PROBLEMS A N D RESULTS IN THE USE
OF FARM ACCOUNT RECORDS TO DERIVE
COBB-DOUGLAS VALUE PRODUCTIVITY FUNCTIONS

By
LOUIS SCHNEIDER DRAKE

A THESIS
Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Agricultural Economics
1952

PBOBLXMS AMD R0ULTS IM THX USB
OF P A W ACCOUNT BBCORDS TO DBIXY1
OGBB-DOUOIUB TALUB PBOXXJCTIPZTT PUMCTXOM

br

Louie Schneider Drain

An Abairaot
of a thaaii praaaaiad to M u .School of Qreduate Stodlee
oP 1H nhtgan St*to College
In partial ftilftU— it of the roqulroaoato
Tor the degree of
DOCTOft OP PHILOSOPHY

Department of Agricultural Xeenoad.ee
Approved

ISOlS M M i i M r

WWlI lit AMD I B D U IiniOII
or rum aooouot w o o m to n m

COIB-DOOOLAS TALUB PhOOSCTIYITI rUSCTIO»
(41 ABSTKAOT)

At

purpose or thia «tod|r is to tort t o t i— f w l w m «f tot C r t W

Dooglas toLtt pvotort&flto f w r t i w ftr trtltrtlip tot gross insoae
of t fora by considering tot Input# or fatton nsrt.

This fowtiat ptr>

alts — t1noting or olsstisitiss of gross lnooao with roopoot to rostors
and tta# norginal ▼olao prodnotlwltiss of rostors*

Tho hypothesis lo sot

up that sop orison tot low nith tho fhaot&on will shoo thot tho lsttor os*
tiaotoo will bo nsoTnl to faraan.

Tho hppothools is adronood that dif-

foronooo botnoon rooorded grooo Inn on a and gross iaooaao ootinotod ftoo
Cobb-Doagloo equations eon bo osooontod for by Torlotloas botnoon f a n s
in prlsos resolved, yields and prodostion rataa, ohoiso of enterprises,
and also of business*

Spoolol ottowtlon is givon to prooodaroo whioh

will noko o totlotlool ly dotomlnod value productivity equations of pros*
tleal value to f a n a r a .
Iho dots a n lpU

ran

5 and 6t Michigan, 1990.

osooont rooords for tjrpo of-famlng oroos

Tho rorns oro cl aoolflod Into too groups,

daliy and othor than dairy.
techniques nagr bo triad on a

This lo dona in order that tho statistical
group of honogonooos f a n s (dairy), as

wall aa on tho noro heterogeneous group of all fams.

Tho statistical

aothod consists oT oonwortlng tho f a n aooount data into logorithno and
solving for aquations ootlnatlng gross 1noons by least oquareo*

M s t ilw iU if Droho
O n of tt« Cobb H w f l M f u w t i N to « U m N |to m I m o m « A mmrglaal rvtarnt to laputa n t N on ttw> prlauy m b u H — >.

H m n ««

Ihti tlto gtltltoNldp Oitw m giM« i M N t and anjr parti— lor input lo
U n i t la tho logarlthno # that grooo lnoono lo o ftwrtioi of laputa* and
that dlfforont fan00 aro ooaonttally trialo «lth ?ai7 U | oomblnatlono of
faotoro* all f«no bolag oa nubotaattally the mono valao produntivitar
funotlon.

Thaoo aooumptloaa aro ooaoldarod*

It lo oheoa that tho Cohb-Oooglae funotlon glvoo a good oatlaato
of grooo lnooao*

Tho ootlaatoo of marginal valao prodnotlTitioo of

faotoro tarn oat to bo about ao ahoald bo orpootod a priori*

For dlf«*

foroat oquatlono ootlaatlag grooo lnooao tho marginal valao prodaotlvlty
of iavootaoat la load vorloo botaoon 0.03b and 0*096 la Ito ootlmatod
▼aluo*

Iho ootlaato of tho marginal roturn to larootaoat la naoMiiarjr»

Inoladlng oboolooooaro and doproolatloa ao amll aa Intoroot* mngoa firsa
0.23 to 0*35*

Thaoo ootlaatoo and alallar onoo rofor to tho rotara to a

aarglaal dollar of larootaoat or ohargo*
Tho roonlta of tho work ouggoot that fara hunt nooo amaljrola roporto
max profitably laelndo tho following Information•
1*

latinotoa of grooo laoomo,
of what ho ahould

00

that a foraor would homo a aotloh

rooolro^ eoaoldorlag hlo lapato la rolatloa

to tho laputo of othor farmora.
2*

Satlaatoa of arorago aarglaal valao prwdaotlrltloo of footoro
for all famo

00

that thoro ahould bo an additional boolo fbr

ronnnaonriotlono for futuro oxpondlturoa on tho avorago farm.
3*

Kotiaatoo of marginal Td.no produotlTltloa on individual farao
to holp dnraoro plan for tho futuro*

[)W«< m Sohnaldar Dpa Ih
k»

fcylaniU o M of

la grota I i m m m a d ast laa a t

ftnoa thatr ootlastod valooo a a r d l n g to tho affooto of jrlalda,
prloM, ohoioo of « W r p r i M i t a d olso of batiain.
In this ttadjr tho ootlaatoo of grooo lnooao aad aarglaal valao
prodaotlvltlaa of 1x9 a to aro baaod oa all farao, ao aro tho ootlaatoo
of tho offoeto of dlfforoaooo In jlolda and prlooo.

Tho standard of

ooaparlooa nood not noooooarllr bo tho •anrorago" f a n bat mmj bo a group
of high profit faroa •

ACKNOWLEDGE ME NTS
The author wishes to express his sincere apprecia­
tion to the members of his graduate committee and to his
teachers Tor their encouragement,

h igh standards,

and gen­

erous assistance and criticism.
Those who helped particularly
this

in the pr ep aration of

dissertation are Professor T. K. Cowden,

who gave

much encouragement and confidence in the worth of study—
in;’; agricxilture at the level of the whole farm as a unit;.
Professor L. V/. Y/itt, who showed h o w to specify the p r o b ­
lem and how to set up the meth od of study; Professor L# H.
Brown, who gave invaluable

suggestions for uses of the

method and the results; Professor V. E. Smith, whose k n o w ­
ledge of production functions provided the author w ith
most

necessary criticisms as the work was in progress;

Professor Leo Katz,

who showed now

to set up and test the

statistical work, w i t h i n the limitations of the data; and
Professor Glen
State College,

T. Johnson,

visiting professor at Michigan

who pointed out basic porblems and n u m e r ­

ous potentialities in the use of the Cobb-Douglas function
and gave m u ch valuable counsel and criticism.
Others who contributed to the work are Professor K. T.
.’/right, who

gave ideas for analyzing the data in consider­

ation of its intended use; Professor C. R. Koglund,

who

suggested ways of finding out how data could be improved
for work with gross income functions in the future;

ii

and Pro

fessors W. D. Baten and L. L. Boger,

both of wh o m h elp ed

generously w ith mathematical problems.
The author acknowledges his debt to Professor E. E.
Peterson,

In whose course In agricultural production econ

omics he came on the application of the exponential gross
Income function to farming.

Professor Peterson helped

him set out the Idea of the work In definite form, and to
coordinate the method w i t h the available data.
Teachers other than those directly concerned with
the dissertation have helped the author to form a p h i l ­
osophy of an economic system and to realize the p o s s ib il ­
ities of service through study of the economics of a g ri ­
culture.

These professors are D. C. Cline, R. W. Llnd-

holn, Leonard Ball,

I.!, F.

Cravens,

and Raleigh 3arlowe.

Thanks are due the farm account clerks for their
willing aid w i t h the computations,

typing,

and diagrams.

By his understanding cooperation In granting the a u ­
thor an extended leave of absence, D ea n Fay L. Partlo of
the Michigan College of Mining and Technology hel p ed to
make the work possible.
It Is Impossible to give

specific credit— —o r enough—

to the anonymous workers who, over the years,
oped the Michigan farm account system,

have devel­

compiled the data,

and thus provided the foundation on w hich this work could
be built.
ill

VITA
Louis Schneider Drake was born in L eel en a u County,
Michigan,

on June 11,

in Traverse City,

1912*

Bellaire,

He attended, public
and Mancelona,

schools

Michigan,

and graduated from the Mancelona High School in 1929*
At M i c h i g a n State College he

specialized in agricultural

economics and farm management,
of Bachelor of Science

and obtained the degree

in 193U- from that institution,

lie obtained a graduate as si stantship

in farm mangement

at Cornell University and pursued graduate
193^4- to 1937*

studies from

At that time he accepted a p o s i ti on as

Land Use Specialist for the Resettlement Administration,
and was 1 ter transferred to the Bureau of Agricultural
Eco no mi cs as Actin g B. A. E. Representative
State.

for New Y o r k

He received the degree of M a s t er of Science from

Cornell University In 1939*

He was w it h the federal

government as A s sistant B. A. E. Representative

for

Michigan and B. A. E. Representative for Connecticut u n ­
til 191^2.

He had meanwhile,

from 1939 until 19U2 been

a partner In a commercial potato farm in Ho ughton County,
Michigan,
l?li-8.

and continued In the farming business until

In 19 -4-6 he accepted a position with the M i c h i ga n

College of Mining and Technology in the department of
engineering administration,
an Instructor,

where he was successively

an assistant professor,

professor of economics,

and an associate

which position he now holds.

iv

INTRODUCTION
A Brief of the Purpose,

Materials, Methods

and Results of the Dissertation

I# Purpose of the study and definition of terms:

the

purpose of this study is to show h o w certain types
of formulas can aid fanners in determining
A* How their gross Incomes compare w ith those of
other farms when investments and expenses are
considered;(estimate of gross i n c o m e );
3, What increase

In gross Income farmers should

expect If they increase any single kind of o u t ­
lay by a "iven proportion (elasticities of prose
income v.'ith respect to factors of p r o d u c t i o n ) ;
C. What the additional returns for additional o u t ­
lays are

(estimates of marginal returns to f a c -

t o rs );
D* Why net Income usually varies from Its expected
amount (effects on net Income of yields,
duction rates,

prices,

pro­

and size of b u s i n e s s )*

II. M a t e r i a l s : Two types of materials are used In this
study;
A. The analytic t o o l s : the Cobb-Douglas gross ina b
k
come estimating equstions--P r Cx y ...z_ •
(Chapter I)
3* The d a t a : observations of I 9 I+ farms in type-offarming areas 5 and 6, HIchipan,
v

1950

(Chapter II)

Ill* Methods (Chapters III and IV)
A* Classificatory methods
1. Inputs: factors of the farm business are
classified into different categories with
different degrees of refinement.
more

Thus

insight is gained into the componental

structure of gross income;
2* Farms:

the farms are divided into two

group s :
a* 86 dairy farms
b* 108 farms other than dairy.
3* Mathematical methods
1. The categories of factors end

*rooc income

are taken off the farm account records;
2* These data are converted into logarithms;
3* Re g re ss io n equations are calculated by the
Doolittle method;
Standard errors of estimate of single co­
efficients are computed in some

cases;

5* The elasticities of gross income w ith r e ­
spect to categories of factors are given by
the coefficients of the terms on the rl;^it—
hand side of each logarithmic equation;
6. Marginal value productivities of categories
of factors are determined by taking the
partial derivatives of the gross income

vi

equations with respect to the categories of
factors.

These equations are given in n u m ­

bers,
IV. Some Applications (Chapter V)
A* Some changes in methods of recording farm account
data are suggested w it h regard to the value of
1. Land
2. Livestock, particularly cows.
B. Suggestions are made

concerning the clasftlfication

of factors of production in farm account records.
C. Farm business analysis reports

can profitably

Include:
1. Estimates of Income to show the average
relationships of inputs to Income on all
farms;
2. Estimates of gross Income for Individual
farms to show each farm er whether he is
running ahead or behind the average rela­
tionships;
3. Statements of average marginal value p ro ­
ductivities of classes of Inputs so that
there can be some indication to farmers in
general whether they would be better off
to Invest more money in particular items,
such as machinery rather than labor;
1;_. The same as 3» only the statements of

vi I

average m arginal value productivities would
indicate to the individual fanner c ompara­
tive marg in al returns on his own farm;
5m E s t i ma te s

of marginal value productivity

of a given single

input at different levels

when other inputs are held constant.

A

farmer thus migh t have an idea of whether
he should put more m o n e y into

cows,

for in­

stance •
The effects of the following upon variations
in gross income bet we en farms can clearly be
seen:
1. Yields,
2. Rates of production,
3. Prices,
ij.. A n d size of business.
Thus a specific explanation that gets d o w n to
crops, yields,

etc.,

can be given for difference

between the net Income recorded in the farm
accounts and the average net Income; or between
the recorded net income and the net income of,
say,
We

the most

can

successful third of the farms.

~ain more insight into the true values of

the v a r iotas factors of a farner's business.
\7e can also find out something of the different
subjective values fanners put up o n the same

viii

factors*

E v e n If two fanners could earn Id ent­

ical labor incomes w i t h identical combinations
of factors,in practice they might still not earn
the same amount because one might be afraid of
becoming "land-poor," for instance.

The reader of this study should always bear in mind
that

the essence of the Cobb-Douglas m e t h o d is comparative

Its value lies in the fact that a great number of relation
ships between various parts of the farm can be found out.
For example,

what bearing will an increased number of

cows have on a decreased proportional expenditure for
labor per cow?
The possibilities of the usefulness of this me th o d
for the analysis of farm accounts are infinite*

Only a

few of the applications of the method can be given in
this study.

But it will be a rewarding field for any one

interested In pursuing further the delicate

croso— in­

fluences between what a farmer puts Into his farm,
what he gets out of it financially.

ix

and

TABLE OF CONTENTS

Page
A c knowle dgeme nt s

ii

V i t a .........

Iv

Introduction.

.

V

Chapter
I. THE GROSS INCOME OF A F A R M A S A F U N C T I O N OF
C AT EGORIES OF I N P U T S ................................. 1
1. A n Es timate of Gross I n c o m e ................. 1
2. The Cobb-Dou^las Statistical Function. • 8
3. The Dependence of Gross Income u p o n
F ac to rs of P r o d u c t i o n ...........
12
Li-. Individual Value Pr od uc ti vi t y F u n c ­
tions and the Cobb-Dougles Function. . . 15
II. DATA F R O M F ARM ACC OU NT RECORDS IN TYPE-OF F A R M IN G A R E A S 5 A N D 6 , MICHIGAN, 1950 .........

21

1. The Value of Land. . . . . . . . . . . .
21
2. The Va l u a t i o n of Dairy C o w s ................ 2 I4.
3. The M ea su re me n t of Categories of F a c ­
tors Other Than Land and D airy Cows. . • 25
Ij-. The Data Concerning Gross Income . . . .
27
5. The E f f e c t s of Price Changes of Crops
H eld in Inventory and of Changes In
"Expected" Prices u pon the R e l i a b i l ­
ity of E s t im at es of Gross Income . . . .
30
..............
32
6 . Co nc lu si o n
III. THE M E T H O D OF CALCULATING GROSS INCOME
E Q U A T I O N S F ROM FARM ACCO U NT D A T A .................. 33
1. Farms In Type-of-Farming A reas 5 and •
6 ......................... ................... 33
2. Procedure for Calculating the Gross
Income E s t i m a t i n g E q u a t i o n s ................. 38

X

TABLE OP C O N T E N T S ( C o n t » d . )
Chapter

Page

IV. GROSS INCOME E Q U A T I O N S A N D THEIR D E R I V A ­
T I VE S ................................................ i+3
1. Gross Income E st i m a t i n g Equations. .
. . 1+3
2. Comparison of Coefficients of Elastlcity and Their Confidence
Intervals for Dairy and Not-Dairy
P a r m s .......................................1+7
3. Interpretation of Gross Income
E q u a t i o n s .................................. 5l
1+. Reasons for E r r o r s In the E s t i m ­
ate of Gross I n c o m e ...................... 53
5. The E f f e c t s of Ov er valuation and
U n d e r va l ua ti on of Categories of
F a c t o r s ....................................... 55
6 . The Constant T e r m In the E s t i m a t ­
ing E q u a t i o n ................................ 56
7« Elasti ci t ie s of Gross Income with
respect to Categories of Factors . . . .
57
8 . Marginal Value P r o d u c t i v i t i e s .............60
9* Est im at e s
of Net I n c o m e ................. 69
10# Es timates of L a b o r Income W h e n
Categories of Factors Are Charged
at Marginal Value Productivities . . . .
75
11# Graphic P re sentation of the E s t i m ­
ate of Gross Income and Costs. . . . . .
77
12* Summary of Chapter I V ......................85
V. USE OF GROSS INCOME F U N CT IO NS IN THE
FAR M MANAGEMENT E X T E N S I O N P R O G R A M ................. 87
1. Suggested Changes in the Data and
Method of A n a l y s i s .......................... 87
2. Selection
of Categories of Factors .. . 89
3. Suggested
Uses of Gross Income
Functions In Farm Du si ness Ana ly si s
R e p o r t s ....................................... 92
Part

I. E st imates of Gross I n ­
come ard L abor Income. . . . 9 2
Part
II. E s t im at es of Marginal
Value Productivities . . . . 9 8
Part III. Effects of Yields, Rates
of Production, Prices, and
Size of Business on V a r i a ­
tions In Gross Income Be ­
tween F a m s ................... 102

xi

T ABLE OF CO NT EN TS (Cont'd.)
Page
Part IV* E x t e n s i o n of A n a ly s is of
Differences In Gross and
L a bo r Income to Two or
More Enterprises* • • • • • •

108

Ij.* The V al uation of Farm B u s i n e s s e s ........... llU
5* Subjective Rates of Charge. • * . • • . .
115
App en di x
A

C OEFFICIENTS OF E L A S T I C I T Y OF GROSS INCOME
WITH RESPECT TO CA TE GO RI ES OF F A C T O R S ........... 118

B

ILLUSTRATION OF RETURNS TO S C A L E .................119

C

THE RELATION OF THE MEAN VALUE P RODUCTIVITY
FUNCTION TO THE EST IM A TE OF GROSS INCOME BY
LEAST S Q U A R E S ....................................... 120

D

THE VALUATION OF L A N D ...............................127

E

A TEST OF DIFFERENCES IN RECORDED VALU ES OF
LAN D A N D I M P R O V E M E N T S .............................. 130

F

THE VALUATION OF DAIRY C O W S ........................ 138

G

E F F E C T S OF CHANGES IN PRICES OF INVENTORIES
A N D OF RELATIVE CHANGES IN SELLING PRICES
ON GROSS I N C O M E .....................................1^3

H

LI NEARITY OF R EL ATIONSHIPS ......................

I

THE PROBLEM OF MULTIPLE SOLUTIONS OF THE
G ROSS INCOME E Q U A T I O N ..............................163

J

E S T I M A T E S OF MARGINAL VALUE PRODUCTIVITIES
FOR IOWA F A R M S .....................................I 67

K

AN ANA LY SI S OF DIFFERENCES BETWEEN E S T I M A T ­
E D A N D REPORTED GROSS INCOME A N D LABOR I N ­
COME ................................................ 169

xi 1

153

INDEX OP TABLES

Number*
1.
2.
3m

Pag©
Income and E xpe ns e Items, Type-of-Farming
A r e a s 5 and 6, 1950. . . . . . . . . . . . .

3U

Income, Expense, and Investment Data R e leted to L a bor Income, Ar e a 5. • • • • • • •

36

Income, Expense, and Investment Data R e ­
lated to L a b o r Income, A rea 6, • • • • • • •

37

U.

Categories of Factors as Given in the
Mic hi g an F a r m Acc ou nt s
............. . 3 9

5«

Ten Gross Income E s t i m a t i n g E q u a t i o n s for
8 6 Dairy Farms, Are a s 5 and 6 . • • • • • • •

1+J+

Six Gross Income E s t i m a t i n g E q u a t io ns for
108 Farms Other Than Dairy, A reas 5
and 6 . . . . . . .

b5

Three Gross Income E s t i m a t i n g E q u a t io ns
for 19U Farms, Areas 5 and 6 . . . . . . . .

1+6

Gross Income E q u a t io ns B a se d on C a t e g o r ­
ies of Productive Factors, 86 Dairy
Farms and 108 Farms Not Dairy, Ar eas
5 and 6 .

I4.8

o.

7.
8.

9.

10.

11.
12.

(Cont'd.) Gross Income E q u a t io n s Based on
Categories of Productive Factors, 8 6
Dairy Farms and 108 Farms Not Dairy,
A r ea s 5 and 6.
.......................1+9
( C o n t ’d. -2) Gross Income Eq u a t i o n s Based
on Categories of Productive Factors, 86
Dairy Farms and 108 Farms Not Dairy,
Areas 5 and 6 . . . ...........
Average Values of Categories of Factors,
Areas 5 and 6 . ................

50
62

Est i ma te s of Marginal Value P r o d u c t i v i ­
ties of Categories of Inputs, 86 Dairy
Farms, Areas 5 and 6 . ......................... 63

xiii

INDEX OP TABLES

(Cont'd. )

Number
13*

ll+,

15«

Page
E s t i m a t e s or Marginal Value P r o d u c t i v ­
ities of Categories of Inputs, 108
Not-Dairy Farms, A r ea s 5 and o. • • • • • • •

61+

E s t i m a t e s of Marginal Value P r o d u c t i v ­
ities of Categories of Inputs, 1914Farms, A r e a s 5 and 6. •

65

Average Marginal Value Productivities of
Categories of F a c t o r s • • • • • • • • . . • •
86 D a ir y Farms, A r e a s 5 and 6

66

lo.

A n alysis of Recorded Labor Income A c c o r d ­
............. 96
ing to E stimate of Gross Income

17»

Amounts and Mean Marginal Value P r o d u c t i v ­
ities of Categories of Factors for Farms
C l a s si fi ed A c c o r d i n g to La bor Income. . . . .

99

18.

E s t i m a t e d Marginal Value Productivites of
Categories of Factors for Different I n ­
vestments in Productive Livestock, 19l+
F a m s , Areas 5 and 6 ............................ 101

19.

Categories of Factors and T heir E s t i m a t e d
Marginal Value P ro du c t i v i t i e s .

103

20,

Data Nee de d to A c c o u n t for Difference in
R ec or de d Labor Income from E s t i m a t e d L a ­
bor Income on a One-Product F a r m ...............10l+

21,

Ex pl an at io n of Difference of L a b o r Income
from its E s t i m a t e d Value, Ac c o r d i n g to
E f f e ct s of Pr od uc ti on Rate, Price, Size
of Business, and C o m b in at io n of Factors,
on a O n e —product F a rm

107

22,

E s t i ma te s of Gross Income and La bor Income
from the C o m b i na ti o n of Factors on a
Michigan Dairy F arm
.............. 109

23.

A n a l y s i s of Yields, Prices, and Units of
M a jo r E n t e rp ri se s on a Mic hi ga n Dairy
F a r m E a r n i n g a Higher Gross Income than
E s t i m a t e d ...........

110

Anal ys is of Difference of Gross Income
from Its Es t i m a t e d Value A c c o r di ng to
Yield, Price, and Size of Business E f ­
fects on a M i c h ig an Dairy Farm, 1950* . . .

112

2l+•

xiv

INDEX OP T A B L E S (Cont'd.)
Number

Pace

25•

Average Values of L a n d r and L and and I m ­
p r o v em en ts p er Acres F a r m A c c o u nt s B e ­
ginn in g in Se lected Years, 1929-1914-9 • . . • 128

26*

The M i c h i g a n F a r m Real E st a t e Index and
an Index of V al u e s of L a n d and I m p r ov e­
m e n t s per Acre of Beginning F a r m Account
C o o p e r a t o r s ....................................... 132

27.

The Reis tion B e t w e e n the Y e a r of Start­
ing F a r m A cco un t Records and the D i s ­
p e r s io n of R e p or te d La n d Values, M i c h i ­
gan F a r m A c c ou nt Farms, 1929-191+9. . • . . • 133

28.

Summary of Inventory Changes for I4.5 Farms,
Areas 5 end 6, 1 9 5 0 ............................... lU3

29.

E nd i n g Crop Inventories Valued at B e g i n ­
ning Inventory Prices on 50 Farms, Areas
5 and 6 , 1950 ..................................ll+l+

30.

Be ginning and E n d i n g Inventories of D i f ­
ferent Kinds of Livestock,
3 8 Farms,
Ar eas 5
and 6, 1950, . , .. .............. lltS

31.

Summary of Gross Income, Change in Gross
Income, and Change In L a bo r Income B e ­
cause of Price Changes in F eed said Crop
Inventories, 1+3 Farms, Areas 5 and 6 , . . . 1 I4.6

32.

C o rr elation Coefficients between Categor­
ies of Factors a n d Repo rt ed Gross In­
come Compared wd th C or re la ti o n C o e f f i ­
cients Between Categories of Factors and
R e v i se d Gross Income, 3 I4. Farms, Areas
5 and 6 , 1 9 5 0 .....................................II4.8

33o

W e i g h t e d Average Annual M i c h i g a n Farm
Prices of Crops, Livestock, and L i v e ­
stock Products, and Ratios of Average
19l+&-$o“191+9 Prices to the Average
Prices of 1950 .................................

3l+.

151

TIntner and Brownlee 's "Estimated M a r ­
ginal Productivities (Per Dollar of
Inputs) and Fiducial Limits (At the 5
Percent L e v e l ) ” .................................. 1 6 7

xv

IITDEX OP TABLES

(Cont»d*)

Number
35*

36.

37.

Page
Heady»s n Marginal Productivities
and Fiducial L i mi t s at the Five P e r ­
cent Level of Pro ba bi li t y (Per
Doll ar of Input). • • • • • • • • • • • •

169

E x p l a n at io n of Difference Between
Rec or d ed and E x p e c t e d Values of
Gross Income and Total Costs. • • • • • •

172

Analysis of Difference in Gross In­
come from Its E x p e c t e d Value by E n ­
terprises, Y i e l d Effects, Price E f ­
fects, and Size of Business . . . . . . .

173

xvi

INDEX OP F I G U R E S
Figure
1.

2.

3.

U*

5*

6.

Page
The R e l a t i o n B e t w e e n L a b o r Income as R e ­
corded and L a b o r Income as E s t i m a t e d
f rom a Value P r o d u c t i v i t y Function,
8 6 D a i r y Farms, A r e a s 5 a n d 6 , 1950. . . .
The R e l a t i o n B e t w e e n R e c o r d e d L a b o r I n ­
come and E s t i m a t e d L a b o r Income w h e n
F actors oT P r o d u c t i o n Are Char ge d
against the 3u siness A c c o r d i n g to
Their M a r g i n a l Value Productivities. . . .
8 6 Dai-ry Farms, A r e a s 5 a nd 6 , 1950

71

76

Gross Income E s t i m a t e d f rom Total F a r m
Eqr en u e and Total Investment, and T o ­
tal Costs, In c luding Interest on the
Investment at % % but Not I n c lu di ng O p ­
erator* s Labor, 8 6 Dairy Farms, A r e a s
5 and 6, 1 9 5 0 .................................

78

Gross Income E s t i m a t e d from Total F a r m
E x p e n s e and Total Investment, 8 6 D a i ry
Farms, Ar eas 5 and 6 , 1950 . . . . . . . .

79

Gross Income E s t i m a t e d from Total F a r m
E x p e n s e and Total Investment, 8 6 D airy
Farms, Areas 5 and 6, 1 950 .................

82

E f f e c t on I s o —Cost Lines of Ch arging T o ­
tal F arm E x p en se at 0.8 of Re po r te d
Amo un t and of Charging Total I n v e s t ­
m e n t at 1 0 ^.

81+

7.

The R e l a t i o n of Theoretical Individual
P r o d u c t i o n F u n c t i o n s to the Co bb -D ou glas F u n c t i o n ...................................123

8.

The C o b b —Douglas Pr o d u c t i o n F u n c t i o n as
a M easure of the R e l a t i o n of Product to
I n p u t ............................................ 125

9.

Stated Values of F a r m L a n d and Buildings
of Be ginning Account Cooper a to rs C o m ­
p a re d with the Mich ig an F a r m Real E s ­
tate Index, S elected Years, 1 9 2 9 —191+9-

xv 11

. • 130

INDEX OP F I G U R E S

(Cont'd.)

Figure

Page

10#

The Relati o ns hi p B e t w e e n Stated Value of
L a n d and Improv em en ts p e r Acre A n d the
Y e a r of Sta rt in g F a r m Accounts, Sele ct ­
ed Years, 1 9 2 9 - & 9 ................................ 131

11.

The Re la t i o n s h i p B e t w ee n the Stated Value
of L a nd a n d Impro ve m en ts per A c r e as of
191+9* and the Y e a r F a r m A c c o u n t s Were
Started, S e l e ct ed Years, 1929-1914-9........... 13U

12.

The R e l a t io ns hi p B e t w e e n the lhl+9-Stated
Value of L a n d p e r Acre, and the Y e a r of
Starting T a m Accounts, S e l e ct e d years,
1929-14-9...........................................

135

13*

The R e l a t i o n B e t w e e n In v e n t o r y Value of
Dairy Cows and the Y e a r of St arting F a r m
Accounts, 32 R a n d o m Farms, Areas 5 and 6 ,
1 9 5 0 ...............................................139

ll+.

Dairy Sales per Cow and Average Value of
C ows per Head at Be ginning Inventory, 32
Ran do m Farms, A r e as 5 and 6 , 1950. . . . . .

lUO

15.

The R e l at io n B e t w e e n Average Value of D airy
Cows p e r Head, Beginning Inventory, and
Pounds of 3 .5% F a t — Co rr ec te d M i l k Sold
Per Cow, 26 R a n d o m Farms, A r e a s 5 and 6 ,
1950 .............................................. lU2

16.

R e l e t i o n B e t w e e n Investment In L a n d and
Gross Income, in Logarithms, 1+0 Farms,
Areas 5 and 6 , 1 9 5 0 ............................

15U

17*

R e l a ti on Between the L o g of Total L a bor
Charge and Residual from E s t im at e of
Gross Income from Investment in Land, 1+0
Farms, A r eas 5 and 6 ............................155

ift.

R e l a t i o n of Log of Investments Other T h a n
L and to the Res i d u a l f r o m the E s t i m a t e of
the Log of Gross Income from the Logs of
Toto.l L a b o r Charge and Investment In Lend,
1+0 Farms, Areas 5 and 6, 1950. . . . . . . .

156

R e l a t i o n of L o g of Investments Other T h a n
Land to Log of Gross Income, 1+0 Farms,
Areas 5 and 6, 1 9 5 0 ............................

158

19*

xv ii i

INDEX OP F I G U R E S

(Cont»d.)

Figvlre
20.

21.

Page
R e l at io n of Log of Investments Other
Than Land to Residual from E stimate
of L og of Gross Income from L o g of
Investment in Land, I4.O Farms, Areas
5 and 6 , 1 9 5 0 ...................................

159

R e l at io n of L o g of Total Labor Charge
to Residual of Estimate of Log of
Gross Income from L o g of Investment
O t he r Than Land and Log of I n v e s t ­
ment In Land, I4.O Farms, Areas 5 and

6 , 1950 ................................... 160
22.

23.

The Log of Gross Income as a Plane D e ­
termined by the Logs of Total Farm
Expense and Total Investment, i+0 F a m s ,
Areas 5 and 6 , 1 9 5 0 .............

162

Three Solutions of the Relationship B e ­
tween Gross Income, Total F a r m Expense,
and Total I n v e s t m e n t ........................... I 6 I4.
I4.O Farms, Areas 5 and 6 , 1950

xix

PROB LE MS A N D RESULTS IN THE USE
OF F A RM A C C O U N T R E C O R D S TO DERIVE
C O BB-DOUOLAS VAL UE PR OD U CT IV IT Y FUNC TI ON S
CHAPTER I
THE G R O S S INCOME OF A F A R M A S A F U N C TI ON OF C A T ­
E G O R I E S OF INPUTS
1. A n Estimate of Gross Income
This study is concerned w i t h e s t i m a t i o n of gross
income o f farms f rom the values of the categories of
factors oT p r o d uc ti on employed.
type

A n equation of the

a b
P - Cx

k
...............z

is u s e d to estimate gross income.^

So used,

this e q u a ­

ls Theoretically, value product functions o f individual
firms for categories of inputs in the economy, If
known, could be used in connection with demand and
supply curves of factors and products to solve the
economic system.
If estimates can be made of the m e a n
values of the coefficients, they m a y have some social
implications.
See Joan Robinson, " E u l e r ’s Theorem and
the Pro bl em of Distribution," Eco no mi c Journal, V. UU
(193U) p. 398.
This equation is sometimes called a "Cobb-Douglas"
function w he n applied to a theoretical statement of
the gross Income of a single firm.
Actually, the
function antedates Cobb *s and Douglas* work, the f u n c ­
tion appearing in the w or k of WIcksell.
(See Knut
V/Icksell, Lectures in Political E c o n o m y , V. 1, pp.
121-3, 127-130; also Martin Bronfenbrenner, "Pro­
duction Functions: Cobb-Douglas, Interfirm, Intraf i m , 11 D c o n o m e t r l c a , V. 12 (Jan., lQljlj.) pp. 37,8).
Furthermore, Douglas actually s o u 3ht average f u n c ­
tions for numbers of firms by statistical methods.
His work was not with Individual firms.
See P. II.
Douglas, The Theory of W a g e s .

2-

tlon Is a value pr od uc ti vi ty fun ct io n of categories of
Inputs.

Categories of Inputs, m e a s u r e d In dollars,

are

factors determining the gross Income, w h i c h Is likewise
m e a s ur ed In dollars.

Gross income is g i ve n by P; x,

•••£ are the values of different categories of Inputs
(factors) used o n farms.

C is a constant term, and a,

b , ..•k are powers to which the respective categories of
factors are raised.

In words,

the e q u at io n says that gross

income equals some number times the Investment I n l a n d (for
example) raised to some power,
than in land (again,

times the investments other

for example)

raised to some power,

and so on.
The terms inputs and factors
be u s e d synonymously.

(of production)

will

On both sides of the equ at io n the

terms are expressed in dollars.

This means that different

physical products and different kinds of factors have been
reduced to their dollar values and combined into dollars'
w orth of gross Income on the left side,

and dollars' w o r t h

of each category of factors on the right.
W h e n equations are u se d to express the physical p r o ­
duct of a n enterprise or of a phase of a total enterprise
in terms of physical quantities of Inputs,

the equ at io n Is

1

called a pr od u ct io n f u n c t i o n .

Pr o d u c t i o n then refers

TZ See ^Input-Output Relationships in M i l k Production,**
U. S. D. A. Technical B ulletin No. 8l5 (May, I9I4.2 );
W. J. Spillman, "Use of E xp on ential Y i e l d Curves in
Fertilizer Experiments," U.S.D.A. Technical B u l le ti n
No. 318 (1933)5 E a rl 0. Heady and Carl W. Allen, "Re­
turns from Capital Required for Soil Conservation F a r m ­
ing Systems," Iowa Research B ulletin 381 (May, 19^1)•

directly to "product**

Thus, a p r o d u c t i o n f u n c t i o n

could be d e r i v e d In w h i c h m i l k c o u l d be a p r o d u c t of
dairy c a ttl e * hay*

a n d co n c entrates*

Or*

the m i l k g i v ­

e n by one cow c o u l d be stated as a function of silage*
hay*

a nd concentrates*
W i t h a value p r o d u c t i v i t y e q u a t i o n of the f o r m g i v ­

e n o n page 1*

t o gether w i t h data r e g arding inputs for a

par t i c u l a r business*

It Is t h e o r e t i c a l l y pos s i b l e to

obt a i n estimates o f the following:
1* The g r o s s Income
2. E l a s t i c i t i e s of gross income w i t h r e s p ect

?

to categories of factors*
3* M a r g i n a l Incomes attrib u t a b l e to categories
of factors*
1+. I n f o r m a t i o n as to w h e t h e r the bu siness o p e r ­
ates a c c o r d i n g to increasing*

constant*

or d e c r e a s ­

ing returns to scale*
5* The net Income*

If the rates at w h i c h cat­

egories of Inputs are cha rg ed against the busi ne ss
are given*
The exponents of the value p r o d u c t i v i t y f u n c t i o n s
TZ See C h a p t e r VI f o r a n i n t e r p r e t a t i o n of the m e a n i n g
of e a c h of the five Items listed*
2* See A p p e n d i x A* and G e r h a r d T i n t n e r a n d 0* H* B r o w n ­
lee* " P r o d u c t i o n F u n c t i o n s D e r i v e d f r o m F a r m R e c o rds*"
J. F* E,* V. 2 6 (Aug., 19Ui+) pp. 566-571.

-1*-

ln C h a pter IV are the elaet l e l t l e e or groia income
wit h respect to the categories of factors*

B y this

elasticity Is m e a n t the ratio of the relative change
in gross Income to the relative change in a specified
category of factors*

C o n t i n u i n g w i t h the symbols g iv­

en on page 1 , we f ind that this elasticity Is, b y f or­
mula:
change i n E

s*

,

s

E .

change In JL
X

- -‘SE *

i

£ = A£._Z_» «£&y

ay

The n o t i o n of e l a s ticity is Important in consider­
ing returns to scale*

A business o perates ac c o r d i n g to

Increasing returns to scale if, w i t h a n equal p r o p o r t i o n ­
al change in all of the categories of factors,

the

gross Income changes in a gre a t e r proportion*

Thus,

if

the prices of the factors do not change as the f i r m
employs more of them, it is apparent that the business;
must become i n c reasingly profitable as its else is made
larger*

If

returns to scale are constant,

gross Income

changes in the same p r o p o r t i o n as the e m p l o y e d factors
are changed*

If returns to scale are decreasing and the

factors are changed in equal proportions,

then the gross

1
Income will change in a smaller proportion*
The sum of the elasticities of gross Income w i t h
respect to the categories of factors indicates the n a ­
ture of returns to scale*
one,

If the sum is g r e a t e r t h a n

the business is operating under increasing returns

I* Knut W i c k sell, op* c l t *.

pp.

lS^-lK)*

5to seal*.

If the sum Is on* exactly,

scale are constant*

t hen returns to

A n d If it is less than one,

to scale are decreasing*

returns

The m a t h e m a t i c a l p r o o f of this

p r o p o sition Is given by E u l e r ' s theorem.

That the p r o ­

p o s i t i o n Is true can be seen simply by considering first
the m e a n i n g of elasticity,

and then the m e a n i n g of the

1

sum of the elasticities*

I f the factor categories are

all Increased In the same proportion,

then the elasticity

of "x" category gives the ratio o f the relative
In gross Income to the relative change In jc*

change

The e l a s ­

ticity of y gives the ratio of the relative change In
gross Income to the relative

change In y, and so on*

If x, y* • ••> are all Increased In the same proportion,
then the elasticity of gross income w ith respect to
eac h category gives the proportional change in gross I n ­
come ascrlbable to each category*
categories of factors Is uniformly

If the Increase In all
for example,

the change In gross income will be \ if the

then

sum of the

elasticities is exactly 1 ; gross income will change by
more or less than % if the sum of the e l asticities Is
greater than o r less than 1, respectively.

It should

be p o i nted out here that the use of the sum of the
elasticities of gross Income w i t h respect to factor
categories to a n s w e r questions of returns to scale
Implies that all of the categories are included In the

T.” IblcT.-----------------------------------------

-6-

value pro du ct equation.

Obviously,

if one of the

cate­

gories is m i s s i n g and the elastic it i es of gross income
w i t h respect to the others are g i v e n correctly,
of the el asticities times the proportional
respective
change

the sura

change in the

categories will give a smaller p r o p o rt io na l

in gross income then w o ul d be in dicated should

all of the factor categories be included.

However,

the

degree of a fu nction may be useful even though not the
true m e a s u r e m e n t of " s c a l e , ” if omit te d factors are not
easily changed in amount.
Es t i m a t e s of the marginal

income a ttributable

to

factors can be determined by differentiating the e s t i m a t ­
ing eqiiation w i t h respect to the factors.

By the m a r ­

ginal income of an input category is m e ant

the i ncre­

mental

change

an incremental

in gross income wnich is a s s o c i a t e d w i t h
change

in the

specified group of inputs.

E s t i m a t e s of net income

can be made from the value

produc t iv it y f un ct i o n provided the rates at w h i c h the
categories of inputs should be charged are known.
bor income

La­

can be given thus:
a b
L r Cx

k
..... z. - (mx / njr / . . .u z ) .

Here l a b or income is indicated by L; m, n,

..... u are

rates at w h i c h the categories of factors are char ge d in
subtracting costs f r o m gross income in the determ in at i on
of labor income,

for example,

in the M i c h i g a n farm

-7account project,

total f a r m expenses are charged at a

rate or 3L; all investments are charged at a rate or
The symbols m, a,
inputs,
a £
Cx y .

. ..n

do not refer to quantities or

xr the factor categories are g i v e n values in
k
w h i c h e x a c t l y equal their u sed up costs,

then in the net income e q u a t i o n m, n, ...a w o u l d all
equal 1.

The cases or Interest and deprec i a t i o n will

call ror values or 9 , n, ...p dirrerent t v o s 1 ir the
ft 1i
factor categories are e n t e r e d in C x y
at their asset
values.

T h e n in the case or ma c h i n e r y deprec i a t e d at

10£, w i t h a 5 % Interest charge against investment, the
value or m wou l d be 0«l5»
it not depreciable,

In the case or land,

assuming

the value or n w o u l d be 0 .05*

The value product fu n c t i o n which has b e e n discussed
in the previous pages will indicate diminishing returns
to a speciried category or factors if the elasticity or
gross income v/ith respect to the category is less than
1.

That is, if the value or

page 1 is i,

g.

in the e quation o n

then gross Income will change as the square

root or the category or ractors called
categories are h eld constant.

x,

ir other

ir all other categories

are held constant and x is Increased,thus,

the relative

change in gross income will be smaller than the relative
change in £•

It is apparent that in the real economic

w o rld the values or the terms a, b, • • .k m u s t generally

be individually less than 1.

F r o m w hat has b e e n said

about returns to s c a l e , it w o u l d a p p e a r that, for a
purely competitive business in equilibrium,

the sum of

these terms must be n e a r 1$ it p a y s the business nei ther
to e x pand nor to contract*
1

•The Cobb-Douglas Statistical
Function:
---------------------The m a t h e m a t i c a l f u n c t i o n P = C x

h a s b e e n u s e d in

efforts to measure average mar g i n a l p r o d u c t i v i t i e s of fac
tors employed for industries w i t h i n regions and national
economies*

P r o f e s s o r Paul H* Dou g l a s has b e e n a s s o c i a t e d
2
w i t h an extensive list of such studies*
The Cobb-Douglas
pr o d u c t i o n function Itself Involves two categories of
factors of production,

labor,

and capital*

Douglas' e s ­

timates of p r o d u c t i o n functions f o r various Industries
TI In some of the l i t e r a t u r e , the Cobb-Douglas f u nction
is taken to m e a n any value p r o d u c t i v i t y or p r o d u c t i o n
f u n ctio n o f the form f m Cx*^fc. usi n g the definitions
of the symbols given o n page 2* W h y the names of Cobb
and Douglas should refer to all u s e s of this type of
functio n w i t h regard to factors and pro d u c t is not
altogether clear* In this study the title Cobb-Douglas
will be interpreted to m e a n the estimate o f product
(specifically, gross income) by multiple r e g r e s s ion
f rom categories of factors* See footnote 1, page 1.
2. The following list Includes part of the studies of
Cobb, Douglas, and associates: P. H* D o u g la s and C.
W. Cobb, "A The o r y of Production," A«E*R* * V* 18,
supplement, (1928), pp. 139-165*
C. W. Cobb, " P ro­
duction in Massachusetts Manufacturing, 31890—1926,."
J*P * E ., V. 39 (1930), pp. 705-7*
P. H. Douglas,
Patricia Daly, and E r n e s t Olson, "The P r o d u c t i o n F u n c ­
tion for Manufa c t u r i n g in the U. S., 19014.," J»P.E..
V* 51 (Feb., 19i4-3)# pp* 6 1 —5«
P. H. Dou g l a s and Grace
T. Gunn, "Further Measurement of Marginal P r o d u c t ivity
Q.* J*E. . V* 514- (19*4-0) , pp399-l4-2.9£ also "The P r o d u c t i o n
F u n c t i o n for A m e r i c a n M a n u f a c t u r i n g for 192l|.»* J*P*B
V. 30 (Aug., 191*2), PP. 595-602.
------

vert b a s e d o n o b s e r v a t i o n s I n v h i o b p r o d u c t v a r i e d
a c c o r d i n g to d i f f e r e n c e s In the w a y In w h i c h l a b o r a n d
1

capital were

combined*

The o r i g i n a l u s e r s of this

m e t h o d of study sought to o b t a i n such o b s e r v a t i o n s by
considering whole

Indust r i e s o ver a p e r i o d of years,

In

w h i c h the records of an I n d u s t r y f o r e a c h y e a r w o u l d
show differ e n t c o mbinations of factors a n d product*
a n o ther m e t h o d u s e d by Douglas,

In

grou p s of f i r m s w i t h i n

a n Industry were t r e a t e d as separate o b s e r v a t i o n s I n the
employment of fac t o r s over a common time period*

One of

p r o f e s s o r Douglas' p r i n c i p a l alms was to determine w h e t h e r
various Ind u s t r i e s were

competitive In f a c t o r m a r k e t s as

well as In the selling mar k e t s *

D o u g l a s I n d i c a t e d that

m a n u f a c t u r i n g b u s i n e s s e s were g e n e r a l l y competitive*
In this study It Is a s s u m e d that f a r m b u s i n e s s e s
are h i g h l y competitive*

The degree of c o m p e t i t i o n b e ­

t w een farms Is not the m a i n Issue here*
use of D o u g l a s ' basic statistical method.

However,

by the

It Is h o p ed

that estima t e s of the gross Income of farms w i l l be u s e #
ful In the d i s covery of p r i n c i p l e s of e f f i c i e n t f a r m
management •
The ob s e r v a t i o n s In this study are of the I 9I4. farms
In t y p e - o f - f a r m i n g a r e a s 5 and 6, Michigan, 1SJ£0.
o bservation s are all for one year*

The

T h i s elimin a t e s the

necessity for y e a r - t o - y e a r ad j u s t m e n t of f a c t o r and proT7 O b j e ctio n s to the C o b b - D o u g l a s f u n c t i o n w h i c h have
b een rai s e d b y o t her e c o n o m i s t s are c o n s idered later
In this chapter, be g i n n i n g o n page 10*

10-

duct values ac cording to changes In prices.
observations,

P r o m these

value p r o d u c t i v i t y functions will be e s ­

timated.
E xp o n e n t i a l

value p ro d u c t i v i t y functions of the

type given on page 1 have been u sed by P r o f e s s o r Hea dy
1

of Iowa and others in the study of farm account records.
In this study we shall likewise

try to determine

the

e xpected average values of the exponents and the constant
terms of the value produ ct iv it y
individual farms.

functions w h i c h face

To attempt this by statistical m e t h od s

implies the assump ti on that the farms in the

survey all

operate on es se nt ia ll y the same value pr o d u c t i v i t y
function.

The farms represent different p d s l tl on s on

the f unction acc or di n g to the combinations in w h i c h farm
operators e m pl oy categories of factors.

The fact that

the positions are different and that the gross incomes
are different w he n they are buying and selling in e s ­
sentially the

same markets from one farm to a n o t h e r c r e ­

ates a pro bl em of choice of factors.
F a r m operators choose different c o mbinations of
factors

in their individual efforts to maximize net

income according to (1) their p a r t i c u l a r circumstances
(with respect

to the a vailability of family labor,

for

Gerhard Tintner, nrA 1Jote on the D e r i v a t i o n of* Prod uc —
tion F un ctions from F a r m A c c o u n t s , ” E c o n o m e t r l c a , V.
12 (Jan., 1 9 h b ) 9 PP» 26-3h*
G e r ha rd T i n t ne r and 0.
H. Brownlee, “P r o d u c t i o n Fun ct i on s Derived f ro m F a r m
Records," J. F. E ., V. 26 (Aug., 19^4-), PP- 566-71.
Earl 0. Heady, ^Production F u n c ti on s from a Rand om
Sample of Farms," J. F. E., V. 28 (19U6), pp. 989100h.

1
ezaaple),

(2) their different concept a of not income,

and (3) their individual appraisals of tho relation
2
of gross Income to different factors.
Thus, granting
that all farms operate o n essentially the same value
productivity function,

farm operators for several rea3
sons will choose different combinations of factors*.

Wh e n the factors are g r o u p e d into categories,

the a v er­

age relation between the categories of faotors a n d
gross Income as stated by account records can be e s t i ­
mated by multiple regression.

The farmer will find

this average relation informative w h e n he Is f a ced w ith
the problem of choosing factors*

If there were one

choice of factors which m a x i m i s e d net Income according
to a common definition h eld by all farmers,

and if all

farmers were completely free to choose this combination,
then all choices of factors would be the same, and all
gross and net incomes would be the same.

There w o u l d

be no set of observations of different choice*,

and there

would be no statistical p r o b l e m of estimating the a ver­
age relationship between stated gross income and cate­
gories of factors*
Several rather serious theoretical objections have
TT F o r example, individuals will have different feelings
about owning land.
One m a y subjectively demand only
a h % return on land, and another m a y demand 8£*
2. Farmers can not, for instance, be in complete a g r e e ­
ment about the returns to ma c h i n e r y investment.
3. See pages

17

and I# following.

12

been raised against the process of estimating value
productivity functions from observations of competitive
businesses..

The equation w h ich has b een stated on

page 1 for one f a r m can be given In logarithmic form
for two factor categories, x and x »
Log

by

= log £ / a log £ / b. log x*

If the gross Incomes earned by different farms result
from different combinations of factors w i t h all operat~
ing on essentially the same value productivity function,
then the value productivity function can be estimated
by multiple regression from the linear equation
(log Jg) » =

log C / a, log x / b log x *

In this equation (log E ) ' is the estimate of the log
of gross Income.

This logarithmic function can be in­

terpreted In numbers as
& £

Z' — Cx

x

•

In logarithms the equation forms a plane; in natural
numbers it will fora a curved surface.
3»The Dependence of Gross Income U p o n Factors of P r o d u c­
tion :
Professor Mendershausen has questioned the justi­
fication of expressing gross income as a surface deter­
mined by categories of factors when the data do not
themselves suggest a surface but rather a m ass of points
1

or a line.

If gross Income should be expressed as a

Y l Horst Mendershausen, w0n the Significance o f Profeasor Douglas* Production F u n c t i o n . " Econometrics.
V. 6 (1938) pp. 143-347*

13

function of total f a r m expense a n d total investment,
for example,

it is p ossible m a t h e m a t i c a l l y to determine

a surface which, gives a best estimate of gross income
by assuming that these lat t e r two are Independent v a r ­
iables and that they are
However,

substitutable for one another.

the v a r i a t i o n s in gross Income not a s s o c i a t e d

wit h changes in f a c t o r s m a y be so great above a n d b e ­
low the statistical " surface" that it can n e i t h e r be
said w i t h certainty that a true g r o s s Income surface
does exist,
this case,

nor be said that it does not exist.

In

if the sums of the squares of the differences

between e s t i m a t e d and actual values of gross Income are
mi n i m i s e d in the directions of total f arm expense and
total investment,
income,

rath e r than in the direction of gross

different "surfaces" will result.

Simply stated,

it is possible to get several estimates of gross Income,
different answers to the same problem,

depending upon

the direction of the m i n i m i z a t i o n of the sums of the
squares of the differences.
I n this l a t t e r case a q u e s t i o n is raised of the
Justification of m i n i m i z a t i o n of the sum of the squares
of the differences b etween e s t i m a t e d and o b served values
of p in the P direction rat h e r t h a n in some oth e r d i rec­
tion.

In the case of the use of multiple r e g r ession to

study the response of gross Income to changes in the
dollar values of input categories,

o u r Justification is

that the gross Income of a f a r m can ba r ightly considered,
1

generally,
case,

to be a f u n c t i o n of the inputs*

In this

the argument resolves itself into a qu e s t i o n of

w h ether the expenditure o f resources on a f a r m arises
out of the p r o d u c t s sold f r o m the farm,

o r whe t h e r the
2
product results f r o m the use of resources*
It is b e ­
l i e v e d that p r o d u c t can generally be considered a resuit of factors*

3

There can be no pro d u c t wit h o u t fac-

1* A t the m i c r o level, the l e vel of the o p e ration o t the
individual farm, this argument is p a r t i c u l a r l y d e ­
fensible*
If the whole economy is considered (the
macro ILevel) then the demand curve faci n g the e n t r e ­
preneurs taken as a group can not be assumed to be
infinitely elastic*
The causal relationship f rom
factors to p roduct is no l o n g e r straight-forward*
A n Increase i n total factors could result in a d e ­
crease in total gross Income, even though, for any
individual farm, an Increase In factors might indic­
ate an increase in gross income*
2* D e t e r min a t i o n of a regression equation by m i n i m i z i n g
the sum of the squares of the differences bet w e e n
e s t imate d and actual values of gross Income in the
gross Income direction Implies that gross Income is
a function of the o t h e r variables in the equation*
It is recognized that the simple a s s u m p t i o n that
gross Income is a function of categories has l i m i t a ­
tions*
However, the data for this a s s u mption m a y
be adequate enough so that a m e a n ingful a n d consis­
tent set of relationships can be determined f rom a
number of farms*

3*

The Cowles Commission has b een Interested in dev e lop­
ing a m e t h o d of statistical analysis w h i c h rests on
supposedly more defensible grounds than m e r e l y o n
the thesis that the variable of interest is a simple
f u n ction of the "independent" variables*
F o r one
thing, w h e n the "Independent" variables are related
to e a c h other, they will take upon themselves p o r ­
tions of the total off e r e d explanations of the d e ­
pendent variable in a n apparently erratic fashion*
Partially to answ e r this objection, w o r k (cont. p . 15)

tors,

but It Is possible that there

can be factors

1

without product*
ij.* Individual Value Product F u n c t i o n s and the C o b b Douglas F u n c t i o n
A n o t h e r ob j e c t i o n to the statistical pr od uc ti on
function is pos si bl y more

serious on theoretical grounds.

It has b een p o i n t e d out by R eder and Bronfenbrenner
that this f u n c t i o n may not represent what it is supposed
2
to represent.
They show that the statistical function
is m e r el y a surface w h i c h describes the relation of
has been done on the "simultaneous equations"
approach, in w h i ch there are as m a ny equations as
unknowns, and each equa ti on is designed to take up
some important phase of the whole set of i nter­
relations.
The use of simultaneous equations m a y
not eliminate the problem w h i c h arises out of cross
r elationships between the independent variables*
They may, however, come closer to a full explanation
of the combined set of relationships than multiple
regression.
F o r the p r o bl em of estimation of the
gross income of a farm from categories of factors
possibly the advantages of simultaneous equations
over multiple regression are not so definite as in
the case of market relations.
Fo r discussions of
the m e t h o d and application of the simultaneous
approach, see M. A. Gershick and Trygve Haavelmo,
"Statistical A nalysis of the Demand for Food: E x ­
amples of Simultaneous E s t i m a t i o n of Structural
E quati on s, " E c o n o m e t r i c a , V. 15 (19^7), PP* 7 9 —111,
particularly™pp"^ 79-^3Also Wass i ly Leontief,
"Introduction to a Theory of the Internal Structure
of Functional Relationships," E c o n o m e t r i c a , V. 15
(19U7), PP« 361-372.
Also A. Wl Prest, "Some E x ­
periments in Demand Analysis," Review of Economics
and Statistics, V. 31 (19^9), ppT 33-^7•
1. See A p p e n d i x A for a more detailed analysis of the
questions of the existence of a logarithmic plane
expressing the relation of gross Income to categor­
ies of factors and the effects of m i n i m i z a t i o n of
the sums of the squares of the differences in di rec­
tions other than the direction of gross Income.
2* M. W. Reder,

"An Alternative Interpretation (cont. p.l6)

gross Income to the

combinations or factors w h i c h the

e ntrepreneurs actually employ.

They do not assume that

the entrepreneurs are operating on essentially the same
p roduction function.

Thus,

if the businesses are in

e quilibrium but all on essentially different pro du ct io n
functions,

then the Cobb-Douglas f unction is nothing

more than an envelope giving the optimum combinations
of factors for the various businesses and has little
analytical

significance for Individual businesses*

E a c h farm operator is simply ma xi mi zi n g his gross in­
come ac cording to his own pa rt ic ul a r p r o d u ct io n f un c ­
tion.

This could be altogether different in all cases

from the function given by considering the earned
-ross Incomes to be observations of results from the
1

employment of different combinations of factors.
As far as the mathematics of the pro bl em are concerned,
the

individual pro d uc ti on functions could be sloping

downward as the factors
labor and capital)

(In D r o n f e n b r e n n e r *s article,

are increased.

Ye t the most f a v o r ­

able positions for the firms could be such that the
loci of the coordinates for labor,

capital,

and p r o ­

duct should tend to move up'/©rd with respect to product
o f 'the Cobb-Douglas Function," Econometric©^ Vh IT
(July-Oct., 19U3)* PP* 259- 20I4.® M a r t i n Bronfenbrenner,
"Production Functions: Cobb-Douglas, Interfirm,
Intrafirm," E c o n o m e t r l c a , V. 12 (Jan., 19UU)# PP* 39-U2.
1. See pp. 11 and 12,

supra.

as l a b o r and capital are increased*
The o b j e c t i o n w h i c h R e d e r and B r o n f e n b r e n n e r raise
to the e s t i m a t i o n or a m e a n In t e r f i r m value p r o d u c t ­
ivity f u n c t i o n from the values o f the Inputs and the
gross Incomes of a num b e r of firms in an industry Is
based on the assumptions that the firms are in e q u i l ­
ibrium and are actually o p e r a t i n g o n different functions*
There is little question that firms in agriculture are
not in equilibrium*

O n the contrary,

the f a r m operators

are more or less continuously in the p r o c e s s of changing
the

structure of their businesses in o r d e r to m eet new

technological and eco n o m i c conditions*

It Is p r a c t i c a l ­

ly impossible to determine for a pa r t i c u l a r farm what
1 ts value p r o d u c t i v i t y function as of a g i v e n m o m e n t
will turn out to be in the light of changes in f a r m i ng
methods*

The process of ad j u s t i n g to a changing e n v i r o n ­

ment and even changing objectives of the farm o p erator
himself is to a great extent one of trial and error*
A n assumpti o n of this study is that the e x p e r i e n c e s of
a numb e r of farmers in the m a t t e r of gross Income re­
ceived from different combinations of factors will be
useful to operators in p lanning their businesses*

These

experiences are condensed to a n average In the e x p o n e n ­
tial value p r o d u c t i v i t y function.
Individually,

farmers seek to maximize p a r t icular

1
personal or family net u t i l i t y functions*

T h i s means

that farmers w i l l have different a t t itudes tow a r d the
risk of capital,

the expenditure o f their o w n and their

families' labor,

and so on*

2
P a r t of these differences

will depend u pon the resources,

including u n m e a s u r e d

resources, w h i c h the farmer h a s at his command*

3

Nei ther

will all farmers m a k e the same estimate of the rel a t ion­
ship b e t w e e n gross Income and factors*

Thus the fact

that farm operators actually choose differing quantities
and combinations of factors of p r o d u c t i o n does not im­
p l y nec e s s a r i l y that these p o s i t i o n s are different b e ­
cause the value p r o d u c t i v i t y functions are different*
The variations in employment of factors can occur b e ­
cause of differences in personal objectives alone,

even

should the farmers be o p e rating o n substantially the
same value p r o d u c t i v i t y function*

There are,

thus,

three reaso n s why the value p r oductivity functions fac­
ing the individual farms do not have to be different in
order to have a result o t h e r than a convergence of all
farms to the

same set of coordinates of factor categor­

ies and value of product*
3T* M a r t i n Bronfenbrenner,

To recapitulate: farms are
op,

c l t*.pp~T 37-3tJ.

2. D. B. Williams, "Price E x p e c t a t i o n s and Reactions to
Uncertai n t y by Farmers in Illinois," J* F* E *, V*
33 (Feb., 1 9 5 1) no. 1, pp. 20-22.

3«

M* Kalecki, "The Principle of Increasing Risk,"
E c o n o m e t r i c s . V* 4 (New series, 1937) PP* I4
J+O-I4.7,
particularly, I44.O-I4.2.

-19

not in equilibrium*

But e v e n a s suming that they were,

the Tact that they select different combinations of
factors and e arn different gross Incomes does not prove
that they are not on essent i a l l y the same value p r o ­
ductivity function*
The value p r o d u c t i v i t y f u n c t i o n g i v e n o n page 1 is
simple*

It does not contain all of the features w h i ch

m a y be need e d to express the true average value p r o ­
ductivity of the categories e m p l o y e d on a particular
farm*

F o r one thing, marginal value product w i t h r e ­

spect to any input category is always decreasing p r o ­
vided the e x p o n e n t of the category (a* b,

***or k) is

less than 1, as it a p p a r e n t l y m u s t be* In reality there
may be ranges in the use of factors- in w h i c h this
margi n a l p r o d u c t i v i t y increases*

O n the other hand,

according to this type of function,

the total product

will always increase w i t h a n increase in the e m p l o y ­
ment of any single factor, h o l d i n g o t her factors con­
stant*

In reality,

the total pro d u c t m a y in some

circumstances decrease w i t h the use of more of one of
the factors*

C o n s i d e r the case of the number of cows

on a 160-acre farm.

W i t h the num b e r of cows carried

by the farm continuously Increased,

a point would

eventually be reached at w h i c h the total value o f m i l k
and beef p r o d u c e d would actually fall should still
more

cows be added.

W i t h i n the range or the use of factors in p r a c ­
tical farm operation,

these are perhaps not serious

1

objections.

The f ora of the e quation does behave

according to several economic concepts.

It is r e l a t ­

ively easy to handle w i t h the use of logarithms.

If

the value of the constant term £_ and the exponents of
the factor categories,

a, b, ...jr are chosen b y the

m e t h o d of least squares,

the form of e q u a t i o n can be

expanded to give a good estimate of gross Income
over the economic range of choices of factors.
1. A functi o n or the type
could be fitted to
any stage of the true p o rductivity f u n c t i o n of a
f a r m or of a group of farms p r o v i d e d the data were
available.
The values of £ , cl, and Jfe. wo uld be
these wh ich applied to the stage.
Data are a v a i l ­
able for the so-called second stage, in w h i c h to­
tal value p r o ductivity is increasing a n d marginal
value productivities of categories of factors are
decreasing.
This is the stage of rational economic
activity in conditions of approximate equilibrium.
It is likely that the f u nction could not be e x ­
tended over all conceivable combinations of f a c ­
tor categories.

CHAPTER II
D A TA P R O M F A R M A C C O U N T REC OR DS

IN

T YP E- OF -FARMING A R E A S 5 A N D 6 , MICHIGAN, 195*0

This chapter p resents an evaluation of deta from
farm account records as a basis for e s t i m at in g gross
Income equations.
F a r m account records of 19U farms In type-offarming areas 5 and 6 , Michigan,
were u se d in the

study.

for the y ear 1950,

These types-of-farming areas

are In the south-central part of the lower peninsula
of Michigan.

One hun dr e d sixty-three farms In area

5 and 35 farms in area 6 kept records in 1950 *
Equati o n s estimating gross income will be given
in Chapter IV.

They will be derived from values of

categories of Inputs as these

categories are

by the farm accounting procedure.

set up

This chapter con­

siders some of the stated values of input categories
in the light of using them for mak in g estimates of
gross income from the categories of factors.
1. The Value of Land
In order to keep land values between farms com­
parable and thus tend to assure

comparability of the

measurement or total charges against the farms,

the

staff of F a r m M a n a g e m e n t E x t e n s i o n at M i c h i g a n State
College has a d o p t e d a poli c y of valuing the land of
new account cooperators similarly to the l a n d of
farmers *1 ready in the project.

The results given

In A p p e n d i c e s D and E Indicate that the p o l i c y has
been effective.

It is r e c o g n i z e d that over a p e r i o d

of years the l and values in the f arm account project
have drifted substantially b e l o w mark e t values.
A n objective of this study,

as has b e e n stated,

Is to estimate gross Income functions f r o m categories
of factors of production.
the estimates m a y arise

Some of the

shortcomings of

simply f rom Inconsistencies

between farms In the value of l a n d as stated in the
farm accounts*
There a p peared to be no way of un c o v e r i n g errors
in the va l u a t i o n of land b e t w e e n farms w h i c h started
records in the same year without a separate appraisal
of each farm.

L a n d values,

as long as the records

were started at e ssentially the same time, were

thought

to be as use f u l as could be o b t a i n e d without prohibitive
field work.

The use of crop yields and types of crops

grown as partial indicators of l a n d values was ruled
out.

Such a procedure would involve the v a l uation of

land after its value h a d been more or less proved*

It

w ould have the effect of arbitrarily a s s igning part
of the residual of gross Income to land,

with no reason

for doing so except that the residual exists*

There

appeared to be no workable m e t h o d o f separating the
differences in l a n d values due to Inherent differences
in the produc t i v i t y of the soil f r o m the variations
due to such differences as in favorableness of the
season,

farming systems,

and abilities of the in d i v i du­

al operators as crop producers*
The hy p o thesis was
ing records
values*.

set up that the y e a r of s t a r t ­

should have a tendency to bias stated l a n d

It was thought that a consistent bias w i t h

respect to the y e a r of starting farm account records
might be f o u n d and that it slight a p p e a r to be w o rth
eliminating by statistical means*
records for the whole

A l t o g e t h e r 176

state were used in tests to d e ­

termine w h e t h e r the y e a r of starting accounts h a d a
significant influence u pon the level at w h i c h farms
1

were valued*.

The conclusion is reached that the

y e a r of starting farm account records h a d no signifi­
cant Influence upon the level of land values or u p o n
the dispersion of the values around the means*
general,

In

land values as stated In the farm account

records for areas 5 and 6 for 1950 seem to be at levels
p r e v ailing In the late 1 9 3 0 * s, e v e n for farms on w h i c h
records were
n

started after the second w o rld war*

See A p p e n d i c e s D and E for the analysis of these
records*

2.The Valuation of D a i r y Cows?
In the case of dairy cows there was a tendency for
the stated values of animals In herds o n farms coming
into the accounting project In the years since the
second world war to be higher than the stated values of
animals in herds coming Into the project earlier*
The effect of this basis for differences in values of
animals will be partially to Invalidate the results of
the analysis of marginal returns o n investment* In
livestock*

A test was made of the relations of the

average stated value per cow to gross income received
1

per cow.

The relation was not significant at

The analysis of cow values does not establish the hypo*
thesis that the stated value of a cow is related to
the gross income which she produces for the farm*
The effect of this will be to reduce the statistical
response of gross income to dairy cattle investment*
W h e n the inconsistencies in the valuations of the fac«*
tors are not similar,

for example,

comparing invest­

ment In cows or land with cash expenditures for fer­
tilizers (with the latter given exactly) the effect
will be to increase the relationship of gross income
to fertilizer expenditures at the e ^ e n s e

of the rela­

tionship of gross Income to Investment In dairy cows.
Eliminating the influence of the time of starting farm
account records on the stated value of dairy cows per
T* Appendix P.

—

—

liea d about doubled the significance of the relations
obtained bet w e e n cow values and the output of 3 »5£
1

fat-corrected milk.

Inconsistencies in the valuations

of dairy cows represent one problem which should be
solved if gross Income estimating equations are to be
calculated as a part of the regular procedure In farm
business analysis.
3*The M easure ment of Categories o f Factors o ther than
Lan d and Dairy CowsT
Inputs other than land could be classified broadly
as labor or capital.
In the case of labor, we shall use the cost of
hired labor,

the charge for family labor,

the o p e r a ­

tor's labor charge, and the total labor charge as
they are reported.

Total labor charge Includes an

allowance for the operator's labor, w h ich was entered
at $ l £ 6 0 for all farms (with m i nor exceptions),

or

$130 per month for the time that the farm operator
worked on the farm.

It is, admittedly,

not a good

measure of the operator's labor and management input.
The operator's labor charge does not include an a l l o w­
ance for management.

The dollar estimate of the total

labor charge can be assumed generally to Include a
differential according to the quality of hired and
unpaid family labor.
I. Appendix F.

That is, more expensive h i red

labor Is p r o b a b l y more valuabl# t han cheaper h i r e d
labor•

The charge for unp a i d family labor takes into

consideration the age of e a c h worker*

A n effort to

charge for the operator's l a bor according to the gross
Income or the measure of net Income w o u l d amount to a
valuing of the operator's labor according to the total
residual*

I n the case of l a bor Income,

the total

residual to all factors above the stated total charges
Is assigne d to the operator*

Thus the o p e r a t o r Is

credited w i t h windfall gains a n d losses*

"Labor In­

come" and the value of the labor and m a n a g e m e n t Input
of the operator are not the

same thing*

In short,

there Is no satisfactory measure of the value of the
operator to the farm*
There are several ways of m e a s u r i n g the capital
input o n a farm*

A l l Items can be considered together

or they can be classified separately.

Investments other

than land can be taken from the farm account records
In such categories as machinery,
crops*

Such expenses as fuel,

livestock,

feeds,

and

fee^£ a n d f e r t ilizer

can be tak e n all together as "total farm expense" or
can be put in special categories.
As far as direct outlays for feed,

fertilizers,

and

so on, as they appear In such items as feed expense a nd
crop expense,

there appears to be no reason to believe

that these figures are bia s e d b etween farms.

The values

of livestock o t her than dairy are re l a t i v e l y u n i m p o r t ­
ant In the records included In the study*

A time e r r o r in va l u a t i o n of m a c h i n e r y arising
from differences in the y e a r s of starting r e c o r d s should
be relatively unimportant*

The m ost expensive types

of equipment generally n e e d to be replaced every few
years either because of obsolescence o r because of
physical wear*

W h e n new o r replacement items appe a r

in the far m inventory,
not always,

they are practi c a l l y always,

e n t e r e d at cost*

if

Since all of the farmers

in this study are buying in e ssentially the same m a ­
chinery market,

there

seems to be little

justification

for a n ass u m p t i o n that m a c h i n e r y investments are not
on substantially the same basis f rom one f a n a to another*
Fee d and crop inventories are comparable because these
items are entered at marke t prices, w h i c h are essentially
the same throughout the area*
U * The Data C o n c e r n i n g Gross Income:
Product is m e a s u r e d by gross income,

dollars being

the best least common denominator available in w h i c h to
express the output of the farm.

A s the data are for a

single year, price changes are not p a r t i c u l a r l y i m p or­
tant*.

F r o m the point of view of the total o p e ration

of a farm,

gross income seems to offer the m o s t access­

ible and meaningful estimate of the volume of product*
If the farms in the area or in the study p r o d u c e d one

specific p hysical product only,

say fluid milk,

w h ich was sold in the same market.

all of

It should be possible

to estimate pro d u c t simply by u s i n g fat— corrected milk*
However, m ost farms In the

study have a number of

enterprises o t h e r than dairy.

O n each farm other e n ter­

prises absorb a share of the total inputs.

Therefore

the e s t i m a t i o n of product f r o m the inputs must account
for the output of other enterprises a n d a least common
denominator must be used*
Several types of error m a y be introduced by u s ing
gross Income from farm records as the value product of
a farm*

(1)

It m a y Include Income from o f f the farm,

the p a y for w h i c h m a y be on some different basis from
the p ay for ope r a t i n g a farm*

(2) It m a y Include

Inventory gains arising out of price changes w h e n In­
ventories are valued at mark e t prices.

(3)

The o r g an­

ization of the farms in the survey is based on e x p e r­
ienced relations between prices of alternative products
and alternative inputs.
lationships

If In any one y e a r these r e ­

should be appreciably out of line,

then

gross Income Includes a windfall type of g ain or loss
which is not necess a r i l y related to operational e f f i ­
ciency*

(I4.) Gross income will include the effects of

events such as windstorms,

floods, unusual seasons,

and so forth.
W i t h these objections to the use of gross income

as the m e a s u r e o f gross product,
ask:

"Why use gross Income?"

reasons:

one oould logically

There are Tour m a i n

(1) X n the first place,

the objections a r i s ­

ing out of the first pr e c e d i n g item can be p a r t i a l l y
answered by eliminating unusual cases.

That is,

the

farmer who earns a large part of his Income from doing
custom work for other farmers can be exc l u d e d from the
study.

We can at least make the

statement,

then,

that

the results a p ply to the typical farm in the area of
the survey, not to farmers who are p r i marily contrac­
tors,

shop workers,

o r dealers in m erchandise*

(2)

It is possible to eliminate the effects u p o n gross in­
come of price changes of goods h e l d in inventory and
of changes from expected prices o f crops sold.

A

study was m a d e of a sample of farms in o r der to e v a l u ­
ate the importance of the e f f e c t s of these variables
upon gross incomes in the areas 5 and 6 for the y e a r
1

1?50.

(3) In the third place,

it would be extremely

difficult to find a satisfactory basis for adjusting
values of product to meet the qualifications of the
long run.
relative

No one can be

sure, for instance,

whether a

shift in the price of one product in comparison

to another is a temporary fluctu a t i o n from a long-run
average,
continue.

or whether it represents a trend w h i c h will
The relative fall of the m a r k e t price of

I. See page 30.

cotton a n d p o t a t o e s in r e l a t i o n to farm p r o d u c t s in
general o f f e r s i l l u strations of cases in w h i c h "de­
viations'* t u r n out to be secular trend*

(L|.) The use

of gross income p e r m i t s study o f the m a r g i n a l returns
to the factors u n d e r

the c o n d i t i o n s w h i c h face the

f arm

operator*

not o p e r a t e d in terms o f l ong

r un

averages;

F a r m s are

the costs a n d returns f rom far m i n g consist

of a series of short-run c o n d i t i o n s in the b u y i n g and
selling m a r k e t s *

If

the returns f o r a cer t a i n type of

o p e r a t i o n are low in

a g i v e n year,

the farmers w h o have

been a d v e r s e l y a f f e c t e d have no altern a t i v e except to
take their losses*

It is m o r e p e r t i n e n t that the re*

turns to factors s h o u l d be studied continuously in or *
der that changes in mar k e t s and techniques can be n o t e d
and responses of f arm operators to these changes observed.
R e g a r d i n g a specific area for a g i ven year, w i t h g i v e n
economic conditions,

t e c h n o l o g y a n d climate,

it is

important to m a k e a p r a c t i c a l a c c o u n t i n g of w h a t a c t u ­
ally happened,

not of what w o u l d have h a p p e n e d if the

w e a t h e r a n d p r i c e s h a d been "normal*"

Thus the idea of

a value p r o d u c t i v i t y fun c t i o n o r relation of categories
factors to gross Income h a s the advantages of b e ing
simple and realistic*
5 * The E f f e c t s o f Price Changes of Crops H e l d in In v e n tory
and of c h a n g e s In ^Expected" P r i c e s U p o n the R e l i a b i l i t y
of E s t i m a t e s of Gro s s I n c o m e :
The hy p o t h e s i s has b een stated that price

changes of

goods h e l d in I n v entory and changes in the m a r k e t p r ic e s

of f a r m p r o d u c t s f r o m " e x p e c t e d ” values constitute
a f o r m of r a n d o m e r r o r In gross income f r o m Its e x p e c t ­
ed value •

The f a r m e r a p p a r e n t l y can not count o n v a r i ­

ations in g r o s s income from these

sources in the p r o ­

cess of d e v e l o p i n g the f arm business*
the em p l o y m e n t of factors

If this is true,

should tend to be m o r e

closely

related to a gross Income w h i c h does not Include these
"r a n d o m e r r o r s * "

If it should be practi c a b l e

nate annual v a r i a t i o n s in crop yields*

to e l i m i ­

a p p a rently the

relationship o f gross Income to fac t o r s should be i m ­
p r o v e d even further*
average experience,

A f a r m e r p l a n s a c c o r d i n g to
but receives returns a c c o r d i n g to

p a r t i c u l a r c o n d itions of prices a n d weather*.
It was found that e l i m i n a t i n g the effects of v a r i a ­
tions in p r i c e s from "expected" values

(on goods sold

and goods h e l d in i n v e n t o r i e s ) did not result in an
increase

In the relation of produc t i v e factors to g ross

1

income*

N e i t h e r did simply e l i m i n a t i n g price

changes

of inventories result in an increase of the r e l a t i o n
b e t w e e n factors a n d gross income*

This mea n s that,

if

a fanner typically does have a " p l anning function" in
mind,

the "planning function" in this case a c c o m m o d a t e d

moderate

changes from "expected" pri c e s as well as it

could have a c c o m m o d a t e d normal o r exp e c t e d prices*
Obviously,

the results of the w o r k in A p p e n d i x G can

1* A p p e n d i x G.

-32be co nsidered to bea r only upon m o d e r a t e v a r i ati ons in
prices f r o m e x p e c t e d v a l u e s . ’*’

<S.

Conclusion
In this chapter questions have been r a ised r e g a r d ­

ing the a d e qua cy of data from farm account records for
areas 5 and 6 for calculating equations es t i m a t i n g gross
Income*

It has b een found that the y ea r of starting

records did not bias
between farms.
undervalued.

stated values of l and p er acre as

However,

as a whole,

the farms

appeared

The y e a r of starting records did not bias

the values of dairy cows sufficiently to justify a
systematic adjustment.
It is concluded that var iations in the bases of
valua tion of factors from one farm to another are I m ­
practical to determine
m a n a g e m e n t , Is not

at this time.

included in the data,

does not appear to be m u c h
In general,

The factor,
but there

that can be done about it.

the input data from the farm accounts In the

two areas are as suitable as could be obtained without
extensive additional

field work.

U n a d j u s t e d gross income values were as m uch related
to factors as were adjusted gross Income figures.
justments were for price

changes

Ad­

In inventories and

changes in prices from "expected** values.

Thus there

appeared to be no reason for a general adjustment of
gross Income

to make allowance for shifts In prices

from "normal."

Tl

Sec A p p e n d i x cf, p V

-13-

chapter

III

THE M E T H O D OF CA L CUE AT I ITG GRO SS INCOME
E Q U A T I O N S F R O M F A R M A C C O U N T DATA

T his chapter contains two m a i n headingss 1* a
des cr ip ti o n b a se d o n farm account records of farms in
areas 5 a n d 6j and 2* a statement of the m e t h o d s of
using these data in calculating gross Income equations*
1* F a r m s In T y p e — o f - F a r m i n g A r eas 5 a n d 6
A r e a s 5 and 6 are in the m l l k s h e d s of Detroit,
Flint,

Pontiac, Lansing,

3 I4.) summarizes

and Jackson*

Table 1 (page

income and expense data for all farm

account farms in areas 5 a n d 6, 1950*.
A r e a 5 is described as "dairy and general farming* **
A r e a 6 is called "dairy and cash c r o p s * ”^

That the sale

of dairy products m a k e s up a somewhat lar ge r p r o p o r t i o n
of the total income for farms in area 6 than in area 5
is evident*

There is more e m p h a s i s on ot her types of

l i v e st oc k than dairy cows in area 5*
The m a i n single product sold in b oth areas is whole
milk*.
YZ

There are a few farms w h i c h m a i n t a i n beef herds,

"Farm Business Analysis,.™ P u b l i c a t i o n of the D e p a r t —
m e nt of Ag ri cu lt u ra l Economics, M i c h i g a n State
College, A. Ec. 1+77# May, 1951*

Table 1. Income a n d E x p e n s e Items,

Type-of-Farming
1

A r e a s 5 a n d 6, 1950
A v e r a g e s f o r 190 F a r m A c c o u n t F a r m s
Item

Area

5

Area

6

Income
Crops
Dtiry products
Hogs
Cattle
P o u ltry a nd e g g s
Sheep
O t her
G r o s s Income

♦ *wU31

U,5l7
233
2,ll*7
709
361

* S»ZZ3
5.562
313
1,835
955
179
118

♦12,131

*11,655

♦ 1,335

* l.olt-9
2,023
983
572
1,1^7

Expense
F e e d bought
Machinery
Hired Labor
F a m i l y l a bo r
Crop expense
Imp rovements
Taxes
O ther
Total
Net F a r m Income
Int. o n invest. @ 5 %
L a b o r Income**

2,017
632
1*86

1,069

6 05

216
7 06
♦ 6,801*.

* 7,301

♦ 5,327

*

t

3t'6&7

1.U98
#^856

■» D oes not Include a charge for the o p e r a t o r ' s labor.
«-* L a b o r Income Is what the farm e r h a s left for his
l a b o r and m a n a g e m e n t after cash expenses, fami l y l a ­
bor, depreciation, a n d Interest o n Investment are d e ­
ducted f r o m total Inc o m e — It Is not the total value
p r o d u c t i v i t y of the operator*a l a b o r Input In a p r e ­
cise conceptlonal sense.
It Is the total of all
residuals r e m a i n i n g w h e n charges a gainst the f a r m
are made a c c o r d i n g to dol l a r costs or a c c o r d i n g to
a s sumed rates.
1. Ibid..

areas 5 and 6, p. 57

and some w h i c h specialize in corn and hogs,
ularly in the southern part of area 5*
specialize in poultry*

partic­

A few f a r m s

Important cash crops are

wheat and, p a r t i c u l a r l y in the n o rthern parts of b o t h
areas,

beans*

A relatively small n u m b e r of farms

specialize in sugar beets,

the center o f the sugar

industry being farther n o r t h in the Saginaw Valley*
O n m u c k lands onions, mint, potatoes,
grown*.

and celery are

There are r e l a tively few m u c k farms*

the farm accounts were
primarily upland soils*

I n 1950

concerned w ith farms containing
The farms in the study are

somewhat larger than the " a verage” and p o s s i b l y are
better m a n a g e d o n the whole.
Investments and expenses are grouped into c ate­
gories for areas 5 and 6 in tables 2 and 3*

In these

tables a classification is made according to labor
income*

F o r area 5 the low third and h i g h third labor

Income farms are compared w i t h the average of all
farms (table 2)* F o r area 6 the h alf of the farms w it h
low labor incomes,
l a b o r Incomas,

the half of the farms w i t h h i g h

and all farms are compared (table 3)*

In both areas the farms w i t h greater Investments in all
categories and greater expenses in all categories tend­
ed to receive larger gross Incomes and larger labor in­
come s *
It is a purpose of this

study to take the invest-

Table 2* Income, Expense,

and Investment Data Related
1
to L a b o r Income

Farms in Type-of-Farming A r e a 5
Michigan, 1950

Average
of all
farms

1/3 high
inoome
farms

163
♦ 3687

* 7068

7,992

12,131

18,020

28,14.56
14-,089
5,14-85
3,056
7,901*.
7,923

32,662

la,iii+

1/3 low
income
Terms
Number of farms
Labor Income

$

Gro ss Income
Inve atments
Total
Productive livestock
Machinery, equipment
Fee d and crops
Improvements
Lan d
Expenses
Total farm expense
Total labor charge
Hired labor
Family labor (unpaid)
Operator
F eed
Crop
Machinery net decreases
Improvement s
Other
TZ

7$

7,ft3 6
2,k29
4-77
1445
1,507
1,020
755
1,859
593

780

i;i&

3,800

55

7,551
7,720

1*,920
10,800

!:»!

10,123

,

1 0 14.28
1:18

632
14.86

i,52U
1,335

890

2,017
637
901

Compiled from tables in publication h7f»

3,1,23

9lj-0

651
1.532
2,020
1,107
2,332

808
1,038
clt» s u p r a.

-3?

Table 3. Income, Expense,

and Investment Data Related
1
to L a b o r Income

Farms in Type-of-Fanning Ar e a 6
Michigan*
1950

A verage
of all
farms

i
low
income
farms
Number of farms
Labor income
Gross income
Investments
Total
Productive livestock
Machinery, equipment
F eed and crops
Improvement s
Land
E xpenses
Total farm expense
Total labor charge
Hired labor
Unpaid family
labor
Operator
Feed
Crop
Mach. net decreases
Improvements
Other

high
income
farms

jt

35
2,856

18
$ 4 , 964

8,7Ul

11,655

14,406

32.911

33, 849
5,071

9,303

33.39a.
5,998
6,569
3,5 ®2
8,603
9,742

3,386
8,470
10,151

7,923
2,827
851

8,625
3 , 063
983

9,288
3,285
1.107

500
1,1*76
959
735

572
1,508
1,049
990
2,023
605
895

1,539
1,135
1,231
2,019
677
941

♦

17
623

k, 92.0

2,027
528
81*7

t

6 >7 ll

639

1. Compiled from "Farm business Analysis, A rea 6,"
Agricultural Economics Department, Michigan State
College, A. Ec. 478, May, 1951.

38

ment and expense data by farms and calculate the average
relationship between cross Income and the outlay in
each Investment and expense category*

This sort of

relationship is not available when either labor income
or gross income is taken as the basis of classification
after which the average value of each Investment and
expense category is given for each labor income or
gross Income class*
In the following procedure the analytic process
will be reversed* so to speak*

and an explanation of

differences in gross income according to each category
of factors will be offered.

This latter process is

more meaningful because we proceed from factors to
income rather than from income to factors.

This

method shows the structure of gross Income from fac­
tors; the former Implies a structure of factors from
gross income.
2.* Procedure for Calculating the Gross Income Estimating
Equations
Of the 19!* farms,
farms.

86 were

classified as dairy

The others were called not— dairy f a r m s .

The categories of factors used In this study are
those from the farm account records which are shown
In tables 1 to 3*

These categories can be combined for

simplicity or listed separately for detail

(table U).

The categories in columns I, II, and III are headings

Table i|.«

Categories of Factors as given in the
M i chigan F a r m Accounts

I
Total invest­
ment*

II
L a n d and im­
provements

III
L and
Improvement s

Inve stments
other than land

Machinery
investment
Livestock
Investment
Feeds and
crops

Total farm e x ­
pense

Total labor charge

Total labor
charge

Total f arm expense

Machinery e x ­
pense
F e e d expense
Crop expense
A l l other
expense

Sum:
All categories
of factors

All categories of
factors

A l l categories
of factors

used in the Michigan system of farm accounts*

Gross

Income equations can be calculated for any desired
amount of detail with i n the limitations Imposed by the
detail of the basic data*

(The calculated equations are

given in Chapter IV)*
The procedure at M i ch i g a n State College Is to take
the data from the f a r m account records and place them
on summary sheets*

The dAta o n the summary sheets are

then punche d o n IBM cards as part of the annual process
of farm business analysis*

The IBM cards and the

summary sheets are then u sed in the preparation of
type-of-farming area reports*
The calculations for this study began at this
point*

They are summarized as follows:

A. k

fc

1* Equ a t i o n s of the form £ = C x j_ • ••*.
appear in log form as log P =
b log y / . . *k_ log z_.
in the logarithms*
logs,

thus,

log Q / a log x / /

The second e q uation is linear

By converting the basic data to

it is possible by multiple linear regression

techniques to determine a m e a n relationship between
factor categories and gross income w h i c h is (a) linear
in the logs and (b) exponential in the natural, numbers*
This is essentially the method of Douglas,

Heady*

Tintner and others as discussed in Chapter I*
The data were converted into logs in four steps:
(a) A n IBM log card was prepared containing the m a n t i s ­
sas of the logs of 100 to 999*

A card was mads for

each number,

100*00, 101.00,.*....999*00.

Columns

1 to 5 wero not u s e d I n the w o r k w i t h logs since these
columns were n e e d e d for ident i f i c a t i o n in subsequent
sets of cards*

In columns 6 to U 5 the natural numbers

were repeated 8 times*

In columns li|.6 to 78 the

mantissas of logs of the respective natural numbers
were repeated 8 times*

Columns

I4.
6, 50,

a n d e v ery

four t h column thereafter were left blank for the char­
acteristics of the logs*

Three significant figures

were given for the mantissas*
(b) D a t a from the IBM cards for the farm accounts
were transferred to columns 6 to J+5 of a new set of
cards, hereinafter called "transfer cards*" (Four
transfer cards were u s e d in recording information u s e d
in this study)*
(c) A n IBM collator was used with the log cards and the
transfer cards to place the m a n tissas of the f arm
account data In columns U-6 to 78 of the transfer cards*
The characteristics were punched by mechanical means.
(d) Columns ij.6 to 78 of the lat t e r group of cards were
reproduced o n a final set of cards on w h i c h the data
were given in log form.
2* The least squares regression equations
were solved by the Doolittle method*

Standard errors of

estimate and confidence limits of coefficients of elas-

ticity were computed in some caeai.

The logarithmic

equations were restated in terns of natural numbers.
Once the above procedure had b e e n carried
through,

the gross income equations and the Informa­

tion w h ich is implied by these equations were ready
for interpretation.
In Chapter IV the equations of gross income and
the information which follows from them,

such as

elasticities of gross income w ith respect to categories
of factors and the marginal value productivities of
categories of factors, are stated and discussed.

CHAPTER IV

GR O S S INCOME E Q U A T I O N S A N D T H E I R
EE R I V A T I V E S

!♦ Gross Income E s t i m a t i n g E q u a t i o n s
E q u a t i o n s estima t i n g gross Income were calculat­
ed (tables 5 to 7)«

The equations are r ead thus:

The coefficients of the factor categories for the
different equations are g i ven In the num b e r e d columns.
Therefore the first equation in table 5# for 86 dairy
farms*

reads:
L o g gross income s. O.U58 / 0 . 5l|-l total farm
expense / 0*322 total In­
vestment*

Translated Into numbers*
Gross I n c o m e s

Altogether,

this Is:
2 * 87(total f a r m expense)
0*322
(total Investment)

10 equations are given for the dairy farms,

6 for the not-dairy farms,

and 3 for all farms.

Tiahle 5. Ten Grose Ihccne Estimating Equations for 86 Deiry A u m
type-of-Amlng Areas 5 end 6 , 1950

Sanations
1
Constant ten

2

3

4

5

6

7

8

9

10

0.458 0.436 0.422 0.400 0.J31 0.149 0.361 0.522 0.119 0.094
(Coefficients)

Caterarv of flieters

1>0.432

Feed expense
Crop expense
Met decrease, mehlnery

1.480 j

0.]
0.387 0.J
-0.(

0.140*
0.370 |o.273 0.271
0.351 [0.383 -0.069
4

Total labor oharge

0.125 0.088 0.090 0.086

Naohlnerj and equipent investment
Productive livestock investment
Feed and crop investment

/ 0.322'

>0.322'

>0.3281

J
Sun of coefficients

0.3281
/

0.863 0.890 0.909

0.172'
0.150 >0.322 0.356
0.312

o o o

0.050 0.Q31 0.108 |0.061 0.020
0.088
0^062

Investment In land
Investment In 1iprevents

0.180
0.153
0.303

A
w

1.035 0.949 0.983 1.066 1.075

Table 6 , Six Gross Income Estimating Equations for 108 Hums Other Than Dairy
Type-of-Faning Areas 5 and 6 , 1950

Equations

Constant tern

1

2

3

4

5

6

0.237

0.180

0.297

0.228

0.189

0.350

Category of factors

(Coefficients)

Total fan expense, not labor

0.577

0.497

0.482

0.582

0.159

0.180

0.150

0.164

|-O.004

0.039
-0.006

J 0.018

>0.346

0.184
0.064
-0.003

| 0.696

0.622

Total labor charge
\

Investment in land
Investment in improvements

jo.002

0.239 i

^ 0.247

Machinery and equipment investment
Productive livestock investment
Feed and crop investment
Sub of coefficients

jo.350

J
0.943

0.974

f 0J“
0.975

1*004

.
1,011

1.009

Table.7.

Ihr«« Qroia Inocne litlaitlBg Sqpatian* for 194 D u n
Tjrpi ftM k a d m Areas 5 end 6, 1950

Equations _

Ccnetant torn
Oateaarv of fhetors

1

.2

3

0.361

0.241

0.359

(Coefficients )

Total Hu b eocponeo, not labor

0.544

0.449

0.544

Total labor eharge

0.149

0.157

0.124

0.101

0.070

lureetaent in land

0.001

Ihveetaent In iaprorennts
Machinery and equipnant Inreetaient

0.256

0.281

0.117

Productive livestock inveetaaant

0.040

Peed and crop Investment

0.106

Sim of coefficients

0.949

0.988
0.

1.002

-14.7-

2, C o M P t r l a o n of Coefficients of E l a s t i c i t y a n d Their
Confidence Intervals for D a i r y a n d Bfro'b-Dairy Farmil
Dairy farms vers ana l y z e d separately from farms
other than dairy for two reasons:

(1) The separation of

the dairy farms f rom all farms permits a more

specific

analysis of the structure o f the farm business*

The

kinds of work done and the classes of Inputs are more
alike from farm to farm.

Thus the equ a t i o n of gross

Income has greater structural

significance*

A n d (2)

the hypothesis was set up that the correlation of gross
Income w i t h the factor categories should be greater for
the more homogeneous group of dairy farms than for the
not— dairy farms*

P r o m this it should follow that the

confidence intervals of the coefficients of elasticity
of gross Income for the different categories of factors
should tend to be narrower*

Better estimates of the

coefficients should be o btained by classification of
farms according to t y p e .
The coefficients themselves were not significantly
different for dairy farms and n otdairy farms (tables 8
to

10)*

The hypothesis that the confidence

intervals

should be narrower for the supposedly more homogeneous
group of dairy farms was not supported by the data*
The relationships between categories of factors and
gross Income are estimated to be stronger in the case
of the 1Q6 not-dairy farms*

E x a m i n a t i o n of the columns

Table 6. Gross Incase Equations Based on Categories of Productive factors
86 Dairy fans and 106 fans let Dairy
fypo-of-faning Areas 5 and 6, 1950

Category of
factors

Symbol

Coefficient

10* Confidence

50* Confidence

dairy tans to
86
Dairy
fans

106
fans
Not
Dairy

0.458
0.5a

10* ocnfldenoe
other fans
86 Dairy
fans

106 Farms
lot Dairy

66 Bairy
fans

106 Fans
lot Dairy

Constant ten
(c)
Total fan expense W
Total lmrestaent
(b)

0.322

0.237 0.392-0.522 0.22L-0.253 0.431-0.484 0.230-0.243
0.696 0.269-0.813 0.561-0.652 0.431-0.652 0.6a-0.751
0.247 0.093-0.551 0.101-0.393 0.229-0.414 0.188-0.306

S m of coefficients

0.863

0.943

Correlation coefficient

Ratio 10* confi­
dence interval

0.833 0.914

Standard error of estimate 0.112

0.094

4.1
2.0
1.6

Table 9* Gross Income Equations Bated on Categories of Produetiye Factors
86 Daily fens and 108 Fluv lot Daisy
Type-of-flai*ing Areas 5 and 6, 1950

.Ratio I0J( ooafl*
Category of
Symbol
Coefficient
106 Confidence
50% Confidence
deuce interval
Factors______________________________ Halts_______________ Limits_____ daisy fkrmt to
10% oenfldnee
86
106
other fhxas
Dairy Turns
Ferns
lot
86 Dairy
108 Ifcrmt
66 Dairy
106 h a s
__________________________BtilZ___ h m ____ M M n . ____ KttDslnr____________________
Constant ten
(o)
Total fkn eapense,
not labor
(»)
Total labor efaarge (b)
Total investment (d)

0.422

0.297 0.356-0.466 0.292-0.302 0.396-0.446 0.295-0.299

12.6

0.432
0.145
0.332

0.577 0.191^0.673 0.471-0.663 0.377-0.467 0.534-0.620
0.159 0.021-0.311 0.019-0.299 0.096-0.194 0.103-0.215
0.239 0.124-0.542 0.132-0.346 0.261-0.405 0.196-0.262

1.2

Sub of ooeffloients

0.909

0.975

Correlation coefficient

0,876

0.916

Standard error of estimate 0,106

0,089

2.3
2.4

labls 10. Grots Inocae Natations Based on Categories of ftrcductira Factors
86 Dairy fciH and 108 Runs lot Dairy
fype-of-Pandng Areas 5 and 6, 1950

Category of
Faotors

Symbol

Coefficient

86
Dairy
A w

108
A w
Not
Dalxy

105 Confldanoa
Halts

86 Dairy
Aims

108 A w
Not Dairy

Batio 105 eenfi505 Confldanoa daiae lntarml
Halts
dairy tu r n to
105 ocnfldsnoa
othar ftws

86 Dairy
Ikns

108 Ains
Not Dairy

0.228 0.380-0.421 0.213-0.243 0.392-0.409 0.222-0.234

1.4

(b)
(d)

0.497 0.159-0.615 0.358-0.636 0.295-0.479 0.441-0.553
0.130
0.160 -0.064-0.3U 0.052-0.268 0.043-0.217 0.116-0.204
0.088 — 0.004 -0.088-0.264 -0.108-0.100 0.017-0.159 0.041-0.047

1.6
2.0
1.7

(•)

0.328

0.351 0.228-0.528 0.227-0.475 0.247-0.409 0.301-0.401

1.9

S m of eoaffieiants

0.933

1.004

Standard en*ar, estimate

O.107

0.089

Constant tara
Total Dun expanse,
not labor
Total labor obarge
Ifarsstnait in land
Ibnrastasnt othar
than land

(•)

0.400

(a)

0.387

called Ratio 103& confidence interval dairy farm* to the
10S& confidence Interval other f a m e
shows this*

(tables 8 to 10)

Calculation or the confidence Intervals

was designed to take into consideration interdependence

of the categories of factors; that is, interdependence
1

among the "independent" or causal variables*
3. Interpretation of the Gross Income E q u ations
The equations given In tables 5 to 10 can be
considered as gross Income estimating equations or as
value productivity equations facing the Individual
farms.

Knowing the values of the input categories on

a particular farm,

the gross income of the farm can be

predicted within the limits implied by the standard
error of estimate of the gross Income*

Farmers who

combine categories of factors in one proportion are
assumed generally to be able to combine them in differ­
ent proportions and earn gross Incomes accordingly.
In other words.

It is assiaed that farmers in a p a r t ­

icular area at a certain time period are operating
on value productivity equations which are essentially
comparable.
The objection implied in the work of Reder and
2
Bronfenbrenner is borne in mind.
However, this obTl

See P. Crajfcn^r, liathematical Methods of Statistics.
pp. 53+ 5 to 51+8.

2. Melvin W. Reder and

iMartln Bronfenbrenner,
c it., p. 1 5 * footnote 2.

opera

Jection oould be mad* to the b u l k or statistical re­
search in f arm management*

F o r example, according to

the long—established fans "success fac t o r s , ” farmers
obtaining h i g h crop yields and h i g h rates of pr o d u c t ion
of lirestock also e a r n relatively h l g b labor incomes.
It Is theoretically possible that certain farmers with
low product i o n records w o u l d earn e ven lower labor
incomes should they seek to raise the level of p r o ­
ductivity of their farms.
possibility,

But, while a theoretical

this proposition Is not ordinarily taken

to m e a n that analysis of the experiences of a group
of farms in use of resources Is not valuable to the
farm operator,

and that, as a rule,

he will increase

his net return if he can increase his yields.
One of the basic premises in farm management r e ­
search and extension work is that farm operators are
able to le arn by the experiences of others and are
capable of duplicating these experiences to some de­
gree.

Naturally, not all Individuals will get exactly

equal results from the same procedures.
of p r oof to the contrary,

In the absence

gross income will in general

respond according to the values of the categories of
Inputs as specified In the equations.

In this case,

the functions can be called value productivity equations
for the farms In the study.

That Is,

should the r e ­

sponses of gross income to the values of categories of
inputs be the same for all farmers,

assuming that the

effects of weather* prices*
out as random variables*

and so on could be averaged

then the gross Income equations

are value p r o d u c t i v i t y functions*
U* R e a s o n s for E r r o r s In the E s t i m a t e of Gross Income
Gross Income is p r e d i c t e d by estimating equations
w i t h i n certain limits

(tables 8 to 10)*

Clearly*

more accura t e l y gross Income can be predicted*
useful will be the m e t h o d of prediction*

the

the more

There are

at least four causes of variations of actual gross In­
come f rom p r e d i c t e d gross income*

These differences

m a y arise f r o m the f o l lowing causes:
(1)
be Inadequate.

The m e t h o d of statistical analysis m a y
Inadequacies of this type Include

failure to use proper types o f equations*

failure to

combine factors of pr o d u c t i o n Into appropriate cate­
gories* and m u l t i c o l l l n e a r l t y .

B y m u l t l colllnearlty

Is m e a n t significant relationships between the inde­
pendent variables (categories of factors)

themselves*

W h e n the Independent variables are correlated*
fulness of mu l t i p l e regression is impaired*
w hen

the use­

In effect*

the independent variables are themselves correlated

as well as e a c h being correlated w i t h the dependent
variable*

It Is extremely difficult*

If not Impossible*

to e s t a blis h at)hypothesis that r e g r ession analysis
attributes a " p r o p e r ” p o r t i o n

of the combined rela-

tlonship- to o a c h Independent variable*
(2) Va r i a t i o n s in gross Income m a y arise
from windfall events*

Increases or decreases I n gross

Income f r o m ex p e c t e d values because of weather, prices*
new diseases of crops* accidents*

and so forth* fall

Into this group*
(3) Gross income will be different f rom Its
p r e d i c t e d value If the factors of p r o d u c t i o n are not
valued on similar bases f r o m farm to fans*
(L|_) There will be differences In gross i n ­
come because of m a n a g e m e n t ♦
in far m account systems,

M a n a gement Is not m e a s u r e d

and thus the quantity of

m a n a gement u s e d is not recorded as a n Input*
To make

the estima t i o n of gross incom* useful*

It

Is desirable that these sources of error be cut off
where practicable.

Statistical met h o d s m a y be developed

over a p e ri o d of time w h ich m a y give at least a partial
answer to the first source of e r ror In the p r e ceding
discussion*

The shortcomings of estimation of gross

Income w h i c h arise from windfall types of gains and
losses m a y be eliminated to some extent by the use of
"normal" prices and yields for each farm*

Furthermore*

possibly a more detailed classification of farms by
type and area will p a r tially answer this problem*

The

use of gross income predicting equations w i t h regard to
(3) above

implies a need for constant reeval u a t i o n of

resources in o r d e r to keep average stated values In
line with, markets,

and to keep valuations comparable

from farm to farm.

The differences In gross income

arising from ma n a g e m e n t will remain as residual unless
a workable scheme for p l a c i n g a value o n the m a n a g e m ent
Input Is devised.
5. The Effe c t s of Overv a l u a t i o n and U n d e r v a l u a t i o n of
Categories o t Factors
W h e n a category of factors Is underv a l u e d In the
farm account books,

the effect on the elasticity o f

gross Income w i t h respect to the category and the effect
on the marginal return Is to Increase both.

Under­

statement of the value of land leads to a conclusion
that

the elasticity coefficient f o r land at market

values is lower than the values given f r o m the gross in­
come equations as calculated f rom f arm account data*

The

marginal return to land will be lower than stated.
Similarly,
tors,

the overvaluation of a category of f ac­

such as labor (as this study seems to show) leads

to a l ow coefficient of elasticity.

Should labor be

valued in the farm account books at the rates whi c h are
subjectively attached to it,
operators,

apparently,

by the farm

the stated value will be lower; the coefficient

of elasticity will be higher,

and the marginal return

for a dollar of l a bor charge will approach 1, as explained
in the next section.

6* The Constant T e r m In the E s t i m a t i n g E q u a t i o n
Tlx© constant term (C In P

C x y ) m a y be In a

sense regarded as a regulator of the extent to which.
It pays to e x p a n d the business*

Thus C can be i n t e r ­

preted to indicate, particularly,

the capacity or the

fixed Inputs pr o f i t a b l y to accommodate the inputs w h i c h
can be varied*

W h a t e v e r m a y be the exponents of x and £

(x and 2. designating the categories of factors employed),
ft fe

gross income is some number times the x y
expression.

p art of the

If returns to scale are decreasing and the

subjective rate of charge against
by the oper a t o r remain constant,
more x and 2

stated d o l l a r costs
the larger C is,

will pay to use.

are constant or increasing,

the

If returns to scale

it m a y be e x pected that

considerations of risk and uncertainty in connection
with charges (entered in the accounting sense) by x
and 2 will eventually b r ing e x p ansion to a close in
any case*

But the larger C la in this case also,

the

greater will it pay to expand the business*
F r o m the point of v iew of the individual farm,

in

the long run the really fixed factor is m anagement*
The quantity of this fixed factor,

in turn,

determines

in considerable degree how m u c h of the other factors it
pays to employ.

Thus there is an analogy between the

constant term C and management;

and management is the

prime facto r which is not included in the categories

x, 2*

A s Improvements i n techniques cause the

•••%•

optimum size of f a r m to become l a r g e r , the average
value of C«

should become greater*

7» Elasticities of Gross Income with Respect to Cate­
gories of factors1

The s m o f

the elasticities is not significantly

different f rom 1 in any of the equations (tables 5 to
7)*

Thus returns to scale for b oth dairy farms and

not-dairy farms in areas 5 end 6 for 1950 are indicated
as constant.
The tendency f o r the sum of the elasticities to
increase w i t h more detailed breakdowns of categories
is noted.

The stun of the coefficients Increases

fairly constantly f rom 0.863 to 1.075 as the number of
categories (detail of breakdown)

is increased.

A test

showed that the probab i li t y that the sum of the co­
efficients w o u l d b y chance Increase as consistently
with a n Increase in the number of categories as it did
2
was 0.025* or one i n forty.
This pos s i b l y can be e x ­
plained on the basis that the more detailed b r e a k ­
downs of factor categories imply more nearly p r o p o r ­
tional changes in real inputs.

A more exactly p r o ­

portional change in all factors w o uld elicit a greater
total response in gross Income than would a change
1. fc’or a discussion o f the theoretical m e a n i n g of these
elasticities and their sum* see pp. 3 to <5 .
2. The test is analagous to a coin-tossing experiment.
E a c h time the sum of the coefficients increased w he n
the number of factor categories Increased* the o u t­
come was considered as a "head" and vice-versa.

will eh w a s c o n c e n t r a t e d r e l a t i v e l y h e a v i l y In c e r t a i n
factors*
If not all of the categories to w h i c h e l a s t i c i t y
of gross Income
Included,

can be p r o p e r l y a s s i g n e d have b e e n

then the I n c l u d e d fac t o r s w i l l take credit

for e l a s t i c i t y o f gross income f r o m the e x c l u d e d f a c ­
tor or factors.
Into account,

If important factors are not taken

and Inc l u d e d factors are themselves

less Important but are
o m i t t e d factors,

strongly c o r r e l a t e d w i t h the

It is possible

seriously to o v e r ­

estimate the e l a s t i c i t i e s of the i n cluded factors.
If the i g n o r e d factors are p a r t i c u l a r l y Important,
gross income m a y a p p e a r to have more o r less e l a s t i c i t y
w i t h respect to the factors used in the estimate of
gross Income
together.

than really exists f o r all factors taken

A l l of the e s t i m a t e d e l a s t i c i t y will be

credltfted to only the factors i h i c h are included*
Thus the que s t i o n of the a d equacy of the f u n c t i o n
in the absence of a statement of the value of m a n a g e ­
m ent appears again.
on any farm,

If m a n a g e m e n t Is a fixed factor

but varies bet w e e n farms,

t hen the factors

w h i c h are u s e d In m a k i n g the estimates of gross income
m a y be a c q u i r i n g apparent influence u p o n gross Income
from man a g e m e n t *

Thus the effect of the e x c l u s i o n of

a value of m a n a g e m e n t m a y be to bias u p w a r d the e l a s ­
ticities of gross income w i t h respect to the other

factors*

B r o n f e n b r e n n e r e x p r e s s e s tht

ism

Idas,

bat

In a differe n t way, w h e n he w r i t e s o f d i f ferent fines*
being on dif ferent " p r o d u c t i o n f u n c t i o n s * "

If all of

the factors o t h e r t han m a n a g e m e n t are consistently
e v a l u a t e d f r o m f a r m to farm,

then the elasti c i t i e s of

gross Income w i t h respect to e a c h factor are the same
for all farms*

But differences In the m a n a g e m e n t In­

put I m ply different values of C (the constant term)
~
1
a n d different p o i n t s of equilibrium*
If returns to scale are constant or Increasing,
there are two p ossible answers to the q u e s t i o n of w h y
some f a r m businesses do not become large enough'Influence the m a r k e t price:
scale

to

(1) I n c r e a s i n g ret u r n s to

( f ro m the standpoint o f the farmer, pri c e s and

costs favorable to expansion) m a y be r e g a r d e d as a
temporary phenomenon*

F a r m e r s m a y not try to capital­

ize upo n increasing returns to scale because of the
risks Involved*

T h a t is, after a business becomes

very large, p a r t i c u l a r l y in agriculture,
will tend to value

the o p e r a t o r

commitments of capital a n d e x p e n d ­

itures at greater than their dollar amounts and at
p r o g r e s s i v e l y h i g h e r rates in reckoning costs*
A s a business becomes very large,

(2)

the type of f u n c t i on

1* M a r t i n Bronfenbrenner, fcP r o d u c t I o n Functions: CobbDouglas, Interfirm, Intrafirm," B o n o m e t r l c a V* 12
(Jan*, 19Ml)» PP* 37-6.

u s e d h«r« p r o b a b l y be comas inadequate*
as a dairy f a r a bacomas v a r y large,

F o r example,

e v e n the time w h i c h

the cows spend going to a n d f rom pasture will increase,
or the n e e d f o r d e l e gation of a u t h o r i t y will i n c r e a s e •.
Thus returns to factors m a y e v e n finally be zero at
the margin*
1
8* Marginal Value Productivities
The average values of the different categories of
factors for the 86 dairy farms,
farms,

the 108 not-dairy

and the 1 9 U farms were calculated from the

farm account data (table 11),

A n estiaiate of the

marginal value p r o ductivity of each category of f a c ­
tors w h e n all categories are at their m e a n values is
given by the first derivative of gross Income w i t h re­
spect to the particular category.

These derivatives

for all categories were calculated for 13 equations
(tables 12 to ill).
The marginal value productivity of each category of
factors can be estimated by farms if the values of the
categories applying to each farm are inserted in the
ma r ginal value productivity equation.

This w a s done

for the 86 dairy farms, using as the equation of estimate
of gross Income the first equation in table 9-

This

!• F o r estimates of marginal productivities o n Iowa
farms f rom farm account records (1939) and survey
records (19U )» see A p p e n d i x J.
For reservations concerning the use of the Cobb-Douglas function
in estimating marginal value productivities see chapter 1, parts
3 and U, and appendices C and I*

e q u a t i o n is,

In log font:

L o g gross income =»0*lj.22 / 0«li32*log total f a r m
expense not l a bor / 0*114.5 *log
total lab o r charge / 0 *332*log
total lnve s t m e n t •
The average return on Investment at the m a r g i n was
1 1 *5# by the lat t e r m e t h o d (table 15) •

A v e r a g e returns

for t o t a l l a b o r charge and total f a r m expense not l a bor
at the m a r g i n were $ 0 *56& a n d $ 0* 87l4
>» respectively*
Es t i m a t e s of m arginal returns by different gross
Income equ ations for both the dairy and the not— dairy
farms are shown in tables 12 to ill-*

The ma r g i n a l r e ­

turn to Investment in land Is es t i m a t e d between 0 % and
10#*

The m a r g i n a l ret u r n to l a b o r on the dairy farms

was estima t e d at a b o u t 0* 5# w h i c h m e a n s 0*5 for the
m a r ginal labor dollar*

Thus,

the es t i m a t e d r e t u r n on

labor for the dairy farms Is $ / 0*50 at the margin*
The return at the m a r g i n is estimated to be about $ 0*95
an h o u r for the not-dairy farms.

It is generally r e c o g ­

nized that the m arginal return to l a bor for dairy farms
is comparatively low, but the regularity of employment
is a n offsetting attraction. The m a rginal r e t u r n of $0.50
to about $ 0*70 for a dol l a r of labor charge for all
farms indicates that at the m a r g i n labor does not earn
as m u c h as is shown by the charges entered in the farm
account books* This also is as one might well expect.
E v e n in a relatively good year,

farmers will w o r k at some

phases of their business for substandard returns to
piece out an acceptable income*

Table 11*

Average Values or Categories or Factors
Type -or -Farming Areas 5 and 6, 1 9 5 D

86 Dairy
Farms

108 Farms
not-dairy

19l*
Farms

♦ 1,552
1,017

♦ 1,311

812.
1,91*9
1,600

2,11*7
1 , 553

2,060
1,575

5*369

6,259

5,872

Total labor charge
Total r a m expense

2*73?
8,10U

2,802
9,061

2,769
8,637

Investment, land
Investment, improve­
ments
Investment,land and
imp ro veme nt s

8,615

9,657

9,195

8,l+il*

9,007

8,714*

17,029

18,661*

17,939

6,1*75

6,1*30

6,1*50

3,1*11

1*, ll*8

3,821

F e e d expense
Crop expense
Machinery net de­
creases
Miscellaneous expense
Total farm expense
not labor

$ 1,008

926

Machinery and equipment
inve stment
F eed and crop invest­
ment
Productive livestock
investment
Investment, not land
and Improvements
Total Investment

5,177

5,212*

5,197

15,087
32,116

15,§79
3U,5l*3

15,528
33,^67

Gross Income
Net F a r m Income
Labor Income

11,081*
1*, 515
2,909

13,1*00
5,391*
C,l67

12,373
5,211
3,506

Table 1 2 •

Estimates or Marginal Value Productivities
of Categories of Inputs

86 Dairy Farms, Type-of-Farming Areas 5
and 6, 1950

Each category taken at mean value for all farms *
Equations
Category of Input

1

2

3

h

5

Total farm expense
not labor

0.8 55

0.778

0.773>

Total labor charge

0.561].

0.513

O.lj.93

Total Farm expense

0.705 0.635

Investment land
Investment Improvements

0.039
0.103

Investment land and
Improvements

0.052

0 .056

0.232

0.235

Machinery and equipment
Inve stment
Productive livestock
lnve stment
Investment feed and
crops
Total Investment not
land and buildings
Total Investment

1,

0.106

0.231

0.110

The arithmetic mean is used because the values of the factors at
which the marginal value productivities are estimated can then be
taken from the farm busineaa analysis reports as they stand.
Use of the arithmetic means of the factors tends to bias the mar*
glnal value productivities of factors downward. The arithmetic
mean is larger than the geometric mean in all cases and the data
(factors) are subject to diminishing returns.

Table 1 3*

E s t i m a t e s of M arginal Value Productivities
of Categories of Inputs

108 N o t - D a l r y Farms, T y p e —of-Farming
A r e a s 5 and o, 1950
E a c h category t a k e n at m e a n value for all farms

Category of Input

1

Equations
2
3
k

5

Total farm expense
not labor

1 .216 1.143 1.081

Total labor charge

0.9U2 0.952 O. 89I4.

Total farm expense

0.999 0.981

O.O3I4.

Investment land
Investment Improvements

0.005

Inve stment l a n d and
Improvements

0*002

0*003

Machinery and equipment
investment
Productive livestock
Inve stment
Investment feed and crops
Total Investment not
land a nd buildings
Total Investment

0.290
0.093

0*318 0*2^2
0.091

-65-

Table ll*..

E s t i m a t e s o r M a r g i n a l Value P r o d u c t i v i t i e s
or C a t e go ri es or Inputs

19l^ Farms,

Type s-or-Farming A r e a s 5
a n d 6, 1950

Each, c ategory ta ken at m e a n value for all farms

1

Equations
2

3

Total r a r m expense
n ot l a b o r

1.112

0 . 91 6

1.150

Total l a b or charge

0. 681

0 .700

0 .573

Category or Input

Total r a r m expense
Investment l a n d
Investment Improvements

0.100
0.002

Investment la n d and
I mprovements

0.069

M a c h i n e r y and e qu ipment
inve s tme nt
Productive live st oc k
inve stment
Investment f ee d and crops

0 . 228
0•060
0.3J+0

Total investment not
l and and bu il di n gs
Total investment

0.217
0.092

66-

Table 15*

Ave r a g e Ma r g i n a l Value Produ c t i v i t i e s or
C a t e gories of Fac t o r s

86 D a i r y Farms, T y p e - o f - F a r m l n g Ar e a s
5 a n d 6, Michigan, 1950
Margin a l value p r o ductivity or e a c h category cal­
culated separately for e a c h fann
Ave r a g e M a rginal
Val u e Produc t i v i ty
of a D o l l a r of

Category of Factors

L a bor

C a s h o u t l a y o r charge # 0 * 5 6 6

Total f a r m e x ­
pense, not labor

Cash outlay o r charge $ 0 * 8714-

Total Investment

Investment

#0* 1 1 5

E s t i m a t e s of m a rginal returns to factors other
than lan d and labor are not m u c h different from fhat
one

should expect.

F o r total farm expense,

and for

the components of total farm expense, m a rginal returns
appear to approach a ratio of 1 to charges.

This

could be Interpreted to m e a n that the farmers in the
study have been able to equate direct outlays to re­
turns at the margin.

If the estimates of elasticity

coefficients are tj^sed upward,

then the mar g i n a l value

productivity coefficients will actually be less than
stated.
Some rates of return p e r dollar of outlay,which
theoretically should be 1 at the margin,

are estimated

-67-

to be less than 1; 1, 6 ., total farm expense not labor*
There are two possible

reasons for thisr

(1)

There

are charges included in total farm expense w h i c h ere
fixed,

such as depreciation on machinery,

and improvements.

equipment,

The marginal return per dollar of

stated charge on these items m a y tend to be low; that
is, the stated charges are too high.

The reason for

this is that conservative bookkeeping demands that
the rates be fully high enough to cover losses.
(2)

Because of federal income taxes farmers m a y reason

that a dollar of cash outlay for running the business
actually costs something less than a dollar.

Expense

items can be charged to the farm business and the
amount deducted from taxable
spent for feed,

fertilizer,

cost only $0.80 or $0.90,

income.

Thus a dollar

or fuel conceivably could

depending upon the per cent

of the marginal personal income which is taken by
income texes.
The marginal returns to categories of factors show
a general tendency to agree in the different formula­
tions from the same data (tables 12 to lU). The m a r g i n ­
al value productivity of labor is always less than the
comparable figure for total farm expense not labor.
Part of this may be due to a comparatively inadequate
evaluation of the labor input.
and the response of gross income

The charge for labor
to changes in the

labor input are poorly correlated.

If gross income is

-o b -

m or a h i g h l y o o r r a l ^ a t a d w i t h tha ‘
t otal stated f a m
expense not labor,

then the expense not l a b o r will take

credit Tor p art of tbe Influence of the labor Input
on gross Income.

T h u s the difference m a y lie In a

p o o r e v a l u a t i o n of labor,

not In a real difference

b e t w e e n the m a r g i n a l value p r o d u c t i v i t i e s of the two
categories of factors.
The e s t i m a t e d m a r g i n a l returns to m a c h i n e r y I n ­
vestment are h i g h e r than c a l l e d for by considerations
of Interest and d e p reciation.

The r e t u r n to m a c h i n e r y

Investment at the m a r g i n Is e s t imated for 19U farms
at 2 3 % (table 1I4.)«.

If 1 0 £ Is a l l o w e d for depreciation,

then the Interest retu r n to m a c h i n e r y a n d equipment
Is h i g h In c o m p a r i s o n to o t h e r categories of factors.
This could be I n t e r preted I n these ways:
(1) G r e a t e r u n c e r t a i n t y a s s o c i a t e d w i t h the o w n e r ­
ship of m a c h i n e r y and equipment than w i t h the ownership
of other ass e t s such as l and m a y account in part for
the h i g h m arginal r e t u r n to m a c h i n e r y and equipment.
(2) Because of r i s k a n d uncertainty,
be short relative to labor.
rationing of capital.

In this case there is

The estimates of m a r g i n a l r e ­

turns to l a n d and livestock,
than w o u l d be expected,

capital m a y

however,

are not high e r

considering p r e v a i l i n g Interest

rate s.
(3) The ret u r n to Investment in machinery,

because

or c o r r e l a t i o n w i t h m a c h i n e r y e x p e n s e s a n d net decreases,
m a y be taking part o f the statistical "credit" for the
latter,
(I4
.) The ret u r n s to Investment In m a c h i n e r y at the
m a r g i n may a c tually be high,

as the estimates

show.

In

this case the Inference Is that f u r t h e r increases in the
relative
expected.

Importance of m a c h i n e r y and e q u ipment m a y be
A c c o r d i n g to this Interpretation, m a c h i n e r y

and equipment as part of the assets on farms are b e low
their equ i l i b r i u m quantities,

and adjustment continues,

9, E s t i m a t e s of Net Income
The accuracy o f the estimate of g r o s s income Is
shown by the c orrelation coefficients b e t w e e n factor
categories a n d gross income a n d the standard errors of
estimate of gross Income

(tables 8 to 10)•

Estimates

of gross income m a y be close e n o u g h that part of the
variations in m e a s u r e s of net Income of a f a r m can be
explained o n the basis of the way In w h i c h the factors
are combined.

If this Is true,

then one

step In i n c r e a s ­

ing the net income in the usual case w o u l d be to change
the combination of resources e m ployed o n the farm.
Practically,

this could m e a n that there

could be too

m u c h Invested in l a n d in rel a t i o n to the investment in
livestock a n d the charge for labor,

and so forth.

"Net farm income" Is e s t i m a t e d by subtracting
from the estimate of gross the "total f a r m expense"

o t h e r than the o p e r a t o r ' s labor*

T h i s estimate of

net far m Income for the 66 dairy farms was c o r r e l a ted
w i t h the net f a r m income g i v e n by the f a r m account
books.

The

simple

co r r e l a t i o n coefficient was 0.U9*

T h i s c o r r e l a t i o n does not tell much.
if one

It states that,

should choose to e a r n a l a r g e r ret u r n to b oth

the o p e r a t o r and the farm,
iable factor,

the farm.

he

should increase the v a r ­

N o t h i n g is said about w h e ther

the a d j u s t m e n t will b r ing in e n o u g h a d d i t i o n a l Income
to cover the a d d i t i o n a l Interest o n the i n c r e a s e d in ­
vestment in farm.
If one will accept "labor income" as a m e a s u r e of
what the f a r m e r seeks to maximize, the f o l l o w i n g q u e s ­
1
t i o n m ay be raised:
Is the estimate o f gross Income
close e n o u g h that,

w h e n charges are deducted,

estimate of l a bor Income is also obtained?

a useful

This h y ­

pothesis is r e jected for the 86 dairy farms w h e n l a b o r
income as g i ven in the f a r m accounts is u s e d as the
standard of c o m p a r i s o n (figure 1).

The va r i a t i o n s

in gross income w h i c h are not a c c o u n t e d for b y the m a n ­
ner of the combining factors are

so great that the re­

sidual of the es t i m a t e d gross income less charges by
the defini t i o n of labor income bears no significant
re l a t i o n to the l a b o r Income g i v e n in the f arm accounts.
Til L a b o r income is gross Income less to^al f a r m expense
'
ora t o r ' s l a b o r less Interest o n

-71
Figure 1*

llj- "

The R e l a t i o n b e t w e e n L a b o r Income as R e c o r d e d
a n d L a b o r Income as E s t i m a t e d f r o m a Value P r o ­
d u c t i v i t y Function*
8 6 D a i r y Farms,
1950.

T y p e - o f - F a r m i n g A r e a s £ and 6,

13
12
11

10

Labor

InooJs
as
tfecordsdl
($

1000]
5
h

3
2

•

*

r ~ 0*114.7

»•

•* o1
•

°

•

1

*
o

•

•

•

O

•

0•

•

o°
.°

*
•

O
©

-1
-2

2

*

3

L a b o r Income as E s t i m a t e d

U
(81000)

The failure of this relati o n s h i p m a y arise f r o m
one of at least three sources:

(1) The estimate of

gross Income is n o t good e n o u g h f o r the p urpose i m ­
p l i e d b y the h ypothesis*

(2) W i n d f a l l s tend to e x ­

aggerate d i f f e r e n c e s in the f arm account l a b o r Income
b e t w e e n farms,

and i n erratic ways*

(3) Some

standard

charges w h i c h are given by the d e f i n i t i o n of l a b o r in­
come a nd by a c c o u n t i n g p r o c e d u r e s are not appro p r i a te*
F o r example,

the charge against the Invest m e n t is

P o s s i b l y this charge should vary f o r the average of
fami

b e t w e e n types of Investment,

tween farms for Investment of the

a n d vary a g a i n b e ­
same types if a more

accurate mea s u r e of what the f a r m e r seeks to max i m i ze
is to be obtained*
"Labor Income" is c onditioned by the three items
above*

The effect of the definition is to charge items

o t her than the o p e r a t o r 1s labor at cost,
or a s s umed rates,

or at standard

and then to a s s i g n all of the differ­

ence b e t w e e n charges and gross income to the farm o pe r ­
ator*

In competitive equilibrium,

w o u l d be its m a rginal

the return to labor

return times the quantity employed*

"Profit" w o u l d be zero*

The returns to all factors,

the others also being valu e d at ma r g i n a l value p r o d u c t ­
ivity times quantity used,
income*

w o uld abs o r b the whole gross

Fac t o r s w o u l d all be charged at their m a r k e t

prices as the factors would be u sed in such quantities
and propor t i o n s that the mar g i n a l value produ c t i v i t ies

would equal the respective prices.
The business or farming Is competitive* but It
Is never In equilibrium from the standpoints of weather,
prices, technology, and biology*

Thus* the marginal

returns to some factors on a farm are greater or less
than Implied by their prices or stated costs*

All of

the residuals resulting from disequilibrium are thrown
together In "labor income.”
This Is not to say that labor Income Is not an
extremely useful concept.

Xt Is the pay of the opera­

tor plus the residuals from the actual costs and returns
attributable to all other factors.
u n m e a s u r e d Input.

That

Management Is an

some fan ne r s are able to earn

h i g h l a b o r incomes over a p e r i o d of y e a r s

suggests

that the factor of management often Is Important enough
as a contributor to all of the elements in labor Income
that there is actually a high correlation between
labor Income and the returns to the operator In the
theoretical sense*
Let It be assumed,

for the moment* that labor In—

come fairly represents the co t£P t which farmers try to
maximize*

Then we are Interested In whether differences

In labor Income can be explained In part by the "struc­
ture" of the farm business.

By structure in this case

Is me ant the quantities and p r o p o r t i o n s in w h i c h cate­
gories of p r o d u c t i v i t y factors have b e e n combined*
It has

Just b e en shown that the estimate of gross Income

for the 86 dairy farms was not close enough,
that, w h e n costs were

generally,

subtracted a c c o r d i n g to the d e ­

f i n i t i o n of l a b o r Income,

the "structure" a ctually

h e l p e d to e x p l a i n p a r t of the differences in l a b o r in­
come *

The failure of correlation, however,

can not be

a s s u m e d to be caused a l t o g e t h e r b y shortcomings in the
estimate of gross income from the gross income function*
In o t h e r words,

there is e r ror in the estimate of

l a b o r income by the f u n c t i o n and by the

statement of

labor income a c c ording to the farm account record.
E s t i m a t e s of the l a b o r Incomes of the 108 n o t -dairy
farms were m a d e f r o m the gross income e q u a t i o n g i v e n in
the second column of table 6.

These est i m a t e s were

correlated at the 5 % level of confidence w i t h the
recorded l a b o r Incomes,
c orrelation being 0,38*

the simple coefficient of
Thus,

part of the d ifferences

in the labor income o f the 108 n o t - d a i r y farms can be
e x p l a i n e d o n the basis of the m a n n e r in w h i c h the c ate­
gories of factors are combined*

It Is to be n o t e d in

this connection that the correl a t i o n coefficients b e ­
t w e e n the rec o r d e d gross incomes and the estimates of
gross income are slightly h i g h e r for the no t - d a i r y
farms than for the dairy farms

(tables 8 to 10)*

10* B»tlaat«a of Labor Income Whin C t U g o r i M of Paciors trt C h a w < i at Marginal Vtlut ^rodttetlvltl»»
Tht h T p o t h e i l s w a s

set up that Termers h a v e d i f fer­

ent concepts o f m a r g i n a l costs d e p e n d i n g o n pe r s o n a l o b ­
jectives an d circumstances*

This should result In the

use of r e s o u r c e s In ways w h i c h w ill tend not to max i mise
labor income as d e f i n e d b y the
ing.

standards of f a r m a c c o u n t ­

That Is, farmers will have d i f ferent concepts of

the m a r g i n a l costs of the same factors.

The m a r g i n a l

costs for categories i n the f o l l o w i n g e s t i m a t i n g e q u a t i o n
were calcul a t e d for e ach of the 86 dairy farms:
L o g _P = 0*lj.22 / O.I4.32 log total f a r m expense not
l a b o r / 0 . 1 log total l a b o r charge /
0 *332: log total I n v e stment (table 0).
This is the same e q u a t i o n as u sed i n the p revious estimate
of l a bor income*

W h e n labor income was ca l c u l a t e d by

charging factors against the business acc o r d i n g to their
e s t i m a t e d m a r g i n a l value p r o d u c t i v i t i e s a p p l y i n g to the
p a r t i c u l a r farm,

the r e lationship b e t w e e n stated labor

Income and e s t i m a t e d l a b o r income was significant at
5%

(figure 2)«

By charging factors a c c ording to their

e s t i m a t e d m a rginal value p r o d u c t i v i t i e s rather than at
standard rates,

the e s t imates of costs are changed*

Costs are thus subtracted at different rates f o r each
farm f rom both the farm-account r e corded gross Income and
the estimate of gross income from the function.
costs are v a l u e d in this way,

When

the differences b e t w e e n

recorded and e s t i m a t e d gross income are not so great but

-76-

Figure 2.
1$-

The Relation between Recorded Labor Income and
Estimated Labor Income when Factors of Production
Are Charged against the Business According to
Their Marginal Value Productivities.
86 Dairy Farms, Type—of —Farming Areas

5

and 6 ,

1950.
lU-

Re corded
Ltbor
Ineomt*
Revised*

(#1,000)
10

-

©O
8•©

r rn 0 .59

••

.
»

2-

- *.• o
• 0 •1
bo

o

-

o O

• o
o
<
O

-2

-31

0
Estimated Labor Inoo:

(#1,000)

« See title

1

I

_r

6

8

310

, Revised*

that the two m e a s u r e s of g r o s s l e s s costs are

completed.

The a v e r a g e value p r o d u c t i v i t y f u n c t i o n thus e x p l a i n s
p a r t of the d i f f e r e n c e s

in this m e a s u r e of n e t b e t w e e n

f anas.
11. G r a p h i c P r e s e n t a t i o n o f the E s t i m a t e o f G r o s s Income
and C o s t s
“
If the n u m b e r of c a t e g o r i e s o f f a c t o r s
gross

the

Income e s t i m a t i n g e q u a t i o n can be s h o w n in three

dimensions
face

Is two,

(figure 3).

The e q u a t i o n of the

curved s u r ­

is the f i rst e q u a t i o n i n table 5 e x p r e s s e d in n a t u ­

ral numbers.

D e d u c t i o n s f r o m gross

Income f o r the e s t i ­

m a t e of l a b o r income are g i v e n b y the t o tal cost plane.
Par t of the a r e a o n the c u r v e d surface lies a b ove the
plane;

gross Income is l a r g e r t han total

ing the o p e r a t o r ' s labor.
vestment,

f o r example.

charges,

includ­

A t the $ 1 0 0 , 0 0 0 l e v e l o f in­

Income l i e s above e x p e n s e b e t w e e n

the $ 5 , 0 0 0 a n d a p p r o x i m a t e l y the $ 3 0 , 0 0 0 l e v e l s of total
f a r m expense.
This c o n c e p t c a n be
map

(figure U)*>

seen m o r e

G r o s s Income

cle a r l y o n a c o n t o u r

is shown as a f u n c t i o n of

total f a r m exp e n s e a n d total i n v e s tment.
indicate

d i f f e r e n t l e v e l s of gross income,

parallel lines

The

contour*

a n d the

indicate l e v e l s of total r e c o r d e d costs.

T o tal r e c o r d e d costs are
the total invest m e n t .

total

f a r m e x p e n s e p l u s 5/6 o n

The f a r m o p e r a t o r who h a s a g i v e n

amou n t o f total re s o u r c e s is I n t e r e s t e d in u s i n g t h e m in
s u c h f o rms as to m a x i m i s e

the d i f f e r e n c e b e t w e e n the

78-

Ftgure 3*

Gross Income Estimated from Total Farm Expense and Total
Investment, and Total Costs, Including Interest on the
Investment at $ % but Hot Including Operator»s Labor.
86 Dairy Farms, Type-of-Farming Areas 5 and 6, 1950*

60
Total oost - total farm expense less opera
/
labor plus interest^-—
/
5*
_ -

50

ko

30

20
Gross Income

10

50

50.>-* -

, ),

111 000 2
expense ^

Total investment (♦1*000)

0.51a
1. Gross income is estimated by 2.87(total farm expense)
times
0.322
(total investment)
. The contours show 110*000, 120*000, and
$30,000 levels of gross in come.

I

Figure i|. Gross Income Estimated from Total Farm Expense and Total Investment
86 Dairy Farms
Type-of-Farming Areas 5 and 6, Michigan, 1950
Total fann
expense

Contours show levels of gross income,
Iso-cost lines _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Scale line _ _ _ _ _ _ _

Farms • f

Area of positive
labor 1ntwMi

oe

100
Total investment ($1,000)

-80

curved gross Income

surface

(contours) and the parallel

lines (the plane or total costs)*

The "scale line" In­

dicates the amounts or "total farm expense" and "total
Investment" w h i c h would on a n average maximize the net
return to the farm and the operatorfor dirrerent values or
total outlays plus charges*

The positions of the rarms

with respect to Investment and totsl farm expense are
plotted*

They do not rollow the "scale line" which Is

developed from taking "total farm expense" as charged,
and taking interest on investment at
emphasize expense more,
Furthermore,

Instead,

they

and investment less*

at the £ 1 0 0 , 0 0 0 level or investment,

the area of net return to both the farm and the operator
has hardly,

IT at all,

were to be continued,

begun to diminish*
in fact,

If the rigure

It would show that the

area of a positive net return would not be closed off
altogether w ith respect to investment until a level of
$ 3 0 0 , 0 0 0 was reached*

common observation*

This result is not consistent with
E i t h e r the gross income estimating

equation o r the definition of charges or both are In
error*
It Is recognized that farmers may not necessarily
have the charges for total farm expense items which are
entered in the farm account books in mind when they u n d e r ­
take outlays for factors In the total farm expense
gory*

cate­

Nor are they necessarily equating the marginal

return on Invested capital to the standard rate of 55®*

-81

The evidence on m a r g i n a l value productivities of these
two categories of factors Indicates that at the m a r g i n
the return to total farm expense Is in the n e i g h b o r h o o d
of 80^ for a dol l a r of outlay or charge e n t e r e d In the
farm account books

(table lLj.) •

F o r all Investments taken

together the r e t u r n at the m a r g i n Is about 10?£,
ers equate

If f a r m ­

subjective costs and returns at the margin,

then

these margi n a l rates of ret u r n may be Interpreted as m e a s ­
ures of the p roportions of the

stated values o f the

factor

categories w h i c h they have in m i n d in the organization
and operat i o n of the farm business*
If total farm, expense is subjectively valued at only
0*8 of the amount entered In the f arm account books at the
m a r g i n and total investment is charged at a rate of 1 0 ^
at the margin,

a new set of iso-cost lines appears,

w ith

new points of tangency w i t h the g r o s s income contours.
Therefore a new "scale line" or line of best a l l o c a t i o n of
total resources between permanent investment and outlays
(including charges for n o n-cash expense items) appears
(figure 5)*

O n figure 5 it appears that u n der average

conditions the area of net r e t u r n above all costs is closed
off in the n e i ghborhood of $70,000 of total investment.
Plotting of positions of the 86 dairy farms w i t h respect
to total farm expense and total investment shows that
they more closely follow the new scale line than the scale
line arising f rom taking expenses at dollar value and

Figure 5.

Gross Income Estimated from Total Farm Expense end Total Investment

86 Dairy Farms
Type-of-Farming Areas 5 and 6, Michigan, 1950
Contours show estimate of gross income.
Iso-cost lines - - - - - - - . Total farm expense charged at 0.8,
investment charged at 10$.
Scale line

Total farm
expense

(51,000)
504

Gross income

line

Iso-cost (total

12.5

Scale linee
Area

bor ini

0

e•

7.5

zo

50

40

5o

<

Total investment ($1,000)

10

charging interest at

%% (figure I*) *

In other words,

charging total farm expense at 0*00 on the dollar and
charging Interest on the investment at 10£ are consis­
tent with the actual business behavior or farmers*

It

may be clarifying to show the positions of iso-cost lines
under the two hypotheses in a diagram which is not com­
plicated by the presence of the gross income contours*
This is done in figure 6*.
As was stated earlier in this chapter, two of the
factors which cause variations in gross income (and hence
in labor income) are windfalls and management*

Gross

income and labor income as defined by the standard pro­
cedure include the former as a contribution to gross*
Theoretically the r esidual due to management appears in
both measures also*

If the estimate of gross income is

derived from a gross income equation, the windfall gains/
and losses and the difference due to management are both
eliminated*

Management may be regarded as a random element

affecting the value of gross income as calculated by ac­
counting*

When gross income is estimated from a regression

equation, the influence of management upon gross income
is lost* Possibly the definition of labor income that is
commonly accepted includes irrelevant element^ which tend
to cause the labor income figures between farms to show
greater variations than In a real sense exist*

plt^uro o.
Total farm
expense
($>1 ,000)
50

s m t . .»

Of .reported Amount and of Cnar-in,. xoxax

Explanation:

E a t l m a t e ^ o f ^ ^ i ^ ^ g w / S d

10 cents

on a dollar of Investment^

-------

ko

30

Total investment (51#000)

-85-

11* Summary o f C h a p t e r I V
Tla© I n f o r m a t i o n in tills chapter can be
under six headings,

summarized

which are as follows:

1* By u s i n g categories of factors given In the far m
accounts,

the l o g o f gross Income of a farm Is e s t i m a t e d

with a standard e r r o r of about 0*1*
2* The farms are est i m a t e d to have b e e n o p e rating
at constant returns to scale.

The sums of the elasticities

of gross income w i t h res p e c t to categories of factors
are slightly more

than one w h e n the num b e r of categories

of factors is larg e r than four*
3* The confidence limits of the coefficients of
elasticity are relatively wide,
reliability.

e ven at the 10^ level of

The ranges of the confidence limits of the

coefficients are not narrower for 86 dairy farms than
they are for all other farms,
U* Marginal
estimated.

regardless of type*

returns to categories of factors are

The estimates are generally reasonable,

al­

though the confidence intervals of the elasticity c o e f f i ­
cients are wide*
5* L a b o r income as defined by farm a c c o u n t i n g p r o ­
cedure can not be p r e dicted from the gross income equations,
subtracting total farm expenses and Interest on the in­
vestment at 5%.

The estimate of gross Income is useful in

explaining part of the differences in Income between farms
when charges are made at the marginal
each farm b y the general gross

rates implied on

income equation.

P r o m this

it is inferred that individual farm operators seek to
maximize different forms of "income," and these are to an
extent revealed by the marginal returns demanded from the
use of the

categories of factors*

6* In the matter of the choice between total farm
expense and Investment,

the farms are operated according

to charges for categories of factors at calculated m a r ­
ginal rates of return*

That is, the d isposition of total

resources between investment a n d expense o n actual farms
agrees better with charges against total farm expense at
0*8 on the dollar,

and a charge on total Investment of

- 67-

CHAPTER V

USE OP GROSS INCOME F U N C T I O N S IN THE
F A R M M A N A G E M E N T E X T E N S I O N PRO G R A M

The work In the p r e c e d i n g chapters shows that c o n sist­
ent r e g r e s s i o n equations es t i m a t i n g gross income can be
derived f r o m f a r m accovint data.
income estimates
account work*

It follows that gross

could be included as a phase of farm

But the pr e c e d i n g part of this study has

shown that there are pitfalls of the r e g r ession analysis
w h ich m a y lead to w r ong conclusions*
part of this chapter will be

Thus*

the first

concerned w ith m e t h o d s of

increasing the reliability of e s t imates and confidence
in the meaning of estimates.

The second part will take

up uses w n l c h might be made of gross Income equations in
the farm account program.
1. Sug ge st ed Changes in the Data and M e t h o d of A n a l y s i s
It has been shown that there may be biases in the
valuations of factors used In computing gross income
equations.

These biases enter into

the calcul a t i o n of

labor Income and the cross sectional analysis of labor
income*.

By the usual cross sectional analysis*

come is related to crop yield*
size of business*

labor in­

efficiency of m a n labor*

and o t her factors of success*

88—

The valuations should be as free from biases as practic­
able If greatest use Is to be made of gross Income equa­
tions*

In the cross sectional analysis explaining differ­

ences In labor Income according to farm success factors,
it is ordinarily taken as conclusive If a general rela­
tionship between labor Income and success factors Is
shown.

In the computation of gross Income equations, how­

ever, and in the estimation of marginal value productiv­
ities of factor categories, biases lead to wrong numerical
estimates of definite concepts.

Thus, the undervaluation

of land, for example, has the effect of overstating the
marginal value productivity of land at Its market value In
direct proportion to the undervaluation*

The undervalua­

tion of land ordinarily would not affect the conclusions
indicated by cross sectional analysis.

But the bias Is

more serious In the case of analysis by the gross Income
function*
In addition to general biases In the data, there may
be random inconsistencies between farms.

These inconsist­

encies have the effect of causing the confidence Intervals
of regression coefficients to be wider than Is actually
called for by the nature of the underlying data.

Thus,

there are two reasons for a general consideration of ab­
sence of bias and consistency of valuations of factor in
the farm account books.
not an easy problem*

It Is recognized that this is

In the case of land, elimination

of bias w o u l d call for p e r i o d i c rev i s i o n s in the values
of farms.

The m a i n p u r p o s e

in a v o i d i n g such changes is

to p r e vent changes in land value f r o m c o n t r i b u t i n g to
labor income.
and changes

F e e d s and crops are

in their p r i c e s

r egarded as more

liquid,

in inventories are p e r m i t t e d

to affect labor income.
As a n a l t e r n a t i v e

to the k e e p i n g of land a n d l i v e ­

stock values m ore o r less in line w i t h markets,
in the farm a ccount books could be

the values

specially a d j u s t e d

for c o m p utin g gross Income equations.

T h i s p r o c e d u r e has

the d i s a d van t a g e of im p l yi n g two schemes of val u e s a n d
w o u l d result in some loss of confidence
lytic work.

Furthermore,

the data for one purpose
not a n other
would

in the whole a n a ­

the procedure of a d j u s t m e n t of
(gross Income equations)

and

(analysis of l a bor Income by success factors)

call for a co m p l i c a t e d i n t e r p r e t a t i o n of both r e ­

sults ,
2, Se l e c t i o n of Categories of F a c t o r s
C o n s i d e r a t i o n could be given to the types of data
o b t ained by the f a r m account records and to the m a n n e r of
summarizing the data.

The categories of factors u s e d in

computing gross income equations in this study are not
the "factors'* of economic

theory,

land and capital.

L a n d and capital in economic theory must be c o n s idered as
factor categories themselves,

as well as items

m a c h i n e r y w h i c h a p p e a r in farm account books.
of interest

such as
The q uestion

concerns the m a n n e r of combining factors into

into categories to obtain m e a n i n g f u l equ a t i o n s of gross
income.

The two broad categories,

farm expense and total Investment,

for example,

total

are concerned w i t h a

practical problem--the p r o b l e m of the ratio of the p e r ­
manent Investment in the business to the outlay in one
peri od of operation.

I n order to avo i d combining expense

and investment factors,

this two-way bre a k d o w n should be

m a i n t a i n e d as a fundamental div i s i o n of factors into
classes.

W i t h i n these two classes expenses a n d i n v e s t ­

ment can be b r o k e n down into further detail.
W i t h i n e a c h of the two broad classes of factors there
can be several categorlea of factors.

These categories

should be de s i g n e d so that the factors w i t h i n e a c h are
nearly perfect complements or nearly perfect substitutes.
W h e n factors are
value

complementary,

in the use of the other.

and oil are complements.

the use of one implies a
F o r example,

Factors are

one can be used to replace the other.

gasoline

substitutes w h e n
Fami l y labor and

hired labor are substitutable and could be Included in
the same category,

labor.

L i v e s t o c k f eed and labor are

neither complements nor substitutes and should be In
1
different categories.
The Mi c h i g a n farm account system
factors are g e n erally combined into categories w h i c h
thus consist largely of complements and substitutes.
W h e n gross income Is estimated f rom categories of factors,
it is important that this be so.

This k ind of scheme of

1. At the level at which factors ordinarily are cowbined for account*
ing purposes*

classification makes it possible to determine elasticities
of gross Income and marginal value productivities by
factor categories which are easily understood.

Further­

more, it implies mathematical results which should dis­
tinguish between categories of factors according to elas­
ticity of gross income and marginal value productivities.
The estimates of marginal returns to investment and
expenses are sometimes confused or duplicated.
Improvements fcnd machinery and equipment.

Consider

Depreciation

on both items is Included at standard rates in "improve­
ment expense" and "machinery and equipment net decreases"
in the farm accounts*

Thus depreciation is Included as

an expense for which the return at the margin would the­
oretically be one in equilibrium*

That is, the farmer

will equate marginal costs and marginal returns of mach­
inery and improvements at the margin.

Marginal costs

will include depreciation.
At the same time the farmer will equate marginal
costs and marginal returns of investment in machinery
and improvements.
clude depreciation.

The marginal costs of these items in­
Therefore, when depreciation is in­

cluded as an expense item of machinery in the estimation
of gross Income, it is included twice, or duplicated.
Thus, depreciation on Investment items should not be in­
cluded in expense in the computing of gross Income equa­
tions, as was done in the previous work.

There are uses

of gross income equations which call for data outside

oT that necessary for pure accounting of Income and ex­
pense.

These data Include acreages and yields of crops*

production rates of livestock* and average prices of pro­
ducts sold.

Some of this Information Is already available.

This type of Information would be useful in connection with
gross Income equations when the equations ©re used to
account for differences between estimated and stated gross
income and labor income by enterprises.

The place of

physical units of production* average yields or rates of
production, and prices of prdducts sold will be made evi­
dent in part three of the second section of this chapter.
Suggested Uses of Gross Income Functions in Farm Busness Analysis Reports
This section will show how analysis of gross income
as a function of factors can be included in publications
based on farm account records.
Part I. Estimates of gross income and labor income
One of the main purposes of farm business analysis
is to explain differences in the profitableness of farms.
When such explanations are based on groups of farm account
records, a long—used procedure has been to show correla­
tions between farm success factors and labor income.
This procedure is based upon the assumption that labor
Income as defined is usually a good estimate of what
farmers seek to maximise.

It was pointed out in Chapter

IV that labor income is not a perfect measure of income*

and furthermore,

that It Is a measure of the sum of all

the residuals above recorded costs. Including, of course,
the residual above the recorded charge for the operator's
labor.

Thus, its use *s an absolute measure of the return

to the operator can be questioned.

However, labor Income

has long been used, is widely understood, and is easily
defined.

If labor income is to be treated as if it were

the measure of returns to the operator,

gross income

equations can be considered from the viewpoint of con­
tributing to the explanation of differences in labor income.
If the estimate of gross income is precise enough,
part of the differences in labor incomes can be explained
on the basis of the way in which factors are combined in
the operation of the farm.

Following this, further differ-*

ences can be explained on the basis of the farm success
factors.

The logic of this order is that first the effi­

ciency of the organization of the farm business as a
whole (the general layout) is considered.

Then more de­

tailed questions are taken up. These questions are con­
cerned with crop yields, efficiency of labor, and so on*
This does not imply that one phase of the problem of
differences in labor income is more important than enother.
It implies that first there must be a general plan (proper
combination of resources) of a farm; secondly,

the effec­

tiveness of the plan will depend in part upon how effi­
ciently the combination of resources performs, once given*

-9k

Specifically,

a h i g h crop yield,

f ar m is to be profitable,

T o r example,

if the

m u s t be o b t a i n e d subject to tbe

r e s t r i c t i o n i m p os ed by a p r a c t i c a l limit to tbe ratio of
tbe out la y f o r g r o w in g tbe crop to tbe Investment in tbe
land u p o n w h i c h it is jrown.
income,

In o r d er to Increase l abor

crop y i e l d s are not inc re as ed regardless of the

direct outlay Involved.

Tbe balance b e t w e e n l a n d i n v e s t ­

ment and crop expense which. Is m o st favorable
Income in the general

case

to labor

can be de t e r m i n e d by the use

of a gross Income f u n ct io n b a s e d on factors of p r o d u c ­
tion.

Tbe factors

Include l and a n d crop expense*

A s a n o t h e r illustration,

a gross Income f un c t i o n

m a y be u se d to define h o w tbe n u m b e r o f productive m a n
work u n its p e r man will r e a c h an optimum.
m e nt in a f a r m and tbe outlays o t h e r thsn

Tbe

invest­

for la bor o b ­

viously can not be In c re as ed w i t h o u t limit while h o l d i n g
the am o u n t of labor constant.
work units p e r m a n are

That is, productive m a n

subject to diminishing returns,

If la bor Is h e l d constant.

The available l a b o r Is spread

too thin In r e l a t i o n to o ther factors.

Tbe gross Income

f u n c t i o n will

resources should

show how,

for most farms,

be divided b e t w e e n labor,
and so forth,

land, machinery,

livestock,

In order to m a x im i ze l a b or income*

It is rec og ni ze d that the earning of a high labor
income Is contingent u pon the f a r m ’s h a vi ng gen er al l y
favorable ratios in regard to mo s t or all of the f arm
success factors.

It is also recogni ze d that tbe farm

success f a c t o r s are In soma ways in conflict w i t h ona
another•

To lllustrata,

w ork u n i t s p e r

a large n u m b e r of p r o d u c t i v e m a n

m a n m a y be a c h i e v e d at the e x p e n s e o f the

cere w i t h w h i c h the w o rk is done.

The gross income e s t i m ­

ating e q u a t i o n will lay down the general

conditions for

the concurrent m a x i m i z a t i o n of l a b o r income w i t h respect
to all of the farm success factors.
A l l of this is a w a y of p o i n t i n g out to farm a c c ou n t
cooperators h o w one of the reasons for a h i g h l a b o r Income
or a low one m a y be a s s o c i a t e d w i t h the balance b e t w e e n
the fact or s of production.

This m e a n s that there m a y be

too little o r too m u c h labor in r elation to the investment
in land,

too little o r too m u c h outl ay for direct o p e r a t ­

ing e x p e n s e s in rel at io n to the total Investment in the
farm,

and
The

so on.
results of the fo re go i ng an al y si s could be p r e ­

sented in a table,

as on page 96.

N o t w i t h s t a n d i n g crop yields,
livestock,

and so on,

(table 1 6 ).
p r o d u c t i o n rates of

the l a b o r income of this farm is

e s t i m a t e d to be 0 307 h i g h e r than the average l a b o r income
w h i c h could be e x p e c t e d from 0^-6,092 Qf all costs,

simply

because of the way in w hich total f ar m expense and in­
v estment are combined.

If the r e l a t io ns hi p b e t we en the

e s t i m a t e d l a b o r income and the
statistically

significant,

stated la bor Income is

the table on the next page

(table 1 6 ) will be of interest to the coopera ti ng farmers.

Table 16.

A n a l y s i s o f R e c o rd ed L a b o r Income A c c o r d i n g
1
to E s t i m a t e of Gross
Income

Your Farm

I tem

Recorded

Estimated

Gross income

$ 22,362

$19,820

12,602

12,602

T otal f ar m
expense

Gross income at
average returns
p e r dollar of to ­
tal f ar m expense
plus Interest 0 5 %

*19,513

16,092
5/o interest on

inve stment

3,14.90

3,14-90

L a b o r income

7,830

3,728

G ross income

2,814-9

307

L a b o r income

14,1409

307

3,1421

Difference from
e x p e ct ed r e ­
turn

1# Tbe est i mating equation is the first one in table 6 .
The farm is in Jac k s o n County.

-97-

The e x p l a n a t i o n of differe nc e in l a b or income b y
tbe c o m b i n a t i o n o f re so ur ce s ad m i t t e d l y is sma ll —
$ 30 7 of a total d i f f e r e n c e of $lj.»lj.09 ($7 #830 m i n u s
$3# 14-21)*

T h i s explanation,

however,

is legiti ma t e

that it does not rest u p o n crop yields,
dollars income p e r dollar of expense,

in

p r o d u c t i o n rates,

and so on, w h i c h

are t h e m s e lv es p r e j ud i ci al of gross income a n d hence of
l a bo r income*

T h i s table

(16)

is m o s t important,

however,

because it gives a statement of the gross Income a n d of
the l a b or income w h i c h the farmer should h ave e x p e c t e d ,
con si de ri n g his

i nvestments and outlays*

w i t h low l a b o r incomes in particular,
stimulate

F o r farmers

the table

should

inquiry into w h y gross income and l abor i n ­

come do not m e a s u r e up to standard.

This phase of the

study is d e v e lo pe d in Part III of this section*.
In C h a p t e r IV It Is shown that for the 86 dai ry
farms the c o r r e l a t i o n b e t w e e n r e c or de d l abor Income and
l a bo r income f rom the estimate of gross Income is not
significant at

F o r the 108 n ot — dairy farms the

coefficient is 0« 38, w h i c h is significant at S % a low correlation,
meaningful
small part)

as is expected*

correlation.

However,

It shows that part

T h i s Is

it is a
(albeit a

of the differences In recorded labor Income

can be e x p l a i n e d fr o m e qu ations es ti ma ti ng gross income.

P a r t II,

E s t i m a t e s of M a r g i n a l Value P r o d u c t i v i t i e s

E s t i m a t e d m a r g i n a l value p r o d u c t i v i t i e s can be

cal­

c u l a t e d Tor all fa rms Tor categories of factors u s e d in
the gross e s t i m a t i n g equation.

One pur po se o f e s t i ma te s

of mo* glnal value p r o d u c t i v i t i e s of categories of f a c to rs
is that they show c o n d i t i o n s of imbalance in the use of
resources o n the in di v id ua l farm*

Consider a farm which

uses too m u c h l a n d in rel at io n to the amount of m a c h i n e r y
a nd equipment,
This will

ac co rd in g to the

gross Income function.

show up in the form o f a h i g h marginal

value

p r o d u c t i v i t y for m a c h i n e r y a nd e q u i p m e n t a n d a relatively
low m a r g i n a l value p r o d u c t i v i t y for land.

The

im pl ic at io n

is that the farm is u n d e r e q u i p p e d for its size,

or is too

large for the amount of equipment.
E s t i m a t e s of m a r g i n a l value p r o d u c t i v i t i e s at the
m e a n values of factors for farms cl as si fi ed a c c o r d i n g to
l a bo r Income

can be

shown (table 17)•

The b r e a k do wn is

m ade by la bor income groups for this reason: m a r g i n a l
returns to factors may v ary w i t h l a b or income because
of v a r i a t i o n s in the q u a n t i t i e s of factors e m p l o y e d at
different levels of labor income.
marginal
come

It is to be n o t e d that

value p r o d u c t i v i t i e s tend to fall as l a bo r in­

(and the expense and Investment categories)

larger.

become

This w a r n s the p e r s o n whose l abor income and

o utlays are a l r e a d y h i g h that he can expect r e l a ti ve ly
smaller returns at the margin.

Table 17 indicates the

lines of investment and expense which,

a c c o r d i n g to

Table 17*

A m o u n t s a nd M e a n Marginal Value P r o d u c t l v l t i e s
of Categories of Factors f or Farms Class1
lfied A c c o r d i n g to La bor Income

A ll Farms,

Area 5, 1950

1/3 lowest
farms

Category of
Factors

A l l Farms

l/3 highest
farms

2

Amt.
L a n d and
improve­
ment 8

MVP

Amt.

MVP

Amt.

MVP

7#

#17*1x14-6

7%

#20,923

1%

Total l a ­
bor charge

2,1)29 # 0 . 8 3

2 , 6241

#0.70

3,123

#0 .75

Total farm
expense o t h ­
er than l a ­
bor

5,007

0.91

5,730

0.92

7,305

0.88

Inve stments
other than
land and i m ­
provements
12,629

23,^

15, 215

22#

20,191

21#

L a b o r in ­
come

# 15,827

7 I 4 .6

3,687

7,068

The es t im at in g equation is 2_ In table 7# Chapter IV.
2. Margin©! Value Productivity

average experiences should be expanded or contracted*
The a v e r ag e m a r g i n a l r e t u r n s to machin er y,
stock,

equipment^live­

T e e d a n d crops as a group are l a r g e r for all l a bor

Income groups than called for by a c c e p t e d inte re st and

The m a r g i n a l p r i n c i p l e

can be d e m o n s t r a t e d by a series

of tables in e a c h one of w h i c h all factors but one are
h e ld constant*

S u c h tables w i l l show h o w the m a r g i n a l

value p r o d u c t i v i t i e s o f the

constant f a c to r s increase as

the a mo u n t u s e d of the v a r i e d f a c t o r bec om es larger.

The

m a r g i n a l value p r o d u c t i v i t y of the v a r i e d fac to r will de­
crease as its qxiantity is i n c r e a s e d (table 16)*
ample g i v en in table 16 was

The e x ­

selected at random*

T a b l e s 17 a n d 18 show h o w e s t i m a t e s of m a r g i n a l value
productivities
by farm*

can be e x p r e s s e d with ou t c o m p u ta ti on s f arm

If the clerical h e l p is available,

computations

can be r e d u c e d to a routine w h i c h c a n be h a n d l e d readily
by clerks with no statistical training*
perience h as b e e n that the clerks,
ed as to the m e a n i n g of the work,
because

they are e a g e r to get

The writer's e x ­

if ca r efully I n s t r u c t ­
are m o s t cooperative

to the results w h i c h they

themselves can anticipate.

The p o s s i b i l i t y that c o o p e r a t ­

ing farmers m i g h t calculate

the es ti ma te s of m a r g i n a l

value p ro du c t i v i t i e s for th eir farms m a y be considered*
This m i g h t be too difficult from the standpoint o f the
m a t h e m a t i c s involved.

However,

the p r o p o r t i o n o f farm

operators who have been t h ro ug h h i g h school Is constantly
increasing,

and m os t of these people have had some t r a in —

Table 18» E s t i m a t e d M a r g i n a l Value P r o d u c t i v i t i e s of C a t e ­
g or i e s of F a c t o r s for D i f f e r e n t In
ve s t m e n t s In Pr od u ct iv e Li ve st oc k^
163 P a m s ,

T y p e - o f - F a r m i n g A r eas 5 a n d 6, 1950

(Other C a t e g o r i e s of F a c t o r s H e l d Constant at T h e i r A v e r a g e
Value s )

Category of
Factors

Charge or
Investment
(average)

Investment in Productive Llvestock
$1,000

$ 2,500

$ 5 ,000

$7, 50 0

*8,77S

0 . 008

0.091

0 .09U

0.096

Improvements

8,578

0.001

0.001

0.001

0.001

Total labor
char ge

2,610-

0 .525

0 . 5U5

0.560

0.569

Total farm
expense oth­
er than labor

5,780

1.061+

1 .10I+

1 . 13 5

1.153

0.257

0.107

0 . 05 5

0 . 03 6

Land

Productive
livestock in­
vestment

(varied
factor)

Machinery and
eq u i p m e n t In­
vestment

6,26 6

0.209

0.217

0 .223.

0.227

I n v e stment In
feed an d crops

3,607

0.306

0.318

0. 327

0.332

TZ The e s t i m a t i n g e q u at io n is
for 19U faras*

*-n table d, Cha pt er IV,

-102-

ing In algebra*

Furthermore,

tbe Salg|hjt— Hughe a p r o g r a m In

the h i g h sch oo ls

could well afford to

time to t e a c h i n g boys how to estim&te

devote considerable
what t h e i r f arms

ou~ht to e a r n c o n s i d e r i n g t h eir inputs.
C o m p u t a t i o n s o f the estimated m a r g i n a l value p r o d u c t ­
ivities oT c a t e go ri es or ract o rs ror i n d i v i d u a l f arms
can be s h o w n as i n table 19
short o n land, machinery,

(page 1 0 3 ) *

and reed a n d crop inventories*

Its m a r g i n a l r e t u r n s to these factors
high.

This r a r m is

are c om p a r a t i v e l y

" E x p e n s e s " a n d Investment in p r o d u c t i v e l i v e s t o c k

are high, a n d the marginal

re turns a r e l o w e r t h a n average*

It is not n e c e s s a r y that the a v e r a g e or all rarms be
u s e d as the

standard or comparison.

w i t h the h i g h e s t l a b or Incomes can be
be c la ssiried a c c o r d i n g to size, and
Tor the v a r io us

size

The

o n e — third rarms

used; or rarms can
s ta ndards e s t a b l i s h e d

groups.

Part III* E f r e c t s or Yields, R a t e s o f P r o d u c t i o n ,
?rlces, a n d Size of Business on V a r i a t i o n s in G r o s s
Income B e t w e e n Farms
The r e c o r d e d jross incomes of f a m s
below their e s t i m a t e d values.

vary above and

It is o f interest to account

specifically f o r these variations by y i e l d s a n d rates of
production,

p r i c e s received,

a n d the

size of business.

A s i m p l if ie d example will be c o n s i d e r e d first.
pose

Sup­

that the bus in es s of a f arm consists of one e n t e r ­

prise,

the p r o d u c t i o n of whole milk.

The ess en t ia l data

-103-

Table 19*

Categories of Factors and Their Estimated Mar­
ginal Value Productivities

163 Area 5 Farms
Category

of
factors

Average
amount

Estimated

Your Farm'*’
Amount

marginal
value pro­
ductivity
at average
amount

Estimated
marginal
value
productiv­
ity

$8,775

0.096

$10,614-5

0 . 138

Improvements

8,578

0.001

16,09k

0.0 0 1

Total labor
charge

2, 6I4.I

0.561

U,075

0 .521+

Total farm
expense oth­
er than labor

5,780

1 . 13k

10, 9 8 3

0.857

Productive
livestock

5, 2I4.3

0.053

19,2 79

0.036

I.Tachinery and
equipment

6,266

0.223

6,180

O.I 4I4-O

F e e d a n d crops

3, 607

0.327

3,197

0.569

Land

Investment in:

■ "Sv —
—T. -A
A--Calhoun
County
Farnu

or the fa r m are g i v e n In the table below:
Table 20#

D a t a N e e d e d to A c c o u n t for Difference I n R e c o r d ­

ed L a b o r Income fr o m E s t i m a t e d L a b o r I n ­
come o n a O n e —pix>duct F a r m

This F a r m
Recorded
gross
income

Production
R e c o rd ed
total
of m i l k
far m e x ­ p e r cow
pense
(c w t .)
plus 5/
o n In­
vestments

$ U ,000

$3,000

50

Price of
m i l k p er
cwt.

A verage
number
of cows

L a bor
income

$U.oo

20

$1,000

Average
number
of cows

L abor
Income

16 2/3

1,900

All F a r m s
Estima­
ted
gross
income

T otal
Production
f a r m exof m i l k
pense plus p er cow
5 % o n In-,
(cwt.)
v es tments

Price of
m l & k per
cwt.

$5,000

$ 3 ,1 0 0

$5.oo

60

Difference in l abor income on this farm from exp e ct ed value considering Inputa:

$

-900

1* F r o m the average relationship (for all farms) between
average gross income and total f arm expense plus 5 % on
investment*
The ratio expressing this relationship Is
100/ 62. Because of the arrangement of factors on this
farm, the gross income Is e s timated at £5,000.
Accord­
ing to the average relationship bet we en gross income
and total farm expense plus 5 % on Investment, a farm
w ith $5, 0 0 0 gross income is estimated to have a total
cost (total farm expense plus S % o n the Investment) of
$3,100.
In other words, structurally this farm is
somewhat "better" than average.

105-

W l t h average milk p e r cow at 6,000 pounds and the average
price of m ilk at $5*00 p er hundredweight,

in general a

farm would need an average of 16 2/3 cows to e a rn the gross
income estimated for this farm*

It is observed that the

total farm expenses plus five percent on investment are
different in order to obtain equal estimates of gross i n ­
come for the average of all farms and for this farm.

The

particular farm is in a more favorable pos it io n on the gross
income function from the standpoint of m a x i mi zi ng labor in­
come than is the average of all farms.
There Is a difference of Ol>000 In gross income to
be accounted for.

The average rates of production,

prices,

and number of cows whi c h a farm would need In order to
earn the estimated gross income can be taken as the stand­
ard of comparison.

Then the difference in gross Income Is

given by:
1. Number of cows (other farms) times average price
times difference in production rate for this farm
plus
2. Average production times number of cows (other farms)
times difference in price for this farm plus
3. Average production times average price times
difference In number of cows for this farm plus
U* Three cross-products involving two differences and
one average value plus the cross-product Involving
only the three differences.^
Numerically this Is given as follows:

lT See Appendix 15#

- 106-

16 2/3 ( cows)x$£.00x(-10( cwt. ) p l u s
60(cwt.)xl 6 2/3( cows)x$ 1.00 plus
60( cwt• )x$ 5*OOx 3 1 / 3 (cows) p l us
(-1 0 )x(-# 1 .0 0 )xl 6 2/3 p lu s

(-10)x3 l/3x$5*00 p l u s

(-4U00)x

3 1/3x60 plus 3 l / 3 x ( - 1 0 )x(-#1.00)
w h i c h equals
1. -*833,

the rate of p r o d u c t i o n eff ec t p l u s

2. >41000 , th© price effect plus
3* $1000,

the size of b u s i n e s s e f f e c t plus

lj.. -§l67,

the

cross effects.

The sum Is —$1000.

T h u s the difference

in gross income

f r o m its e x p e c t e d value is a c c o u n t e d for by the price received,
the rate of production,

and the

size of business.

The p r o ­

cedure can be simplified and a f a i r a p p r o x i m a t i o n of the full
difference

can be o b t a i n e d by conside ri ng yields,

prices,

and numbers of units at the m i d p o i n t s between values given
f o r the f arm a n d the av erages for all farms.

In this case

there are the three p r i m ar y effe ct s l i s t e d above.
1
e ffects can be ignored.
The difference in labor

The c r oss—

income from its e x p e c t e d value

is a n a l yz e d a c c or di ng to the e f f ec t s of combina ti on of factors,
pr o d u c t i o n rate, price,
1. A p p e n d i x S

a n d size of business

(table

21).

Tabl® 21*

E x p l a n a t i o n of Difference of Labor Incoma from
Its E s t i m a t e d Value, According to Effects of
P r o d u c t i o n Rate, Price, Sice of Business, and
C o m b i n a t i o n of Factors, on a One-Pro duct Farm

Thi s f ar m

Item

Gross Inco me

$!p, 000

Expected,
b a s e d on
all farms
1
$ 5,000
2

Total f a r m e x p e n s e
plus 5,^ i n t e r e s t

3,000

3,100

Labor I nc o m e

1,000

1,900

Milk p e r cow

(cwt.)

Price o f m i l k p e r

A v e r a g e of
all farms

50

60

cwt.

♦5
20

Number o f cows
Difference of g r o s s
income f r o m e x p e c t e d ,
consisting of the
followings
Price e f f e c t (6 0 x
16 2/3x($5-$l4-) )
Rate of p r o d u c t i o n
effect (16 2 / 3 x $ 5
xlO )

$-1,000

-1,000

-833

Size of h e r d e f f e c t
( 60 x$ 5x (2 0 - 1 6 2 / 3 ) )
Cross-effe ct s

16 2/3

1,000
-167

Difference In l a b o r income f ro m e x p e c t e d value be cause
of d i f f e r e n c e I n g r o s s Income
$■*1,000
Difference in l a b o r
of c o m b i n a t i o n o f
Net difference

Income from ex p ec te d value because
fac to rs to e a r n an e s t i m a t e d gross

100

in l a b o r income from its exp ec te d value

—909

1. The $ 5 * 0 0 0 Is o b t a i n e d from e st imating the gross Income
for t his f a r m f r o m P a C x 6 y b .
2. The $ 3 , 1 0 0 Is g i v e n by the average relationship f o r all
farms b e t w e e n g r o s s Income and total farm expense plus
5% I n t e r e s t o n t h e investment.

Part Hr* E x t e n sion of A n a l y s i s of Differences In
Gross and Labor Incoma •to Two or Mora Enterprises
I n p r a c t i c a l w o r k It w o u l d not ba f easible to Include
all o f the en t e r p r i s e s on the average farm.

However,

b u l k of di ff er en c es in gross Income and l a b o r income

the
can

be a c c o u n t e d for by considering the m a j o r enterprises*
The analysis consists of two pr i n c i p a l operations,

w i t h two

parts each:
1* D i f f e re nc es from e x p e c t e d expe ns es are calculated.
a* G r os s income
tors,
b* The

is e s t i m a t e d f r o m the e m p l o ym e nt of f a c ­

u sing a gross income e q u a t i o n d e ri ve d f r o m all farms,
total of cash outlays plus Implicit charges w h i c h

w o u l d be

Incurred on an average

g r o s s are determined*
(table 21). Thus,

to o b t a i n the e s t i m a t e d

T his is the next to the last Item

al t h g e t h e r the first

step shows how

m u c h labor income is g ai n e d or lost by the m a n n e r in
w h i c h factors are
2* Di f f e r e n c e s

combined*

from e x p e c t e d gross are

calculated*

a. U s i n g the same en t e r p r i s e s In the same proportions,
and u s i ng average yields and selling prices,

the acres

of crops and the numbers of li v e s t o c k n e e d e d to o b t a i n
the e s t i ma te d gross are

computed.

b. The differences b e t w ee n e x p e c t e d and r ecorded gross
are divided into p r o d u c t i o n rate,

price,

and size of

b usiness effects.
T a bl es 2C^riand 21 were for a h yp ot he ti ca l farm, p r o d u c ­
ing only whole milk*

Tables 22-25 give the complete a n a ­

lysis for an actual farm w i t h several enterprises*

Table 22* Estimates o r Gross Income and Labor Income from
the Combination of Factors on a Michigan
Dairy Farm1
G r os s income
T otal f a r m e x p en se
Less: o p e r a t o r ' s
labor
I n t e r e s t o n invest*

01*6,31*1

$ 23*960
$15,058
1,1*30

013,628

© 5^

2*317

T o t a l charges l es s
operator's labor
L a b o r income
E s t i m a t e _ o f gross
income^
E x p e c t e d total charges**
$17,1*20 -f- l.ll*l5
Less operator's
labor
E x p e c t e d total charges
less operator's
labor
E x p e c t e d l a b o r Income
f r o m *17.1*20 of
g r os s at ave ra ge r e ­
l a t i o n ^ of charges
to gross?

15, 9U'
r§7oT!

17, 1*20

15 ,262

13,739

3,681

L a b o r income from
$17,1*20 of g r o s s
w i t h c^Brges o t h er
than o p e r a t o r as
of this f a r m

1,1*75

L a b o r income

8,015

Part of l a bo r income to be e x p l a i n e d
b y yields, prices, size of business,
and choice of e n t e r p r i s e s

6,51*0

1* A f a r m in C a l h o u n County*
In d e al in g w i t h c o m p l ex e n —
terprises, it is n e c e s s a r y to include one more fac to r
w h i c h affects gross besides yield, price, and size of
b u s i n e s s — the s e l e c t i o n of en terprises*
It Is Imp or ta nt
that, w h e n this type of a n a l ys is is used, the farms in
the study be comparable r e g a r d i n g k i n d s of p r o d u c t s sold
T h e n the f or c e s a f f e c t i n g gross income are p r a c t i c a l l y
the same as f o r a o n e - p r o d u c t situation*
2* E s t i m a t e d f r o m e q u a t i o n 1, table 6, f o r 8 6 dairy farms.
3* The average r e l a t io ns hi p b e t w e e n gross Income and total
charges fer the 86 dair y farms is l«ll*l5 to 1*0000«

xxu—

Table 23.

An alysis of Yields, Prices, and Units of M a ­
jor Enterp ri se s on a M i c hi g an D airy F a r m E a r n ­
ing a Higher Gross Income than E s t i m a t e d
R ec or d e d values for
this farm

E x p e c t e d values
for all farms
------------------j

Enterprise
Dairy cattle
Cattle in­
come
M ilk sales
Poultry-*
W heat
Corn
Oats

Y i e l d Price

Units
ne e de d
to earn
e stlmated
gross

($139.00)
05.1 $l+*52
13*0
0.1+0
28
1.95
1+2
1.57
1+3
0.81+

28.9
28.9
191.0
30.6*
7.6*
11+.7*

Yield

Price

Units

($150 .80) ,
106.7 $3.51+
12.2
0.40
1.92
1.60
1+1+.1+
1+0.0
0.91

31+.0
31+.0
225.0
3£.o
9.0
17.3

*Acres sold
Gross Income by enterprises
Enterprise
Dairy, cattle
Cattle Income
Milk sales
Poultry
7/he at
Corn
Oats
Total

(yield x price x units):

E xp ec te d gross
income

ol+o
8,1+90
1,000

1,670

5oo

530
$15,230

Recorded gross
income
$ 5,126

12,858
1,099
1,965
61+2

630
$^2,320

Dlfference

/ -'$1, 086
/ U,368
/
99
/
295
/
142
/
100
/ $6,6 9 0

1. Units refer to m i l k cows for dalry^ acres for crops,
e t c . E xpected values of units are calculated by m u l t i ­
plying the units for this farm by 0.81+9* This es tablish­
es expected units so that w hen multiplied by average
prices and average yields, the gross income is equal to
the gross estimated for this farm.
2. Cattle income consisted of sales of animals of different
types from the dairy herd.
It is Impracticable to attempt
to separate yield and price effects.
3* Average yields and prices of these crops are estimated.

How this farm earns a ^ross income of $22,230 from
its five major enterprises is compared with how an "aver­
age" farm would earn the estlmated gross ($1 6 , 2 2 8 ) from
five identical enterprises (table 23)•

The "average" farm

would receive average yields and average prices for products
sold.

It would have in production of output for sale either
1
more or less units than this farm in each enterprise*
The
five major enterprises account for $ 6 , 0 9 0 of the $ 6 , 5 U 0 of
gross Income to be explained by yields,
business,
ners,

and choice of enterprises

prices,

size of

(lower right-hand cor­

tables 22 and 23). Thus, $7,565 of a labor income of

$ 8 , 0 1 5 are accounted for by the combination of factors and
the yields, price,

and volume of the five m a jor enterprises.

The effects upon gross income by the five enterprises
are separated according to y i e l d , p r i c e , and size of busi­
n e s s . (taole 21+).

It was impracticable with the data to

attempt to show a separation between price and yield
effects for dairy cattle income. Dairy cattle income con­
sists of sales of calves, heifers,

bulls,

and cows from

1* The expected number of units producing for sale is
computed as follows: Multiply each unlt-figure for
this farm by a common factor.
Compute the factor in
this way:
(x) times cows this farm times average sales
per cow times average price of milk, plus (x) times
cows this farm times average cattle income per cow, plus
(x) times hens this farm times average eggs per hen
times average price of e g g s , ••••*••equals estimated
gross income.
Thus a factor of size is derived which
will fix the expected numbers of units In such a way
that the sum of average yields times average price
times expected units will equal the estimate of gross.

112

Table 2l** An al ys is or Difference of Gross Income from
Its E s t i m a t e d Value A c c o r d i n g to Yield, Price, and
Size of Business E f f e c t s on a M i c h i g a n D a ir y F a r m , 1950
Effects accounting for difference in gros
Enterprise
Dairy
Cattle

Yield

Price

Size o f 2
business
t

. . .*£3U5 • • • • •

Total

#71*1

$ 1 ,0 8 6

$5,260

$-2,61*0

1,71*8

1*,368

— 70

0

169

99

Wheat

30

-280

5U5

295

Corn

30

10

102

11*2

Oats

14-0

50

10

100

Total, not Including
cattle
05, 290

$ - 2,860

Milk
Poultry

(eggs)

Uns ep a ra te d between
y i e l d and price
Cattle
&3U5__________________________________
Total___________________ $2,_775___________ $ 3 ,3 1 5_________ $6,090
1. The effects are calculated as follows:
Y i e l d effect = difference in y i e l d from average times
price f o r this farm plus average price,
the sum divided by two, times units for
this farm plus expected units, the aim
divided by two.
(See table 23 f o r data)*
F o r the y i e ld effect of milk, thus;
Y i e l d effect * (106*7 - 6 5 * 1 )

£ .34? ?

a $5* 260
2* The size of business and choice of enterprise effects are
together called size of business In this case*
The com­
parison is with a group o f Bo ctairy farms*
This farm is
typical*
Thus it is impossible for gross income to be a f ­
fected by unusual prices in any line*
Should the group
of farms be heterogeneous the meaning of the size of busln—
ess column should be expanded to include selection of enterprise effects.
The column can be computed as in (1), p r e ­
ceding or as a residual*

-113-

the herd*

There is no common denominator of output*

I.Tost of the price and y i el d effects are accounted for by
the dairy enterprise, whi ch is dominant.
milk is more

The y i e l d of

than adequate to offset the adverse price,

$5# 260 compared with $-2,614.0*

A lt og et he r prices and

yields account for $ 2 , 7 7 5 of the difference

in gross*

The size of business accounts for $3#315*
The p r e c ed i ng analysis will contain information of
interest aside from the estimate of the causes of varia*
tion in gross income from expected values*

Average yields

of crops and production rates of livestock for all farms
and for the particular farm will be compared directly.
Average

selling prices will also appear alongside prices

received by this farm.

As in other analyses calling for

the comparison of a particular farm with some standard,
there i3 no compelling reason here why the comparison
should necessarily be with the mean of all farms.

The

comparison could be with the hi g h third labor income
farms,

in which case the majo ri ty of the farm account

cooperators would have an opportunity for direct compar­
isons of their farms with the more successful units,
enterprise by enterprise*
The analysis does not depend upon a significant re­
lationship between expected labor income and recorded
labor income.

E v e n if the correlation in this matter is

not significant,

the analyst can account for the differ­

ences in expected from recorded gross income and labor
income according to the factors wh ich in this case must

must account for the differences in labor income*
factors are yields,
of enterprises*

prices,

size of business,

These

and choice

The computations involve no mathematics

other than arithmetic once the estimate of jross income
is obtained*
The V al uation of F a r m Businesses
It will turn out in some cases that it will not pay
a farmer to increase the yields of crops or perhaps the
product io n rates of livestock in order to eliminate ad­
verse yield effects shown by the preceding analysis*
Possibly because of the location of a farm or because of
the effect of certain types of soil on the quality of
crops grown,

it m a y not pay to seek to eliminate unfavor-

aole price effects.

In the case of plans designed for

the stabilization of the price and quantity of milk,

it

m a y not p ay a farmer to reduce the production of m i l k in
the surplus season even though he could raise his average
price by doing so.
Thus the price effects, yield effects,

and possibly

the size of business effects may all be adverse,
it may not pay to do anytning about them*

and yet

The conclusion

in this case must be that the farm and the factors e m ­
ployed upon it are overvalued in relation to other farms*
It Is difficult to find any way by which the Influence of
the farm operator can be separated from the effects of the
farm Itself and its appurtenances*

In the case of the

farm which consistently returns more than It should
according to the gross Income estimating function,
or all o f the primary factors are undervalued*
tion in the valuation In the first Instance,

some

A re duc­

and an In­

crease in the valuation in the second case will tend to
cause the estimates of gross income to agree more nearly
with a general gross income estimating equation.

No

specific suggestions ere made here regarding how the im­
plications of persistent non-conforaity to gross income
estimating equations should be considered in connection
with stated values of factor categories*
5# Subjective Rates of Charge for Categories of Factors
The estimates of marginal returns to categories of
factors could be used to make revisions of rates of charge
against the farm business*
estimating equations were

In this study, gross income
calculated for 19U farms*

sample was broken into two parts,
dairy farms,
of farms.

The

dairy farms and not—

two independent samples from the universe

The

statistical instability of the coefficients

of elasticity of

toss

income w it h respect to factor

categories Is shown in tables 8-10 of C a r t e r IV.

If

similar analyses should be conducted by type-of-farming
areas,

the coefficients of elasticity and the marginal

value productivities of categories of factors will cluster
around central values*, Then tables of marginal returns to cate

“1X0*

gories or factors for the state as a whole could be pre­
pared with confidence that the marginal rates would be
nearly what farmers received for the use of the factors.
Thus the marginal return to investment in land could
be shown to be, say, five per cent; the marginal return
to investment in productive livestock could be another
figure, and so on.

These marginal retes of return should

be indicative of the charges against the factors which
fanners actually had in mind in the organization of farm
businesses.

If the factor categories were charged accord­

ing to marginal rates of return, the effect should be
to approximate more nearly a type of net income which
farmers in practice seek to maximize.
It is recognized that various dlsequillbrla can account
in part for marginal rate of return to categories of fac­
tors.

For Instance, the proportion of investment In

macninery and equipment on Michigan farms has been in­
creasing for some time.

The results of tnls study indicate

that the marginal returns for machinery and equipment con­
tinue to be relatively high.

The increase in investment

in this category can thus be expected to continue.
Nevertheless farmers have not cid up the prices of avail­
able machinery and equipment to a point where the marginal
returns would be similar to those for land and improvements,
for example.

In general, the subjective rate of interest

return demanded for machinery and equipment remains high.

and farmers appear to maximize a labor income which calls
for a differential between the interest rate on machinery
and equipment and the interest return on land investment*
It is possible that the definition of labor income may
come a little closer to conformity with, the behavior of

the entrepreneurs in the field if the rates of charge
are adjusted to be more in the line with marginal value

productivities.

Theoretically, of course, all entre­

preneurs should borrow money and invest money in the
factors of production up to where the marginal rate of
return is equal to the rate of interest.

However, if

farmers do not actually follow such a procedure with
respect to all categories of factors, it is likely that
there are valid reasons for their not doing so, and
a more accurate report of earnings above charges is
possible if charges are made according to practice.
A summary of this chapter has already been made in
the Introduction, page vii.

118

APPENDIX A
COEFFICIENTS OF ELASTICITY OF GROSS INCOME WITH RESPECT
TO CATEGORIES OF FACTORS
Proof that .a, b, ••• k are the elasticities of P
with respect to x, y, ♦..«z. respectively, in
o d
k
P. ■
C3C V
.... Z
For an Increment of x the change In P Is

The elasticity of P with respect to x,

a-1 b
P

a
cx y

, la

119

A pp en di x B

Illustration of R s turns to Seals
If there

is an equal proportional change In fill of

the factors t h e n the proportional

change I n product w i ll

be g r e at er than, equal to, or ]e a s t han the change I n the
factors.

L e t the e q u a t i o n for p r o d u c t be £

iAj 3A *

s

The s u m of the elasticities of p r o d u c t w i t h respect to
the fac to rs Is 1,
x and x

R e t u r n s to acaLle are constant*

If

SLZ*e assumed to be 16 eacti, a n d are then raised

to 61 e a c h It will be found that
f r o m 6I4. to 32i|.,

product will Increase

The ratios of clxange are equal*

If the e q u a t i o n Is given by ^
x a nd y are equal to 16, product

iAx 1/2

* I4
-X

wi l l be 32.

a n d b oth
If they

are b o t h r a i s e d to 81 the new p r o d u c t w ill be 10 8*
The ratio of change of product I s Is ss t ha n the ratio
of change of factors*

I f the ex p o n e n t s of

s ad x

are

more than 1 In their sum the Increase In product Is r e l —
a t l v el y l a r g e r then

the Increase

In factors*

-120Appendlx C
The R e l a t i o n of the M e a n Value P r o d u c t iv it y
F u n c t i o n to the E s t i m a t e of Gross Income
b y Least Squares

F o l l o w i n g is a study of the behavior of the sta­
tistical "value productivity" function,

ass u m i n g that

the actual value productivity functions are known*

F or

this purpose three arb i tr ar y value product iv it y functloni
are cho sen :
F a r m 1.

F a r m 2*

F a r m 3.

£

2

. !/3 1 / 3
— c_
= hx_
_y
2
2
2

P

-c
3

=
3

ox
1

y
_1

1/2

1
£

»

lA

- c^
1

1/2 ylA

6x
3

3

«» (x
1
- (x

/ y
1

)

j* y )
2
"“2

— (x / y )
"3
3

Ret ur n s to scale are given as 3aess than 1 in order
that o p t i m u m inputs can be determined without specifying
a risk function.

The three farms are operated for the

purpose of earning as large a net Income as possible. To
determine w hat quantities of _x and _y will be empl oy ed
the partial derivatives of £ w i t h respect to x a nd £
are set to zero.

The supply curves of factors and the

demand curve of product are ass um ed to be Infinitely
elastic.

A s these are assumpt io n s of perfect competi­

tion they are not unreasonable for m o s t farm operations.

The n ext p r o b l e m la to solve Tor the plane w h i ch
will result If the coordinates expressing the optimum
positions for the three farms are to be joined, as is
inherent to the idea of the statistical value productivity
function.

This plane turns out to be,

in logs:

log J* - 0.1+86 / 0.1+88 log x. / 0.1+88 log jr
The coefficients of

and i

are ©qual because of the

values of the exponents chosen in the three value p ro ductiv­
ity functions.
There are n ow four planes altogether (figure 7)«

All

of the individual value productivity planes cut through
the statistical plane from above.

Furthermore,

it follows

from the construction of the latter that the optimum p o s i ­
tion of each firm must lie u pon the line where the individ­
ual plane cuts below the plane common to the three b usin­
esses.

This will be true if the average value productiv­

ity is greater than the marginal value productivity.

The

average will be greater than the marginal if the exponent
of the factor in the function is Ifi as than 1.

If average

value productivity equals marginal value productivity in
the function then value productivity in total for the case
of constant returns to scale can equal the value of only
one factor.

This is a useless case for analysis.

The statistical plane is fixed by the condition that
the derivatives of net income w i th respect to the factors
are zero.

As long as the plane for the firm lies above

the statistical plane these derivatives are larger than

122.

called Tor by the conditions of equilibrium.

A s soon as

the plane for the firm drops below the general plane the
marginal value productivities of the factors become less
than their costs.
It is worthwhile to express the gross income equations
for the three farms in their natural numbers and then deter­
mine marginal value productivities of factors according to
(1) the value productivity plane on w h i c h each f a r m Is as­
sumed to operate and,

(2) the statistical plane. This a-

mounts to

differentiating two functions using

values of

the factors according to one of the

the optimum
functions.

The results are given below;
Farm

Factor

1
2
3

Marginal value
productivity according
to the "true"
value
productivity f u nction
of the
firm

Marginal value
productivity
according to the
statistical
function

x
7
x
y
x

1.00
1*00
1*00
1.00
1.00

1.9U
0.97
1.1+6
1.1+6
0.97

y

1.00

1.91+

The errors In the estimation of marginal value produc tivity
are biased upward.

This Is to be expected as the slope

of the common plane is greater than the slopes of a n y of
the Individual planes (figure 7)
The above difficulties do not apply If one Is Interested
only In an estimate of gross income from the employment of
factors.

The trouble

starts w h e n the function describing

the positions of the firms is interpreted to be a n average

Figure 7»

The Relation of Theoretical Individual Production
Function* to the Cobb-Douglas Function

P (c-d) * °*1+86 r 0,l|88x /
P

a 0.788 / 0.25x / 0.50y

P3 a 0.778 / O.SOOx / 0.2

f 0 = 0.602 ^ 0.333x / 0.333y

P: log of gross income, for Cobb-Douglas function, farms 1» 2
and 3.
x and jr: logs of the two factors of production.

of the value p r o d u c t i v i t y p l a n e s r a c i n g all or the rirms.
At worst the C o b b —D o u g l a s r u n e t I o n can at lea s t provide
a measure or estimate or g r o s s Income considering the e m ­
ployment or factors.
In agriculture there are m a n y Tirms In the Industry
a n d the rirms are confronted w i t h substantially similar
t y p e - o f - r a r m i n g area.

conditions
therraore, who is to

Pur—

say w hat are the value p r o d u c t i v i ty

runctions ror the dirrerent r a r m e r s t

This would Involve

a n appraisal or the w o r t h or the m a n a g e m e n t contri b u t ion
or the individual rarmer.

A procedure ror such an e v alu­

a t i o n has not been developed*

P e r h a p s the best that can

be done in the m a t t e r is to use a runetion

such as the

C o b b —Douglas and qialiry its use w i t h the statement that
it does show the r e s u l t s o b t a i n e d v h e n dirrerent m e a s u r ­
able factors are u s e d in d i r rerent combinations.

T hat not

all or the rarmers w ill be ab^le to take rull advantage or
the runctlon Is recognized.
In the m o s t ravorable

light the r u nction can be consid­

ered as one w h i c h corrects erroneous Impressions or the
nature or response of gross Income to employment or f a c ­
tors.

This is shown in rigure 8.

The three points A,

B,

and 2. represent the p o s itions u h l c h rarmers believe that
they should reach in o r d e r to m a ximize net incomes.

The

points a, b, and jc represent the gross Incomes according to
relationships bet w e e n ractors and product tested by e x p e r ­
ience.

Figure 8.

The Cobb-Douglaa Production Function as a Measure
of the Relation of Product to Inputs

P^ s 0.l|.86 / 0,ij.88x

2.5r

4 £«lj$8y

0.36ly

2.0

P

P:

log of gross income.

x and y: logs of
factors

P^: for plane formed by farmers* estimates of
optimum positions.
P/ d )r ?or
formed by the Cobb-Douglas
function derived from gross income a*

- 126 -

It cannot be c l a i m e d f o r the C o b b - D o u g l a s f u n c t i o n
that t h e o r e t i c a l l y a n d in eqi i l i b r i u m that it gives an
average of the value p r o d u c t i v i t y f u n ctions f a c i n g the
individual firms.

However,

farms are n o t in equilibrium.

The f a r m o p e r a t o r w i l l be I n t e r e s t e d in the o u t c o m e s of
c ombinations o f f a c t o r s on o t h e r farms.

-127-

Appendix D
The V a l u a t i o n of L a n d
It w a s a x p e c t e d that s t a t e d l a n d v a l u e s i n 1 9 5 0 w o u l d
be m o r e or l e s s I n l i n e w i t h the l e v e l o f l a n d p r i c e s p r e ­
v a i l i n g w h e n r e c o r d s were
po the sis,

started.

A c c o r d i n g to this h y ­

a c c o u n t c o o p e r a t o r s w h o b e g a n r e c o r d s i n 1 93 5

a n d m a i n t a i n e d t h e m c o n t i n u o u s l y s h o u l d t e n d to give l o w e r
stated v a l u e s o f l a n d p e r acre
o r d s in,

say,

t h a n f a r m e r s who b e g a n r e c ­

19i4-8.

The v a l u e s of f a r m l a n d i n the f a r m a c c o u n t b o oks are
n o t c h a n g e d to con f o r m w i t h c h a n g i n g l a n d prices.

This pro­

cedure a v o i d s the i n c l u s i o n of changes I n l a n d v a l u e s in
m e a s u r e s of net Income.
One h u n d r e d s e v e n t y — six a c c o u n t s w e r e u s e d I n a study
of the effe c t o f the y e a r o f s t a r t i n g r e c o r d s u p o n s t a t e d
v a l u e s of l a n d p e r acre.

R e g a r d l e s s of w h e n started,

these

r e c o r d s h a d to be c a r r i e d s t raight t h r o u g h to a r e c e n t y e a r
(191+9) in o r d e r that c o n c l u s i o n s c o u l d be drawn.

Table

25 g i ves the n u m b e r of a c c o u n t s b e g i n n i n g in s c a t t e r e d y e ars
since 1929*

the o r i g i n a l t a l u e s o f i m p r o v e d 1 and a n d total

l a n d p er acre,
acre.

The

a n d the v a l u e s o f l a n d and i m p r o v e m e n t s p e r

s t a n d a r d d e v i a t i o n s of the m e a n s o f samples

were

e s t i m a t e d by a v e r a g e range in s u b s a m p l e s of two.^
There was no definite
1. See 2. L. Grant,

t e n d e n c y f o r fan n e r s b e g i n n i n g

S t a t i s t i c a l Q u a l i t y C o n t r o l , pp.

103/-112.

-128

Table

25* A v e r a g e V a l u e s o f Land, a n d L a n d a n d I m p r o v e ­
m e n t s p e r A c r e : F a r m A c c o u n t s B e g i n n i n g in
S e l e c t e d Years, 1 9 2 9 — 191+9

Y e a r of
Number
starting
of
recor ds
farms

Improved land Total land
L a n d a nd 1m—
p e r acre
p e r acre
^ provements per
M e a n value
2 d ^ M e a n value 2 6L
acre
^
x M e a n value 2 o £
$60
50

$10.3
9.9

16

k*

14-3

9.6
9.1

1+2
37

9.5
6.9

73
62

1939
•

16

38

1+.2

36

2.9

53

6 .I4.

191+2
*

16

37

1+.3

3l+

3.6

62

9.1

1914,.
•

16

1+2

37

10.2

60

191+7
•

17

1+1+

6.5

39

7.0

67

10.U

191+9

17

l+o

8.5

31+

6.6

65.

17.1

9

193U
1935
•

1* Two

standard deviat i o n s

•
o
H

#7.3
10. h

•
H
CM

$3.0.2

$62
58

1929
1930
•

(e stimated)

a c c ount s in recent y e ars

16.8
19.3
12.6

of the mean.

to give h i g h e r initial figures

p er acre than far m e r s b e g i n n i n g a c c o u n t s in the
The ranges of two

$ 9 2 .7
75

e a r l i e r years*

standard deviat i o n s o f the m e a n indicate

that the differences bet w e e n years are not significant.
The h y p o t h e s i s is set up that farmers b e g i n n i n g records
since

the second VTorld W a r

can be d i v i d e d Into

two groups:

those kho value land at m a r k e t p r i c e s and those who value
It a c c o r d i n g to some e a r l i e r scale*

In this case there

should be a tendency toward gre a t e r dispersion of values

-129-

p e r acr« In l a t « r years.

Th« f a r m r eal esta t t v a l u e s

o b t a i n e d in tills s t udy were
w i t b 1 929 t a k e n as llj-O.

c o n v e r t e d to a n Index*

L a n d v a l u e s r e p o r t e d by b e g i n ­

n in g a c c o u n t c o o p e r a t o r s did n o t rise c o m p a r a b l y with.
1
the M i c h i g a n f a r m real e s t a t e I n dex (figure 9).
There
w as not a si g n i f i c a n t t e n d e n c y f o r l a n d v a l u e s to v a r y
more a b o u t the a v w r a g e
2
y e a r s (figure 10).
The p r i m a r y

in the b o o k s

s t a r t e d In r e c e n t

c o n c e r n is w i t h the In f l u e n c e of y e a r

of s t a r t i n g r e c o r d s o n real est a t e v a l u e s as o f J a n u a r y
1* 1950.

N e i t h e r the ave r a g e value of l a n d a n d I m p r o v e ­

m e n t s p e r acre n o r the ave r a g e value of l a n d p e r acre
w a s a p p r e c i a b l y a f f e c t e d by the y e a r of s t a r t i n g records.
T ha t Is*

the f i g u r e s f o r 1950 tend to be

l ess o f w h e n r e c o r d s were

1. See table 26* p. 132
2. T a b l e 27, P- 133.

started

comparable r e g a r d *

(figures 11 a n d 12).

r l^Ui'O 7®

juat/ou

v al u es

vi

i uiui uunu

**
**
*
* *****»**..*q~ —
— **»0---------------o

Compared with the Michigan Farm Real Estate Index
Selected years, 1929-191*9

250

Michigan farm
real estate index

«£t

200

Index

Upper

%

confidence li»*

Index based on values
by beginning account
cooperators
Lower
limit
X

0 1—

1925

1930

1935

x

4

%

confidence

X

1950
Year cooperator atarted recorda

1955

Figure 10.

200f

The Relationship b0tween Stated Value of Land and Improvements
*
^ per Acre and the Year of Starting Farm Accounts
^
^
,
Selected years, 1929-1 9k9^
/
'

*

*

\

/ \
/ \

'*

125

/

H loo

150"

V alue
p er

acre

Index
based
on

75

100
50

50
25

t

0
#25

1925

1930

1935.

19l|0

19^5

Year of starting recorda
1* Values as of year recorda were started*

I

1950

o

xex

I

recordb

I

-132-

Table

26.

The M i c h i g a n F a r m Real Est a t e I n d e x a n d an
I n d e x o f V a l u e s of L a n d a n d Improv e m e n t s per
A c r e of B e g i n n i n g F a r m A c c o u n t

Year

Michigan
in dex
(1935— 39 * 10O)

1929
1930
19311932
1933
193 4
1935
1936
1937
1933
1939
1940
19Ul
1942
a

s

1945
19S 6
1947
1946
1949
1950
1951

140
137
130
109
91
93
94
95

104
10L
104
103
106
119
130
152
164

190
219
224
229
225
256

I n d e x of v a l u e s
of l a n d a n d
im—
pr o v e m e n t s of b e —
ginning f a r m
account cooperators

Cooperators

confidence
intervals of
farm account
index

5%

140

125 - 1 55

113
-----

—

110
94
—
—
—

GO
-------

93
-----

91
— —
— ——

88 - 139
— - — —
--- — ---

81 - 139
75 - 113
----

-

-------

—

-------

70 -------

—
-

—

90
-—
— .

79 - 107
----

—

—

69 - 123.
---— —

—

———

100

85

-

115

100

74

_

126

—

—

..--— —

— —

___
—

-133-

T able 27* The R e l a t i o n B e t w e e n the Y e a r or S t a r t i n g F a r m
A c c o u n t R e c o r d a a n d the D i s p e r s i o n o t R e p o r t e d
L a n d Values,

Michigan Farm Account Farms
1 9 2 9-19U9
Land and Improvements

Year

Number
of farms

1929
1930

V 1

a
*

193141935
•
•

14
•
•
•
9

16
•
•

Ave r a g e
value
p e r acre

Index,
1929 = ll^O

93
75
•
•

1I4.O

73
62
•
a

a-

S t a n d a r d deviation
T e rms
Terms “
dollars
of index

b7

110
9U
•
a

I4.0
a

*
38
a
a

*80

<

18

1939
•

16
•

53
•

19U2

16

62

19W4a

16

60

•
a
17
a
17

a

*93
a
91
•

67

100

31

66

100

52

19>+7
19U9

27

,
19
1*2
a

113
•
•

a

a

a

29

28

U9
a
a

60
a
+
31
a

52

Figure 11.

The Relationship between the Stated Value of Land and Improvements
per Acre as of 19b9t and the Year Farm Accounts Were Started

$200

Selected jreare, 1929-19l|9

150Value
of land
and im­
prove­
ments
per
acre

• •

10Q
(X9l»9)
Average
So

/\
/
\
/
\

v
\
19*5

1930

1935

191*0

19l*5

1950

Year started records
1. Two standard deviations. 2. Two standard deviationa of the mean*

Figure 12*

The Relationship between the 194.9-^ated value 01
Land per Acre, and the Year of Starting Farm Accounts
Selected years, 1929-19^-9

nso

100

Value
per
Acre

/
^

50 ’

k *

f\*

.,

,| -

oh— """"

'■

1

1925

t

1930

•" ■—

—

—

_

«

—

+ -

1* Two standard deviations*

f

— ....... - " . A j y * " 1

- . . .

1935

J9

.

t - | -

j

L

'

.

.

---------------------

i

19^0
19U5
Year of starting recorda

2

w ,

— -.a
-

\

1950

2* Two standard deviations of the mean.

-136-

Appendix E
A Test of Differences

in R e c o r d e d Values of

L a n d a n d Improvements

The pr i ncipal conclusion of a p p e n d i x D,

that the yea r

of starting f a r m account records h a d little o r no effect
u p o n r e c o r d e d values per acre of l a n d a n d improvements,
was test e d independently.

The hypoth e s i s was

set up that

w i t h i n a partic u l a r county the 19l4-9- g i v e n values w o u l d
not be a f f e c t e d by whe t h e r records began in the p e r i o d
1932 to 1 9l+2 o r in the p e r i o d 19U3 to 1 9 h 9 »

I n o r der to

minimize

the number of observ a t i o n s a sequential probability
1
ratio test was used*
The counties were chosen at random.

A pair of observations was chosen at rand o m for e a c h county.
Ond of the observations b e g a n records in the earlier period
and one began in the later period.
The

sub-hypothesis that 0.5 of the records in the

m a t c h e d pairs b e g inning in the 1932-ij.2 p e r i o d should show
h i g h e r values per acre than that 0.7 should show higher
values was a ccepted w i t h 20 obeervaticns.
of b e ing correct is 9/l0ths.

The probability

This initial hypothesis con­

cerned values of land per acre,

not l and and buildings.

Twenty-four observations were n e e d e d to e s t ablish w i th 9 / l 0ths
probability of being right a similar hypoth e s i s w i t h respect
to l and and improvements*
1.

See Paul G. Hoel,
pp. 12U, 125.

I n t r o d u c t i o n to Mathematical Statistics.

-137

A

second, sub-hypothesis was set up*

T his was that

0.5 was a bett e r estimate t han 0.3 of the p r o p o r t i o n of
comparisons of values w i t h i n the same county In w h i c h
the farms b e g i n n i n g r e c o r d s In the earlier p e r i o d would
show v a l u e s p e r acre w h i c h were h i g h e r than values for
the farms b e g inning In the later period.
was a c c epte d w i t h 12 observations,
alone,

This hypothesis

w i t h respect to land

a n d l and a n d b u i ldings together.

The results of the

sequential p r o b a b i l i t y ratio test

show that as far as the p e r i o d of

starting records Is con

corned, w h e t h e r p e r acre val u e s are h i g h e r or low e r Is es
sentially similar to coin-tossing.

136
Appendix P
The Valuation or Dairy Cows

I n a ran d o m sample of* 32 farms in areas 5 and 6 there

Is

some r e l a t i o n b e t w e e n the y e a r or s tarting records a n d the
average value of dairy cows p e r h e a d (figure 13)*

The

sample

shows little o r no r e l a t i o n s h i p b e t w e e n average value per
h e a d a n d dairy sales p e r cow (figure llj.) •

A v e r a g e value p er

h e a d is d e t e r m i n e d o n the basis of the b e g i n n i n g inventory*
D a i r y sales p e r cow is d e t e rmined on the basis or "cow units",
a rigure o b t a i n e d w h e n the rarm acc o u n t books are checked in*
One source or e r r o r in u s i n g sales as a m e a s u r e or p r o ­
ductivity or cows is that n o t all f a r m e r s have equally good
markets*

Therefore,

for 26 rarms the m i c r o f i l m e d records at

M i c h i g a n State College were u s e d to convert p o u n d s or m i l k
sold o v e r into p o u n d s or r a t — c o r rected milk*

The

simple

correlation coefficient b e t w e e n average value per cow and
p o u n d s or f a t - c o r r e c t e d m i l k

sold p e r cow is 0*22*

This

coefricient was t e s t e d a c c o r d i n g to its theoretical dls—
1
t r i b u t i o n should a series of similar trials be made*
If
It is a s s u m e d that the actual co r r e l a t i o n coefficient is
0*00,

It turns out that the

trials is e s t i m a t e d at 0*21*

standard de viation of similar
Thus an absolute value of the

correl a ti on coefficient equal to or greater t ha n the 0*22
1*

See Paul G* Hoel,
tics, pp. 88— 90*

I n t r o d u c t i o n to M a t h e m a t i c a l S t a t i s ­

-139

F i g u r e 13* The R e l a t i o n b e t w e e n I n v e n t o r y V a l u e or D a i r y
C o w a a n d the Y e a r of S t a r t i n g F a r m A c c o u n t s
32 R a n d o m Farms, A r e a s 5 a n d 6
1950

$30or

#

2.00
•

t

Av.
Valu >
p er
H ead

I

.

100
r

0*
1930

1935

xy

= 0.32

19U0
1914-5
1950
Y e a r of Sta r t i n g F a r m A c c o u n t s

- 114.0-

Figure 1I4..
♦I4
.OO

D a i r y Sales p e r Cow and Ave r a g e Value of Cows
per H e a d at B e g i n n i n g I n v entory
32 R a n d o m Farms,
A r e a s 5 and 6, 1950

•

•

300
Da iry
sale s
per
cow

••

200

100
r

*y

S 0.01

200
Average value p e r h e a d

300

-11*1

o b t a i n e d In th.ls trial c o uld be e x p e c t e d to o c cur about
one time

In three b y chance*

The results of the test or

the r e l a t i o n s h i p b e t w e e n T a t - c o r r e c t e d m i l k s old p e r cow
a n d the average value of cows In the h e r d are shown In
figure 15*
The h y p o t h e s i s w a s
time o f

set up that If the Influence of the

starting r e c o r d s u p o n the value of cows p e r h e a d

should be e l i m i n a t e d * t h a t the r e l a t i o n b e t w e e n f a t - c o r r e c t e d
m i l k p e r cow a n d average value p e r h e a d should be s t atistic­
ally

significant*

The par t i a l

c o r r e l a t i o n coefficient im—
1
p i l e d by the above h y p o t h e s i s Is 0*29*
Thus the degree of
relationshi p b e t w e e n p r o d u c t i v i t y a n d r e p o r t e d value Is In­
c r e ased somewhat*

However,

the significance test still

shows that a value of r x y • z as large as 0*29 could be exp e c t e d about one time In seven by chance*

1* Let y be the value of dairy cows p e r head;
p o u n d s of 3*I?/6 f a t - c o r r e c t e d m i l k sold p e r
be the y e a r of starting record**
Then r
* 0.32, r _ z = -0*l6, a n d r
= 0.29.
^
Hoel, op. cit., pp. 110-110*

let x be the
cow; let _z
- 0*22, r
S e e Paul G.

-lij.2

Fig ure l£«

Tine R e l a t i o n b e t w e e n A v e r a g e V a l u e of D a i r y Cows
p e r Head, B e g i n n i n g Inventory, a n d Pou n d s of
3»5/S F a t — C o r r e c t e d M i l k
Sold per Cow
26 R a n d o m Farms, T y p e - o f - F a r m i n g A r e a s
5 a n d 6, 1950

12|

10

*

8
Thou­
sand
Poundfc
p er
Cow
r * 0.22

-

a

100

$150

200

Ave r a g e V a l u e p e r Head

250

300

-114-3Appendlx

G

Effects of Changes in Prices of Inventories and of Relative
Change a in Selling Prices on Gross Income
Changes in Prices of Inventories
A study was made of effects on gross income of price
changes which could presumeably not be anticipated and their
effects on gross income.

One such price change la concerned

with goods held in inventories.
o f invento ri es were
Table 23.

Simple changes in the values

c on s id er ed first

(table 2 8 ).

S u m m a r y of I n v e n t o r y Changes for 1+5 Farms,
T y p e - o f - F a r m i n g A r e as 5 and 6, 1950

Item

A v e r ag e change p er
G reatest change for a n y
farm, Jan. 1-Dec. JL
one f a r m
_______ Increase
Decrease

Crops
Dairy cattle

*272

$ 2,810

$1,776

723

3,360

700

3,100
2,070
628

3,293
700
102
190

Dther livestock
Beef cattle
Hogs
Sheep
Poultry

g

Total livestock

1,025

5,900

3,293

Total crops and
livestock

1,296

5, 201+

1,809

The range of changes a m o n g farms Is large relative to
the average effect.

The average gross Income for the 1+5

farms in table 28 Is $13,32l+.

Thus the total inventory

H4+-

change a c c o u n t e d o n a n average for about 10£ of the groai
Income •
I n v e n t o r y changes
erent group o f 50 farms.
v e n t o r y w as $ 2 , 8 6 0

o f crops were a n a l y z e d f o r a difThe average b e g i n n i n g crop i n ­

(table 2$X

average I n v e n t o r y w a s $2,907*

A t the e n d of the y e a r the
W h e n e n d i n g I n v e n t o r i es were

a s s i g n e d b e g i n n i n g p r i c e s the average e n d i n g In v e n t o ry w a a
$ 2,144-8.

Thus o n an average the In c r e a s e s in pri c e s o f crop

I n v e ntories a c c o u n t e d f o r $ 14-59 of gross income,

a n d the

same a m ount of l a b o r Income.
T a b l e 29*

E n d i n g C r o p I n v e n t o r i e s V a l u e d at B e g i n n i n g I n ­
v e n t o r y P r i c e s o n 50 Warms, A r e a s 5 a n d 6, 1950
Average
beginning
Inventory

Crops

Corn
$1,227
Oats
371
Wheat
34-9
Ha y
7144Beans
75
Potatoes
1+2
S u gar beets 21
Barley
25
Soybeans
0
Total $ 2 , 8 6 0

Average
ending
inventory

$ 1»193
503

250

Ending
G r o s s Income
In v e n t o r y a c c o u n t ed f o r
valued
P r ice
Quantity
at b e g i n ­
change s change s
ning pric­
es

$896
1+12
222
813
27

strips

28

2

21+
32
21
1

60
3
-8
-7
-6
1

$2, 907

$2,14+8

$1+59

873
30

16
25
15

It has b e e n shown that the average
dairy cattle

$297
91

69

-1+8

-18
11
- c

-5

increase in the

income for 1+5 farms was $723*

(microfilm)

$ - 331
Ul
-127

By use of film

o n w h i c h sections of the f arm account

-iUS-

books Tor M i c h i g a n T o r 1 950 w are r e c o r d e d it was possible
to analyse changes in dairy cattle inventories Tor the e f ­
fects of changes in prices*

The gre a t e r p a r t of the in­

crease in dairy cattle inventories
er numbers

is a c c o u n t e d for by l a r g ­

a n d values p e r h e a d of dairy calves and h e i f ­

ers (table 30).

A

c o m p a r i s o n of average values of dairy

cows at b e g i n n i n g a n d e n d i n g i nventories s h o w e d no a p p r e c ­
iable

difference.

Table 30.

B e g i n n i n g a n d E n d i n g I n v e n t o r i e s of D i f f e rent
K i n d s of Livestock, 38 Farms, T y p e — o f - F a r m i n g
A r e a s 5 a n d 6, 1950

Class of l i v e s t o c k

A v e r a g e inven t ory____
beginning
Ending
N u m b e r Val u e N u m b e r Value

Gain
in
inventory

Dairy cattle
7.2
0*9
9*3
1.9

718
226
1+11
221

Beef cattle

1.2

189

Hogs
Sows
Gilts
Boars
Other

2.1
1.5

116

Sheep
Ewes

Rams
Ot her
Poultry
Total

♦81
319
1+2
305
57

138

•51

2.0
1.1+0.2
19.2

122
71
18
1+20

6
•3
86

2.9
0.2
0.9

55
6
ll+

-3
-5
6

261 222.0

302
♦6,9 8 9

Ip♦885

67
21

0.3
17.3

33b

3.0
0.3
o.U

58
11
8

208.0

19.1 ♦ 3 . 5 M +
8.7
0.9
716
11.4
278
2.3

0

20.1 ♦3,1+63

Heifers
Bulls
Calves
Other

•
H

Cows

♦ 6 , 10l+

k

-ll+6-

The p r i c e s or a n i m a l s o t h e r t h a n d a i r y cows of com­
p a r a b l e age, weight,

kind,

etc. were n o t s ignificant­

l y different In the e n d i n g I n v e n tories f r o m the b e g i n n i n g
Inventories.

Therefore,

as far as p r ice changes In all

crop a n d l i v e s t o c k I nventories are concerned,

the p r o b l e m

la r e d u c e d to one of changes In the prices of crops.
F o r 1+3 farms g r o s s Income w a s r e c a l c u l a t e d to d eter­
m in e wha t It w o u l d have b e e n h a d there b e e n no price

chang­

es In the crop I n v e n t o r y (table 31)*
Table 31*

S u m m a r y of G r o s s Income, Change In Gross Income
a n d Change In L a b o r Income because of Price
C h a n g e s In F e e d and Crop Inventories, 1+3 F a r m s
T y p e - o f - F a r m i n g A r e a s 5 end 6
1950

Numb e r of
farms

Increase In gross
Income because of
price changes In
feed a n d crop In­
ventories

R a nge of
gross
Income

Range of
labor
Income

2

$3^?00 a n d more

$3-9,785 to
$23,780

$ 7 , 8 3 0 to
$9,8t2

6

$ 1 , 0 0 0 to $ 1 , 5 0 0

$ 8 , 3 9 3 to
$20,523

$ 2 , 7 6 2 to
$7,672

0

$500 to $ 1 , 0 0 0

to
$ 15, 89I+

$ 1 , 3 5 9 to
$5, 739

J 53'!01 to
$ 23,811

$6** 8 0

26

L e s s than $ 5 0 0

$2ftL to

Averages
1+3

$532

$11,000

$3,1+82

Th« tendency for farms with larger gross Incomes to
show greater effecta of price changes in Inventories
suggests that the relationships of gross Income to fac­
tors will not be Increased by eliminating price changes
In Inventories.

A n average of $532 of the labor Income

for the I4.3 farms Is accounted for by price differences
between beginning and ending Inventories of crops.
5£ confidence Intervals are # 3 6 7 and $697*

The

In the state

of Michigan as a whole labor Incomes Increased from 19 I4.9
to 1950.

In type-of-farming area 5 labor Incomes In­

creased, on an average, $l,0l(.0.
Increase was $322.

In area 6 the average

The sample of I4.3 farms Included 37

farms In area 5 and 6 farms In area 6.

According to this

evidence price gains In feed and crop Inventories can not
explain all of the Increase In the average labor Income*
Price gains of Inventories of livestock have been shown
to be negligible.
The h y p o t h e s i s w a s set

up that price changes in in­

ventories constitute a r a n d o m type of c o n t r i b u t i o n to
gross income.

The r a n d o m c o n t r i b u t i o n should not be i n ­

cluded In p l a n n i n g the o p e r a t i o n s of the farm.
a r e v i s e d figure f o r gross income,

Therefore

a figure f r o m w h i c h

this type of change o r v a r i a t i o n h a s b e e n eliminated,
should bear a stronger r e l a t i o n to the employment of p r o ­
ductive factors*

R e v i s e d g r o s a Income figures were cor­

r e l a t e d w i t h total f a r m expense a n d o t her classes of f a c ­
tors.

The gross Income as r e p o r t e d w a s similarly correl­

ated.

None of the differences Is significant at the 5%

—11+8—

level of confidence
Table 32.

(table 32).

Correlation Coefficients between Categories
of F a c t o r s and R e p o r t e d Gross Income C o m p a r e d
w i t h C o r r e l a t i o n C o e f f i c i e n t s b e t w e e n Cate?
gor l e s of F a c t o r s and R e v i s e d Gross Income
3 U Farms,

C a t e g o r i e s of
factors c o r r e l —
ate d w i t h gross
Income

Type-of-Farming Areas 5 and 6
1950
Total correla t l o n coefficient w i t h
gross Income
as r e p o r t e d

Total cor rel­
ation coeffi­
cient w i t h
gro s s Income
%a r e v i s e d

Total f a r m expense
a n d total investment

0.855

0. 778

Total l a b o r charge,
total far m expense
not labor, and total
inve stnent

0.81+0

0.900

Total f a m expense, in­
v e s t m e n t In l a n d and
Investmen t s o t h e r than
in land

0.052

0 .857

1.

G r o s s Income Is revised by v a l u i n g ending Inventories
at b e g i n n i n g i n v entor^ prices, for crops.
F r o m the fact that none of the c o rrelations Is

sig­

n i f i c a n t l y i m p r o v e d by e l i m i n a t i n g price gains f r o m crop
inventories,

it follows that there can be no general n a r ­

r o w i n g of the confidence limits of the coefficients in
the gross income equation.

Other elements w h i c h cause

v a r i a t i o n s of gross income from its p r e d i c t e d value are
so important that the effects,

if there are any,

of c h ang—

-1*9-

es In Inven t o r y prices are obscured.
The h y p o t h e s i s was

set up that if p r o d u c t s sold and

h e l d in inventories should be v a l u e d at " n o r m a l 1* prices
the relationship between factors of p r o d u c t i o n and gross
income

should be stronger.

As a first appro x i m a t i o n to

normal values the average prices for different crops and
livestock pro d u c t s for the period 19U6-19U9 were
ed.

A ratio was calculated,

c o nsider­

e x p r e s s i n g the relationship

of the 19I4-6—19 U-9 prices to the prices for 1950.

This ratio

has the effect of b r i n g i n g the prices of the different farm
products into line w i t h eafch other in accordance w i t h the
li--year period,

from 19^.6 to 19^4-9• (See table 33)*

It w o u l d be simple enough m e r e l y to value
ties of products sold at the M i chigan average
prices for the years 191+6 to 1 9 h 9 •

However,

the q u anti­
season farm

equal pricing

for all farms w o u l d Imply no differences in quality of
products,
tices.

per cent butterfat of milk,

For example,

and m a r k e t i n g p r a c ­

the va l u a t i o n of the physical quantity

of m i l k sold at the Michigan season average price
corrected for butterfat)

(even if

w o u l d deny to a farmer sfoo deliv­

ers mil k the part of gross Income arising from the delivery
service.

Income differences caused by m a r k e t i n g practices

should remain In the data after "normalizing" prices.
In order to accomplish this,

and at the same time make an

allowance for the possib i l i t y that s ome p r i c e s may be "out

- 1 50 -

or line",

the dollar sales of* crops and l i v e s t o c k p r o ­

ducts and inventory changes or crops were m u l t i p l i e d by
factors w h i c h c o n verted 1950

sales and inventory changes

o ver to 1 9 U 6 — 19U9 "expected"

sales and inventory changes*

Beginning and end i n g inventories were both v a l u e d a c ­
cording to average y e a r - e n d figures for 19U6 to 19U9*
There is little doubt that the length of the p e r i o d
n e eded to develop a set of expectations varies between
enterprises.
price

Furthermore,

the responses of T a m e r s

changes 'of a single product will vary,

u p o n the l e n g t h of run considered.

to

depending

For these and other

reasons it is a d m i t t e d that the Tour years p r e ceding 1950
do not Torm a perfect basis Tor the development oT a set
of expectations.

The hypothesis was, however,

that the

price relat i o n s h i p s w h i c h pr e v a i l e d in those Tour years
should be more related to the o r ganization oT the a v e r ­
age farm th a n the prices and

their relationships

tiiich

happened to occur in 1950.
The h y p o t h e s i s was not

supported by the det%.

The

correlation coefficients between categories of Tactors
and r e p o r t e d gross income were not significantly different from the coefficients w i t h respect to "normal­
i z e d ” income.
It can be concluded that the reliability of the
value p r o d u c t i v i t y functions w o u l d not be increased by
adjustment of the f arm account data in this study,

to

Tabl9 33*

Weighted Average Annual Michigan Farm prices 01 urops, Livesi/uuu.,
and Livestock Products, and Raii<os of Average 191*6 to 19l*9 Prices
to the Average Prices of 1950.
(Farm products important on farm account farms in areas 5 and 6)

T/eighted average iannual prices
Farm product

Unit

mo

'to w

.....
•

Ratio of average of
191*6 to 19l*9 to
average price for

mo
Corn
Wheat
Oats

fcu.
ti

t»
n
tt

Barley

Rye
Sugar beets
Potatoes
Field beans
Soybeans

Sheep
Lambs

Chickens
Milk cows

§ 1 .65

2.0?
0.8U

2.01
0.81*

1.66

1.20
1.2 7

o.yj)
l.ol*

1.00
1.20
1.32
1.1 5
1.38

T.

13.12

11.1*0

bu.

1.38

cwt.
bu.

8.78
2.5U

1.00
6.90
2.50

1 .2 7
1 .0 2

T.

18.1*5

20.70

0.89

cwt.

23.78
20.98
18.67
8.15

28.80
18.10

0.83

21.80
10.50

0.86

20.78
0.30
169.18

25. 1*0

All hay
Veal calves
Hogs
Beef cattle

51.63

tt
tt
tt
tt

lb.
each

0.25
208,58

Milk, wholesale cwt/
I*.17
doz.
o.kii
Eggs
Wool
lb.
O .46
1* Adapted from Michigan Agricultural Statistics,
Agriculture

3.61*
0.37

0.50

________

1 .1 6
0*80

1.19

0.81
1.15
1.21
0.92

19i?Of Michigan Separtment of

152

eliminate price changes in inventories,

or to eli mi n­

ate relative changes in the pri ce s of farm products*
U n de r circumstances comparable to those of this
in o t h e r words,

study,

the m a n i p u l a t i o n of farm account data

in o r d e r to o b t a i n greater reliability of gross income
est im at in g equations is not worthwhile*

If a planning

f u n c t i o n b a s e d on p r e v i o u s experience is implied in the
layout of e a c h farm,

noth in g has been accomplished in

this case by inventory and price adjustment in discov­
ering what the f u n c ti on is*

-*5>-

APPENDIX H
L i n e a r i t y of R e l a t i o n s h i p s

F o r t y farms were
(in lo"s)

of

selected to

study the relationships

the factors to g r o s s income*

The farms were

chosen at r a n d o m w i t h the c o n d i ti on that the gross i n ­
comes
This

should t he mselves form a logarit hm ic distribution*
c on dition was n e e d e d because,

chosen at random,

the gross

a ro u n d the average.

if the farms should be

incomes w o u ld be d i s t r ib ut e d

There w o u l d then be less o pportunity

to study the b e h av io r of gross Income o ver its range*
Figu re 16 shows the r e l a t i o n b e t w ee n the log of invest­
m e nt in land and the log of gross income.
ship a p p e ar s to be approximately linear*

The r e l a t i o n ­
In figure 17

the d e v i a ti on s of the estimate of the log of the gross
income

(figure 1 6 ) f rom the logs of gross income are

p l o t t e d against logs of the total l a b or charge.
second r e l a t i o n appears

This

to be linear.

There Is a h i g h degree of i n t e r c o r r e l a t i o n between
the indepen de nt v ar iables
f (x,

z ). Thus It

in the general eq u at io n P =

can be e x p e c t e d that the relationship

b e t w e e n the present residuals and a dd it i on al ly Introduced
factors will disappear*

In figure 18 the residuals from

figure 17 are related to the log of the investments other
than in land.

There is no significant relationship.

The order of p l o t t i n g residuals against additional factors

-1514--

F i g u r e 16.

R e l a t i o n b e t w e e n I n v e s t m e n t in L a n d a n d
Gross Income, In Logarithms, I4.O F a r m s
Type-of-Farming Areas

5 and

6, 1950

L o s of
g r os s Income
i n * 1 , 0 0 0 ts

i*5r

U13 / 0.731x

1.2

1.0

0.8

0.6

0.8
1*0
L o g of Investment In la n d in $ 1 , 0 0 0 's

0.6

-155

F i g u r e 17.

R e l a t i o n b e t w e e n the L o g of Total L a b o r
Charge a n d Re s i d u a l f r o m E s t i m a t e of Gro ss
Income f r o m I n v e s t m e n t In Land, I4.O F a r m s
T y p e - o f - F a r m i n g A r e a s 5 a n d 6, 1950

D e v i a t i o n s of Gross
Income fro m gross I n ­
come e s t i m a t e d by In­
ve s tment In l a n d
(logs of $l,000 - u n l t s )

0J+

X = - 0.169 / 0 .I4J

a.o

0.2

o.U

0.6

L o g of total labor charge In $ 1 , 0 0 0 'a
x

- 156-

Flgure 18*

Rela tlon of L o g of Investments other than
L a n d to the R e sidual from the Estimate of
the L o g of Gross Income f r o m the Logs of
Do tal L a b o r Charge a n d Investment In L a n d
1+0 Farms, T y p e — of-Farming Areas 5 and 6,
1950

Residual from the
estimate of the log
of gross Income
(Figure 17)
0 .1+

0+ 2

• •.

.

• • •

0 .0

1
“

*

-

*

0.1

— 0 * 1+

0.8

1.0

1.2

l.U

JL
1.6

1.8

Log of Investments other than land In $ l , 0 0 0 * s

can be cha ng ed w i t h different results*

I n figure 19 the

l og of In ve s tm en ts othe r than l a n d is r e l a t e d directly
to the l o g of the gross income,
is a good l i n e a r relation.
ship is not apparent.

and,

as expected,

In figure 1.6 this rel at io n­

This is because

17 two o t he r independent variables,

in figures 16 and

w h i ch aro themselves

r e l a t e d to inv es tm e nt s o t h e r than land,
all o f the r e l a t i o n of the

there

three

have taken about

categories to the gross

income•
In figure 20 the residuals from the estimate of the
log of gross

Income from the log of investments in l and

(figure 1 6 ) are p l o t t e d against the log of the investments
o t he r than land*

I n figure 21 the log of the total labor

charge Is p l o t t e d against r e s i d u a l s In the log of the
gross Income w h i c h were

(1) not e x p l a i n e d by the

r e l a t i o n b e t w e e n the log of the
the log of the gross Income,

and

simple

investment in land and
(2) were not e x p la i ne d

by the r e l a t i o n b e t we en the residuals of

(1) previous and

the log of the Investment other t ha n In land, a gainst the
l og of the total labor charge*

In figure

21, wh ich shows

the residual errors from figures 19 and 20 p l o t t e d
against the log of the total la bor charge,
r el at io ns hi p

an observacle

still remains*

F i g u r e s 16 through 21 bring out two facts:

(1) V i s u ­

ally it appears that the gross income of a f arm Is related
l o g a r i t hm ic a ll y to the factors of production.

Simple

-158

Figure 19*

R e l a t i o n of L o g of Investments oth er than
L a n d to L o g of G r o s s Income, I4.O Farms
Ty p e - o f - F a r m i n g Are a s 5 and 6,

1950

Log of
gross Income
In
, 0 00's

1.0

0.8''

0*6

o*U

0.8

1*0

1*6
L o g of Investments other than land

-159
F i g u r e 20.

R e l a t i o n of L o g or I n v e s t m e n t s o t h e r than
L a n d to R e s i d u a l f r o m E s t i m a t e of L o g o f
G r o s s Income f r o m L o g of I n v e s t m e n t in L a n d
1+0 Farms,

T y p e - o f - F a r m i n g A r e a s 5 a n d 6,1950

D e v i a t i o n s of g r oss Income
f r o m g r o s s income e s t i m a t ­
e d by I n v e s t m e n t in l a n d
0.*>r
(from figure 16)

0.273 / 0
0.0

0.8

1.0

1.2

1.6

1.8

L o g of Investments o t h e r than l a n d in # 1 , 0 0 0 * a
x

•160

F i g u r e 21*

R e l a t i o n of L o g of Total L a b o r Charge to Re«
aidual of E s t i m a t e of L o g of Gross Income
f r o m L o g of I n v e s t m e n t o t her than L a n d a n d
L o g of I n v e s t m e n t in Land, 1+0 F a rms
T y p e - o f - F a r m i n g A r e a s 5 and 6,

1950

0.30‘

0 . 080.06-

OJDU
o.oet
X

0,00
- 0 .0 2|*

%

•o.oU

- 0*06
- 0.08fc

-

0.10

0.2

0.3

.+

0 1

0.5

0.6

0.7

L o ^ of totall l a b o r charge In $l,000»s
X:

R esidual f r o m the estimate b y the log of Investment
other than l a n d of residual f ro m the estimate of
the l o g of gross Income by the log of the invest­
men t in land, (from figure 20).

a n a lysis of variance
of the relations.
more

tests

shows l o g a r i t h m i c l i n earity

That is,

closely e stimate

the lines of best fit will

the value of gross income than will

the m e a n of g r oss income,

a n d (2) the factors a f f ecting

g r oss income are intercorrelated.
income

In figure 22 gross

is taken as a f unction of the total farm expense

a nd total Investment,

The lj.0 farms do not in these cases

fall on a line as far as the relative amounts of the two
categories are concerned.
the "causal" variables,
l a b o r charge,
plane.

It

The points of intersection of

investment in land and total

do not fall o n a line in the horizontal

can be concluded that there Is some selection

of categories of factors b etween farms.

Figure 22.

The Log of Grose Income as a P l a n e ^ D e t e r m i n e d b y the Logs of
Tfttui Farm Expense
Exoense and Total In
I n ^ ^ vestment, Q0 Farms
Total
T y p e - o f - F a m i n g Areas
5 and 6, 19$0

Plane: P s 0.10 / 0.75x /
l!

I!

I i
'5.5

0.7
0.9
l.l
l»3
Log of total farm expense in $l,000's
y: Log of total investment in 01,000’s.

APPENDIX I

The P r o b l e m o f M u l ti pl e S o l u ti on s of the
Gross Income E q u a t i o n

One of the criticisms of the work of Douglas Is that
he d i d not have

st at istically d if ferent o bs er va t io ns of
1
the e m p l o y m e n t of factors.
M e n d e r s h a u s e n has shown that
the r e g r e s s i o n surfaces w h i c h D o u g l a s
significant

since

calcula t ed are not

the o b s e r v at io ns in three dimensions

f o r m a line r a t ho r than a surface,

Douglas*

answer to

this cr iticism was that in repeated studies the c o nc lu ­
sions

(values of C — the constant t e r m — and of

the ex po ne nt s In o ur work)
The

b, —

tend ed to be consistent.

c o ns is te nc y of conclusions

from one study to another
2
was o f f e re d as a v i n d i c a t i o n of the met ho d *
F i g u r e 23 gives a n o t he r o pp o r t u n i t y to evaluate
degree

to w h ic h the I4.O fains selected for the

l o ga r it hm ic

lin ea r it y (Appendix H)

r ep resent independent o bs er va ti on s
c a t e go ri es oi factors.

can be

the

study of

considered to

in combinations of

The dis pe rs io n of the observations

in the dir e ct io ns of the total

farm expense

and the total

investment axes indicates that w i t h i n the range of pracH o rs t Mendershausen, "On the Significance of Professor
D ouglas* P r o d u c t i o n F u n c ti on ," B c o n o m e t r i c a . V. 6 (1938)
pp. lij.3-53.
2. See footnote 2, p.

8,

supra.

Figure 2 3 .

Three S<
come, T<
I4.O Farmi

1.5

1.3

0,7

P: Log of gr
“ " ir
y:
to
x:

Figure 23.

Three Solutions of the Relationships between Gross In­
come, Total Farm Expense, and Total Investment
lj.0 Farms, Type-of-Farming Areas 5 and 6, 1950
P • -1.11

2-.0

XO

1.2

1.0

1-

w

//
//.

16
1,6

1.8
2.0
P: Log of gross income in $l,000's
investment
"
"total farm exp."
"
x:

ti c a b i l i t y or farm o p e r a t i o n s there Is some s u b s t i t u t i o n
b e t w e e n factors*
However,

the e x p l a n a t i o n of gross Income f r a m total

far m expense alone

is p ra c t i c a l l y

as complete

as the e x ­

p l a n a t i o n of gross Income by both total f arm expense and
total investment.

The fol lo wi ng table

gives the c o r r e l a ­

tion coefficients between the va riables
Expression,

e ^c *

0.926

r

0.851

Xz

0.856

yz

r *.yz
If a true plane existed,
to min im iz e

ences in all three

Value of r

xv„

r

siole

shown in figure 9.

° * 931
however, it should be pos-

the sums of the squares of the d i ff er ­
directions and get ap p r o x i m a t e l y the

same e q u a t i o n on the same plane.

However,

w h e n this is

a t t e m p t e d in figure 23 it is seen that divergent equ a ti on s
result.

The m e a n i n g of the d i f e r e n c e s

in the re gr es s io n

planes on figure 23 can be resolved into the ques ti on of
vhat are the dependent and the
an e q u a t i o n relating the

independent variables in

-ross income of a farm w i th the

categories of factors u sed on the farm.
is inclined to agree w i t h Eouglas

3r on f e n b r e n n e r

Is a fu nction of outlays,

that the gross Income
1
in effect.
In this case the

TI See footnote 5^ pT 15,

supra

-166-

plane which minimizes

the sums of the squares of the

d i f f e r e n c e s b e t w e e n actual and e s t i m a t e d values of gross
Income
plane*

in the gross income d i r e c t i o n is the relevant
Thi s plane

is the one r e f e r r e d to on page 163#

a n d it is the olane of special interest in this study.
Prac t i c a l l y , this means that
duct

is a result

of factors.

s ed in part 3 of c h a p t e r I.

it is assumed that p r o ­

The as s u m p t i o n is d i s c u s ­

U U X ttA

Land
Total
Mean
Upper limit
Lower limit
Hogs
Mean
Upper limit
Lower limit
Beef feeders
Mean
Upper limit
Lower limit
Dairy
Mean
Upper limit
Lower limit
Crops
Mean
Upper limit
Lower limit
General
Mean
Upper limit
Lower limit

JjlIU X U P

Labor

\jjW

uiaw

B

0.098
0.118
0.077

0.087
0.123
0.051

O.OlA
0.014+
0.092
0.092
-0.002
-0.002

0,002
0.090
0.033

0.079
0.130
0.027

0.037
0.037
0.102
-0.026
-0.026

0. 131+
0.179
0.089

0.123
0.202
0. 01+1+

0.081
0.081
0.188
0.188
0.027
0.027

0.083
0.003

0.030
0 . 121+
- 0.063

0.138
0.202
0. 071+

0. 151+
0.257
0.059

0,132
0.221
0 . 01+2

-0.073
0.128
- 0.275

j g* w * ¥

Cash
Improve- Liquid
Working
operating
ments
assets
assets
expenses
C_ _ _ _ _ _ D_ _ _ _ _ _ _ E_ _ _ _ _ _ _ _ F
._

A

0.103

j

0.102

0.008
0.008

0.179
0.179
0 . 21*3
0.21*3
0.115
0.115
0.265
0.265
0.370
0.370
0.159
0.159
0.196
0.198
0.298
0.298
0.099
0.099
0.325
0.325
0.685
0.685
- 0.033
-0.033
--O.ll+l
0 .1U1
091+
00..09
I*
--0.378
0.378

0.211;
0 . 211+
- 0.197
-0.197
0.007
0.007
0.021
0,021
-0 .0 0 6
-0.006
0.366
0.010
0,366
0.010
0.226
0. 61+1
0.226
o.6ia
0.091
- 0,205
0.091
-0.205

0.158
0 . 1^8
0.289
0.289
0.027
0.027

0.280
0 . 391+
0.167

0.119
0.119

0 . 4+1
O.I+98

0.235
0.285
- 0 . 01+7
-0.0l*7
--0.115
0.115
0 . 1+37
-0.1*37
-0.1*37
- 0. 1+37
00.16L
,16k
O.55o
0.55S
-0.230
-0.230
0.190
0.190
0.570
0.570
-0.189
- 0.189
0.298
0.298
0.850
0.850
--0.252
0.252

0.181+

0.082
-0.110
-0.110
0.901
1*580
0.223
0JJ1O
0.866
0.012
0.578
1.092
0.063

Taken from Gerhard Tintner and 0. H. Brownlee, , "Production Functions Derived
from Farm Records," Journal of Farm Economics, V. 26 (1914+).
The data are for

Iowa farms, calendar year 1939,

rable 35* Marginal Productivities and Fiducial Limits at the
Five Percent Level of Probability (Per Dollar of Input)
Mean
LiveMisc.
Upper")
Land
Labor
Equipstock,
ope rat.
Lower/
ment
Teed
expense
limits
A
B
C
D
E
farms------ m 7------751+5---- 707?----- 7501------ 7839-----7393—
U.L.
.057
.232
.285
1 . 077
.5-32
_______________ L.L.
.035
-.0714.117______ .600
.35U
Northeast
m7
.033
.697
.11+8
.658
.5?8
D a iry
U.L.
.051
.335
.285
1.072
.1+1+9
Area
L.L.
.0ll+
-.11+1
.001
.21+5
.307

m

TasH

m

7

705T----7To5---- 7TB0----- TCT7---- 735?—

Grain
U.L.
.086
.ll+6
.357
.079
.1+66
Area
L.L.
.036
.067
.001+
.155
.272
Ve stern
M.
.038
.030
751+1
7713
.1+03
Meat
U.L.
.079
.320
.393
1 . 207
.1+73
A r e a ______ L.L.
- .002
.260
.089 ___ .218
.333
Southern
M. ~
.013
-. lo9
.313
2 •61+1
.1+02
P a sture
U.L.
.01+8
.196
.563
3 . 812
.516
Area
L.L.
-.011
-.l+ll+
.063
1.1+71
.298
eastern”
TH
7 0 3 7 “----705§-----72Th------ 770T---------- —
M eat
U.L.
.065
.089
.1+1+7
.973
.1+15
Area
L.L.
.0ll+
.0 I+7
-.017
.028
.265
Jrop~
3!7
70T7+---^31+9'
7379----- 755J------7 5 5 3 “
U.L.
.073
.215
.503
.786
.873
L.L.
.156
-.713
-.11+5
-.260
.373
_
__
_orri
778?----- 751+3—
U.L.
.030
.390
-.017
1.189
.637
_
__
L.L.
-.007
-.272
-.1+31
.388
.1+51+
Dual an d
MU
.02o
7021
7lJ8
.538
71+63
D a ir y
U.L.
.0l+3
.310
.298
.91+1+
.531
L.L.
-.003
-.268
-.028
.133
.360
General
M7
7050---- 75BI----- 751+0------ 7991-----751+5—
U.L.
.056
.572
.1+98
1.601
.1+82
_______________L.L.
.006
.009
.182
.380
. 21+2
special
m7
70T5
-.o2o
.Oi+o
1.593
.1+1+5
u.l.
.111
.371
.009
2.631
.585
______________ L.L.
.01+0
-.1+13
.011______ .555_____.31+1
'Large"
Wl
7051
701+0
7539
THSl
.1+16
U.L.
.071
.291+
.1+31+
.677
.1+71
L.L.
.031
-.293
.083
.205
.361
TSmall"------ m 7---- 70E5--- 7T03----750B--- 1.1+6$--- 7375—
U.L.
.065
.267
.312
1.986
.1+32
______________ L.L.
.028
-.061
.103______ .952_____.318
?aken f r o m E a r l 0. Heady, "P r o d u c t i o n F u n c t i o n s fr o m a R a n ­
dom Sample of Far m s, " Journal of F a r m E c o n o m i c s , V. 28 (191+6),

>p .O 8 9 -IOOI+.

The data are for Iowa farms, calendar year

1939.

APPENDIX K

A n analysis of D i f f e r e n c e s B e t w e e n E s t i m a t e d
and R e p o r t e d G r o s s Income and L a b o r

Income

The m e t h o d of a c c o u n t i n g for d i f f e rences b e t w e enreported and e s t i m a t e d gross income

can be e x t e n d e d to a business

involving two or more enterprises.

Let the data for a given

farm and for the average of all farms be

given b y the f o l l o w ­

ing table:
Enterprise

Averages, all farms
Yield
Price
Acres

Yield

Acres

1

a

b

c

(a / A a )

(b / ^ b )

(c / ^ c )

2

d

e

f

(d /zid)

(e / A e )

(f / A f )

F o r two enterprises,

u s i n g a numerical example:

1

15

1

2

20

1

(9.6)

(15/

5)

(1

/ 0.2)

(10)

(28.8 ) (20-

8)

(1

/

(3 0 )

The gross income on this farm is $600.
m a t e d gross income is $720.
units of enterprise 1,
to earn 0720.
are

This F a r m
Price

The

0 .0 )

Suppose that the e s t i ­

The average f ar m w o u l d require 12

and 3 6 units of enterprise 2 in order

"expected" values of the number of units

Inserted in the table preceding.
The difference between expected and re ported gross In-

Is approximated by three primary effects as follows:

/A d
2

.

2a

/ A a
£
-

* 2c

2d / A d
2 " '
2b
2e

_£&>
/ Ae

.

/A e
•A f
2 .........
/ A c

2f

. 2c M c
2
.

2

2e

2r

Ac

/Af

2---

- 1 7 .5

•

$

1.1

- 16.0 • $ 1 . 0

•

o .U

-

•

1 .2

3 $ 1 9 .2 0

$ 7 .7 0

•Ab

=

1 7 .5

•A e

=

16.0

• Aa

m $ 1 .1

•

•A d

m O i. o

• 29. k *(-8.0) = 0-235.20

9 .8
* 2 9 . l|
9 .8

be total o f size of busi ne ss e f f e c t s

•$0*2

a
1

2d

•

0

.

O

4 ^

•

o .

5

* $ 5 3 .9 0

•

2 b< i A h

2a

•

Dine

$

0 .0 0

( c) is $26*90;. the

o tal p r i ce a n d y i e l d e f f e ct s are $ 3 U . 3 0 a n d $ - 1 8 1 * 3 0 repec ti ve ly *

The p rice and y i e l d e f f e c t s have

significance

h e n c o n s i d e r e d w i t h respect to individual e nt erprises.
he size of b u s i n e s s e f f e c t is m o s t m e a n i n g f u l w h e n the
arm as a whole

is considered*

The a n a l y s i s o f the
e c o r d e d gross income

differ en ce b e t w e e n e x p e c t e d and

can be u s e d

in c o n n e c t i o n w i t h the

o sts o f o p e r a t i n g the f a r m in an e x p l a n a t i o n of the differnce^ of e s t i m a t e d labor income f r o m B e c o r d e d l a bor income.
There will be a r e l a t i o n of total f a rm expense plus
.nterest on the i n v e s t m e n t at 55 and gross
’anas*

Suppose

’he gross

that this is

income fo r all

given by:

C r o s s income = 5 / U (total farm expense / 55 o n
i nve s tme n t ) •
income of e ac h farm is e s t i m a t e d by a function:

Gross Income « £ ( total farm expense)^( i n t e r e st on
inve stment )-k*
;n tills m o d e l let the e q u a t i o n be:

jet It be a s s u m e d tbat the

total farm expense and the total

.nve stment for the p a r t i c u l a r farm be ing co n s i d e r e d are # 14-55
ind #I|.096, respectively.

T h e n the f o l l o w i n g I n f o r m a t l o n is

ivailable:
Total

costs, that is,

total farm expense plus

>n the investment for the average
fc720 and #51+0 •

The

lyzed is #660,

It Is now possible

form the

interest

f arm w i t h a gross income of

comparable figure for the f arm being a n a ­
to pre s e n t

in tabular

sources of the differences b e t w e e n e x p e c t e d and r e ­

corded gross Income and e x p e c t e d and recorded l a bor Income,
fable 3 6 . E x p l a n a t i o n of Difference B e t w e e n D e ­
cor ded and E x p e c t e d Values of Gross I n ­
come and Total Costs
This farm
M e a n of all farms
Cross Income
Recorded
Expected

720

Difference in gross in ­
come f rom e x p e c t e d values
Costs
Total f ar m expense
Int er e st on i n v e s t m e n t —
5 # x # 14.096
Total costs
E x p e c t e d total costs
Difference In costs
from e xp ec t e d amoun t

#720
720

#600

#120

1+55
205
660

576

81+

T his f a r m
(cont.)
L a b o r Income
Recorded
Expected
D i f f erence
Difference
come due
ture ;
due
In gross

M e a n of all farms
(oont.)
O 1Ml
ll4+

$ -60
1I+J4.
— 20L|.

0

in L a b o r I n ­
to cost s t r u c ­
— 8I4.
to difference
income

0

-120

0

Table

37* A n a l y s i s of D i f f e r e n c e in G r o s s Income f rom its
E x p e c t e d Value by Enterprises, Y i e l d Effects,
Price Effects, and Sl>e of Business
_________________________________E f f e c t __________________________________
Gross Income

Difference A t tributable

Enterp ri se

Re corded

E stimsted

(■1 )

$ 2Uo'

fcll^li-

(2 )

360

576

-235

v^OO

0720

-£lfll

Total

Yield

Price

$ Zk

$ 3*4
-

Size of
buslne ss

to:

Total
Difference

8

$ 96

0

19

-216

03k

$27

-0120

$

The p r e c e d i n g tables have a c c ounted for differences from
e x p ected values of gross income and l a bor income according to
the

structure of the use of resources,

size of business*

The difference

yields,

prices,

and

in gross Income attributable

to yields and p r i c e s are ac c o u n t e d f o r by enterprises*

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