study.

37

Table 3. Hypothetical data demonstrating the computational
procedures of least squares regression.

 

 

 

 

Observations of X Observations of Y X2 XY
4 3 16 12
2 7 4 l4
6 4 36 24
5 6 25 30
3 q 8 9 24
2X = 20 BY = 28 2X2 = 90 ZXY = 104

 

All the observations of X and Y are listed under headings
?X? and ?Y?. Then each X observation is squared and entered
below "X2?. Each X Observation is multiplied by the
corresponding Y observation and entered in the XY column.
The summation of each column is represented by 2X; 2Y;£X2
and ZXY. This means the sum of all X's, sum of all Y's
and so on.

With these values calculated, we can proceed to find

the values of a and b, with the aid of the following

formulas:

ZXY - nZXZY

2

b
2x -n (2302

a = My — be

38

Where

n number of observations (five in our example)

Mx arithmetic average of X's (four in our example)
My = arithmetic average of Y's (5.60 in our example)

Using the values in our example we can solve for b as

 

 

follows:
b = 104 - 5 . 202. 28 = 104 - 2800 = 1.41
90 - 5 (20) 90 - 2000
a = 5.60 — (+1.41) (4) = 5.60 - 5.64 = —.04

Therefore the equation of the straight line determined by
all the observations is

Y = -.04 + 1.41 X
This line is called the line of_best_fip. The nature of this
best fit is explained by the fact that if the differences
between each of the £p§1_observations and the estimated
values given by this equation are computed, squared and
summed, this sum will be smaller than it would be if any
other straight line were used. Because with this method
the line with the smallest possible sum of the squared
deviations is determined, this line is knoWn as the I'least
squares" line of regression and the process of its computation
is called the "method of least squares.“ In addition, a
normal distribution of the residuals or u's produced by

uncontrolled variables is assumed.

39
The determination of the constants for the linear
equation, when certain data or observations are given, is
known as 'fitting' the equation to the data. The linear
equation is popular because of its simplicity in ?fitting."
Hewever, since it describes only a straight line, its use is

limited.

iii) The curvilinear functions.

Curvilinear relations may also be fitted to a set of
data. The number of curves that can be described with equations
is infinite.

In statistical analysis, some useful and familiar
equations for describing different kind of curves or types

of functional relationships are shown below:

1) Y = a + b X + c X2
2) log Y = a + b X
3) log Y = a + b log X

y = a + b log X

1
4) Y — a + bX
2 3
5) Y = a + UK + CX + dX
1
6) Y = a + bX + c (“£79

Equation 1) is found to be the equation of a parabola:

4O

1) Y = a + bX + cX2
Assuming values of the unknown constants a = l; b = 0.5
and c = -0.1, and entering these in the formula, we have,
2
y=l+005X-001Xo
Now if X takes different values, we can work out the value
of Y. For instance, if Y is computed at successive values

of X from 0 to 6, we have the following results:

1 + 0.5(0) — 0.1(02) = 1

X=0 Y=

x = 1 Y = 1 + 0.5(1) - 0.1(12) = 1.4
x=2 Y=l +0.5(2) - 0.1(22) =1.6
x = 3 Y = 1 + 0.5(3) — 0.1(32) = 1.6
x=4 Y=l +0.5(4) — 0.1(42) =1.4
x=5 y=1 910.5(5)- 0.1(52) =1
X=6 Y=1+0.5(6) - 0.1(62) = .4

When the above values are graphed, we have a parabola.

Y

1 I l

1 L
O I a 5 4L 5 e

 

><

Figure 3. Graph of the function Y = a + bX + ch
with given values for a, b and X.

41
One of the characteristics of the curve in Figure 3 is that
it is always symetrical on both sides with respect to the
highest point (H in our case).

If the value of b were negative and the value of c
were positive, the curve would be concave from above instead
of convex and would be symetrical with respect to its lowest
point.

This curve has great flexibility in that many other
curves with different shapes can be represented by this
parabola or by some of its segments. On the other hand,
the parabolic shape is so simple that the real relationship
between the variables may not be describable.

When log of Y is used instead Of Y, our curves are
mathematically modified. For example, a straight line
function like Y = a + bX can be transformed into a curvi-
linear form by using log Y = a + bX.

In using log Y = a + bx + ch instead of
1) Y = a + bX + cX2 the top of the bend is lengthened if
b is positive. The botton of the dip flattens out if b is
negative. When considering a cubic parabola the results
are pretty much the same. The above logarithmic equation is

graphed in Figure 4.

42

e——— IOB Y = o+bX+ch

 

 

Figure 4. Graph of the function log Y = a + bX + c X2

when a = 0, b > 0, c < 0.
If X is replaced by log X in the straight line
formula Y = a + bx, we have Y = a +'b log X that becomes
convex from above if b is positive and concave from above

when b is negative; this is Shown in Figure 5.

\.

     

«)YEQ+bIOBX

 

 

Figure 5. Graph of the function Y = a + b log X.

43
A third case is found When logarithms are for both Y
and X (the Cobb—Douglas case). The curve log Y = a + blog X
is concave or convex when b is positive, being always concave

from above if b is negative;1 this can be seen in Figure 6.

?Ysa’fb \OBX

K‘ >b70

uto>b21

 

 

Figure 6. Graph of the function log Y = a + b log X
when b < 1.
It can be stated that the curves described by logarithmic
equations maintain certain characteristics similar to equations
without logarithms. For example, a and b are constants in

both forms.

 

1When b > 1 the curve is concave from above. When

b < l the curve is convex from above and when b = l the
curve becomes a straight line.

44

It can be stated that a formula with logarithms of
observations can only be used when zero and negative values
are absent from the observations. Unlike other functions
which are able to show both positive and negative values, the
logarithmic curves described by the formula tend to but do
not necessarily become asymptotic to a constant value for
Y as X approaches infinity. In other words, they tend to
become parallel with the X axis for extremes positive values

for X. .

CHAPTER IV

CHARACTERISTICS OF THE DATA HERE USED

In this chapter an experiment on fertilizer use
performed in Colombia will be described. In the first place,
an ecological description will be drawn and then irrigation,
plant population, fertilizer rates used and corn yields

obtained will be presented.

The Colombian Experiment

The experiment on fertilizer use analyzed in this
thesis was performed in the Cauca valley, of Florida, Colombia,
South America. Because of certain characteristics of the
Cauca Valley, it seems worthwhile to discuss, in brief, some
of its details.

It has been said that much of the Cauca valley area
is one of the most fertile pieces of land in Latin America
comparable with the ?Pampa" of Argentina or the Central valley
of Chile, to mention only a few other fertile lands.

The Cauca valley is located in the southern part of
Colombia, running between the western and central ranges of
the Andes cordillera (mountains). Its total area is about

.8 million acres (320,000 hectars) covering an area 100 miles

45

46
long and 8 to 20 miles wide. This Valley is an old lake bed,
fairly flat with some broad terraces and a general altitude
of 3000 feet.

The weather conditions associated with wet and dry
seasons, make it possible to obtain two crops, corn and
beans. When irrigation is present, sugar cane is a permanent
crop. The years are divided into nearly two periods each
with 3 months dry and 3 months wet. About 40 inches (1000 mm)
of precipitation is the annual average. The average temperature
is around 75°F (24°C) in the wet periods and 77°F (25°C) in
the dry months from July to September and January to March.

The above description provides a general background
on the area in which the experiment here under analysis was
performed.

In March of 1958, Lawton and Patifiol started an
experiment to study the effects on corn yield of irrigation,
number of plants per unit of area and different fertilizers.
The soil chosen was a well drained, loam to clay loam with '

pH 6.5, with 3 to 4 percent organic matter, located two

miles east of Florida. The available phosphorus was 16 Kgs.

 

lKirk Lawton, Ph.D., Professor of Soil Science at M.S.U.;
Edgardo Patifio, M.Sc., Colombian agriculturist in Florida,
valle del Cauca, Colombia.

47
per hectar (about 14 lbs/acre) and 218 Kgs. of exchangeable
potassium per hectar (about 194 lbs/acre).

The following variables were studied:

1) Irrigation.

Irrigation was given to half of the area where the
experiment was conducted. The need for water was determined
by a soil moisture test, plant appearance and the frequency
of rainfall. The experimental plots were designed in such a
way that water could not pass from irrigated to non-irrigated

plots.

2) Plant populations.

Three level of plant populations were chosen: low,
medium and high, or 11,200, 16,000 and 18,500 plants per
acre respectively (about 27,500; 39,300 and 45,500 plants
per hectar).

The corn seed used was a hybrid called Diaco H—203‘
(with yellow grain). The seed was sown four inches deep
because moisture was scarce near the surface. Each plot
had four rows 16.35 yards long separated by a 1 yard inter-
lane (15 mts. x .90 mts.). One hundred twenty plots were
treated including 6 check plots (3 with and 3 without

irrigation).

3)

48

Fertilizer used..

Three fertilizers were used: nitrogen, N; phospho—

rus, P; and potassium, K. The N was applied as sulfate of

ammonium,

the P as concentrated superphosphate and the K as

muriate of potash.

The amounts used were 50, 100 and 150 kilos (kgs.)

of N per hectar; 50 and 100 kilos of P 0 per hectar and 50

 

 

 

2 5

kilos of K20 per hectar. The combinations used were as

follows:

Table 4. Use of N, P, and K on irrigated and non-irrigated
experimental plots with three levels of plant
population in the Cauca valley of Florida,
Colombia, 1958.

Plant popu-
lation per
Irri— acre:
gation: 1 = 11,200
Number N P205 K20 Yes + 3 = 16,000
of plots (Kgs/Ha) (Kgs/Ha) (Kgs Ha) No - 4 # 18,500
3 0 0 0 - 1.3.4
3 O 0 0 + 1,3,4
3 50 0 0 - 1,3,4
3 50 0 0 + 1,3,4
3 0 50 0 — 1,3,4
3 0 50 0 + 1,3,4
3 0 0 50 - 1.3.4
3 0 0 50 + 1,3,4
3 0 50 50 - 1,3,4
3 0 50 50 + 1,3,4

49

Table 4.--Continued.

 

 

 

Plant popu-

lation per
Irri- acre:

gation: l = 11.200

Number N P205 K20 Yes + 3 = 16,000

of plots (Kgs/Hé) (Kgs/Ha) (Kgs/Hfi) No — 4 = 18.500

3 50 50 0 - 1,3,4
3 50 50 0 + 1,3,4
3 50 0 50 - 1,3,4
3 50 O 50 + 1,3,4
3 50 50 50 - 1,3,4
3 50 50 50 + 1,3,4
3 100 0 0 - 1,3,4
3 100 O 0 + 1,3,4
3 0 100 0 - 1,3,4
3 0 100 0 + 1,3,4
3 100 100 0 - 1,3,4
3 100 100 0 + 1,3,4
3 100 0 50 — 1,3,4
3 100 0 50 + 1,3,4
3 0 100 50 - 1,3,4
3 0 100 50 .+ 1,3,4
3 100 100 50 - 1,3,4
3 100 100 50 + 1,3,4
3 150 0 0 - 1,3,4
3 150 0 0 + 1,3,4
3 150 50 0 " 1,3,4
3 150 50 0 + 1,3,4
3 150 0 50 - 1,3,4
3 150 0 50 + 1,3,4

50

Table 4.—-Continued.

 

 

 

Plant popu-
lation per
Irri- acre:
gation: l = 11,200
Number N P205 K20 Yes + 3 = 16,000
of plots (Kgs/Ha'l) (Kgs/Ha') (Kgs/Ha‘i) No — 4 = 18,500
3 150 100 0 - 1,3,4
3 150 100 0 + 1,3,4
3 150 50 50 - 1,3,4
3 150 50 50 + 1,3,4
3 150 100 50 - 1,3,4
3 150 100 50 + 1,3,4
120 plots
in
total

A A A A

4) Corn yield.

The yield of each plot was computed by the researchers
on the basis of shelled corn with 15 percent moisture.

When the corn was cropped without irrigation the
highest average yield of 75.8 bu/acre (equivalent to 20.3
"cargas/fanegada?l or 4,757 Kgs/Hfi.) was obtained from the

medium level plant population (16,000 plants/acre).i

 

A special appendix is presented with the weight and
measure equivalents used in this thesis.

51
With irrigation, the highest average yield of 94.8
bu/acre (25.4 cargas/fanegada or 5,953 Kgs/Ha.) came again
from the medium plant population.
In both cases, irrigated and non-irrigated, all the

plots under experiment were considered.

CHAPTER V
FITTING THE DATA TO THE HYPOTHESIZED MODEL

In this chapter, data from the field experiment are
fitted to the hypothesized model and the results evaluated
from a statistical standpoint. The function used was an
incomplete second degree polynomial.

A. Seeking the estimatedgproduction
function.

In the present work, the hypothesized model is one
which treats the yield of the corn crop as a function of the
amount of fertilizer used (in this case N, P and K measured
separately) irrigation and number of plants per acre.

In addition, the interaction effects of nitrogen
and potassium (NK), nitrogen and number of plants per
acre (NS) and phosphorus and number of plants per acre (PS)
were also considered.

12 was not included because of the nature of the
data, i.e., only two levels of I were used. It was
decided not to study the interaction terms IN, IP and IK.
Use of I2 variable implies a curvilinear relation between
I and Y5. To test the hypothesis that a curvilinear relation

existed between Ye and I (by use of 12) at least three levels

52

53

of I were required. When only two points are drawn on a
two—dimensional diagram, one and only one straight line of
regression fits these data perfectly. No residual error is
possible if the line fitted minimizes the squared unexplained
residuals. However, an infinite number of vastly different
curved lines of regression would also fit these two obser—
vations with no unexplained residuals. Thus, in order to
obtain even a gross measure of the curvilinearity which may
exist between two variables, at least three sets of obser-
vations are required. More would be desirable since the
reliability would thus be increased.

NK, NS and PS were included because it was decided
to study interaction effect of these variables. N, N2, P, P2,
K, S, and 52 were included in the analysis in order to study
their effects on Y.

The final unspecified function used (two previous
ones were tested) was of the following type:

1? = f (N,P,K,N2,P2,NK,I,S,SZ,NS,PS).

The above formula was specified as a second degree polynomial
in order to estimate the parameters as follows:
2 2

I: + + + + +
Y a blN b2P b3K + b4N + bSP b6NK

2
+ + + +
b7I b8S bgs blONS bllPS

54

Where:

E estimated yield

a = a constant, representing here a yield independent of
the effect of the input variables
bl to bll = parameters to be estimated

N, P, K, S and I = symbols for nitrogen, phosphorus,
potassium, number of corn plants per
acre, and irrigation, respectively.

With the help of the electronic computer (called

MISTIC at Michigan State University), the a and b constants

were calculated, giving the following results:

§ = 35.1085 + .047638 N + .247890 F + .062468 K

(.0463) (.06365)* (.03926)

— .0001341 N2 — .001698 P2 - .00042 NK + 9.17084 I

(.000254) (.000542)* (.0004082) (.6145)*
2

+ 29.265 S - 5.18672 S + .00948 NS - .009273 PS
(3.356)* (.6637)* (.00819) (.01188)
* = significant difference from 0 at the .05 percent
level of significance;

Numbers in parentheses are standard errors of the estimated

coefficients.

55

B. Some statistical remarks.

For this experiment and equation, the variables
N, P, K, S, I were associated with approximately 79 percent
of the sample variance. This is shown by the coefficient
of multiple determination, R2 = .79.

The standard error of estimate was 6.73. This measures
the error about the fitted regression line. In addition,
the location of the regression line is subject to a related
error, part of which is indicated by the standard error of
the b's.

The coefficient of multiple determination corresponds
to the square of the coefficient of multiple correlation;
this latter coefficient may be measured by dividing the
standard deviation of the estimated values by that of the
original values.

By the standard t-test, the regression coefficients
of P, P2, I, S and S2 were significantly different from 0
at the .05 percent level. For N, K, N2, NK, NS and PS, the
hypotheses that the coefficients were different from 0 was
not accepted at the same level of significance. HOWever,

there is strong a priori evidence that changes in these

terms are, in fact, associated with changes in crop yields.

56
The marginal physical productivity of an input X is
found by taking the partial derivative of the production
function with respect to that input. The reliability of

this estimate of MPP is, of course, influenced by the

X(Y)
standard errors of the estimated coefficients contained in

the partial derivative. However, the standard error associated

with the estimated MPP

X(Y) is some undefined (in this theSis)

linear combination of the standard errors of the bi's.
Thus, we cannot conclude that non—significant b-coefficients
imply non-significant MPPX(Y) estimates.

In this experiment, the following equation describes
how expected yield Changes with small change in N. Numbers

in parentheses refer to associated standard errors of the

 

coefficients:
$3?
= = o - o - o + o
MPPN(Y) AN 0476 2( 000134)N 00042K 00948 s

(.0463) (.000254) (.000408) (.00819)
The partial derivative of the production function

with respect to P is:

&

MPP =‘—%%;‘ = .24789 - 2(.001698)P - .009273 S

P(Y)
(.06365) (.000542) (.01188)

The partial derivative of the production function

with respect to S is:

MPP = i:- = 29.265 — 2(5.18672)S + .00948N - .OO9273P

s(y)
(3.356) (.6637) (.00819) (.01188)

 

57
It is impossible to make concrete statements regarding
the significance of the above MPP estimates. Future work
will be necessary in order to test the hypothesis that the
individual MPP's are or are not significantly different than

zero.

C. Analysis of the Residuals.

The residual figures are actual yields minus the
predicted yields obtained from the polynomial formula
presented above. The residuals (Y - E) = u represent the
effects of uncontrolled variables. The u's are assumed to
be randomly and independently distributed in relation with
the variables under study. Assuming these circumstances
are met, the effects of the uncontrolled and unstudied
variables which generate the u's can be averaged out with
statistical procedures.

Other requirement is that the u's be small enough
to make the estimates of Y5 usable.

Unexplained residuals in experimental data are
themselves partial functions of uncontrolled variables such
as between-plot variation in soils, insects, disease, experi—
mental errors, hail, weeds, past soil treatment, etc.

It is important that experiments be designed to

insure that unexplained residuals are reasonably random with

58
respect to the inputs treated as experimental variables.

The factors that cause unexplained residuals can be
divided into three main types:

1) Errors in recording the data.

2) Those due to the omission of certain variables.
This may be because the analyst failed to think of them,
because no data were available or, perhaps, because they were
so minor as not to be worth including in the study. This
is the kind of random error normally assumed in a least
squares analysis. It is the type of error allowed for in
simultaneous-equation "shock? models.

3) Those resulting from the use of wrong types of
curves, incorrect lags, and similar factors.

Johnson2 relates the importance of the unexplained
residuals and farmer's estimate of uncertainty as follows:
"Both the size of unexplained residuals in experimental data
and the correspondence between the causes of unexplained

residuals under experimental and farm conditions are crucial

as farmers form their subjective estimates of the uncertainty

 

1USDA, Agricultural Handbodk No. 146, Agricultural

Marketing Service, 1958, p. 172.

2G. L. Johnson, "Discussion: Economic Implications
of Agricultural Experiments," Journal gijarm Economigg, XXXIX,
(May, 1957), p. 394.

59

involved in using experimental results. Large subjective
uncertainties relative to the objective uncertainties involved
slow up adoption of experimental results unduly. Biased
estimates of yields and of partial derivatives mislead
farmers. Similarly, inaccurate adoption results if subjective
uncertainty is less than objective uncertainty. The problem
is to help bring a farmer's estimates of expected yields and
the derivatives of uncertainty into line with those he
actually faces."

The occurrence of u's may be reduced, in part at
least, by (1) using procedures able to reduce errors in
measuring Xj and YE, (2) better control on non-studied inputs
and factors, and (3) randomization of the incidence of unstudied
and uncontrolled variables in the experiment.

Finally, some assumptions1 commonly made about
unexplained residuals may be summarized as follows:
1. u s are random variables.
2. The variance of u's is constant over time.
3. The u's are normally distributed.

4. The u's are not correlated with any predetermined

variable.

._.A_4 L;

1S. Valvanis, Econometrics, An Introduction to Maximum

Likelihood Methods (New YOrk: McGraw—Hill Book Co., Inc.,
1959); also see p. 24 f. this thesis.

 

60

In order to study the characteristics of unexplained
residuals for the experiment and fit here analyzed, a number
of graphs were drawn. In these graphs the actual and the
predicted yields at different levels of inputs were compared.

The result of this graphic analysis indicated that
in the majority of cases observed no correlation existed
between the residuals and the independent variables in the
equation. The conclusion was that the functional form used
in this analysis was adequate and that the statistical tests
used above were valid.1

The actual and predicted yields plus the calculated

residuals for each one of the 120 plots in the experiment

are given in Appendix A.

 

lAgain, se p. 24 f. this thesis.

CHAPTER VI

ECONOMIC ANALYSIS OF THE DATA AND

STATISTICAL ESTIMATES

Because one of the chief purposes of the farmer is
to maximize profit on crops, one of the first answers that
the farm economist can give to him on using a production
function such as above is the profit maximizing amounts of
fertilizer and plants per acre for different prices of corn,
N, P, K and S.

In order to evaluate profit maximizing alternatives
under varying price levels, seven different prices for corn,
three different price levels for N and P, and four for S were
set. The amount of K was maintained constant.

‘ The combinations of prices used to compute high
profit amounts of the inputs are shown in Table 5.

A. High_profit point analysis under
different price assumptions.

Knowledge of the input-output coefficients, such as
those obtained from estimation of the parameters in the
production function from the previous section, permits
determination of optimum fertilizer inputs under varying
assumptions regarding input-output price ratios.

61

62

 

 

 

Table 5. Different price levels of corn, N, P, and S;
the amount of K being constant.
A O
PY(bu) PN(Kgs) PP(Kgs) Ps(bu) K(Kgs/Ha)
$ $ $ 5
item (1) 1.00 .068 .045 3.90 40
ll (2 ) 1 . 10 II ‘II II II
II (3) l . 2 O II II II n
vary ll (4) l. 30 II II n n
PY ll (5) 1 . 40 II ll 0! II
II (6) l . 50 II II II .II‘
II (7) l . 60 II II II II
" (8) 1.30 .054 .045 3.90 40
vary ll (9) ll . 068 II II II
PN ll (10) ll . 082 II II II
" (11) 1.30 .068 .036 3.90 40
vary II (12) II n . 045 II n
PP n (13) II II . 054 II II
" (14) 1.30 .068 .036 4.20 40
Vary (15) 4.50
PS (16) 4.80
‘0 (1 7) III I: II 5 . l 0 ll

63

Optimum inputs are determined by the profit maximizing

 

 

 

 

principle:
X X X. X
l _ 2 _ .. _ i _ . m _ l
P P P P
X1 X2 X1 Xm
where,

X. = the X.th input
1 i

Xi = the marginal value product of the Xith input.

For expository purposes let us consider the profit
maximizing principle given one variable input X in the
production of a single commodity Y. The quantity of X at
the HPP (high profit point) in this simple case is given
by solving the following equation for X.

X

PX

 

= 1

The MVPx in this case is given also by multiplying the
marginal physical product, MPP, of a unit of X by the unit

1 .
price of the product PY. Hence, we may rewrite the above

formula as:

 

N
K
ti o
N

l MPP
or X

N
K

 

lProviding that unit price of the product is constant
throughout all levels of output.

64
In words, this principle says that maximum net profit
is attained where the addition to total product forthcoming
from an added unit of the X input is equal to the input-
output price ratio.
For the case of several variables as shown previously,

the quantities of the input at the HPP are given by equating
P

 

(MPP ) to one, and solving simultaneously for the

x1 Px.
l

Xi’s, where,

i = l ,..., m.

Thus both price ratios and the marginal physical
products derived from the production function are involved
in estimating optima or high profit combinations and amount
of inputs and of production.

Actually the procedure described above to determine
the HPP levels of the several inputs in this study follows
a mathematical principle. This principle says that given a
functional equation in several variables having a maximum
but no minimum in a specified range, the values of the several
variables which maximize the functional value in that range
can be determined by: 1) taking the partial derivatives
of the function with respect to each independent variable;

2) setting these derivatives equal to zero and 3) solving

65
these equations simultaneously for the unknown values which
fall in the stated range.

Thus, in order to maximize profits we may follow the
procedure outlined above on the profit function which is
W = PY Y - PN N - PP P - PS S - PK K - PI I - PC
for our polynomial function.

A Change in N is known to affect the output Y by the
amount given by the marginal physical product of N or MPPN(Y)°
Hence, the partial derivative of the profit equation with
respect to N is:

—§§f- = PY (MPPN) — PN = PY [.047638 — 2 (.000134) N - .00042 K
+ .00948 S] - P = 0

N

Similar reasoning leads to the two other equations:

 

6w _ _
Js — PY (MPPP) Pp — PY [.24789. 2 0001698) P .009273 S]
- PP = 0
ELL : _ = _ +
63 PY (MPPS) PS PY [29.265 2 (5.18672) 3 .00948 N

— .009273 P] - PS = 0

These equations were set equal to zero and solved
simultaneously for N, P and S with the use of MISTIC.l The
several different sets of input-output prices presented on

page 57 were used and the analysis rerun in each case. The

results follow_in Tables 6 and 7.

l . . . .
Michigan State Integral Computer.

66

Table 6. HPP quantities of N, P and S, at various prices

of corn PY_and PN (values for PP, PS and K are

held fixed).

A A A LA

 

 

 

PY PN
$ Bu $ .054 (Kg) $ .068 (Kg) $ .082 (Kg)
*
item 53.P
(1) 2.346 s
1.10 - - 31 N —
item 54 P
(2) 2.402 S
1.20 - — 10 -
item 55
(3) 2.499
* , *
1.30 49 8 - 34
(8) 56 items 56 (10) 56
2.527 (4) = (9) 2.489 2.451
(5) 56.6
2.523
(6) 58
2.554
1.60 — 48 -
(7) 58
2.578
P = $ .045 * = HPP's quantities
P . . .
P _ $ 3 90 With data given in
S - ° Table 5 from the
K = 40 Kgs/Hé. different items ( ).

67

1) The effect of various prices of nitrogen, Pn' and

 

of Py, on optimum quantities of N to use.

 

From Table 6 we can see that three price levels of
N have been chosen: $ .054 Kg., $ .068 Kg., and $ .082 Kg.

Here the prices of corn, P take on values of $1.00;

Y'
$1.10; $1.20; $1.30; $1.40; $1.50 and $1.60 per Bu. Fixed
values were given for PP at $ .045 Kg.; PS at $3.90 Bu.
and K at 40 Kgs/Ha.

From Table 6, it is possible to observe that when PN
is maintained at $ .068 Kg., with PY running from $1.00
to $1.20, the use of N was not profitable and only negative
N's quantities were obtained as HPP ( - 56; - 31 and - 10

Kgs/Ha). However, when PY became $1.30 per bushel, the use

of 8 Kgs/Ha. of N was most profitable.

When P $1.40 Bu., 23 Kgs/Ha. of N was most profitable;

Y

5' M $ 1 . 5 0 ‘ll ' 3 6 II II II II

'II II $ 1 . 6 O _" ' 48 II II II II .

Thus, between corn prices of $1.20 and $1.60, for every
increase of $ .05 per bushel of corn, profits would have
been maximized by increasing nitrogen applications about
6.5 Kgs/Ha. providing prices of other inputs remained fixed
at the levels shown. These results can be visualized in

Figure 7.

68

Om:

 

.wm\mmm 08 u M
om.m n mm M
moo. u m .2 mo mHo>OH Eofiflumo co m mSOHHm> mo muoowmm .n onsmwm
z
woo. w u m
"Cong
.Hmuown Hem madam on Gomouuflz mo mpawx mo gonads manmufimoum umoz
om ow om om OH 0 0.7. om: om: owl
a _ 4 _ _ _ _ w a a
10m.
noo.aw 1111111111
1111111 .. hmuuwwiiniuiu
nom.aw
Ems.aw

 

69

2) The effect of various prices of nitrogen, P ,

N
on optimumgguantities of N to use.

P

If PY is maintained at $1.30 per Bu. while PP, S

and K remain fixed as above, but PN is changed, the following

results are observed:

When PN = $ .054 Kg., 49 Kgs/Hé. of N was most profitable;
" " = .068 " , 8 " . ? ? " :
" " = .082 ? , the most profitable amount of N to

use becomes negative, — 34 Kgs/Ha. In other words, at the

high PN, application of N is not recommended for economic
reasons. The decrease of the amount of N Which maximizes

profits is consistent with increase of P In general, this

N.
means that for an increase in PN of $ .05 the quantity of N

should be decreased around 15 Kgs/Hé. in order to maximize

profits. In Figure 8 this result is shown.

3) The effect of various prices of corn, Py,

on gptimum quantities of P to use.

From Table 6, we observe the effects of changes in

the optimum quantity of P to use when P is

price of P P

Y
fixed at $ .045 Kg/Hé. It can be noticed that the optimum
quantity of P per H5. goes up when PY goes up from $1.00 to

$1.60; however, when PY is $1.30 and $1.40, the resulting

owl

 

.wm\mmm og u x
O Cmom H m
7
moo. n m z
om.H w I m .2 mo mam>ma Eofiflumo co m m50flum> mo mpommmm .m whomflm
Cong
.Hmuoon mom madam ou :Omouufiz mo OOHHE mo Hones: manmufimoum umoz
om ow om om 0H 0 0H: om: oml
Ni 4 _ a _ _ ooo. _ _ a E
-oao.
1omo.
Jomo.
Lovo.
iomo.
iomo.
1:--1--ubbbh-
-omo. -1---:----:---

 

omo. w
2m

71

optimum amounts of P is 56 and 56.6 Kgs/Ha., respectively.

P
When Y

at 58 Kgs/Ha.

The effect of various prices of corn, PY’

to use.

Figure 9 shows this result.

on optimum quantities of seed,

is $1.50 and $1.60 the quantity of P use remains

In regard to the different number of plants per acre,

different amounts were found to be associated with the HPP's

for the various price levels of P , P , P

This is shown in Table 7.

Y

N P

and P .

S

 

 

 

Table 7. High profit number of plants per acre at various
prices of corn and with P , PP, P and amount of
K held at specified levels.
Coded Actual PY PN PP PS K
plants/ plants/ $ per $ per $ per $ per Kgs/
acre acre Bu. Kg. Kg. Bu. acre
2.346 16,865 1.00 .068 .045 3.90 40
2.402 17,005 1.10 " " ? “
2.449 17,123 1.20 ” " " "
2.489 17,223 1.30 " " " ?
2.523 17,308 1.40 " " " "
2.554 17,385 1.50 " " ? "
2.578 17,445 1.60 " " " "
2.527 17,317 1.30 .054 .045 3.90 40
2.451 17,127 1.30 .082 .045 3.90 40
2.457 17,143 1.30 .068 .036 4.20 40
2.435 17,087 " " " 4.50 "
2.408 17,020 " " " 4.80 "
2.386 16,965 " " " "

5.10

n 65ng 08 u m
om.m H mm
moo. H mm
moo. w u zm .m mo mam>ma ESEHDQO Go Wm mooflhm> mo muommmm .m musmflm
"cmnz

.Hmuomn Mom madam ou monosmmonm mo moaflx mo Hones: wanmuwmoum Duo:

00 om ow om ON OH o
_ _ _ q _ .

 

 

.064 w.»

73

The original data were coded as follows:

S = l = 11,200 plants/acre;
s = 3 = 16,000 " ;
S = 4 = 18,500 " .

The optimum quantity of S to use goes up when PY
increases from $1.00 to $1.60 Bu. Taking into consideration
the different values of S from Table 7, we obtain a range

of 16,865 to 17,445 plants per acre.

5) The effect of various prices of seed, PS,
onjoptimum quantities of N, P and S to use.

Four price levels ($4.20, $4.50, $4.80 and $5.10 Bu)
of hybrid seed corn were used. The result can be expressed
as follows: When PS is varied under the assumed conditions
all figures for N are negative suggesting that no N be used;
under other assumptions positive amounts of N would have been

profitable. When the price of seed went up, the optimum

number of plants per acre decreased.

B. Comparison of somegpredicted_yields, Y,
with and without irrigation under
varyingpprices of corn and fixed

prices of N, Bigfig and a fixed
(quantity ong.

In order to obtain some predicted yields, Y, from

given prices and quantities of fertilizers, S, PY and K,

74

a number of computations were worked out with the help of

the estimated production function given on page 54. As

this production function gives the value of Y under irrigation,
in order to get the result without irrigation a simple
transformation was made as follows:

Y = (ai9.17084I) + .047638N + .247890P + .062468K - .0001341N2

- .001698P2 - .00042NK + 29.2653 - 5.1867282 + .00948NS_
- .009273PS.
where
I = + l, irrigation
I = — 1, no irrigation.

l) The first computation of Y'was worked out using price data

from Table 5, item (1) and using quantity data from Table 6,

item (1).
Given: Using HPP quantities:
PN = $ .068 Kg. -56N
PP = $ .045 53P
PS = $3.90 Bu. 2.346s
PY = $1.00 Bu.
K = 40 Kgs/Hé' .

we obtain,

1?

90.710 Bu/acre with irrigation

m»
u

72.370 ? with no irrigation.

75
It can be observed that the effect of irrigation is positive
giving a higher Y. Unfortunately the net profit from this
increase of about 20 percent in yield cannot be compared with
that for non-irrigated yield because of the lack of cost
data on irrigation. chever, the difference in total revenue
between irrigation and no irrigation can be determined by
computing Y . PY. This difference is the amount that is
available to cover the cost of irrigation and, in this

example, amounts to $18.34.

2) If we take PY at $1.10 Bu. but PN, PP' PS and K are
fixed as above, the HPP quantities of fertilizer given in
Table 6, item (2), are as follows:

N = - 31; P = 54 and S = 2.402.

With this information we have the following result:

Y 93.917 Bu/acre with irrigation

Kit
ll

75.577 ? with no irrigation.
Again the use of irrigation gives a higher yield of corn

with our estimated production function.

3) When PY moves to $1.20, other things equal as 2) above,

from Table 6, item (3) we get:

N = - 10; P = 55 and S = 2.449.

76

The predicted yields are:

4)

Y 94.034 Bu/acre with irrigation

K)
II

75.694 ? with no irrigation.

If PY goes to $1.30, other things equal as in 2) above,

from Table 6, item (4) and (9) we obtain:

N = 8; P = 56 and S = 2.489.

The predicted yields were:

5)

6)

7)

h

Y

95.177 Bu/acre with irrigation

Y

76.837 ? with no irrigation.

When PY is $1.40, from Table 6 item (5) we get:

N = 23; P = 56.6 and S = 2.523.

predicted yields are:

Y 96.032 Bu/acre with irrigation

K)
II

77.692 ? with no irrigation.

At PY $1.50, N = 36; P = 58 and S = 2.554.
associated predicted yields were:

a
Y

96.788 Bu/acre with irrigation

K)
n

78.448 ? with no irrigation.

If PY is $1.60, the highest price that we have assumed

in this thesis for corn, from Table 6, item (7) we obtain:

N = 48; P = 58 and S = 2.578.

77
The predicted yields obtained are:

Y

97.375 Bu/acre with irrigation

I-<}
ll

79.035 ? with no irrigation.
The recommended amounts of fertilizer increases
with the increases of Py. The same thing is true with the

recommended quantity of S.

CHAPTER VII

EVALUATION, RESULTS AND IMPLICATIONS

A. Evaluation.

In the first place, a general evaluation of "goodness?
of the experiment here analyzed shows that it was well done.
For the purposes of this experiment, 120 plots were considered
adequate. Three replications of each treatment were performed.
However, it is necessary to keep in mind that this experiment
contains observations from only one particular year, for a
particular type of soil under particular environmental
conditions.

The experiment, even with the limitations listed
above, is a very promising starting point. In addition, the
experience obtained will permit still better future experi-
ments to be planned. Future work should involve more fertilizer
combinations and levels of irrigation to take into account
the further problems of the area. With a more complete set
of data it would be possible to use more flexible functions
to give better answers to a number of fertilizing problems.

It should be recognized that those elements ?fixed"
in an experiment (such as soil type, management of the soil

and slope) may not be controllable by the farmer. Important

78

79

barriers between plot experiment and commercial farms include
(1) the difference in levels at which controlled variables
are fixed and (2) the exercise of controls by experimenters
which cannot be maintained by farmers. Unique characteristics
sometimes associated with an experiment can be very important
in determining the results from an experiment. On applying
such results to a large number of farms, several reservations
would be required. Though reduction of variance in experi-
mental results is highly desirable, the experimental situation
should be made similar to those on the farms expected to use
the results.

When the characteristics of the unexplained residuals
for the experiment here analyzed were studied graphically,
no correlation was revealed between residuals and the
independent variables in the equation. Under these circumr
stances, it was possible to assume that the function used
fitted well enough to justify use of the common statistical
tests.

From a general statistical viewpoint, the experiment
has shown a high coefficient of multiple determination
(R2 = .79) that can be taken as a measure of confidence in

the application of results. The standard error of predicted

yield (6.73) can be considered satisfactory and the average

80

yield of 76.87 Bu/acre is a realistic result. Here again,
the limitations discussed above are important to consider

if an extension program is eventually worked out.

B. Results.

Coming now to the results, the optimum quantities
of fertilizer and plants per acre used were found to increase
when price of corn was increased. Hewever, when the fertilizer
and seed prices also increase different answers were obtained:

a) When the price of nitrogen went up the optimum
quantity of N to use decreased. In general, it was estimated
that for an increase in PN of $ .05, the optimum amount of
N which should have been used to maximize profits decreased
about 13.3 lbs/acre.

b) When the price of seed corn went up the number of
plants per acre, S, tended to decrease. However, the
influence of changing seeding rates on profits was slight
and of little practical importance so long as around 17,000
plants were maintained on an acre. It may be added here
that ?better seed corn? is much more impOrtant than changes
in corn seed prices as seed costs per acre are low.

The predicted optimum yields when PN'PP' P and K

S

were fixed, but P was increasing, increased, as indicated

Y

81
above, while the optimum quantities of N, P and S increased.
This was shown in Chapter VI, item B. In more detail,
computations indicated that if the price of corn increases,
while fertilizers and seed prices are fixed, the optimum
amount to use of these imputs should be increased in order
to obtain the higher most profitable yields. For example,
when corn price, Py, was $1.00 Bu., and the HPP quantities
for N, P and S were - 49.8 lbs/acre, 47.2 lbs/acre and
16,865 plants/acre, respectively, other things equal; this
results assumes PN = $ .068, PP = $ .045, and PS = $3.90.
The predicted corn yield was 90.710 and 72.370 Bu/acre
with irrigation and non-irrigation, respectively. But when
PY increased to $1.60 Bu., HPP quantities were 42.7 lbs/acre
for N, 51.6 lbs/acre for P and 17,445 plants/acre; the
predicted corn yield was 97.375 and 79.035 Bu/acre for
irrigation and non-irrigation, respectively.

If we remember, for instance, that the best corn
yield reported from the experiment results was obtained from
medium level plant population (16,000 plants/acre), for
both irrigated and non-irrigated plots, we can say that
farmers still have wide possibilities for higher yields by

increasing the amount of plants per acre from around 14,000

82
plants per acre, which is common in Colombia, almost
regardless of seed corn prices.

Because the experiment was performed at only two

levels of irrigation, any statistical inference regarding
changing marginal returns to irrigation would be inaccurate.
Increased yields were obtained on irrigation plots. However,
three or more irrigation levels are needed to measure the
marginal influence of irrigation on yield, fertilizer—irrigation

interaction and hence, optimum rates of irrigation.

C. Implications.

In general for Colombia and the rest of South
America, more experiments such as analyzed would help
farmers find the most profitable combinations and amounts
of fertilizer to apply in producing different crops under
varying price levels. In addition such experiments have
methodological value. These methodological values direct
researchers in attaining better estimates to help farmers
maximize profits. The development of workable solutions
to practical problems is one of the ultimate goals of
fundamental researchers. Fundamental research dealing with
more efficient use of fertilizer may provide, eventually,
the best help on practical recommendations for individual

farmers.

83

In Latin America, the fertilizer information received
by farmers often indicates that the best fertilization level
is that at which the highest yield per hectar is obtained.
It is almost forgotten that maximum profits obtained from
a given fertilizer application are seldom found at the
highest yield. It seems highly desirable that more emphasis
be placed in profit maximization than on maximizing yields.
The data here analyzed have shown that in the particular
year under study, the estimated HPP quantities of fertilizers
varied with prices and exceeded common rates of application.

Although one year's data are limited, experiments
over a longer period would remedy this difficulty.

Fertilizer recommendations should be based on experi-
mental data for a period of years. Such data would permit
an average production function to be derived which would
average out between year variations. It would also be
possible to estimate probable deviations from expected
returns as well as expected deviations from the recommended
amounts of fertilizer to use as it has been shown above.
With this information farmers may adjust fertilization
programs in relation with particular capital levels in order

to minimize the risk and uncertainty involved.

84

Similar conclusions may be drawn concerning irrigation
data. In this study irrigation plots have proved much more
productive than non-irrigated plots.

It seems important to stress that the economic
optimum conditions are related both to physical function
relationship and the input-output prices in a particular
period of time. Input—output price changes implies a new
optimum amounts and combinations of fertilizer inputs to
use. This point has a practical implication when recome
mendations are being extended to farmers.

From the point of view of farm planning, it sounds
logical to suggest that a farmer attempting to make the best
use of limited resources in spending money on fertilizers,
should use fertilizer inputs until a point is reached where
a greater return can no longer be obtained from fertilizer
than elsewhere in the business. If it were possible to
obtain reliable information on the returns farmers in the
Cauca Valley are earning from other than fertilizer inputs
in their businesses, the opportunity cost principle could
be used in making marginal productivities comparisons to
maximize profits. Still better decisions could then be
reached. A well designed farm management survey or record

keeping system would be suitable means of obtaining some of

85
the necessary information on farm business returns.

It should also be remembered that general management
levels seems important in determining economic use of ferti—
lizer. ”Intangible as management measures are, fertilizer
is apparently more productive under superior management
which includes efficiency in timeliness of operations, choice
of varieties, and other recommended cultural practices.
Increased use of fertilizer is most effective on many farms
only if improved cultural practices are used at the same
time. Management considerations also involve adjustments
to risk."1

The use of fertilizer is badly needed in most of the
Latin American farms. The research reported here on levels
of fertilization to maximize profits should be of great
interest among farmers, providing these kinds of results
can be transmitted in such a way that farmers can understand

their real meaning and benefit.

 

1R. C. Woodworth, ?Organizing Fertilizer Input-Output

Data in Farm Planning,? Economig_Analysis of FertilizergUse
Data, edited by Baum and others (Ames, Iowa, 1956), p. 158.

Fertilizer use experiment, FloridaL Colombia,

APPENDIX A

residuals.

1958:

actual yields, predictedgyields and calculated

A.‘

 

Number of ‘Actual Predicted Calculated
plot yield yield residual
Y Y u
1 57.6 50.015 + 1.584
2 66.5 67.052 — .552
3 47.8 60.010 -12.210
4 59.0 68.357 — 9.357
5 86.0 85.393 '+ .606
6 75.0 78.351 - 3.351
7 53.9 52.536 + 1.363
8 68.3 70.520 — 2.220
9 63.0 63.952 - .952
10 69.7 70.878 — 1.178
11 102.0 88.862 +13.l37
12 90.5 82.294 + 8.205
13 63.6 57.700 + 5.899
14 77.0 73.809 + 3.190
15 66.0 66.303 - .303
16 67.2 76.042 - 8.842
17 78.3 92.151 -13.851
18 93.7 84.645 + 9.054
19 61.7 53.139 + 8.560
20 73.2 70.175 + 3.024

86

 

Appendix A.--Continued

87

 

4A

 

Number of Actual Predicted Calculated
plots yield yield residual
Y Y u
21 65.4 63.133 + 2.266
22 58.5 71.481 ~12.981
23 90.3 88.517 + 1.782
24 92.3 81.475 +10.824
25 65.1 60.823 + 4.276
26 83.0 76.932 + 6.067
27 67.9 69.427 — 1.527
28 77.4 79.165 — 1.765
29 95.0 95.274 - .274
30 84.3 87.768 - 3.468
31 63.6 60.221 + 3.378
32 72.1 77.278 - 5.178
33 55.1 70.246 —l4.746
34 76.3 78.562 - 2.262
35 109.8 95.619 +14.l80
36 93.7 88.588 + 5.111
37 55.1 54.609 + .490
38 72.3 72.594 — .294
39 58.5 66.026 - 7.526
40 74.4 72.951 + 1.448
41 88.2 90.935 - 2.735
42 90.0 84.367 + 5.632
43 54.8 62.294 - 7.494
44 83.6 79.351 + 4.248
45 88.2 72.319 +15.880
46 75.7 80.636 - 4.936

 

Appendix A.—-Continued

88

 

 

Number of Actual Predicted Calculated
plots yield yield residual
Y Y u
47 92.6 97.693 — 5.093
48 94.2 90.661 + 3.538
49 64.5 54.386 +10.113
50 73.0 73.318 - .318
51 55.4 67.224 —ll.824
52 65.2 72.728 - 7.528
53 91.3 91.660 - .360
54 85.2 85.566 — .366
55 68.0 56.892 +11.107
56 78.1 72.074 + 6.025
57 50.8 64.105 -l3.305
58 73.3 75.234 - 1.934
59 94.3 90.416 + 3.883
60 97.1 82.446 +14.653
61 61.1 61.263 - .163
62 72.9 78.341 - 5.441
63 73.1 71.319 + 1.780
64 74.1 79.605 — 5.505
65 95.5 96.682 — 1.182
66 95.9 89.661 + 6.238
67 46.6 55.410 - 8.810
68 71.7 74.342 - 2.642
69 70.2 68.248 + 1.951
70 74.2 73.751 + .448
71 96.8 92.683 + 4.116
72 82.7 86.586 - 3.889

89
Appendix A.--Continued

44‘ A “41—-

 

Number of Actual Predicted Calculated
plots yield yield residual

Y Y u
73 ‘75.1 60.016 +15.083
74 79.0 75.197 + 3.802
75 60.2 67.228 - 7.028
76 70.0 78.357 - 8.357
77. 88.7 93.539 - 4.839
78 70.5 85.570 -15.070
79 63.1 62.286 + .813
80 78.4 79.364 - .964
81 71.9 72.343 - .443
82 83.0 80.628 + 2.371
83 96.3 97.706 - 1.406
84 91.7 90.684 + 1.015
85 62.8 55.566 + 7.233
86 71.5 75.446 — 3.946
87 57.2 69.826 —12.626
88 80.6 73.908 + 6.691
89 99.5 93.788 + 5.711
90 91.5 88.168 + 3.131
91 65.1 63.251 + 1.848
92 77.0 82.203 — 5.203
93 79.0 76.120 + 2.879
94 78.3 81.592 - 3.292
95 94.0 100.545 - 6.545
96 100.0 94.461 + 5.538
97 63.9 55.539 + 8.360
98 70.8 75.419 - 4.619

90
Appendix A.——Continued

 

 

Number of Actual Predicted Calculated
plots yield yield residual
Y Y u
99 70.4 69.799 + .600
100 75.2 73.881 + 1.318
101 89.0 93.761 — 4.761
102 94.6 88.141 + 6.458
103 59.0 62.443 - 3.443
104 81.1 80.468 + .631
105 70.9 73.921 - 3.021
106 70.2 80.785 ~ 1.585
107 109.0 98.810 +10.189
108 91.5 92.263 - .763
109 63.0 63.224 - .224
110 80.4 82.177 — 1.777
111 82.5 76.093 + 6.406
112 78.9 81.566 - 2.666
113 100.6 100.518 + .081
114 92.3 94.435 - 2.135
115 63.1 62.416 + .683
116 81.1 80.442 + .657
117 72.5 73.894 — 1.394
118 80.0 80.758 - .758
119 91.7 98.783 - 7.083
120 97.0 92.236 + 4.763

 

We ights “and measure 5 :

H Ia ha F‘ H Id pa F4 H ta Pd Ha H

APPENDIX B

used in

centimeter
meter

hectar (Ha)
acre

sq. kilometer
sq. mile

sq. hectar
bushel (Bu)
hectoliter
kildgramo(Kg.)
pound (1b.)
Vcargafi

”fanegadaF

metric equivalents

this thesis.

0.3937
39.37
2.47
0.4047
0.386
2.59
10.000
0.3524
2.8375
2.2046
0.4536
150
6.400

91

inches
inches
acres
hectar

sq. mile
sq. kilometer (Km.)
sq. meters
hectoliter
Bu.

lbs.

Kg.

Kgs.

sq. meters

 

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1U
‘33”

 

 

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