ENERGY-RELATED INPUT DEMAND BY CROP PRODUCERS

quite well (Levy and Markowitz). Katoaka has shown
that E-V efficient solutions are also E-S (expected in-
come∕standard deviation) efficient solutions.

Activities and Resource Constraints

Farm size is assumed to be 400 acres with all labor
supplied by the operator and family. Machinery and
equipment complements are assumed to be compara-
ble to those available on a typical 400-acre Missouri
crop farm. Crops produced are those common to Mis-
souri—com, sorghum, soybeans, and wheat. Tillage
practices considered are chisel plowing, disking, and
planting for wheat, while conventional, reduced, and
no-tillage are possibilities for the other crops. Conven-
tional tillage involves plowing, two diskings, harrow-
ing, and planting. Reduced tillage involves use of a
field cultivator, disking, and planting. No-tillage does
not require field cultivation because planting is with a
no-till planter. For soybeans, both 15- and 30-inch rows
are considered as alternatives. Only 30-inch-row ac-
tivities are included for com and sorghum. Altogether,
there are 13 production activities: 3 for com, 6 for soy-
bean, 3 for sorghum, and one for wheat. Input coef-
ficients for fuel, seed, fertilizer, chemicals, labor, and
so forth were obtained from crop budgets, farm man-
agement specialists, agricultural engineers, agrono-
mists, producers, and farm management specialists
(Workman and Kirtley).

Estimation of Objective Function Coefficients

The experimental design adopted for this study re-
quires the use of many different expected income vec-
tors and VarianceZcovariance matrices. A base expected
income vector and a base VarianceZcovariance matrix
are computed from prices and yields for the period
1963-79. Prices are annual average prices for the re-
spective crops (Missouri Crop and Livestock Report-
ing Service). Crop-yield series are based on yields at a
Central Missouri experiment station (Minor et al.
1979a, 1979b, 1979c; Sechler et al.). Use of experi-
ment station yields eliminates much of the variation due
to management associated with other sources of yield
data. Data appropriate for determining the effects of
alternative tillage practices on yields are sparse. Dis-
cussions with crop-production specialists suggest that
yields from reduced tillage are about the same as those
from conventional tillage, but yields associated with
no-tillage are about 5 percent less. They also suggest
that 15-inch-row spacing for soybeans gives a 6 per-
cent greater yield than does 30-inch-row spacing. These
suggestions are used to construct yield series for re-
duced and no-tillage activities.

Gross returns are generally greater and more vari-
able for the years 1973-79 than for the 1963-72 pe-
riod. Adams, Menkhaus, and Woolery concluded that
E-V frontiers obtained using expected returns and var-
iances based on short, recent time series do a better job
of approximating farmer behavior. Our approach is
consistent with the spirit of their findings. The study
period is divided into two subperiods, 1963-72 and
1973-79. Separate average-returns vectors are esti-
mated for each subperiod. The base expected-returns
vector is a weighted average of the average-returns
vectors for the two subperiods. A greater weight (0.55)
is assigned to the average-returns vector for the second
sub-period. The VarianceZcovariance matrix is com-
puted using the formula:

10  _    _  17  _    _

Σ (R_t - R₁) ' (R_t - R,) + Σ (R_t - R₂) ' (R_t - R₂)
_{v =} t = I                    t = ll

15

where V is the estimated variance matrix, R_t is the re-
turns vector for year t( = 1 for 1963), and R_j (j = 1,2)
is the average-returns vector for subperiod j. Since the
variation in net returns is greater in the second period,
the estimated-variance matrix is much more like the
matrix for the second subperiod than the first.

Both the base expected-returns vector and VarianceZ
covariance matrix are “rounded” to simplify data en-
try. Even though the prices and yields for each crop
have a correlation coefficient slightly greater than zero,
it is convenient to think of the expected gross returns
for any crop as being the product of the average yield
and an “average” price. The expected-returns vector
is adjusted so that these “average” prices are multi-
ples of 0.05. The portion of the base-variance matrix
associated with conventional (and 30-inch-row soy-
beans) tillage activities is:

Due to data limitations, net returns series were con-
structed for the activities associated with other tillage
options by assuming that these returns are perfectly
correlated with those for the conventional tillage activ-
ity for the same crop. Thus the balance of the base-var-
iance matrix can be inferred by the reader. Klemme has
recently shown that the perfect correlation assumption
may not be completely valid. More research is needed
on this issue.

Experimental Design

As is the case for most programming models, it is
not possible to find a globally valid closed-form
expression for the energy-demand function implied by
the E-V analysis model. Even the expression for a given
basis is more complex than usual because (as noted be-
low) changes in crop prices affect both linear and
quadratic components of the objective function. The
energy-demand function is approximated by fitting
linear, quadratic, and cubic functions to solutions cor-
responding to many combinations of prices and de-
grees of risk aversion. The experimental design used
to generate the data involves varying six variables: en-

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