AN ABSTRACT OF THE THESIS OF
WADE LEWIS GRIFFIN, SR. for the DOCTOR OF PHILIOSPHY
(Name)
(Degree)
in Agricultural Economics
(Major)
Title:
presented on ^^<^( / / <^7JZ—^(DaVe)
The Relationship Among Income, Labor Productivity,
Property Taxes, and Migration, U. S. Agriculture,
1957-1970.
Abstract approved:
//
John A. Edwards
The main focus of this thesis was to quantify the relationship
among relative income, relative labor productivity, relative property taxes, and out-migration so that the hypotheses derived from
a theoretical production economy model regarding these relationships could be empirically tested.
In the theoretical production economy model, if out-migration occurred, the relative income position of the farm operators
increased.
If relative labor productivity increased, relative income
decreased and for farm operators to regain their original relative
income position, out-migration had to occur.
Relative property taxes were introduced into the theoretical
production economy model as a net transfer payment from the
farm households to the nonfarm households.
This left the farm
operators in a lower relative income position.
It was assumed
that the farm operators would have to improve their relative
income position by adopting output-increasing technology which
would only make their situation worse.
Here again, out-migration
would have to occur if the farm operators were to regain their
original relative income position.
The hypothesis that an increase in relative labor productivity
would decrease relative income was rejected and an alternative
hypothesis, that increases in relative labor productivity would
increase relative income, was accepted.
By the rejection of this
hypothesis, it was concluded that the statistical results did not
support the theoretical production economy model.
A reason for
this inconsistency between the theoretical production economy
model and the statistical results was presented.
The hypothesis that an increase in relative property taxes
would cause out-migration was rejected and it was concluded that
relative property taxes had no significant, direct effect on outmigration.
Increases in relative property taxes were significant
in increasing relative labor productivity.
The statistical results
suggested that relative property taxes, working indirectly through
relative labor productivity, cause out-migration and increase
relative income.
The Relationship Among Income
Labor Productivity, Property Taxes
and Migration, U.S. Agriculture,.
1957-1970
by
Wade Lewis Griffin^ .'Sri
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
June 1973,. .
APPROVED:
Professor>of Agricultural Economic
in charge of major
Head of Department of Agricultural Economics
'^—^
yi" «■- ■-—-—-/' •
Dean of Graduate School
Date thesis is presented
Qu^/P^^a^'
/ /'?73—
Typed by Gail Griffin for Wade Lewis Griffin,.. Sr.
ACKNOWLEDGEMENT
There are many individuals to whom this author is indebted
for making this thesis possible.
I am particularly grateful to:
Dr. John A. Edwards for his constructive counsel and
guidance in every facet of this study.
Dr. William G. Brown, Dr. Timothy M. Hammonds, Dr.
Gene Nelson, and Dr. Roger G. Petersen for their assistance
in the statistical procedures.
The graduate students who served as a sounding board and
Pal Moon for his assistance in the statistical procedures.
The Computer Center which provided a grant for the computations in this study.
My wife for her patience, her sacrifices, and her typing of
this thesis..
TABLE OF CONTENTS
Chapter
Page
DEVELOPMENT OF A PRODUCTION
ECONOMY MODEL
Introduction
A Production Economy Model
Farm Household-Firm Sector
Nonfarm Household Sector
Nonfarm Firm Sector
Aggregation Within Sectors
Permanent Market Equilibrium
Additional Assumptions for
Solving Market Equilibrium
Values
Conclusion of PE Model
Transfer Payments in the
PE Model
Objectives
Hypotheses
Outline of Thesis
II
III
1
1
3
7
11
13
16
17
■20
21
28
31
32
33
THE MODEL AND STATISTICAL
PROCEDURE
The Model
The Statistical Procedure
Method of Estimating
Coefficients
Introduction of Lagged Variable
Adjustment of Data
38
43
45
STATISTICAL RESULTS AND TESTING
THE HYPOTHESES
Statistical Results
Testing the Hypotheses
Ho: #1
Ho: #2
Ho: #3
Ho: #4
Ho: #5
Ho: #6
Ho: #7
62
62
69
69
70
70
71
71
72
72
34
34
38
,
IV.
IMPLICATION OF STATISTICAL RESULTS
The PE Model Versus the
Statistical Results
Relative Property Taxes and
Rejection of Ho: #2
Short-run Versus Long-run Equations
Conclusion
BIBLIOGRAPHY
74
74
79
83
92
9?
APPENDIX A - TABLES
100
APPENDIX B - EQUATIONS
107
APPENDIX C - DESCRIPTION OF DATA
Percent Farm Operator Household, N
Relative Income, R
Relative Labor Productivity, B
111
111
117
118
APPENDIX D - COMPARISON OF AGGREGATE TIME-SERIES DATA AND CROSSSECTIONAL TIME-SERIES DATA
119
APPENDIX E - COMPARISON OF STATIC
AND DYNAMIC MODELS
126
APPENDIX F - COMPARISON OF ORDINARY
LEAST-SQUARES AND TWO-STAGE
LEAST-SQUARES ANALYSES
129
LIST OF FIGURES
Figure
Page
1. 1
Iso-relative income curves.
22
1.2
Iso-relative income curves with net
transfer payments.
27
2. 1
Iso-relative income curves.
35
4. 1
Iso-relative income curves, statistical
results..
75
The upward bias in relative labor productivity caused from the omission of
the capital variable.
80
Out-migration as a function of relative
income with B
= 4. 08 .
84
Out-migration as a function of relative
labor_productivity with R 1 = 5. 18
and T
= 1. 46 .
85
Relative labor productivity_as a function
of relative income with T
- 1.46 .
86
Relative labor productivity as a function
of relative property taxes with
R
= .518 .
87
Relative income as a function of outmigration with B = 4. 22 .
93
Relative inconae as a function of relative
labor productivity with N = 6. 21 .
96
4. 2
4. 3
4.4
4. 5
4. 6
4.7
4. 8
LIST OF TABLES
Table
3. 1
4. 1
4. 2
A. 1
A. 2
A. 3
A. 4
A. 5
A. 6
Page
Period of adjustment for relative income,
R ; out-migration, N ; and relative
laoor productivity, B .
67
Projection to 1980 of relative income, R ;
out-migration, N ; and relative labor
productivity, B , with relative property
taxes, T , held constant.
90
Projection to 1980 of relative income, R ;
out-migration, N ; and relative labor
productivity, B , with relative property
taxes, T , increased 0. 06 units each
year.
91
Number of households in the United States,
farm and nonfarm, 1957-1970(1,000).
101
Personal income of the United States, farm
and nonfarm, 1957-1970 (Million Dollars).
102
Per household personal income of the United
States, farm and nonfarm, 1957-1970.
103
Property taxes for the United States, farm
and nonfarm, 1957 - 1970 (Million Dollars).
104
Property tax as a percent of personal income
for the United States, farm and nonfarm,
1957-1970.
105
Property tax per household for the United
States, farm and nonfarm, 1957-1970
(Dollars).
106
Table
C. 5
D. 1
D. 2
Page
Relative income, R ; percent of households
in agriculture, N ; relative labor productivity, B ; and relative property taxes,
T, United States, 1957-1970.
116
Diagonal elements of the inverse simple
correlation coefficient matrix: aggregate
tinae-series analysis (ATS) and crosssectional time-series analysis (CSTS) for
three equations.
120
Simple correlation coefficient matrix: aggregate time-series analysis (ATS) and crosssectional time-series analysis (CSTS) for
three equations.
122
D. 3
Regression coefficients (RC), t values (t),
and S. E. of regression coefficients (SE):
aggregate time-series analysis (ATS) and
cross-sectional time-series analysis (CSTS)
for three equations.
124
E. 1
Comparison of static and dynamic analyses:
regression coefficients (RC), t values (t),
and multiple correlation (R^) for three
equations.
128
Comparison of ordinary least-squares and
two-stage least-squares: relative income
equation, cross sectional time-series data.
131
F. 1
THE RELATIONSHIP AMONG INCOME, .'■'
LABOR PRODUCTIVITY, PROPERTY TAXES,
AND MIGRATION, U.S. AGRICULTURE . .
1957-1970- • .
I.
DEVELOPMENT OF A PRODUCTION
ECONOMY MODEL
Introduction
The low returns to human effort in agriculture have been a
major problem for several decades.
Many agriculture economists
have advocated or hypothesized that out-migration would be the
basic solution to the low income problem [ Boyne, 1965; Brandow,
1962; Committee for Economic Development, 1962; Hathaway and
Perkins, 1968; Heady, 1956; Heady, 1969; Johnson, 195 6; Nikolitch, 1962; Tweeten, 197 0] .
The fact is that out-migration has
occurred during the past decades, yet there has been little, if
any, improvement in the income position of individuals remaining
in agriculture relative to individuals in nonagriculture [ Boyne,
1965,' Gallaway, 1967; Hathaway, 1960; Hathaway and Perkins,
1968] .
For the 1957-1970 period, the number of farms changed
from 4, 372, 000 to 2, 924, 000, a reduction of 1, 443, 000 in this
14-year period (Appendix, Table A. 1).
Relative income changed
from 43.4 percent in 1957 to 53. 2 percent in 1970 (Appendix,
Tables A. 2 and A. 3).
Thus, even though the number of farms
were reduced by one-third, relative income increased only by
one-eighth, and still remains low.
The rate at which out-migration from agriculture has
occurred has been sufficient only to permit farm operators to
achieve a slightly larger gain in income compared to the per
household income of nonfarm households.
The major explana-
tion of why out-migration has not improved the relative position
of individuals in agriculture is that production per man-hour in
agriculture has increased substantially because of farm operators
adopting new technology [ Bauer, 1969; Brandow, 1962; -Heady,
1969,* Nikolitch, 1962; Tyrchniewicz and Schuh, 1969].
The
index of production per man-hour (1967 = 100) for 1957 was 53
and for 1970 was 113.
This is an increase of 113 percent [ U. S.
Department of Agriculture, 197 1] .
This rapid technological
advance in agriculture is considered to have a negative impact
upon income.
In a purely competitive market, farm operators
will employ output-increasing technology and will produce a
larger volume of outputs for a given dollar of inputs.
The first
individual to adopt the output-increasing technology profits from
it.
As more and more farmers adopt the output-increasing
technology, the macro effect shifts the supply curve for agriculture commodities to the right at a greater rate than population
increases shift the demand curve to the right.
This depresses
prices and causes incomes to fall.
A Production Economy Model
Edwards,
in a course at Oregon State University, con-
structed a theoretical production economiy (PE) model of which
the conclusions of the model are the sanne as the above described
condition of agriculture.
into:
In the PE model, the economy is divided
a farm household -firm sector, a nonfarm household sector,
and a nonfarm firm sector.
The assumptions
2
on which the PE model is built are as
follows:
All individuals in the economy have a different
utility function; however, within each sector,
their utility is a function of the same variables.
Between sectors, utility may be a function of
different variables.
John A. Edwards, Professor of Agriculture Economics,
Oregon State University.
2
The foundation for the assumptions of the PE model are
based on Patinkin's [ 1965] exchange economy.
2.
The marginal propensity to consume is the same
for individuals within a sector but may differ for
individuals between sectors.
3.
Individuals in the farm household-firm sector and
in the nonfarm household sector are endowed with
a given number of labor hours which they may use
for work or leisure.
4.
Individuals in the farm household-firm sector and in
the nonfarm firm sector have production functions
which are a function of capital and labor.
The farm
household-firm sector produces food and the nonfarm
firm sector produces nonfood.
Production functions
are identical within sectors but differ between sectors.
The two sectors each have a given endowment of
capital.
5.
Time is divided into discrete, uniform intervals.
6.
All individuals start with an initial endowment of
cash balances which may be carried to the next time
period.
7.
The market is perfectly competitive.
8.
Individuals in the economy consume food, nonfood,
and leisure.
9.
There are no lags in the economy.
Commodities are
produced and consumed within a given time period.
10.
Individuals make no consumption plans beyond the
current time period.
11.
Individuals want to start the next time period with
adequate cash balances, assuming prices in the next
time period will be the same as in the current period.
The variables used in the PE model are defined as follows:
t
-
time period
q
-
quantity of food consumed by an individual
in farm household-firm sector (i = f), nonfarm household sector (i = h), and nonfarm
firm sector (i = n) in time period t .
q^
-
quantity of food produced by an individual
in the farm household-firm sector in time
period t
q .
nit
-
quantity of nonfood consumed by an individual
in farm household-firm sector (i = f), nonfarm household sector (i = h), and nonfarm
firm sector (i = n) in time period t ,
q
nt
-
quantity of nonfood produced by an individual
in the nonfarm firm sector in time period t
m.
it
-
end of period cash balances held by an
individual in farm household-firm sector
(i = f), nonfarm household sector (i = h),
and nonfarm firm sector (i = n) in time
period t
m
-
beginning period cash balances held by an
individual in farm household-firm sector
(i = f), nonfarm household sector (i = h),
and nonfarm firm sector (i = n) in time
period t
1.^
it
-
hours of leisure consumed by an individual
'
in farm household-firm sector (i = f) and
nonfarm household sector (i = h) in time
period t
W
-
wage rate of labor purchased (sold) by farm
household-firm sector in time period t
W
nt
-
wage rate of labor sold by nonfarm household
sector in time period t
P
P
h
nt
-
unit price of food in time period t
-
unit price of nonfood in time period t
-
hours of labor purchased (if positive) or
sold (if negative) by an individual in farm
household-firm sector in time period t
h ^
nt
-
hours of labor purchased by an individual
'
in the nonfarm firm sector in time period t
F
-
total supply of labor services owned by an
individual in farm household-firm sector
(i = f),and nonfarm household sector (i = h)
in time period t
H
-
total quantity of labor services supplied to
a firm for use in production by an individual
in farm household-firm sector (i = f) and
nonfarm household sector (i = h) in time
period t.
The three sectors will first be discussed separately so
as to develop supply and demand equations for individuals in each
sector.
Then, the supply and demand equations will be aggregated
in each sector.
Permanent market equilibrium conditions will
be defined yielding equations from which all equilibriuna values
for the three sectors can be determined.
Farm Household -Fir m Sector
The individual in the farm household-firm sector is assumed
to have a utility function of the following form:
a
„
"if
U
q
ft - fft
2f
q
nft
a
Q
3f
'ft
4f
ft *
I
.. u
(1 1)
m
-
That is, the individual's utility in time period t is a function
of the quantity of food (qff. )> nonfood (q
Land leisure (1 ) that
he consumes and the amount of cash balances (m ) that he holds
at the end of period t .
The af (i = 1, 2, 3, 4,) are positive
constants less than one.
The individual in the farm household-
firm sector maximizes utility with respect to two constraints:
ft
H
£t
ft-1
M
ft
fft
(1.2)
- P
2
-
q ,
nt ^nft
F
£t
" 'ft
- m,
ft
"
H
ft =
- W, h, = 0
ft
ft
0
-
(1 3
- >
The first constraint says that the monetary value of farm
production plus cash balances at the beginning of the period, less
the monetary value of food and nonfood consumed, less cash
balances at the end of the period, less the monetary values of
hours of labor purchased (if hri > 0) or labor sold (if h.^ < 0)
ft
ft
must equal zero.
The second constraint says that total labor
hours owned less hours of leisure consumed, less hours of labor
employed must equal zero.
The production function is assumed to be of the following
form:
<H
<t - Pof V
where k
ft
+
U 4)
V
-
is the endowment of capital available to the farm at the
beginning of time period t .
Notice also that if (H
+ h ) > H ,
then the individual in the farm household-firm sector employs
some nonfarm household labor.
If (H. + h, ) < H , then the
individual in the farm household-firm sector sells some labor to
the nonfarm firm sector.
It is assumed in this PE model that
the individual decides how much he is going to produce during a
period and produces and sells it.
The LaGrange equation for maximizing the individual's
utility with respect to the two budget constraints is:
v
Y
ft
= U
ft
+ X. , (P,
qf + mr
If v ft Hft
ft-1
- P
nt
q
H
e
nft
- mr
ft
- P, q,.,
ft Hfft
- Wr h )
ft
ft
.
_.
( 1. =>)
Taking the partial derivative with respect to those
variables that the individual in the farm household-firm sector
has control over (qff.» q
f.»
lf.»
respect to the two multipliers (X.
m
f.»
H , and h ) and with
. and \
), then one can solve
10
to get the following equations for individuals in the farm household-firm sector.
Demand for food
q_
lfft
-
Q f
mft
—l
.a4f
M
(n1. 6)
S
P
ft
Demaiid for nonfood
(1
Srft - "^7'"PT
4f
nt
)
Demand for labor
P.
ft
Supply of labor
a
3f m£t
H
£t = Fft - -— Vt4f
ft
"•9»
Demajid for cash balances
m
ft =
A
4f[mft-,+WftFft
(1. 10)
+ ( 1. 0 - P2f) Pft q£ ]
11
where
A =
^f
4f
a1{ + a2£ + a3f + o.^
Nonfarm Household Sector
The individual in the nonfarm household sector is assumed
to have a utility function of the following form:
a
a
ih
U
ht " *fbt
q
2h .^h
nh
Sit
Q
4h
"hit
.
*
,.
(1 11)
-
That is, the individual's utility in time period t is a function
of the quantity of food (q^ ), nonfood (q
), and leisure (1
)
that he consumes, and the amount of cash balances (m, ) at the
end of period t .
The individual in the nonfarm household sector
maximizes utility with respect to two constraints:
1.
m.
^
+
W
t "ht -Pft *fht
- P
2
-
F
nt
ht - V- "h.
q
M
,
- m
=0
nht
ht
=
0
-
(1. 12)
(1 13)
-
The first constraint says that cash balances at the beginning of the period plus the monetary value of labor services sold,
less the monetary value of food and nonfood consumed, less the
12
cash balances at the end of the period equals zero.
The second
constrsiint says that total labor hours owned less hours of leisure
consumed, less hours of labor employed must equal zero.
The LaGrange equation for maximizing the individual's
utility with respect to the two budget constraints is
\t -
u
ht
+
Sh
{r
\t.i
+ w
t
H
ht
" P£t "fht " Pnt "nht " "V
+
^h
< 1- I4)
(F
ht - v, - V
Taking the partial derivative with respect to those variables
that the individual in the nonfarm household sector has control
over (q, ,., q , ., 1, ., m, . and H ) and with respect to the two
hit
nht ht
ht
ht
multipliers (X.
and X.
), then one can solve to get the following
equations for the individual in the nonfarm household sector.
Demand for food
a
ifKt
^
=
ih
T.a
4h
^t
P—
P
,
^
(1
*
ft
l5)
Demand for nonfood
a
Vxt
=
2h
""ht
ir-pT
4h
nt
.
M
(1 l6)
-
13
Supply of labor
H., =
ht
F
ht
-
-^L ^L
a.,
4h
(1.17,
W^
t
Demand for cash balances
-ht
=
A
<1 I8)
«h •-ht-i + W
-
where
4h
4h
a , + a_, + a0, + a.,
lh
2h
3h
4h
Nonfarm Firm Sector
The individual in the nonfarm firm sector is assumed to have
a utility function of the following form;
_
nt
in
fnt
2n
nnt
4n
nt
(1. 19)
That is, the individual's utility in time period t is a function of
the quantity of food (q, ) and nonfood (q
) that he consumes and
^
'
fnt
nnt
the amount of cash balances (m
time period t .
) that he holds at the end of the
Leisure does not appear in his utility function
since he does not have any labor, only capital.
The individual
in the nonfarm firm sector maximizes utility with respect to
the following constraint.
14
m
nt-1
+ P
P
nt
q
M
- -P, H
q,
ft fnt
nt
(1.20)
—P
nt
q
nnt
-W
h
nt
nt
-m
=0
nt
The constraint says that money at the beginning of time
period t plus the monetary value of its production, less the
monetary value of its consumption of food and nonfood, less the
monetary value of labor employed in production, must equal zero.
The production function is of the following form:
p
„
, in
qTit«. - PnOn knt«.
where k
nt
, r2n
nt«.
, _ .
(1.2 1)
h
is the endowment of capital available to the individual
in the nonfarm firm sector at the beginning of time period t .
As in the case of the individual in the farm household-firm sector,
the individual in the nonfarm firm sector decides how much he is
going to produce during a period and produces and sells it.
The LaGrange equation for maximizing the individual's
utility with respect to the budget constraint is
Y
=U+\
(m
+P.qP
'nt
nt
n
nt-1
nt nt
ft
W
nt
q
fnt
h
nt
"
nt
- m )
nt
4
nnt
(1.22)
15
Taking the partial derivative with respect to those variables
that the individual in the nonfarm firm sector has control over
(q, ^,0
., m , and h ) and with respect to the multiplier (X.n),
"nt
nnt
nt
nt
then one cam solve to get the following equations for the individual
in the nonfarm firm sector.
Demajid for food
-
q
a
m ,
nt
in
fnt " TT
PT"
4n
ft
.
.
(1 23)
'
Demand for nonfood
a
Suit
m
2n
a.
4n
P
nt
(1.24)
nt
Demand for labor
h
P
-^W _,
nt
P
nt
= p_
q
2n nt
(1.25)
Demajnd for cash balances
ni
nt
= A. [L m ^
+ (1. 0
4n
nt-1
)
P?n
^n
P
n 1
nt«. ^t
t
(l 26)
-
16
where
A
4n
a
+ a_
+ a.
In
2n
4n
4n
Aggregation Within Sectors
It is assumed that the marginal propensity to consume is the
same among individuals within a sector and that the production
functions are identical within the sector.
Therefore, to aggregate
the individuals' supply and demand equations within sectors is
simply a matter of multiplying the equations of the various sectors
by the population of that sector.
For example, let N
farm household-firm sector population.
be the total
Then, the demand for food
by the farm household-firm sector would be
= Nft •
%t
=
.
"if
m
ft
N
ft •
a
2f
"if
M
a
P
2f
ft
P
ft
(1.27)3
ft
3
Note that capital letters are defined the same as small
letters except capital letters refer to the sector and small letters
refer to the individual within the sector, i.e., N
• m
= M
where M,. is the total end of period
cash balances held by all
r
ft
'
individuals in the farm household firm sector. This generalization
applies
to all variables except F_ and H.^ .rir
it
it
17
The population variables are defined below I
N
-
total farm household-firm sector population in time period t
N
-
total nonfarm household sector population in time period t
N
nt
-
total nonfarm firm sector population
in time period t
(Nr + N, )ft
ht
number of households in labor force in
time period t
N
ft
~
(———-rj—)ft
ht
proportion of households in the farm
household-firm sector (N ) in time
period t
N
-
total population of the economy in time
period t.
Permanent Market Equilibrium
Permanent market equilibrium will be defined by the
following equations:
VT
ft
M..
it
= W
= M
nt
? W^
t
it-1
(1.28)
i = .£„ h, n
(1.29)
18
^ft ^fht
+
^nt
Q
=
(l 30)
ft
-
Qdft + Q*
+ Qd
= Qpt
nft
nht
nnt
nt
(1.31)
M
(
£t
+
+
<
M
M
nt =
t •
'•
32
>
Equation (1. 28) is a simplification of the model which states
that wage rates must be the same for all individuals in the economy
for a permanent equilibrium to exist.
Equation (l. 29) states that
at permanent equilibrium, individuals' beginning of the period
cash balances must be equal to the end of the period cash balances.
Equations (l. 30) and (1. 3 l) state that aggregate demand must equal
a
ggregate supply for food and nonfood, respectively.
Equation
(1. 32) states that the demand for cash balances must equal the
total stock of money.
Thus, Equations (1.30), ( 1. 3 l), and (1.32)
represent a system of equations which must be solved simultaneously7 for P, .
ft
P
nt
.
and W .
t
Solving this system we have
. fLiV^l^JklM ^L
p
£t
"
A
5
N
ft
F
f«
+
A
6
N
ht
F
h«
QP
,. 33,
19
p
A0 N
F, + A, N
F,
3
ft
ft
4
ht
ht
Ac N, F, + A. Nn
F,
5
ft
ft
6
ht
ht
=
nt
M
t
^p
Q*'
nt
( 1
'
'
M
W
t
=
"•35)
Ac N
F, \ A, N
r
5
ft
ft
6
ht vht
where
M
-
total stock of money in time period t
A.
-
(i =
1, 2, 3, 4, 5, 6) a mixture of the
parameters from all the utility functions of
the three sectors and the labor coefficients
of the two production functions.
4
From these three equations, all the long-run equilibrium
values for the three sectors can be determined.
Thus, given
values of the parameters, population of each sector, labor services
available from individuals, and total stock of money, unique values
can be determined for the variables in the system.
When discussing the problems of agriculture in the introduction, it was suggested that technology had a negative return
in agriculture and out-migration had a positive affect.
In Equa-
tions (1.33), (l.34), and (1.35), the populations of the farm
4
For derivation of the A.'s, see Appendix B.
20
household and nonfarm household sectors appear on the right
side of the equations.
By changing the proportions of the house-
holds in the farm household and nonfarm household sectors, the
affects of out-migration on all variables in the system for all
three sectors can be determined.
equations also contains
(3
(3
The right side of the three
hi
.
If a change in -—:— by changing
2n
can be used as a change in relative labor productivity between
the two production sectors, the affects of a change in relative
labor productivity on all variables in the system for all three
sectors can be determined.
Additional Assumptions for Solving Market Equilibrium Values
For the production of commodities, it is assumed that total
endowment of capital in both the farm and nonfarm sectors is
constant within the sectors and over time.
Total population in
the economy will remain constant but movement between farm
and nonfarm households is possible.
When out-migration occurs,
the total capital in agriculture will be divided equally between
those remaining in the farm sector.
That is, if K
endowment of capital in agriculture, then
«
N£t
is the total
21
Also, for nonfarm firms
nt
where K
and N
nt
N ^
nt
is the total endowment of capital in the nonfarm firms
remains constant through the analysis.
Conclusion of the PE Model
Estimated values
5
were assigned to the coefficients in the
system of equations and the equilibrium values were obtained.
Different equilibrium values of the PE model were obtained by
varying the parameter on the labor coefficient of the farm production function ((3, J> thus varying the ratio
hi
, and by varying
2n
proportions of households in the farm household and nonfarm
household sectors (N ).
The basic reason for varying these is
to find their effect upon farm household firm sectors1 income
relative to the rest of the economy.
Figure 1. 1 shows the results of the PE model and is a
graphical representation of the condition in agriculture as described on page 1 and 2 of this chapter.
5
The vertical axis in
Production coefficient estimates were taken from a study
by Paul Zarembka. Consumption estimates were taken from
research by John A„ Edwards (unpublished) and from educated
guesses.
I-"
CO
<
a
H
G
n
(t
3
0
o
•-•
3
(6
<
(t>
i—'
I-J
O
l
CO
t-i
i->
*-*
n
e►i
TO
CO
1
o
a
CO
o
cr
o
i—
t—'
O
rt-
0
3
4
p>
l-K
o
o
t-fl
{"
JO
Index of relative labor productivity
td
23
Figure 1. 1 measures the relative labor producitivity (B )
farm sector to the nonfarm sector.
of the
In the PE model, the ratio
of the parameters of the labor variable of the agriculture production function to the labor variable of the nonagriculture production
function was used as the relative labor productivity indicator.
The horizontal axis measures the percent of farm households to all households in the economy (N ).
hold is equivalent to one farm operator.
One farm house-
From this point on
in this study, out-migration will be defined as a decrease in N ,
i. e. , as a decrease in the percent of farm households in the
economy.
Out-migration in U.S. agriculture has occurred
during 1957-1970 because the number of farm operators in
agriculture has decreased and because the total number of
households has increased.
Each curve in Figure 1. 1 is an iso-relative income curve
since it represents a constant level of relative income (R ).
Relative income is defined as
6
B
= "«
r
Zn
24
p
ft
Q p
ft -
w
t
h
ft
N
ft
R
t =
W.t H.nt + P nt. Qf.
- W t h nt.
ft
N,
+ N
ht
nt
where if h
> 0, then it is included in the equation;
it is excluded from the equation.
otherwise,
The numerator is gross income
from farm production less wages paid to labor from the nonfarm
sector employed by the farm sector, which yields net income from
farm production to the farm sector.
Off farm income to the farm
operator is excluded since h
Net income from farm pro-
> 0.
duction to the farm sector is divided by the number of households
in the farm sector to get net income per farm operator household.
The denominator is wages paid to nonfarm households plus gross
income from nonfarm production, less wages paid to labor employed
by nonfarm firms,, which yields net income to the nonfarm sectors.
This is divided by all households in the nonfarm sectors to get
the average net income of the nonfarm sectors.
The closer the
iso-relative income curve to the origin, the better off farm
operators are relative to the rest of society, i. e., R. < R < R?.
7
7
Notice that if B remains constant and the number of farm
operators in agriculture remains constant, but the total number of
25
Suppose the economy is at some point, say A, on isorelative income curve R
.
Since farm operators are in a
perfectly competitive market, suppose they employ outputincreasing technology (increase relative labor productivity)
with the expectation that income will rise (increase relative
income).
However, as explained in the introduction, the in-
elastic demand for agricultural products causes incomes to fall
and hence relative income to decline.
This is shown in Figure
1.1 as a movement from point A to point B.
Farm operators
are now on a lower iso-relative income curve, R
.
Farm
operators will want to regain their relative income position,
R
.
They could revert back to their old level of technology,
g
thus reducing relative labor productivity.
However, given
that farm operators operate in a competitive industry, reducing
labor productivity is not a realistic alternative.
If farm operators
increase relative labor productivity above point B, they will
households increases, then N would decrease causing farm
operators to be relatively better off. This is logical since an
an increase in total households would increase the demand for
agriculture products.
8
Notice that relative labor productivity could also decrease
if farm operators held technology constant in the farm sector
while technology increased in the nonfarm sector.
26
become worse off.
The only other alternative under these given
conditions is for out-migration to occur (decrease N ) leaving
those in agriculture better off.
This is shown as a movement
from point B to point C in Figure 1. 1.
Suppose the economy is at point A, again on iso-relative
income curve R .
Now suppose some exogenous force causes
the iso-relative income curves to shift toward the origin as in
Figure 1. 2.
R ' , where
Point A is now on a lower iso-relative income curve,
R' < R
.
It cannot be determined from Figure
1. 2 if the households in this economy are absolutely better off
or worse off.
However, it is obvious that the farm operator
households in agriculture are relatively worse off than before.
Farm operators will now want to regain the relative position
they had, i. e., to move back to iso-relative income curve R .
According to Figure 1.2, farm operators should adopt output-*
decreasing technology causing relative labor productivity to
decline (point A to point D) or out-migration must occur (point
A to point E).
Historically, however, they have adopted output--
increasing technology causing relative labor productivity to
increase (point A to point B), moving them further from isorelative income curve R .
Thus, under the present conditions,
out-migration must occur if farm operators want to be on isorelative income curve R .
27
Ratio of farm to nonfarm households
Figure 1. 2.
Iso-relative income curves with net
transfer payments.
28
Transf er Payments in the PE Model
In the analysis of the preceding section, it was argued that
relative labor productivity and exogenous forces aifect out-migration by changing the relative income position of individuals in the
farm sector.
One kind of exogenous force which will change the
relative income position in the manner described is represented
by a net transfer payment between the farm and nonfarm households.
Transfer payments were introduced into the PE model
in the form of net taxes and net subsidies (T ).
They were put
into the budget constraint as follows:
Farm household-firm sector
P
ft
qP+m-P
4
ft
ft-1
ft
q
4
fft
-P
q
4
nt
nft
(1.36)
- rn
ft
- "W
h
ft
ft
- T
t
= 0
Nonfarm household sector
m,
+ W ^ H ^ - P, M
q
- rn ^
Tit-l
nt
ht
ft rl
fht
ht
(1. 37)
T
N,
29
If T
is positive, then it is a tax on the farm household-
firm sector and a subsidy to the nonfarm household sector.
T
If
is negative, then it is a tax on the nonfarm household sector
and a subsidy to the farm household-firm sector.
Figure 1. 2
may be used to illustrate the results of this net tax and subsidy.
If, as in Figure 1. 1, point A is the initial equilibrium position,
the iso-relative income curve R
shifts toward the origin as
in Figure 1. 2, leaving the farm operators on a lower iso-relative income curve R1 .
That is, the net income of farm operator
households relative to the net income of other households in the
economy has decreased.
The results of such a net transfer out
of the farm sector is the same as the exogenous shift discussed
in the previous section.
Farm operators may attempt to improve
their relative position by adopting output-increasing technology, ,
but this will only make their situation worse.
Out-migration must
occur under the present conditions if the farm operators1 relative
income positions are to improve.
A net transfer from the nonfarm sector to the farm sector
would shift the iso-relative income curve away from the origin
so that farm operators would be on a higher iso-relative income
curve.
position.
They would want to maintain their new iso-relative income
Thus, net transfer payments effect out-naigration by
shifting the iso-relative income curve.
30
Transfer payments may be in the form of public goods and
services where one sector pays a larger share of the costs than the
other sector, or where one sector makes a direct monetary payment
to the other sector.
Property taxes are an example of the first
type of transfer payments and are the type of transfer payments
dealt with in this study.
L/Ooking at the U. S. aggregate, individuals
in the farm sector pay a higher property tax than individuals in the
nonfarm sector, whether considering property taxes as a percent
of aggregate sector personal income or on a per household basis,d
and this difference has been increasing over time.
For the period
1957-197 0, property taxes as a percent of income increased from
9. 14 percent to 14, 77 percent for the farm sector, an increase
of 62 percent;
whereas, property tax as a percent of income
increased from 3.49 percent to 3.96 percent for the nonfarm
sector, an increase of 13 percent (Appendix, Table A. 5).
Considering property taxes on a per household basis, they
increased from $248 to $1024 for the farm operator household
and from $250 to $515 for the nonfarm operator household, i. e.,
relative property taxes on a per household basis almost doubled
(Appendix, Table A. 6).
When broken down on a state basis, pro-
perty taxes range from less than 3. 0 percent of the net farm
31
income to more than 30.0 percent of the net farm income.
9
If one
can assume that all social services provided by property taxes
are distributed equally among all individuals in the economy, then
a net transfer is being made from farm operator households to
nonfarm households through local and state governments.
Objectives
The overall objective of this study is to quantify the relationship between out-migration, relative income, relative labor productivity, and relative property taxes so that the conclusions of the
PE model, which describe the condition that some economists say
exists in agriculture today, may be empirically tested.
More
specifically, the objectives are:
1.
to formulate into hypotheses the conclusions of
the PE model;.
2.
to develop a conceptual framework which relates
out-migration, relative income, relative labor
productivity, and relative property taxesj
9
From the table prepared by Grant Blanch, Professor of
Agriculture Economics, Oregon State University, entitled,
"Frequency Distribution of Agriculture States by Percent Farm
Property, Real and Personal, Represent of Adjusted Total Net
Income from Farming, 39>States" (unpublished).
32
3.
to empirically estimate the relationships between
these factors and test the hypotheses.
The information provided can be utilized to ascertain whether
local and state decision-making in the past has been important
in achieving socially desirable objectives of agrarians in our
society.
Better understanding of the past will be useful for
future decision-making.
Hypotheses
Hypotheses to be tested are:
1,
Out-migration has a significant effect on increasing
the net income per farm household relative to the
net income per nonfarm household.
2.
An increase in the labor productivity of the farm
sector relative to the nonfarm sector causes a
significant decline in the net income per farm
household relative to the net income per nonfarm
household.
3.
A decrease in the net income per farm household
relative to the net income per nonfarm household
has a significant effect on causing out-migration.
33
4.
An increase in the labor productivity of the farm
sector relative to the nonfarm sector has a significant
effect on causing out-migration.
5.
An increase in property taxes paid per farm household
relative to the property taxes paid per nonfarm household has a significant effect on causing out-migration.
6.
A decrease in the net income per farm household
relative to the net income per nonfarm household
will increase the labor productivity of the farm sector
relative to the nonfarm sector.
7.
An increase in property taxes paid per farm household
relative to the property taxes paid per nonfarm household causes labor productivity of the farm sector
relative to the nonfarm sector to increase.
Outline of Thesis
The remainder of this thesis is divided into three main
parts.
Chapter II is devoted to developing the statistical pro-
cedure used and statistical problems involved in obtaining a
"best" estimate of the coefficients in the model.
Chapter III
deals with the empirical results and testing the hypotheses.
Chapter IV is a discussion of the statistical results.
34
II.
THE MODEL AND STATISTICAL PROCEDURE
The Model
In the discussion of the theoretical PE model as represented
in Figure 1. 1, adjustments were assumed to take place instantaneously.
However, in the farm sector, the production of farm
products takes a certain length of time which will be called a
production period.
convenience.
Figure 1. 1 is reproduced in Figure 2.. Lfor
At the beginning of the production period* there is
a given percent of total households in the economy whicfe are farm
operator households (N ) and a given labor productivity ratio
(B ).
Both are assumed to remain constant during a given
production period and determine the relative income position
(R ) of the farm operators at the end of the production period.
Thus, there is the functional relationship
Rt = f (Nt, Bt)
where
R is the net income per farm operator household relative
to the personal income per nonfarm household in pro-
lielative property taxes (T ) is not included in this functional
relationship since there is an additive relationship between R
t
and T .
35
Figure 2. 1.
Iso-relative income curves.
36
duction period t .
N is the percent of total households in the economy that
are farm operator households in production period t .
B is the index of relative labor productivity of the farm
sector to the nonfarm sector in production period t.
Only between production periods can the number of farm operators
or relative labor productivity change.
The percent of households that are farm operator households at the beginning of a production period t are a function of
the relative income during the previous time period,
t - 1, and
a function of the variables that affect relative income during that
period.
For example, if farm operators are at point A on iso-
relative income curve R , an increase in relative labor productivity above B
will move point A to a lower relative inconae curve.
Or, if property taxes of the farm sectors relative to the nonfarm
sectors increase, the iso-relative income curves will shift to the
right, leaving point A on a lower relative income curve.
In both
cases, out-migration must occur if those remaining in agriculture
are to regain the relative inconae position they had before relative
labor productivity or relative property taxes increased.
Since
farm operators naigrate at the end of a production period, the
percent of farna operators in agriculture during production period
37
t is determined by factors in production period t - l.
The
functional relationship may be written as:
N
where
T
t
= Bg v(R
,
t-l'
B
,
t-l'
T
)
t-r
is the property taxes paid per farm operator household relative to the property taxes paid per nonfarm
household in production period t-l.
Relative labor productivity may increase because farm
operators want to increase their relative income position or because a change in relative property taxes causes a decline in
their relative income position.
An increase in the relative labor
productivity cannot occur during a production period but can only
be employed at the start of the next production period.
The
functional relationship may be written a.sl
B
t
= h (R
, T
t-l'
)
t-r
The model used in this study may be explicitly expressed
in the following equations:
Relative income equation
t
10
lit
12
t
^It
(2. 1)
38
Farm operator equation
N
t
=
%
+
0
R
2I
t-1
+
a
22
B
t-1
(2.2)
+
^
T
+
t-1
^t
Relative labor productivity equation
B
t
=
Q
30+
a
31
R
t-1
+
Q
32
T
t-1
+
^t
(2 3)
where the a..'s are the parameters of the three equations
(i = 1, 2, 3 ;
and the (i. 's
j = 0,
1, 2, 3)
are the error terms of the three equations
(i = 1, 2, 3).
The Statistical Procedure
Method of Estimating Coefficients
The type of procedure used to estimate the coefficients
in Equations (2.1 ), (2. 2), and (2. 3) will depend on the type
of model the equations form and the assumptions made about
the relationships of the model.
A "causal chain" is established in the model because of
-
39
the following relationship:
the exogenous variables,
|i,
;
N
R
in Equation (2. 2) is determined by
,
B
, and T
, and the error
B in Equation (2. 3) is determined by the exogenous var-
iables,
R
and T
, and the error \i
',
while R
in
Equation (2. i) is deduced from N and B , and the error
|JL
.
The model may be rewritten more generally as:
Ax
where
x
+ Cz
+ |j,
= 0
(2.4)
A is the coefficient matrix on the endogenous variables,
is the matrix of the endogenous variables,
matrix of the exogenous variables,
error terms.
and u
C is the coefficient
is the matrix of the
Equation (2.4) may be considered to be the struc-
tural form of a simultaneous equation model.
However,
The model is said to be recursive if there exists
an ordering of the endogenous variables (i = 1,
2, . . . , n) such that the matrix A
is triangular
and if the covariance matrix of (j,
is diagonal,
that is, if
a.. = 0 for all j > i
E
(M^, ^t) = 0 for all j/i
The original quote was a B.
The original quote was a
f\.
[Malinvaud, 1970, p. 612]
40
The elements in Equations (2. l), (2. 2), and (2. 3) may be
arranged as;
Endogenous Variables
R.
R
N.
B.
a
a _.
12
11
Exogenous Variables
B
t-1
T
t-1
a
u It u 2t u3t
t-1
a
22
^l
Disturbances
23
l
32
31
The coefficient matrix linking the endogenous variables to one
another has no coefficient below the major diagonal and the model
is therefore recursive.
Up to now, it has been assumed that there
is no correlation between |j,
,
|JL
,
could be correlated in the population.
and |i
;
however, they
If the error terms in the
three equations are not correlated, then the coefficients can be
estimated optimally by least squares.
If, on the other hand, the
error terms are correlated, then another method such as twostage least squares will provide an unbiased estimate of the
parameters [ Malinvaud, 1970; Fox, 1968] .
It is interesting and important to know the difference
between the two estimating procedures.
Estimation of the
coefficients of Equations (2. 2),and (2. 3) will be the same regardless of whether ordinary least squares or two-stage least squares
is used because N
and B
If
are functions of exogenous variables
t
41
only.
However, for Equation (2. 1) the results are different.
Re-
writing Equation (2. 1) in a more general form
(2.5)
R = Xa + (a
where
1 N1 B1
R.
1
R =
N
2
^ 11
B
10
2
, X
a.
12
^i
=
11
R
1 N
n
n
B
a
n
12
M- In
and using ordinary least squares
'
-1
■
a = (X X)
X R
o
+ (X
X)"1 X
p.
Using two-stage least squares, Equation (2.5) is rewritten as
R = Xa
where
1 N1 B1
1
N
2
B
2
X =
1 N
n
B
n
+ (j.
42
A
and N
A
t
and B
are the estimated fitted values of N
t
t
and B
t
from the corresponding regression Equations (2. 2) and 2. 3),
respectively.
Therefore,
a
*
"
= (X
X)
_ 1 ■
X
' ..
R
/\' A - 1
= a1 + (X X)
It is obvious that if E
,
(JJ.
|JL
A'
X
^ .
(2.7)
) = 0 , where (i = 2, 3), then
both estimates are consistent estimates of a
since
E (a ) = a
1
1
and
E (a
However,
1
) = a
1
a is a best linear unbiased estimate of a
it has the smallest variance.
13
If there is other than zero
correlation between the error terms, i. e. ,
#
where (i = 2, 3), then a
'1
because
E (u.
It
,
u. ) ^ 0
T.t '
still tends to be the true value of a
1
(since M and Bt are exact functions of the exogenous variables,
13
For proof see Johnston [ i960, pp. l6-]7]
43
they are independent of the error terms) while a
estimate of a
is a biased
since
1
E (Xji )
/
0
As the error terms are unobservable, it is uncertain as to
whether or not they are correlated.
However, to avoid "specifi-
cation error" two-stage least squares will be used in the
analysis.
Introduction of Lagged Variable
The structural equations in the model as developed here
can be termed static equations and the relationships among the
variables within an equation represent
"a curious mixture of
short- and long-run adjustments" [ Nerlove, 1958a, p. 86 l] .
Nerlove has suggested a technique that disentangles the two
types of adjustments [Nerlove, 1958a^ Nerlove and Addison,
1958] .
Consider the long-run function of relative income to be
R
t
where R
=
"10
+
a
i1
N
t
+
a
i2
B
t
<2-8>
represents the long-run equilibrium value of rela-
tive income.
The long-run equilibrium value of relative income,
(R.) is not observable so Equation (2.8) cannot be estimated
44
directly.
According to Nerlove, the relation between the observed
value, (R ); and the equilibrium value ; ( R ).
in time period t ,
is given by the following difference equation:
Rt - R^
where
=
Y(Rt
- R^)
(2.9)
y is called the coefficient of adjustment.
Substitution of
Equation (2. 8) into Equation (2. 9) yields:
t
10
Y
11
Y
t
12
Y
t
(2. 10)
+ (1.0 -
Y)
^^ + v^ .
This is an equation which can be estimated statistically
and
v
lt
is a randomly7 distributed residual term.
The coeffi-
cients of the long-run relative income Equation (2. 8) may be
derived from estimates of the coefficients of N
Equation (2. jO).
coefficient of R
(y) is obtained.
B
in
By subtracting the statistically estimated
t-]
from one. the coefficient of adjustment
J
'
Then by dividing the coefficients of N
by the estimate of
is obtained.
and B
Thus,
and
y,, the long-run relative income equation
-y determines the relation among short-
run and long-run coefficients.
Based on the above procedure,
Equations (2. l), (2.2), and (2. 3) may be rewritten as:
45
R
t
= a . + a
N + a , 13
10
11
t
12
t
(2. 11)
+ a
L
t
=
13
R
, +
t-1
v
2t
i-.rt + ci_
R,
+ a,., B
20
2]
t-1
Z2
t-1
(2. 12)
+
t
CL^T
^3
30
t-1
31
+ a.,, N
+v^
24
t-l
2t
t-1
32
t-1
(2. 13)
+ a
where the
33
B
t-1
+
v
3t
a., 's are now short-run coefficients.
The model now is dynamic rather than static.
When analyses based on dynamic models are contrasted
with those based on the more traditional static approach,
we find that the former analyses explain the data better,
that the coefficients are more reasonable in sign and
magnitude, and that the calculated residuals indicate
a lesser degree of serial correlation" [ Nerlove, 1958,,
p. 301],
Adjustment of Data
There are other statistical problems in obtaining the best
estimates of the parameters in Equations (2. ll), (2. 12), and
(2. 13) .
Since cross-sectional time-series data is available.
46
it seems plausible to assume that this type of data has more information than aggregate time-series data and would provide better
estimates of the a., 's .
Brown and Nawas [ l97l] in their paper, "Improving the
Estimation and Specification of Statistical Outdoor Recreation
Demand Functions, " were concerned with using all individual
observations versus traditional "distance zone averages. " They
found,
Gains in efficiency of several hundred percent over
traditional procedures are possible. „^» Chief reason
that the traditional 'zone average1 regression analysis
gives such poor results in... [ Brown and Nawas, , 197 1, ,
P. l] ■
their example was because it greatly increased the correlation
between the explanatory variables.
Using cross-sectional time-series data, however, puts the
regression in a classification problem.
All variables in Equa-
tions (2. 11), (2. 12), and (2.13) are a function of states, i.e.,
within each state there are factors that are characteristic of
that particular state.
problem.
Thus, there is a one-way classification
A better estimate of the parameters in Equations
(2. ii), (2. 12), and (2. 13) can be obtained if the influence of
state is removed from the data.
Covariance analysis is a
statistical procedure where regressions can be studied in a
47
classification problem.
However, one of the basic assumptions
of covariance analysis is that the independent variables in the
regression equation be independent of the row effect (state effect).
As stated earlier, all variables are a function of the row effect;
therefore, covariance analysis cannot be used.
An alternative way to estimate the parameters in Equations
(2. 1 i), (2. 12), and (2. 13) is to throw away all the information
available in the cross-sectional data and use aggregate timeseries data.
The results of this alternative is presented below
for Equation (2. 12).
N
=
0.4157
(0.56)
+
(0.7379)
+
0.2464 T
(1.11)
(0.222 1)
0.6785
(1.30)
(0.5226)
+
R
t 1
"
0.2l6i B
t 1
(-1.22)
"
(0. 1768)
0.9159 N
t 1
(19.05)
"
(0.0481)
R2 «= .999 .
The first numbers in parentheses below the coefficients
are the t values and the second numbers are the standard error
of the coefficients.
The correlation matrix for the variables
in Equation (2. 12) are given below;
48
N.
N
t
t
R.
t-i
t-1
t-1
t-1
1.0
-0.7479
• 0.9822
-0.9516
0.9996
0.7864
0.6486
-0. 7517
1.0
0.9658
-0.982 1
1.0
• 0.9523
Kt-1,
B
1.0
t-1
"t-1
1.0
N
t-1
The t values are all insignificant except for the lagged
variable,
N
t *• 1
.
Given the insignificant t values and looking
at the correlation matrix, one would suspect that there is a
serious multicollinearity problem.
To be more certain that there
is a problem of multicollinearity, the following is calculated.
(X* X)..|
ii
x* xl
where (X X) is the matrix of simple correlation coefficients of
the independent variables and (X X).. is the correlation matrix,
excluding the ith variable which will be called X..
diagonal element of (X X)
r
ii
is the
corresponding to the ith variable.
If X. is orthogonal to the other independent variableSj.m X,
1
^
then
(X X)..
ii
X* X
49
and
r11 =
1.0
If X. is perfectly dependent on the other members of X,
(X X) is singular and the denominator equals zero.
then
The numer-
ator keeps the same value since it does not depend on X. .
Thus,
r
will be one, if no multicollinearity exists.
greater r
The
is than one, the more serious the problem of
multicollinearity.
Since
Var (b) = o-2 (X* X)"1
or
1
2 ii
Var (b.) = o- r
1
(X
t
where b
is the normalized regression coefficient, then as the
matrix approaches singularity, because of multicollinearity,
the variance of the regression becomes increasingly larger.
Also, since
b
= (X* Y) (XtX)"1
the regression will be biased when multicollinearity exists in
the independent variables [Farrar and Galauber, 196?] .
Thus,
the diagonal elements of the inverse simple correlation matrix are;
50
R .
t-1
B
T
5.20
79.09
t-1
,
t-1
N
t-1
28.76
29.18
The large values of the diagonal elements for B
and N
,
T
,
are evidence that a serious multicollinearity problem
does exist.
Thus, given the multicollinearity problem between the
independent variables, the separate influences of the independent
variables may be so tangled that the coefficients in the equation
are not reasonable estimates of the effect of the independent
variables on the dependent variables.
Estimating the parameters
in Equation (2. 11) with aggregate time-series data does not
provide meaningful results for testing the hypotheses of the
PE model.
Using aggregate time-series data to estimate Equation (2. 13)
gives a similar results:
B
=
t
0.8524 - 0. 1363 R
(2.33)
(-0.087i6);
*"1
(0.3663)
(1.5557)
+ 1. 1450 T
t 1
(1.7 1)
"
(0.670 1)
+
0.4337 B
(1.09)
(0.3964)
.
The correlation matrix for the variables in Equation (2. 13)
51
is given below.
B
t
1.0
B
t
R
t-1
R
t-1
T
Vl
t-1
0.6974
0.9772
0.9715
1.0
0. 6486
0.7864
1.0
0.9658
Vi
1.0
Vi
The t values of the coefficients are all insignificant except
for the constcint term which is of no interest.
However, the pro-
blem of multicollinearity exists for Equation (2.13), as it did
for Equation (2.12 ), which is seen in the very large diagonal
elements of the inverse simple correlation matrix given below.
R
T
B
5.28
28.64
43.47
t-1
t-1
t-1
Thus, estimating the parameters in Equation (2. 13) with
aggregate time-series data does not provide meajtiingful results.
Estimation of Equation (2. 11) does not give any better results
than the estimation of the coefficients of the previous two equations.
Estimation of Equation (2. 11) is given below.
52
R
=
1.5 180 - 0.0724 N
(2.50)
(-1.9 1)
(0.6082)
(0.0378)
- 0. 094 B
(«a»ai)
(0.0776)
-
0.2813 R
t 1
(-0.94)
'
(0.3003)
The correlation matrix for the variables in Equation (2. 11)
is given below.
N
R
t
R
1.0
N
t
-0.6229
1.0
B
B
R
t
t-1
0.5462
0.34 n
-0.9730
-0.7481
L0
R
0.7094
1 0
t-i
-
The diagonal elements of the inverse simple correlation
matrix are given below.
N
t
21.52
B^
t
R
,
t-1
19.10
2.31
Again, the diagonal elements indicate serious multicollinearity
problems.
From the statistical results obtained in Equations (2. n),
(2.12), and (2.13), it is obvious that aggregate time-series
data is inadequate and that a statistical method must be used
53
to adjust the cross-sectional time-series data for states so that
better estimates of the parameters in Equations (2. 11), (2.12),
and (2.13) cam be obtained.
The problem with covariance
analysis is that the state effect is estimated simultaneously with
the coefficients in the regression equation.
Since the independent
variables are related to the state effect, the regression coeffi*
cients are meaningless.
A statistical method must be used
which adjusts the data for state effect independent of estimating
the parameters in the regression equation.
This statistical
method must produce coefficients that are unbiased estimates
of the parameters in Equations (2. 11), (2.12,), and (2.13).
For later convenience. Equations (2. 11), (2.12), and
(2.13) will be rewritten as follows :
X
^ = a,_ + a
X0 ^ + a10 X0
1st
10
ii
2st
12
3st
(2.14)
+
a
13
X
ist-1
X^
= a^ + cu
2st
20
2
X
+
€
ist
lst-1
+ a.^ X
22
3st-l
(2.15)
+ a
X.
+ a„ „ X_
+ €„
23
4st-l
24
2st-l
2st
54
X
=
3st
a
+
30
a
3i
X
ist=l
+
a
32
X
4st-1
(2.16)
+ a.
where
X
33
3st-l
s =
1, 2, . . ., 48 states
t
1, 2, ... , 14 years
X
=
1
N
X
=
B
X
=
T
2
3
4
3st
—' R
=
X
+ e
To adjust the data, the following method will be used.
Each observation of each variable in the data has a relationship
to the state in which the observation occurred.
The relationship
of each observation to the state will be defined as follows:
X
=
kst
X
+
k
P
ks
+
\st
(2. 17)
where X,
is the observation of the kth variable in the sth
kst
—
—
state in the tth year,
(3
X.
is the overall mean of the kth variable,
is the state contribution of the kth variable,
the error term.
and u
is
The adjusted data may be written as:
U
kst
=
X
kst " \ • :
"
B
ks
(2. 18)
55
where
B,
KS
= X,
ks •
- X
k ••
Adjusting the data in this way will allow better estimates to be
made of the parameters that describe the relationship between
the variables in the system.
(2. l6)
Equations (2. 14), (2. 15), and
maybe rewritten as:
U
1st
=
vY
U,
+
2st
ll
vY , U,
l2
3st
(2. 19)
+
U
2st
=
vY „ U
+
13
lst-l
U
X2i
lst-1
+
v
1st
U
\Z
3st-1
(2.20)
Y
2-
U
3st *
U
^3 1
Y
4st-l
lst-1
24
+
^2
2st-l
2st
U^t.i
(2.2 1)
Y
33
where
(2. 14), (2. 15).
It should be noted that Equations (2. 19), (2. 20), and
satisfy the same assumptions as Equations (2. 14), (2. 15),
and (2. l6)
other.
3st
v.. is an estimate of a., in Equations
and (2. l6) .
(2. 2 l)
3st-l
since all U
's were solved independently of each
Equations (2. 19), (2. 20), and (2. 2 l) are a recursive
56
model as were Equations (2. 14), (2. 15), and (2. 16),
estimated by two-stage least squares.
and may be
Notice also that there
are no constant terms in Equations (2. 19), (2.20), and (2.2l)
since theoretically they should pass through the origin.
It must now be shown that
a.. .
To show that
v.. is an unbiased estimate of
v.. is an unbiased estimate of a...
a simple
example of one dependent and one independent variable is used.
Suppose we have the following relationship:
Y =
f (S, X)
(2.22)
X = g (S)
(2.23)
where Y is the dependent variable,
variable, and S is the state effect.
X is the independent
The structural equation is;
Y . = a. + a X
+
st
0
1
st
A meaningful estimate of a
X are related to S.
€
st
•
(2.24)
cannot be obtained since Y and
The observations of Y and X are defined
as follows:
Yit = \.+p+ii
K
rJ.
st
1
is
st
(2.25)
X
(2 26)
st =
X
2
+
hs
+
V
st
-
57
where X.
and \- are the overall mean,
12
and rB . and (3^
r
is
2s
are the state contribution of the observation Y
respectively.
c
X
,
u.r
st
and v
respectively.
st
and X
are the error terms of Y
st
st
st
.
and
Using B to indicate estimates, they are
defined as follows!
B
=
Y
B.
=
Zs
X
Is
s •
s .
Each observation of Y
Y
X
St
st
= Y
=X
- Y
-
X
and X may be written as.
+(Y
• •
+ (X
• •
-Y
S"«
-X
s •
• •
• •
) + U
) + V
st
st
or
U
V
st
st
= Y
st
- Y
(2.27)
s •
= X ,- X
st
s •
.
(2.28)
Thus, the data has been corrected for the state effect and
Equation (2. 24) may be rewritten as
U
st
= y
Yn +
0
v
Y
l
V
st
+
6
st
(2.29)
'
58
It must now be shown that y
a
.
is an unbiased estimate of
Estimating Equation (2.29) by ordinary least-squares gives
A
=
1
v
5
Jt
u
at*
v
^
St
st+
t
(2>30)
sti.
the small letters imply mean corrected.
Equation (2. 30) can be rewritten as.,
/\
_
Yl
"
V
S
st
U
st4.
U
st4.
Z
st+ st*
2
st
st
st*
2
st
st
V
v
at
st
st
^.JI;
since
S. v
st st
=0
where
v
w
st
st
^2
v
^
st
st*
Notice that since the mean of a residual is zero, that
V
w
St
Z
st
V
st4.
v2
st
2
st
st♦
V2
st
59
and
E (w
st
) = 0
since
E (v ) = 0
st
Also,
st
st
2 W
st4.
V
4.
st
:::
st
st
2
st
2.
W .
st
t
1
„
st
V
st
2
st
=1
st
Now, substitute Equation (2.27) into Equation (2.31).
0
'I
=
S
st
w
=
Sw
st
st
st
(Y
- Y
st
s •
)
Y-Sw
st
st
st
Y
s •
•
(? ■\?\
^- ^
Substitute Equation (2.24) into Equation (2. 32).
v
Y
l
=
2w
(a„ + aX
+e
)
st
st x 0
1
st
st '
2
st
a
1
w
st
Y
,
s .
2 . w ^ X .
St :
St
St
.+ :2
:
St
:
w
. € .
St ■
st
(2. 33)
2
st
w
st
Y
s
Substitute Equation (2.28) into Equation (2. 33).
60
0Y
l
= a
+
= a
2
st
1
2:
st
w
w
+ a
1
(X
st
€
st
st
2.
st
1
+ V
s •
st
- 2
st
w .
st
X
)
w i' X
st
s
+2.
st
s •
w
st
e
st
(2.34)
+
2
st
w
st
Y
s •
.
The expected value of Equation (2. 34) is
E (yY ) = a
l
1
since
E (w
and w
and e
SL
st
) = 0
are independent by assunaption.
Sir
Therefore,
y
is
an
unbiased estimate of a
statistical method described above,
.
Using the
the data can be adjusted for
state effect and estimates can be made of the parameters in
Equations (2. 14),
(2.15), and (2. 16).
In summary, the model developed in this chapter is a recursive model and will be estimated by two-stage least-squares.
A lagged variable was introduced into each equation which changed
it from a static model to a dynamic model.
The dynamic model
allowed estimation of both short- and long-run coefficients.
61
Aggregate time-series data was inadequate for the estimation of
the coefficients since it contained serious multicoUinearity problems.
Therefore, cross-sectional time-series data will be used which will
be adjusted for state effect.
62
III.
STATISTICAL RESULTS AND TESTING
THE HYPOTHESES
Statistical Results
The period of analysis for this study is from 1957 to 1970
since for this period a complete set of cross-sectional time-series
data was available.
14
The results of the empirical estimation of
the coefficients in the dynamic model of Equations (2. 11), (2. 12),
and (2„ 13) are presented below.
15
Relative income equation (short-run)
R
= 0.3231
1
R
2
- 0.0064 N + 0.0293 B + 0.2278 R
(-1.74)*
*
(2.30)** '
(5.58)*****
f3'1*
= 0. 1235
14
A comparison of the cross-sectional time-series data and
the aggregate time-series data is made in Appendhc^D.
15
The static model was also estimated and the results are
presented in Appendix E.
*
**=
***
****
*****
Significant
Significant
Significant
Significant
Significant
at
at
at
at
at
5. 0 percent level.
2. 5 percent level.
1.0 percent level.
0. 5 percent level.
0. 05 percent level.
63
Out-migration equation (short-run)
N
= 0.5987 + 0.2646 R
(2.81j*****
- 0.1039 B
(-4. eo)*****'"
(3.2)
- 0. 0142 T
(-0.67)
*"
+ 0.9092 N
(141.48)*****"
R2 = 0.9826
Relative labor productivity equation (short-run)
B
= 1.0621 -
1.3778 R
(-8. 46)*****"
+ 0.1867 T
(5. 07)***** ~
(3.3)
- 0.8821 B
(26. 00)*****
R2 = 0. 6305
The numbers in the parentheses below the coefficients are
the "student - t" values.
The constant terms in Equations (3. 1),
(3. 2), and (3. 3) were calculated using the means from the aggregate
time-series data.
One reason for calculating the constant terms
in this manner is that fitting the regression equation by the adjusted
cross-sectional time-series data fbrces the equation through the
origin.
Another reason for calculating the constant terms in this
manner is that the overall average of the cross-sectional time-,
series data does not necessarily equal the average of the aggregate
64
time-series data.
All coefficients in Equations (3. 1), (3. 2), and (3. 3) are different from zero at least at the 5 percent level in a one-tail
"student - t" test, except for the coefficient on relative property
taxes
(T
) in the out-migration equation [ Equation (3. 2)] which
is not significantly different from zero at any acceptable level.
The coefficient on the lagged
terms
55
R
,,
t-l
N
, , and B
t-1
t-1
in Equations (3. 1), (3. 2), and (3. 3), respectively, are positive and
significantly greater than zero at the 0. 05 percent level.
suggests that the time it takes for R ,
N ,
and B
This
to make full
adjustment for a changed condition exceeds the interval of observation.
For example,
consider Equation (3. 1).
Suppose there was a
A
change in relative labor productivity (B ) in the tth time period, all
other things held constant.
The full effect on relative income (R )
is not all felt in the tth production period (interval length of one
year) but only after several production periods have elapsed.
Thus,
the coefficients in the three equations above are the partial adjustment that takes place in the dependent variable in the first production period after a changed condition.
16
A one-tail test is used since all the coefficients are hypothesized to be either positive or negative.
65
The adjustment coefficients for calculating the long-run
coefficients of the variables in Equations (3. 1), (3. 2), and (3. 3)
are
0.7722,
0.0908,
and 0.1179, respectively.
17
All the adjust-
ment coefficients are between zero and one as expected.
If any of
the adjustment coefficients had been equal to one, then the long-run
coefficients would be the same as the short-run coefficients.
If
any of the adjustment coefficients had been zero, then the long-run
coefficients would be equal to infinity.
The relative income Equation (3. 1) has a large adjustment
coefficient, implying a short period of adjustment; whereas, the
out-migration Equation (3. 2) and the relative labor productivity
Equation (3. 3) have small adjustment coefficients, implying a
fairly long period of adjustment.
The full adjustment takes an
infinite length of time, however, anything less than full adjustment,
say 95 percent, takes place in a finite length of time.
the number of periods for 95
If M is
percent of the adjustnaent to occur,
then M may be determined by the equation below .
(1.0.- y)M =
(1.0
-
.95)
or
17
See Chapter II, page 44, for the calculation of the longrun adjustment coefficients and the long-run coefficients.
66
M
where
= ttlfr-y
y is the adjustment coefficient.
In considering the period required for nearlyfull adjustment, one should bear in mind two
factors: (l) The figure 95 percent is somewhat
arbitrary and so is the dynamic model under consideration. Thus the median adjustment period
(i. e. , the period under which 50 percent of the
adjustment occurs) may be a reasonably accurate
figure, whereas, the time required for 95 percent
of the adjustment to take place may be inaccurate,
... [ Nerlove, 1958a, p. 874],
even though the estimated long-run coefficients are correct.
. . . (2) Furthermore, there are strong reasons for
suspecting that the coefficients. . . of adjustment are
subject to a greater extent than other parameters
to what is known as specification bias. . . . That is,
the estimated coefficients of adjustment are partic- 1R
ularly senstivie to the omission of relevant variables
[Nerlove, l958a, p. 87 4].
The values of M for each of the three equations with 50 and 95
percent of adjustment is given in Table 3. 1.
Adjustment is very rapid for R
Equation (3. 1).
in the relative income
Over 50 percent of the adjustment occurs in one
year and over 95 percent of the adjustment occurs in 3 years.
This rapid adjustment seems logical since income in time period
18
For a discussion of omission of relevant variables, see
the discussion between Brandow [1958] and Nerlove [ 1958b] .
67
Table 3. 1.
Period of adjustment for relative income, R ; outmigration, N ; and relative labor productivity, B .
Equation
Length of Period of Adjustment, M
50%
95%
Rt
1
3
N
8
32
B
6
24
t should depend on factors of production during that time period and
not previous time periods.
N
Adjustment of 50 and 95 percent for
in the out-migration Equation (3.2) takes 8 and 32 years, respec-
tively. This slow adjustment in N
seems logical since it usually
involves some farm operators changing their way of life by moving
from the farm sector to the nonfarm sector and/or it involves an
increase in total population.
for B
The adjustment of 50 and 95 percent
in the relative labor productivity Equation (3. 3) takes 6
and 24 years, respectively.
This slow adjustment process is
expected since it usually involves changing the techniques of production and could require new, large capital investments.
The long-run equations are given below.
68
Relative income equation (long-run)
R
= 0.4184 - 0.0083 N + 0.0379 B
t
*
(-1.73)*
(2.38)*** t
(
'
Out-migration equation (long-run)
N
= 6.5936 + 2.9141 R
*
(2.87)****
-
1.1443 B
(-5.26)*****
(3.6)
- 0. 1564 T
(-0.67)
Relative labor productivity equation (long-run)
B
= 9.0085 -
11.6862 R
(4.32.)*****
+ 1.5835 T
(3,41)*****
As implied in the above discussion, there is little difference
between the short- and long-run coefficients of the variables in
the relative income Equations (3. 1), and (3. 5).
There is, however,
a substantial difference between the short- and long-run coefficients
of the variables in the out-migration Equations (3. 2) and (3. 6) and
the relative labor productivity Equations (3. 3) and (3. 7).
A "t"
test 19 was made on the long-run coefficients and the results were
19 The following was used to calculate the variance of
69
the same as those for the short-run coefficients.
All coefficients
are statistically different from zero at least at the 5. 0 percent
level in a one-tail test, except for the coefficient on T
in the
out-migration Equation (3. 6).
Testing the Hypotheses
Each of the seven hypotheses stated in Chapter I will be
restated along with a brief statement, based on statistical evidence,
as to whether the hypothesis is or is not rejected.
Ho: #1
Out-migration has a significant effect on increasing the net
income of the farm sector relative to the nonfarm sector.
each of the long-run coefficients.
a = f (pi, P2.-.-(3n)
n
var (a) = 2
k=l
2
a f
(-£-£- )
9
P
k=l
var ((3, )
k
k
"k
r
j
J
[Gujarati, 197 0]
That is,
70
the coefficients on N
in both the short- and long-run relative
income Equations (3. 1) and (3. 5), respectively, should be negative.
The short-run coefficient is
is
-0. 0083.
-0. 0064 and the long-run coefficient
Both are significantly less than zero at the 5. 0 percent
level and, therefore, the d^tta strongly support the hull hypothesis.
Ho: #2
An increase in the labor productivity of the farm sector
relative to the nonfarm sector causes a significant decline in the
net income of the farm sector.
That is, the coefficient on B
in
both the short- and long-run relative income Equations (3. 1) and
(3. 5), respectively, should be negative.
The short-run coefficient
is 0. 0293 and the long-run coefficient is 0. 0379.
Both are sig-
nificantly greater than zero at the 2. 5 percent level and., therefore, the data stronly contradict the null hypothesis.
Since both are significantly greater than zero, the implied
alternative hypothesis, an increase in relative labor productivity
increases relative income, is strongly supported by the data.
Ho: #3
A decrease in the net income per farm household relative
to the net income per nonfarm household has a significant effect
on causing out-migration.
That is, the coefficients on R
.
in
71
both short- and long-run out-migration Equations (3.2) and (3. 6),
respectively, should be positive.
The short-run coefficient is
0. 2646 and the long-run coefficient is 2. 9141.
Both are sig-
nificantly greater than zero at the 0. 5 percent level and, therefore,
the data strongly support the null hypothesis.
H&j. #4
An increase in the labor productivity of the farm sector relative to the nonfarm sector has a significant effect on causing outmigration.
That is, the coefficients on B
in both the short- and
long-run out-migration Equations (3.2) and (3. 6), respectively,
should be negative.
The short-run coefficient is
long-run coefficient is
-1. 1443.
-0.1039 and the
Both are significantly less than
zero at the 0. 05 percent level and, therefore, the data strongly
support the null hypothesis.
Ho: #5
An increase in the property taxes paid per farm household
relative to the property taxes paid per nonfarm household has a
significant effect on causing out-migration.
on T
in the short- and long-run out-migration Equations (3. 2)
and (3. 6), respectively, should be negative.
ficient is
That is, the coefficients
The short-run coef-
-0. 0142 and the long-run coefficient is
-0. 1564.
72
Although both have the correct sign, both are only significantly less
than 50'
jpercent
level, therefore the data are rather inclusive with
regard to the null hypothesis.
Hence, based on the data used in this
study, it is concluded that relative property taxes have no significant,
direct effect on out-migration.
Ho: #6
A decrease in the net income per farm household relative to
the net income per nonfarm household will increase the labor productivity of the fam sector relative to the nonfarm sector.
is, the coefficients on R
That
in both the short- and long-run
Equations (3. 3) and (3. 7), respectively, should be negative.
The
short-run coefficient is -1. 3778 and the long-run coefficient is
-11. 6862.
Both are significantly less than zero at the 0. 05 percent
level and, therefore, the data strongly support the null hypothesis.
Ho: #7
An increase in the property taxes paid per farm household
relative to the property taxes paid per nonfarm household causes
labor productivity of the farm sector relative to the nonfarm sector
to increase.
That is, the coefficients on T
in both the short-
and long-run Equations (3, 3) and (3. 7), respectively, should be
positive.
The short-run coefficient is 0. 1867 and long-run
73
coefficient is
1. 5835.
Both are significantly greater than zero
at the 0. 05 percent level and, therefore, the data strongly support
the null hypothesis.
74
IV.
IMPLICATION OF STATISTICAL RESULTS
The PE Model Versus the
Statistical Results
The PE model, illustrated in Figure 1. 1 and 1.2, which
describes the situation that is believed to exist for the agriculture
sector in the United States, is not supported by the statistical
results.
In the PE model, as illustrated in Figure 1. 1, an increase
in relative labor productivity (B ) moved farm operators to a lower
iso-relative income curve.
For farm operators to move back to
the original iso-relative income position they had before B
increased, out-migration must occur.
The statistical results reject null Ho: #2 and imply that the
alternative hypothesis, an increase in relative labor productivity
(B ) increases relative income (R ), should be accepted.
!
the long-run Equation (3.5),
with R
A
B
A
was solved for in terms of N
held constant at three different levels (0.50,
and 0. 55).
0.525,
A
The results are plotted in Figure 4. 1 with B
the vertical axis and N
on the horizontal axis.
represents a constant value of R .
on
Each curve
The distance between the two
horizontal lines represents the observed range of B
vertical lines represent the observed range of N
1958-1970 period.
Using
and the two
during the
In Figure 4. 1, if out-migration occurs, farm
75
Figure 4. 1.
Iso-relative income curves, statistical results.
76
operators move to a higher iso-relative income curve, as they
did in Figure 1. 1.
A
In Figure 4. 1, if B
is increased, farm
operators move to a higher iso-relative income position.
In
Figure 1. 1, on the other hand, they move to a lower iso-relative
income position.
Thus, there is an inconsistency between the
PE model and the statistical results.
The logical place to check for this inconsistency is in the
statistical model to see if it was correctly specified.
A misspeci-
fied statistical model may cause a serious bias in the estimated
regression coefficients.
Comparing the PE model with the
statistical model, it was found that in the PE model relative labor
productivity (B
PE
) was defined as the ratio of the labor coefficients
of the farm and nonfarm production functions ( B
PE
?f
=
).
2n
g
In the statistical model, relative labor productivity (B ) was
defined as the ratio of labor productivity of the farm sector to the
labor productivity of the nonfarm sector.
Labor productivity, in
either sector, changes because of a technological change or because
of an increase in capital (labor held constant).
a technological change; whereas,
B
s
B
PE
reflects only
may be a result of a
technological change and a capital change.
Therefore,
using the
simplest functional form possible, the relationship between B
and B
may be written as
PE
77
s
B
where K
= a B
PE
+ b K
(4. 1)
is the ratio of capital in the farm sector to capital in
the nonfarm sector, and
PE model,
K
a and b are positive constants.
was assumed to be constant and, therefore,
Equation (4. 1) would be zero, leaving B
s
= B
PE
In the
b in
with a = 1.0.
Thus, in the relative income equation
R
t
=
a
l0
+
a
ll
N
t
+
a
PE
B
l2
t
+
>V
(4 2)
-
This is the relationship that was thought to have been estimated
since with constant K ,
B
In the real world,
Equation (4. 1) for B
s
K
PE
= B
PE
is very possibly not constant.
Solving
gives
and substituting Equation (4. 3) into Equation (4. 2) gives
R=a
+a
t
10
11
N+a
(
t
12 v
1
' 0 BS
a
t
a
R
= a, „ + a, , N +
t
10
11
t
— BS
at
- — K ) + K
u.
a
t'
lt
a - b
- ——— K + a,
a
t
^lt
which is the relationship that should have been estimated.
(4.
4)
v
'
The
omission of the relevant variable (K ) from the statistical model
may have caused serious bias in the estimated regression
,.,
78
coefficients of the model.
To determine the bias,
Equations
(4. 2) and (4. 4) are written in matrix notion as
Y = X a +
(4.5)
|JL
and
Y = Xa
+ Z y + (x ,
(4.6)
respectively, where Y is a column matrix of the dependent variable (R ) ;
and B
g
) ;
X is a matrix of the two independent variables (N
and Z is the column matrix of K .
Using ordinary
least-squares estimates of regression coefficients in Equation (4. 5)
yields
a* = (X X)"1 X Y .
{47)
Substituting Equation (4. 6) into Equation (4.7) for Y gives
a* = (XX)"1 X X a + (X X)"1 X Z y + (X X)"1 X ^
a + (X X)"1 X Z y + (X X)"1 X
1
The bias in a* is (X X)
- 1
'
X Z V
According to Brown [ 1968]
which a* will be unbiased.
since
\x
E (^ = 0.
there are only two cases in
The first case is if
y equals zero,
i
and the second case is if all the elements of X Z
On the basis of the above discussion, since K
are zero.
is probably not
'
79
constant,
b is not equal to zero;
therefore,
V is not considered
i
to be zero.
Brown shows that X Z will be zero if Z , the omitted
variable, has zero correlation with all the included X variables.
Unfortunately, labor productivity and capital are probably highly
i
correlated causing X Z not to be zero.
The direction of the bias would have to be upward on the
coefficient of relative labor productivity in the relative income
equation for the statistical results to agree with the PE model.
This type of bias is explained in Figure 4. 2.
The vertical axis
measures relative (R ) and the horizontal axis measures relative
labor productivity (B ).
The negative sloped curves represent the
true relationship between R
(K
K , and K ).
and B
at constant levels of capital
But, since capital has been increasing over time,
the curve a b represents the observed relationship between R
and B .
With capital omitted from the relative income equation,
the statistical results are an estimation of the curve a b.
Relative Property Taxes and Rejection of Ho: #2
The rejection of Ho: #2 has affected the relationship of
relative property taxes with respect to some of the variables in
the model.
In the PE model, an increase in relative property
taxes puts farm operators at a lower relative income position.
When farm operators attempt to regain the relative income position
80
R.
B
Figure 4.2.
The upward bias in relative labor
productivity caused from the omission
of the capital variable.
81
they had by increasing their labor productivity, they become worse
off;
hence, out-migration must occur.
From the statistical re-
sults, however, an increase in relative labor productivity would
cause their relative income to increase (Figure 4. l) which would
reduce the need for out-migration.
This reduced need for out-
migration may be the reason Ho: #5 was rejected.
The rejection of Ho: #5 implies that the direct relationship
between relative property taxes and out-migration is insignificant.
However, there is an indirect relationship that exists.
An increase
in relative property taxes increases relative labor productivity,
which in the next time period causes out-migration.
This may be
expressed as
a
N
8 N
8
T
8B
t-2
•B.-l
t-l
9
<
0
T
t-2
and this value is equal to -1. 812 for the long-run equations.
Thus,
a unit increase in relative property taxes causes a -1. 8121 unit
change in N
in the long-run.
20
Relative property taxes have a direct additive relationship
with relative income and also an indirect relationship with it.
20
The value -1. 8121 is not the total long-run, indirect
effect since the model is a recursive model.
82
The indirect relationship may be expressed as
a R
9 R
a N
a R
a B
9 T
9
9
9
9
t-1
N
t
T
t-1
B
t
T
t-1
In the PE model, the first term on the right side is positive
9 R
t
( —r=- < 0 ,
8 Nt
(
8
R
Q
p
t
9
N
3
T
t
■
< 0 ) and the second term is negative
t-1
9
8B
< 0,
,
^—
8
T
negative, or zero.
> 0).
t-1
Thus,
R
—=
9
t
may be positive,
t-1
In the statistical results, however, both terms
on the right side of the above equation are positive (since
a Rt
> 0 ) I
9
therefore, relative property taxes have an indirect
B
t
a Rt
positive effect on relative income ( ——
9 T
t-1
ilong
^
- run).
= 0. 0713 in the
21
From the above discussion, the statistical results suggest
that transfer payments, in the form of relative property taxes,
are an important economic factor affecting the farm sector, even
though relative property taxes have a different relationship to
some of the variables in the statistical results than described
by the PE model.
21
The value 0. 0713 is not the total long-run, indirect
effect since the model is a recursive model.
83
Short-run Versus Long-run Equations
There are several ways the short- and long-run equations
can be compared.
First, the long-run coefficients in the out-
migration Equation (4. 6) are approximately 11 times larger than
the short-run coefficients in Equation (4. 2).
The long-run co-
efficients in the relative labor productivity Equation (4.7) are
approximately 8. 5 times larger than the short-run coefficients
in Equation (4. 3).
This is illustrated in Figures 4. 3 through
4. 6 by the difference in the steepness of the short-run curves
(solid lines) and the long-run curves (dash lines).
The large
difference in the slope of the short- and long-run curves reflects
the slowness with which out-migration and relative labor productivity adjust, as is evident in Table 3. 1.
It takes at least eight
years for out-migration to make a 50 percent adjustment due to
a change in one of the independent variables;
whereas, it takes
relative labor productivity six years.
Second, suppose there was a one unit change in one of the
independent variables, all other variables held constant, then one
would expect that in the first year the change in the dependent
variable would be equal to the value of the short-run coefficient.
However, over time, the total change in the dependent variable
would approach the value of the long-run coefficient.
For example,
84
0)
>
>
u
tn
N
CO
t
0
■8
CJ
tf
ao
g
i
9 4
15
s >
JMaocimum
observed value
of N
8 .-
7 --
Short-run
6 --
.Minimum
observed value
of N.
4 --
__ .. —— Long-run
3 --
.4
Figure 4. 3.
.5
R
t-1
Out-migration as a function of relative income
with B
= 4.08.
85
T3
(0
>
>
u
<u
U
(0
CO
N.
0
9 --
•a 3
XI
O
ffl
C
ffl
(V)
>
_Maximum observed
value of N
8 .-
7 --
Short-run
6 ,.
5 ..
Minimum observed
"value of N.
-V-
\
4 --
\
\
\
s.
v
N
\
3 ..
N
\
\
S
\
N
Long-run
t-1
Figure 4.4.
Out-migration as ei_function of relative_labor
productivity with R . = .518 and T
= 1.46.
86
B,
\
\
■g
\\
>
6 ..
u
5% -r ^
0
4->
e 0
'Ss
"8>
\x
*
sIt
0
\
S 0
XI s
^
\
>
\
A
\
5 .:
>
Maximum
observed
value of B
\
\
■
+J
\
\
\
t
\
\
\
\
1
—-
\ Long-run
Short-run
4 ..-
Minimum
observed
value of B^
t
1
1
.4
.5
Figure 4.5.
_l
.6
Relative labor productivity as a function of
relative income with T
= 1.46.
't-1
R
t-1
87
B.
TJ
;>
>
^
S2 -*i
6 -
53 /
» ^i
.or
§0
/
/
5•J
s
T*
/
/
/
*y
/
/
—■
/
Long-run
a0
/
—-—-
Maximum observed
value of 13
,
/
Short-run
4 _ i-
Miniraum observed
value of B
t
1
3
—1
1.5
Figure 4. 6.
"t-1
Relative labor productivity as a function of
518.
relative property taxes with R
t-1
88
in the out-migration equation, a one unit change in B
cause a change of -0. 1039 in N
the total change in N
in the first year, but over time
would approach -1. 1443.
If it could be
expected that there would be a one unit change in B
N
would
each year,
would approach a change of -1.443 units each year, all other
variables held constant.
The last interesting feature to be discussed is the vertical
distance between the short- and long-run curves in Figures 4. 3
through 4. 6.
The long-run curves in these four figures lie almost
completely out of the range of the observed data.
In Figures 4. 3
and 4. 4, the short-run curves of the out-migration Equation (4. 2)
pass through the mean value 6.21 for N ;
whereas, the long-run
curves pass through the value 3. 21 for N
(with p.
B
= 4. 08 ,
and T
=
1.46 ) .
= . 518 ,
This implies that if the
independent variables are held constant at their mean values,
out-migration (N ) would approach the long-run equilibrium value
3. 21.
In Figures 4. 5 and 4. 6,
the short-run curves of the rela-
tive labor productivity Equation (4. 3) pass through the mean value
of 4.22 for B ,'
whereas, the long-run curves pass through
the value 5.27 for B
(with R
= .518 and T
= 1.46).
This implies that if the independent variables are held constant
at their mean values, relative labor productivity (B ) would
approach the long-run equilibrium value 5.27.
89
This last comparison of short- and long-run equations suggests
that the agricultural sector still has a significant amount of adjusting
to do.
Using the short-run equations and the 1970 values of all of the
variables in the statistical model, projections are made forward to
1980 in Tables 4. 1 and 4. 2.
These projections to 1980 are conditional
on the assumption that the relationships between the independent
variables which existed during the study period continue to exist in
the future.
In Table 4. 1, relative property taxes are held constant at
the 1970 value (T . = 1. 99) and in Table 4. 2, relative property taxes
/
22
are increased 0. 06 units each year.
In Table 4. 1, relative income
would seem to stabilize around 0.5 92 which would be an 11 percent
increase from 1970.
Out-migration would decline to 3. 02 by.1980
which would be a 35 percent decrease.
Relative labor productivity
would stabilize at 5.25 which would be a 4 percent increase over 1970.
In Table 4. 2, when relative property taxes are increased in equal
increments each year, relative labor productivity would not stabilize
but would continue to change.
These unit increment increases in
relative property taxes would cause relative labor productivity to
increase 11 percent by 1980, instead of 4 percent.
The effect
of relative property taxes on out-migration and relative income
would be felt indirectly through relative labor productivity.
The
proportion of households in agriculture would decrease by 38 percent
22
The value 0. 06 was the average yearly increase in relative
property taxes during the period 1957-1970.
90
Table 4. 1.
Year
Projection to 1980 of relative income, R ; outmigration, N ; and relative labor productivity,
B , with relative property taxes, T , held constant.
R
t
N
B
t
t
T
t
1970
.532
4. 65
5. 04
1.99
1971
.567
4.42
5. 15
1.99
1972
.578
4.20
5.20
1.99
1973
.582
4.00
5.22
1.99
1974
.584
3.82
5.23
1.99
1975
.586
3.64
5.24
1.99
1976
.588
3.49
5.25
1.99
1977
.589
3.35
5.25
1.99
1978
.590
3.23.,
5.25
1.99
1979
.591
3. 12
5.25
1.99
1980
. 592
3. 02
5.25
1.99
91
Table 4.2.
Projection to 1980 of relative income, R ; outmigration, N ; and relative labor productivity,
B , with relative property taxes, T , increased
0. 06 units each year.
Year
R
N
1970
.532
4. 65
5. 04
1. 99
1971
. 567
4. 42
5. 15
2. 05
1972
. 578
4. 20
5. 21
2. 11
1973
.583
4. 00
5. 26
2. 17
1974
.587
3. 81
5. 30
2. 23
1975
.592
3. 63
5. 34
2. 29
1976
.593
3. 46
5. 38
2. 35
1977
.596
3. 31
5. 43
2. 41
1978
.599
3. 17
5. 48
2. 47
1979
.602
3. 03
5. 53
2. 53
1980
.605
2. 90
5. 58
2. 69
t
t
B
t
T
t
92
instead of 35 percent by 1980, and relative income would increase
by 14 percent instead of 11 percent by 1980 .
The important thing to note here is that the statistical results
suggest that transfer payments in the form of relative property
taxes are an important policy variable.
The statistical results also
suggest that before an equilibrium can exist in agriculture, relative property taxes must be constant.
At present, property taxes
are assessed and collected at the local level and, therefore, are
not what could be considered a controlled policy variable.
If, in
the future, the federal or state governments should take over property tax collection, then it could become an important policy tool.
Conclusion
One of the important empirical questions to be answered by
this study pertains to the cure for low returns in agriculture.
Since Ho: #1 could not be rejected at the 5. 0 percent level, does
this imply that out-migration will be the solution to low returns in
agriculture?
This is very doubtful.
Consider Figure 4.7 .
The vertical axis measures relative income (R ) and the horizontal
A
axis measures out-migration (N ).
short-run Equation (4. 1) with B
mean values.
(4.5) with B
The solid line represents the
and R
substituted for their
The dashed line represents the long-run Equation
substituted for its mean value.
The flatness of both
93
R.
'V
0)
(U
7 --
>
u
<u
>
to
w
S>
4
o
<u
>
6 --
Maximum
observed
value of R
Short-run
5 --
Long-run
Minimum
observed
value of R
N
Figure 4.7,
Relative income as a function of out-migration
with B = 4.22.
94
the short- and long-run curves implies that as out-migration
occurs, relative income increases very little even though the
increase is statistically significant.
Another way of looking at
this is by calculating the elasticity of relative income with respect
to out-migration.
The short- and long-run elasticities are -0. 0757
and -0. 0978, respectively.
23
During the last 13-year period,
43 percent decrease in N .
efficient,
24
1958-1970, there was a
Using the long-run elasticity co-
this would imply approximately a 4. 2 percent in-
crease in the relative income over this 13-year period, all other
things held constant.
It is doubtful that a 4. 2 percent increase
in relative income from an average relative income of 0.525 would
make a farm operator feel that he had gained much since he may be
still far below an equitable relative income distribution.
Thus,
although out-migration has been statistically significant in increasing relative income during the 195 8-1970 period, it has not been a
cure for low returns in agriculture during this period.
The rate
at which out-migration has occurred-has done little more than allow
23
Both short- and long-run elasticities were calculated at
the mean value of N .
24
The reason for using the long-run elasticity is that R
makes 95 percent adjustment within 3 production periods.
95
net income of farm operators to keep pace with the net income of
the nonfarm households.
Since Ho: #2 was rejected and the alternative hypothesis,
an increase in relative labor productivity increases relative income,
has been accepted, does this imply that technology could be the
solution to low income in agriculture?
as it was for out-migration.
Figure 4. 8 is set up in the same
manner as Figure 4.7, except with B
and N
The analysis is the same
equal to its mean value.
on the horizontal axis
The two curves in Figure 4. 8
are relatively flat as were those in Figure 4. 7.
The short- and
long-run elasticities of relative income with respect to relative
labor productivity are 0.2355 and 0.3036, respectively.
During
the time period 1958-1970, there was a 44 percent increase in
relative labor productivity.
Using the long-run elasticity
coefficient, this would imply an increase in relative income of
approximately 13. 6 percent over the 13-year period, all other
variables held constant.
A 13. 6 percent increase in relative
income from an average relative income of 0. 525 would not
make the farm operator feel he had gained much since he is still
far below a relative income position of 1.0.
Thus, increases
in relative labor productivity have not been a cure for low returns
in agriculture.
96
R.
TD
0)
7--
0)
>
>
1-4
<u
U
<u
m
<n
X>
o
fi
m
d >!
o CQ
a
1—1
>
s >
Maximum observed
^v-alue
, „ ^> Long-run
e
of R ^
Short-run
5 --
Mininaum observed
value of R,
B
Figure 4. 8.
Relative income as a function of relative labor
productivity with N = 6.21.
97
BIBLIOGRAPHY
Bauer, Larry L
1969. The effect of technology on the farm labor
market. American Journal of Agriculture Economics 51:
605-618.
Bishop, C. E., ed. 1967. Farm labor in the United States.
York, Columbia University Press. 143 p.
New
Boyne, David A. 1965. Changes in the income distribution in
agriculture. Journal of Farm Econonaics 47: 1213-1224.
Brandow, G. E. 1958. A note on the Nerlove estimate of supply
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Brandow, G. E. 1962. In search of principles of farm policy.
Journal of Farm Economics 44: 1145-1155.
Brown, William G. 1968. Effect of omitting relevant variables
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conducted under Project 845. (unpublished)
Brown, W. G. and F. Nawas. 1971. Improving the estimation
and specification of outdoor recreation demand functions.
Oregon Agriculture Experiment Station Project 85 0, August,
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Committee for Economic Development. 1945. Agriculture in an
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Committee for Economic Development. 1962. An adaptive program
for agriculture. A Statement by the Research E^nd Policy
Committee. 74 p.
Farm Income State Estimates, 1949-1970.
218 Supplement, August, 1971.
1971.
USDA-ERS-FIS,
Farrar, Donald E. and Robert R. Galauber. 1967. Multicollinearity in regression: the problem revisited. The Review
of Economics and Statistics 44:92-107.
Fox, Karl A. 1968. Intermediate economic statistics.
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New York,
Gallaway, Lowell E. 1967. Mobility of hired agriculture labor:
1957-1960. Journal of Farm Economics 49:32r-52.
98
Gujarati, Damodar. 1970. Use of dummy variables in testing for
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a generalization. The American Statistician 24: 18-22.
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379-391.
Hathaway, D. E. and Brian B. Perkins. 1968. Farm labor mobility, migration, and income distribution. American Journal
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ed. by Earl O. Heady, Howard G. Oiesslin, Harold R.
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Johnston, J. I960.
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Econometric methods.
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Malinvaud, E. 1970. Statistical methods of econometrics, 2nd ed.
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99
Nikolitch, Radoje. 1962. Family labor and technological advance
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APPENDICES
100
APPENDIX A
TABLES
101
Table A. 1.
Number of households in the United States, farm
and nonfarm, , 1957-1970 (1,000).
Year
United States
Farm
Nonfarm
1957
50,934
4,372
46,562
1958
51,821
4.233
47,558
1959
52,711
4. 105
48,606
1960
53,531
3,962
49,569
1961
54,471
3,821
50,650
1962
55,308
3,685
51,623
1963
56, 137
3,561
52,576
1964
56,942
3,442
53,500
1965
57,251
3,340
53,911
1966
58,092
3,239
54,853
1967
58,845
3, 146
55,699
1968
60,444
3,054
57,390
1969
61,805
2,991
58,834
1970
62,874
2,924
59,950
From U. S. Department of Commerce, Statistical Abstract
of the United States, Bureau of Census.
U. S. Department of Agriculture, Number of Farms and
Land in Farm Statistical Reporting Service, SpSy. One farm
operator is assumed to be equal to one household.
Column 3 subtracted from column 2.
102
Table A. 2.
Personal income of the United States, farm
and nonfarm, . 1957-1970 (Million Dollars).
United Statesa
Year
Farm
b
Nonfarm
1957
346,871
13,596
333,275
1958
359,139
15,967
343, 172
1959
382,889
13,866
369, 023
1960
400,290
14,591
385,699
1961
416,291
15,693
400,598
1962
440,968
16,031
424, 937
1963
464, 307
16,162
448, 145
1964
496,277
15,322
480,955
1965
534,572
18,258
516, 314
1966
585,758
19,803
565,955
19 67
628, 095
18,462
609,633
1968
686,593
18,647
667,946
1969
747,583
20,969
726, 614
1970
799, 659
20,274
799,385
From U.S. Department of Commerce , Statistical Abstract
of the United States, Bureau of Census. Adjusted by subtracting
net rent to nonfarm landlords.
From U. S. Department of Agriculture, Number of Farms
and Land in Farms, Statistical Reporting Service, SpSy. Includes
government payments, value of farm products produced, gross
rental value of farm dwellings, net change in farm inventories,
farm property taxes, and net rent paid to nonfarm landlords.
c
Column 3 subtracted from column 2.
103
Table A. 3.
Year
Per household personal income of the United States,
farmland nonfarm, 1957-1970.a
,
United States
Farm
($)
($)
Nonfarm Farm x 100. 0
Nonfarm
($)
1957
6,810
3, 110
7, 158
43.4
1958
6,930
3, 772
7,216
52, 3
1959
7,264
3,378
7,592
44. 5
1960
7,478
3,683
7,781
47. 3
1961
7,642
4, 107
7,909
51.9
1962
7,973
4,350
8,232
52. 8
1963
8,271
4,539
8,524
53.2
1964
8,715
4,451
8,990
49. 5
1965
9,337
5,466
9,577
57. 1
1966
10,083
6, 114
10,318
59.3
1967
10,674
5,868
10,945
53.6
1968
11,359
6, 106
11,639
52.5
1969
12,096
7, 011
12,350
56.8
1970
12,718
6,934
13,001
53.2
Calculated from Tables A. 1 and A. 2.
104
Table A. 4.
Property taxes for the United States, farm
and nonfarm,, 1957-1970 (Million Dollars)
b
c
Nonfarm
Year
United States3"
Farm
1957
12,864
1,242
11, 622
1958
14,047
1, 306
12,741
1959
14,983
1,401
13,582
1960
16,405
1,502
14,903
1961
18,002
1,597
16,406
1962
19,054
1, 684
17,370
1963
19,833
1,763
18,070
1964
21,241
1,833
19,408
1965
22,583
1,943
20, 640
1966
24,670
2, 108
22,562
1967
2 6,047
2,275
23, 772
1968
27,747
2,515
25,232
1969
30,673
2, 761
27,912
1970
33,848
2,994
30,854
From U.S. Department of Commerce.
of the United States, Bureau of the Census.
Statistical Abstract
From Farm Income State Estimates 1949-70, USDA-ERSFIS, 216 Supplement, August, 1970.
Column 3 subtracted from column 2.
105
Table A. 5.
Property tax as a percent of personal income for
the United States, farm and nonfarm, 1957-1970.
a
Farm/Nonfarm
Year
United States
Farm
Nonfarm
1957
3. 7E
9. 14
3.49
2. 62
1958
3.91
8. 18
3.71
2.20
1959
3.91
10. 10
3.68
2. 75
1960
4. 10
10.29
3.86
2. 66
1961
4. 32
10. 18
4. 10
2.48
1962
4.32
10.50
4.09
2.57
1963
4.27
10.91
4.03
2.71
1964
4.28
11.96
4.04
2.96
1965
4.22
10. 64
4.00
2. 66
1966
4.21
10. 64
3.99
2. 67
1967
4. 15
12.32
3.90
3. 16
1968
' 4.04
13.49
3. 78
3.57
1969
4. 10
13. 17
3.84
3.43
1970
4.23
14.77
3.96
3. 73
Calculated from Tables A. 2 and A. 4.
106
Table A. 6.
Year
Property tax per household for the United States,
farm and nonfarm, 1957-1970 (Dollars). a,
United States
Farm
Nonfarm
Farm x I00..0
Nonfarm
1957
253
284
250
113.6
1958
276
309
268
115.3
1969
284
341
2 79
122.2
1960
306
379
301
125.9
1961
330
418
324
129.0
1962
345
457
336
136.0
1963
353
495
344
143.9
1964
373
533
363
146.8
1965
394
582
383
152.0
1966
425
651
411
158.4
1967
443
723
42 7
169.3
1968
459
824
440
187.3
1969
496
923
474
194.7
1970
538
1024
515
198.8
Calculated from Tables A. 1 and A. 4.
107
APPENDIX B
EQUATIONS
a
ln
a,
+ a_
In
2n
In
Q
2n
a,
+ a_
In
2n
2n
A.
4n
a,
+ a
in
2n
A.
lh
A
2h
"ih
,
+
a
2h
'
"
"
3h
"J*
a
4h
a
lh
+
a
2h
+
a
3h
a
Llf
0
a,, + a
+ a_,
lh
2h
3h
4h
+
if
a
lf
+
a
2f
+
Q
3f
a
2f
L
2f - alf + Q2£ + a3f
a
^ "
4f
a
lf
+
Q
2f
+
Q
3£
108
B1
=
1.0
- p2f
B.
2
=
1.0
-
1
1.0
"If
- A1£ B
2
1.0
"lh
- A
3
A.
B,
In
2
1.0 - Alf B1
'4
(3_
Zn
B
4f
- A,
B,
2n
2
1.0
A
2f Bl
1.0 - A,
B,
2n
2
'5
c
C,
6
A,
1
=
=
1.0
+ c
c
-
C.
C,
c1 + S
c c4
1.0
-
C,
-5
C_
D
l
2 + C3 C6
1.0 - C, Cc
L
4
1
5
1.0 - C, C_
_
2
3 '
C
109
C
6 + CZ
A. =
4
1.0- C,
5
C
5
Cc
D
Ac = A.. + A.. B, A, + A,
B, A,
5
4f
4f
1
1
4n
2
3
A, = A.
+ A^ B, A, + A,
B^ A,
6
4n
4f
1
1
4n
2
4
lio
APPENDIX C
DESCRIPTION OF DATA
The data used in this study was cross-sectional time-series
data.
The data was obtained for the 48 states for the period 1957
to 197 0;
Alaska and Hawaii were excluded since the data did not
exist prior to I960 for these two states.
Tables C. 1 through C.4.
This data is given in
Aggregate time-series data was also
obtained for this period and is given in Table C. 5.
Each variable
and its calculations will be discussed separately.
Percent Farm Operator
Households,.
4
N
N£
is the percent of total households in the economy that are
farm operator households, i. e.,
Tt
N
is the number of farm operator households and is assumed
to be equivalent to the number of farms.
The number of farms,
by states, is taken from Statistical Reporting Service estimates
as reported in Number of Farms and Land Farms, which uses
the same definition of a farm as the Agriculture Census.
N
is the total number of households as reported, by states, in the
CC
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Table C. 5.
Year
Relative income, R ; percent of households in
agriculture, N ; relative labor productivity, B ;
and relative property taxes, T , United States,
t
1957-1970.
R
t
N
t
B
t
T
t
1957
433
8.58
3.19
1. 14
1958
521
8. 17
3.49
1. 15
1959
444
7.79
3.46
1.22
1960
472
7.40
3.73
1.26
1961
518
7. 01
3.83
1.29
1962
527
6.66
3.91
1.36
1963
531
6.34
4.09
1.44
1964
494
6.04
4.09
1.47
1965
569
5. 83
4.35
1.52
1966
591
5. 58
4.51
1.58
1967
535
5. 35
4. 67
1.69
1968
524
5.05
4.65
1.87
1969
570
4.81
5.01
1.95
197 0
532
4. 65
5.04
1.99
116
Statistical Abstract only for the years 1950, 1960, 1965, 1968, and
1970.
The missing data is estimated by taking a straight line pro-
jection of persons per household for each state between the years
for which the data is available and dividing this into total population per state.
Relative Income,
R
R
is the net income per farna operator household relative
to the net income per nonfarm household,, i. e. ,
Y
F,
R.
Ft
=
Nt
Nt
where
Nt
Tt
Ft '
and
Y
Y
Nt
= Y
Tt
- Y
Ft '
is the total net income of farm operators as reported, by
states, in Farna Income State Estimates, 1949-70.
It includes
cash receipts from farm marketings, government payments, value
of home consumption, and gross rental value of farm dwellings
less all farm production expenses.
Total net income is adjusted
117
for net change in farm inventories and includes taxes on farm
property and net rent paid to nonfarm landlords.
Y
is personal
income, by state, as reported in the Statistical Abstract.
Relative Labor Productivity,
B
B
',
is the index of relative labor productivity of the farm
sector to the nonfarm sector, i. e.,
T.
Ft
T
Ft
Nt
Nt
where
T1VT = T^ - T
.
Nt
Tt
Ft
T
Ft
is the total property taxes paid by farm operators as reported,
by states, in Farm Income State Estimates, 1949-70.
T
is
the total property taxes paid by all individuals in the economy as
reported, by states, in the Statistical Abstract.
118
APPENDIX D
COMPARISON OF AGGREGATE TIME-SERIES DATA
AND CROSS-SECTIONAL TIME-SERIES DATA
It was shown in Chapter III, page 45 through 52,
that the
aggregate time-series data had high multicollinearity problems
among the independent variables.
Multicollinearity was shown to
exist in the aggregate time-series data by the very large diagonal
element of the inverse simple correlation coefficient matrix of the
independent variables.
The multicollinearity problem for the
aggregate time-series data was also characterized by the low t
values, the large variance on regression coefficients, and the
large simple correlation coefficients between some of the independent variables.
The above indicates the low information content
of the aggregate time-series data and the need for more informa tion such as is available in the cross-sectional time-series data.
Table D. 1 shows the diagonal elements of the inverse
simple correlation coefficient matrix of the independent variables
for the two types of data for the relative inconae equation, the
out-migration equation, and the relative labor productivity equation.
The diagonal elements using the aggregate time-series
data for the three equations range from 2. 31
to 79. 9;
whereas,
using the cross-sectional time-series data, they range from 1.22
119
Table
D. 1.
Diagonal elements of the inverse simple correlation
coefficient matrix: aggregate time-series analysis
(ATS) and cross-sectional time-sexies analysis (CSTS)
for three equations.
Diagonal Elements of the Inverse Simple
Correlation Coefficient Matrix
Equation
ATS
CSTS..
21.52
1.74
19. 10
1.95
2. 31
1.22
R
5.20
1.77
B
79.09
2.59
28.76
1. 32
29. 18
1. 67
5.28
1.70
28. 64
1.28
43.47
1.87
Relative Income, R
B
t
R
t-i
Out-Migration,
N
,
t-i
t-i
T
t-i
Vi
Relative Labor
Productivity,
Vi
T
,-i
B. .
t-1
B
,
120
to 2. 59.
It is obvious that the additional information in the
cross-sectional time-series data has virtually eliminated the
multicollinearity problem.
The near elimination of the multi-
collinearity is characterized in the other statistical values.
Table D. 2 shows the absolute values of the simple correlation
coefficients for the independent variables.
These coefficients
are smaller for the cross-sectional time-series data than for the
aggregate time-series data.
The largest absolute value for the
aggregate time-series data is .9821 ; whereas, the largest absolute
value for the cross-sectional time-series data is . 6511.
In Table D. 3 only two of the regression coefficients have
a significant t value for the aggregate time-series analysis;
whereas, nine of the regression coefficients have a significant t
value for the cross-sectional time-series analysis.
The two
estimating procedures produce markedly different results in the
regression coefficients,.
tion, in Table D. 3,
For example, in the relative income equa-
all three regression coefficients of the aggre-
gate time-series analysis are negative; whereas, only one is negative for the cross-sectional time-series analysis.
The standard
error of the regression coefficient ranged from 5. 6 to 18.2
times as large for the aggregate time-series analysis as they did
for the cross-sectional time series analysis.
Thus, it is clear that
an analysis based on aggregate time-series data would be meaningless.
121
Table
D. 2.
Simple correlation coefficient matrix: aggregate
time-series analysis (ATS) and cross-sectional
time-series analysis (CSTS) for three equations.
Simple Correlation Coefficient Matrix
of Independent Variables
Equation
Relative Income, R
A
B
N
R
t-1
ATS
N
1. 0
t
-0..9737
1.0
B
-0.7481
0.7094
1.0
R.
t-1
CSTS
1. 0
N
0. 6511
1.0
B.
R
0.4219
1.0
t-1
Out-Migration, N
-0.2778
R
t-1
N
t-1
t-1
t-1
0.7864
0. 6486
-0.7517
1.0
0.9658
-0.9821
1.0
-0.9523
ATS
R
B
t-1
1. 0
t-1
t-1
1. 0
t-1
CSTS
t-1
l
t-l
t-l
N
t-1
1.0
0. 6378
0.3657
-0.2767
1. 0
0.4593
-0. 6021
1.0
-0.3880
1. 0
122
Table D. 2.
continued
Equation
Simple Correlation Coefficient Matrix
of the Independent Variables
Relative Labor
Productivity, B
R
t-1
t-l
B
t-l
ATS
t-l
1.0
T
t-i
0.6486
0.7864
1.0
0.9658
1. 0
B
t-i
CSTS
R
t-l
t-l
B
t-l
1.0
0.3 657
0.6378
1.0
0.4593
1. o
Table D. 3.
Regression coefficients (RC), t values (t), and S. E. of regression coefficients (SE);
aggregate time-series analysis (ATS) and cross-sectional time-series analysis (CSTS)
for three equations.
CSTS
t
ATS
t
SE
-0.0724
-1.91*
0. 0378
-0.0064
-1. 74*
0. 0037
-0. 0941
-1.21
0.0776
0.0293
2. 30*
0. 0127
-0.2813
-0.94
0.3003
0.2278
5.58*
0. 0408
R
0. 6785
1. 30
0.5226
0.2 646
2.81*
0. 0940
B
0.2161
-1.22
0. 1768
-0.1039
-4.60*
0. 0226
0.2464
1. 11
0.2221
-0. 0142
-0. 67
0.0212
0.9159
19.05*
0. 0481
0. 0902
141.48*
0. 0064
Equation
RC
Relative Income,
A
N
t
B
t
R
t-1
RC
SE
R
t
Out-Migration, N
t-i
t-i
Vi
00
Table D. 3.
continued
Equation
CSTS
ATS
RC
t
SE
RC
t
SE
Relative Labor
Productivity, B
R
-0. 1363
-0.09
1.5557
-1.3778
8.46*
0. 1629
T
1. 1450
1.71
0. 6701
0.1867
5. 07*
0.0368
B
0.4337
1. 09
0. 3964
0.8821
2 6.00*
0.0339
t-i
t-i
t-1
* Significant at least at the 5. 0 percent level.
125
APPENDIX E
COMPARISON OF STATIC AND
DYNAMIC MODELS
The statistical results of the static and dynamic models are
presented in Table E. 1 .
This table includes:
regression coefficients for the static model;
the estimated
the estimated regres-
sion coefficients for the dynamic model, both short- and long-run;
the square of the coefficients of multiple correlation of the static
and dynamic models; and the t value of the static and dynamic
models.
Comparison of the results of the static and dynamic analyses,
presented in Table E. 1,
reveals that the square of the multiple
correlation coefficients in two of the equations is markedly lower
for the static analysis and that all lagged variables are highly significant.
All but two of the seven regression coefficients estimated
on the basis of the static approach lie outside of the range of the
short- and long-run regression coefficients estimated in the dynamic
analysis.
Some are smaller than the short-run regression coeffi-
cients and some are larger than the long-run regression coefficients
estimated by the dynamic analysis.
One would suspect that the static
regression coefficients should lie somewhere between the short- and
long-run regression coefficients of the dynamic analyses since the
126
static analysis is a mixture of short- and long-run.
Thus, the
dynamic analysis explains more of the variation than the static
analysis and appears to give better estimates of the regression
coefficients.
Table E. 1.
Comparison of static and dynamic analyses: regression coefficients (RC),
(t), and multiple correlation (R ) for three equations.
Static Analysis
Equation
Dynamic Analysis
Short-run
Long-run
R2
RC
R2
RC
N
B
t-1
t-1
t-1
N
2.8601
5.37
0.2646
2.81
2.9141
2.87
-1.7944 -16.17
-0.1039
-4.60
-1.1443
-5.26
-0.5537
-0.0142
-0.67
-0.1564
-0.67
-4.59
0.9092 141.84
t-1
631
228
B.
R
t-1
t-1
B
1.0284
5. 31
-1.3778
-8.46
-11. 6862
4. 32
0.4887
9.68
0.1867
5.07
1. 5835
3.41
0.8821
26.00
t-1
124
136
R.
B
R
t-1
RC
983
418
R
t values
-0.0099
-1.79
0. 0064
-1.74
-0.0083
-1.73
0. 1237
6.30
0. 0293
2.30
0.0379
2.38
0.2278
5.58
-J
128
APPENDIX F
COMPARISON OF ORDINARY LEAST-SQUARES
AND TWO-STAGE LEAST-SQUARES ANALYSES
It was argued in Chapter III, pages 38 through 43, that
ordinary least-squares and two-stage least-squares give unbiased
estimates of the regression coefficients if there is no correlation
of the error terms between equations.
The difference of the two
results in the case of no correlation is that ordinary least-squares
has smaller variance.
If on the other hand, there is correlation of
the error terms between equations, then ordinary least-squares
will give a biased estimate of the regression coefficients; whereas,
two-stage least-squares will still give an unbiased estimate of the
regression coefficients.
Table F. 1 presents the results of the
two statistical procedures which includes the regression coefficients,
the t values,
and the standard errors of the regression coef-
ficients.
A
"t" test was made between the regression coefficients of
the two estimating procedures given in Table F. 1.
are
The results
129
Regression Coeffici ents
TSLS**
OLS*
N
vs.
B
vs.
,
.
R
t-i
vs.
t Values
5. 16
2. 37
t
■
1.87
R
t-i
* Ordinary least squares.
** Two-stage least squares.
The regression coefficient on N
for the ordinary least-
squares approach is positive, while the regression coefficient on
A
N
for the two-stage least squares approach is negative.
They
are significantly different from each other at the . 05 percent level.
This means that Ho: #1 would have been rejected with the ordinary
least squares approach instead of not being rejected as it was with
the two-stage least-squares approach.
on B
and B
have the same sign; however, they are significantly
different at the .5 percent level.
R
The regression coefficients
The regression coefficients on
for the two approaches are significantly different at the
5. 0 percent level.
Thus, if ordinary least-squares had been used,
it appears that there would have been a significant bias in the
regression coefficients of all three variables in the relative income
equation.
Table F. 1.
Comparison of ordinary least-squares and two-stage least-squares-; relative
income equation, cross-sectional time-series data.
Item
Ordinary Least Squares
N
t
Regression Coefficient
0.0174
t Value
6.17
S. E. of
Regression Coefficient
0.0028
B
t
0.1365
18.25
0.0075
R
^ Two-Stage Least Squares
,
t-1
N
t
B
t
R
,
t-1
0.1313
-0.0064
0.0293
0.2278
4.10
-1.74
2.30
5.58
0.0127
0.0408
0.0320
0.0037