xt7bnz80mm3b https://exploreuk.uky.edu/dips/xt7bnz80mm3b/data/mets.xml   Kentucky Agricultural Experiment Station. 1959 journals 084 English Lexington : Agricultural Experiment Station, University of Kentucky Contact the Special Collections Research Center for information regarding rights and use of this collection. Kentucky Agricultural Experiment Station Progress report (Kentucky Agricultural Experiment Station) n.84 text Progress report (Kentucky Agricultural Experiment Station) n.84 1959 2014 true xt7bnz80mm3b section xt7bnz80mm3b \ Changes in the Weather and Production Functions
for a Sample of West Kentucky Farms  
R I .
By A. N. HALTER ¤nd G. L. BRADFORD
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AGRICULTURAL EXPERIMENT STATION

 I

 3.
CHANGES 1N THE WEATHER. AND PRODUCTlON FUNCTIONS
FOR. A SAMPLE OF WEST KENTUCKY FARMS
By A N Halter elnd G L Bredf0rd>?‘
Farms an thhee TVA ·:‘o·m.t1ee of We etetn Kentucky thjt weve surveyed
in 1953 by Jensen .»m.d Sundquietl weve tevisited in 1957 to dscertair the change?
y that occurred. in #·==po-==e to tnowt end output pricee Specifically we wanted ¢·_.
l deter  11} whether the p··0du1
inputs amd the adjustrnente of inputs toward than leger coe? c·on1hin;t1=m xxzene-
; nqeasuted Aklodgl tP·i= leltet pkjv **° tenont pvenerlts the lrY1poVt¤ntime,+ll;‘r;# ofon.-· ,,4] lg.
This report dxscueeee ·‘*#·e_t. rloengee in the sample;. serzonu. e,t1n·lm.x ,·
of the produqtior. :"¤,r~c**on lT`V`l*J.d`1T'lg + ll eeperiflratlon of lnpute land in¤;o»ne 1mi
lb) rcwrertworw of d»t¤ fop <"l·w.gee ln pwre level wd differences in weather; h.l'lC,L
third. the n1¤j0~‘tee—·‘te *t·< mdwtg ;M—+ttr:g for .1 Change in the- ptoduction
function Lind *b} rnqrgw Ll p··Vc>d=1<‘t‘·=t1¤»; of mputs {tom the ut;—Cor:·<=ct.ed and
we=¤.tl*e   ·¢.dj··.lsted {uns tion;
CHANGES 1N THE SA1\/[PLE
T11-F Wetleet "d$*‘ r’d<1·.‘~*i -·:· ev p*·o· wded We been dnl.; lot @*52 ll Tn
:1.I'lT`L];)n‘3C.1 u'€:. Vi/1S 'i"'.'·'(?"C T", ll]   €·("11" (O"-’wl1Vlg O{.:.[;)];)" Wllllltlklx   j,
cient G*•enl;d.¤ Jnd 1% p——··=’e·w C M s:~~:v wet *r`»¤rr1ec>w;# o· TTi;:.r»=r‘·g\¢ or .; Sump —- or W·»¤t#= ··n ].<•rT'll!1wrng rlrme by D G P0‘w— *<,»=‘ l’¢·»— l?"· D
diseertdtaor A.pp·e<¤;t·<‘·n te doe l*w21’nfo‘ Ad ¤nc1¤·.lg;;·.——ll<’» ~ M pt o ~ wr
of this veprvt
3J'et1$c=*rw.1m*lS`1.»r;dq~.:»*. on nit pp 7 8
4B.j_;,1< dn; rm furled the qll.,v1:1y‘*)q·r·~# glrl pvc ¢r - {rv tlw 1·;])·l1 lrgrl
output items Otlw *t;l'<>*rn¤t‘ot· ws l’>‘r sn· rl {toni U S D<¤p·~·1m·’nt of
AgTlCl§1lTl,l”`v* tn,lbliC.¤r*on> TVA ptllyttr Atoms and Kenlm ley /\.g~·· _1lllJ,l”lv,l
Experintent Stnzon §t'\\lr'(. 

 _.4,.
Marshall counties. Uniformity in output combinations is irnplied by the sample 1
being from a contiguous homogeneous area. A block random sampling method
was used, and data were collected from 144 farms.
Whenever possible, the same farmers were revisited in 1957, and basic ,
data were collected for the same input and output items for 1956. Thus, the
sampling considerations apply to both years. If a farmer had moved to another
farm in the area specified above, he was interviewed again However, it was
found that 27 of the 144 farmers had either died, moved out of the area, or
declined to be interviewed In these cases it was necessary to replace them ,
with units of as similar characteristics as possible
Any great error in replacing these 27 units would have distorted the
representativeness of the sample Since predictions were to be made using the
1952 data as a basis, it was important to notice that the sample of 144 farms I
was representative for both years. The 117 farms visited in 1952 and 1956 had
an average gross income of $4,437 and $6, 562 respectively. 5 The 27 farms
sampled only in 1956 had an average of $7, 196 The difference in gross income '*
between the two years for the 117 farms was $2, 125 and for the 27 farrns is _
$2, 453 The change in gross income between the two years was not significantly
different when the) "t" test was applied to the two sets of farms With this
information it was concluded that the sample was not materially changed by the
new farms surveyed in 1957. _
ESTIMATION OF THE PRODUCTION FUNCTION
The estimation of the production function or the input-output relationship
that existed on this sample of farms for 1952 was carried out in a previous
. . 6 . . .
productivity study. However, since a number of new techniques were used in
attempting to reduce the high intercorrelations between inputs that were present
in the Jensen and Sundquist study, the function estimation was re-examined, and
therefore details of the analysis are presented here Also, since the 1952
function coefficients were going to be used to test if the function had changed,
i. e., if technology had changed by 1956, a number of adjustments for price
level had to be made in the data These are also presented in this section.
The general nature of the production function is Y = f(X1, X2, X3, X4, W, U)
where'
Y denotes gross incorne,
X1 denotes acres of land and associated inputs,
Xg denotes days of labor, I
X3 denotes services from forage machinery, forage and nonforage
consurning livestock, and livestock buildings 1
 
5The 1952 items were enumerated in 1956 dollars as will be explained later
bJensen and Sundquist, gp cit.

 -5-
X4 denotes purchased inputs,
W denotes weather, and
U denotes unexplained residuals due to omitting variables from the equation,
Specification of Input Categories
All the items of gposs income (Y) and input categories X3 and X4 were
enumerated in dollars, Thus, the observations (for each farm) for these three
variables were determined by adding the value of each item in dollars Quantity
figures for some of the items were determined in the survey (for 1952), while
other items were listed in total dollars sold or paid 8 The procedure was to `
enumerate items in 1952 dollars and then to convert to 1956 dollars by a price
index or price relative 9 The corresponding items for 1956 were, of course,
1 enumerated in 1956 dollars,
{ The following formula was used to adjust the items
X l = X where
k l< k "
(G) (n) ln)
X signifies the input or output item, 1 the index or price relative, and the sub-
scripts k, o, and n respectively denote the kth item, the base year (1952), and
the nonbase year (1956) The price relative (1) was computed by dividing the
1956 price by the 1952 price and multiplying by 1OO Each item of the three
categories (Y, X3 and X4) was adjusted for each farm for 1952 This amounts
to the same as adjusting the entire category by a weighted aggregative index of
the form
1 = Fl(n) X(n) where,
 V CMO) Xm)
q denotes the price, X denotes the input or output item, and the subscripts o
and n denoting the base year and the nonbase year ln general, the prices of
input items were higher in 1956 than in 1952, and the prices of gross incorne
items had declined This is what would have been expected since the overall
index of prices paid by farmers has been increasing, and the prices received
fell during the period Thus, the importance of stating the 1952 items in 1956
dollars cannot be overemphasized since the effect of a change in technology
l (if any) would have been obscured by the price changes.
See Appendix A for a discussion of how the items were first enunierated
and a list of the prices used if the item was not already scheduled in dollars
8Had quantity figures been available for all items, then the iterris could
have been evaluated in 1956 dollars simply by multiplying the quantity by the
1956 price.
9See Appendix A for a list of indices

 -5-.
Land and Associated Inputs
In general it is desirable to group input items that are complements or
substitutes together. However, in actual practice many of the input items
considered are neither perfect complements nor perfect substitutes or near-
substitutes. The input categories were specified with the idea that they were
not complements or substitutes, i.e. , the correlation between categories was
assumed to be fairly low. 10
Specifying X]
Most of the items in this category were assumed to be used by farmers
in more or less fixed proportions with land (operating acres). For example,
machinery depreciation increases as operating acres increase and decreases
as operating acres decrease. These proportions were not necessarily constant,
and the exact relationship was not known when the items were grouped into the
category. However, preliminary analysis and past experience gave good indication T
that the items were complements or near—complements. 11 Thus, the input could
be specified as a single item or as an index or "set number" representing all the ‘ V
items. For example, gross income is a function of operating acres and machinery
depreciation, i e. , Y = f(L, D), and if D = g(L) then operating acres andmachinery
depreciation may be specified as one variable, thus, Y = h(L) The problem is the
same conceptually when more than two variables are considered.
Using the above logic as a guide, eight items of the X1 category were cor—
related with an index of gross income which was to reflect the level of use of this
input category 12 A linear function of the form was used `
Y =a·l·b1V1+.... . +b.8V8+U. where, .__\
Y denotes the index of gross incomelj and the following items were represented
by the Vi variables (l) operating acres which included all farm acreage except
woodland, wasteland, and unused lots; (2.) tobacco acreage which included burley, _
IOSee Appendix B for a list of intercorrelation figures for the current
study and for the Jensen and Sundquist study
UA similar problem will occur when a majority of the farm firms are
combining the inputs in least cost combinations High intercorrelation will exist I
when a majority of the sampled farrn firms are producing with least cost com-
binations This high correlation occurs because the scatter of points will lie
along the scale line or expansion path, e g., when one input is increased the »
other is increased in direct proportion. This phenomenon may make it impossible I
to derive parameters for the input categories since there will be no scatter of
points to which to fit the function. `
12·Any other variable highly correlated with these items could have been
used to construct the index and reflect the level of combined usage.
l3This index was computed for each observation by dividing the dollars of
gross income for each farm by the arithmetic mean of all farms and multiplying
the result by 100.

 -7-
dark~·fired, and dark air—cured tobaccos; (3) machinery depreciation which
included tractors, trucks, cornpickers, grain drills, and combines; (4) machinery
repairs, (5) tobacco barn area; (6) fence depreciation; (7) fuel expense; and (8)
crop seed expense, 14 This procedure was followed for both 1952 and 1956, since
the 1956 inputs were also needed in testing the hypothesis that the technology had
remained constantt
Coefficients derived for each of these items are presented in Table 1, The
"set number" was estimated for each observation by evaluating the function using
the regression coefficients and the input observations for that farnri, The range
of the "set numbers" for 1952 was 14 to 442 and from 1956 was 16 to 402 1
Table 1,. — Coefficients of proportionality for items of variable land and
associated inputs, 8*
1
ltem Coefficient
 
V1 —~ Operating acres 17397
V2 — Tobacco acreage 6 64
V3 · Machinery depreciationb . 06153
V4 - Machinery repairs , 08171
V5 — Tobacco barn area it 01376
Vg, — Fence depreciation .00381
V7 - Fuel expense 01916
Vg - Crop seed expense 10156
 
aBased upon 1952 data only
blncludes only tractors, trucks, cornpickers, grain drills, and combines
Labor, Livestock-forage lnput, and Purchased Inputs
The labor variable (category X2) was specified in total days per farrn including
hired labor, family labor, and the manager‘s labor The observations were
determined for each year by determining the number of days of labor on each
farm, No adjustment was necessary on the 1952 data since it was enunierated in
physical units,
The variable input X3 denotes services from forage machinery, forage con-
suming livestock, grain consuming livestock, and livestock buildings Machinery
items included hay balers, forage harvesters, trailers and wagons, ensilage
cutters, feed grinders, mowing machines, nnilkers, and milk coolers, Livestock
The items which were in 1952 dollars were first adjusted to 1956 dollars.

 1,3-
included dairy and beef breeding animals, sheep breeding animals, swine breeding
animals, and beginning inventory value of home -produced feeder animals. _
Buildings included silos, dairy barns, and cattle sheds.
The variable input X4 denotes purchased inputs which included purchased
feeder livestock; fertilizer and lime; electricity, telephone, and automobile
expense (farm share); spray materials; seed treatment; breeding fees; baby
chicks; custom hired machinery services including corn picking, bulldozing, and
trucking; and sawdust and slab wood
Specifying the Weather Variable
The general practice in productivity studies has been to disregard the effects
of the exogenous variable weather However, in this study this variable cannot be
ignored since the results depend directly upon taking account of the major changes [
or differences between 1952 and 1956 in the variables influencing production.
Generally speaking, 1952 was a dry, hot year while 1956 was much. more season—-
able. More specifically the weather variable may be considered or specified. in .
the following mannerv
1 Consider again the function for 1952 in the general form
Y = f(X1, X2, X3, X4, W, U).
2 The equality still holds when manipulated, thus
-3 I f(X1, X2, X3, X4, W, U)
W W
Yi : f(X1,X2,X3, X4, U)
The weather variable was considered by this method, i e., an adjustnnent factor
for weather was computed for 1956 (1952 being 100) and multiplied by the 1952
gross income, or division by the reciprocal of the factor as in 2 above,
Derivation of the Weather Adjustment Factor A
The steps in deriving the factor were" (1) correlation was established
between weather and crop yields. and (2) farms with different crops and acreages
were weighed such that the separate effect of differences in weather could be U
determined for each farm Specifically, crop yield data for corn, wheat, and
hay were regressed upon weather data in an attempt to specify correlation between
the two The crop yield data were taken from the Mayfield Soil Experiment Field
for the years 1927 through 1954, excluding 1943 and 1944. 15 The weather data
 and Harold F Miller, A Summary of Kentucky Soil
Fertility Experinients, Ky Agr Exp Sta. Bul 663 (Lexington, June, 1958),

 ig.,
used were the number of drought days for April through October for the corre·~
sponding yearsr
Computation of Drought Days
Drought days, for April through October, from the Paducah weather station
for the years 1927 through 1954, excluding 1943 and 1944, were used as the inde·
pendent variables in the correlation of crop yield data, 16 Rainfall and temperature
data alone ignore certain plant and soil characteristics which in a large part
determine crop yields, However, in the number of drought days several variables
directly or indirectly make up the weather variable The number of drought days P
is an index determined by several relevant clirnatic and agronomic factors which
bring about drought conditions
,, The number of drought days for each month are computed frorn rainfall and
V evapotranspiration data by a moisture-balance method The daily evapotranspirzr
tion was calculated by the Penman formula by Knetsch and Smallshaw. ll The
° different water—holding capacities and the number of drought days had to be con
sidered for each level, since the exact water~holding capacity of the soil in the
area is also a variable For each of five levels of irioisturelg the rainfall is
added for each day it occurs during the month, and the calculated inches of evap
transpiration are subtracted from the available amount of soil moisture for each
day For example, if the available soil moisture is 31 inches for the l— inch
moisture level and 55 inches for the 3—inch level on the first day of the month,
and the evaptranspiration was 17 inches per day, then within two days of ntioisture
would be exhausted for the l·inch level, and within four days the moisture would
be exhausted for the 3—inch level, Every day after the moisture has been depleted
is ra drought day until it rains. Rainfall. when it occurs, is added to the availaole
soil moisture There will naturally be more drought days for the l-inch level
than for the higher levels, since the base amount of available soil nioisture is l»<,·—,,
Regression of Yields on Drought Days
Yields of corn, wheat, and hay were regressed upon the number of drought
days and time lto establish trend, if existing) for the years 1927 through 1954,
excluding 1943 and 1944 The number of drought days for each month, April
p through October, were the independent variables in this regression lg The core
I6For a tabulation of data for the Paducah station see Jack L Knetsch and
p James Smallshaw, The Occurrence of Drought in the Tennessee Valley, Report
T 58-2 AE, lKnoxvi1le‘ Tennessee Valley Authority, June, 1958), p 47
Ng , pp 5 7
18The quantity of inoisture available to crops was determined by multiplying
the available 1rioisture—holding capacity of the soil, in inches per foot, by the effective
_ X rooting depth of a crop Thus, a soil is said to have 4 inches of available soil moisture
or 6 inches, or 7 inches The five levels used here were l, 2, 3, 4, and 6 inches
19Knetscha.ndSmallshaw, Q, iii, p 47

 -l0»· p
wheat, and hay yield figgures were obtained from plots 4, 5, and 6 of the Mayfield
Soil Experiment Field. O This regression was performed for all five moisture _
levels, regressing the yield of each plot and the average of the three plots upon
the drought days and upon time Drought days for August through October were 1
lagged for hay. For example, drought days for July through October of 1930 and
` drought days for April through June of 1931 were paired with 1931 yield data. Thus,
in the first regression 2.5 observations since 1943 and 1944 were omitted and the
results were lagged leaving out 1927 Specifically these functions may be syrnbolized:
1 C I 3. + b1Z1(t)-l-b2Z2(t)+ b3Z3(t) ·l· b4Z4(t) ·l· b5Z5(t) ·l‘
b6Z6(t)1 b"/Z7(t)1b8t
Z W Z B. ’l‘   ·l‘   ‘l‘   ‘l‘   ‘l‘   ‘l‘  
b6Z6(t 1) t bvzm. 1) t bet _
3 H = a ·l b1Z1(t) + bZZ2_(t) + b3Z3(t) + b4Z4(tg1) + b5Z5(t_l) »l·
    +   *l’ b8t W1'1€I‘€,
C denotes corn yield in bushels, W denotes wheat yield in bushels, 1-1 denotes hay
yield in pounds, Z1 through Z7 denote drought days of April through October
respectively, and t denotes the current year with t 1 denoting the previous year. .
The parameters of these equations were calculated for each of the five moisture
levels and for each of the yields of the three plots (4, 5, 6) and their average.
The regression coefficient bg was significant for corn and wheat; hence,
the effect of time was removed from the yield datagzl hay showed no trend. The
deviations from the trend line for corn and wheat were converted to indices. Yields
lying above the trend line were entered as an index above 100 and vice versal 1`hen,
the corn and wheat indices of yield and the hay yields were again regressed upon
the drought days data; only the average of the yields from the three plots were used
in this case as the dependent variable 2*2* After this second regression, one level T
Zvliarraker and Miller, gp, cit. , pp 19 Z1. Also from records of the
Agrononiy Department, Kentucky Agricultural Experiment Station. These plots 4
had relatively siniilar treatments
21The moisture levels which had the highest R2 value were selected as the
function to be used for each crop Then, for this moisture level only the regression
Coefficients which were significant were used
&ZThe average yield of the three plots was used since the R2 value was higher 1
than for the individual plots

 ..1 1-
of moisture and the appropriate months were chosen for each of the crops in order
that an index or crop yield could be predicted for each year (1952 and 1956). For
corn, moisture level three (3 inches) had the highest R value, and the regression
coefficients for the months of July and August were significant. For wheat, moisture
level three had the highest R2 value, and coefficients for July for the present year
and August of the previous year were significant. For hay, moisture level one was
used (highest R2), with coefficients of July of the present year and October of the
_ previous year. These variables were used to calculate the following values: (1)
corn index 98.8;; in 1952 and 109. 9 in 1956, (2) wheat index 108. 7 in 1952 and 100. 0
in 1956 and (3) hay,4, 618. 8 pounds per acre in 1952 and 8, 139. 5 in 1956 The ·
adjustment factors were obtained by expressing the 1956 index (yield in the case
of hay) as a percentage of the 1952 index (yield). These factors were 110 for corn,
92 for wheat, and 176 for hay. .
vp Adjustment Factor for Each Farm. — The final step in the adjustment
" technique consisted of determining weights to give to each of the adjustment
factors for each farm. The method used was to determine for each of the 144
* farms the percentage of the total operating acres devoted to (1) corn, including
popcorn and field corn, (2) small grains, since they are all similar to wheat, and
(3) pasture and hay acreage. On most farms these three categories included over
90 percent of the total operating acres. Each of these percentages was respectively
multiplied by the adjustment factors determined above and the results added together,
giving a weighted adjustment factor for each farm This was multiplied by the 1952
gross income, and the resultant product was the 1952 gross incoxne in terms of 1956
weather. It is interesting to note that for almost all of the 144 farms the income
was adjusted upward; most of the adjustment indices were between 140 and 150.
This is quite consistent with the idea that 1952 was a dry, hot year while 1956 was
much more seasonable. Hence, regardless of the crops grown on the various farms,
income was adjusted upward.
TESTING FOR A CHANGE IN THE PRODUCTION FUNCTION
The 1952 function in general notation now appears as follows:
Y, = f(X1, X2, X3, X4, U) where the variable inputs represent the items
defined above and YI represents the adjusted gross income for 1952. Least squares
regression was used to derive the parameters of the Cobb·-Douglas and the tran-
J scendental equations in their logarithmic fornis This method is standard procedure
and need not be discussed here, _
The Cobb -~Douglas equation which was fitted to the adjusted 1952 dataZ3 allows
( for either increasing, constant, or decreasing marginal returns throughout the
production relationship. The transcendental equation, also fitted to the data, can
  . . . bl bz b3 - b4
The form of the equation with U Z residuals is Y 1aX] Xg Xg X4 U
and in logarithms is lnY = lna + b11nX1 + b2lnX2 -1- b3lnX3 + b41nX4 -1- lnU. The
residual is assumed to be due to errors in the equation

 -1;-
exhibit nonconstant elasticity, i. e. , increasing, decreasing, and negative _
marginal returns, singularly, in pairs, or all three simultaneously. 24 Thus,
if the data clearly indicates three stages, then the parameters when fitted will ·
show three stages Regression coefficients and related statistics are shown for
the transcendental and the Cobb—Douglas equations in Tables 2 and 3. The
coefficient of determination (RZ) was .70 for the Cobb-Douglas and .72 for the
transcendental function Thus, approximately 30 percent of the variation in
gross income was left unexplained by the five independent variables (including
weather) of the 1952 function
The major hypothesis of this phase of the research was that the production
function had not changed between 1952 and 1956. To test this hypothesis, the
predicted gross income for 1956 was compared to the actual 1956 gross income.
The predicted gross income was calculated by evaluating Y (i e , 1952 parameters)
with the 1956 input observations This is made explicit in the following formula E,
for the logarithniic form of the transcendental equationr
lnY‘=inc +a 1nX +b X +a lnX ·I— ‘
lol llc) llnl lm) lin) $(0) $(0)
l » X + : l X i- b X + · l X +
Dglo) $(0) $$(0) n $(0) $(0) $(0) $@0) n 4(0)
b X
$(0) 4(0)
where the subscripts 1 to 4 indicate the input category, o denotes the base year
(1952) and n denotes the nonbase year (1956) The predictions were made by the
s.·imc· rnethod for the Cobb Douglas equation
The predicted and the actual grossincome for 1956 were compared. The
magnitude of this difference was determined by a chi—square statistic. The chi-
square test provided a means of discovering if the deviation was larger than what
would be expected to be due to chance alone This test was performed by the
following forniulai
SUM (a e)2 where,
e
Z"I’The form of this equation with U = residuals is ·
-7 al blX1 az b2pX2 ag b3X3 all 134 X4 · ' ·
Y c ck) e X; e X5 e X4 e U and in logarithms is
1l'lY Y 1IlC *1* éillllxl ‘i‘   ‘i‘   ‘i‘   i‘ El31I'1X3 ·l—   ‘l‘ &41HX4 i`   ‘i‘ 1HU.
For a detailed discussion of the niathernatical properties of this function see A.N.
Halter, H O. Carter, and.] G Hocking, "A Note on the Transcendental Production
Function. " Journal of Farni Econoniics, XXXIX (November, 1957), pp. 966—~974..

 -13-
TABLE 2. · Regression coefficients and the standard errors of the mean niarginal
productivities for the 1952 transcendental function adjusted for weather.
 
. Standard Error of
Regression Mean Marginal Mean Marginal
Input Category Coefficientsa Productivity Productivityb
lnX1 » Q .43938
X1 (index) .00091 33. 73 .92
1nXZ . 23018 _
X2 (days) -.00007 3. 91 L 14 b
l1'1X3 . 03232
X3 (dollars) .00011 1.13 .09
lnX4 . 19159
X4 (dollars) .00002 1.05 ,05
{ c __ (constant) 39. 73760
W aThe function coefficient or the "returns to scale" is 1. 04 at the mean level
• of inputs.
bThe definition of the variance of marginal productivities is
NI (MPi=_ - "1\T§)2·
  where,
i=1  
MP is the computed marginal productivity for the observed level of input, MP is
the marginal productivity computed at the mean level of inputs, and d. f. stands ·
for the appropriate degrees of freedom. For this function the arithmetic means
of the inputs were used. N = 135. The standard error is the square root of the
variance divided by the square root of N.
TABLE 3. » Regression coefficients, their standard. error, and standard error
of the mean marginal productivities for the 1952 Cobb—Doug1as
function adjusted for weather
 
StandardError Mean Standard Error of
Regression of Regression Marginal Mean Marginal
lnput Category Coefficientsa Coefficients Productivity Productivityb
Xl (index) .51226 .02345 31.79 .71
Xg (days) .24420 .02236 4.27 .16
X3 (dollars) .05178 .00632 1.25 .12
X4 (dollars) , 24321 .01788 1.39 .06
5,;; (constant) 19. 34317
9The function coefficient or the "returns to scale" is 1.05
l bFor this function the geonietric means were used in the definition given in
footnote b of Table 2 N 4 135.

 -14 _ A
a denotes the actual 1956 gross income and e the theoretical observation or predicted
1956 gross income , 25 Q
When the test was performed for the predicted logarithms of gross income V
· and the actual logarithms of gross income for 1956, a chi—square value of 4. 56
was obtained for the transcendental equationt A similar test using the parameters 6
from the Cobb—Douglas function yielded a value of 4. 37. These small values of chi-
square indicate how closely the predicted agreed with the actual. With only one
sample, one cannot attach a probability statement to the credulity of the hypothesis.
However, the probability of finding the predicted deviating from the actual in repeated
sairipling from the same population is extremely small By knowing this, it maybe
concluded that the hypothesis is true or it has been confirmed until further notice.Z6
SUMMARY OF MAJOR RESULTS
s
The results given in Tables 2 and 3 are of particular interest to researchers
and to those who want to know the productivity of inputs. First, notice the contrasts
between the two functions While the Cobb-Douglas shows constant returns to scale `
throughout the range of inputs, the transcendental gives a similar result only at the
mean level of inputs ln addition the estimates of marginal productivities are ’
different; however, considering the size of their standard errors these differences ma
not be significant from a statistical standpoint From the standpoint of predicting
farme·rs' action in respect to the combining of inputs in least cost combinations, it
remains to be seen which function provides the rnost accurate predictions.
Second. notice for the transcendental equation that only the standard errors
of the marginal productivities are meaningful, i. e. , it takes-both coefficients to
show the contribution of the variable Thus, a test of significance on either A
regression coefficient would provide no information.
Third, notice that for the Cobb—Douglas the standard errors of the mean
marginal productivity fo'r inputs X2, X3, and X4 are higher than for the-transcendenta
function. Although the rnean marginal productivity for these three inputs are higher
for the Cobb Douglas the testing of predictions in the second phase of this study
will provide a more powerful criterion upon which to judge the two equations.
Another inciportant result from the standpoint of research methodology is the
adjustrnent of the gross inconie for differences in weather ln this study gross
incoine was first predicted without considering weather as a variable The same
Z—;George W Snedecor, Statistical Methods, (Ames' lowa State College
Press, 1946), pp 16-18
‘2bCoiiiparing actual 1956 gross incoirie with the antilogarithmic value of the
above predicted gross income will give exactly the same chi-square value.

 -15l
equations as given above were fitted by least—squares regression to the unadjusted
data. The chi—square test was used as it was above using the unco1·rected 1952
coefficients and 1956 inputs to predict 1956 gross income. The chi—square value
was 7. O2 when comparing predicted with observed. This compared to a value of
4. 37 when the corrected 1956 coefficients were used to predict 1956 income.
Thus, the unexplained residuals were cut almost in half by considering the weather
variable; this gives strong support to the adjustment procedure used, and credit is
due those who developed the concept and measurement of drought days as an indi-—
cator of the weather variable.
To demonstrate the effect of omitting the weather variable on the estimation
· of marginal productivity of inputs, Table 4 presentsthe marginal productivities -
before and after adjusting for weather at the arithmetic mean levels of the four
input categories given in Table 5.
l TABLE 4. - Marginalproductivities for arithmetic mean levels of input categories
for a sample of Western Kentucky farms, 1952 (income adjusted and
’ unadjusted for weather). a
` Marginal Productivity of
Input Category X1 X2 X3 X4
Before weather corrected 22. 98 3. 30i .27 .71
After weather corrected 33. 73 3. 91 1.13 1.05
 
derived by the formula:
5-}; = Y (E + b-)
§ X1 Xi 1
In this study the problem of adjusting for weather was particularly crucial
i since predictions over time were being niade 27 ln addition predictions of changes
in combinations of inputs will be made, and when these are based upon the level of
unadjusted marginal productivities serious errors are likely to result. For
example, since the uncorrected marginal productivity of X3 is extr