Table 3. BTU Equivalents of Energy-Related In-
puts
|
Inputs (Units) |
BTUls∕Unit |
|
Diesel Fuel (Gallon) |
135,000 |
|
Propane (Gallon) |
84,613 |
|
Nitrogen (Pound) |
25,000 |
|
Phosphorus and Potassium (Pound) |
5,000 |
|
Chemicals ($1.00 at 1979 prices) |
120,000 |
In equation (3), μ is expected income, β is the risk-
aversion coefficient, σ is the standard deviation of in-
come, and δ is the exponent of the income-variability
measure. The three members of this family considered
in this study are those for which δ= 1.0, 1.5, or 2.0.
The member for which δ = 2 is simply the quadratic-
programming objective function. For this member,
equation (3) can be recognized as a common definition
of certainty equivalence. For δ = 1, the function is the
“safety-first” criterion suggested by Katoaka. The
member corresponding to δ = 1.5 Waschosenbecause
it implies a treatment of risk intermediate to the other
two. The alternative risk-aversion measures used as
independent variables in the regressions are equal to a
(for δ = 2), ασ 5∕.75 (for δ = 1.5), and 2ασ (for δ =
D-6
To obtain regression coefficients of manageable size
without changing the analysis in any meaningful way,
the risk-aversion coefficient for δ = 2 is multiplied by
1,000, and the one for δ = 1.5 is multiplied by 100.
When these two risk-aversion measures are employed,
observations associated with the arbitrary starting value
of α = 0.05 are included in the set of observations used
to estimate the response functions. For δ = 1, the risk-
aversion measure associated with the arbitrary starting
value and the first basis change are the same for any
given price combination. However, the energy use is
quite different. Thus, an estimated response surface
based on this risk-aversion measure cannot adequately
explain the change that occurs between these two ob-
servations. Therefore, only observations associated
with basis changes are used to estimate the response
surface when δ = 1.0.
RESULTS AND IMPLICATIONS
The estimated coefficients for the quadratic ver-
sions of the response surfaces are presented in Table 5.
The quadratic functions provided much better approx-
imations than the linear functions. They provided a
better fit (smaller standard errors of estimate) to the
data. Cubic functions provided only slightly better ap-
proximations than those of the quadratic versions. The
quadratic functions (rather than the cubic functions) are
presented because they provide comparable approxi-
mations and are simpler to present and interpret.
Ordinary least squares is used to compute the re-
sponse-function coefficients. Although the usual
regression assumptions about the random errors and so
forth are not satisfied in this study, the standard error
of the estimate provides some indication of the ade-
quacy of each approximation.
Rather than attempt to interpret the quadratic func-
tions directly, we illustrate some of their implications
by presenting energy consumption elasticities for a
farmer with a high degree of risk aversion. A diesel fuel
price of $1.00, corresponding prices of other energy-
based inputs, and average com, soybeans, sorghum,
and wheat prices of $3.00, $6.90, $2.77, and $3.90,
respectively, are used. For the estimated response
function associated with δ = 1.0, a risk-aversion coef-
ficient of 2.0 is selected. Comparable risk-aversion
coefficient levels of 0.9698 and 0.0529 are used with
the estimates associated with δ = 1.5 and 2.0. The re-
sults presented in Table 6 suggest that energy con-
sumption by a crop producer is only moderately
responsive to energy price changes. Energy demand
elasticities with respect to most of the crop prices are
larger. As expected, increases in soybean and wheat
prices would reduce total farm energy consumption,
while increases in com and sorghum prices would in-
crease total farm energy consumption. This is true for
all three risk-aversion measures.
The findings of Brink and McCarl, and Dillon and
Scandizzo suggest that most farmers are less risk-
averse than the farmer considered above. The estimated
Table 4. Per Acre Energy Requirements by Crop and
Tillage Options
Crop/Tillage Option
Corn
Conventional
Minimum Till
No-till
Sorghum
Conventional
Minimum Till
No-ti11
Soybeans (30 inch rows)
Conventional
Minimum Til1
No-till
Soybeans (15 inch rows)
Conventional
Minimum Till
No-till
Wheat
BTUs∕Acre
7,071,963
6,664,263
7,773,796
6,163,507
5,755,807
6,890,094
2,634,300
2,226,600
3,447,900
2,634,300
2,226,600
3,447,900
2,013,100
6 Regardless of the value of δ, a solution is optimal only if the trade-off between expected income and the standard deviation of income implied by the objective function equals the trade-
off implied by the curve describing the E,S efficient solutions. The trade-off implied by the objective function can be obtained by total differentiation of the objective function. Setting df =
0 gives
dμ∕d<r2 - δβ(δ)σδ- “/2
Setting dμ√dσ2 equal to α (the trade-off when 8 = 2) results in the risk-aversion measures shown in the text. The fact that these functions imply somewhat different attitudes toward risk may
be confirmed by noting the effect of doubling all activity levels. For δ = 1, this would double the ‘ ‘risk premium’ ’ but for δ = 2, the ‘ ‘risk premium’ ’ would be quadrupled.
66