The problem is not so straightforward when considering field and/or
year random effects on the yield plateau. However, the optimal input
level for the stochastic plateau model can be derived using theorems
developed for Tobit models. In this case, the plateau is the censoring
point since we only observe yields that are on or to the left of the
plateau. Taking expectations in equation (1) leads to E([y.sub.it]) =
E(min [[beta].sub.0] + [[beta].sub.1] [x.sub.it], [[mu].sub.m] +
[v.sub.t]]) since [[epsilon].sub.it] and [u.sub.t] appear in the
equation linearly, while [v.sub.t] occurs nonlinearly. Then, using
theorem 20.3 in Greene (2000, p.907), it can be shown that if yield
plateau [y.sub.m] ~ N([[mu].sub.m], [[sigma].sup.2.sub.v]), and
E([y.sub.it] | [y.sub.m] [greater than or equal to] [[beta.sub.0] +
[[beta].sub.1] [x.sub.it]) = [[beta].sub.0] + [[beta].sub.1] [x.sub.it]
and E([y.sub.it] | [y.sub.m] [less than or equal to] [[beta].sub.0] +
[[beta].sub.1] [x.sub.it]) = [y.sub.m], then
(6) E([y.sub.it]) = (1 - [PHI])a + [PHI])([[mu].sub.m] -
[[sigma].sub.v][phi]/[PHI])
where a = [[beta].sub.0] + [[beta].sub.1] [x.sub.it], [PHI] =
[PHI][(a - [[mu].sub.m])/[[sigma].sub.v]] = prob ([y.sub.m] [less than
or equal to] a) is the cumulative normal distribution function and [phi]
= [phi] [(a - [[mu].sub.m])/[[sigma].sub.v]] is the standard normal
density function. The term (1 - [PHI]) in equation (6) gives the
probability of being on the plateau and the term [PHI] ([[mu].sub.m] -
[[sigma].sub.v] [PHI]/[PHI]) gives the contribution to the expected
value when below the mean plateau yield.
Substituting equation (6) into equation (4) yields
(7) E([[pi].sub.it] | [x.sub.it]) = p[(1 - [PHI])([[beta].sub.0] +
[[beta].sub.1][x.sub.it]) + [PHI}([[mu].sub.m] -
[[sigma].sub.v][PHI]/[PHI])] - r [x.sub.it].
Equation (7) describes the profit-maximizing decision maker's
utility function under conditions of a linear response stochastic
plateau function.
The first-order condition for profit maximization can be obtained
by differentiating equation (7) with respect to [x.sub.it] and setting
that derivative equal to zero. That is,
(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
By the chain rule, noting that [partial derivative][PHI]/[partial
derivative][x.sub.it] = [phi][[beta].sub.1]/[[sigma].sub.v] and [partial
derivative][phi]/[partial derivative][x.sub.it] =
z[phi][[beta].sub.1]/[[sigma].sub.v] with z = [([[beta].sub.0] +
[[beta].sub.1] [x.sub.it]) - [[mu].sub.m]]/[[sigma].sub.v] (McDonald and
Moffitt 1980), equation (8) reduces to
(9) [partial derivative]E([[pi].sub.it] | [x.sub.it])/[partial
derivative][x.sub.it] = p[[beta].sub.1](1 - [PHI]) - r = 0.
According to (9), the decision maker determines the level of input
by equating the value of marginal expected product, p[[beta].sub.1] (1 -
[PHI]), to the input price, r. Notice that the second-order condition
[[partial derivative].sup.2]E([[pi].sub.it] \ [x.sub.it])/[partial
derivative][x.sup.2.sub.it] = p[[beta].sup.2.sub.1][phi]/[[sigma].sub.v]
< 0 is satisfied at expected profit maximum. This implies that
imposing a stochastic knot on a linear response function yields a
strictly concave expected profit function. Berck and Helfand (1990) drew
a similar conclusion with regard to the switching regression model.
Rearranging terms in equation (9) leads to
(10) [phi] = 1 - r/(p[[beta].sub.1]).
Because 0 [less than or equal to] [phi] [less than or equal to] 1,
(10) applies only if the condition
(11) [[beta].sub.1] [greater than or equal to] r/p
is satisfied. Using equation (10) and noting that [PHI] =
[PHI][([[beta].sub.0] + [[beta].sub.1][x.sub.it] -
[[mu].sub.m])/[[sigma].sub.v]], the profit maximizing level of the input
([x.sup.*]) can be expressed as
(12) [x.sup.*.sub.it] = 1/[[beta].sub.1][[[PHI].sup.1]
[[sigma].sub.v] + [[mu].sub.m] - [[beta].sub.0]]
where [[PHI].sup.-1] = [[PHI].sup.-1][1 - r/(p[[beta].sub.1])] is
the inverse standard normal cumulative distribution function.
With [y.sub.m] ~ N([[mu].sub.m], [[sigma].sup.2.sub.v]), standard
normal probability tables available in statistics and econometrics
textbooks or functions in statistical software can be used to
approximate [[PHI].sup.-1] in (12). This can be done by converting
E([y.sub.m] | [x.sub.m] = [x.sub.it]) into a standard normal variate
[Z.sub.[alpha]]. That is
(13) [Z.sub.[alpha]] = [[beta].sub.0] + [[beta].sub.1][x.sub.it] -
[[mu].sub.m]/[[sigma].sub.v]
where [alpha] = 1 - [PHI] = r/(p[[beta].sub.1]) is the observed
probability in the right-hand tail of the N(0, 1) distribution and [PHI]
= [PHI] [([[beta].sub.0] + [[beta].sub.1][x.sub.it] -
[[mu].sub.m])/[[sigma].sub.v]], the standard normal cumulative
distribution function of [y.sub.m] evaluated at [[beta].sub.0] +
[[beta].sub.1][x.sub.it], is defined in equation (10). The optimum input
level can then be determined by rearranging terms in equation (13) as
(14) [x.sup.*.sub.it] = 1/[[beta].sub.1]([[mu].sub.m] +
[Z.sub.[alpha]][[sigma].sub.v] - [[beta].sub.0]).
Equation (14) indicates that the optimum input level increases with
the variance of the plateau. Our long-term goal is to adapt equation
(14) for use with the optical sensing and variable rate technology of
Raun et al. (2002). In that case, [[beta].sub.0], [[beta].sub.1],
[[mu].sub.m], and [[sigma].sub.v] could all be made a function of sensor
readings. Note that if these measures are made with error, the
additional parameter uncertainty would need to be included.
To complete the computation, r and p can be replaced with data from
input and output markets, and the parameters, [[beta].sub.0] and
[[beta].sub.1], can be replaced by their statistical estimates. Because
x cannot be negative and [[beta].sub.1] [greater than or equal to] 0,
equation (12) is valid only if
(15) [[PHI].sup.-1][[sigma].sub.v] + [[mu].sub.m] - [[beta].sub.0]
[greater than or equal to] 0.
Thus, both conditions (11) and (15) need to hold for the optimal
input level to be nonzero.
Under what circumstances will the optimum nitrogen level for the
linear response stochastic plateau model be equal to the optimum level
of nitrogen for the linear response plateau model? In other words, when
will [x.sup.*] in equation (12) equal the deterministic parameter
[x.sub.m] in equation (5)? First note that equation (12) is similar to
equation (2). If [[PHI].sup.-1][1 - r/(p[[beta].sub.1])] = 0, then
[x.sup.*] would equal E([x.sub.m]). So, again, we can rephrase the
question to say when will the inverse cumulative density function (cdf)
of the standard normal random variable that has been derived from
[y.sub.m] equal zero, its expected value? In the case of a symmetric
distribution where mean and median coincide, this will occur when 1 -
r/(p[[beta].sub.1]) equals 0.5. For values below this level, i.e., when
r/(p[[beta].sub.1]) > 0.5, the optimum level of nitrogen under the
stochastic plateau model will be lower than the one obtained with a
nonstochastic plateau model. Conversely, if r/(p[[beta].sub.1]) < 0.5
our model predicts that there will be a tendency to apply more nitrogen
than what the nonstochastic plateau model formula suggests, assuming all
other parameters are the same. The tendency of the quantities of
nitrogen actually applied by the farmers to fall below and above the
recommended rates based on the nonstochastic plateau is in fact what has
been observed. Given historical output/input price ratios observed in
the United States, farmers tend to apply more nitrogen than what the
certainty case predicts (Babcock 1992).
Generalizing to Multiple Input and Higher-Order Plateau Functions
with Nonnormality
The procedures developed for the univariate LRSP can be extended to
multiple inputs, higher-order plateau functions, and nonnormality of the
plateau random effect. While we only have the data to estimate a
univariate model, later researchers may be interested in estimating a
more general model. Suppressing the subscripts for individual plot i and
year t, a general plateau response function can be expressed as
(16) y = min ([h.sub.1]([x.sub.1]), ..., [h.sub.K]([x.sub.K]),
[y.sub.m]) + [epsilon]
where [x.sub.k] is the level of the kth factor and [h.sub.k](*) is
an increasing function of [x.sub.k], [for all] k = 1, ..., K. The
plateau [y.sub.m] is assumed to be distributed with cumulative
distribution function F and density function f. The random error term,
[epsilon], may also include a random effect as with the univariate
model. Note that under normality, the multivariate model has three
variance parameters just like the univariate model. With the switching
regression approach, the number of parameters expands exponentially as K
increases. If it is assumed that the elasticity of input substitution is
zero, then the implied separability permits separate augmentation of the
plateau with [h.sub.k]([x.sub.k]), [for all] k. By applying the result
in equation (6) to equation (16), it can be shown that
(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where x is a vector of all K factors, the [x.sub.k]'s. Let
[x.sub.j] [subset] {[x.sub.k] | k = l, 2, ..., j, ..., K} be the most
limiting factor. Then the expected profit maximizing decision
maker's problem is
(18)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
To maximize expected profit, the solution will need to determine
only the necessary quantities of all inputs. Thus, the decision maker
will solve equation (18), subject to [h.sub.j]([x.sub.j]) =
[h.sub.k]([x.sub.k]), or
(19) [x.sub.k] = [h.sup.-1.sub.k] ([h.sub.j]([x.sub.j])), [for all]
k [not equal to] j
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