I,j
K!
A routine for the IBM 650 computer was used which selects the significant
variables on the basis of the significance of the b value subjected to a
t test.
The equation with the highest R2 value was:
(2) =449.7 + 430.6 X2 + 14.9 X X2 0.46 X12 36.9 X2
The R2 is .423. For this equation when X2, precipitation, equals zero,
Y is negative for all values of X1. With one inch of precipitation Y is
positive but small from 0 to about 27 animal units. With 2 inches of pre
cipitation Y reaches a maximum at 32 animal units, and with three inches
reaches a maximum at 48 animal units per section. For X2 = 4.7 inches,
brF C)
OPTIMUM RATES OF GRAZING VARIOUS CLASSES AND COMBINATIONS
OF LIVESTOCK ON THE EDWARDS PLATEAU
Ce te er E. I\ Jl F Irmd
The data upon which this paper is based was taken from seven years of a
grazing study at the Texas Ranch Experiment OW$Station, SL Sonora, Texas.
The grazing studies included grazing the single classes of livestock alone
and grazing cattle, sheep and goats in various combinations. Also, the
grazing rate was varied from 16 to 32 to 48 animal units per section. The
livestock was weighed every four months, and the gain or loss in weight re
corded.
In order to make a first estimate of the production functions involved,
the data for cattle grazing alone was fitted to a quadratic function of the
form,
2 2
(1) Y = a+bI X1 + b2 X2 + b X1 X2 + b X1 + b5 X2 + b X i + b7 X2
where Y = pounds of beef produced per section per four months,
X1 = the number of animal units p cattle per section year
long, and
X2 = the precipitation in inches for the four month period.
A
Y reaches a maximum at 76 animal units per section.
The first partial derivitive of Y, with respect to X1, that is the
marginal physical product (MPP) of X1 is a linear function in X1 and X2 and
equals:
(3) IL = 14.9 X2 9.92 X1
bxl
Multiplying by the price of cattle yields the marginal value product (MVP)
of X1. Figure I shows the MVP curve of X1 when X2 is valued at the mean or
4.7 inches.
MA,
I,,
fPt
_/ I ILC
^i(/Scc
FIGURE I CArE
The maximum profit rate of grazing for cattle alone can be determined
by equating the MVP of X1 with the marginal factor cost (MFC) of an additional
animal unit of cattle. An estimate of the MFC is $8.5, which is shown also
in Figure I. The estimated MFC is based on a charge of 750 tax per head,
$1.00 for veterinary services and supplies, a death loss of $3.50 based on 2%
death loss, and an interest charge of $2.80 based on 6% of a $140 investment
for a four month period. Notice that the MVP is equated with the MFC at
32 animal units per section.
Because the original production function is based on data for a seven
year period, the physical production functions should reflect changes in
the pasture over this seven year period. That is, any effect of the heavy
rate of grazing on the pasture and on the production from the pasture will
be reflected in the physical data.
Miller and Merrill, however, have determined that the change in value
of the pasture based on the capitalized change in carrying capacity amounts
to a gain of 200 per acre for the stocking rate of 16 animal units, no change
for 32 animal units, and a loss of 54# per acre per year under the heavy
rate of grazing with cattle alone. Beginning at 32 animal units, this
amounts to a gain of 500 per animal unit as the grazing is decreased, and
a loss of 500 per animal unit when increasing from 32 animal units up. The
adjusted MFC curve in Figure I takes into account this additional cost. Based
on this Figure, the optimum grazing rate for cattle alone remain sat 32
animal units for the average rainfall.
The optimum grazing rate figured in Figure I is based upon a buying
and selling price of $20.00 per hundred. Figure II shows the effect on op
timum grazing rate of 4 levels of purchasing price and 4 levels of selling
price. The heaviest optimum rate of stocking would be when the rancher
bought at a price of 15 dollars a hundred, and sold at a price of $30.00
per hundred. The lowest rate of grazing is when the rancher purchases
his cattle at $30.00 and sells at only $15.00. In fact, it does not pay
to graze at all under these conditions, as shown in Figure II.
FIGURE II
Because MVP is sensitive to precipitation as well as price it would
be well to consider the effect of basing decisions upon different expected
levels of rainfall. Figures I and II are based un the arithmetic mean or
4.7 inches. In many situations the geometric mean is a more descriptive
measure of central tendency than is the arithmetic mean. The sample geometric
mean of rainfall for a four month period is 2.93 or about 3 inches. Consider
ing the unadjusted MFC, the optimum grazing rate using the geometric mean
is only about 5 animal units per section. When considering the adjusted
MFC, optimum grazing rate increases, because the grazing is at less than 32
animal units. Thus, the optimum rate of grazing, considering the adjusted
MFC, is about 25 animal units per section. This is shown in Figure III.
It can be seen that in this situation, basing the decision on the geometric
mean, rather than the arithmetic mean *~Mm1ef isAmore conservative A Game
theory could be used as an approach to determine grazing strategies with at
i MVP
~vP
J 6/ o I 5
FIGURE III
least two more different levels of *naa In this situation, stocking
rates would be the ranchers strategy and amount of precipitation would be
nature's strategy. The payoff would be based on the net income which would
be obtained for each combination of stocking rate and precipitation. Another
game matrix might take into consideration the relationships in Figure II.
The rancher would know at time of purchase what the price would be. Nature's
strategy would be the selling price.
It is a fairly simple procedure to determine optimum stocking rate for
a single class of livestock such as for cattle alone, as has been done. To
determine the optimum stocking rate for combinations of livestock,,is n
^ .. . .'."' (.
more difficult, and to determine the optimum combinations yet more difficult.
In considering the problem of optimum stocking rate for a combination, say
cattle, sheep and goats, it is necessary to convert the products into some
common denominator. For the purpose of illustration, the products for cattle,
6
sheep and goats were valued as follows in dollar terms: beef at 200 a pound,
lamb at 180, wool at 500, and mohair at 800. The value from each of the
livestock products was then added together to form the total value product
(TVP) from the complete combination.
The function fitted using the same routine as previously was of the form,
(4) Y = a+b1 X1 + b2 X2 + b3 X1 + b X2 + b X1 X2
where Y = the TVP from all three classes of livestock together L the
period from July to June of each year,
X1 = the stocking rate in animal units per section, and
X2 = the annual precipitation from July to June.
Notice that the square root of the independent variables was omitted from
the equation. These were not significant in ha the original equation and
also their omission facilitates the taking of derivitives and working with
the functions. Thus, it was beneficial to omit them from this equation.
2
The function selected, had an R value of .90. The equation was,
A 2
(5) Y = 5.03 + 57.37 X1 0.52 X1 + 0.78 X1 X2.
Notice here that the TVP is primarily a function of the stocking rate.
Precipitation enters only as a positive interaction term. This is desirable
over the range of the data, because one would expect no decrease in product
as precipitation increased within the range of the data. In fact, one would
not expect a decrease in product as precipitation increased until precipitation
2
was at a very high level. For this reason, it is desirable to have no X2
term in the equation, unless it is negative and of a very small absolute
2
value. It is doubtful that the b value for X2 would be significantly
different from zero at any rate.
Once again, the partial derivitive of Y with respect to X1 is a linear
function in X1 and X2. It is,
(6) = 57.73 1.04 X1 + 0.78 X2
Because is already in total value terms this need not be multiplied by
the price of the product. Thus, this equation itself represents the marginal
value product of X1.
It is interesting to note that with the combination of livestock,
(including sheep and goats), Y is positive even'though X2 is zero. In fact,
Y increases to 55 animal units before it begins to decrease. In placing X2
at its arithmetic mean 14.13 inches, Y increases to 65 animal units before
it starts to diminish. With X2 at 25, the maximum Y is reached at about 74
animal units per section.
40
MVP
iM F
FIGURE IV /kAe<
Figure IV shows the MVP of X1 for the arithmetic mean value of X2. The sample
geometric mean is 12.08 and does not vary significantly from the arithmetic
mean in this case. Also in Figure TV are the unadjusted and adjusted MFC
curves. Equating adjusted MFC curve with the MVP shows that optimum stocking
8
rate for the combination of cattle, sheep and goats is at 47 animal units
per section. It should be noted that due to combining the value of the
livestock products, the MVP will be sensitive to relative changes in the
prices for these products. The MVP for a slightly different estimate of the
relative prices may well be considerably lower, or at least different, than
the MVP shown.
(A
//0
FIGURE V
Figure V shows the projection of the prouIction surface in the factor
factor dimension. If X2 is fixed at, say, 2 inches one can see that production
reaches a maximum and then declines, which is a desirable characteristic of
the function. This characteristic would hold for all expected levels of X2.
W*e letting X2 vary while X1 is held constant, at say 30, it can be seen
that production never reaches a maximum. This situation would be expected
for all expected levels of X2. The curves eventually become asympotic to the
X axis.
2
