Perhaps a more useful form of curve is the parabola,
or quadratic function (Fig. VI-lb). This type of curve is
representative of part of the law of diminishing returns.
It has a maximum and can demonstrate a decline in
production resulting from high or toxic use of an input if
that phenomenon is present. A similar form is the square
root function (Fig. VI-lc). The square root function also
has a maximum but it is less sharp than the parabola. In
some cases, data represent surfaces which are more complex
than can be represented by the previous curves. If the
response surface contains all three economic stages of
production, a form such as a cubic function (Fig. VI-ld)
is required. A cubic function can have a portion that
increases at an increasing rate (Stage I), another part
that increases at a decreasing rate (mostly in Stage II),
and finally a portion that decreases with increases in the
level of the input (Stage III). Another useful function
is the logarithmic function (Fig. VI-le), called by
economists the Cobb-Douglas function. This function does
not have a maximum and is therefore useful for responses
that display this characteristic (for example, herbicides
and insecticides if they do not induce decreasing
production). It is also an easier function to work with
than are some other forms.

Economic Analysis

 The analysis of data by response surfaces is related
to analysis of variance, but it is more efficient in
describing the relationships and results in more directly
applicable information for researchers and farmers. The
least-squares method, which is used to calculate the
statistical values of the surfaces by regression, is based
on all usable observations and is therefore efficient in
the use of the data. Also, once a researcher has a
mathematical function, it is possible to interpolate and
predict responses for input levels not included in the
original experimental design. With care, estimated
responses can be extrapolated beyond the range of the
data. Also, with a curvilinear function it is possible to
find the quantity or the combination of inputs which
result in maximum physical production by equating the
first derivative of the response surface to zero.
 Perhaps the most useful reason to calculate response