ESTIMATING ENPUT-OUTPUT RELATWNSHEPS
50R WHEAT IN MICHIGAN USENG
MUN—G DATA, Wfiz-M

111093: fie»: tho Dams a! M. S.
MICHEG-AN STATE UNWERSH‘Y
Hsiang Hsing Yeh
1955

ESTIMATING INPUT-OUTPUT REIATIONSHIPS FOR WHEAT IN

MICHIGAN USING SAMPIING DATA, 1952-511

HSIANG HSINGYEH

AN ABSTRACT
Submitted to the College of Agriculture of Michigan
State University of Agriculture and Applied
Science in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE
Department of Agricultural Economics

1955

AWN“?d by W

Hsiang Hsing Ieh
ABSTRACT

The purpose of this study is to test a technique of estimating
input-output relationships for wheat in Michigan using sampling data.
1 better knowledge of these input-output relationships are needed to
nake recommendations to farm Operators regarding the Optimum amount of
fertilizer of a given nutrient combination that they should use in
order to maximise their profit from producing wheat.

The survey data was a portion of a Fern Management Survey
Questionnaire collected from hll farms by interviews and covered the
crop years 1952-51». The input-output relationships were derived by
fitting a Cobb Douglas function. this is an exponential equation,
linear in logarithmic form and in that for: is expressed as

log 11 - log a 4- b2 log 12 4- b3 log 13 + .... + b22 log 122
there 21 is the dependent variable and X2 .... 122 are groups in
independent variable inputs and the bi's are elasticities of 12 ....
122 with respect to dependent variable. The equation was fitted
by the least square regression technique to find the bi's.

The fertilizer data were fitted in two different forms. In the
first form, fertilizer application were classified into total pounds
of nitrogen, total pounds of phosphorus, and total pounds of potassium

The variable for nitrogen top dressing as one independent variable.
The second for: of fertilizer application were classified into nitrogen
applied at the planting tins and nitrogen tap dressing as two independent

variables.
ii

Hsiang Being Yeh

The data from the field survey were fitted in four different equa-
tions. The first two equations included five and six variables and
the other two equations included seventeen and eighteen variables.

They then were compared with data which has been available from con-
trolled experiments on plots on the University experimental farms.

The tentative results of this study indicate that comparison of
the most profitable rate of fertilizer application between survey and
experimental data and the least cost combination which obtained from
survey data might be as follows:

The most profitable rate of 3-12-12 fertilizer application obtained
from the results of the five variable equation was approximately 220
pounds and from the seventeen variable equation was approximately 260
pounds while the most profitable rate of 0-12-12 fertiliser plus 10
pounds nitrogen tap dressing in the six variable equation was approxi-
mately 1:8 pounds and of 0-30-10 fertiliser plus 10 pounds nitrogen
top dressing in the eighteen variable equation was approximately £16
pounds. The 0-12-12 and 0-30-10 fertilisers were the least cost
combination that were derived from six and eighteen variable equations.
If these figures were compared with the experimental results, it appears
that the application rate of 3-12-12 fertiliser indicated by the survq
data might be slightly more than one-half those indicated by the experi-
mental results. The most profitable rates of 0-12-12 and 0-30-10 ferti-
lizer plus 10 pounds nitrogen tap dressing was not available from
experiments to compare with results from the survey data, but the

2 111

Hsiang Hsing lab

results derived from the survey data appeared low. Nitrogen tap dressing
was used as an independent variable in the survey data and as a fixed
variable in experimental design

The least cost combination for N-P-K fertilizer analysis were
approximately 2-2-1 in the five variable equation and h-lO-l in the
seventeen variable equation while the least cost combination for N-P-K
fertilizer plus nitrogen tOp dressing were approximately 0-1-1 in the
six variable equation and 0-3-1 in the eighteen variable equation. If
these combinations were compared with 1-11-11 in experimental data, it
appears that an increase in nitrogen and a decrease in potassium in the
fertilizer analysis used by farmers might be recommended in the future
years. It also suggests that the future research projects for determin-
ing the optimum fertilizer applications should give more attention to
least cost combinations than has. been the case in the past.

Thus, while the results of this analysis indicate that the use of
survey data to determine input-output relationships is feasible, further
testing and methodological research is indicated before it can become

a useful research tool.

iv

ESTIMATING INPUT-OUTPUT RELATIONSHIPS FOR WHEAT IN

MICHIGAN USING SAMPLING DATA, 1952-511

By
HSIANG HSING'IEH

A TIESIS
Submitted to the College of Agriculture of Michigan
State University of Agriculture and Applied
Science, in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE

Department of Agricultural Economics
1955

ACKNOWLEDGMENTS

The author wishes to express his appreciation to all who aided in
the preparation of this manuscript. Sincere gratitude is expressed to
Dr. D. E. Hathaway for his guidance and criticism throughout the study.

Thanks are due Dr. Gt 1» Jehnson for his helpful suggestions.

The aid of Dr. K. J. Arnold of the Mathematics Department in
deve10ping the statistical analysis was appreciated.

The author also wishes to express his thanks to Mrs..Marcia Bungo
who aided in the analysis of the survey data.

Particular thanks are due Dr. L.‘W.'Witt for his generous guidance
and assistance. I

Financial aid in the form of a graduate research assistantship
provided by Dr. T. K. Cowden, Dean of the College of Agriculture and
Dr. L. L. Boger, Head of the Department of Agricultural Economics,
which made it possible for the author to complete his study at
Iflichigan State university is deeply appreciated.

The author assumes responsibility for any errors in this manuscript.

v1

361671

CHAPTER

II

III

TABLE OF CONTENTS

INTRODUCT ION O O O O O O O O O O O O O O O O O O O O O O

THEORETICALBACKGROUND.................

TheProductionFunction. ee eeeeeeeeeee

Substitutability and Complementarity Relationships.

The Cobb-Douglas Production Function .
Value Productivity and Profit . . . . .
METHODS OF SAMPIINGAND ANALYSIS . . . . . .
Methods of Gathering Data . . . . . . .
MethodofAnalysis . . . . . . . . . .

ANALYSIS OF INPUT-OUTPUT RELATIONSHIPS FROM
EIPH‘IIMENTALDATA...........

The Pricing Of Qitput and Input e e e e

The Most Profitable Rate of Application

of

Fertilizer eeeeeeeeeeeeeeee

AN EVALUATION OF THE STATISTICAL RESULTS AFTER
FITTINGTHEFUNCTION . . . . . . . . . . .

The Results of the Five and Six Variable Equations.

The Results of the Seventeen and Eighteen Variable

PAGE

1

h

5
11
13
1h
16
16
18

23
28

28

32
33

hut10DBeeeeeeeeeeeeeeeeeeeeehb

SUMMARI.AND CONCLUSIONS . . . . . . . . . . . . . . . . 57

Limitations of Using Survey Data for Input-

OutputRelatiothipa...............6O

SomeAdvantagesofSurveyData . . . . .. . . . .61

Conclusions and Recommendations for Future Studies. 61

711

TABLE OF CONTENTS (Continued)
Chapter Page

APPENDIX A: A PORTION OF THE QUESTIONNAIRE IN FARM
MANAGEMENT SURVEY USED INTHIS STUDY . . . . . . . 63

APPENDIX B: CONVERSION RATES FOR LIVESTOCK TO
STANMRDANMLUNHSOOOOOOOOOO<000066

BIBHM O O O O O O O O O O O O O O O O O O O O O 6?

Viii

TABLE

II

III

VII

VIII

LIST OF TABLES
PAGE

The Effect of Increasing Rates of Fertilizer
Application «1 the Total, Marginal, and
Average Physical Product for Wheat . . . . . . . . . 2h

The Effect of Increasing Rates of Fertilizer
Application on the Marginal Value Product
for WhCQt in.1952'5h e e e o e e e e e e e e e e e e 29

Regression Coefficients, Standard Errors of Regression
Coefficients, Coeffioimts of Multiple Correlation,
Coefficients of Multiple Determination and Standard
Errors of Estimate in Five and Six Variable Equa-

tions e e e e e.e e e e e e e e e e e e e e e e e e 3h

The Effect of Increasing Rates of 3-12-12 Fertilizer
Applications on the Total, Marginal Physical
Products and Marginal Value Products Per Acre
For Hheat with Different Intensities of Animal
Units in the Five Variable Equation . . . . . . . . 38

The Effect of Increasing Rates of the Least Cost
. Combination or 0-12-12 Fertilizer Plus 10 Pounds
Nitrogen Top Dressing Application on the Total,
Marginal Physical Products and Marginal Value
Products Per Acre for Wheat with Different
Intensities of Animal Units in the Six Variable
Equation ee e e e e e e e e e e e e e e e e e e e e 39

The Most Profitable Rates of 3-12-12 Fertilizer and
0-12-12 Fertilizer Plus 10 Pounds litrogen Top
Dressing Application and the Most Profitable
Yields for Hheat with Different Intensities of
Anilfll Uhitl, 1952-5h e e e e e e e e e e e e e e e hl

Marginal Value Products of N , P2 , K20, and Nitrogen
Top Dressing in Five ens Six ariable Equations. . . M;

The Least Cost Combination for N , P205, and K20 in
Five and Six.V‘r13b19 Equat ODE e e e e e e e e e 0 NS

Regression Coefficients, Standard Errors of Regression
Coefficients, Coefficients of Multiple Correlation,
Coefficients of Multiple Determination, and Standard
Errors of Estimate in Seventeen and Eighteen Vari-
able Equation, e e e e e e e e e e e e e e e e e e e h?

ix

LIST OF TABLES (Continued)

TABLE PAGE

I. The Effect of Increasing Rates of 3-12-12 Fertilizer
for the Seventeen.Variable Equation and 0-30-10
Fertilizer Plus 10 Pbunds Nitrogen Top Dressing
for Eighteen Variable Equation on the Total,
‘Marginal Physical Products and Marginal Value
ProductsPerAcreforNheat........... . 51

XI Marginal Value Products of N2, P205, K20, and
Nitrogen Top Dressing in Seventeen and Eighteen
Variable Equations, 1952-514 e e e e e e e e e e e e 5’4

XII The Least Cost Combination fork!
in Seventeen and Eighteen Vzariaglean Equations,
1952.51‘0OOOOOOOOOOOOOOOOOOOOO56

LIST OF FIGURES
FIGURE PAGE

1 Relationship of Factor Input to Total, Marginal,
and Average Physical Product, Stages of
Production and Rational Use of Resources . . . . . . 7

2 A Production Surface Showing lee-Product Lines,
Iso-Cost Line, Ridge Lines and Scale Line . . . . . 10

3 A Production Surface Showing Substitute and
Complementary Resources e e e e e e e e e e e e e e 12

)4 Relationship of Factor Input to Total, Marginal
and Average Value Production and location of
the Right Profit Point Using Variable Factor
Xi With.12, Xa,eeee,xh Fixed e e e e e e e e e e e 15

5 Map of Michigan Showing Counties in Which Surveys
“Ere Taken’. 0 e e e e e b e e e e e e e e e e e e e 17

6 The Effect of Increasing Rates of 3-12-12 Fertilizer
Applications on the Total, Marginal, and
Average Physical Product for wheat . . . . . . . . . 25

7 The, Effect of Increasing Rates of 3-12-12 Fertilizer
Plus 20 Pounds Nitrogen Tap Dressing Applications
on the Total, Marginal and Average Physical Product
for Wheat in Experimental Data 0 e o e e e e e e e e 26

8 Equating Marginal Value Product of wheat with Marginal
Factor Cost of Fertilizer to Determine Optimum
Rate of 3-12-12 Fertilizer and 3-12-12 Plus
20 Pounds Nitrogen Top Dressing Application with
1952 Price. 0 e e e e e e e e e e e e e e e e e e e 31

9 A Comparison between Experimental and Smey Data of
the Effect of Increasing 3-12-12 Fertilizer
Application on the Total Physical Product and
Marginal Value Product for “heat with 1953
Prices Using the Five Variable Equation . . . . . . 1:2

10 A Comparison Between Piperimental and Survey Data
of the Effect of Increasing 3-12-12 Fertilizer
Application on the Total Physical Product and
Marginal Value Product for Hheat with 1953
Prices Using Seventeen Variable Equation . . . . . . 53

xi

CHAPTER I
INTRODUCTION

‘Hheat is Mdchiganfis asst twportant cash grain crop. Since 1866,
when the first crap estimates were:lade, the acreage of wheat has
fluctuated between the high of 1,985,000 acres in 1882 and in 18811
to the low acreage cf’661,000 in l9h3. The acreage of’w‘heatl has
been reduced in.recent years by the acreage allot-ent.prograa. But,
the1yield of'wheat per acre has had.a definite upward.trend. The
first five-year (1866-70) average was 111.8 bushels per acre. The
19h5-h9 average was 26.3 bushels per acre. The averagejyield was
30.0 bushels per acre in.l95h. The upward.trend.inxpart is due to
iaprcving acne practices and.increasing the rate of fertilizer

application.
no objective of this study is to test a technique of estilating

input-output relationships for wheat in Michigan using sampling date.
A better knowledge of these input-output relationships are needed to
make reccuendations to farm operators regarding the optima amount
of fertilizer of a given nutrient cubination that they should use
in order to maximise their profit from producing wheat.

 

1
the data on acreages and yields was obtained from Michigan
Agricultural Statistics, Lansing, Michigan.

Up to the present time there has been very little input-output data
available for most Michigan crops. The data that has been available has
come from controlled experiments on plots on the University experimental
far-s. Most of the experiments on crop response to fertilizer appli-
cations have not been designed to answer economic questions. Due to
limitations of cost they gena'ally have included a relatively limited
range of applications of fertilizers. In nearly all cases they have
been done using one or two different analysis of fertilizer which
have seemed that the nutritive ratios tested were the least-cost
combination under all conditions. In addition, the application of
results from experimental plots to actual farm conditions makes
several assqutions regarding the relationships between fertiliser
applications and other production practices.

It was felt that survey data on input-output relationships on
farms might provide one method of obtaining increased amounts of
input-output data for sue crops.

The surveydatainthis casewas aporticn ofanraManagault
Survey mesticnnaire collected from lull farms in 19514 which included
yield and ilput data for wheat for the crop years 1952-51; in four
areas of the state.

It was felt that this input-output data from this survey night
meet a desirable degree of reliability of results at a lower cost
than the information could be obtained by other methods. In addition,
the survey data is related to sane actual farm conditions and if use-
able tight overcome some of the problems of applicability of snori-
mental data to far- conditions.

Sue econcuc theories relevant to these problems are discussed and
related to the farm business in Chapter II.

Chapter III describes the survey data used in this stuw and method-
oloy uplcyed in measuring the input-output relationship by fitting a
Cobb-Douglas production function.

file effects of fertilizer application of various rates on total
and marginal plwsical product of wheat ham available calorimental
data are presented and discussed in Chapter IV. These data are used
bothasacheck onthe surveydataandtoindicate theoptinumanoumt
of fertilizer that should be used according to the WM results
currently available.

Chapter V deals with the fitting of the function to the survey
data and an appraisal of the statistical results. Comparison of the
results between experimental and survey data are indicated in this
chapter.

The final chapter includes the general conclusions obtained frm
the research and suggests possible future research that might be done
in this area.

CHAPTER II
TWICE“. RAM

1 statement of a useful classification which covers nest of exist-
ing economic theory divides it into two broad categories. The first
category of economic theory is the static economic theory covering
micro and macro-static economic theory. The theory usually considered
when the word static is used is a theory of a given unbu- of exact
relationships along the sane given nunber of economic variables. in
exact relationship, as used herein, is one which has a standard devi-
ation of zero. In a theory of mast static relationships, the magni-
tudesofcertainvariahles canandarepernittedto ohangeasthe
theory is used to explain the changes which occur when the value of
one or a set of variables is (changed. The second category of economic
theory is the dynamic econonie theory covering nicro and nacro ironic
economic theory. This theory is concerned with the tine relation-
ships cf economic variables.

There have been mm. levels of develop-ant and intep‘ation
achieved within and between these various categories. Both micro-
static and macro-static econaic theories have been rather well
developed and integrated. On the other hand, the dynsnic cyst-s
of theory have yet to achieve a cellparable develcpent and inte-

cation.

A naJor difficulty associated with static economic theory has been
the nature of the assumptions nade. wassmingas fixednost ofthe
changes or variable factors, the system has been silplified at the
ecpenses of reality. Teohnolog, wants and preferences, asset distri-
bution, and institutions have been posited as unchanged. it the sale
tine, individuals have been assmed both rational and as having perfect
knowledge.

Dynamic systens of econo-ic thought have arisenwhenone ornere
of the static assumptions have been relaxed. Although the dynsnic
cyst-s nay be nore realistic in their assumtions, the attendant
coqlexities have ifliibited their usefulness to a large extent.

the of the areas of static promotion theory is factor-factor and
factor-product analysis. This area is concerned with problens of
‘ resource conbination and allocation in the production of a single'
product. In this thesis, static factor-factor and factor-product
theory is applied to both experimental data and sampling date by the
Joint use of nathnatics (as represented by the u(Babb-«Douglas function")
and statistics.

The Hoduction Junction
the production function has been enpressed as the concept of the
factor-product relationship or the input-output relationship. This
concept refers to the relationship between the input of factor and
the output of product.
Ihe problms for which static factor-factor and factor-product
analysis are useful are one, the determination of proportions in

which inputs should be combined to yield a maximum of a product for a
given input or outlay; and, two, the determination of how much of the
various inputs, combined in optinun proportions are required to yield
a maxim profit.

1 production function nay be messed as an algebraic equation of
the fern

I - f (5'5, 13, 1“,...”5)

Its meaning is that output I is dependent on a variable factor
Xlwhilexz isfixedalongwith resources 13 throughxn. talcum
variable inputs are held constant and others are allowed to vary, a
special relationship exists betweai input and the rate of output of the
product. his relationship has hem turned the In of Dininishing
Returns. It may be stated as follows:

When output depends upon both variable and fixed inputs, the
addition of a variable input to fixed inputs results first in out-
put which increase at an increasing rate, then increase at a decreas-
ing rate and finally tends to decrease."

The production function is divided into three stages. Stage II
is a rational area of production with technical efficiency while both
stage I and III are irrational area of production with technical
inefficiency. These are shown graphically in Figure 1.

In stage II, narginal physical product decreases continuously and
is always less than average physical product and greater than or equal
to acre. The boundary of the rational area of production extends from
the input denoting a maxim: average plvsical productivity to the one
defining the maxim total pusical productivity. In other words, all

TPP

I
«~— Stage I ——-- Stage II—u-Ic—StechIL.

-------’.‘-

MP?

/ m

x1 ' XZ’ Is,eeee’x‘1

 

b..--
.

Figure 1. Relationship of Factor Input to Total, Marginal, and
Average Physical Product, Stages of Production
and Rational Use of Resources.

resource inputs between the one consistent with a naxiaua product per
unit of variable factor and the one consistent with a nanmm product
fronthefixedfactorfall instage II.

In stage I, aarginal plwsical product is always greater than average
physical product. The average physical productivity of the variable re-
source will increase continuously as acre units are applied to the fixed
factor. is production per unit of fixed factor is pushed to the limits
of stage I, a greater product is forthcoming fro. the fixed factors as
well as no. each nit cf the variable factor. in level of resource
usefellinginstageIisunecononic. InstageIII, narginalplysical
product is less than sore, and additional input reduces total output.

So, stage III is also an area of irrational production.

In most fara aanagenent work, it has been assumed that stage II is
the area in which faras are being operated and that farmers cease appli-
cation of variable resources to fixed factors before the upper limit of
stage II has been reached.

1 production function involving two variable inputs and an indefinite,
but given, number of fixed inputs nu be expressed in the following fora as

I I f (11, 12' 13,....,In)
This equation states that the output of I depends Jointly upon the inputs
ofxlsndlzinsonedefinitenannerwhentheitputsIBtoXhareheld
constant. This function nay be represented graphically as in Figure 2.

The product contour lines I1, 12.0.! are called Ice-product curves

10
because equal accents of I are produced by the various combination of
11 and X2 represented by a given line II. Ice-product line I2 is

pester than I]. until point B is reached indicating the highest possible

output of I. he dotted lines, 00 and OD, are ridge lines and enclose
both stage I and stage II since in that area all positive derivatives
of I as dictated by the function I - r (11, 12 [13,...”15) are in
this area. he nsrginal physical productivity of both factors :1 and
12 are greater than aero falling tithin the area of the ridge lines
while less than zero falling in outside of the ridge lines. Quanti-
ties of I also exist beyond the ridge lines but production outside of
the area requires greater amount of either 11 and/ or 12 for the one
output than the area enclosed by the ridge lines. Ridge lines, the Iso-
product lines, the Iso-cost line and the scale line are shown in Figure 2.
The line AB is known as an Iso-cost line because all conbinations of 11
and 12 represented by it have equal total costs. In Figure 2 the Iso-
cost line is tangent to Isa-product line Ih' Here is the greatest out-
put of I for a given quantity of 11, 12 used at that point G.

it the point of tangency, the slopes of the two lines are equal.
The slope of the Ice-product line is the rate of product substitution
ofxzforllwhichis m2 andthe slopeofthe Iso-costlineisthe
rate of cuth substitution xlwhich is :31. Therefore, the rate of
product substitution is equal to the rate of outh substitution at the
point of tangency. Thisnaybe enpressedinfon suwugfz-iflo

:1

his is the optimu- conbinaticn of resources. If a series of these
points are Joined togefiier the resultant line (EF shown in Figure 2)
is teraed the scale line because it indicates the optima preportions
inwhichto conbinexlandx2 toproduceawgiven output. Fornore

10

 

 

 

 

 

 

 

 

 

 

 

11 HPle < O
c I -, “1......1--1"""" l t 1
c l
‘t
g
a
g
i I
i > o
:9
‘ .
.18 W2<o
5"“ Y7
L ,6
’5
I
F L
B D K?

Figure 2. A Production Surface Showing Iso-Pmduce Lines,
Ice-Cost Line, Ridge Lines and Scale Line.

than two inputs, they may be combined in scale line proportion. The
optima combination of In inputs nay be determined by

x1 :2 1‘n

Substitutabilit; and momenta“; Relationship;
Resources can be either substitutes or complements. hey are conple-

 

 

neuts where they nust be combined in fixed proportions or where a re-
duction in input of one factor cannot be replaced by an increase in other
factor. Factors are substitutes when output can be uintained as re-
sources are reshuffled, when one factor is reduced in quantity, a second
factor nust always be increased. Resources which are conplenents or
substitutes can be defined by the narginal rate of substitution and are
illustrated in Figure 3.

Resources are canplenents when the marginal rate of substitution
is acre or floater} In Figure 3, the ratio of change, AIZ / All,
is zero when output is constant and input of 12 is at the niniaun of OT.
If 11 is increased beyond (It, none of 12 is replaced.

Resources are substitutes when the mginal rate of substitution
is less than aerc. In other words, the sign ofA 12/ All, the
narginal rate at which one factor substitutes for another, must be
negative.

A product contour characterising these relationships is illus-
trated in Figure 3. If output is maintained at the 100 units

1 Ready, m1 0. Economics of Agatha Production and Resource Use.
New Iork: Prent ce- , c., 9 2, pp. list-110.

 

 

 

 

 

P5 (300 Units)

 

Pb (200 units)

 

 

Pi (100 units)

 

 

 

1‘2

Figure 3. A Production Surface Showing Substitution
and Complementary Resources .

13

indicated by contour I1 Pl’ at the 200 units by contour 12 P2 and at the
300 units by contour I3 P3, 11 and 12 serve as substitutes only between
the extentoftheridgelinesasOAandOB. Beyondthese extent, in-
crease in input 12 require that inputs of I1 also be increased.

he Cobb-Douglas Production Function

he Cobb-Douglas production function is an exponential equation,
linear in logarithmic fern. It was developed by a nathenatician,
harles H. Cobb of Amherst College in cooperation with Paul R. Douglas,
an economist now in the United States Senate,2 in 1927-28.

In natural numbers, the equation is written I - albl 1:2 ""1an
where I is the dependent variable, 11 to In are independent categories
of inputs and the exponents (bi's) are elasticities of the dependent
variable with respect to their corresponding independent categories of
inputs and a is a constant.

In logarithmic fern, the equation is linear and is fitted by linear
lultiple correlation using the least square technique to determine the
bi's. he equation written in logarithmic fora becomes:

log I - log a 4- b1 log 11 + b2 log 1215... 4- bn log In

A function of this type allows the diminishing physical productivity
to be reflected in the estimated relationship between the dependent and
independent categories. he sun of the b's indicates the nature of the
productivities to scale for the business as a whole. Diminishing

 

2
Cobb, Charles V. and Paul H. Douglas. “A heory of Production,"
The American Econanic Review, Vol. XVIII, Supplement, (March, 1928).

physical productivity exists if the sun of the b's is less than one,
increasing if greater than one and constant if equal to one.

his function is capable of reflecting am one stage of production,
but for an input or the sun of am combination of inputs, it can re-
flect only one stage for different inputs at the same time.

Value Productivig and Profit

he physical relationships may be converted to value relationships
by multiplying the marginal, average, and total physical products by
the price of the product. ’hus the value of the marginal product, the
value of the average product, and the value of the total product may be
obtained. Under conditions of perfect competition assuming homogeneity
and divisibility of factors and products it is also true that VHP I MVP,
VAP - HP, and VTP - TVP. he value productivity curves are shown in
Figure it.

he determination of the maximum profit point, when the price of
the product and of the variable inputs is known, can be shown graphi-
cally as in Figure I; where at point B the P11 line intersects the
MVP line.

hese economic theories provide the analytical framework used
in this study to measure input-output relationships for wheat with
emerinental (Chapter IV) and survey data (Chapter V). his gave a
basis for estimating the possibility of getting useful information
via the two different methods.

15

NP

Dollars

 

 

 

 

o/ /\ px1

11; 1,, x3,....,x,,

 

Figure 14. Relationship of Factor Input to Total, Marginal
and Average Value Production and location of the
High Profit Point Using Variable Factor

XI with 12, 33,....,Jg1 Fixed.

CHAPTERIII
METHGDSOFWGLNDANMISIS

Methods of Gatheriiiggtg

he farms visited were selected by the random sampling method and
were drawn Rom the wheat listing sheets of the County Ag'iciltural
Stabilisation and Conservation Cos-ittee. The Sampling data were
collected from all. farms in four areas covering nineteen townships in
five counties. he townships within those counties were selected
with an attmnpt to maintain approximately a uniform soil type within
each of the areas. Data on the calendar years 1952 through 1951; were
collected. The farms that were visited were distributed among those
four areas as follows: .

Five townships in the southern part of Kalamazoo County were in-
cluded in area I and 103 farms were interviewed in this area. he
maJor fan enterprises are wheat, corn, dairy, and hogs. he soil
type is well drained sandy loam.1

In area II, two counties, Gratiot and Isabella, were selected.
Three of the townships and 69 fame were in Gratiot County and to
farms were interviewed in two townships in Isabella County. he

 

1
Hill, Elton B. and Russell 0. Hawby. apes of FW°
Michigan Agricultural Emeriment Station, Fast lensing, Special
Bulletin 206, September 1951;.

17

 

 

 

Becca} ! vain-EH.
. \ -
\

I
.4

 

 

 

Figure 5. lap of Michigan Showing Counties in

Which Surveys Here Taken.

18

maJor farm enterprises are cash crops and dairy. he soil type is clq
loam to silty clay loam soils.

Five townships in Sanilac County were included in area III and 99
farms were visited. Cash crops and dairy are important in this area.
he soil type is poorly drained loam, silt loan and clay loam.

Four townships in Livingston County were included in area IV and
100' farms were selected. he maJor farm enterprises are dairy and
general farming in this area. he soil type is well drained loony
sands and sanw loan.

1 portion of the data sheet used in gathering the intonation is
shown in Appendix i. he naJor contents of the questionnaire are
as follows: I

l. Acreage of wheat.

2. Iield per acre of wheat.

3. Seed quality and fertiliser practices for wheat.

h. livestock enterprises on the farm.

This intmiewing was done by four graduate students and one under-
graduate student in the Department of Agricultural Icons-ice, mdiigan
State Univn‘sity. he data were gathered betwemi June 2h and Septaber
16, 195k and were tabulated between October, 1951: and April, 1955.

Hethod 9f Analgis
he estimates for this stuchr were derived by fitting a Cobb-
Douglas function. he equation was of the form: :1 . £2132 13b3,...122b22
andappearedinthelogformas

log 11 - log. a + b2 log 12 + b3 log 13 + ....+ 1’22 log 122.

19

By converting the basic data to logs, it is possible by multiple linear

regression technique to determine a mean relationship between the

independent variable factor categories and the dependent variable,

physical yield of wheat, which is (1) linear in the logs and (2)

exponential in the natural number.

he variables included in the equations are indicated as follows:

11 8

X2:

13:

19:

is the logarithm of total yields of wheat.

is the logarithm of total pounds of nitrogen plant food in-
cluding nitrogen top dressing.

is the logarithm of total pounds of phosphorus plant food.
is the logarithn of total pounds of potassium plant food.
is the logarithm of total pounds of nitrogen top dressing.
is the logarithn of total pounds of nitrogen applied at
the planting time.

indicates seed quality, where log 1 indicates certified
seed was not used while log 10 indicates certified seed
was used.

indicates the animal units per acre. Log 10 indicates
the presence of no animal units per acre while log 1
indicates eowe livestock in 1953.

indicates the animal units per acre. Log 10 indicates
0.01-0.19 animal units per acre while log 1 indicates
the presence of more or less than 0.01--0.19 animal

units per acre in 1953.

xlo‘

20

indicates the aninal units per acre. log 10 indicates the
presence of 0.20 or more units per acre while log 1 indicates
less than 0.20 unite in 1953.

In, 112....122: indicates the relationships between four areas in

113 3

which interviewing was done and three years 1952, 1953 and
1951: as follows:

 

 

 

Areas
Itar .
I II III IV
1952 In 11!: 11., 120

1953 ‘12 I15 ‘18 x21
19511 113 116 X19 122

 

 

Log 10 indicates the observed event occurred in a given area
and year while log 1 indicates that the event did not occur.
Log 10 indicates the observed event occurred in area I in
1952 while log 1 indicates that it did not occur in area I
in 1952.

Log 10 indicates. the observed event occurred in area I in
1953 while log 1 indicates that it did not occur in area

I in 1953.

Log 10 indicates the observed event occurred in area I in
1951; while log 1 indicates that it did not occur in area I
in 1951:.

Dog 10 indicates the observed event occurred in area II in
1952 while log 1 indicates that it did not occur in area II
in 1952.

I17’

58‘

x19:

22

21

Log 10 indicates the observed event occurred in area II in
1953 while log 1 indicates that it did not occur in area II
in 1953.

Log 10 indicates the observed event occurred in use II in
1951. while log 1 indicates that it did not occur in area II
in l95h.

Log 10 indicates the observed event occurred in area III in
1952 while log 1 indicates that it did not occur in area III
in 1952.

log 10 indicates the observed event occurred in area III in
1953 while log 1 indicates that it did not occur in area III
in 1953.

Log 10 indicates the observed event ocean-red in area III in
1951: while log 1 indicates that it did not occur in area III
in 1951;.

Dog 10 indicates the observed event occurred in area IV in
1952 while log 1 indicates that it did not occur in area IV
in 1952.

log 10 indicates the observed event occurred in area IV in
1953 while log 1 indicates that it did not occur in area IV
in 1953.

Log 10 indicates the observed event occurred in area IV in
19514 .while log 1 indicates that it did not occur in area IV

in 1951;.

he data from the questionnaires were arranged in the coded sheets

and were converted into logs and then punched on International Business

22

mchine cards. The exponent (bi's) were computed by using the Doolittle

Method. The computation work was carried out partly on the Widen

Calculator and partly on the International Business Machine. The five

and six variable equations did not include all of the variables and

were cmputed on the Mdm Calculator. The seventeen and eighteen

variable equations using of the all dmm variables were computed on
the International Business Machine.

The fertilizer data were fitted into two different ferns. In the
first form, fertilizer application were classified into total pounds of
nitrogen (N2), total pounds of phosphoric acid (P205), and total pounds
of potassium ([20). The variable for nitrogen in this form included
nitrogen applied at planting tins and nitrogen tap-dressing as one
independent variable. The second fora of fertiliser application were
classified into nitrogen applied at the planting tine and nitrogen top-
dressing as two independent variables.

The total amounts of fertiliser applied by the wheat farmers range
from 8C0 pound to 500 pounds. me nitrogen top dressing range fro-I)
pounds to 250 pounds. In addition, to account for the application of
suns]. manure, the equivalent animal units per acre in 1953 were used to
adjust for differences in resulting soil fertility, assuming this would
have the sane effect on the production of wheat. The conversion rates
for the various types of livestock are shown in Appendix B.

The dumw variables were used in the seventeen and eighteen variable
equations to allow for differences in weather between years and for dif-

ferences in soil type between the four areas.

CHAPTE'RIV

ANALYSIS OF INPUT—OUTPUT RELATIONSHIPS
FRO! EXPEDGENTAL DATA

In order to evaluate the usefulness of survey data, it seared
desirable to compare it with available experimental data. Two con-
trolled merinents using different fertiliser applications were
available for this purpose.1 One fertiliser used was 3-12-12 ferti-
liser. The applications began with sero pounds and increased at 25
pound increnents until they reached a maxim of 550 pounds. The
yield ranged frcn 23 bushels of wheat at zero pound of fertiliser
to 112.8 bushels when using 550 pounds of 3-12-12. The second experi-
nent included a combination of 3-l2-l2 fertilizer and nitrogen top-
dressing as before, the 3.12.12 was increased in increments of 25
pounds until a new of 550 pounds was reached, but in each case
there was an additional application of 20 pounds of nitrogen top-
dressing. The yield ranged frm 27.0 bushels where sero fertiliser
was applied to 113.9 bushels where 550 pounds of 3-12-12 plus 20
pounds of nitrogen top-dressing was applied.

The relationships between the total and marginal physical product
in the two experiments are shown in Table I and are illustrated graphi-é

callyinFiguresband7.

 

1
The fertiliser input data is the result of experimental work done by
the Department of Soil Science, Michigan State University. Fertiliser
applications of differing amounts and various analysis were applied on
Fox sanh loan in 19116 to 19147.

TABLE I

THE EFFECT OF INCREASING RATES OF FERTILIZER APPLICATION OF THE
mm, mom, 1ND AVERAGE PHYSICAL PRODUCT non mm

.._ _M--_—.—_._—.~._-_~—-._-.H-——.M—--.——_.—_———- _1._.—_._.._.—__ .u— —— _...~. —_.

- 2-12
3.12.12 Fertiliser

   

Plus 20 Pounds Nitrogen

 

 

 

 

Pounds of Top-Dressing
Firtiliser TPP‘ nrrb arro IP17i HPPBA 192°
mnn man mnn mnn mnn mnn
0 23.0 27.0
1. 1.
25 2h.5 1 : 1.50 28.h h’ 1.h0
50 26.0 1'5 1.50 29.5 :’1 1.25
75 27.5 ’ 1.50 30.6 ‘1 1.20
1.5 . 1.h
100 29.0 1.50 32.0 1.25
1.u 1.2
125 30.h 1.h8 33.2 1.28
1.h 1.2
150 31.8 1.1;? Blah 1.23
1.3 1.2
175 33.1 1.hh 35.6 1.23
1.3 1.2
200 31M; 1.113 36.8 1.22
1.3 1.2
225 35.7 1.h1 38.0 1.22
1.3 1.1
250 37.0 1.u0 39.1 1.21
1.0 1.1
275 38.0 1.36 no.2 1.20
1.0 0.8
300 39e0 1033 131.0 1.17
5 ho 0 1'0 1 31 h1 8 0'8 1 1h
32 . 0.9 . . 0.h .
350 110.9 1.28 112.2 1.09
0.7 0.h
375 h1.6 0 5 1.2!; h2.6 0 h 1.01;
too .h2.1 0.5 1.19 h3.0 0:3 1.00
h25 h2.6 0 h 1.16 h3.3 0 3 0.96
h50 h3.0 0.0 1.11 . h3.6 0.2 0.92
h75 h3.0 1.05 h3.8 0.88
0.0 0.2
500 1.3.0 -0 1 1.00 1111.0 0 0 0.85
525 1.2.9 ° 0.95 M4.0 ' 0.3:
£01 £01
550 h2.8 0.90 h3.9 0.77
&

TPP: Total physical product for wheat.
bIMPP: Marginal physical product.
° APP: Average ptusical procbiot.

 

 

 

 

Iields of Wheat Per Acre (Bushel)

 

 

 

 

ls '“hm~fl W'M A" ”a: ”N ' APP
I i w . $.V$~N ‘
2+ H N
N.“
i
0 e5 "' MPP ” I
O 1 *J L TJ ‘ ;!l
100 200 300 “0° 500'

Pounds of Fertilizer

Figure 6. The Effect of Increasing Rates of 3-12-12 Fertilizer
Application on the Total, Marginal, and
Average Physical Product for‘Wheat,

Yields of'Wheat Per Acre (Bushel)

ho

30

1.0

0.5

Figure 7.

TPPI + 205‘ N

 

 

 

 

  

 

n_ I
100 200 300 hOO

Pounds of Fertilizer

The Effect of Increasing Rates of 3-12-12 Fertilizer Plus
20 Pounds Nitrogen Top Dressing Applications on the
Total, Mbrginal and Average Physical Product for
Wheat in Experimental Data.

27

where only the 3-12-12 fertilizer was used the TPPx of wheat con-
tinues to increase at a decreasing rate as long as the MPI decreases
but is great» than zero. It reaches a maxim at a point C in Figure
6, 113.0 bushels, as the HP!"x becomes zero at a point B. The MPPx
decreases continuously and is equal to zero more 19 units of 3-12-12
fertiliser were used. 1110 line BC in the Figure 6 indicates the end
of a rational area of production of wheat.

Figure 7 shows the sane relationships for the 3-12-12 plus the
addition of a 20 pound application of nitrogen tap-dressing. Using
the sale nethod of locating the end of a rational area of production
for wheat, it is reached a point B in Figure 7 where 20 units of
3-12-12 fertiliser plus 20 pounds nitrogen top-dressing is used. The
‘l'PPx + 20 pounds nitrogen reaches a maxim at a point C, 1114.0 bushels,
as the 13151"x 4» 20 pounds nitrogen becomes am at a point B.

In order to deterline the point of the highest profit, we lust
take into consideration the price of the input and the price of the
output. Of course, we must realise that recounendation concerning
optima anount of am input which are nade under one set of price con-
ditions will change under a different set of price conditions.

He can derive a fundanental condition which will hold true for any
prohlu concerning the optima use of a variable input. This condition
is that the marginal value product (MVP) of any input (11) must equal
the price of the variable input P . This equation of this statment

can be stated algebraically in the following manner.
MVP;

MVle-leor 15—3; '1

1‘1

28

-

his equation tells us several things:2 (1) If the last unit of
11 does not pay for itself less of 11 should be used, (2) If the last
unit of 11 more than pays for itself more of 11 should be used and
(3) the use of 11 should be stopped at the point at which x1 Just pays
for itself.

‘Ihe Pricing‘jf Output and Input

If we apply a set of prices to Table II, we can determine the aost
profitable rate of production. Let the price of each unit of 342-12
fertiliser be $0.50 in l952, 30.1.? in Ma. and to.» in 1.95133 while the
price of each unit of 342-12 fertiliser plus 20 pounds nitrogen top-
dressing be same as the price of 3.12.12 fertilizer. Because of
additional 20 poun.ds nitrogen toy-dressing are kept constant as a fixed
cost. The price of each bushel of wheat for 1952 would be $1.97, for
1953 would be $2.00 and for 195); would be $2.09.“

The Most Profitable Rate of Application of Fertiliser

According to the equation, MVPx1 - Pxi or H - 1, 3. find the

nest profitable rate of application in 1952 was 1131; pounds of 3-12-12

 

fertiliser and h36 p0unds of 3-12-12 fertiliser plus 20 pounds nitrogen
2

Bradford, Lawrence 1., and Glenn 1.. Johnson. Farm Management flue.
New York: John Wiley and Sons, Inc., 1953, pp. - .

3 Price of 25 pounds increment of 3-12-12 fertiliser was collected fro.
the Davison Chaucal Gcnpaw, Lansing, Michigan. HFC - $0.50 in 1952,
m - 80.1;7 in $53. and m - solo in l95h for 3.12-l2 fertilizer.

wheat prices are the seasonal average price per bushel for crop years
1952-51; received by Michigan farmers as shown in Michigan Agl'iculttmal
Statistics.

29

. TABIB II

THE W or INCREASING RATES or FERTILIZER APPLICATIONS
ON mi: 1112011111. VALUE mower F02
was” IN 1952-51.

 

 

 

3_12_12’ Ferl Ill ' 3-12-12 Fertilizer us 20
Pounds of

Pounds Niu'o en To Dress
Fertiliser W9 119NPSZ 13%: 1%

 

 

 

_.9 u. 2. u. . .97 11.

° 2.96 ‘ 3.00 3.11. 2.76 2.80 2.93
:3 2.96 3.00 3.11. 2.17 2.20 2.30
75 2.96 3.00 3.11. 2.17 2.20 2.30

100 2.96 3.00 3.11. 2.76 2.80 2.93
125 2.76 2.80 2. 93 2.36 2.110 2.51
150 2.76 2.80 2.93 2.36 2.1.0 2.51
175 2.56 2.60 2. 72 2.36 2.1.0 2.51
200 2.56 2.60 2.72 2.36 2.1.0 2.51
225 2.56 2.60 2.72 2.36 2.1.0 2.51
250 2.56 2.60 2.72 2.17 2.20 2.30
275 1.97 2.00 2.09 2.17 2.20 2.30
1.97 2.00 2.09 1.58 1.60 1.67

300
1.97 2.00 2.09 1.58 1.60 1.67

325
350 1.77 LN 1.88 0.79 0.80 0.811
375 1.38 1.1.0 1.1.6 0.79 0.80 0.81.
1.00 0.99 1.00 1.05 0.79 0.80 0.81.
1.25 0.99 1.00 1.05 0.59 0.60 0.63
1.50 0.79 0.80 0.81. 0.59 0.60 0.63
1.75 0.00 0.00 0.00 0.39 0.1.0 0.1.2
500 0.00 0.00 0.00 0.39 0.1.0 0.1.2
525 -0.20 -0.20 -0.21 0.00 0.00 0.00
-o.20 -0.20 -O.2l -0.20 -0.20 -0.21

550

30

tap-dressing while the most profitable rate of application in 1953 was
1.135 pounds and in 19511 was 1136 pounds of 3-12-12 fertilizer and 1.1.1
pounds in 1953 and 1.1.5 pounds in 1951. of 3-12-12 fertilizer plus 20
pounds nitrogen top-dressing. These are shown graphically for the
year 1952 in Figure 8.

In Figure 8, the following evaluations can be made from a study of

the experimental data:

1. The essence of the law of diminishing returns exists in the
experimental data.

2. An increase in the price of wheat and a decrease in the price
of fertilizer from 1952 to 1951.. would increase the amount of
fertilizer which it pays to use and the amount of wheat it
pays to produce.

3. All changes in the use of fertilizer as a result of changes in
price of wheat and fertilizer are limited to stage II in which
MVP<AVPandinwhichMVP> 0.

Thus, it appears that the experimental data provides some basis

for‘ comparison for the field survey data. Some of the limitations
and weaknesses of the mcperimental data will be discussed at a later

point .

 

 

 

 

 

3
5%
(D
3 2%
H
3"? 2
.p
8
§
"6‘
ti)
:1
3
1170 =
$50.50
0 L l 1_ L H 1 \: O
100 200 300 1.00 800“ x

Pounds of Fertilizer

Figure 8. Equating Marginal Value Product of Wheat with Marginal
Factor Cost of Fertilizer to Determine Optimum 'Rate of
3-12-12 Fertilizer and 3-12-12 Plus 20 Pounds Nitrogen
Top Dressing Application with 1952 Prices.

CHAPTER?

AN EVALUATION OF THE STATISTICAL RESULTS
AFTER FITTING THE FUNCTION

The data from the field survey was fitted in four different forms.
The first equation included five variables: total nitrogen input,
total phosphorus input, total potassium input, seed quality, and live-
stock intensity. 1 six variable equation was fitted using the sane
variables except that the nitrogen input was divided into that applied
at planting time and nitrogen applied as top-dressing. Each of these
equations were fitted three. tines using a different dam variable for
livestock intensity per acre to test the effect this had on the co-
efficients for the inputs of nutritive elements.

The other equations fitted included seventeen and eighteen vari-
ables. 1h. seventeen variable equation included the variables of the five
variable equation and an additional variable for livestock intensity and
eleven additional dually variables to allow for variations in area and year.
The eighteen variable equation was the same as the seventeen variable
equation but it included two variables for nitrogen inputs as did the six
variable one discussed above.

The method followed in fitting the Cobb-Douglas function was that
presented by M. Ezekiall for fitting a normal equation and coefficient

1 Ezekiel, Mordecai. Methods of Correlation Lelia“, 2nd Edition.
New: York: John Wiley andSons, c., 19 , pp. 08-212.

 

 

33

of multiple correlation. Hence, the normal equation were solved by
the Doolittle Method. The resultant regression coefficients (elastici-
ties), nultiple correlation coefficients (1;) , coefficients of multiple
determination (£2) for the several independent variables and their
corresponding standard error of estimate (8) , standard error of
reg-ession coefficients ( 0-b1) and constant (a) for the function in
each case were indicated in Tables III and II and are discussed in

the following sections.

he Results of the Five and Six Variable Equations

In Table III, the regession coefficient for nitrogen plant food
(12) in the five variable equation was significant at the five per-
cent level while the regression coefficient for nitrogen top-dressing
in the six variable equation was significant at one percent level by
I't" test. his indicates that increasing quantities of nitrogen top-
dressing night increase the yield of wheat per acre. he sign of the
regresssion coefficient for nitrogen applied at planting tine in the
six variable equation was negative. The regression coefficient for
nitrogen applied at planting tine was not significant by "t' test.
According to Tintner and Brownlee,2 "Negative reg-ession coefficients
within the range of inputs on nost farms are meaningless.“

he regression coefficient for seed quality was significant in
the five variable equation at the one percent level of significance

 

2
Tintner, Gerhard, and D. H. Brownlee, "Production Functions Derived

from Fara Becords,‘l Journal of Farm Economics, Vol. XXVI,
(August, 19%).

 

TABLE III

REGRESSION COEFFICIENTS, STANDARD ERRORS CF REGRESSION COEFFICIENTS,COEFFHIEHTHS OF MULTIPLE
CORRELATION, COEFFICIENTS OF MULTIPLE DETERMINATION AND STANDARD
ERRORS OF ESTIMATE IN FIVE AND SIX VARIABLE EQUATIONS

WE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N9 (x21_ P295#(I3) K20'(Xfij Nitrogen T2? Dressing Nitrogen $pplied at Certified Seed zero AnimalXDnits 0.0l-O.l9 Animal 0.20 or more Animal - _
Cases 21 bl ‘ 0b» b (Tb b GD .5) Planting lme (X6), _ X2) _______ Per Acre ( 8)‘__ Units Per'Acre(Xg) Units Per Acre(Xlo) R R2 S
y m 1-n 1J1 l-n l-n 1°“ bin Vblin bl.n 0‘01-.. T11-.. #0701-.. b1.n 0‘51-.. b1.n 651.11 ban 651...
1 1.38150 0.02673* 0.01103 0.0h777* 0.01851. 0.00967 0.02132 .. - - - 0.01618“ 0.00331 -0.00922 0.00331 - - - - 0.21627 0.01.677 0.106146
1 9
2 1.37869 0.02608* 0.01088 0.01.760* 0.02102 0.01029 0.02102 - - - - 0.01601.“ 0.00328 - - -0.00012 0.00656 - - 0.21319 0.01.515 0.10552
3 11.37798 0.02612* 0.01.163 0.01.898* 0.02130 0.00815 0.02130 - .. ' - - 0.01578“ 10-00332 - - - - 0.01373# 0.009111 0.21797 0.01.751 0.106hl
1. 1.38637 - - 0.05338“F 0.021113 0.01722 0.02133 0.02303“ 0.00709 -0.00521 0.01123 0.01568# 001035 -0.00977 0.00783 - - - - 0.222910.01.969 0.10629
5 1.382111 - - 0.05290* 0.02115 0.01769 0.02136 0.02300M 0.00712 -0.001.35 0.01616 0.0115631? 0.01036 - - 0.00MB 0.00696 - - 0.21955 0.01.820 0.10638
6 1.38264 - - 0.051.27* 0.02185 0.01537 0.02133 0.022118“ 0.00709 -0.001.1.1. 0.011102 0.01532# 0.01035 - - - - 0.01337# 0.009110 0.22387 0.05011 0.10627
Remarks: **: 1 percent.level of significance for regression coefficients by "t" test.

5 percent level of significance for regression coefficients by "t" test.
#: 3256 percent level of significance for regression coefficients by "t" test.

-: Variable not included in this equation.

_A#—_______Vir___‘__._ r 777/ ‘

   

35

and in the six variable equation at the 32-6 percent level. his indi-
cates that using catified seed miylt increase the yield of wheat pa
acre.

It will be noted that negative recession coefficients were obtained
for sac animal units pa acre in the first case and in the fourth case
and for 0.01-0.l9 aniaal units pa acre in the second case. he re-
aession coefficient for animal units were not significant. But, the
regression coefficient for animal units both was significant in the
third case and in the sixth case at 32-6 pacent level of significance.
his suggests that the presence of 0.20 or more animal units pa acre
light increase the yield of wheat per acre.

After taking nitrogen top-dressing and nitrogen applied at plant-
ing tine as two independent variables in the six variable equation,
the sipificance of the regession coefficient for seed quality dropped
from one percent level in the five variable equation to 32-6 pacent in
the six variable equation. he regression coefficients for nitrogen
applied at planting time in the six variable equation were not signifi-
cant. his suggests that the regression coefficients in the five
variable equation were affected by the intacorrelation between the
use of catified seed and nitrogen top-dressing or between good
practices and the use of nitrogen tap-dressing.

The coefficient of multiple determination or 1.12 was approximately
0.05 in both five and six variable equations showing that five pacent
of the variance in the dependent variable or the yield of wheat was
associated with two forms of fertiliser applications. he ruining
95 pacent of the variation of the dependent variable nay have been

a.

36

due to otha input factors such as labor, machinery or otha unstudied
variable factors such as weatha and management which was seemed to
be normally distributed.

he logarithm of the estimated yields of wheat E (I) was approxi-
mately 1.3810 in both five and six variable equations, the antilog of
which is 21..01. bushels. he standard error of estimate (5) of the
dependent variable was found to be approximately 0.1060.

he sum of the regression coefficients (elasticity) were less
than one in both five and six variable equations, and decreasing
returns to scale are indicated in the 1:11 farms.

The multiple correlation coefficients (F) were between 0.21319
and 0.22387 in the five and six variable equations. With 978 obser-
vations and either five or six independent variable and one dependent
variable, this high a multiple correlation coefficient would be
expected in one ample out of a thousand if the true R containing
five variables in five pacent level of significance were 0.097 and
in one pacent level of significance was. 0.1.15.3 Consequently the
correlation is significant.

he TPP pa acre was derived by using various rates of fatiliser
applications to fit in each of five and six variable functions in the
logarithmic fon. he MPP was the additional yield of wheat result-
ing frcn the last 25 pounds increment of fatilisa.

 

3 .
Faba, Robert. Statistical Techni es in Market Research 1st Edition,
new Iork: McGraw-Hill Book W, 53., pp. 520-521

37

Tables and.figures derived from.the data from.the five and.six
variable equations illustrate the essence of the law of diminishing
returns and show the effect of increasing rate of two forms of ferti-
liser applications on the total,:nargina1 physical products and.margina1
value products under different price conditions. these are also related
to three different variables representing animal units and indicate the
effect of using the different analysis of fertiliser such as 3-12-12
fertilizer and.0~12~12 fertiliser plus ten.pounds nitrogen tqp-dressing
on the yield of wheat. The 0-12-12 fertiliser analysis was the least
cost ccnbinatioanhi h was derived.franthe six.variahle equation.and
is discussed in.the following sections.

All TPP figures in Table IV were derived.by fitting'the five
variable equation for 3-12-12 fertilizer with three different animal
unit variables included. In Table V the TPP of wheat per acre‘sas
derived‘hy fitting the six variable equation for 0-12412 fertiliser
plus ten pounds nitrogen tap-dressing with the variables for three
intensities of animal units per acre included.

The relationships between.the total and.nsrginal physical
products in.two forms of fertilizer applications and the ccnparison
on.the;yield of wheat per acre with three different animal unit
variables included were illustrated as follows:

The TH? per acre'was continuing to increase at a decreasing
rate in'both five and.si::variable equations as long as the.HPP
decreased.but was greater than acre. The IPP per acre of wheat
was slightly greater in.0q12-12 plus nitrogen top-dressing than in
3.12-12 fertiliser.

M0 engaged.“ nuance mm .40.“ 03.3 a 0% use Monaco. and 20.3 a sofa $00.5 6:033:00 eoa

38$». 93H Squad-m

3&3 .895 o

 

 

 

 

8 as» use «4044
3 «0 9005.85 0050A ma you 5.0. a 9.5 05 Hannah .Bm 00.3 a 006.3 0.025 .edowpdcnoo 00 Emma .805 a
. .flwwmawuom m7~4un
no 088.83 season 3 .5 8.8 u 0...: e5 423.. .80 34a .. 84.3 08.5 3034008 8 .N m .825 a
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3:00 0:00 “:00 ”N00 ”“0““ 300 £00 “400 NNOO FMeNM 30° 440° “:00 NNOO 80H“ 8“
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39

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0.

 

 

 

 

 

 

 

 

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>348

ho

'me profitable rate of fertilizer application indicated by the survey
data can be derived by fitting the equation: the MVP of any input (X1)
lust be equal. to the price of the variable input (P11). The price of
sheet and the price of fertiliser used yore the same as used previously
in Chapter IV for the crop years 1952 to 1951;.

The most profitable rate of two forms of fertilizer application and
the most profitable yields for wheat per acre with different intensities
of animal units per acre in 1952-51: are shown in Table VI. If these
figures are compared with the experimental results, they appear to‘
indicate that application rates of 3-12-12 fertiliser derived from the
may data are approximately 100 and nore pounds lower than the rates
indicated by the emerimcntal data. The most profitable application
rate for 0-12-12 fertiliser plus 10 pounds nitrogen top dressing appears
to be very low. The TPP and MVP for the experimental and survey data
with 1953 prices are shown in Figure 9.

(hoe the regression coefficients of the various input categories
were determined, it is possible to derive estimate of marginal value
products fro: the for-uh

m _ b1E(I)
‘1 H

where b1 is the regression coefficient (elasticity) of gross incone

 

with respect to corresponding inputs, E (I) is estimated goes incoee,
and Xi is the quantity of the input. I (I) and 11 are measured in
natural numbers. The MVP of each input categories indicates an in-
crease in gross income resulting from an increase in the use of xi.

 

 

 

 

 

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Haas 326 one 0~.0
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3nd 3.0.58.0
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3.39 as o3
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and hobo un- 0~.0
3.2 3 8.2 E .8. as 33 a:
Hulda.- 3.0.3.0
3.3 3 moJH .Bm 98.. and
SSH: Haas chem...
3.2 .3 snug 0mm .8. to $3
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3.8 .3 «baa emu 98. 8a 2:5 «m2
35 3.0.8.0
8.2.. m: on.“ 5N 98¢ .80
and? 155 one“
$3.35 3.53 logy agony
00260.3 98¢ pod—Was 33.99.53 002680 shoe 00.3er 833383
non goons mo modem no 3595 .80 peonséo 3.3.“ Ho 3588

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gnu-3.3133

3.8 no:

 

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ggésgglfiégggfiliggaaaggagg

”—55de

Yields of Wheat Per Acre (Bushel)

value of‘Hheat Per Acre (Dollar)

 

 

 

.MVPX for
ental Data

 

 

 

 

 

K
\.
-.,. S ta
ta
_ Ea?
MFG =
\ $0M
J I I l 0
100 200 300 hOO ‘500

Pounds of Fertilizer

Figure 9. A Comparison between Experimental and.Survey Data of the

Effect of Increasing 3-12-12 Fertilizer Application
on the Total Physical Product and Mhrginal
Value Product for "heat with 1953 Prices
Doing the Five Variable Equation.

h3

111s MVP for the various input categories in the survey data computed
at their geometric mean were shown in Table VII. . In the five variable
equation, the MVP for one pound of nitrogen plant food was approximately
$0.160 with 1952 prices while the MVP were approximately 30.161 and $0.168
with 1953 and 1951; prices. has MVP for one pound of phosphorus plant food
and one pound of potassium plant food were approximately $0.07? and 60.019
for 1952 prices. They were $0.078 and 80.019 for 1953 prices and 80.082
and $0.020 with l95h prices. This indicates that returns for one pound
or nitrogen plant food was probably twice that for one pound of phosphorus
plant food and eight times that for one pound of potassium plant food.

Under 1952, 1953 and 195m. price conditions, the MVP for one pound
of nitrogen applied at planting time was approximately zero or less and
for one pound nitrogen top dressing was approximately $0.68. me MVP
for one pound of P205 plant food was approximately $0.09 and for one
pound of K20 plant food was approximately $0.03. A negative MVP for
one pound of nitrogen applied at planting time is meaningless in this
stuw.

Under a set of price conditions, the least cost combination for
N—P-K fertiliser ratio might be two-two-one in the five variable equation
and sero-one-one in the six variable equation shown in Table VIII. hese
were derived by the formula

MVP MVP MVP
N2 . P205 . K20 I 1

mug ”Orzo; m2.

where will is the marginal value products of the 11 input categories

and M5011 is the narginal .factor cost for the corresponding X1 input.

 

 

 

 

 

 

 

mm0.I 000.0 “0.0 mmod I fiOII 30.0 80.0 30.0 I mn0.I $0.0 80.0 80.0 I 0
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«#0.: 0.230 Mn0.0 M8.0 I 0.40.I 80.0 ano.0 000.0 I Gaol 0&0.0 mm0.0 80.0 I .4
I I A3.0.0 30.0 ‘3.70 I I 0.3.0 000.0 p36 I I 0.8.0 30.0 3.0 n
I I HN0.0 50.0 03.0 I I 08.0 2.0.0 00.10 I I 0N0.0 20.0 09.0 N
I I 08.0 «00.0 may—”.0 I I 0.8.0 08.0 NOH.0 I I 5.8.0 2.0.0 03.0 H
on“... umm amuse—u man—no.6 undone
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«Ema. a.“ E Mum.” a.“ E «mad EH. E

   

£38 I g0
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m8 58E a: .of .mewm an .8 Enema 5» gamma:

EH43

 

 

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.5352 £533 639—8 doe-.6 manta .3 commie... 3: Iona suflfise 2:
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a

mmm.o 2.5.0 Medea 000.0 3.4.0 0m~..0 32... 000.0 5.30.0 omfio 05.0 000.0 0
03m.0 00m.0 QMHJ” 000.0 80.0 mgoo 0002—” 000.0 ~43.0 30.0 30.0 08.0 m
mMMIO mg.0 MON...” 000.0 Hma.0 80.0 Mg...” 000.0 «3.0 mg.0 0mm.0 80.0 4
I «3.0 Mao...” 0mm...” I ammé 0&6 3H...” I 0330 «2.0.0 mac...” n

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film“ 0.. a... a 3mm“ 0.. a... r 3mm... 9... a... a I...

a «ma a

 

 

 

 

 

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E can a: .moum .Na 8.. 323056 .58 snag a:

Hun—Hg

be

It appears that the ratio of the commercial fertilizer used by
farmers was weighted too much on potassium and too little on nitrogen.
In other words, there might be an increase in nitrogen and a decrease
in potassium in the fertilizer ratio used.

The Results of the Seventeen and Eighteen Variable Equations

is presiously mentioned, the seventeen variable equation included
nitrogen plant food as one indepenth variable and dnnmw variables to
represent different soil types and years. The eighteen variable
equation was the sane as in the seventeen variable equation except
nitrogen top dressing and nitrogen applied at planting time were
separated as two independent variables.

In Table II, the regression coefficients for total nitrogen plant
food including nitrogen tap dressing in the seventeen variable equation
was significant at the one percent level while the regression coefficient
for nitrogen top dressing was significant at five percent level by 't'
test. This indicates that increasing quantities of either the nitrogen
plant food or nitrogen top dressing might increase the yield of wheat
per acre.

he regression coefficient for seed quality was significant in the
seventeen variable equation at the one percent level and in the eighteen
variable equation at the 32-6 percent level of significance by "t" test.
this indicates that using the certified seed migxt increase the yield of

wheat per acre.

TABLE IX

REGRESSION COEFFICIENTS, STANDARD ERRORS OF REGRESSION COEFFICIENTS, COEFFICIENTS
0F MULTIPLE CORRELATION, COEFFICIENTS OF MULTIPLE DETERMINATION, AND STANDARD
ERRORS OF ESTIMATE IN SEVENTEEN AND EIGHTEEN VARIABLE EQUATIONS

 

 

Seventeen Variable Equation

Eighteen Variable Equation

 

 

 

 

: are . M

variables and.0thers bl.n 651.n b1.n 651.n
Total Ngnplant food including N top **
dress g X2 0.01277 0.00001 - -
Total P205 plant food x3 0.06805** 0.01700 0.07176** 0.02101
Total K20'plant food X“ 0.00278 0.02596 0.00858 0.02068
Nitrogen top dressing x5 - - 0.01A82* 0.00678
Nitrogen applied at planting time X6 - - -.01001 0.013lh
Seed quality I? 0.01381** 0.00001 0.01339# 0.00972
0.01-0.19 animal units per acre 19 0.00985** 0.00001 0.011u2# 0.00762
0.20 or more animal units per acre x10 0.01771** 0.00981 0.01861# 0.01033
Area I, 1952 I11 0.08262** 0.00981 0.08290** 0.01613
Area I, 1953 x12 0.12301** 0.00981 0.122u6** 0.01562
Area I, 1958 x13 0.08579** 0.00981 0.08585** 0.01536
_Area II, 1952 XII 0.09h63** 0.00981 0.09085** 0.01595
Area II, 1953 x15 0.09272** 0.00981 0.09197** 0.01519“
Area II, 195A x16 0.13h58** 0.00981 0.13312** 0.01538
Area III, 1952 x17 0.00938** 0.00981 0.05031** 0.01627
Area III, 1953 x18 -.00019 0.00981 - -
Area III, 1950 x19 0.09163** 0.00981 0.093ul** 0.01570
Area IV, 1952 x20 0.07307** 0.01388 0.07361** 0.01652
Area IV, 1953 X21 0.09036** 0.01388 0.09311** 0.01586
Area IV, 195A X22 - - -.00055 0.01595
E 0.01055 0.01795
2 0.17185 0.17u68
7 , 0.09923 0.09906
3 1.281426 1.28680
Remarks: **: 1 percent level of significance for regression coefficients by "t" test.
*I 5 percent level of significance for regression coefficients by "t" test.
#3 32-6 percent level of significance for regression coefficients by "t" test.
-: Variable not included in this equation.

 

LR

 

118

After taking nitrogen tap dressing and nitrogen applied at planting
tine as two independent variables in the eighteen variable equation, the
significance for recession coefficient for seed quality drapped from one
percent level in the seventeen variable equation to 32-6 percent in the
eighteen variable equation. The recession coefficient for nitrogen
applied at planting time in the eigzteen variable equation was not signifi-
cant. his again suggests that the recession coefficient were affected
by the intercorrelation between certified seed and nitrogen tap dressing
or between good practices and more nitrogen tap dressing. But the yield
of wheat was not increased by nitrogen applied at planting time.

It will be noted that negative recession coefficients were obtained
for Area III in 1953 (118) in the seventeen variable equation and for
nitrogen applied at planting tine (16) and for Area IV in 19514 (122)
in the eicxteen variable equation. 'nmese indicate that the negative
recession coefficients for nitrogen applied at planting time and for
du-w variables of Aree III in 1953 and of Area Iv in 1951; were lean-
ingless. The recession coefficients for these dumy variables 118
and 122 and for nitrogen applied at planting tine were not significant.
It night indicate that weather or other plusical conditions for Area
III in 1953 and for Area Iv in 1951. decreased the yield of wheat per acre.

The recession coefficients were significant for both 0.01-0.19 and
0.20 or more animal units per acre in the seventeen variable equation at
one percent level of significance and in the eighteen variable equation
at 32-6 percent level of significance by "t” test, indicating a positive
relationship between animal units per acre and the yield of wheat per

acre.

1.19

The coefficient of multiple determination or 132 was approximately
0.17 in both seventeen and eighteen variable equations showing that
seventeen percent of the variance in the dependent variable or the
yield of wheat per acre was associated with the variables included in
the equation. The running eighty-three percent of the variation in
the dependent variable (I) may have been due to other factors such as
labor, machinery or other unstudied variable factors such as weather
and nanaganent which were assumed to be normally distributed.

The logarithm of the estimated yields of wheat E (I) was approxi-
nately 1.2853 in both seventeen and eighteen variable equations, the
antilog of which is 19.29 bushels per acre. The standard error of
estinste (5) of the dependent variable was found to be approxilate]:
0.0991. I

The sun of the recession coefficient (elasticities) was greater
than one in both seventeen and eighteen variable equations indicating
increasing returns to scale on the 1111 ferns. But, the IF? figures
shown in Table I were increasing at a decreasing rate. This nay have
been influenced by the dam variables which were classified as fixed
variables in the Cobb-Douglas function.

The mltiple correlation coefficients (13) were 0.1.1155 end 0.1.1912
in the two equations. Under the conditions of 978 observations with
' either seventeen or eighteen independent variable and one dependent
variable, this high a nultiple correlation coefficimt would be enacted
in one sample out of a thousand if the true R containing five variables

in five percent level of significance were 0.097 and in one percent level
of significance were 0.1.15. Consequently the decee of correlation is

significant.

50

The TPP per acre was derived by using various rate of fertilizer
applications to fit in both seventeen and eighteen variable equations in
the logarithm fern.

In Table I, the TPP per acre was continuing to increase at decreas-
ing rate in the seventeen variable equation for 3-12-12 fertilizer and
in the eighteen variable equation for 0-30-10 fertiliser plus 10 pounds
utrogen top dressing as long as the HP? decreased but was ceeter than
acre. The 0-30-10 fertiliser analysis was the least cost combination
which was derived from the eighteen variable equation. The TPP per acre
was ceater in 0-30-10 fertiliser plus nitrogen top dressing than in
3-12-12 fertiliser. The survey results also show the effect of using
mall amount of fertilizer in higher analysis would produce higher
yields per acre than larger mount of fertilizer with a lower analysis.
For instance, 200 pounds of 3-12-12 fertiliser was required to furnish
the plant food contained in 100 pounds of 0-30-10 fertiliser plus 10
pounds nitrogen top dressing with snaller yields per acre.

The TPP per acre probably the highest in Area II from 1952-5h while
the TPPper acre inlrea Iwas the second highest, inArea IVwas third
higher and in Area III was the lowest from 1952 to 19511. This indicates
that the effect of the good soil types on hicer yields per acre assuming
other factors studied were constant.

The nest profitable rate of 3-12-12 fertilizer application for the
seventeen variable equation were 2M; pounds with 1952 prices, 269 pounds
with 1953 prices, and 288 pounds with 1951; prices. If these figures were
compared with the emperinental results, it would indicate that an appli-
cation rate of 3-12-12 fertiliser in the survey data would be about

TABLEX

THE EFFECT OF INCREASING RATES OF 3-12-12 FERTILIZER FOR
SEVENTEEN INDEPENDENT VARIABLES AND 0-30-10 FERTILIZER

PHYSICAL PRODUCTS AND MARGINAL VAIDE
PRODUCTS PER ACRE FOR WHEAT

0-20-10 Fertiliser Plus 10

PLUS 10 POUNDS NITROGEN TOP DRESSING FOR EIGHTEEN
INDEPENDENT VARIABLES ON THE TOTAL, HARGINLL

51

 

 

 

 

 

:aunds of 3.12.12 Fertiliser Pounds Nitrogen Top Dressing_
mm“:- m: m . _ m m
bu. bu.'b° bu. mad”f
0 0.00 0.00
25 25°“ 1 52 $2.99 $3.01; 3; 17 28°68 1.6}: $3.23 33.28 8313
50 26.” e e e e 30.32 e e e 0
75 7 89 0.93 1.83 1.86 1.9h 1.00 1.97 2.00 2.09
2 . . 2
0.68 1.3!; 1.36 1.h2 31 3 0.7h 1.h6 1.!48 1.55
100 28.57 32.06
0.55 1.08 1.10 1.15 0.58 1.1!: 1.16 1.21
125 29.12 32.61;
0.141: 0.87 0.88 0.92 0.1m 0.95 0.96 1.00
150 29.56 33.12
0.39 0.77 0.78 0.82 0.141 0.81 0.82 0.86
175 29.95 33.53
0.3!; 0.67 0.68 0.71 0.37 0.73 0.71; 0.77
200 30.29 33.90
225 30 59 0.30 0.59 0.00 0.63
' 0.27 0.53 0.51; 0.56
250 30.86
0.25 0.h9 0.50 0.52
275 31'11023 0h5 one 01:8
. am 31.1)... e e e e
0.21 0.1.1 0.h2 0.1a;
325 31.55
" bandcarethesameasfootnotes a, band'zginTable IV.
d Under 1952's price conditions, wheat price - $1.97 per bu. and MFG -
. gnder 1953's price conditions, wheat price - $2.00 per bu. and MFG -
2.10.
1 Under 195h's price conditions, wheat price - $2.09 per bu. and MFG -

$2.00.

52

one-half that indicated by the experimental data. The TPP and MVP for
the experimental and survey data with 1953 prices are shown in Figure
10.

me most profitable rate of 0-30-10 fertilizer plus 10 pounds
nitrogen top dressing were 1414 pounds with 1952 prices, 178 pounds with
1953 prices and 149 pounds with 195h prices. There were no comparable
experimental results, but the indicated application rate of 0-30-10
fertiliser plus 10 pounds nitrogen top dressing appears to be very low.

In Table II, the MVP for one pound of total nitrogen plant food
was approzinately $0.06 in the seventeen variable equation under 1952
and 1953's price conditions while the MVP was approximately $0.07 with
1951.; prices. Under 1952 to l95h's price conditions, the MVP for both
one pound of phosphorus plant food and one pound of potassium plant
food were appronmately 30.090 and $0.00!; in the seventeen variable
equation. his appears that returns for one pound of phosphorus plant
food was probably one-half higher than for one pound of nitrogen plant
food and one pound of nitrogen plant food was probably twelve tines
greater than for one pound of potassium plant food. In the same table,
the MVP for one pound of nitrogen applied at planting time was approxi-
mately zero or less under 1952, 1953 and l95h's price conditions and
for one pound nitrogen tap dressing was appronnately $0.35. The MVP
for one pound of phosphorus plant food was approximately $0.095 and
for one pound of potassium plant food was approadnately 80.013. 1
negative HVP for one pound of nitrogen applied at planting time is
meaningless in this study. It appears that returns from one pound of

phosphorus plant food was probably much greater than for one pound of

53

     
    

 

 

 

 

 

   

 

 

 

 

 

 

C
to +—
,\ TPPx for
g Eacperimental Data
0)
(‘13 30 ,x‘w
9* \ TPPx for
'§ Survey Data
é
e4
0
=3
e 2° C.
0? L j I J J
3 -
MVP.x for
E; Experimental Data
«E!
5
m 2 F-
3 a,
53
5‘ :1
t a
‘3";
1 ..
§
Q-l
3 . a A MFG. =
a 5 ........a... ., , \ $0.147
b
0 l J J l ‘_ - O
100 200 300 boo 500
Pounds of Fertilizer
Figure 10. A Comparison between Experimental and Survey'Data of

the Effect of Increasing 3-12-12 Fertilizer Application on
the Total Physical Product and Marginal Value Product
for, Wheat with 1953 Prices Using the
Seventeen Variable Equation.

 

 

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a. 838‘ 88.32
2.3. 5nd «mas .. .. - mu 288.8 as. 883.2
48.0 «86 98.0 89° 48.0 :86 an own
8.8 8.8 «8.8 48.0 086 806 mm mama
.. - .. 898 «8.3 _.H8.3 an 8» 88.38 Bfimufleaa
has .88

 

 

 

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«mg
g

 

 

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am. «m3 @8388 as
5.583 8: gm an EB. 233.5 94 as: ESE:

sauna m8 3% .oNu .momm 8.: .8 888% n34» gems.

“an.

55

potassium plant food and an increase in gross income would probably be
greatest resulting from an increase in use of nitrogen tap dressing.

The least cost combination for N-P-K fertilizer ratio under a set
of price conditions in 1952-51; man be h-lO-l for the seventeen vari-
able equation and 0-3-1 in the eighteen variable equation shown in
Table III. This again suggests that the ratio of the connnercial
fertiliser used by farmers was weighted too much on potassium plant
food and too little on phosphorus plant food.

 

 

 

 

 

 

 

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03.0 20.0 80.0 03.0 08.0 80.0 02.0 «use. in
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CHAPTER VI
SW AND CCNCLUSIOKS

This study was to test the feasibility of estimating input-output
relationships for wheat in Michigan using sampling data. After the
results obtained from experimental and survey data have been compared,
several conclusions can be drawn.

In general, the results obtained from an analysis of both the survey
data and from the experimental plots appeared to conform to the economic
theory outlined in Chapter II.

The essence of the Law of Diminishing Returns existed in both
experimental and survey data.

The sum of regression coefficients or elasticities were less than
one in all equations derived from the survey data indicating decreas-
ing retm‘ns to scale for the m farms studied. The elasticity of a
Cobb-Douglas production function, however, is constant over all ranges
of independent variables. This means that the Cobb—Douglas function
can approxinate only a segment of the complete function. This appeared
to be the lower limit of stage II in all equations. It indicates, on
the average, that the farmers that were interviewed ceased application
of fertilizer before the upper limit of stage II in the production
function was reached. An increase in fertilizer application migt be

recommended in the future years.

58

In accordance with static factor-factor analysis, technical comple-
ments appeared to exist in N-P-K fertilizer analysis where fertilizer
was obtained in a fixed ratio. The least cost combination for h-lO-lo
fertilizer ratio in the seventeen variable equation and for 2-2-1
fertilizer ratio in the five variable equation indicate that a
reduction in nitrogen cannot be replaced by an increase in element of
phosphorus or potassium in a given fertiliser analysis. The animal
nanure and nitrogen top dressing appear to be substitutes in producing
the minimum nitrogen required in wheat production. When animal manure
or nitrogen top dressing is increased in quantity, total pounds of
fertilizer night be decreased with the same yields of wheat per acre.
For mam, uoo pounds of 3-12-12 fertilizer plus 20 pounds nitrogen
top dressing substituted for 160 pounds of 3-12-12 fertiliser with the
sale yields per acre (10.0 bushels per acre) in the eaqaerimental data,
as shown in Table I, and 250 pounds of 3-12-12 fertilizer with 0.0l-O.l9
animal units per acre substituted for 300 pounds of 3-12-12 fertiliser
with zero animal units per acre on the same Iso-product line (31.87
bushels per acre) as shown in Table IV. The use of more nitrogen top
dressing and animal manure on wheat qaparently should continue to be
encouraged. But nitrogen applied at planting time was not profitable
according to the relationships appearing.

. The most profitable rate of 3-12-12 fertiliser application obtained
from the results of the five variable equation was approximately 220
ponds and from the seventeen variable equation was approximately 260
pounds. The most profitable rate of 0-12-12 fertiliser plus 10 pounds
nitrogen top dressing in the six variable equation was approximately

59‘

1:8 pounds and of 0-30—10 fertilizer plus 10 pounds nitrogen top dress-
ing in the eighteen variable equation was approximately to pounds. The
042-12 and 0-30-10 fertilisas were the least cost combination and
were derived non six and eighteen variable equations.

If these figures were coupared with the eXperimental results, it
appears that the application rate of 3-12-12 fertilizer in the survey
data were slightly more than one-half those indicated by the experi-
nental results. The most profitable rates of 0—12-12 and 0-30-10
fatiliser plus 10 pounds nitrogen top dressing were not available
hem experiments to compare with results from the survey data. Nitrogen
tap dressing was used as an independent variable in the survey data and
as a fixed variable in espaimental design.

lack of capital and some limitations in the farm operation might be
major reasons why the farmers ceased more application of fertilizer
before the maxim profit point in their farm business.

The least cost combination for H-P-K fertiliser analysis was approxi-
mately 2-2-1 in the five variable equation and h-lO-l in the seventeen
variable equation while the least cost caabination for N-P-K fertilizer
plus nitrogen tap dressing were approximately 0-1-1 in the six variable
equation and 0-3-1 in the eighteen variable equation. If these combi-
nations wae compared with l-h-h in experimental data, it appears that
an increase in nitrogen and a decrease in potassium in the fertilizer
analysis used by farmers mith be reconnended in the future years. It

also suggests that the future research projects for determining the
optimum fertilisa applications should give more attention to least
cost combinations than has been the case in the past.

Limitations of Using 311er Data for Input-
Output Relationships

The output response of wheat yields with respect to the correspond-
ing inputs of fatiliser, appears slaller in the survey data than in the
experimental data. In otha words, the regression line in the survey
data was much flatter throughout from the origin than the reyession
line in the expaimental data originate sometmere on the I axis. This
night have been due to several weaknesses in the survey data that was
used. It is possible that farmers with low fertilizer inputs and low
wheat yields had an upward bias in their yield estimates which would
result in underestimating the true coefficients for fertiliser in the
statistical results. It is also possible that the intercorrelation
between fertilizer use and inherent level of soil fertility could also
have resulted in a downward bias in the coefficients.

Biases and arors arising from both the interviewer and the sanple
also were possible using the random sapling method.

In addition, some shortcoming of the nature of the Cobb-Douglas
production function itself which shows yields always starting at the
origin and constant elasticities throughout all ranges of 11 may in
part explain the difference in results between the survey data and
the emerimental data. In the future it might be desirable to make
some adjustment and modification of the function to partially overcome

these obj ections.1

 

1 Carter, Harold 0. "Modifications of the Cobb-Douglas Function to
Destroy Constant Elasticity and Symmetry." Unpublished H.’ S.
Thesis, Department of Agricultural Economics, mchigan State
University, 1955.

61

Different methods of fertilizer applications, diffaent rotation
systans and otha variations in farm practices would exist in the survey
data. These factors might affect the yields of wheat per acre. 0tha
factors such as differences in weatha, moisture, and soil types within
the four areas in this study might have influenced the yields of wheat
pa acre.

There was not enough cases in the survey data with larger appli-
cation of fertiliser. Host of the farmers' fertiliser application
appear to have been concentrated in the area of the Iowa limit of
stage II. To get meaningful input-output data, applications over the
entire range of inputs are needed.

Some Advantaggs of Survey Data

The survey data offers some results of value which are not avail-
able from current experimental data. Althougz they are not quantitative
estinates the positive coefficients for seed quality and intensity of
animals per acre indicate that these contribute to increased yields of
wheat under actual fern conditions. It is also possible to compute
least cost combinations of fertilizer under avaage farm conditions,
something that is not possible with the available expainental data.

Conclusions and Recommendations for Future Studies

In general, it appears to be possible to derive estimates of input-
output relationships from survey data that are not inconsistent with
econoadc theory or existing experimental data. The results obtained
from this analysis has sevaal questionable aspects which have been

62

mentioned. However, these may be due in part to the particular method
of handling the data, particularly in the use of the dummy variables,
and may be due to the statistical function that was fitted. It appears
that furtha work should be done with this or similar data to determine
the effect of using differing methodology.

The data used in this analysis was not gathered specifically for
use in this type of analysis. If data was collected specifically for
this purpose several improvements might be achieved in reliability,
number of input factors outrolled, range of inputs, and otha factors
which would improve the usefulness of the data.

Thus, while the results of this analysis indicate that the use of
survey data to determine input-output relationships is feasible, further
testing and methodological research is indicated before it can become

a useful research tool.

1.

2.

a.
be
ce

1'.

as
be
0e

63

APPENDIX A

A PORTION OF THE QUESTIONNAIRE IN FARM IVIANAG’EII-IEN'JIl
SURVEY USED IN THIS STUDY

 

 

How may acres of wheat will you harvest in 19514? A.
How many acres of wheat did you harvest in 1953? A.
How many acres of wheat did you harvest in 1952? A.

 

(IF WHEAT ACREAGE was REDUCED FROM 1953 To 1951”)

What was the reasons for your reduction in wheat acreage?

 

 

Were there any other reasons for your reduction in wheat acreage?

 

 

How many acres of wheat would you have planted in 1953 (for the

 

 

 

1951. harvest) 1: there had been no acreage allotment? A.

ilhat was your wheat yield per acre in 1951;? (expected) bu.
What was your wheat yield per acre in 1953? bu.
what was your wheat yield per acre in 1952? bu.

 

Now I'd like to ask you about some of your production practices for wheat:

 

 

 

 

Use Tertilizer r ‘ NT—igp Dr. ,
3. Onthe crop Cat. lbs./ Anal. 1138.7 Anal.
Seed? A. A.
a. Planted in 1953 '

HarvestegL in 195).;

 

b.

Planted in 1952
Harvested in 1953

 

Ce

Planted in 1951
Harvested in 1252

 

 

 

 

 

 

 

 

he

a.

Unda the most favorable conditions what is the highest wheat yield
you think you can get on your farm? bu./A.

b.

C.

a.

b.

61;

(IN ASKING QUESTION 3b INSERT THE ANALYSIS OF FERTILIZER THAT FARMER
HAS MOST RECENTLY USED ON HIS WHEAT.)

What is the greatest amount of fertilizer that you can
profitably apply on wheat on your farm? ‘ lbs. /A.

Do you believe nitrogen top-dressing for wheat would be profitable
on your farm?

Yes 3 How many pounds per acre can you profitably use?
‘ , lbs. of N.
No .
D.K. .
Have you made any changes in your livestock numbers because of the

acreage allotments on your camps? Ies No
IF THE ANSWER IS YES, ASK WHAT KIND OF LIVESTOCK HAS BEEN ADJUSTED,
cm THIS CATEGORY AND GET THE DATA FROM THEM. THEN COWIEI'E THE
LIVESTOCK INVENTORY AND LIST THE REASONS FOR ALL SHANGES IN THE
AfPPROPRIATE SPACE BEEN.

 

 

No. on hand No. on hand Direction No. of

Kind of Livestock July 1, 1951: July 1, 1953 of change change

 

1._Dairy cows
2.__Heifers (Dairy)

3.__Beef cows (Breeding)
h.__;Feedem' cattle
5.__Bred sows

6.__Hogs on feed
7.__Laying hens
8.__Pullets
9.__Broilers

10 .__Turkey, geese, etc.

11.__Sheep, ewes
12 ._Feeder lambs
l3.__0ther

 

 

 

 

 

l.
2.
3.
h.
S.

65

(IF THERE HAVE BEEN CHANGES IN LIVESTOCK NUMBERS IN ANY CATEGORY,

ASK WHY FOR EACH ONE AND LIST THE REASONS BELOW BY NUMBER.)

6.

 

7.

 

8.

 

9.

 

10.

 

 

 

 

 

 

11.

 

l2.

 

13.

 

APPENDIX B

CONVERSION RATES FOR LIVESTOCK TO
STANDARD ANIMAL UNITS

The animal units were converted using one cow as a standard unit. It

is primarily on manure produced in one year per 1000 pounds of liveweightl

as follows :

 

 

 

 

 

Head of animals equal Manure produced in one year
to one animal unit per 1000 pounds of liveweight

Cow 1 12.0 T

Steer l 8.5

Horse 1 8.0

Hog 6 16.0

Sheep 8 6.0

Chickens 250 11.5

1.

Source: Illinois Agricultm'e' Handbook, 19149, PP. 206.

67

BIBLIOGRAPHY

Boulding, Kenneth E. Economic Anal sis, Revised Edition. New York:
Harper and Brothers Publisher, 1956, pp. 671-709.

Bradford, Lawrence A. and Glenn 1.. Johnson, Farm Hana ement Analysis.
New York: John Wiley and Sons, Inc., 1933, Chapters 8, 9 and 10.

Carter, Harold 0. "Modifications of the Cobb-Douglas Function to
Destroy Constant Elasticity and Symmetry." Unpublished M. S.
'nlesis, Department of Agricultural Economics, Michigan State
University, 1955.

Cobb, Charles H. and Paul H. Douglas. "A Theory of Production,"
The American Economic Review, Vol. XVIII, Supplaent, March 1928.

Ezekiel, Mordecai, Methods of Correlation Analysis, 2nd Edition,

New York: John Riley and Sons, c. , 9 9, pp. 208-212 and
Appendix 1 and 2.

Ferber, Robert. Statistical Techni es in Market Research, lst Edition,
New York: HcGraw-EII Baok Comparw, I30” pp. 520-521, 19149.

Heady, Earl 0. Economics of A icultural Pr_oduction and Resource Use.
New York: Hentice-Hall, c., 1952, Chapters 2, T, 11, 5 and 6.

 

Hill, Elton B. , and Russell G. Mawby, "Types of Farming in Michigan,"
Special Bulletin 206, Michigan Agricultural Eacperiment Station,
East Lansing, 1951:.

Illinois Agricultural Handbook, 191:9, pp. 206
Michigan Agricultural Statistics, 1950-94.

Tintner, Gerhard and Brownlee, D. H. "Production Functions Derived
from Farm Records," Journal of Farm Economics, Vol. XXVI,
August 1955.

Trant, Gerald Ion, "A Technique of Adjusting Marginal Value Productivity
Estimates for (hanging Prices," Unpublished M. S. Thesis, Depart-
ment of Agricultural Economics, Michigan State College, 1951;.

Nagley, Robert 1., "Marginal Productivities of‘Investments and Expendi-
tures, Selected Ingham County Farms, 1952," Unpublished M. S.
Thesis, Department of Agricultural Economics, Michigan State
College, 1953.

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:e For}! ”in"! . 5 g ‘1‘1 -‘ ... ll. 3 3;

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NOV 14 '57,
Dec 2 ’57
Dec 17 '57
MAY 1 1 5°

 

 

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