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1145519

This is to certifg that the

thesis entitled

A mama) OF ESTDEATING MARGDIAL VALUE PRGJUCTIVI «-
TIES 0? INPUT AND INVESTMENT CATEGORIES ON MUL -
TIPIE ENTERPRISE FARMS

. presented by
Christoph Beringer

has been accepted towards fulfillment
of the requirements for

Mat—degree in W81 Economics

1%.“

Major pr es 1'

Date August. 1815 1955

 

0-169

. _,A4_,

A] . V'-

 

W44“ “rm

«ewe

qu

fi
\ .
A METHOD OF ESTIMATING MARGINAL VALUE PRODUCTIVITIES
OF INPUT AND INVESTMENT CATEGORIES ON
MULTIPLE ENTERPRISE FARMS
‘ V

By
Christoph Beringer

AN ABSTRACT

Submitted to the School of Advanced Graduate Studies of
Michigan State University of Agriculture and
Applied Science in partial fulfillment
of the requirements for the degree of

DOCTOR OF PHILOSOPHY
Department of Agricultural Economics

1955

Approved M WM»

1' H ESlS

 

 

Christoph Beringer
ABSTRACT

The purpose of the analysis was to modify presently used methods of
productivity estimation so that the* can be applied to the analysis of in-
dividual enterprises on multiple enterprise farms. Three multi-equational
approaches were suggested as possible ways of solving the problem. These
three approaches consisted of (l) a system of equations fitted by the method
of simultaneous equations, (2) a system of equations, one equation for each
major enterprise fitted independently to enterprise input-output data and
(3) a system of equations each equation fitted independently to data from
SpeCializing farms where the results of these estimates are applied on
multiple enterprise farms.

Methods of grouping products into output categories and productive
factors into input categories were considered. The conclusion was reached
that generally products which are produced jointly can be grouped into one
output category while products competing for resources should be analyzed
separately.

Regarding the fitting of enterprise functions, it was concluded that
fitting of one function to vertically integrated enterprises such as crops
and hogs or crops and dairy'is insufficient if it is desired to compare the
productivity of various factors between crops and the livestoc? enterprises.
Consequently, three separate functions were fitted, one to dairy, one to

hog and one to crop-enterprise input-output data.

Christoph Beringer

Regarding the grouping of inputs into input categories, it was concluded
that an input classification which keeps intercorrelation among the inde-
pendent variables and the errors of the regression coefficients at the low-
est possible level is most desirable. This can be accomplished by choosing
the sample purposively, thus, increasing the variance of the observations and
by recombining input categories which are highly correlated. Furthermore,
it is necessary to distinguish clearly in the accounts between investments
and expenses as well as productive and nonproductive inputs.

In order to test the proposed methodologies independent enterprise
functions and one aggregate function were fitted to detailed enterprise input-
output data from 27 dairy-hog farms in northwestern Illinois. The Illinois
records contained more detailed information than similar records kept at
other experiment stations contacted in connection with this study.

A statistical analysis of the resulting enterprise functicn was carried
out by testing the MVP of each production:factor in each function against a
minimum or reservation MVP which should have been earned by these factors in
northern Illinois in 1950. Comparisons of the geometric mean organizations
with these minimum MVP's revealed no serious maladjustments on the farms studied
while a comparison of individual farms whose organization deviated from that

on the geometric mean showed very serious madadjustments.

3-."

The productivity estimates carried out indicated that on the aVerage

farm in the sample the returns to labor in hogs are significantly below the

{Jiq' "

price which has to be paid for labor indicating that less of this factor

should be applied in hogs.

 

 

Christoph Ber-lager

Regarding the productivity of feed, it was concluded that when
compared at the geometric mean the returns in both livestock enterorises
are just equal to the cost of feed.
The returns to land are high indicating that a oossible expansion of
c i u . on L, , o. ...... 15.71;; ' , '. b -..‘. o
the o eret ons on t‘ese farms m1 ”t be profitatle
a i i c man i--. i th, i divicu _ -o- ‘ in i.” .i - u; _
A s+at st cal 0 per son of e n 7 al en‘ernr ce funct ons Tnth
-,, u i . '1 ' as o one pi, i, - o " f1 ui-e i:,' I _ - -
the a presets functior indie tel ii * *‘e metho‘ 01 at* n lleldual enter
prise functions furnishes more reliable information regarding individual
enterprises than does the method of fittinf one a:*refate function to data

from the entire farm rusfi"ess.

 

A METHOD OF ESTIMATING MARGINAL VALUE PRODUCTIVITIES
OF INPUT AND INVESTMENT CATEGORIES ON

MULTIPLE ENTERPRISE FARMS

By

Christoph Beringer

A THESIS

Submitted to the School of Advanced Graduate Studies of
Michigan State University of Agriculture and
Applied Science in partial fulfillment
of the requirements for the degree of

DOCTOR OF PHILOSOPHY

Department of Agricultural Economics

1955

 

ACKNOWLEDGEMENTS

The writer wishes to express his sincere appreciation to the chairman
of his guidance committee, Dr. Glenn L. Johnson, for suggesting the problem
and rendering help and encouragement at all stages during the development
of this thesis.

The empirical part of the analysis could not have been carried out with-
out the generous permission of the Department of Agricultural Economics,
University of Illinois to use data from the Illinois Detailed Cost Account
Project. In particular, the author is indebted to Dr. Roy Wilcox, Professor
of Farm Managment who developed the Illinois Detailed Cost Account Project
and willingly explained necessary details to the author during his stay at
the University of Illinois. Professors P. E. Johnston, John E. Hills, Earl
Swanson, Franklin Reiss and Paul Mueller all at the University of Illinois
gave valuable assistance and helped make data available for use at Michigan
State University.

The financial aid in the form of a research assistantship which was pro-
Vided by the Department of Agricultural Ebonomics, Michigan State University
Successively headed by Dr. Thomas K. Cowden and Dr. Lawrence L. Roger is

deeply appreciated. 'fiithout it the study could not have been carried out.

-_ 3?

The writer is also indebted to Mrs. Carol Izzo and Miss Jeanne Troyer

(347"

for extending much effort in typing the final manuscript.
Last, but not least, the writer wishes to express his thanks to staff

Inembers and fellow graduate students at Michigan State University, in

particular Messrs. Ray Hoglund, Ingram Olkin, John Hocking, Karl T. wright,
William A. Cromarty, Albert Halter, Harold Carter, Wesley B. Sundquist and
Gerald I. Trant, who through formal and informal discussions contributed

much to the content of this dissertation.

TABLE OF CONTENTS

Chapter
I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . .

II AN EXPOSITION OF THE THEORY SERVING AS THE CONCEPTUAL
FRAMEWORK FOR THE EMPIRICAL INVESTIGATION. . . . . . . . . .

A. The Relevant Aspects of Static General Equilibrium
Theory. 0 O O O C C 0 O I 0 O O O O O O O I O O O O O O O

l. The Theory Regarding Optimum Adjustment of Pro-
duction Factors within One Enterprise. . . . . . . . .

2. The Theory Regarding Optimum Adjustment of Pro—
duction Factors Between Enterprises. . . . . . . . . .

a. The theory of enterprise combination in gra-
phic tems . I O O 1 O O O O Q 0 O O I O C C O O O O

b. The generalized theory of enterprise
combinationucoooaoooooooooooooo

B. The Empirical Procedures of Estimating Production
FI-lnc t ions 0 O I O O O O I O O I I O I O O I O O O I I O O

l. The Mathematics of the Cobb-Douglas Function . . . . .
2. The Ibrginal Value PTOdUCtS. . o o o o o o o o o o o 0

III PAST ETFIRICAL STUDIES ESTIMATING HARGIRAL VALUE PRODUCTS
OF INPUTS AND INVESTMENTS IN AGRICULTURE

A

Reasons Why Past Productivity Analyses Have Concentrated
Largely on Single Enterprise Farms. . . . . . . . . . . .

l. The Errors Introduced by Fitting an Aggregate
Elnction I I O O I l I O O O D O O I C O O O O O O O O

N
o

The Availability of Accurate Input-Output Accounts . .

3. The Conceptual Difficulty of Proving that Production
Functions of Individual Enterprises are Independent
0f Each other. 0 O O O O t O O O O O O O O 0 0 O O O O

B. Review of Past Empirical work Estimating Marginal Value
Productivities in Agriculture . . . . . . . . . . . . . .

Page

12
1h
16

l?

18

20

21

22

7’17, ..

Chapter

 

1. Studies Which Confined Themselves to Special—
iZin-g Fams C I O I O O O O O I O O O O O O O O O I O

2. Productivity Studies Dealing with Multiple Enterprise

anus . a I O O O O 0 0 o o l t o o o 0 o o o o 0

IV METHODS OF ANALYSIS AND THE PROBLEM OF ACCOUNTING . . .

A. Analytical Methods Available and the Choice of the Most
ApprOpriate Procedure on the Basis of Existing Rela-

o

tionships on Multiple Enterprise Farms. . . . . . . . .

1. Definition of "Multi-Equational Approaches". . .

2. The Applicability of the Three Approaches. . . .

3. The Existing Relationships Among Enterprises . .

a.

b.

e.

Input-input complementarity . . . . . . . . .
Product-product complementarities . . . . . .
True complementarity. . . . . . . . . . . .

Supplementarity due to better utilization
of fixed resources. . . . . . . . . . . . .

Byproduct - complementarity ...... .. . . .
The effects of economies and diseconomies
upon production functions on multiple enter-

prisefarms.....c...........

Differences between production functions on
Specializing and.on multiple enterprise farms

Conclusions from the foregoing arguments . .

B. Empirical Problems in Estimating Production Functions
01'). I'lultiple filterrprise FarmS. . o o o o o o o o o o o

1. Output Classification. . . . . . . . . . . . . .

a.
b.

C.

Joint products. . . . . . . . . . . . . . . .
Products competing for resources. . . . . . .

Horizontally versus vertically fitted produc-
tionfunCtionSooooooo-aoaooooo

Page

22

28

3h
35

38

39

ho
ho
hi

N2

 

Chapter Page
2. Problems of Input Classification. . . . . . . . . . . NB
NE

a. Pricing of outputs . . . . . . . . . . . . . . . . N6

3. Problems of Pricing Inputs and Outputs.

0
I
o
O
O
o
O

b. Pricing of inputs. . . . . . . . . . . . . . . . . N6
h. Management and Unexplained Residuals. . . . . . . . . N7
5. Problems of Sampling. . . . . . . . . . . . . . . . . M8
a. Random sampling. . . . . . . . . . . . . . . . . . NB
b. Purposive sampling . . . . . . . . . . . . . . . . ha
6. The Choice of the Fitting Procedure . . . . . . . . . 50

V DISCUSSION OF THE DATA USED FOR THE EMPIRICAL PART OF
THIS S A1.‘T~I])'YI O C O O I O O Q 0 O O O O I O C I I O O O O O O O S3

A. The Organization of the Records. . . . . . . . . . . . . 5N
B. The Functions Fitted from the Data . . . . . . . . . . . 5h
55

2. The Variables Included in the Crop Function . . . . . 59

l. The Variables Included in the Livestock Functions

3. The Variables Included in the Aggregate Function. . . 60

VI INTERPRETATION OF FOUR PRODUCTION FUNCTIONS FITTED TO ENTER-
PRISE INPUT-OUTPUT DATA FROM 27 DAIRY-HOG FARMS IN
NOILTI’I‘JJESTERN ILLDJOIS o o o o o a c 0 o o o o o o o o O o O 62

A. Intra Enterprise Comparison of Regression Coefficients
and IVTVP ‘ s 0 O O O O O O I O O O O 0 Q C I O O O C O O O O 63

l. The Hog Function. . . . . . . . . . . . . . . .

a.

0"
o

The regression coefficients and the FWP

eS tjmates o O O I I o o o o o o O O O o 0 O 0

Testing the regression coefficients against

regression coefficients which would yield
minjmm ETVP'S. o o o a o a I o o o o a o

The errors of the regression line and the
coefficients of multiple correlation and

determination. . . . . . . . . . . . . . . .

67

72

TEE'Tgé'.

Cthter

ZoTheDaiI'yIMCtj—onoc0000....oooooo

3.

a. The regression coefficients and the MVP
estmates O I O O O O O O O O O C O O O I O O I

b. Testing the regression coefficients against
regression coefficients which would yield
minimumMVP'S...o.........o...

c The errors of the regression line and the

coefficients of multiple regression and

determination. . . . . . . . . . . . . . . . .

The Crop metion O O o o n a o 0 a o o O o o o D

a. The regression coefficients and marginal
value productivity estimates . . . . . . . . .

0"
0

Testing the regression coefficient against
regression coefficients which would yield
minijJ'nMVP'S...ooooooo.......

c. The standard error of the regression line and
the coefficients of multiple correlation and
detemination. O O O I O O O O I O O C O l O O

B. Inter Enterprise Comparison of Production Functions
and Regression Coefficients. . . . . . . . . . . . .

l.

2.

Comparison of the Slopes of the Production
Fme tion. I O I O O O 0 O I O O O O O O O O O O 0

Comparison of Individual Esthnates. . . . . . . .

C. Comparison of the Three Enterprise Functions with
theAggregatemnC-tionoo0.00.0000...c

l.

3.

The Estimates of Regression Coefficients and
Marginal Value Productivities in the Aggregate
metion. 0 a O O O O O O O O O O O O O O O I I 0

Comparison of Marginal Value Productivity
Estimates in the Aggregate Function with the
Corresponding Estimates in the Individual
Enterprise Functions. . . . . . . . . . . . . . .

Comparison of the Constant "a", Standard Errors
of Estimate and Coefficients of Multiple Correl-
ation and Determination in the Aggregate Function
with the Corresponding Estimates in the Enter-
prise Functions . . . . . . . . . . . . . . . . .

Page

73

76

78
80

80

82

85

86

86
88

91

91

93

95

 

 

 

. -
. . . . c e .
. e . . . e
e .

. ., . a

. . . . . a n
. . . A . . . . o
. . w . _ . . c . t _
. a . . o n . . . . x
. . e . . _ x . . w . n e
w e . . . a e . e a e
. n . . . a . ~ . e . . _.
. n . . . e r . e 1 , c v
. n . n . v . . n . . n , . t
a a . . ~ _ . . . . h c A o .
. r . _. i ,. . . , n . . p . e
. . . h . .. . _ . . a a a c . e .
. e n . . . a . w u . a .. t c
. . . w . . . . a a a 1 . .
. .. . . . ,. . ~ . . n a t .
. . n a . . u . . a A . . q .
n . a ~ . .. . . o o u a a n w

. . o c e n . o . . . c u n

 

Chapter

VII

EVALUATION OF THE METHODOLOGICAL AND EHPIRICAL RESULTS
OBTAINED IN THE FOREGOIEG ANALYSIS. . . . . . . . . . . .

A. Evaluation of the Conclusions Reached in the Con-

ceptual Part of the Analysis. . . . . . . . . . . . .

1. Applicability of Various Fulti-Equational
Approaches . . . . . . . . . . . . . . . . . . . .

2. Methods of Grouping Outputs and Inputs into
ca-tegories 0 0 o O O o o o o o o o o o o o o O O o

3. Hethods of Sampling and Accounting Procedures . . .

Evaluation of the Conclusions Reached in the Empirical
Part of the Analysis. . . . . . . . . . . . . . . . .

1. Comparison of an Aggregate Function with Individual
Enterprise Functions . . . . . . . . . . . . . . .

2. Inter and Intra Enterprise Comparisons of Individual
Productivity'Zstimates . . . . . . . . . . . . . . .

3. Conclusions Tega ding the Comparative Efficiency
of Production Factors on Eultiple Enterprise Farms
in Northern Illinois . . . . . . . . . . . . . . . .

Page

98

98

100

101

:_J
O
[\3

102

103

10h

.......

.......

 

:1

TABLE

II.

IIIa.

 

PACE

Regression Coefficients and Marginal Value Productivity

Estimates Obtained from a Cobb-Douglas Function

Fitted to Hog Enterprise Data from 27 Illinois Farms,

1950. . . . . . . . . . . . . . . . . . . . . . . . . . . oh
MVP’s for Various In‘f-uts and Investments ‘.-.'.1ich are

Considered Miniwum Expected Returns or Reservation

Pflces in Illinois, 1950. O ' O C O O O C O O O C O C O C C ()9

Comparison between the Estimated b and the b.*
l

*J-

Necessary to Equate Z'IVPT with IFC in the Rog

Enterprises of a Sample 8f 2? Northern Illinois

Farms, 1950. . . . . . . . . . . . . . . . . . . . . . . . . 70

Statistical Comparison betnexn the bi* Necessary to Yield a
Minimum IV? on a Poorly Adjusted Farm in the Sample and

the Estimated bi3 Dairy Enterprise, 1950. . . . . . . . . . 71

Regression Coefficients and Marginal Value Productivity
Estimates Obtained from a Cobb-Douglas Functiwn
Fitted to Dairy Enterprise Data from 27 Illinois
Farms, 1950. . . . . . . . . . . . . . . . . . . . . .

Comparison between the Estimated bi and the bi%
Necessary to Equate NVPKi with L2G” in the Dairy
Enterprises of a Sample of 27 Northern Illinois

77

Fart-115, 1950 o o o o o O o o 0 o I o a o o o o o c o o o t I o

 

TABLE

VI.

VII.

VIIa.

VIII.

Statistical Comparison between the bi* Necessary to Yield a

Minimum MVP on a Poorly Adjusted Farm in the Sample and

the Estimated bi5 Dairy Enterprise, 1950 . . . . . . . . . . . 79
Regression Coefficients and Marginal Value Productivity.

Estimates Obtained from a Cobb-Douglas Function Fitted

to Crop Enterprise Data from 27 Illinois Farms, 1950 . . . . . 81
Comparison between the Estimated bi and the bi* Necessary

to Equate MVPXi with MFCXi in the Crop Enterprises of a

Sample of 27 Northern Illinois Farms, 1950 . . . . . . . . . . 83
Statistical Comparison between the bi* Necessary to

Yield a Minimum MVP on a Poorly Adjusted Farm in the

Sample and the Estimated bi Crop Enterprise, 1950 . . . . .'. 3h
Comparison of Individual Estimates of Marginal Value

Productivities between Enterprises . . . . . . . . . . . . . . 90
Regression Coefficients and Marginal Value Productivity

Estimates Obtained from a Cobb-Douglas Function Fitted

to Aggregated Data from 27 Illinois Farms, 1950 . . . . . . . 90
Comparison of Marginal Value Productivity Estimates

from Three Enterprise Functions with the Corresponding

Estimates Obtained from an Aggregate Function . . . . . . . . 9h
Inter—Functional Comparison of the Constant "a",

Standard Errors of Estimates and Coefficients of

Multiple Correlation and Determination . . . . . . . . . . . 96

 

 

PAGE
Distribution of Gross Income by Enterprises on the
27 Dairy-Hog Farms Included in the Analysis . . . . . . . . . 113
Tractor Operating Cost by Drawbar Horsepower Ratings

and Hours Used Iwring 1950, Blackhawk Area. . . . . . . . . . 115

 

LIST OF FIGURES

FIGURE PAGE
1. Hypothetical Production Function for Enterprise Yl . . . . . . 10
2. Hypothetical Production Function for Enterprise Y2 . . . . . . lO
3. Hypothetical Surface of 130 Cost Lines for Various

Level of Xi"""Xn . . . . . . . . . . . . . . . . . . . . 10
h. Graphic Determination of the Optimum Combination

of Y1 and Y2 . . . . . . . . . . . . . . . . . . . . . . . . 11

5. Hypothetical Surface of Iso Cost Curves Showing
Areas of Complementarity and Competitiveness . . . . . . . . 3h

6. Hypothetical Production Surface for Corn (Y1)
using M and K20. . . . . . . . . . . . . . . . . . . . . . . 36
7. Hypothetical Production Surface for Hay (Y2)

using N and K20. . . . . . . . . . . . . . . . . . . . . . . 36
8. "ypothetical Iso Cost Curve between Corn and Hay . . . . . . . 3s

9. T3_ica1 Distribution of the Eanple Observations
in the Case of a Randomly Chosen Sample. . . . . . . . . . . h?

10. Typical Distribution of the Sample Observations

in the Case of a Purposively Chosen Sample . . . . . . . . . E9

 

CHAPTER I
INTRODUCTION

A brief reflection on the 2,000 year history of western civilization
reveals that Specialization has been recognized as the basic principle for
achieving greater technical and intellectual skill and thus greater economic
wealth for the individual as well as for the society as a whole.

Advocation of this principle has, however, not been equally strong at
all times. Plato1 was its most emphatic promoter in Greece of hOO B.C.
making the principle of specialization the foundation on which he constructed
his "Ideal State". While the Greek Empire fell, its intellectual and cul-
tural heritage went into a hibernation lasting for over 1,000 years. The
Renaissance ending the "dark" middle ages reawakened greek ideals of which
the idea of specialization was at least an important part. Men like Adam
Smith, David Ricardo and.dohn.Stuart Mill later extended the concept which had
at the time of Plato referred only to the life within one city or a city state
such as Athens, to cover various nations and even continents. Developments
in the physical sciences through the discovery of physical laws also permit—
ted the principle of specialization to become increasingly operative and

effective. ikwi

 

l Plato, The Republic, Jowett Translation, New York: The Modern Library,

p. 60 ff.

 

Thus, it is not entirely unjustified to look upon the Renaissance and
ancient Greece as the intellectual father and grandfather of the present
age of Specialization with its rapid scientific and economic development.
While specialization is a powerful tool of achieving increasing wealth,
it is not omnipotent. Even in economies as highly developed and specialized
as those of the United States and Canada a number of industries,which are
usually regarded as highly Specialized, operate most efficiently When several
products are produced in the same firm. Thus, Shoe factories produce shoes
of various sizes, colors and styles, automobile factories maintain a series
of lines producing different models of cars of differing colors, shapes and
sizes with the goal of using fixed resources in a way that the greatest total
profit is achieved.
In agriculture, where considerable seasonal variation is present and
where land, labor and/or other factors may be fixed, a greater total profit
is often achieved if several enterprises are combined to permit full util-
ization of these fixed factors. In general, even in the most Specialized
economies, diversification will be present almost always although not to
the same extent as and often for different reasons than in the economies which
are less highly developed than those of North America and Northern Europe.
When enterprise combination is profitable, the economists major concern
is to find in each particular case an enterprise combination which will max- ‘L
run

imize profit for each given quantity of resources. '

”M
”'1'
\.

This thesis deals with the problem of enterprise combination in agricul-

ture. Its particular ahn is to find a method with which it is possible to

(a) determine whether a farm Should specialize in the production of one pro-
duct or combine several enterprises and (b) determine the kind of enterprises
which should be combined and the relative size in which they should be combin-
ed.

It is believed that this distinction is in line with the thinking of
farm managers who are not only concerned about allocation of resources with-
in one enterprise but also with the allocation of resources between enter-
prises. In fact, one can frequently observe that farmers are more concerned
about inter- than intra- enterprise resource allocation which means essen-
tially that their decisions are influenced more by output prices than by in-
put prices. One possible explanation for this behavior,w1ich is here mere-
ly suggested as an hypothesis, is that to the farm manager input prices vary
on the average less widely than product prices.

This shall not imply that factor adjustments within one enterprise are
unimportant or unnecessary; it merely means that the required adjustments
due to factor price changes might on the average be considerably less signi-
ficant than the changes required due to product price variations.

Past empirical workers estimating marginal productivities have concerned
themselves almost exclusively with finding methods with which they could deter-
mine the best allocation of resources within one enterprise and have neglec-

ted to broaden their analyses to study inter-enterprise resource allocation.

x-‘xA
-$
"1’

The lack of empirical work in this area suggests the need to extend

CDg‘

“'""’."'

marginal productivity analysis to the problem of enterprise combination on

multiple enterprise farms.

The following plan has been adopted to guide this investigation:

In Chapter II the static theory of production economics which furnishes
the conceptual guide for the study will be explained. The relevant princi-
ples are presented in graphical as well as in mathematical terms, the latter
permitting a generalization of the argument from one to several input vari-
ables used in one or several different enterprises.

The empirical part of the dissertation will be concerned mostly with an
application of the theory of production economics to empirical estimation.

Chapter III containsezreview of pioneer research studies dealing with
the estimation of marginal productivities of inputs and investments in agri-
culture.

In Chapter IV, possible methods of estimating marginal value productive

ities on multiple enterprise farms and the conditions under which these methods

are applicable are discussed.

In Chapter V and VI, the suggested.methodologies are applied to actual
farm enterprise data taken from the Illinois detailed Cost Account Records.
With the help of the resulting estimates, a critical evaluation of the Sug-

gested methodologies is possible.

CHAPTER II

AN EXEOSITION OF THE THEORY SERVING AS THE CONCIPTUAL

FRAMEWORK FOR THE EMPIRICAL INVESTIGATION

The present chapter will be concerned with the conceptual bases upon
which the empirical analysis rests. The chapter is divided into two sections,
one dealing with the relevant aspects of static general equilibrium theory,
the other discussing the function used to obtain the empirical estimates and
its mathematical characteristics. A discussion of sampling, combining of
inputs into categories and combining of outputs into categories is reserved

for Chapter IV.

A. The Relevant Aspects of Static General Equilibrium Theory

The theory guiding the empirical part of this study is the general equil-
ibrium theory as developed by walras,l Marshall,2 Hicks3 et. al. This theory
is deduced from the laws of diminishing marginal returns and diminishing mar-
ginal utilities.

The static theory of production economics which is an essential part of
general equilibrium theory deduces from the law of diminishing returns under
a set of static assumptions the conditions which have to be met if a firm.wishes

to maximize profits. Since the empirical estimation of production functions

 

 

 

1 'Walras, Leon, Elements of Pure Economics; or, The Theory of Social Wealth,
Translated by William Jaffe, London: Allen and Brown, l95h.

2 Marshall, Alfred, Principles of Economics, New York: The HacMillan Company,
Eighth Edition, 19H9.

3

Hicks, John R., Value and Capital, Oxford: The Claredon Press, Second Edition
19%.

 

in succeeding chapters will be undertaken to establish whether a group of
farm firms would meet these maximum.profit conditions, it is necessary to
discuss briefly the origin and nature of these conditions. First, the
simple case of the firm with only one enterprise will be discussed. Then
the case for the firm producing several outputs witheaseries of inputs

will be considered.

1. The Theory Regarding Optimum Adjustment of Production

Factors within One Enterprise

C.
I
L.

From the law of diminishing returns, it :ollows that profit in any

h
enterprise Y, can be increased as long as the marginal value product (MVP)
i

of any factor Xi ceteris paribus, is not equal to the cost of this marginal

unit (NFC).S The proof for this proposition is obtained from the simplest
form of the profit equation:

:1
(2.1) I = P r. — ,2 a, x.
I Yi 1' 0:, J'Lj 3

ll‘

which differentiated with reSpect to kj yields

D Ti
( 2. 2 ) --. = IIPPV 1 :9 .. ,
Dxa .‘;j(Ij-) _fi_ -Lj

Setting ( 2.2 ) equal to acre and assuming perfect competition

 

( 2.3 ) w r = BECK. where j = l,....,n
‘J J
the profit maximiz ng condition for the firm producing one product using one
iEK
5“,
h . .

Marginal value product is defined as the value of the marginal product
plus or minus the change in the value of the original total product which
was caused by the marginal output.

Marginal factor cost is defined as the value of the marginal factor
applied, plus or minus changes in the value of the original total quantity
of the input used which resulted from the application of the las unit.

variable factor given one or several fixed factors is obtained.

2. The Theory Regarding Optimum Adjustment of
Production Factors Between Enterprises

The static theory of enterprise combination can be explained both in
graphical and in mathematical terms. Since this theory is used directly in
the study, it will be explained first diagrammaticallgr where it is limited
to the analysis of two enterprises and then mathematically in the form of a
generalized equation showing the optimum adjustment for any number of inputs
in any number of enterprises.

a. The theory of enterprise combination in graphic terms. Two enter-

“V

rises are assumed each one using a set of factors X. .... A fixed for
3 - l, 3 1

n
the farm as a whole but variable between enterprises. The return obtained
for these factors in both enterprises is smaller than the replacement cost
of these factors and larger than their salvage value; for these factors the

condition (Replacement) ) MVPX

P a v
Xi,oooo,)x.n i,oooo’-¥\n<Y1-)

MVP v 1 P: v Salvsce hold"
Xi,oooo,J\-n(f2) 7 1(i,...,’1\.n ( it» ) 90
Another set of factors in + 1,....,Xq are fixed in enterprise Y .

For these factors the condition PX (Replacement) )

-n + iao--o.Xq

Li 1 1 P _ ton]— we 1 _S
VPXn + l’OOOO’Xq(Yl) ) Xn + 1).. ..,2{q (_) Va‘) ) ho d .
. 7 -7 ’ o . 'i _
Analagously there is a set of factors to + 1:....,xz Uhlch are fixed in Emu
‘ 5
, . . r
enterprise Y . For these factors the condition PV 7 (Replacement )’ 3‘
IL 9
2 “LG + 1,oooo,.xz * I“

MVPX (Salvage) holds.

1 P1
q + l,oooo,4{z(Y2) > "{q + 1,0000,XZ

 

'-

Finally there are variable inputs ha,...

in one enterprise nor fixed for the

condition PX (Replacement) (T.

Aa,oooo,X.h

(Salvage) )1 '

VI’J

Aa”

‘f
.DJiaj o o o o ,Jlll

Pr

holds.
ka,OOOO’Xh

The two hypothetical production fuctions shown in Fi

refer to enterprises Y1 and Y2 respectively.

factors assumed fixed for the

. . -0 1 ._L . 1
prises, is varied along Wlbh factors Xa,....,

and Y2, the surface of transformation

A complication is introduced

are priced in terms of theiro 0p

farm while the iUPUtS Xa:‘°"aXh are priced at rarket value.

farm as a whole.

crrves shown in
by the fact

pc rtunity costs since thev are

.,Xh which are neither fired

For them either the

v 3 or th e cond tion

000,--h

gure l and Figlre 2
If Xi , . . . . ,in, the group of

farm as a whole but variable between enter-

Kw between th as enterprises Y1

“‘1

riaure 3 results.

that the factors Xfi,....,Xh

.0'
_L

1} (ed on the

Thus, the

condition specifyurzthe best combination of enterprisesfor a given output is

 

 

 

 

 

 

MVP}. Tr-TVT NV?
Aa,ooo,Xh(Yl> )Ca,oooo,—’Lh (Y9) _ . .— X' 1,0000,J:n (Y1?
P x B-.. a 11,
Xa,ooo,1{h JLa,OO ,KI] Aj,oooo,:)<n
\' - , ,
INPXi,....,J{n(12) ‘ 1 I”. .r . . ‘ J-
cannot flOlO since A1,....,An are priced in the product-
P ' _
Xi’"ooo,xn
Tflfljr i; v v
o n - --— O - 1 a -‘. -. >Li’ . . . ,:n(Yl>"1,I./‘L:L’ . . O "‘n (Y 2)
ion 01 Y1 at their MVP in I, maxing =
" l3 , 1?-r
-Lj-,oooo,Xn _'._i,oooo’J\/:n
= 1 while the corresponding ratios for Xa2°°":Kh: do not have to equal one.
Consequently, there are two equations which have to hold if the the.
i
. . . . . . . . . . . . . p
flflm.is in equilibrium assuming the iso cost line or opportunities line in !m

a product-prod_uct dimension is Specified.

 

v'za

These equations are

iCVPK BRIE: -.(- lNPwr r MVP -r
y A (
(2M an) -a:<2> : ..-hil> _ (2) C
P: HY “j§:“"‘ “—-T%r-~' =
'a ”‘a ‘11 J}1

a constant and

 

 

 

ENE-'10? ) WK: (7‘? )
( 2.5 ) ‘X:L l -....= n l _ l
I-JP, r 251%. ,
)xi(-'-2) {fix-12)

The whole surface of iso cost curves, shown in Ti'ur re ;, shifts with

each change in the relative prices of Xa,....,2h. Thus, each additions

1‘

iso cost curve drawn in Figure refers only to 01% part biCL W18 @701 o 21=

[—1.

= P1? :Ca,oooo,1‘v'y\r‘ LO}
ta ‘ii‘

The iso revenue line which is tanrent to the iso cost line in Figure h
shows the relative prices of Y1 and Y2 and connects the various possible
combinations of Y1 and Y2 yielding equal revenues. The is o revenue l:Ines
increase from the origin in the same way as the iso cost lines. The point
at which an iso cost line is tangent to an iso revenue line represents that
combination of Y1 and Y2 which will yield the greatest revenue and profit
given a certain level of i. and a certain set of relative prices among the

Xa,....,Xh. T_n_ Inchrn h an amount A of Y combined with an amount B of

l
6

Y2 represents the high profit corhination.

 

6 In Figure b, any change (—AY-,) in Y1 must be equal to -(T'l)'-‘(-:: v (+-
-_. ,. O O O O ,— an) 1 >
.- r? (A? \. '2 "‘7
(A (A1,,,,,,;;n )> and any chem e . 2, ll l, mu at be equal to
(hr? )7 y; ) 'v \ {A( v >).1‘t the point of tangency bsti-Ieen
Lo,oooo, n l2) (J: i,oooo,.'s.fl ‘ 1‘,
9%“
the iso cost and the iso revenue line their slepes are apprOXimately equal 5
{-AYl /4 Y2). The eCjI.I.::tion it" thehise revenue line is R = T:— 1“ + PT v El.
L’W‘ ‘r ”l 1 *2”2 ‘3.
which, if solved for Y] = i» _ "d 2 and differentiated with respect
- '1) ""
*‘ii,’ tr
“1 “l
le FY _
t0 Y2,_ _ produces the slope of the iso—revenue line. Aquating
dY2 I’ll

~

the expressions for the slope of the iso revenue line and the iso cost line

 

 

 

 

Pig. 1 Hypothetical Produc-v Fig. 2. Hypothetical Produc-
tim Function for Enterprise tion thctign for Enterprise
Y
1

 

 

 

 

i1 5-2.1 3 i ’4:
. Ifa
Fig. 3 Hypothetical Surface 01'

150 Cost Lines for Various 129--
V61 0f x1002me

2.526

10

 

xa"'."xh311! 0. ..,xnjxn,,1,,. ' 'xq 1a»: "15931, " "KAXQQI’ ”.9le

A “' M ’ 'A‘

ll

 

 

 

 

' I
Fig. 2;; Graphic Determination of the2

Optimum Combinatim of Y1 and Y2 7

. nan-y

(cont.)

- MPP(Xi’99” Kn)(Yl)( (Xi oooo,Xn)) - PYZ
MPP " " """"
(x1,eooe,xn)(Y2) (xi’°”"xn) FY].
one obtains through simplification and division by MFG the equality

x1
PYz (WP(X13....;Xn)(Y2) .. PYi WP<X1j,....,Xn)(Y1)
(xi,oeoopxn) m(xi’°""xn)

This eXpression can be generalized for any number of outputs and op-
timmn adjustment for any given level of g is reached when the follow-
ing condition. is fulfilled

PY1(MPP (Xi, 9 o o 0 9311) (Y1) ; PYZ (MPP (X1, 3 9 9 ’ 34x11) (Y2)
(Xi; eesegxn) "(x1)" - .,Xn)

1 i" ’3’?“ 1 , Ultimate adjustment cannot be reached until

 

 

 

3-0000,

 

(xi’eceo’xfi) .
the factors X1, . .. . ,Xn now fixed for the farm as a whole and variable
between enterprises can become variable for the farm as a whole. Only
then can they be freely bought or sold in such a way that the condition p.

xi - x3 , , xn 1
P2 (MPP ) ‘- ‘P (MP ) ny (mx ) P2 (Winn
2 X Y2 Y2 Pxn Y2 k :5 Y1: k k _
X1 ’0...) .

,0000’ W’...., T 1
o Xn

is fulfilled.
A mathematical derivation of the equation of the iso cost line is prea-
ented in Appendix A‘.

12

The foregoing argument refers only to two enterprises and gives a graph-
ical, intuitively appealing explanation of the theory of profit maximization
in the farm with more than one major enterprise. A generalized theory of
optimum.combination of enterprises can be deduced from the profit equation just
as this was done for the one variable case under (A) above.

b. The generalized theory of enterprise combination. Equation ( 2.h )

 

is a generalized profit equation covering m outputs (Y1), each one produced

by any or all of n inputs (Xj)'

t1.

( 206 ) Tl. = i PYY. "’ Z P1 X o
.__ l .L . 4{-j '
1"]. 3:]. t 3

Assuming that changes in inputscu‘outputs which are made by the firm do not

influence PY- or P, the system of differential equations ( 2.? ) results
i

l.
J

upon successive differentiation of ( 2.6 ) with respect to the amount of each
Xj used in producing each Yi‘
(2.7) 33:" = MPPXW.) PY - PX. .
“J :J *1 i J

Setting each of the m n ( 2.7 ) equal to zero and assuming the second order
conditions satisfied on the basis of the law of diminishing returns, results
in m n profit maximizing conditions which can be summed into

' m n
(2.8 )' i=1 I-IPPXjWi) PYi = 351 PXj .

Equation ( 2.8 ) is equivalent to the final equation in footnote 6 above.

B. The Empirical Procedures of Estimating Production Functions ;

 

The development of the theoretical concepts of marginal productivity

analysis induced empirical workers to look for equations which would approxi-

l3

mate actual production functions. Marginal Value productivities could be
obtained from these equations through partial differentiation.

Generally, five names are associated with this work, all of them hav-
ing suggested and used certain equations approximating the law of diminishing
returns. These men are: Uicksellg, Nitscherlich9, Spillmannlo, Cobb and
Douglas.11 The function originally developed by Cobb and Douglas was used
at first to measure the respective contribution of labor and capital upon
the gross national product of the United States economy. It is a power-
function of the form P= aCkLl k.12 Cobb and Douglas did not use this
function to estimate the NVP's for individual firms but confined themselves
to the analysis of macro-economic relationships.

TintnerlB, Brownlee and Headylh, following a alggestion made by Durandlg,

introduced to the analysis of input and investment productivities on farms

 

8 'Wicksell, Knut, Lectures on Political Economy, Volume 1, London; G.
Routledge and Sons Ltd., l93h-3S, pp. 121-3 and pp. 127—30.

 

9 Mitscherlich, Eilhard A., Bodenhmde fuer Land und Forstwirte, Berlin;
Paul Parey, 1905.

 

10 Spillman, W. I., Use of Erponential Yield Curves in Fertilizer Experiments,
U. S. D. A. Technical Bulletin No. 318, 19 33.

11
Douglas, P. H. and C. W. Cobb, "A Theory of Production", American Economic
Review, Vol. 18, 1928 supplement, pp. 139 ff.

 

 

l2 . . . . . .

"P" in the original Cobb Douglas function represents Gross National Product,

"C" represents Capital and "L" stands for Labor. E?“
13 Tintner, G., and O. H. Brownlee, "Production Functions Derived from Farm h

Records", Journal of Farm Economics, Vol. 26, l9hh, pp. see ff.

 

Heady, Earl 0., "Production Functions from a Random Sample of Farms”,
Journal of Farm Economics, Vol. XXVIII, No. h, November l9h6, pp. 989-100h.

 

15

Durand, David, "Some Thoughts on Narginal Productivity with Special Refer-
ence to Professor Douglas' Analysis", Journal of Political Economy; Vol. XLV,
December 1937, pp. 7hO-58.

 

1h

a power function, which resembled the one used originally by Cobb and Douglas
but did not require the sum of the etponents to equal one. The equation was

of the form

b1 b2 b
(209) Y = a-Xl ,00003X2 ,oooogxnn

where Y represents output, the dependent variable, and K1,....,K represents

n
the independent inputscn’investments which determine output. The following
section will analyze the mathematics of this function which is commonly re-
ferred to as a Cobb-Douglas function. A treatment of the contents of past

to the sub-

.0“
o

empirical studies using the Cobb—Doug-as technique and pertainin

ject matter of this stucy is reserved for Cha,ter Ill.

1 I

l. The Hathematics of the Cobb-Douglas :unction

The Cobb—Douglas ginct_on

[JO

.3 a power function and has certain u ique
mathematical characteristics; some are helpful, others detrimental for em-
pirical work. Among its positive features, the following are the most im-

portant: (l) TLe function is easily fitted to empirical da a. In logarith-

nic form
, :: '1 N r? ‘7 N -r — .- V
( 2.10 ) Log Y log a + b1 100 ll + b2 log a2 F ,...., + bn log in
the function is linear and can be fitted by .he method of least squares. The

C

only transformation required where the function is to be written in exponential

I) 3

form is the conversion or tne constant a bee; into the natural numbers. The

\ rm 1 - I

regression coefficients (bi) are already in natural numbers and merely have
to be raised to the exponent position. {2) The function gives immediately
elasticities of the product Y with rCSpect to the factors of production.

(3) The power function allows the phenomenon of decreasing marginal returns

to come into evidence without having to use up too many degrees of freedom.

15

If instead of the power function a quadratic function were used, decreasing
marginal returns could also be shown. Iowever, the number of regression
coefficients which would have to be obtained increases substantially, thus
reduCing the number of degrees of ° reedom and TGCQ1T_D” an increased number
of observations. (h) If the errors in the data are small and nor mall; dis-
‘ributed, a logarithmic transformation preserves the normality to a substan-
tial degree. Even if the errors are not normally distributed and not in—
dependent the best linear estimate is obtained by the application of the
method of least squares. Tests of significance, however, are no longer

16

valid.

1“)

Against these advantages stand a series 0 necative aSpects which need

to be understood: (1) the function can show increasing or decreasing re-
turns but not both at the same time. Thus, the function increases at an
increasing rate when the i bi ) 1, it increases at a constant rate when the

2 bi = 1, it increases at a decreasing rate when thei bi < l, and it daecre ises
absolutely when the i'b,<LO. (2) The function never reaches a maximum.
(3) The function has constant elast ticity throu_ghoot, which means that the
scale linesvfill.be strairht l:n s resulting in the nurse istic feature that
the optimum proportion in wnica the production factors are combined never
changes as the enterprise increases in size. (h) The function in the form
it is presently rsed must start at the origin and intersects the output-4 —

'5
l u
put planes at Y = O. 7

 

16

Tintner, Gerha‘dt, ”A Note n the Derivation of Production Functions
, No. 1, January, l9hh, p. 26.

 

l7 . . .
Certain modifies ations of the function which would do away wii h some of

its negative features have been develop ed two T. 0. Carter at 'Lchican 3tate
Un.ivers: tv ano are the subject matt<r of a wthscoming h.S. Dissertation.

16

2. The Marginal Value Products
The MVP's are obtained upon partial differentiation with respect to the
factor for which the MVP is sought.

a y ‘ bi b2 , bi-l. b1+1 bn
( 2.11) m = ax]. A2 ’....,bj-1{i JCj+l ’oooo,Xn

Multiplying top and bottom of the right side of the differential equation
( 2.ll ) by Xi, equation

b b b. .
Lr 2 1 117' .—
AY ~ 8—K]- A2 ,oooo,bi"{i Ai+l 3000.,Xnn

37E _ x

 

( 2.12 )

is obtained which reduces to

 

(2.13) JY -—- 3152
6X1 X,

This partial derivative represents the empirical estimate of the MVP at the
geometric mean and has the advantage that it can be computed for any level
of Y and Xi which is not beyond the range of the data from which the original

function was estimated.

17

PAST EMPIRICAL STUDIES ESiIHHTIhG MARGIEAL VALUE -ROD JCTS OF

INPUTS AND x'w.Tn:TTS IL ACRICIU TUBE

The present chapter relates past empirical work in the firld of pro-

ductivity analysis to the present investigation. host 0

H)
Cl‘

he::tudies which

have been conducted in t1 e past crncerned tlm selves with estizm ting pro—

V

ductivities of inputs and investments on specialising farms. Only two

research worlie Ts attempted to obtain “corucLiv‘tV stimates of production

0

factors in individual enterpr es 0 multiple enterprise farms.

The chapter is divided into two sections, one of these will be con-
cerned with the reasons why the analysis of multiple enterprise farms has

been neglected, one other will niseuss tne more irnortant studies union have

been conducted ane point out their relevance to tie present investigation.

 

A. Reasons wnv rasi: Productivity finalyses Hive Concentrated

T ‘ ~r "I‘ -. II V' Y WA .
harsel\ on single huge; rise

, L' .
l l
.3 c. J ‘L ' '- ‘

 

There appear to be three r {sons why the analvsis oi indivinual enter-

“W

prises on multif le en"e rprise :arms has been neg-ccted: (l) It was recog-

nized by a number of Gllpirl ical workers that fitting of one aggregate func-

s
. fl ,., . . ‘ . 3??!
tion to data irom Tiliiple (ciarriice :arms would yield a general function :
tr

waich is frequently not applicable in any on: enterprise on that farm, (2) *

accounts which were detailed enough to allow fitting of independent enter-

' L_J

prise function were not "enerally available, and ”3‘ cc centual knowledge

‘\ J, / ’-

l8

regarding the dependence or independence of individual production functions
was vague.

l. The errors introduced by fitting an aqbregate function. An aggregate

O

 

f

function in this contex is defined as a production function estimated or

the entire farm firm from input—output data taken from the entire farm, in
contrast to a production function estimated for an individual enterprise on

"‘ I. ‘ ‘. " "‘ ' ‘2‘ “" " ’ "n‘ "
a f“. ..'3 «a? i_-_"\_,-T; Lu. :0. o en 1.1817347 5 Q9 Hilly.

*I-

a farm, this function beiig fitte

(‘4
J
.l.
'3
I-

Assuming that such an aggregate function is fitted to inyut-output data

from farms with severa.~ ecually important enterprises, unrersonable results

I” '3

must be expected: (a) If the nature of the returns no scale is di ferent
in the individual enterprises such that some of ta;m would show increasing
others decreasing r turns to scale, the resulting estimate would he an average
function which is not sneei-ieallv annlieaole in anv one of the ente-3rises

H U _ a ”P 9
(b) the estimates of the regression coefficients we 1d be mraningless if the

true HVP of the factors for which the reg coefficient is estimated is

s
a
L)
U)
z

o
m

not equal in the various enterprises on the farms studied. estimates 0

.0

marginal productivity resulting from such reersssion coeificients would not

yield specific information about where in the -irm the use of the laCtOT should

.5‘

be changed, and (c) the errors or the regression coefficienbs which deiend

‘

partially upon the correlation among the independent variables and the standard

error of the regression line will sometimes be increased thus making the result— 1
[Vi-1s; . ;

ing MVP estimates more unreliable. In particular, fitting of an aggregate func- i
r5:

van
tion should be avoided if the aggregation of inputs and outputs increases the “

standard error of the regression line and/or the correlation among the in-

dependent variables.

1

Two recent studies, one conducted oy Glenn L. Johnson1 in Galloway
County, Kentucir, the other by Gerald Ion Trant2 in northern hichigan
may serve as illustrations of the complications which may result from
itting aggregate functions: Johnson fitted a function using 3h farm records
from.Calloway County farms. host of these farms produced livestock, dark
tobacco and popcorn in some combination. All the inputs and outputs were
added and the resulting regression coefficients appeared unrealistic when

compared to other estimates which were obtained from more specialized farms

,‘
.1.

operating under similar production conditions as those found in Galloway
County. In the Calloway County study, the MVP of land was unreasonably high,
while those of forage-livestock investment and labor appeared unusually low.
Johnson attributes these biases to a high intercorrelation among the afore-
mentioned factors and suggests that some of the productivity of labor and
livestock investment might have been reflected in the earni g power of the
land variable. nis is in agreement with what has been pointed outcearlier
in this chapter. If two input categories are highly correlated, as might
well be the case for forage-livestock investments and land, it is di fieult

to determine exactly the proper share of the product which is attributable

to each one of these factors.

 

1 Johnson, Glenn L., The Earning Power of I nuts and Investments on tpland

 

 

Galloway County Farms, 1951, Progress Report No. b, R ? NA60, Kentucky the
Agricultural TXperiment Station, Wnivereity of Kentucky with Tennessee {
Valley Authority Cooperating. L:

.Trant, Gerald Ton, A Technique of Adjusting harginal Value Productivities
Estimates for Changing Prices, Unpublished h.S. Dissertation, hiehigan
State College, 19gb.

 

 

20

Trant fitted two functions for thirty and thirty three farms respective-
ly, both samples taken from 12 counties in northern lower Michigan. The re-
gression coefficients for land and labor in the first sample turned out to
be — .232073ei .235095 and — .1305b’h i .225911 respectively. The co-
efficient for labor in the second sample was - .1738h55 i .lhl70lO. These
appeared unreasonable in view of a general knowledge of agriculture in that
area. Trant points out "... while it is possible that increased quantities
of land and labor might decrease gross income, it was not believed probable

that they would do so."3

A number of the farms in both samples produced a
highly heterogeneous product. A function fitted only to those farms in the
sample which derived at least hO% of their total gross income from dairy pro-
duced regression coefficients which were positive and appeared to be more
meaningful. Also the regression coefficients were associated with consider-
ably smaller standard errors.

From the foregoing considerations supported by the two studies mention-
ed, it appears that in many cases fitting of one aggregate function to data
taken from a series of enterprises on a farm will not likely be a successful
way of obtaining meaningful marginal value productivity estimates on multi-

ple enterprise farms.

2. The availability of accurate input-output accounts. The second reason

 

why empirical warkers have avoided working with individual enterprises is the 1.
ms- .

 

P ;
Ibid, p. 37.

Ibid, p. 39.

difficulty of getting accurate input and output data. Detailed enterprise
cost accounts are difficult to obtain by the survey method and farm account
records ordinarily fail to contain enough detail by enterprises to fit such
functions.

3. The conceptual difficulty of proving that production functions of

 

individual enterprises are independent of each other. One solution for the

 

multiple enterprise problem which appears, prima facie, to be a plausible
oneis that of fitting one independent function to each enterprise a1d then
compare the resulting coefficients. This comparison would enable one to
determine in.which enterprise the application of any factor, Xi, is most
profitable. However, there are conceptual and empirical questions which
have to be answered before such a solution can be suggested as a reasonable
one.

The conceptual question centers around the problem of determining the
presence or absence of functional relationships among individual production
functions on multiple enterprise farms. The empirical Question centers around
the accounting problem and asks if it is possible to devise an accounting
scheme which is able to separate inputs which were used in enterprise Yi from
those used in enterprise Yj in case production functions on multiple enter-

prise farms are independent of each other. The answers to these two questions

will be sought in Chapter IV.

22

B. Review of Past Empirical work Estimating Marginal

 

Value Productivities in Agriculture

 

Empirical work using the Cobb-Douglas technique to estimate Marginal
Value Productivities of input and investment categories in agriculture was
begun during'World war II.5 This section will discuss some of the pioneer
studies which can be classified according to whether they analyzed single or
multiple enterprise farms. The first group of studies is by far the largest
and within it the studies can be grouped according to the types of data upon
which the analysis was based.

1. Studies which confined themselves to specializing farms. The design

 

of most of the studies which estimated hVP's reflect the realization that data
from multiple enterprise farms, if aggregated, yield unreliable results. Thus,
in order to have only specializing farms in the sample, it was usually spe-
cified that a certain minimum percentage of the gross income on the farms
included in the sample had to come from one enterprise.

The first applications of the Cobb—Douglas technique were made by Tintner-
and Tintner and Brownlee7 at Iowa State College. Farm business records from

5 types of Iowa farms were used: "Dairy", "Hogs”, "Beef Feeders", "Crops"

and "General". The independent variables were (A) Number of acres; (B) Labor

 

 

5 Tintner, Gerhard, "A Note on the Derivation of Production Functions from
Farm Records" Econometrica XII, No. 1, January, l9hh. Lg.
Ibid, pp. 26-3h. is

-

Tintner, Gerhardt and O. H. Brownlee, "Production Functions Derived from
Farm Records", Journal of Farm Economics, Vol. 26, August, lth.

 

months; (C) Farm improvements (buildings, fences, etc.); (D) Liquid assets
(livestock, feed, seed, fertilizer, etc.); (E) Wbrking assets (farm machinery,
including the farm share of the automobile, breeding livestock, equipment other
than buildings and fences); and (F) Cash operating expenses (equipment repairs,
fuel, oil, feed purchased). Management was not included as an input.

The standard errors of the regression coefficients of the farms called
"General" were compared with those of all farms included in the study and
were found to be higher than the average for each of the various input cate-
gories on the ”General" farms. However, it is difficult to establish that
these larger errors came about as a result of the aggregation of inputs and
outputs from several enterprises. Tintner and Brownlee's sample included
only 20 "General Farms" which reduces the number of degrees of freedom to
13 and the larger errors might be due to the smallness of the sample.

Drake? using Michigan Farm Account records fitted two general functions,
one for dairy farms, the other for general farms. The reason for this sep-
aration is given as follows; (1) ”the separation of the dairy farms from all
farms permits a more specific analysis of the structure of the farm business..."
(2) "The hypothesis was set up that the correlation of gross income with the
factor categories should be greater for the more homogeneous groups of dairy
farms than for the not-dairy farms. From this it should follow that the con-

fidence intervals of the coefficients of elasticity of gross income for the

F't:r .

3 n

 

8 Drake, Louis 8., Problems and Results in the Use of Farm.Account Records
to Derive Cobb-Douglas Value Productivity Functions. Unpublished Ph.D.
Dissertation, Michigan State College, 1952.

 

 

 

2h

different categories of factors should tend to be narrower..."9 Drake does
not analyze in detail the reasons for his hypothesis. Presumably, however,
his argument was similar to that developed in the preceding section of this
chapter where it was shown that aggregation of input and output data can
under certain circumstances increase the correlation between the independent
variables and thus, make the estimates more unreliable. It turned out that
Drake was unable to support his hypothesis with the data he used.10 However,
he was the first one who recognized this difficulty and tried to establish the
empirical proof.

Headyll fitted functions to random samples of farms and was the first
one to use survey sampling data for Cobb-Douglas productivity analysis. A
total of 13 functions were fitted to data which were grouped according to
the enterprise which was most prevalent in the regions wherefrom the data
were chosen.

The errors of the regression coefficients for the group called "General
Farms" are not significantly higher than the errors which were found fer the

regression coefficients on Specializing farms.

 

Ibid, p. MY.

10 Ibid, p. m. .
s
11 Heady, Earl 0., "Production Functions from a Random Sample of Farms", ’W“
Journal of Farm Economics, Vol. 28 (19h6), pp. 989-lOOh.

 

 

25

Johnson12 was first to apply purposive sampling in devising sample sur-
veys. The notion here is to obtain data covering the largest possible area
of stage II on the production surface. Johnson also developed rules for in-
put classification. Both the various possible sampling procedures and the
rules for input classification as developed by Johnson will be discussed in
the next chapter.

2. Productivity studies dealing with multiple enterprise farms. There

 

are two studies which attempted to analyze individual enterprises on multi-
ple enterprise farms.

13

Fienup fitted a crop and a livestock function to data from 150 Dry
Land Crop and Livestock Farms in Montana. Fienup does not justify, concept—
ually, the fitting of independent functions nor does he attempt to treat the
accounting problem in any systematic way. The data to which the individual
enterprise functions are fitted are not the result of individual enterprise
input-output accounts but are, in several cases, obtained by arbitrarily ap-
portioning of the inputs used on the farm as a whole.
1h 15’

French , leaning heavily on a paper by Marshak and Andrews employed

simultaneous equations to estimate production functions for crop and livestock

 

 

 

 

 

12 Johnson, Glenn L., Sources of Income on Upland Marshall County Farms,
Progress Report No. l, and Sources of Income on Upland McCracken County
Farms, Progress Report No. 2, Lexington, Kentucky Agricultural EXperiment .‘
Station, 1952. «w.
13 Fienup, Darell, Resource Productivity on Montana Dry Land erp Farms, i
Mimeograph Circular 66, Montana State College Agricultural Experiment ”
Station, Bozeman, Montana, 1952.
lb

French, Burton, Estimation by Simultaneous Equations of Resource Produc-
tivities from Time Series and Cross Sectional Fann Observations, Unpub-
lished Ph.D. Dissertation, Iowa State College, 195

 

 

15 Marshak, Jacob, and'William Andrews Jr., ”Random Simultaneous Equations
and the Theory of Production", Econometrica, Vol. 12, l9hh, pp. lh3-205.

 

26

enterprises on 15 Iowa farms. French treated these systems (crops and
livestock) independently although he assumes that "... in actual operation
we have two production functions each dependent upon the other."

The reason.why Marshak and Andrews suggest using simultaneous equations
and not the single least squares approach to estimate production functions is
mainly that the Ki in the production function are not considered to be truly
independent variables since they arealfunction not only of prices - (Marshak
and.Andrews do not make the assumption of perfect competition) - but also of
uncontrolled factors such as management for which no Specific assumptions
regarding their probability distribution can be made. Marshak's and Andrews'
suggestions refer to the estimation of production functions not only for
agricultural but also for industrial firms.

Since this study is concerned with production functions for agricultural
firms only, it is believed that the essential assumptions for perfect com-
petition can be made; thus Marshak's and Andrews' criticism of the single
least squares approach is at least partially invalidated in this particular
instance. The assumptions which were made in this study regarding the dis-
tribution of unexplained residuals such as management will be discussed in
Chapter IV.

It should be pointed out also that Marshak's and Andrews' suggestion to
use simultaneous equations is made because it is believed that by this method,
the best fit of individual production functions is obtained. The simultaneous aw.
equations approach is not suggested to explain the relationships among pro-

duction functions in multiple enterprise firms.

——___

16

French, Burton, op. cit. p. 55.

27

Analagously French' study is concerned with the best way of fitting in-
dependent equations and does not attack the most important problem of multiple
enterprise analysis namely the question of functional relationships between
individual production functions.

The following chapter will investigate two very closely allied questions:
(1) Can the production functions within agricultural firms assumed to be in-
dependent of each other or are they functionally related? (2) If these func-
tions are actually independent, is it possible to guard against pseudo-de-

pendencies through accurate accounting methods?

rigs!"

CHAPTER IV

METHODS OF ANALZSIS AND THE PROBLEM OF ACCOUNTING
The present chapter will consider ways of analyzing enterprises on
multiple enterprise farms. In particular, possible methods of estimating
the parameters statistically and problems of input and output accounting

will be considered.

A. Analytical Methods Available and the Choice of the Nest Appropriate
Procedure on the Basis of Existing Relationships

on Multiple Enterprise Farms

 

It has been shown in the previous chapter that many past empirical studies
have been able to circumvent the multiple enterprise problem but have not
employed the necessary analytical apparatus to cope with it adequately. The
‘ problem was circumvented by fitting functions to data from specializing farms;
the results in these instances were accurate but revealed nothing about opti-
mum.combinations of enterprises. No attempts were made in earlier studies
to work out the conditions under which functions fitted to data from special-
izing farms could be used in handling problems on multiple enterprise farms.

Studies Which fitted an aggregate function to farms with a variety of
enterprises yielded results which revealed nothing about enterprise combina- ~sv
tions and often produced meaningless estimates.

As a result, it is concluded that the single equational approach is not

generally applicable to the solution of the multiple enterprise problem. The

alternatives which are suggested consist of (1) fitting a series of equations,
one for each major enterprise, to data from multiple enterprise farms or

(2) fitting a series of equations to enterprise data from specializing farms,
ascertaining the situations in which synthesis of such fits is possible

on multiple enterprise farms.

1. Definition of Hulti-Equational Approaches

By multi-equational approach either one of the following is implied:
(a) a system of equations in which all or several of the individual functions
are functionally related to each other. This system would have to be solved
by the method of simultaneous equation which, if the system is "just identified"
involves a transformation of the equations into the "reduced form" where each
endogeneous variable is expressed in terms of exogeneous variables only. The
equations are fitted in this form by the least squares procedure and recon-
verted into their original "non-reduced" form. If the system is "overidenti-
fied" the limited information, maximum likelihood method yields more accurate

. 1 .

estimates of the parameters. (b) a system of equations, each one fitted in-
dependently to data from farms on which those enterprises are combined, and
(c) a system of equations each fitted independently to data from farms special-
izing in those enterprises which are combined on the multiple enterprise

farms studied.

 

l Klein, Lawrence, A Textbook of Econometrics, Evanstonzlllinois and White

Plains, New York: How, Peterson and Company, 1953, p. 25h.

 

 

30

2. The Applicability of the Three Approaches

The applicability of any one of the three methods of analysis out-
lined is not immediately evident and.depends upon the presence or absence of
structural interrelations between enterprises.

Approach (a) a system of equations, one for each technica1.production
function, solved by the method of simultaneous equations may be applicable
in those instances where the physical production functions of the individual
enterprises are functionally related to each other. In less technical terms,
this means that the physical input-output relationships and therewith the
parameters of the production function, in one or all of the enterprises
Y1,....,Yh, would always change asziresult of changes in inputscu'outputs
occurring in enterprise Tm.

Physical interdependence of production functions implies that the pro-
duction parameters in one enterprise change as changes in inputs or outputs
in other enterprises occur even if all the inputs are measured in terms of
physical units. Another kind of interdependence is economic and occurs if
all or some of the inputs are measured in terms of their opportunity costs
in other enterprises making the production functions involving inputs so
measured dependent upon each other. The remarks Concerning the applicability
of simultaneous equations apply only if this first type of interdependence
which is of a technical nature exists. Economic interdependency can be
handled through avoiding pricing of inputs or investments in terms of their
opportunity costs for, if no external economies or diseconomies are present,
market prices will be constanusin the sense that the farmer has no influence
upon them. If, however, internal economies and diseconomies exist and on-

farm opportunity costs are used to price inputs, biases of the quantities of

31

inputs actually used would result. Accounting in physical units or constant
dollar terms is not affected by the existence of internal economies or dis-
economies.

Approach (b) a system of equations each fitted independently to data
from multiple enterprise farms, appears applicable if the physical production
functions of the individual enterprises are not related to each other. In
those cases where physical dependencies among production functions seem to
exist, accurate accounting can determine whether a pseudo-dependency or a
structural dependency is at hand.

Approach (c) appears useable if the production functions along which
enterprises on multiple enterprise farms operate are independent, the same
as those on specializing farms and involve the same inputs.

In the remainder of this section a number of conceivable relationships
on multiple enterprise farms will be investigated in an effort to determine
under what conditions the production functions should be expected to be
physically related or unrelated and/or the same or different from those of
specializing farms. After these investigations are completed, it will be
possible to choose from among the approaches outlined above the most appro-

priate one.

3. The Existing Relationships among Enterprises

a. Input-input complementarity. Input—input complementarity exists if

 

two inputs used in producing a certain product have to be combined in rather
definite proportions. A great number of inputs used in agriculture are com-
plements to each other although the prOportion in which they are optimally

combined varies from enterprise to enterprise. Complementarity is due mostly

to physiological requirements which permit substitution of factors only within
a rather narrow range. For purposes of illustration, a beef-hog farm is
assumed. The inputs are hay and grain. The supply of both these inputs is
fixed on the farm at least within one year. If the output of hogs is reduced
due to lower anticipated pork prices, both grain and hay (pasture) will be
released which can now be used in the production of beef feeders. However,
since the hay/grain consumption ratio for hogs is, in almost all cases, lower
than that for beef feeders it is apparent that grain will now be available

in relatively larger quantities and a beef feeding system leaning more heavily
towards dry lot feeding would be indicated. This is especially true in those
instances where hay and grain are fixed on the farm, i.e. their MVP's are
higher than the price which would be obtained if they were sold on the market.
The question relevant to this investigation is whether or not shifts such as
the one described change the parameters of the production function of the enter-
prise in which the substitution has occurred. If the inputs are measured in
dollar values based on alternative earnings, this question must be answered

in the affirmative since each change in the relative sizes of the enterprises
would be connected with a re-valuation of the total quantity of the factor
used in the two enterprises. Thus, in the example above, after the shift of
grain from hogs to beef has occurred the opportunity cost of grain in beef
would have fallen while that of hay would have risen. If the total quantity
of grain fed to feeders were valued at this new (lower) opportunity cost a
downward bias in the grain variable would result producing an apparent

upward shift in the production function of beef.

If, however, the factors are measured in terms of constant dollar values
or in terms of physical units, changes such as the ones described above will
appear to be what they really are, namely movements along the same physical
production function. It appears on the basis of the foregoing argument that
the simultaneous equations approach is not necessary as a result of shifts
involving differences in input-input complementarities.

b. Producteproduct complementarities. Two subcases of product-product

 

complementarity can be distinguished: True complementarity and byproduct

complementarity. True Complementarity exists if variable factors of pro-

 

duction i.e. factors whose MVP is larger than their replacement cost or
whose MVP is smaller than their salvage value can be varied between enter-
prises in such a way that the output of one enterprise is increased without
decreasing the output of other enterprises. Each one of the curves in Figure
5 which shows different combinations of two products Y1 and Y2 obtainable
with a certain quantity of these variable factors is called an iso outlay
curve and shows ranges of complementarity and competitiveness. Thus, if a
farmer decided to apply g ton of fertilizer worth ;B0.00 on two acres of
land, one of them producing corn the other oats, he might well be in stage
III if he applied all the fertilizer in corn. If he began to reduce the ap-
plication in corn and.applied some 0f it in oats both his corn and oats
yields might be increased. A different, higher or lower, iso cost curve exists
for each total quantity of the variable factors used between the two enter-
prises.

There is no reason to assume that movements along the scale line of each

individual production surface occurring as factors are shifted between enter-

3h

%

Y2

 

 

Fig. 5 , Hypothetical Surface of
150 Cost Curves Showing Areas
of Complementarity and Com—
petitiveness

prises or movements along the scale line of the iso cost surface in the
product-product dimension would make the individual production functions de-
pendent upon each other. This is true only as long as inputs are measured
either in constant dollar terms or in terms of physical units.
Supplementarity due to better utilization of fixed resources can be
looked upon as a subcase of true complementarity closely allied with the one
discussed above. The only difference is that part of the factors previous-
ly assumed to be variable have now become fixed inputs. While they are fixed
for the farm as a whole they remain variable between enterprises. In such
a case enterprises may be added and their size expanded until the returns of

the fixed factors in the two or more enterprises in which they are used are

equal to each other. The important source of complementarity here is that
the MVP of the fixed factors which might have been zero before additional
enterprises were added has now increased thus reducing the average cost per
unit of output. Again there is no reason to believe that this type of
complementarity makes the physical production functions dependent upon each
other.

Byproduct complementarity is a form of pseudo-complementarity occurring

 

when part of the output coming from one enterprise is used as an input in the
other enterprises in the firm without being considered in the accounting
process. Failure, in accounting procedures, to credit such byproducts to
the producing enterprise and charge them to the utilizing enterprise creates
an apparent complementarity. An example of this is the situation which exists
between corn and legume enterprises when the byproducts, nitrogen, humus and
soil tilth are ignored. It appears as if, after the addition of the legume
enterprise, the same amount or more corn can be grown in addition to the
hay crop with the same total outlay used when no legumes were grown. Act-
ually, however, the inputs to the two enterprises have increased if one
considers the nitrogen added as well as the improvement in soil tilth which
was brought about as a result of the addition of legumes. Figure 6, 7, and
8 show the situation which exists as a result of the presence of byproducts.
Assume that the total quantity of factors available are (a) of nitrogen
and (b) of potassium. If these quantities were used in corn production, an
amount A of corn would be produced. Assume now that a part of the total
quantity of N and KéO is applied to legumes. Theselegumes will build up
nitrogen in the soil which might in succeeding years grow a larger quantity

of corn with a smaller quantity of the commercial inputs N and.K20)than was

used previously in this enterprise. The conclusion might then be reached

 

36

 

\\\\A.
\\\‘

 

 

 

 

 

 

 

 

 

\ \
\\\
N\
b M!
. K20 . _ K20
Fig. 6 Hypothetical Pro- Fig. 7 Hypothetical Pro-
duction Surface for Corn duction Surface for Hay
(Y1) using N and K20 (Y2) using N and K20

 

 

_._ A

2'3. i 2 £3 E— h
. Y2
Fig. 8 Hypothetical 150 Cost Curve
' between Corn and Hay

that corn was produced in stage III since the production of both corn and legumes
has been increased through a shift of the productive agents between enterprises.
Figure 6 illustrates the situation for corn. If the total quantity of N and

K 0 were applied to corn, a quantity A (Figure 8) might be produced. After the

2

application of N and K20 in corn has been reduced to a' of N and b' of K20
an amount B of legumes is produced. However, since the legumes grown left
nitrogen in the soil, the actual quantity of nitrogen available to corn might,

in the following year, be a " which combined with b' of K O produced a quantity

2
A' of corn which is more than the original quantity A. Here than a peculiar
type of discontinuity exists in the iso cost function of Figure 8, which can
be explained if it is recalled (Chapter II) that a different iso cost curve
in the product-product dimension exists for each new quantity of the factor
N which becomes available. Thus, the true transformation curves 2 1’ E. 2,....,
J$»n are intersected by a pseudo-iso cost line AB which appears to be the rele-
vant curve if no proper charges are made for the byproducts utilized. There
appears to be no reason for expecting the production functions for byproduct
producing and utilizing enterprises to be dependent upon each other if the
byproducts are apprOpriately measured, charged and credited in either physical
of constant dollar terms. Again, the conclusion is reached that simultaneous
equation techniques are not required to fit the functions.

At this point in the investigation, it is concluded that production func-
tions of individual enterprises on multiple enterprise farms can be regarded
as being independent in a technical sense. Apparent dependencies are due to

pricing in terms of on-farm opportunity costs and/or neglecting to charge and

credit byproducts in a proper way.

38

Aside from the technical relationships discussed so far there are econ-
omic influences usually referred to as external economies and external dis-
economies. Possible changes of production functions due to the influence of
these factors still have to be considered.

c. The effects of economies and diseconomies upon production functions

 

on multiple enterprise farms. Examples of external economies are reductions

 

in factor prices due to larger quantities bought and increases in product
prices resulting from a larger volume of products making it possible to enter
a better market. If such economies exist, it is possible that as one enter—
prise is expanded, it becomes profitable to replace certain inputs by differ-
ent ones which can be bought more profitably in larger quantities. In such
cases it may be profitable to change the production processes in a number of
enterprises.

In the case of external diseconomies, a similar argument applies, namely
that the production processes will change as a result of relative increases
in the price or Opportunity cost of using one of the factors or a relative
decrease in the price of the product.

Whether or not the individual production functions can be looked upon
as being dependent upon or independent of each other depends upon whether or
not the changes in the production functions‘are continuous changes or whether
the new function, once determined, remains the same. If the changes appear
continuous, stratified sampling might handle the difficulty without the use
of simultaneous equations. If the change is not continuous, simultaneous
equationsareunnecessary.

The next step is to investigate the nature of production functions on

specializing farms and compare them.with those on multiple enterprise farms

in order to determine whether functions on these two types of farms are,in
general, the same or different from each other.
d. Differences between production functions on specializing and on multi-

ple enterprise farms. The third approach which was outlined above would

 

utilize functions fitted to data from Specializing farms to make inferences
about the productivity of factors in multiple enterprise farms. This approach
is not likely to be successful because the utilization of byproducts which

are frequently wasted on Specialty farms comes into play on multiple enter-
prise farms. Thus, when several enterprises are combined on the same farm,
production processes may be altered substantially to provide for byproduct
utilization. These alterations exist whether inputs are measured physically,
in constant dollars or in terms of opportunity costs. An example is straw
bailing which might occur on a wheat livestock farm, but might not on straight
livestock or wheat farms.

Regarding the influence of external economies and diseconomies it has
been shown in the previous section that these factors may be of considerable
importance on multiple enterprise farms. Consequently, it should be expected
that these factors also cause production functions on multiple enterprise farms
to differ considerably from those found on Specializing farms.

On the basis of these arguments, it is concluded that functions Should
not be fitted to data from Single enterprise farms for use on multiple enter-
prise farms.

e. Conclusions from the foregoing arguments. In summary, the following

 

conclusions are reached on the basis of the foregoing analysis:

 

 

 

hO

Input-input relationships, product-product relationships and internal
economies and diseconomies are unlikely sources of dependency among physical
production functions on multiple enterprise farms if all inputs are measured
in physical or constant dollar terms. Consequently, it appears unnecessary
to use the Simultaneous equations approach in fitting production functions.

From.the investigation of the physical and economic relationships on
multiple enterprise farms and from a comparison of the nature of production
functions between specializing and multiple enterprise farms, it appears that
functions fitted to data from specializing farms for use on multiple enter-
prise farms should not be eXpected to be appropriate. This leaves the approach
of fitting each equation of a system of equations for multiple enterprise
farms independently to data from such farms. This is the approach which has

been used in the empirical part of this study.

B. Emgrical Problems in Estimating Production Functions

 

on Multiple Enterprise Farms

 

The remainder of this chapter will investigate problems which arise in
connection with the empirical estimation of production functions. Problems
of output and input classification, problems of bfproduct accounting and
pricing of fixed factors, methods of sampling and alternative fitting pro-

cedures are considered.

1. Output Classification
In multiple enterprise analysis, it often becomes necessary to combine
enterprises into categories because even in the Simplest farm organizations
many products are frequently produced which cannot and need not be analyzed

separately. Combination of products into output categories often makes the

analysis easier and more meaningful. Output classification should be undertaken
in View of the objectives of a productivity analysia'which on multiple enter-
prise farms, are (a) determination of the optimum.relative size of the present
enterprises on the basis of the relative efficiency of the productive agents

in these enterprises; (b) conclusions regarding the profitability of the
enterprises presently found on the farms which are studied in comparison'with
enterprises not found on the farms under similar conditions; and (c) con—
clusions regarding the profitability of feeding crops to livestock versus
selling these crops for cash.

a. Joint products. Bradford and Johnson define a farm enterprise as "... a

 

line of production necessitating individual and distinct production treatment".
Following this definition, it appears that all products requiring the same pro-
duction treatments can be called joint products even though these products differ
in their nature. Examples are milk and calves, mutton and wool, or pork and
lard. These pairs of products compete very little for resources and can thus

be considered as one enterprise; they need not be analyzed separately in a
productivity analysis.

b. Products competing for resources. All products requiring different

 

production treatment can be ccnsidered as competing for resources and should
be analyzed separately if it is the objective of the analysis to determine

the relative efficiency of factors in the competing enterprises.

 

Bradford, Lawrence A. and Glenn L. Johnson, Farm Management Analysis,
New York: John Wiley and Sons Inc., l953, p. 153,

 

c. Horizontally versus vertically fitted production functions. A multi-

 

ple enterprise farm can be looked upon as consisting of a series of major
enterprise groups. These groups are (a) the livestock enterprises and (b) the
crop enterprises producing feed and/or cash crops. Each one of these enter-
prise groups contains a series of sub-enterprises which, as it was shown pre-
viously, can sometimes be treated as joint products not competing for resources
and do not have to be analyzed separately.

In addition to integrating the enterprises within the major categories
it is conceivable to combine the feed crop enterprises with the livestock
enterprises and consider the proper share of the inputs in crops, this share
being determined on the basis of the proportionate amount of feed consumed by
each livestock enterprise, as direct inputs in the livestock enterprises. In
such a function, livestock output would be considered as the dependent vari-
able and the inputs used in crops and the livestock enterprise would be con-
sidered as the independent variables. It is apparent that in such a function
feed produced on the faxm.cannot be charged as an input since this would
amount to a double accounting of the factors used in producing feed.

This, in brief, is the approach which has been used in most productivity
studies conducted on farms producing crops and specializing in the production
of one type of livestock. The approach could be extended to the analysis of
multiple enterprise farms by fitting a series of such vertical functions.

If, on the other hand, it is desired to obtain a productivity estimate
for all major groups of enterprises and if it is to be determined whether
feeding of crops rather than selling them is profitable the vertical connec-
tion between crops and livestock enterprises has to be interrupted and inde-
pendent functions have to be fitted to the major livestock and crop enter-

prises. Since it was desired in this study to arrive at conclusions regard-

b3

ing the profitability of feeding crops versus selling them and conclusions
regarding the efficiency of other factors in crops as compared to their
efficiency in livestock it was decided to fit one independent function each

to the dairy enterprise, the hog enterprise and the crop enterprises.

2. Problems of Input Classification

The need for classifying inputs into groups in such a way that the
categories are economically meaningful was expressed in every product-
ivity study undertaken so far. The reasons for attempting to classify inputs
into categories are the same as those given for output classification; the
actual production processes are too complex to allow each input to be handled
separately. Farm account data which were used in early productivity studies
are not set up to permit meaningful groupings of inputs into categories be-
cause investments are not clearly distinguished from expenses and productive
inputs are not kept separate from nonproductive inputs. Many of these Short-
comings have been overcome in studies based on survey data which can be
designed to permit a more meaningful grouping of the variables. Bradford
and Johnson3 partly through learning from errors made in earlier studies,
partly through analytical reasoning developed a set of rules which proved use-
ful in the present investigation. In addition to these rules, new ones were
developed in connection with the problem of input classification on multiple

enterprise farms.

 

3 Bradford and Johnson, 0 . cit., p. lhh.

The ideal in input classification is to have the inputs within one cate-
gory combined in scale line proportions. Since this ideal is often difficult
to attain, Johnson and Bradford suggest 3 rules which assure that it is at
least approximated "... the inputs within one category be as nearly perfect
substitutes or perfect complements as possible". If inputs within one cate-
gory are perfect complements they can only be used in one Specified combin-
ation and the estimated MVP will apply to them jointly. If the inputs with-
in the category are good substitutes and measured in terms of their least comp
mon denominator the estimated MVP applies to the least common denominator which
caused them to be good substitutes.

"... that categories made up of substitutes (a) be measured according to
the least common denominator (often physical) causing them to be good substi-
tutes and (b) be priced on the basis of the dollar value of the least common
denominator unit."

"... that categories made up of complements (a) be measured in terms of
units made up of the inputs combined in the proper proportions (which are
relatively unaffected by price relationships) and (b) be priced on an index
basis with constant weights assigned to each complementary input..."

Concerning the relationship of input categories among each other, Bradford
and Johnson specify ”... that the categories of inputs be neither perfect com-
plements nor substitutes relative to each other”. If input categories do not
fulfill this condition the correlation among them will be high thus making
the resulting MVP estimates for the individual categories unreliable. Comp

bining such categories into one is the appropriate procedure in such a case.

"... investments and expenses should be kept in separate categories." The
reason for this suggestion is that one expects to get a dollar in return for
each dollar Spent on items not depreciated, i.e. expenditures which have to
be met annually. Examples are expenditures for fertilizer, animal care etc.
Investments in turn are not expected to pay for themselves in one year but
have to be replaced over a period of years. Thus, if a machine is expected
to be worn out after 5 years one would be satisfied if each dollar invested
in this machine returns approximately 20¢ plus interest on the investment.
Finally, Bradford and Johnson suggest "... that maintenance expenditures and
depreciation be eliminated from the expense categories because of the diffi-
culty encountered in preventing duplication. This means that the earning of
the investment categories must be large enough to cover maintenance and/or
depreciation". The implication is that depreciation, if included as an ex-
penditure and the depreciated item is also included as an investment wauld
cause the bi for both categories (expenditures and investment) to be biased
downward.

Regarding the treatment of interest and taxes similar rules apply. It
is erroneous to regard these factors as productive since really no return is
expected to accrue from them. Consequently, such items have to be excluded
from the input categories because they would bias the resulting marginal value

productivity estimate downward.

3. Problems of Pricing
Pricing of inputs and outputs presents difficulties in those cases where

either no good market for the factors exists or accounting in physical terms

is impossible.

a. Pricing of outputs. Most outputs produced by the various enterprises

 

on a multiple enterprise farm can be priced rather easily because a well es-
tablished market exists for them. Difficulties exist in pricing byproducts
which are used only on the farm. These are often not handled commercially

in large enough quantities and no well established market exists for them.
Examples are manure, leguminous nitrogen etc. Since most of these items

are substitutes in one way or another for commercially handled production fac—
tors, it seems reasonable to price them in terms of the least common denom-
inator which causes them to be good substitutes.

b. Pricing of inputs. Difficulties similar to those discussed in the

 

previous section exist with reSpect to pricing of fixed inputs. Some of
these problems are eased since the Cobb-Douglas technique admits accounting
of inputs in physical terms. Thus, items like labor, land and building space
can be measured in phySical units and the resulting MVP's state the returns
for labor hour, acre of land, housing animal units rather than the dollar value
of the investment. The problem is more difficult for such items as machinery
investment which cannot be accounted in terms of physical units but have to
be priced. Machinery investment can be established fairly accurately by
resorting to Farm Implement Bluebooks which give current prices for most
types of new and used machinery. Appropriate discounts or additions can be
made depending upon the condition of the machine. Major repairs (new motors,
tires, etc.) are added to the investment since they prolong the life of the

machine.

h?

h. Management and Unexplained desiduals

Not all of the factors contributing in the production process can be
included among the independent variables. It is for this reason that the
theory of regression analysis speaks of unexplained residuals which reflect
the influence in the case of this study of such items as management, weather,
different production systems in hogs and dairy. degression analysis works
with.the assumption that these errors or unexplained residuals are normally
and independently distributed with respect to the independent variables.

How justified is the aforementioned assumption regarding management
on multiple enterprise farms?

Management obviously plays an important role in combining enterprises
and farms which are well adjusted will have, in general, a high quantity
and quality of management while the opposite is true for farms which are
poorly adjusted. The implication here is that management is inversely re-
lated to the magnitude of the MVPXi ratios in the different enterprises.
Since the percentage profit on well adjusted farms is higher than on poorly
adjusted farms, a profit index of management could be incorporated as an
independent variable in a productivity study. This, however, is only true
if one is willing to accept monetary profit as the major objective and the
sole criterion of management. If, however, a broader definition of manage-
ment is adopted which bases itself not only upon profit maximization but
upon maximization of satisfactions as well, the assumption that management

is distributed randomly, normally and independently of other factors is well

justified.

 

I:
o:

5. Problems of Sampling

that most farm account data

.’3

It was pointed out in a previous sectio
are not well suited for use in productivity studies in the way they are
presently organized. Usually there is no clear distinction made between ex-
penses and investment repairs and between productive and nonproductive inputs.
As a result, several analysts preferred the survey sample and the data thus
gathered proved to furnish more reliable estimates. In general, two different
types of samples were used (a) the random sample, (b) the purposively chosen

sample.

Random Sampling appears at first sight to be statistically a more ac-

 

curate and commendable procedure. However, if the majority of the farms in
the area are relatively well adjusted economically, a random sample would
provide observations which are mostly concentrated around a small section of
the scale line as indicated in Figure 9.

Such random samples might not provide good estimates of the entire pro-
duction surface which of course extends beyond the small area in which a
randon sample might fall. In addition, the errors of the regression.coef-
ficients might be large due to the fact that the variances of the independent
variables are small. Since only a small area of the surface is covered,
the estimates of the iso product lines extend only over a small area thus

preventing reliable conclusions about the least cost combination.

Purposive Sampling can overcome some of these shortcomings of random

 

samples through selectiny observations covering the largest possible area
of Stage II of the production surface. Iurposive sampling thus approximates

closely the design of controlled field eXperiments where observations over

 

 

 

Xi

Fig. 9 Typical Distribution
of the Sample'Observations
in the Case of a Random-
ly Chosen Sample

 

 

xi

Fig. 10 Typical Distribution
of the Sample Observations
in the Case ’of a Purposi-
vely Chosen Sample

the entire production surface can be obtained. In addition, purposive
sampling increases the variance of the independent variables; thus, a tend-
ency to reduce the errors of the regression coefficients exists. The reduc-
tion of the inter—correlation among the independent variables which can be
effected through purposive sampling is another positive feature of this type
of sample design. Figure 10 ill‘strates such a purposive sample design.

A problem common to all sampling designs is that of finding enough farms
which are on the same production function. For this reason, restrictions re-
lating to the equality of land quality, equali‘y of the types of milk produc—
tion, climatic and market conditions are usually imposed when the samples are
drawn. If these conditions are .et, it is safe to assume that the estimated

. . . . . . , . . . h . -
production function is intrafirm rather than interfirm. In selecting a sample
of multiple enterprise farms, the sane restrictions are applicable and care
should be taken that the farms throughout the sample are on the same produc-

tion function with respect to each enterprise.

‘

6. The Choice of tne Fitting Procedure
There have been discussions regarding the most expedient approach to use

for fitting regression eucations to non—experimental data. The principal

 

cf: Bronnfenbrenner, hartin, ”Production Functions: Cobb-Douglas, Inter-
firm, lntrafirm," Econometrica, Vol. 12, January l9hh, pp. 3* ff. Brenn-
fenbrenner's argument is that the Cobb-Douglas function is an interfirm
rather than an intrafirm function. His argument might well have been valid
in those cases where a function was fitted to data from different industries
throughout the United States Lconouy. (cf. Cobb and Douglas, op. cit.)
However, since in arricultural productivity studies controls of the type
explained above are used in drawing the sample, it is not likely that the
estimated functions are interfirm functions.

 

 

antagonists are the methods of single least squares and the simultaneous
equations technique the latter using both the li ited information technique
and the reduced form methods. The method of least squares has been applied
in all but one agricultural productivity study, mainly because it requires
only two assumptions, namely that the dis turbsnce factors are normally dis-
tributed and uncorrelated with the regressors.S It has been pointed out pre-
viously that there are grounds on which it can be assumed that thi is assump-
tion is justified.

The second method was sugcested by Earschak and Andrews and has been

-0

applied to agricultural productivity studies by Burton FLrench. French fit—
ted equations with both the least squares method and thel imite d information
method. The computational methods used in the limited information method

0
were those given in Bronnfenbrenner andC hernoff and Anderson and Rubin.
The results obtained from the limited information method compare unfavorably
with estimates obtained from the same data by the simple least squares re-

. 11 D , ‘. A _. n,, L., A .
greSSion method, as La as the economic meaninc oi the eSLimates is concerned.

"S

L)

 

.,

wold, Herman and Lars Jureen, Demand Analysis, hen fork: John'Tiley and Sons

In0., 1953, p. 56.

 

 

Supra p.
MarEChak and Andrews, 2p. cit.

8 "m'
French, Burton, op. Cit.

9 Bronnfenbrenner, Jean and Termann Chernoff, “Cor nputational Kethods Used
in Limited Information Treatment of a Set of Li_near Stochastic Difierence
Equations", Cowles Commission Discussion Paper: Statistics: No. 328.
Cowles Commission for Research in Economics, University of Chicago,
Chicago, Illinois, Februrary 22, l9h9.

10

Anderson, T. V; and J. Rubin, "The asymptotic Properties of Estimates of the
Parameters of a single {ouatien in a Complete System of Ctochastic Equations,
Annals of Mathematical Statistics 21: pp. 570-b2, 1950.

l .
French, Burton, Op. Cit., p. 58.

The choice of the method of simple least squares in this analysis is

only incidentally in agreement with French' conclusion since one empirical

application can not establish the superiority of one method over the other.

Rather, the choice has been made on the basis of the treatment given the sub-

<

ject by Hold and Jureen which, after comparing the arious approaches, con-

cludes that simple least squares regression is still the most efficient

p

method since it can be employed with a minimum of assumptions.

53

CHAPTER V

DISCUSSION OF TUE DATA USED

FOR THE EXPIYICAL PART

Since no detailed input output accounts for multiple enterprise farms
are compiled in Michigan it was necessary to obtain data elsewhere in order
to apply the foregoing analysis.

Among the experiment stations keeping detailed enterprise input - out-
put accounts, Illinois had information which was well suited for the purposes
of this project. To set up a questionnaire and obtain the data here in
Michigan through a sample survey would have involved excessive costs in view
of the exploratory nature of the study. Furthermore, farmers can not be
expected to remember the required detail on how much of each input they used
in each enterprise during the past year.

The Illinois detailed cost account project is part of the much larger
Farm Bureau - Farm Management cost account project and uses data from multi-
ple enterprise farms only. The project is moved every second year to a dif-
ferent area of Illinois. The number of farms studied varies between 30 and
50; however, not all of these farms have the same enterprise combinations.

In choosing one area to use for the empirical part of this study it was
necessary to select a sample area which had a large enough number (at least
25 - 30) multiple enterprise farms with the same enterprises on them, was
not subject to unusual weather conditions and had approximately the same

types of land.

The 1950 data from the Blaclchawk areal appeared to meet these condi-
tions. In this area, financial and.production records were kept by 37
farmers, 27 of which were dairy-hob - farmers. T ass 27 dairy-hog farmers
constitute the sample which was used in this analysis.

On the average, these farmers derived he per cent of their gross income
from dairy sales, 36 per cent from hogs, 3 per cent from poultry and 13 per
cent from feeds and grains sold for cash while 2 per cent came from work
off farm and miscellaneous receipts.2 Thus, the 27 farmers in the sample
derived 89 per cent of their total gross income from the ho og and dairy enter-

prises.

A. The Organization of the Records

Each cooperating farmer kept separate sheets for each of the enter-
prises on his farm. A fieldman supervising the project helped the coop-
erating farmers in summarizing the individual entries. A summary sheet
was prepared annually by the T -"cr53t" of Illinois.

Most of the entries in the input and investment categories met the
requirementscu‘innut classification cu lined in Chapter IV; the entries
which had to be ac djr sted were mostly ,xpense items which included nonpro-

ductive expenses such as ta-es, deprecr. ation, interest and similar items.

B. The Functions wrich were Fitted from the data
Following the theoretical arguments outlined in the previous chapter
it was decided to fit three independent functions one for dairy, one for

hogs and one for the crOp enterprises. To permit a more extended evaluation

 

l . . i
The Blackhawk area includes Carrol, Jo DaViess, Ogle, btephenson,
Wuitesidc and' innebago Counties.
2

A table showing the distri bution of income it sources is included in
Appendix B.

 

of the suggested procedure namely that of fitting individual enterprise
functions, it was decided to fit on: a:gr regate function to iata from the
entire farm and compare the results wit th those obtained from the enterprise
functions.

Regarding the crop functi :n, it was assumed that the aggregation of
data from various crop enterprises would not increase the intercorrelation
among the inde ese ndent variables as the major crop enterpri_ses on these farms

J-

are on approximately the same production function. Thus, an aggregate function
might still permit valid inferences about each particular enterprise includ-

ed in the crop function.

1. The Variables Included in the Two Li estock Functions

(X1) Dairy Gross Income was used as the dependentv riables in the dairy

 

function. It is determined in the Illinois Detailed Cost 3Accounts by sub-
tracting the sum of the opening inventory and purchases from the sum of sales,
closing inventory and value of dairy animals and dairy products consumed by
the household or fed to hogs. The dair" en' cerprise is also credited for the
manure produced by the animals.

(X1) Ho og Gr ros 3 Income was used as the dependent variable in the hog

 

function. It is determ ned inslcr0131r to the determination of gross income
in the dairy entergjrise by subtractin: the sum of the op hing inventory and
purchases from the sum of sa es, closing inventory and value of pork consumed

by the household.

 

3 This definition of Dafi y Gross Income is taken from Detailed Cost Report
for Northwestern Illinm , l9h9 and 1950, Department of A? ricultural
Economics, University of Illinois. Publication (AE2871) Ap ril 1952, p. 17.

 

 

 

56

The following variables were considered as independent and causal
variables in the analysis.
X2 Labor. The Illinois cost accounts give "direct" and "indirect"
labor hours spent in each enterprise. "Direct" labor includes time for
milking, feeding and caring for the animals while "indirect" labor includes
time Spent in repairing barns, milking equipment etc. For the purposes of
this analysis "direct" and "indirect" labor hours were added to obtain the
total amount of labor hours used in each of the enterprises. Although some
of the "indirect" labor is used in repairing investment items such as barns
and should properly be added to the appropriate investment categories there
was no possibility of making appropriate subtractions to eliminate this
type of labor from the indirect labor categories.

X Feed. Since feed is one of the most important inputs both in the

3

dairy and hog enterprises feed expenses were separated from other expenses

 

and the MVP of feed was estimated separately. The total value of the feed
fed (both home grown and purchased feed) is included plus a 13 cent charge
for each day the animals were pastured.

XLl Expenses. "Power and Machinery Expense", "Equipment Expense”,
"Building Expense”, ”Cash and other Expense", "Interest on Investment" and
"General Farm Expense" are items which are classified as cash expenses in
the Illinois as well as in most other traditional farm accounts. Since not
all of these items can be considered productive inputs for the purposes of

productivity studies machinery and equipment expenses as given in the Illinois
records were adjusted to eliminate the depreciation charges. The adjustment
was made on the basis of the depreciation rate used by each individual farmer

for his machinery. "Building Expenses" were excluded entirely because they

consist mostly of depreciation charges and/or expenses for improvements
which are properly classified as building investments since they prolong
the life of the building. "Cash and Other Expenses" were included while
"Interest on Livestock Investment and "General Farm Expense", the latter
including such items as taxes, magazine subscriptions etc., were not in—
cluded. It is believed that the resulting figure for cash expenses re-
flects more closely the actual productive cash expenses incurred in the
LL

different enterprises on these farms.

X Machinery Investment. Determination of Hachinery Investment in

 

5

each enterprise was not particularly dilliCUlt since the Illinois cost

1

data list each machine one its value in the machinery i:ventory.

Only beginning inventories have been used for this study. In two
cases where new tractors were bought during 1950 appropriate credits
were made. The values of the individual machines were taken from the ac-
count data at the beginning inventory value stated for them. This proced—
ure was follow d for all maelines except tractors which UCF: priced on the
basis of the used tractor value stated in the Farr Implement Bluebook for
19505. Admittedly these valu s are not entirely accurate because the actual
value of one and the same tractor type varies from farm to farm depending
upon the machine's condition. Nevertheless it is believed that tTis value

is closer to the actual value of the machine since it is not based on a

constant depreciation rate.employed by most farmers in determining the values

 

Appendix C shows formally how these adjustments were made.

National Tractor and Farm Implement Bluebook, Chicago, National Market
Reports Inc. Publisher.

 

of their machines for purposes of taxation. Determination of the share
of the tractor investment in each enterprise was done on the basis of the
number of tractor hours used in each enterprise. General farm equipment,
i.e. items which were not separately listed, were als, apportioned on this
basis.

Silo and silo equipment was included under "Dairy Equipment". Fences
and water systems were apportioned on the basis of pasture days which were

consumed by dairy cows and hogs reSpectively. No fencing charges were made

for crops. Tilage was charged to creps only.

 

X6 Livestock Investment. Livestock investment was determined on the
basis of the beginning inventory value of the animals plus proportional
allowances for animals sold during the year less proportional charges for
animals bought.6 No charges for forage investment were made since the value
of "Pasture Days" consumed was included in the feed variable.

X7 Housing. Following a suggestion made by”agley7 it was decided to
circumvent the difficulties of pricing farm buildings by using physical
capacity measures. One dairy housing animal unit was defined as 1500 cu.,ft.

This includes space requirements for one mature dairy cow plus replacement

. 8
-e animals. The

7
l

and storage space for a one year supply of feeds for t

 

If a cow was sold in February at $300.00, 10/12 of the beginning inventory
value of this cow was subtracted from the livestock investment. If a cow
was bought in July at $300.00, 6/12 of this value, filb0.00, was added to
the livestock investment.

7 'fiagley, R. Vance, Marginal Productivities of Investments and Expenditures,
Selected Ingham County Farms, 1952, pp. hSlho. Unpubl. M.S. Dissertation,
Michigan State College, 1953.

 

This estimate of the Space requirement is based on Farm Management Facts
and Figures, Department of Agricultural Iconomics, Michigan State College,
1953, pp. 68-69.

 

 

S9

dairy housing available on each farm included in the study was known in terms
of cubic feet of barn space. For hog housing no physical data were available,
consequently the values of permanent and temporary hog housing as indicated

in the building inventory had to be used.

2. The Variables Included in the Crop Function

It was pointed out earlier that it is possible to fit an aggregate
function to a number of enterprises if these enterprises can be assumed to
be on approximately the same production function and if the aggregation of
inputs from several enterprises does not cause the intercorrelation among
the independent variables to increase. This assumption was made for the crop
enterprises on the 27 farms and inputs as well as outputs from each crop
enterprise were aggregated and one functionvas fitted to them. Almost all
of the farms in the sample raised corn (both grain and silage), oats, and
legumes as their major crops. Other crops were soybeans, barley and wheat.
Pasture was not included in the function because no input data for permanent
pasture were available.

The following variables were included in the crop function:

 

X1 Gross Income from Crops. The yields obtained in each enterprise
were multiplied by the appropriate prices which were stated by the farmers
in the sample to have prevailed in their market. Thus all crops were valued
at market prices.

The following variables were treated as independent (causal) variables:

X2 Labor. As in the case of livestock he records indicated the num—
ber of labor hours which were used "directly" and "indirectly“ in each crop
enterprise. The total amount of labor used in all the crop enterprises was

thus easily obtained through addition.

 

60

X3 Lapd. Land was measured in acres rather thaL in monetary terms.
The total number of acres in crops was obtained by adding the number of
acres used in each enterprise. It should be pointed out here that not
all land was of the same quality. Thus the resulting VVF estimates apply
to the average quality land included in the sample.

Xh Expenses. Regarding crop expenses considerable adjustments were
made in the original data by applying a "depreciation factor" to eliminate
machinery depreciation from the expense categories. Building expenses and
general farm expense were left out entirely because they contain mostly de-
'preciation charges or nonproductive expenses such as taxes, interest charges

4.

and magazine subscriptions.

r7

XS Machinery Investment. -ne value of crop machinery and equipment

 

had to be determined on the basis of the values given in the machinery in-
ventory since neither age nor make of these machines were known to perrit
the use Farm Implement ?luebooks in determining their values. The values of

tractors on the other hand for whtch this information was available were

I

1

16 help of Tara lrnlsment Dluebooks. The cron share of the

ct

determined with
tractor was computed on the basis of the number of hours the tractors were

used in crops.

/{

.,
I

The variables Included in the Aggregate Function.

LL)
0

f)

In order to furnish an additional basis for an evaluation o~ the in—

0. fl

dividual enterprise fun tions it was decided to fit an aggregate lunction
to data from all the enterprises on the 27 farms.

In this a; re ate function (T ) was used as the dependent variable. It

H
(D
"U
"5
(D
U)
3

d"
U)
*‘5
O
O
b
O
(7
1
.3
3

_J
..J

I

(
O
l

=2: received from dairy, income received

 

 

61

fro hogs and M.:comc received from cash crop sales.

The following variables were treated as independent in the spare? ate
function: (X2) labor, including all the la nor hours spent ”directly” and
"indirectly" in dairy, hogs and the various crop enterprises; (X ) land,
measured in acres and including all crop acres used in producing feed and
cash crops (pasture acres were not included si11ce no ingut — outptt date
were available for pasture; a pa s1;ure charee has made, however, under ex-
penses); (Kb) cash expenses, includinp all productive cash expenses in
livestocx and crops, feed purchased and pasture consumed but not feed pro-
duced on the farm (non [reinct“ve e:;”e“s s C:clrded tinclude, taxes, insur—
ance, and depreciation); (:§> maclinerf invrstnent, including the value of
machinery ande Alitment used in livestock anl crop enterprises; (K ) livestock
investment including dairy cows, bulls and breeding ho , (K barn space and
space in permanent he: housed”, measured in dairy hOU_Siu 2 animal unitsgsince
no physical data for housinf were avzu -la :le in the cas or hogs the building
inventory value indicated for hogs was converted to<iairy housine animal
united using the rate of 1 Housing Animal Unit 3 fih03.00. This is the rate

.

v - O . 1 i "h .
Whlch Nagley’ determined as tie es;abli§1ncnt 035p 01 one d iry housinp

animal unit.

 

7‘)

Q " o
’ Wagiey, Vance, Op. Cit. p. 82.

62

CHAPTER VI

INTERPRETATION OF FOUR PRODUCTION FUNCTIONS FITTED TO AGTERPHISE
INPUT-OUTPUT DATA FROM 27 DAIRY-HOG FARMS IN
NORTHTISTEPN ILLINOIS

The present chapter examines results obtained from the statistical
analysis of input-output data from 27 multiple enterprise farms in north-
western Illinois. Four separate functions were fitted, one for the hog enter-
prises, one for the dairy enterprises, one for the combined crop enterprises
and one aggregate function for the entire farm.

In Part A of this chapter, the regression coefficients from each func-
tion and their economic meaning are discussed independently of the other
functions fitted. In Part B the functions and the regression coefficients
for like factors are compared. Inter-enterprise comparison of coefficients
is the method by which the firm is brought towards internal adjustment. If
each factor is used to equal advantage in the various enterprises, a maximum
profit from a given amount of resources will be achieved.

In Part C the results obtained from the individual enterprise functions
are compared with results obtained from an aggregate function in order to

evaluate the methodoloa suggested and emplored in this thesis.
g. -5) .L 3

 

63

A. Inter-Enterprise Comparison of Regression Coefficients

 

and Marginal value Productivity Estimates

0...

l. The Hog Function

a. The regression coefficients and the MVP estimates. The function fit-

 

ted to the hog enterprises of the 27 farms included in the sample yielded the
regression coefficients and marginal value productivities, estimated at the
geometric mean, indicated in Table I.

The regression coefficients reported in column 2 of Table I are the
elasticities of gross income with respect to each of the six inputs or in-
vestments which were used as independent or explanatory variables in the

1 T‘\ n n r‘. 0 v c o
mach one 01 these coeiiiCients indicates the percen-

regression analysis.
tage change in gross income associated with a one per cent increase in the
independent factors. Thus, if the feed input in the hog function were in-
creased by one per cent, the hog output would be expected to increase

.702hl7 per cent.

The areas under the normal curve corresponding to the "t" values in
column 3, indicate the probability with which it can be assumed that regress-
ion coefficients obtained from a ifferent sample of the same population would
be as large or larger than the estimated regression coefficient if the true
population coefficient were zero. Thus, a "t" of .066780 indicates that a

regression coefficient as large as or larger than .OOSblB would be obtained

 

l

The following is a simple proof that the regression coefficients in the
Cobb-Douglas function are the elasticities of gross income (Y) with re—
Spect to the independent variable (Xi):

ASY: v Y

.. Z _ “ia since AY bub ..

let e -.2§7r7 " YF-:f- -Zr—;— = LVPX. = -3%:- it follows that
1 AA. Ai ‘ l iii

J o l -

‘ . CC?"

- JClbj-JJJ. — 1
v X Qi

 

 

 

 

 

 

6h
TABLE I
REGRESSION COEFFICIENTS AND MARGINAL VALUE PRODUCTIVITY ESTIMATES
OBTAINED FROM A COBB-DOUGLAS FUNCTION FITTED TO HOG
ENTERPRISE DATA FROM 27 ILLINOIS FARMS, 1950
(l) (2) (3)

Production Regression coefficient t value MVP at geometric
factor mean
Labor (hours) .005o18 i .08u127 .06o780 Mean .0n53

Upper .72h5
Lower -.6338
Feed (dollars) .702A17 i .131202 5.353706 Mean 1.0087
Upper 1.1970
Lower .8203
Cash +
expense (dollars) .106hl3 - .083u7h 1.27uo0u Mean 2.6n26

Upper b.7100
Lower .5696

Machinery (dollars) .O39h36 i .053057 .7h3276 Mean 1.3693
Upper 3.2113
Lower -.h73O

Breeding

invest. (dollars) -.0592os i .097u10 .608u08 Mean -.3un3
Upper .2216
Lower -.9lO3

Housing (dollars) -.0028h8 i .018586 .15323A Mean -.0875

Upper .u839
Lower -.659l

 

66

in 95 out of 100 cases if functions were fitted to different samples from
the same population and the true population regression coefficient were zero.

Of greatest interest are the marginal value productivity estimates re-
ported in column h. The MVP's are a direct function of the bi and indicate
the return which might be expected from the last unit of each of the inputs
used in the hog enterprise, ceteris paribus. The marginal value productivi—
ties reported here are estimated at the geometric mean of gross income and
the inputs in the sample. Upper and lower confidence limits for the MVP'S are
computed to correspond to the upper and lower confidence limihsfor the re-
gression coefficients.2 It should be mentioned that MVP's can be estimated
for any size of the input and output. For purposes of estimating MVP‘s for
any particular farm, values of Y and Xi appr0priate for this farm have to
be used. Such a comparison requires (1) that the farm be within a proba-
bility distribution about the same production function as the farms in the
sample and (2) that the data are not outside the range from which the func-
tion was estimated.

The "t" values in column 3 indicate that the regression coefficients
for labor, machinery, breeding investment and housing are not significantly
different from zero at the 68 per cent level. The relatively large standard
errors are caused mostly by large intercorrelations of these variables with
other independent variables.

The MVP estimate for labor at its geometric mean indicates thatthis

factor returned about 5 cents per hour. This MVP varies from between 72 cents

 

2 The upper and lower confidence limits of the MVP are computed using
[bfL io'bngOf)
IWP = X o
i

 

 

67

per hour when estimated at b2 +<Tb2 to -63 cents per hour when estimated
at b2 -C7b2 . Obviously the mean estimate is very unreliable. The situa-
tion is similar for the estimated MVP'S of Machinery Investment, Breeding
Investment and Hog Housing.

The regression coefficients of feed and cash eXpenses are significantly
different from zero at the 75 per cent level. The corresponding MVP's esti-
mated at the geometric mean.are 1.0087 dollars for feed and 2.6h26 dollars
for expenses. The MVP estimate of feed is not surprising since one would
expect 1.00 dollar of feed input to return at least one dollar. The estimate
for cash.expenses indicates that at the geometric mean a dollar spent would
return 2.6h dollars. Several factors aside from actual productivity of cash
eXpenses on these farms might have caused this relatively large MVP; the
following two appear to be the most likely causes: (I) the amount of cash
expenses other than feed is relatively small in the hog enterprise, therefore,
small errors in computing this figure might have affected the MVP estimate
considerably. (2) Cash expenses (Xh) are very highly correlated with both
feed (X3), r3h = .85, and labor (X2), r2h = .68, and might thus reflect some
of the earnings of these categories.

b. Testing the regression coefficient against regression coefficients

 

which would yield minimum MVP's. The only test of regression coefficients

 

'dixmwsed so far tests the bi against zero. A more meaningful test from an
economic vieWpoint compares the bi of each input or investment against a

"standard" bi*, capable of yielding a minimum or reservation MVP. The stan—

_ b E(Y)
dard bi* is obtained by solving the equation MVP = __EX____ for bi after

i

the required minimum MVP has been determined. The MVP's assumed to be mini-

68

mum expected returns of inputs and investments in hog enterprises on northern
Illinois farms are given in Table II, column 1.

The resulting bi* necessary to yield these F 's on Illinois hog farms
are indicated in Table III, column 3. In column h, the estimated bi is sub-
tracted from.bi*. Column 5 indicates the size of this difference in terms
of the standard error of the bi' Column 6 indicates the probability with
which the biw-would be eXpected to assume that or any value farther away
from the estimated bi if the latter were the true population regression
coefficient. It appears from this test that when compared at the geometric
mean only the regression coefficients of labor and.livestock investment are
significantly lower, at the 68 per cent level, than the corresponding bi*
yielding a minimum MVP. Consequently, it is concluded that a reduction in
the use of these two inputs in the hog enterprise could be suggested.

While the farms appear to be in relatively good adjustment with reSpect
to the remaining inputs if the comparison is made at the geometric mean, this
does not necessarily hold for all the individual farms in the sample. Con-
sequently, individual farms have been selected and their organization has
been compared with an organization which would yield minimum or reservation
MVP'S for the various inputs aid investments. Table III a makes a compar-
ison using a faivn'whose hog enterprise appeared to deviate considerably
from the scale lxhae adjustment. The regression coefficients of labor and
feed in column I appear to be significantly lower, on the 80 per cent level,
than the bi* necessary to return a minimum MVP for these two inputs, indicat-
ing that too much labor is used and too much or too expensive feed is fed to

hogs on this farm.

69

TABDE II

MVP'S FOR VARIOUS INPUTS AND INVESTMENTS WHICH ARE
CONSIDERED MINIMUM EXPECTED RETURNS OR

RESERVATION PRICES IN ILLINOIS, 1950

 

 

Input or investment ; (l) ; (2) ; (3)
category . Hogs : Dairy : Crops

Labor : $200.00/month = $200.00/montn = $200.00/month
= = 7h¢/hour = = 7h¢/hour = = 7h¢/hour

Land (acres) 3 ----- 3 ----- 3 6%% = $1h.501
3 3 ' per acre

Feed (dollars) : 1.00 ‘ 1.00 i -----

Cash 3 ‘ ~

expenses (dollars) i l.OO ' 1.00 . 1.00

Machinery —

invest. (dollars) : 20% 20% ' 20%

Livestock ~

invest. (dollars) _ 50% 35% — h5% -----

Housing (dollars) 5% - 8% 5% - 82 -----

 

This figure is based upon an average land value of $223.00 for the sample.
Source: Illinois Schedule of Land ValuestmrSoil Productivity Ratings,

Department of Agricultural Economics, University of Illinois, 1952.

 

 

 

70

 

 

 

 

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I2.

72

The forego: Ing an wl sis sugrests that comparisons made at the gefnetric
mean do not necessarily give reliable information regarding the adjustments
of individual farms or the statistical and. econorj_c Siénwficaqce of produc-
tivity estimates at the geometric :nean; however a. stati st:ical comparison be-
tween the MV;P 's on a particu.lar farm and rni ini1v um or reservation NVP'. showed
significant differences with respect to a number of variables for which the
comparison ati:be geometric mean succested no seri ..... s malaoqustoent This
result is not surprising i. one considers that the farm at the geometric mean
tel {en from. a moderately representative sample should be expected to be re-
latively‘well adjuste eo.

c. The errors of the regression line and the coeffic ie . sof multip_e

correlation and deterrination. Regarding the error of the repression equa-

q

tion and the relationships oet ween. dependent no. independent variables, the

fMll owing qu -an tities have been corr‘te'°

The standard error of estimate Pal . 667 was cornuted to .066573 which
c .QlL ’

V

indicates in logarithms t1e size of the standard error of hog output when

a
estimated from th e indepen nd.ent variaoles 32,....,X7.“ prressed in natural
numbers, this meass that the probability that the mean of he true popu-
lation income falls in the interval fih,8h3.10 to $6,73b.00 is .68. She
geometric mean of Kl in this sample was $5,6h5.fi
The adjusted coefficient of multiple correlation R1 21hg67was .950682 while

—21.23h567the Coef1.icient of Multiple Determination was .903796 indicating

 

3 Exekiel, Mordecai, Methods of Correlat ivn Analysis, New John Wiley

 

and Company, Second Edition, 1953, p. 208.

73

that 90 per cent of the variation in the dependent variable X1 is associated
with variations in the independent variables X2,....,X7.

An evaluation of the hog function with the aim of making specific recom-
mendations to the farmers included in this sample or to farms working under
similar production conditions suggests that the farms are relatively well
adjusted as long as the comparison is made at the geometric mean. Only the
productivity of labor is significantly below the wage which has to be paid
to a worker in agriculture. If, on the other hand, individual farms are
investigated regarding their organizations it appears that a number of farms
could be found whose organization deviates considerably from an optimal one.
Consequently, it should be emphasized that in evaluating Cobb-Douglas pro-
ductivity estimates comparisons of individual farms reveal more than com-

parisons made at the geometric mean.

2. The Dairy Function
A function using the same independent variables as the hog function was
fitted to input-output data from the dairy enterprises of the 27 farms includ-
ed in the sample. The only major difference between the two functions was
that dairy housingifis measured in physical terms while hog housing was meas-
ured in terms of it‘s inventory values indicated by the farmers.

a. The regression coefficients and the 1"1V? estimates. The regression

 

coefficients which were obtained for the variables X2,....,X7 as well as the
marginal value productivities, estimated at the geometric mean, are presented
in Table IV. The area under the normal curve correSponding to the t values

in column 3 indicates the probability with which it can be assumed that re-

 

TABLE IV

7h

REGRESSION COEFFICIENTS AND VARGINAL VALUE PRODUCTIVITY

ESTIMATES OBTAINED FROM A COBB4DOUGLAS FUNCTION FITTED

TC DAIRY ENTERPRISE DATA snow 27 ILLINOIS FARNS, 1950

 

<1) i <2) i (3) (u)
Input or invest- : Regression ° Marginal value
ment category : coefficients t values products

j 3 Mean .6867
Labor (hours) j .221717 i .1u1106 : 1.571280 Upper 1.12bo
‘ ’ Lover .2h99
: Mean 1.0187
Feed (dollars) .730h61 i .192u97 . 3.79h662 Upper 1.2829
' Lower .7h78
Mean - .lhO?
Cash expense -.01L169 : .lSOhh3 .O9h18? Upper 1.3532
(dollars) Lower -l.63h6
. Mean .hBIO
Machinery .Oéléhé 1 .096077 7 .6b1631 Upper 1.1026
(dollars) Lower - .2hO7
Livestock Mean .02h5
investment .009030 : .098352 .0913h9 Upper .2922
(dollars) Lower — .2h33
Housing : Mean 2h.6796
(dairy housing : .125801 1 .128098 ; .982068 Upper h9.8100
animal units) . Lower - .hSOO

 

 

75

gression coefficients obtained from a different sample of the same pop—
ulation would be as large or larger than the estimated regression coefficient
in column 2 if the true population coefficient were zero.

The marginal'value productivities estimated at the geometric mean are
reported in column h. Three values of the MVP are computed, one at the
mean value of the regression coefficient bi’ one at bi +(Tbi representing
the upper limit and one at bi -<Tbi, representing the lower limit. The
choice of plus and minus one standard deviation for computing the upper and
the lower limits reSpectively is quite arbitrary. If a 95 per cent rather
than a 68 per cent confidence interval is desired the lower and upper limits
would have to be recomputed at plus or minus l.96(7bi.

The regression coefficient of labor is significantly different from
zero at the 88 per cent level owing to the relatively small error associated
with the estimate. The MVP of labor in dairy estimated at the geometric mean
is 69 cents per hour or 186.30 dollars per month.

The t value of 3.79h662 associated with the regression coefficient for
the feed variable indicates that this regression coefficient is different
from zero at the 99 per cent level. The MVP of this input estimated at the
geometric mean indicates that one dollars worth of feed returns 1.02 dollars.

The only other regression coefficient significantly different from zero
is that of dairy housing. The "t" value of .982068 in column 3 indicates that
a regression coefficient of .125801 or larger would occur in 67 out of 100
cases if functions were fitted to different samples from the same population
and the true population coefficient were zero. The MVP of housing estimated

at the geometric mean indicates that one ”dairy housing animal unit" returns

7o

2h.68 dollars. Assuming that the replacement cost of one "dairy housing
animal unit" is about hO0.00 dollars, this MVP would correSpond to a return
of 6 per cent on the investment in dairy housing. The estimated percentage
returns would increase correSpondingly as one lowers the value of one hous-
ing animal unit to adjust for the age and condition of the buildings. The
regression coefficients for cash expenses, machinery investment and live-
stock investment were not significantly different from zero as large errors
were associated with these estimates. Regression coefficients, ”t" values
and MVP'S of these inputs are indicated in columns 2, 3, and h of Table IV
respectively.

b. Testing the regression coefficients against regression coefficients

 

which would yield minimum HVP'S. As in the case of the hog function, the bi

 

were tested against regression coefficients which would yield a certain
minimum expected MVP. These minimum or reservation returns have been pres-
ented in Table II column 2 above. \The bi* necessary to yield these minimum
marginal value products are presented in column 3 of Table V. Column h of
Table V presents the difference between the bi* and the estimated bi' Column
restates this difference in terms of the standard error of the bi’ The
area under the normal curve corresponding to this standardized difference
indicates the probability with which it can be expected that a regression
coefficient which differs from bi by more than bi* - bi would occur if bi
were the true correlation coefficient for the population. It is seen that
only b6 the coefficient of livestock investment can be eXpected to be sig-

nificantly different from b6* (at the 80 per cent level) which makes these

farms appear to be well adjusted when they are analyzed at the geometric mean.

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As before a farm whose organization appeared to deviate considerably from the
scale line adjustment has been selected to determine if the sample regression
coefficient bi differs significantly from the bi* yielding a certain minimum
or reservation MVP on this poorly adjusted farm. The comparison is presented
in Table V a, and it indicates that on this particular farm the MVP's of sev-
eral important variables differ considerably from the corresponding minimum
MVP'S. On the particular farm which was chosen for comparison it appears
that the use of labor in dairy should be reduced while the use of feed, for
which the MVP is high, could be increased or more expensive feed mixtures could
be used. The MVP of dairy housing is significantly below the MVP which should
be expected from one dairy housing animal unit suggesting that an over-invest-
ment in buildings exists on this farm.

Again the conclusion is reached that the analysis of individual farms
on the basis of the estimated production function reveals considerably more
about the location of these farms with respect to the scale line than do the
comparisons which are made at the geometric mean.

c. The errors of the regression line and the coefficients of multiple

 

 

correlation and determination. The Standard Error of Eatimate €1.23h567
was computed to be .088679. It indicates in logarithms the size of the
standard error of dairy output (X1) when estimated from the independent
variables X2,....,X7. Expressed in natural numbers this means that the
probability that the geometric mean of gross income in the population falls
in the interval so050.9o to $9102.75 is .68. The geometric mean of gross

income in this sample was $7h21.50.

 

 

 

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80

The Adjusted Coefficient of Multiple Correlation Rl.23h567 was com-
puted to be .87658h while the coefficient of multiple determination
-2
R 1.23h567 was .768hOO. The latter measures the amount of variation in
X1 which was associated with the independent variables included in the an-
alysis. Again the omission of management, a factor which is very important
in dairy farming, might be a serious diortcoming of the presently used input

classification. However, no satisfactory measure of management is available

so far.

3. The Crop Function

The function fitted to data from the crop enterprises of the 27 Illinois
farms is an aggregate function in the sense that inputs used in several crop
enterprises as well as outputs produced by them have been aggregated and one
function was fitted to the resulting data. Crop output (X1) valued at market
prices was the dependent variable, Labor (X2), Land (X3), Cash expenses (Xh)
and Machinery investment (X5) are the explanatory or independent variables.

a. The regression coefficients and marginal valuekproductivigy estimates.
The regression coefficientswhich resulted from the crop function are indicated
in column 2 and the marginal value productivi y estimates are shown in column
3 of Table VI. The test against zero indicates that only no regression co-
efficients namely those of land aid machinery investment are significantly dif-
ferent from zero at the 68 per cent level. The regression coefficients for
labor and expenses are not significantly different from zero indicating that

large standard errors were associated with these estimates.

 

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TABLE VI

 

81

REGBESSIOH COEFFICIENTS AND “AEOINAL VALUE PRODUCTIVITY ESTIMATES

OBTAINED FRTF A CORY-DOVGLAS FUNCTION FITTED LO

CROP

"Y‘. ‘m*'-n.j" " \
lay-1 l Lulu-.’RISL

DATA FROM 27 ILLINOIS TAPES

, 1950

 

 

 

 

(1) 3 (2) 3 (3) : (h)
Input or : Regression : : Marginal value
investment : coefficients : t value : products
; : - :
I Mean .7182
Labor (hours) : .128762 1 .166021 .773??? Upper 1.6hh9
. Lower - .2078
. Jean 3h.6680
Land (acres) 2 .5326F6—1 .1hll29 3.77h2h7 Upper h3.8§37
Lower 25.h826
: Mean .hBOF
Crop expense .128;77 3 .218881 .588801 Upper 1.1616
(dollars) Lower - .3006
* Jean .1830
Machinery : 008h76 - .007,92 1.011130 Upper .36h0
(dollars) Lower .0020

 

._ " ._._--..

82

b. Testing the regression coefficients against regression coefficients

 

yielding a minimum MVP. Again as done previously with the estimates of

 

the livestock functions the estimated bi in crops have been compared sta-
tistically with the corresponding bi* which would yield certain minimum or
reservation returns. These minimum returns considered applicable for crOp
farms in northern Illinois in 1950 are indicated in column 3 of Table II
above. The resulting bi* necessary to yield such minimum hVP's are indi-
cated in column 3 of Table VII. The difference between bi* and bi is
presented in column A. Column 5 states this difference in terms of the
standard error of the bi. Only the regression coeffiCient of land is signif-
icantly higher than the bi yielding a minimum return suggesting that invest-
ments in land of the quality found on the average farm in the sample can be
very profitable.

Again a farm has been selected whose crOp organization appeared to
deviate from the geometric mean organization and the estimated bi have been
compared with a bi* yielding a minimum or reservation MVP on this farm. This
comparison is presented in Table VII a. On the particular farm which was
selected the MVP of land is significantly higher, at the 98 per cent level,
than the minimum MVP eXpected for land in this area, indicating that an addi-
tion of crop acres on this farm might be profitable. Regarding the other
factors, it appears from the t test that this farm deviates considerably more,
although not significantly at the 68 per cent level, from an optimum.adjust-

ment than did the farm with the geometric mean organization.

83

 

 

 

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V9

85

c. The standard error of the regression line and the coefficients of

multiple correlation and determination. The Standard Error of Estimate

 

§1.83h5 of .080566 indicates in terms of logarithms the size of the standard
error of crop output Xl when estimated from the independent variable X2,....,X5.
A standard error of this size indicates that the probability of the true geom—
etric mean of the population gross income falls in the interval $5,713.60 to
$8,278.80 is .68. The geometric mean of gross income in this sample 96,996.20.
The Adjusted Coefficient of Multiple Correlation E12315 was .832389
while its square the coefficient of multiple determination R21.23h5 turned
out to be .692871. The latter measures the amount of variation in X1 which
is associated with the independent variables X2,....,XS. It is seen in this
case that 69 per cent of the variation in the dependent variable is associa-
ted with the independent variables in the analysis. The omission of crop
storage as an explanatory variable might have caused this coefficient to be
considerably below the coefficient of multiple determination of the other two

functions.

86

B. Inter-Enterprise Comparisons of Production

 

Finctions and Regression Coefficients

 

Up to this point, the three production functions have been analyzed in-
dividually and nothing has been said about the possibility of shifting part-
icular production factors from one enterprise to another one or expanding one
enterprise in favor of another one. Since determining optimum inter-enter-
prise resource allocation is one of the major purposes of analyzing multiple
enterprise farms, this section will be concerned with comparing the three
functions in general and the marginal value productivity of like factors in

particular.

I. Comparison of the Slopes of the Production Functions

It has been pointed out in Cahpter II that fer a power function such
as the Cobb-Douglas, the sum of the regression coefficients detennines
whether the function shows increasing, constant or decreasing returns de~
pending upon whether thEgi bi is greater than, equal to, or smaller than one.
For the economist, increasing returns to scale mean that each additional unit
of production factors X2,....,Zn (combined in scale line proportions) returns
more than the previous unit. Increasing returns to scale must always be
uneconomical because they imply that some other production factor not in-
cluded in the analysis yields negative margiral returns."L Constant returns
to scale mean to the economist that each additional unit of production
factors X2,....,Kn (combined in scale line proportions) returns the same

amount as the previous unit; in such a case profit also increases uniformly.

 

Lerner, Abba P., The Economics of Control, New York, the Machillan Company,
l9h7, pp. 155-56.

 

 

87

I“)

Decreas ing returns to scale teen that seen a QltiOnP unit 0
factors 32,....,Xn (corbined in scale line propo. fiions) returns less than
the previous unit. In agriculture WHLTG the law of diminishing returns is
expected to come into play fairly quickly +91: 19 the only part of
1 . 1‘

diction luncticn alone which an entrroci e whould be operated.

_)

In order to prove t:at a certain sum of refresrion COGlllClSDLS is

O
—.. .HCC

(0'
tw
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J

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D

_: o p o _ -L ‘_ _ .L _ / _'_1 f 1" ‘ .L 1 .A _'_
signiiicanolj greaocr KSfiullw7> than oh-, a see

whether he sum of the re resrion coefficients in a particular sample is

S I
I.)
(4
9
r3
ioJ
C+
4
{i
(D
<1
(D
1...:
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FL

significantly larger than one. Clli Ol Licnigan State

a test usin: the T-statistic w"ic1 “or“its statiStical testing of theibi

l"
.‘3
V n- _‘ r‘- J. L eh r : D - arlnl'
again an* COASUaflC C.’ :LC test as ell as ‘ e ricianios o- o 'wteoien

8
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Cl-
(0
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5

' ”'0: _' .e-- ‘- .'. -.. 4-- h . '.... . ‘T - :_ .. ..
TGET€5330 N? COCZLIOIGIVS mein~ areted afai st 1. none of ;ee sunr o

' ”'1' O I - _o_fio ‘-__ — I. .. 1‘ ‘ ’- V -- H- ‘
regreSCion coeiiiCient: were Flffill103»blf differeih from one. For tne
daivv functioniib. was coral to l .l: .tn, the eo‘w ted U ratio wes

u r ; a _, .-- . i , H _ .
.L

1.09229h which was below the ff CC\ 21 of 2.27. for the hog function
/_j 1. e

\ O
.L, A P’I-‘v 1“.“ --1 _ ”‘1‘ c . Or, q‘ \ ‘11-. F1, "1 . J. 4' '1 ,_ 0 ., .L
the la; Lure .7;J_({l, sin? A ixioia inn? ._;,igll In i;fl.lmlefl t€mitet EUteQSu

{(4.
,3

l—J.
U)

.’3

O

C

hog functo significantly diffrrent from 1.

 

S Olkin, Ingreri, ”Unpubl isLed report about a problem in testin g sums of
regression coefficients of linear nrlti‘le regreSLLQn lines agaiist a
T‘ ‘ ' tiC'l “o p of :he

3 o _ _ _ _ ‘ _ . M_‘_ 1 'n ‘_ _g_1r _ -_-o

conStant”. ills report has been nude or the stat;
- . 4-, .L Y‘ r .0. , . a-.- 1

mathematics rear -cnt to rrotes.o“ Zlv-- I, -

\vo . ‘ ~.- 0
1

O
o w - Q ~ ' vs" -r- (1““f‘“ ~ If - T 11 .J--r~,‘
Agricultural Scone ice, Michigan “CQCC IHUJCtrlU .

88

For the crop functionwi bi was .888772, the F ratio was .6982h8 indica-
ting that the lepe of the crop function is not significantly different from 1.

Thus, it is concluded that constant returns to scale prevail for the
three functions included in the analysis.

In addition to testing the sum of the regression coefficients against
one the 2 bi of dairy (1.13hh86) was tested against thei bi of hogs (.791771).
The resulting F ratio was 7.093510 which indicates if compared with F(.9S),

(1 21) = h.33 that the slope of the dairy function is significantly differ-
’

ent at the 95 per cent level from the slope of the hog function.

2. Comparison of Individual Estimates
It was Shown previously that the reallocation of resources between en-

terprises is complete when the ratios MVPX./ MFC are equal in the differ-
i

Xi

ent enterprises.

Direct comparisons of MVP'S are difficult to make because the errors of
the regression coefficients and thus also the errors of the MVP's differ from
enterprise to enterprise. To circumvent this difficulty, it was decided to
assume minimum values for the marginal factor costs and then determine in each
function that bi% which would have yielded a. MVP equating the ratio

MVPX / MFCX . Using the t test it can be determined whether the estimated
i l

bi is different from bi*. By this method, regression coefficients obtained
in several enterprises become comparable even though they assume different

absolute values and have different standard errors attached to them.

 

 

89

The comparison is carried out in Table VIII. Column 1 lists the various
production factors used in the three enterprises. Column 2 states the mini-
mum MVP which should have been expected on Illinois farms in 1950. Sub-

column a, b and c show the values for bi, b.* and the t values for each of

i
the three enterprises. t values below "one" indicate tlat the bi* is not
significantly different from bi at the 68% level. Thus, when the comparison
is made at the geometric mean there are only two input categories which sug-
gest possible readjustment between enterprises. The first is the bi* for
labor employed in hogs which indicates that less labor should be used in the
hog enterprise. The second is the bi for land in crops Which indicates that
the application of land to grow crops is highly profitable and should pro-
bably be expanded.

There appear to be two reasons why the foregoing comparison of regression
coefficients of like factors between functions did not suggest many signi-
ficant changes. (1) The geometric mean organization of a sample of farms not
purposively chosen is expected to be fairly close to the scale line adjust-
ment. (2) The errors of most regression coefficients were large causing large
confidence intervals for regression coefficients and marginal value produc-
tivity estimates.

A comparison of MVP's for one particular farm has not been undertaken,
however, it is exiected that here too, several.farms could be detected on

which significant inter-enterprise reallocations of factors would be required.

 

6

For detailed description of how these values are determined see Appendix E.

 

u
'

AmmeHopv

 

 

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. a r i. Pcoebmobg.“
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u h , m ‘ mmnmflflopv
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H.HHBHQ muses>HBQDDCfim flggdb

B<LHBmfl Q<SQH>HQEH m9 HQmHm<gon

91

C. Comparison of the Three Enterprise Functions

with the Aggrggate Function

 

In order to evaluate the methodology suggested in this thesis (namely
to fit separate functions to individual enterprises) an aggregate function
was fitted which employed the input-output data used in the three enterprise
fits without regard to the enterprise with which they were associated. Re-
gression coefficients as well as other statistics were then compared with

those for the three enterprise functions.

1. The Estimates of Regression Coefficients and Marginal
Value Productivities in the Aggregate Function

Regression Coefficients and marginal Value Productivity estimates at
the geometric mean for the aggregate function are presented in Table IX. The
regression coefficients in column 2 are the elasticities of gross income with
respect to each of the 6 independent variables listed in column 1. Column 3'
lists the "t" values correSponding to each one of the regressioncpefficients.
Areas under the normal curve correSponding to the "t" values indicate if the
regression coefficients are significantly different from zero. The coeffi-
cients of land and expenses are significantly different at the 95 per cent
level, the coefficient of labor is significantly different from zero at
the 88 per cent level while the coefficient of livestock investment is sig-
nificantly different from zero at the 70 per cent level. The remaining co-
efficients are not significantly different from zero at the 68 per cent level.

The marginal value productivities estimated at the geometric mean in-

dicate that labor returns 70 cents per hour, land y33.80 per acre, eXpenses

REGRESSION COEFFICIENTS.AMD MARGINAL VuLUE PRODUCTIVITY ESTIMATES
OBTaINED FRQN’A COBB-DOD

AGGREGATED DLTA.FROM 27 ILLINOIS

TABLE IX

(1

J

7—}

Finns, 1950

' .”.CTICH FITTED TO

 

 

(1) (2) (3)
Input or Regression Larginal value
investment coefficients t values products

Mean .7056
labor (hours) .228812 f .1h6651 1.5602hh Upper 1.1579
Lower .253h
Land (acres) .2‘89lh : .121503 2.130926 Upper h9.6626
Lower 17.9387
Vean 1.2965
Ekpenses .509587 f .175216 2.90833h Upper 1.7u23
(dollars) Lower .8507
Ma chinery Jean . 20014
:investment .075081 i_.09282h .80885h 'tpper .hh8l
Giollars) nower .0h73
LiVestock :eai .38}?
llnrestment .110b22 + .093060 1.126071 cppcr .7362
«dollars) - 'ower .CflBé
U . H'ean S.l7hO
nOUS131g .01u59u i .087396 .166992 Lpper 36.1560
(animal unit 5) Lower ~25 . 8085

x

$1.29 for each dollar Spent, machinery investment 20 per cent, livestock
investment 39 per cent and housing investment 35.17 per housing animal unit

which is equal to 1.29 per cent if one housing animal unit is valued atth0.00.

2. Comparison of Marginal Value Productivity Estimates in the Aggregate
Function with the CorreSponding Estimates in the
Individual Enterprise Functions

Table X compares the MVP estimates obtained in the aggregate function
'with the corresponding estimates in each one of the individual enterprise
functions. The coefficients of "expenses" are not comparable since this
variable in the aggregate function necessarily included feed bought while in
the livestock functions feed was considered as a separate variable.

‘With respect to the labor variable, the returns estimated in the aggre-
gate function are the same as those estimated in crops and dairy. However,
the low returns which this factor earned in hogs are not reflected in the
estimate obtained through the aggregate function.

The returns to the land variable is the same in both the aggregate and
the crOp function. The standard error of this estimate is considerably smaller
in the crop function indicating that the aggregation had a negative influence
upon the reliability of this estimate.

Returns to machinery investment vary widely in the four functions. It
appears that the estimate for this variable in the aggregate function is a
weighted average of the returns estimated for this input in the three individ-

ual enterprises.

 

 

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memeflpmm: how mpcmpw =9: mpcmeHmmooo cowmmcummh one mo :mram> cmoe one pm omeapmms pom mesmpm =5:

 

9h

 

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x aflm<a

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The estita es for livestocfi investment and ani”al housing cannot be
considerei separatel“. -t is easil' seen that the estina es var moth vari-

ables are iL complete disarrsement with the correcdond‘r” estirates ootainet

N
6
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in the livestock functions. In the a¢5;~“

stock investment are ver" hich (75 per cent) while the retrras for housing
m‘,‘ . -,«-- ' J- .' . J- - .- ° . 1.1 " _‘ ,i .0 1m .1. ° . i .. 1-1 _ .1- - . _'_
are low. lue oppos_cc 15 tr e IQ the cair; lHanlOn Where .ne ret‘-ne to

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.3
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dairy investment are unusuallf low Uh le t1 _

to dairy housing are hirh.
- '1 - i _._ 7 a: 4 .'— -., —-. ti: v. 1-: -' . -. .L - .1. v .,
It is very limelj that'tie hi(h inCELCOTLQldLlTA <-1st1af between phase two

1

ables caused tie resulti ; estinates in both f he ions to oe unreliahl;

<1
{—1.

er
a e n r ' I" :- -:. ' - r- . I 'n '. - -' I r J- u ~1— ' -'-— ‘ T‘ --- . v '
beherall; the lore;o_ng COYDQTJSOH inwicates that the IV: estimates

obtained from the indiviiual enterprise functions are statis+ically as re—

{7'
9.
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H
(D
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"egate function and are

from an economic point of view 9 perior inasmrch as thev permit statements

"I

Q ‘ J... — - _c -‘ n.- I ’ __ I ._ ‘R _ ‘ .. __. o -A _ "_ “.1 >0 ~‘ _. _ _I -_a . - ’ _ Q
aoout the relative GillClGNCg ox grew CLICH lac,ors l1 individual enteracises

,
.—

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tiereas bflG ar:reya e inaction i1 CJuSJ auwlicaole in inlVihUEl €1E€_-

prises would result in several i stances in fault? inferences re ardinf the

.1 _

.ars in individual enterprise

w. —t -- f“ - 1 ..L J. - -1 - - M .
3. Cr parisons c the COhStant a, Standard Errors of
Ls+iwate and Coefficirn‘s of Iultiwle
Cor1€7ati’1 and Esterminatien in

Function chara_teristics other than regression coefficients and mar-
ginal value productivity estimates are presented in Table II. It is seen that

the "a" values ei_ier considesably arena the ? enterprise functions iniicatins

96

 

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Hx mqm<9

97

tuat the steepness of the refression surfaces are not tne ears. The dairy
function shows the least steepness, the crop
the "a" value estirafed for the agfre ate: wnC“1ou wolld "iel: erroneous
results if avulied ,o crops or hogs.

The Sta 119 ard jrror of Lstitate El oatsé7 has not increased as a result

 

of the aggregation process.

The Adjusted Coefficients oi hultiple Correlaiion R1 23hf€7 and “ultiple

E?
I

1 ghg67 anoear to be higher in the acrre ate function than in the

/

letermination
dairy and the h): functions they are, however, somewhat lower than the

corresponding values in the hog function.

1

From this co"13 misox, it is concluded that tie acrrcoatjon of data did

0

not affect the errors 01 the regression li;oe which might also be the reason

I“)

a

why the errors of the re ression coef1 ic ients in toe ah lefate fu1 ction have
not increased conpared to toe corres,ondinc coefficients in the individual

enterprise functions.

CHAPTER VII

EVALUATION OF THE W T.nODOLOC}TCAL ANDH PIIiICAL
PJ SULTS OBTAI"ED IN THE
FO.EGOll IG ANALYSIS
The present chapter evaluates the results of the methodological and

empirical investigation in the light of the major purpose for which the
analysis was undertaken, namely to improve methods of estimating resource
productivities of inputs and investments on multiple enterprise farms.

In addition to eva lua.ting the re esults, surgestions will be made regarding

areas which need further investigation.

_. the Conceptual

Q:
I_J.
'3

A. Evallation of the Conclusions Reache'

 

Part of the Analysis

 

M

1. Applicability of various rulti - equational ap.iroache8. ihe con~

 

ceptual part of t‘e analvsis began wi h the assumntion that] is oossible

to estimate productivities of production factors in individual enterprises

{"1

and suggested three multi - equational approaches. 1heir applicability was

made dependent noon the e mist Ling relationsnins among individual enterprises
on multiple enterprise farms. The three approaches were (l) a system of
equations solved ov the method of simultaneous equations, (2) a system of
equations in which each function is fitted independently to data from multi-
ple enterprise farms, and (3) a system of equations each one fitted to enter-

prises on specializing farms the results to be applied in individual enter-

prises on multiple enterprise farms.

99

The suggested approach seems general and does not appear to depend
upon the type of function chosen. Thus, if it were decided to use a func-
tion which is different from the Cobb Douglas, such as the quadratic, the
same arguments would probably apply.

If the physical production functions on multiple enterprise farms are
dependent upon each other approach (I) is the most appropriate one since it
would determine the functional relationship existing between production
functions of different enterprises.

If the functions are independent in a technical sense but different
from those on specializing farms, approache (2) is applicable.

If the utilization of bypro‘ucts and existing external economies and
diseconomies do not influence the shape of the production function on multi—
ple enterprise farms and these functions appear to be the same as the cor—
responding functions on specializing farms approach (3) seems to be the
most expedient one.

A detailed investigation of the relationships among production functions
on multiple enterprise farms revealed that most production functions can be
regarded as being technically independent and that apparent dependencies
caused by economic factors and the presence of byproduct complementarity can
be resolved through accounting in physical units or constant dollar terms
and through appropriate accounting procedures involving charges and credits
made for byproducts utilized and produced on the farm. At the same time it
was concluded that due to byproduct utilization production functions on multi-
ple enterprise farms should be expected to be different from the corresponding
production functions on specializing farms. Thus approach (2) fitting of in-

dependent functions to enterprise data from multiple enterprise farms evolved

100

on an a priori basis as the most appropriate method.

2. Methods of grouping outputs and inputs into categories. Since most

 

farms produce a series of outputs and use a great number of different inputs
which cannot possible be treated individually in a productivity analysis it
was concluded that grouping of outputs and inputs into meaningful categories
was necessany.

The investigation revealed that in general all products which are pro-
duced jointly, e.g., silk and calves, mutton and wool, or lean and lard, can
be grouped into one output category and treated as an individual enterprise.
Products which compete for resources Should be treated as separate enter-
prises and should be analysed individually. With respect to the combinations
of vertically integrates enterprises such as crops and hogs or crops and
dairy it was decided to depart from the traditional procedures of fitting
one aggregate function to such a vertical chain of enterprises. In this
study, mainly in order to permit comparisons of the earning powers of feeds
and other injuts in crops as compared to the livestock enterprises a separ-
ate function was fitted to the major livestock enterprises and the crop
enterprises.

The question'was also investigated if it is at all possible to fit one
aggregate function to a group of enterprises even though these enterprises
compete for resources. On the basis of a theoretical analysis of this pro-
bkam it was concluded that if the enterprises included in the aggregate
function can be ass‘med to be on the same production function Showing the
same nature of returns to scale and if the aggregation of the inputs used
in the individual enterprises does not increase the intercorrelation among

the input and investment categories,valid productivity estimates for individual

lOl

factors used can be expected.

Regarding the grouping of inputs intoczategories it was concluded that
thecastimates obtained from the Cobb Douglas analysis are unreliable if se-
parate regression coefficie-ts were computed for individual inputs which are
highly correlated and should have been combined properly into one category.
Thus Bradford's and Johnson's suggestion to use that input classification
which keeps intercorrelation among independent variables low appears to be

a useful guide 1.

3. Methods of sampling and accounting procedures. The empirical

 

evaluation of the estimates revealed several important conclusions regarding
sampling methods and accounting procedures.

Most important (mong these conclusions is that in order to obtain a
better estimate of the production functions it is necessary to use a pur-
posively chosen sample which would insure that a larger area of the production
surface is covered than if a random sample were used. Besides, the inter-
correlation among the independent variables would likely be reduced through
the use of this type of a sample.

To reduce the amount of disturbances and to insure their independence
of the factors included in the analysis it is necessary that tight controls
on non-studied variables be maintained when the sample is chosen. These
controls can be imposed by choosing farms whose land quality, climate, milk
and hog production systems and market location are approximately equal. The
independence of management from the variables included in the analysis can

be approximated by choosing farmers whose managerial ability and intellectual

 

Bradford and Johnson, op. oi .

102

habits are approximately the same.

Controls such the ones mentioned could not be imposed upon the sample
which was used in this analysis since it was necessary to take the largest
sample of multiple enterprise farms with the same enterprise combination which
was available. Future detailed accounting projects which are set up with
the objective to be used for productivity studies should provide a controlLad
sample in the sense outlined above.

It should be pointed out that the sirvey interview method is not likely
to be a successful way of obtaining accurate enterprise input output infor-
mation since the farmers interviewed can hot be expected to remember enter-
prise inputs and outputs in sufficient detail. Consequently it is concluded
that farm accounts will be necessary in order to obtain reliable information.

Regarding the accounting procedures used to obtain individual enter-
prise input output data the following remarks appear relevant: The Illinois
enterprise aCCOunts give reliable information in the way they are presently
set up with the following exceptions: (l) The expense categories should in-
clude only productive expenses and not such items as interest, depreciation
and taxes. (2) Investments and Investment repairs should be kept Separate
from operating expenses. (3) Physical measures of buildings including the

number of animals which can be housed should be given.

B. Evaluation of the Conclusions Reached in the Empirical

 

Part of the Analysis

 

1. Comparison of an aggregate function with individual enterprise

 

 

2 This suggestion is taken from a paper by Glenn L. Johnson entitled

Certain Classification, Accounting and Sampling Problems in Fitting
Production Functions to Farm Record and Survey Data. A paper precented
Conference on Resource Productivity and Farm Size, Chicago, Ill., Oct.
19 - 2o, 195u.

 

 

103

functions. In order to make it possible to evaluate the suggested pro-
cedure, namely that of fitting individual enterprise functions, it was de-
cided to fit an aggregate function to the entire farm business and compare
the resulting estimates with respect to their statistical reliability and
economic meaning. On the basis of this comparison it is concluded that the
estimates obtained from the aggregate function did not permit valid infer-
ences regarding the profitability of factors in the individual enterprises.
For example the profitability of labor in the hog enterprise was signifi—
cantly below the minimum.NVP which should have been expected for this factor
while the aggregate function showed a labor return which was approximately
the same as the minimum MVP expected for this factor. In general, MVP's and
other function characteristics such as the multiple regression coefficient
£1.23h567’ the standard error of the estimate €1.23h567 and the sum of the
regression coefficients appeared to be weighted averages of the corresponding
estimates in the individual enterprise functions. Thus it appeared that

the method of fitting independent functions represen s a significant improve-
ment over the method of fitting one aggregate function to the entire farm
business. This result was anticipated on the basis of the theoretical dis-
cussion of this topic in the conceptual part of tfliS analysis.

2. Inter-and intra—enterprise comparison of individual productivity

 

estimates. A considerable part of the analysis was devoted to developing

 

methods with which productivity estimates could be evaluated both within

-'- - i- ~ ' T “If ‘ ‘0 -~ (1 ic' ."JC‘ 4— wwx t- " , 9‘”?
and between enterprises. -he MGEJOQ Oi comparison was to compu e a legrecs_on
coefficient which would yield a certain minimun.or reservation MVP and using
the "t” Test establish whether this computed regression coefficient is sign-

ificantly different from the regression coefficient estimated for the function.

10h

The comparisons were made at the geometric mean of each function as
well as for individual farms whose organization appeared to deviate from
that of the geometric mean organization. The conclusion was reached that
the comparison at the geometric mean revealed no serious maladjustments
suggesting that in this sample the organization at the geometric mean did
not deviate significantly from an optimum organization. The comparison
made for individual farms evidenced very serious maladjustments on these
farms suggesting that this last type of comparison should be given consid-

erably more weight in future evaluations of productivity estimates.

3. Conclusions regarding the comparative efficiency of production

 

factors on multiple enterprise farms in northern Illinois. Although the

 

principal aims of the thesis were of a methodological nature the estimates
obtained reveal several conclusions which can be of practical use to the
farmers in northern Illinois.

From the ”F Test" which was used to determine whether the returns to
scale of the individual production functions are increasing, constant or
decreasing it appears that none of the sums of the regression coefficients
is significantly different from one. This result suggests that returns to
scale which are not different from constant return to scale prevail on these
production functions.

'Uith respect to the individual estimates it appe rs that when compared
at the geometric mean the farmers in the sample used feed to equal advaitage
in the dairy and the hog enterprises. Also the returns obtained for each
dollar's worth of feed are equal to one dollar if estimated at the geometric

mean indicating that farmers are correct in feeding rather than selling the

 

crop for cash.

With respect to the utilization of labor it is concluded that the
farmers in the sample get relatively high returns for this factor in the
dairy and crop enterprises while the returns of labor in hogs were signi-
ficantly below a minimum wage indicating that a reduction of this factor
in hogs is necessary.

The returns to land and labor in crops are high indicating that an addi-
tion of crop acres might be profitable. On this basis as well as on the
basis of the nature of the returns to scale in the livestock enterprises it
is concluded that the farms studied could profitably expand their operations
in all the enterprises studied.

In concluding this evaluation of the results of the foregoing analysis
it should be pointed out that additional valuable conclusions could probably
have been reached if the analyst would have been personally acquainted with
the farms included in the sample. Only practical inferences going beyond a
mere statistical comparison would fully exploit the results of an investi-

gation such as the one carried out in the latter part of this thesis.

 

 

106

BIBLIOGRAPHY

Anderson, T. W. and.H. Rubin, "The Asymptotic Properties of Estimates of
the Parameters of a Single Equation in a Complete System of Stochastic
Equations“, Annals of Mathematical Statistics, 21, 1950.

 

Bradford, Lawrence A. and Glenn L. Johnson, Farm Management Analysis, New
York: John Wiley and Sons Inc., 1953.

 

Bronnfenbrenner, Martin, "Production Functions: Cobb-Douglas, Interfirm,
Intrafirm,” Econometrica, Vol. 12, January, 19h).

 

Bronnfenbrenner, Iran and Herman Chernoff, "Computational Methods Used in
Limited Information Treatment of a Set of Linear Stochastic Difference
Equations", Cowles Commission for Research in Economics, University of
Chicago, Chicago, Illinois, February, 19h9.

 

Douglas, P. H. and C. U. Cobb, "A Theory of Production", American Economic
Review, Vol. 18, 1928 Supplement.

 

Drake, Louis 8., Problems and Results in the Use of Farm Account Records to
Derive Cobb-Douglas Value Productivity Functions, Unpublished Ph.D.
Dissertation, Michigan State College, 1952.

 

 

Durand, David, "Some Thoughts on Marginal Productivity with Special Refer-
ence to Professor Douglas' Analysis”, Journal of Political Economy;
Vol. h5, December, 1937.

 

Ezekiel, Mordecai, Methods of Correlation Analysis, New York: John Wiley
and Company, Second Fdition, 1953.

 

Fienup, Darell, Resource Productivity on Montana Dry Land erp Farms, Mimeo-
graph Circular 66, Montana State College Agricultural Experiment Station,
Bozeman, Montana, 1952.

 

 

French, Burton, Lstimation by Simultaneous Equations of Resource Productivities
from Time Series and Cross Sectional Farm Observations, Unpublished Ph.D.
Dissertation, Iowa State College, 1952.

 

 

Heady, Earl 0., "Production Functions from a Random Sample of Farms", Journal
of Farm Economics, Vol. 28, No. h, 19ho.

 

Hicks, John 2., Value and Capital, Oxford: The Claredon Press, Second Edition,

19u6. C

 

107

Johnson, Glenn L., Sources of Income on Upland Marshall County Farms, Progress
Report No. 1, and Sources of income on Upland KcCrac1<en County Farms,
Progress Report 110. 2, Lexington: Kentucky Agricultural Imperiment Station,
1952.

 

Johnson, Glenn L., The Farning Power of Inputs and Investments on Upland
Galloway Count; Farms, 1951, Progress Report No. A, R ? MAoO, Kentucky
Agricultural Experiment Station, University of Kentucky with Tennessee
Valley Authority Cooperating.

 

Klein, Lawrence, A Textbook of Econometrics, Evanston, Illinois and White Plains,
New York: Row, Peters on and Company 1953.

 

Lerner, Abha P., The iconomics of Control, New York, the Kacmillan Company,

19u7.

 

harschak, Jacob, and U.i11iam Andrews Jr, "Random Simultaneous Equations and
the Theory of Prodr _ction", Aconometricm, Vol. 12, 19hh.

 

Marshall, Alfred, Principles of loonomics, kew York: The Kac Iiillan Company,
Eighth Elition, 19h9.

 

Mitscherlich, Eilhard A., Podenhnvde fuer Land und Fors twirte, Berlin:
Paul Parey, 1905.

 

 

Olkin, Ingram, ”Unoxblisned em3 ort About a Problem in Testing Sums of Re-
gression Coeifi ients of Li rKultiple Regression Lines Against a
Constant”. A reporth sbee n dmade by the statistical group of the
mathematics department3 to Professor Glenn L. Tohnson, Department of

Agricultural “conomics, hichijan )LT e University.

Plato, The Republic, Jowett Irans lation, New York: The hodern Library.

 

Spillmann, “. I.

, se of L_oorenti 1 Held Curves in Fertilizer Experiments,
U.S.D. A. 1ecu ic

cal Irllrwnrilki. 31K, 1933.
Tintner, C. and C. P. Rrownlee ”Production Functions Derived f*am Farm
Records”, Journal of 1arm£conomics,“uol.26,19hh.

 

.0

Tintner, Gerhardt, ”A Uote on the Drivation 01 Production Functions from
Farm Records", Aconometrica XII, K0. 1, January, 19hh.

 

Trant,Cera1d Ion, A Technieue of Adjusting M irg nal Value Productivities
Estimates for Charoinc Prices, Unpuolished U.S. Dissertation, Alohi"an

‘m-u.

State College, 195D

 

 

Ualras, Leon, Elements of Pure Lconomics; or, The Theory of Social Uealth,
Translated by filliam Jaffe, London: Allen an.d Brown, 1951.

 

‘Uicksell, Knut, Lectures on Political Iconomy, Volume 1, London: G. Routledge
and Sons Ltd., 193A-352

 

hold, Herman and Lars Jureen, Demand Analysis, New York, John Uiley and
Sons Inc., l953.

 

 

 

 

APPENDIX A
A MATHEMATICAL DERIVATION OF EQUATIONS FOR THE ISO COST ( TRANS -
FORMATION ) CURVES FRCM TWO COBB - DOUGLAS EQUATIONS FIT -
TED TO INPUT - OUTPUT DATA FRCM TWO DIFFERENT ENTER -
PRISES

   

 

 

t WTICAL DERIVATION OF EQUATIONS FOR THE ISO COST ( TRANSFORMATION )
‘ CURVES FRO! TWO COBB_- DOUGLAS EQUATIQIS FITTED TO INPUT - OUTPUT DA -
TA FRCM TWO DIFFERENT ENTERPRISES .

Referring to sectim B of chapter II where the iso cost curve was ex -
plained conceptually as derived from two-production functions, it can now be
shown how the equations for the iso - cost curves can be obtained from two
independently fitted Cobb Douglas Equations. For purposes of illustration,

a function with only two independent variables will be used. The total quan «-
tity of each factor ( X1 , 12 ) available cm the fern is designated byi 1 o
11 is the quantity of the factor which is used in enterprise I, i i - 11

is used in Enterprise II. Equations I and II with the constants a written as
/\ 1 and A 2 read 8 . ' i I

( I ) Y1 -A1 If“ 12b?
(in) i2=A2(21-xl)b1-(22-xz>b2'_

Since alcng the iso - cost lines the factors in each of the enterprises
have to be combined in scale line proportions and since these proportions re -
main the same along the scale line of any Cobb Douglas function, the factors

12 and. 2 2 - 12 can be expressed as functions of 11 and Z 1 - 11 respective-
ly. Along the scale line the condition maul) we 120:1) holds .

 

 

 

 

 

 

1‘11 P12
In empirical terms the foregoing ccnditicn can be written as
B
7—4131 ‘9 Pxi b1 2(2) P“ m) z
11 ' ,, x1 b1 PXi 2 blPx112 1
a I .—. -- A ' = ' ’
b2 (_ ) ' 9x2 b2 ‘(I) P12 b2 3(Y) P12 X1 b2 P12 11

111

b P
Fran the foregoing it follows that 12 z £512.. . similarly
' b1 P11

 

' u - )
22-42: 12—55% 1 x1

b1. PIll

Substituting these expressions back into equations I and II, equatials
I'andII'areobtained: ‘ I.

 

 

b 1 b2
(1') 1r1= A13‘1b1 [2% l 1
" 1’1le J
b ' r' (2 x ) b‘z'
' 2 I 1" l 1
av) 22=Ath- ‘ 2 ,
' ’ 3
b1, 131 T l
21 brbz

 

 

Solving for X1 in eQuation I' 8 X1 ' 1(b2PXZ
" I ' A P I;
I. 701 x

and substituting into II',equation III' is obtained

._ r

..ifl'

3

 

(III') I

 

 

 

 

 

 

From equation III' the transformation curve can be plotted in the 11 and

T2 dimension. The Optimum point is obtained upon solving equation III'
simultaneously with an iso revenue equation. In the example above the e-
quation for the transformation curve has been worked out for two variables
only. The extension to :1 variables is apparent however computationally diffi-
cult. '

APPF'i‘t-DDI B
JISI‘R.-.BU'1‘IW\‘ CF CROSS II‘JCOME R: EITxITEFPTiISE-S ON THE 27 DAIRY-

“1“

HOG FARE/IS II‘ECLUDID IN THE ANALYSIS

DISTRIBUTION O: CROSS INCOME BI ENTERPRISES Oh THE 27 DAIRY-

ROG PARKS INCLUDED In THE ANALYSIS

TAP-LIE XII

D '1
.L

ieturns per Farm by Systems 6 farming, Blackhawk Area, 1950

 

 

 

 

: Dairy-hog
: ‘Soild : ’SoiI
Item : rating : rating
: 2.0-3.9 : h.0 and over
Feed, Grain, and Seed f f .
Inventory increase : g 1,028 : S 772
Consumed in the home ; 57 : 76
Sales ; 1,7SE : 1,1h3
Total Feed, Grain and Seed Returns 5 a 2,869 ‘ $ 1,991
Livestock 7
Sheep ’ A 17 ’ S _-
Poultry . A10 7 602
Beef herd ‘ -- ' --
Dairy , 9,166 ; 7,718
Feeders : -- . A2
ther cattlel i —— j --
Hogs 7,359 : 5,602
Total Livestock Returns ? $16,952 ' 313,965
Labor and Machine Work of: t*e Perm. 21b 3 137
All Other Receipts ‘ 186 f 120
Total Returns $20,221 f 516,213

 

1 Other cattle include dual-purpose herds, mixed herds of breeding animals,
and mixed herds of breeding and feeder cattle.

DCTEFI-UNATION OF THE. DEEMED TUITION RAT- USE, TO ADJUST

I'LACHII‘IFIRY FL-IPEIISES .N THE CASH I'CPENSE ACCOUNTS

A

115

DETERMINATION OF THE DEPRECIATION RATE USED TO ADJUST

MACHINERY EXPENSES IN THE CASH EXPENSE ACCOUNTS

In order to eliminate depreciation from the machinery expense item

adjustment factors were used which were determined on the basis of the

1
following table stating the operating cost for different types of tractors:

TABLE XIII

Tractor Operating Cost by Drawbar Horsepower Ratings and

Hours Used During 1950, Blackhawk Area

 

Drawbar horsepower rating

 

10.0 through 20.9 21.0 through 31.9

 

 

 

 

Item Used fewer. Used 500: Used fewer: Used 500
than 500 ' or more 1 than 500 I or more
hours hours hours hours
Number of tractors 2h 3 2O : . 10 : 15
Average drawbar horsepower 16.9 17.2 ' 26.3 26.2
Cost items per tractor :
Fuel, oil, and grease $128.38 $2h9.77 $196.32 : $295.9
Repairs h7.72 7h.18 38.80 107.5h
Labor b.35 8.13 5.h9 b.08
Shelter 9.07 lO.h6 1 10.65 20.55
Depreciation 108.h3 109.36 2 205.50 183.99
Interest on investment 27.8h 27.87 56.01 : 57.23
Miscellaneous 1.h5 3.09 h.16 : l.h5
Total cost $327.2h fih82.86 $516.93 : U670.79

Total cost minus depreciation
and interest on investment

Hours tractors used
Drawbar work
Belt work

Total hours used

Cost per hour of use

' 190.97=58%

=3hb.86=7l%= 255.h6=h9%= h29.57=6h%

 

 

302 685 : 357 - S96
13 32 - 9 59
3.5 717 366 1 655
a 1.0h .67 A 1.h1 1.02

 

 

1 Taken from "Detailed Cost Report for Northwestern Illinois l9h9 and 1950",

Department of Agricultural Aconomics, Agricultural Experiment Station
UniverSIty of I linois College of Agriculture, Urbana, Illinois, Apri

AE 2871, p. to.

i, 1952,

116

The adjustment factor for depreciation in each one of the four classes
was the percentage which "Total Cost Linus Depreciation and Interest on

T. .

Investment" were of the "Total Cost”. nus, if a farmer's machinery cost in
crops was given as 01,000 and his tractors were used 500 or more hours and
fell in the class 10.0 through 20.9 drawbar horsepower, the machinery cost

included under cash expenses was $710.

APPENDIX D
UH OF THE REGRE.SION COEFFIC

THE F TIS' FOR TESTING THE S

IN A LINEAR REGRESSION EQUATION AJATNST A CONSTANT

118

TIL F TEST FDR TESTING THE SUM OF IHE IFGRESSION

EQUATION AGAINST A CONSTANT

The following is a method of testing the sum of the regression coefficients
of a regression line against a constant. l/ The test, developed by Dr. Ingram
Olkin, Associate Professor of Statistics at Michigan State University, is appli-
cable in all fitting procedures which use an (n - 1) x (n - 1) matrix when
n parameters (including the (a) value) are to be estimated. The Doolittle
method which was used in this study is a particular example of this kind of

fitting procedure.

The Test: Consider a regreSSion equation of the form

y: £22. +/j?-:{2+ 000.00 #13 :{p + a

where 6 is normally distributed with mean 0 and standard deviation 0" .

A sample of N independent observation is taken and the hypothesis

 

 

 

 

 

P
H0 : ‘1 flé _ c (same constant) is to be tested.
r 1
Solution: P r ‘3 r
V J kll-“l°°°”‘ln'z1
*1 " ‘ _ _
0 :‘IL - :{h 0.... I; — 3‘:
Let Y __ . X __ 21 2 2n 2
V T x " J _ 3;
L‘71""[ Ln1 ‘ Xn Xnn ‘nJ
" I.
l/ , It is recalled (Chapter II) that the sum of the regression coefficients

in the Cobb Douglas function determines whether increasing, constant, or
decreasing returns to scale are present depending upon whether the sum
of the regression coefficients is greater than, equal to, or smaller
than unity.

v’
A — Kn.)

the least squares estirates of the/#3 's,

b
(l)
where N
F
c

v4.
k.

FJhn'
U‘
{-h

ad

then A is a p X p matrix.

(

I-p) (c-

119

The normal equations lead to

namely
b
l
X', where b = i . The test to be used is :
bu

(l , N - D)

 

II

II

' 9
z a 13 Si

number of observations in the sample

nimber of regression coefficients (excluding a ) which

are estimated

some constant ( c = l in case of linear hypothesis)

.-_ i'
matrix. The a*J are the Cij values

obtained in the back solution of the Doolittle method.

elements of the A

(r — 3:11;») 1 (r - Klb)

Observe here afain that Y is the column vector of the
adjusted values of the dependent variables , i.e.,

v _ 1'
“:1
) = I
- v - V
.Ln -- _ .-
X - :: oooo Tr "' I:
ll 1 nl n
Similarly, for the matrix 31 =
7; - T E” _ 7
ln. ‘“1 °°'° ‘Tnn. “n

In the case of the Cobb Douglas function, the I and Y matrices contain

logarithms rather than natural numbers.

The statistic (1) has an 1“

distribution with 1 degree of freedom in the numerator ane U - p derrees

of freedom in the denoninator. Large values of F are critical.

APPENDIX E

THE ORIGINAL CQQERVATIONS WHICH UNDERLY TH? THREE

EhTERPRISE PRODUCTION FUNCTIONS

THE ORIGILAL OBSERVATIONS KHICH UNDERLY THE THREE

ENTERPRISE PEODUCTION FUNCTIONS

121

 

 

HOGS

Farm i1 X2 13 in AS X6 17
No. Gross Labor Feed Exp. hach. Breed. Perm. hog
inc. hrs. inv. inv. housing
1 3903.00 667.75 2686.70 171.12 126.16 938.50 18h.00
2 3723.00 703.67 2570.02 231.86 69.32 315.00 1.00
3 6132.00 2118.01 3921.86 276.99 201.60 1020.00 325.00
6 10395.00 866.92 6271.63 629.77 812.12 1519.36 926.00
5 u335.00 873.55 £688.68 258.72 352.67 1065.00 220300
6 6692.00 766.39 6209.89 291.81 122.36 935.00 501.00
7 20127.00 27h5.92 15663.08 1132.56 755.68 2635.00 h95.00
8 10630.00 1666.92 812u.83 862.87 562.33 2396.00 6359.00
9 6006.00 55h.00 h338.00 253.77 77.28 12h6.00 878.00
10 7228.00 510.92 5167.63 610.35 206.h7 770.00 656.00
11 10667.00 1070.6h 6229.16 769.06 939.96 1500.00 160.00
12 5962.00 369.39 h130.08 156.62 287.1h 870.00 80.00
13 2626.00 hOo.7O 2006.33 96.61 6.00 800.00 975.00
16 2063.00 696.5 1323.13 36.25 7.85 567.00 160.00
15 6956.00 657.96 h078.28 287.70 396.86 1651.00 158.00
16 5137.00 561.15 3657.50 96.27 252.10 1060.00 1.00
17 h0h0.00 778.18 2h72.oo 100.98 60.53 707.50 70.00
18 -6660.00 h22.he 2375.0h 150.21 80.30 1100.00 330.00
19 8675.00 725.68 5805.22 702.23 856.71 990.00 358.00
20 h288.00 900.39 3556.77 170.52 187.16 1191.00 680.00
21 7577.00 555.78 8823.09 183.32 153.72 013.50 987.00
22 1356.00 369.01 3659.71 69.32 63.36 560.00 90.00
23 E717.00 399.37 2358.72 10b.08 191.55 688.50 1.00
2h 2726.00 333.66 1606.39 106.79 37.27 555.00 227.00
25 6691.00 839.38 5035.71 319.72 239.21 1353.50 518.00
26 7076.00 1087.32 6u16.§6 298.92 279.57 982.50 568.00
27 7603.00 8h6.h0 5501.61 269.97 180.18 1531.50 525.00

 

 

 

 

mm!

m 11 12 13 It. x5 x6 ‘7
No. Gross Labor Feed Exp. Mach. Breed. Dairy

inc. hrs. inv. inv. . housing

1 60118.00 1985.30- 11109.22 710.16 1002.141: 578.38 111.11?
2 5201.00 23011.80 117111.07 1069.08 25112.00 1013.110 21.09
3 1935.00 1500.60 5297.68 6h9.7h 1667.00 866. 28 20.13
h 95117.00 27118.00 8079.88 1018.13 6226.32 1860.82 52.10
5 5828.00 2078.80 3573.85 690.53 3792.77 1386.03 32.53
6 13350.00 11120.90 87014039 1251. 53 71:36.75 1010.10 36.11?
7 18115.00 18157.10 9358.93 11187.07 11187.27 23111.12 37.
8 155116.00 51.85.50 8231.80 1630.25 7361.92 1625.68 106.67
9 6619.00 23117.00 6296.91 1250.86 11390.82 981.12 51. 37
10 111601.00 14131. 70 8571.00 989.80 2317.73 2582.73 59.62
11 36011.00 1713.70 181114.116 1.91.63 602.86 1398.02 811.32
12 85911.00 1205.00 5633.05 5011.00 5156.81: 16711.13 50.53
13 13798.00 2686.110 9632.10 1568.05 8815.00 1269.00 63.20
11; 3528.00 12611.80 30112.11: 307.58 183h.16 11.2.35 9.39
15 5790.00 2923.50 11515.81 792.72 2357.00 923.78 112.03
16 7627.00 1853.70 5013.11: 528.06 2079.93 131111.26 311.33
17 6206.00 18811.80 38115.51: 11511.86 1855.61 1821.59 1111.80
18 7h38.00 1298.80 6169.06 799.12 1810.98 1727.19 119.27
19 10578.00 3362.30 6530.37 832.15 11.69.23 14611.57 21677
20 8195.00 2630.30 5001.20 525.29 21:88.66 1337.09 35.20
21 5093.00 3110.70 1971.90 7111.02 2619.18 1082.52 33.60
22 6681.00 2272.30 6511.98 830.03 2908.hh 1162.61; 36.39
23 7903.00 32115.60 71171.32 630.63 3272.81; 11405.16 50.83
21. 6529.00 2201.50 3955.26 517.115 2078.65 372.10. 35.06
25 67118.00 2655.00 141183.91 993.78 1526.99 1021.91 30.71
26 6386.00 1713.70 3919.83 619.73 1613.95 1210.79 30.01
27 21.26.60 1.1310117 310.10 1677.89 87h.37 21.16

5012.00

 

 

 

CROPS
No. Gross Labor Land Exp. Mach.
inc. hrs . inv.
1 3798.90 967.51 70.70 1236.09 3022.66
2 7296.03 956.61 120.60 1690.66 3926.28
3 7972.06 1297.22 95.60 1738.28 2732.12
6 8718.81 1606.78 121.30 1882.98 5780.06
5 5701.72 1200.77 68.00 2111.60 3656.50
6 6958.75 766.21 58.50 2360.90 5531.56
7 16258.00 2693.07 315.20 6228.13 8866.60
8 9350.20 2152 .58 136.30 2803.60 6605.99
9 7636.81 1185.93 106.00 2125.62 6813.60
10 7502.62 1396.80 153.80 2865.66 8769.80
11 7576.06 773.86 120.50 1296.29 2053.62
12 5386.96 862.55 106.60 1970.52 5609.73
13 8236.50 1357.65 83.60 2586.67 8361.00
16 3588.00 1236.56 53.00 965.92 756.80
15 5379.02 820.66 71.00 1723.20 2285.60
16 5670.90 782.66 78.50 1876.86 1655.36
17 5376.90 1559.68 138.60 1982.66 2632.88
18 11331.56 1367.19 136.00 2592.95 5116.51
19 11017.00 1666.25 158.80 2635.67 6736.72
20 5769.00 1123.55 91.00 2598.89 5699.75
21 9672.96 1767.33 129.50 2630.79 3917.76
22 5236.25 1212 . 61 93.70 2186.63 2007.00
23 8907.83 1166.36 127.00 1919.71 7070.16
26 6305.15 1002.88 69.00 1679.28 6067.00
25 7956.50 1157.10 167.90 2193.02 3615.88
26 6650.00 1303.56 107.60 2008.65 2707.96
27 7306.38 2375.75 116.20 2198.97 1328.00

 

 

ROOM USE 03M

ROQM sJSE {EEK-ELY

 

 

" ifi HT

2E33

Icwxonw STATE UNIV L R0
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31293100996