Crop input response functions with stochastic plateaus.

Estimating crop yield response to fertilizer and determining economically optimal levels of fertilizer has been of interest for many decades. Early efforts to estimate crop yield response functions recognized the intuitive appeal of plateau-type functional forms (Spillman 1933; Johnson 1953; Heady and Pesek 1954; Heady, Pesek, and Brown 1955). Spillman (1933) developed and applied what has come to be known as the Spillman functional form to reflect the von Liebig law of the minimum. Heady and Dillon (1961) wrote, "... most production functions probably have a von Liebig point ..." (Heady and Dillon 1961, p. 10).

Linear response plateau models have been estimated by several researchers (Ackello-Ogutu, Paris, and Williams 1985; Cerrato and Blackmer 1990; Llewelyn and Featherstone 1997). Perrin (1976) and Lanzer and Paris (1981) both concluded that linear plateau models performed as well or better than polynomial specifications. Grimm, Paris, and Williams (1987) concluded that the linear response plateau models explained crop response to fertilizer at least as well as if not better than polynomial forms. Data obtained in a 1952 experiment and published by Heady, Pesek, and Brown (1955) have been used by a number of researchers who conclude that a plateau function is a more appropriate fit than polynomial specifications (Frank, Beattie, and Embleton 1990; Paris 1992; Chambers and Lichtenberg 1996). However, Berck, Stohs, and Geoghegan (2000) disagree. They argue that plateau response functions fit the data significantly worse than an unrestricted regression, and call for further research.

Some past research has found that yield plateaus shifted across years when separate plateau models were estimated for each year (Cerrato and Blackmer 1990; Babcock and Blackmer 1994; Backman, Vermeulen, and Taavitsainen. 1997). Sumelius (1993) included annual dummy variables and thus allowed the intercept to shift across years. Although not estimated econometrically, the optimal sensing and variable rate application technology used by Raun et al. (2002) is a plateau function where the plateau varies by year and field. Given that past research suggests that yield plateaus are stochastic, including random effects for year and/or field could provide a more realistic model of producers' profit expectations.

The best-known way of estimating a stochastic plateau is Maddala and Nelson's (1974) switching regression approach used by Berck and Helfand (1990) and Paris (1992). There is a need to extend the switching regression approach to include random effects, but doing so presents a formidable estimation problem. In this article we propose an alternative to the Maddala and Nelson approach, which includes year random effects and a stochastic plateau. The ultimate goal of our research is to incorporate an economic optimization routine into the wheat plant optical real-time sensing and variable nitrogen rate fertilization technology as developed by Raun et al. (2002).

A Linear Response Model with a Stochastic Plateau

Our linear response stochastic plateau has two random effects. One random effect shifts the whole production function up or down, which might be due to hail, poor stand, insects, disease, growing degree days, freeze damage, or weed pressure. The second random effect shifts only the additional yield potential from applying more nitrogen, which is most likely due to weather such as rainfall during critical growth periods (figure 1). The function was designed specifically to match the nitrogen response function used in Raun et al.'s precision sensing algorithm. Raun et al. put in a nitrogen-rich strip, which lets them measure the yield potential of the plateau. They also measure the yield potential of an unfertilized strip. Their recommendation is then based on the difference between measurements on the nitrogen-rich strip and the unfertilized strip.

To simplify the presentation and to match our empirical model, we initially derive the model assuming a single input, a linear response, and normality. In a later section, we discuss how to extend the model to consider multiple inputs, a nonlinear response, and nonnormality. A univariate linear response stochastic plateau can be expressed as

(1) [y.sub.it] = min([[beta].sub.0] + [[beta].sub.1][x.sub.it], [[mu].sub.m] + [[v.sub.t]) + [[epsilon].sub.it] + [u.sub.t]

where [y.sub.it] is the response variable (in this case yield) in the ith plot at time t, [x.sub.it] is the level of the limiting input, [[epsilon].sub.it] ~ N(0, [[sigma].sup.2.sub.e]) is a random error term, [[v.sup.t] ~ N(0, [[sigma].sup.2.sub.v]) is the plateau year random effect, [u.sub.t] ~ N(0, [[sigma].sup.2.sub.u]) is the year random effect, [[mu].sub.m] is the average plateau yield, and [[beta].sub.0] and [[beta].sub.1] are intercept and slope parameters to be estimated. The three stochastic variables in the model ([[epsilon].sub.it], [v.sub.t], [u.sub.t]) are assumed to be independent. The discussion here is with year random effects in order to match the empirical model. A field random effect would be needed instead of a time random effect if cross-sectional data were available with several plots within each field. If the data were cross-section time-series data with multiple plots within each field then either a field-year random effect or nested random effect for field within year could be used.

[FIGURE 1 OMITTED]

To ensure continuity at the threshold, maximum yield is often defined as

(2) [[mu].sub.m] = [[beta].sub.0] + [[beta].sub.1][x.sub.m]

where [x.sub.m] is the level of the input necessary to reach the plateau. Thus, we can define ([x.sub.m], [[mu].sub.m]) as the knot point at which the response and plateau portions are splined. A linear response plateau function is a special case of (1) where [[sigma].sup.2.sub.v] = 0.

Linear response stochastic plateau and the Maddala and Nelson (1974) switching regression model are nonnested due to different covariance assumptions. To see why, let [[kappa].sub.it] = [[epsilon].sub.it] + [u.sub.t] and [[omega].sub.it] = [[epsilon].sub.it] + [v.sub.t] + [u.sub.t]. Then, the switching regression model can be written as

(3) [y.sub.it] = min([[beta].sub.0] + [[beta].sub.1][x.sub.it] + [[kappa].sub.it], [[mu].sup.m] + [[omega].sub.it]).

Thus, [[sigma].sup.2.sub.[kappa]] = [[sigma].sup.2.sub.[epsilon]] + [[sigma].sup.2.sub.u], [[sigma].sup.2.sub.[omega]] = [[sigma].sup.2.sub.[epsilon]] + [[sigma].sup.2.sub.u] + [[sigma].sup.2.sub.v], and cov([[kappa].sub.it], [[omega].sub.it]) = [rho][[sigma].sub.[kappa]][[sigma].sub.[omega]]. The switching regression model does not include year random effects because it imposes cov([[kappa].sub.it], [[kappa].sub.i't])= 0 and cov([[omega].sub.it], [[omega].sub.i't]) = 0 [for all]i [not equal to] i'. The linear response stochastic plateau model does allow for year random effects because it considers cov([[kappa].sub.it], [[kappa].sub.i't]) [[sigma].sup.2.sub.u] [for all]i [not equal to] i' and cov([[omega].sub.it], [[omega].sub.i't]) = [[sigma].sup.2.sub.v] + [[sigma].sup.2.sub.u] [for all]i [not equal to] i'. Further, the covariance between [[omega].sub.it] and [[kappa].sub.it] in the linear response stochastic plateau is cov([[kappa].sub.it], [[omega].sub.it]) = [[sigma].sup.2.sub.[kappa]], which leads to [rho] = [[sigma].sup.2.sub.[kappa]]/[[sigma].sup.2.sub.[omega]]. In the switching regression model, the likelihood function is relatively flat with respect to changes in [rho], which leads to large standard errors. For example, Paris (1992, p. 1024) found that "the error terms of the estimated model are independent," which may be a result of the high standard errors rather than the errors truly being independent. The fact that [rho] is not a free parameter in our model may be an advantage over trying to estimate a poorly identified parameter. The two models are nonnested models because neither is a special case of the other.

Determining the Profit-Maximizing Level of the Input

Consider a risk neutral decision maker whose behavior can be adequately described by expected profit maximization. Such a decision maker's objective function can be expressed as

(4) E([[pi].sub.it] | [x.sub.it]) = pE([y.sub.it]) - r[x.sub.it]

where p and r are output and input prices, and E([[pi].sub.it] \ [x.sub.it]) is expected profit. With the usual assumption of a nonstochastic plateau, the optimal input level is either the plateau level or zero. Increasing x beyond [x.sub.m] will generate negative marginal returns, equal in absolute terms to the price of the input. Therefore, with the linear response model with nonstochastic plateau, there are only two possibilities with respect to optimum input level. That is

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, the profit maximizing and yield maximizing levels of input are the same except in the case where the value marginal product of nitrogen is less than its marginal factor cost.

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

where equation (19) describes the locus of points at which the crop receives none of the K inputs in excess amounts. Note that, although equation (18) seems to indicate that [x.sub.j] has to be known in advance, imposing constraint equation (19) on the optimization process makes this requirement unnecessary. Furthermore, since equation (19) is an equality constraint, this problem can be converted to an unconstrained equivalent by substituting the constraint (equation 19) into equation (18). Thus, the objective function becomes

(20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first-order condition for expected profit maximization can then be obtained by differentiating equation (20) with respect to [x.sub.j]. Using the chain rule and the Liebnitz integral rule (McDonald and Moffitt 1980), the first-order condition is

(21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Notice that equation (8) is a special case of equation (21) and arises when [h.sub.k](*) is linear and k = {1}.

In general, if [h.sub.k](*) is linear in [x.sub.k], [for all] k, equation (21) will also be linear and could be solved analytically for [x.sub.j, as long as [F.sup.-1](*) can be computed, analytically or numerically. To illustrate this, suppose

(22) [h.sub.k]([x.sub.k]) = [[beta].sub.0k] + [[beta].sub.1k][x.sub.k], [for all] k

where [[beta].sub.0k] and [[beta].sub.1k] are the intercept and slope for the kth factor. It can be shown, by substituting equation (22) into equation (21) and rearranging terms, that

(23) [x.sub.j] = 1/[[beta].sub.1j]{[F.sup.-1](1 - 1/p [summation over (k)] [r.sub.k]/[[beta].sub.1k]) - [[beta].sub.0j]}.

Solution values for each of the other inputs can be obtained by substituting the optimal [x.sub.j] into equation (19):

(24) [x.sub.k] = 1/[[beta].sub.1k]([[beta].sub.0j] + [[beta].sub.1j][x.sub.j] - [[beta].sub.0k]), [for all] k [not equal to] j.

If [h.sub.k](*) is nonlinear in [x.sub.k], equation (21) will be nonlinear and its solution will typically require nonlinear optimization techniques. As an example, suppose [h.sub.k](*)= h(*) is a univariate quadratic function:

(25) h(x) = [[beta].sub.0] + [[beta].sub.1]x + [[beta].sub.2][x.sup.2].

Then equation (21) becomes

(26)

[partial derivative]E([pi] | x)/[partial derivative]x = p([[beta].sub.1] + 2[[beta].sub.2]x)[1 F([[beta].sub.0] + [[beta].sub.1]x + [[beta].sub.2][x.sup.2])] - r = 0

which does not have an explicit analytical solution and can only be solved numerically.

Maximum Likelihood Estimation

This section gives an overview of the procedures used to estimate equation (1). Year random effects associated with the intercept [u.sub.t] and the random error term [[epsilon].sub.it] enter equation (1) linearly but year random effects associated with the plateau, [v.sub.t], enter nonlinearly. We propose maximizing the marginal log-likelihood function directly using the theory of nonlinear mixed-effects models (Wolfinger 1993; Wolfinger 1999; SAS Institute Inc 2004).

Let [f.sub.y]([y.sub.it] | [x.sub.it], [[beta].sub.0], [[beta].sub.1], [[sigma].sup.2.sub.[epsilon]], [v.sub.t], [u.sub.t]), [f.sub.v]([v.sub.t] | [[sigma].sup.2.sub.v]), and [f.sub.u]([u.sub.t]|[[sigma].sup.2.sub.u]) denote the conditional probability density function (pdf) of [y.sub.it], the pdf of v, and pdf of u in equation (1). Let the symbol [theta] represent the vector of unknown parameters, defined as [theta] = [([[beta].sub.0], [[beta].sub.1], [[sigma].sup.2.sub.[epsilon]], [[sigma].sup.2.sub.u], [[sigma].sup.2.sub.v])].sup./]. Then the joint probability density function is

(27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [v.sub.t] and [u.sub.t] are assumed to have mean zero and variance [[sigma].sup.2.sub.u] and [[sigma].sup.2.sub.u]. The marginal likelihood function of [y.sub.it] is obtained by integrating (27) with respect to [v.sub.t] and [u.sub.t] and taking the product over t and i

(28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where a and b are the limits of integration for the distribution of [v.sub.t], c and d are the limits of the distribution of [u.sub.t], t is the number of years under consideration, and [I.sub.t] is the number of cross-sectional observations in year t, [f.sub.u](*) is the pdf of u, and [f.sub.y](*) is the pdf of [y.sub.it] conditional on both [v.sub.t] and [u.sub.t].

Theoretically, (28) is the function whose logarithmic transformation is supposed to be maximized in maximum likelihood estimation. However, because [v.sub.t] enters nonlinearly, equation (28) has no closed form and can only be approximated numerically. As with most nonlinear optimization, convergence of equation (28) is not assured. Making sure that convergence occurs at a global maximum rather than a local maximum poses an additional challenge. Several approaches are available for approximating such integrals, including Monte Carlo integration and Gaussian quadrature. Of all the numerical integration techniques, Gaussian quadrature is believed to offer the highest degree of accuracy (Liu 1997; Stiegert and Hertel 1997; Ghomi and Hashemin 1999).

We tried various combinations of starting values, optimization algorithms, and quadrature methods available in the SAS NLMIXED (SAS Institute 2004) procedure. Scaling was used so that the diagonal elements of the Hessian did not differ too much. The largest likelihood value was obtained using first-order approximation starting values, and then non-adaptive Gaussian quadrature. Convergence was slow with nonadaptive Gaussian quadrature, but with adaptive Gaussian quadrature the estimation failed.

Data and Empirical Procedures

Data are from a long-term experiment conducted at the North Central Oklahoma research station located near Lahoma, Oklahoma in the Southern Great Plains of the United States. The study was established in 1970 to investigate winter wheat grain yield response to fertilizer application, using a randomized complete block design (Westerman et al. 1994; Raun et al. 1998). The treatments include a control (no nitrogen) and five levels of nitrogen (20, 40, 60, 80, and 100 pounds per acre). Each treatment was replicated four times. Data from 36 years (1971-2006) were used for estimation. Parameter estimates of the stochastic plateau, nonstochastic plateau, and the switching regression models are estimated using SAS NLMIXED. Then, these estimates are used in MAPLE (2002) to determine the maximum expected profit for the stochastic and nonstochastic plateau models. The analytical solution was verified using a grid search. The expected profits for the switching regression model were estimated using Monte Carlo integration and grid search in SIMETAR (2006) and verified with MAPLE.

The data from this experiment are from plots that are relatively close together and selected for soil uniformity. With other data sets, it might be appropriate to further decompose the random effects to include spatial random effects within a field. Adding such a spatial random effect to the entire equation would not affect the derivation of the optimal level of fertilizer in (14). But the standard deviation of a spatial random effect in the plateau error would need to be added to [[sigma].sub.v] in (14). The SAS NLMIXED procedure used here cannot currently estimate a nonlinear model with multilevel random effects, so a different estimation procedure would need to be used. The possibility of spatial random effects is important in developing nitrogen recommendations from precision sensing. Raun et al. (2002) have developed algorithms that make recommendations for applying the same nitrogen level to the whole field as well as applying a different level to each square meter. If there is spatial variability in the plateau, then the whole field precision-sensing recommendation would have some remaining variability in the plateau that should be considered in determining the optimal amount of nitrogen to apply.

Results

The estimation results for the linear response plateau function, linear response stochastic plateau function, and Maddala and Nelson switching regression models are reported in table 1. All parameters and variance components are significant at the 1% level, except the variance correlation p in the switching regression model. The hypothesis that the plateau is non- stochastic ([H.sub.0] : [[sigma].sup.2.sub.v] = 0) is rejected at the 1% level of significance based on a likelihood ratio test. The likelihood dominance criterion for testing competing nonnested models (Pollack and Wales 1991) indicated that the stochastic plateau model is about 668 times more likely than the switching regression model.

The expected plateau wheat grain yield is about 42, 41.8, and 39.7 bushels per acre for the nonstochastic plateau, stochastic plateau, and switching regression models. The threshold level of nitrogen is 70.6, 57.7, and 38.7 pounds per acre for the switching regression, nonstochastic plateau, and stochastic plateau models. The key difference is that the estimated marginal productivity of nitrogen is higher with the stochastic plateau model and so less nitrogen is needed. Nitrogen productivity is the lowest with the switching regression model and so using it would suggest more nitrogen is needed. There may be some attenuation bias in the nonstochastic plateau and switching regression models that causes the low estimates of nitrogen productivity.

[FIGURE 2 OMITTED]

The optimum level of nitrogen when the plateau is nonstochastic is either zero or 58 pounds per acre. With wheat price assumed to be $3 per bushel, the value of marginal productivity of nitrogen is $0.81 per pound. The optimal choice of nitrogen remains at 58 pounds per acre as long as the price of nitrogen is above zero and is less than the value of marginal productivity of $0.81 per pound.

For the stochastic plateau and switching regression models, the optimal level of nitrogen changes with the price of nitrogen. Figure 2 contains the optimal level of nitrogen for three price ratios for the linear response stochastic plateau, linear response plateau, and switching regression models (nitrogen prices at $0.01, $0.2, and $0.6 per pound and wheat price at $3.0 per bushel). The optimal level of nitrogen at these three prices is 114, 69, and 38 pounds per acre with the stochastic plateau model, and 217, 102, and 0.0 pounds per acre with the switching regression model. Thus, the models lead to quite different optimal levels of nitrogen.

Notice that when r = $0.2 per pound, which is close to historical prices of nitrogen, the optimal level of nitrogen is less under the linear response stochastic plateau model than it is under the linear response plateau and switching regression models. The major reason for this difference is the greater marginal productivity of nitrogen with the linear response stochastic plateau model. As figure 2 shows, fertilizer recommendations with the nonstochastic plateau and switching regression models can be either less than or greater than recommendations with the stochastic plateau depending on relative prices. This may explain the seemingly contradictory empirical observations, with some researchers arguing that farmers applied less nitrogen than recommended (de Janvry 1972; Ryan and Perrin 1974) and others arguing otherwise (Babcock 1992). Figure 2 offers a potential explanation of the differing findings. Current recommendations from Oklahoma State University's Cooperative Extension Service are to apply two pounds of nitrogen for each bushel of yield goal. With a yield goal of 42 bushels per acre, the advice would be to apply 84 pounds of nitrogen per acre. Thus, recommended rates exceed those obtained with either plateau model.

Table 2 shows expected profits for each of the cases shown in figure 2. Again, profits will vary according to the value of the output/input price ratio. The losses from using a nonoptimal level of nitrogen are small. Thus, it should not be a surprise to observe successful farmers using a range of nitrogen levels. The wheat yield linear response to nitrogen stochastic plateau function provides an example of what Pannell (2006) calls flat earth economics.

The perfect information case provides the upper bound of the benefits that can be attained using the true "optimal" nitrogen level if it could be determined. The difference between the expected profits with the perfect information scenario and the stochastic plateau is $9.56 per acre (with r = 0.2), which represents all benefits that can be captured from using information to guide nitrogen application. So, the benefit of a perfect information precision system for applying nitrogen would be $9.56 per acre, which is similar to the estimates found by Biermacher et al. (2006).

Raun et al. (2002) use a similar production function, but they estimate the marginal product of nitrogen based on the quantity of nitrogen in the harvested wheat. Our estimated marginal product of nitrogen is less than what Raun et al. (2002) assume. In addition, Raun et al. (2002) treat their plateau as nonstochastic and do not consider the additional nitrogen needed due to remaining uncertainty about the plateau.

Conclusions

A number of researchers argued that crop-response-to-nitrogen functions should include a yield plateau. In prior work, the plateau has usually been assumed nonstochastic. However, agronomic research suggests that yield plateaus can vary across fields and/or years. Available models that consider a stochastic plateau, including switching regressions, are not readily extendable to consider field or year random effects.

We develop a linear response stochastic plateau model with random effects that shift the intercept and the plateau. Our model and the Maddala and Nelson (1974) switching regression model used in previous studies are nonnested. An additional advantage of our model is in estimating the correlation between the yield response and plateau errors, which is treated as a free parameter in the switching regression model. This correlation is poorly identified in the switching regression model, which leads to large standard errors. Our approach avoids this identification problem. Of the six discrete treatment levels of 0, 20, 40, 60, 80, and 100, the 60-pound treatment has the largest average profit of $112 per acre. The expected profit of $108 per acre estimated with the stochastic plateau model is much closer to this actual average profit of $112 per acre than is the expected profit of $89.9 per acre calculated with the switching regression model. With current prices, the optimal level of nitrogen is lower with the stochastic plateau than with the nonstochastic plateau and switching regression models.

The use of a stochastic plateau provides insight into why farmers may apply more or less nitrogen than would appear optimal. The optimum level of nitrogen for the linear response stochastic plateau model can be lower or higher than that of linear response plateau and switching regression models depending on the output/input price ratio as well as differing parameter estimates. This may explain the seemingly contradictory empirical observations, with some researchers arguing that farmers applied less nitrogen than recommended (de Janvry 1972; Ryan and Perrin 1974) and others arguing otherwise (Babcock 1992). Also, the expected profit function is relatively flat with current prices and so the optimal level is likely difficult for farmers to determine. Results also showed that the highest benefit from using additional information to guide nitrogen application is $9.56 per acre. Because information-enhancing technologies such as precision farming are costly, any investment cost needs to be carefully weighed against potential benefits of about $10 per acre.

[Received January 2007; accepted September 2007.]

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Gelson Tembo is lecturer I, Department of Agricultural Economics and Extension Education, University of Zambia, Lusaka; B. Wade Brorsen is regents professor and Jean & Patsy Neustadt chair; Francis M. Epplin is Charles A. Breedlove Professor, Department of Agricultural Economics at Oklahoma State University; and Emilio Tostao is assistant professor, Division of Agricultural Economics at Universidade Eduardo Mondlane, Maputo, Mozambique. The authors thank professor William R. Raun, for providing access to data and information regarding the experiment, and Francisca G.C. Richter for helpful comments and technical assistance. Journal paper AEJ-245 of the Oklahoma Agricultural Experiment Station, Projects H-2237, H-2403. Table 1. Summary of Regression Results for Wheat Yield Response Functions

Estimates and

Standard Errors by

Type of Response

Function (a)

Linear response Statistic Symbol Stochastic Plateau Intercept [[beta].sub.0] 26.32

(0.93) Level of nitrogen (lbs) [[beta].sub.1] 0.40

(0.02) Expected plateau yield

(bus) [[mu].sub.m] 41.78

(0.82) Nitrogen at expected

plateau (lbs) [x.sub.m] 38.32

(2.70) Variance of plateau

yield [[sigma].sup.2.sub.v] 163.92

(28.85) Variance of year random

effect [[sigma].sup.2.sub.u] 75.81

(10.39) Variance of error term [[sigma].sup.2.sub.

[epsilon]] 28.51

(1.46) Variance correlation [rho] 0.39

(0.041) Log-likelihood (b) -2,722.45

Estimates and Standard Errors

by Type of Response Function (a)

Linear response Switching Statistic Plateau Regression Intercept 26.31 24.54

(1.51) (0.77) Level of nitrogen (lbs) 0.27 0.22

(0.02) (0.02) Expected plateau yield

(bus) 42.06 39.73

(1.45) (0.91) Nitrogen at expected

plateau (lbs) 57.71 70.56

(3.54) (6.48) Variance of plateau

yield 205.22

(16.46) Variance of year random

effect 68.93

(17.01) Variance of error term

53.49 53.70

(2.66) (1.82) Variance correlation 0.28

(0.17) Log-likelihood (b) -2,920.45 -3,390.8 (a) Standard errors are in parentheses. (b) The null hypothesis that the nonstochastic plateau is the correct model (i.e.. [H.sub.0]: [[sigma].sup.2.sub.v] = O) is rejected at any conventional level of significance based on a likelihood ratio test. The calculated value of the likelihood ratio statistic is 396, which is considerably above the [[chi square].sub.(1,0.01)] critical value of 6.63. Pollack and Wales's (1991) likelihood dominance criterion for testing nonnested models indicated that the stochastic plateau model is about 668 times more likely than the switching regression model. Table 2. Maximum Expected Profit Per Acre, Assuming the Linear Response Stochastic Plateau Is the Correct Model and the Price of Wheat Is $3 Per Bushel

Profit by Price of Nitrogen (r) Model $0.01 $0.02 $0.06

[lb.sup.-1] [lb.sup.-1] [lb.sup.-1] Linear response stochastic

plateau (a) 124.08 108.12 87.01 Linear response plateau (b) 118.39 107.42 84.34 Switching regression (c) 116.78 89.86 70.46 Perfect information (d) 124.96 117.68 102.35 (a) For the stochastic plateau model, the optimal quantity of nitrogen is 114.40 lbs/acre, 69.20 lbs/acre, and 38.65 lbs/acre when r is equal to $0.1 [lb.sup.-1] 11. $0.2 [lb.sup.-1], and $0.6 [lb.sup.-1]. This is translates into an expected yield, E(y|x), of 72.47 bu/acre, 54.23 bu/acre, and 41.91 bu/acre. (b) For the nonstochastic plateau model. at all three prices, [x.sub.m] = 57.71 bu [acre.sup.-1], which translates into E(y|x) = 42.58 bu/acre, and 24.54 bu/acre. (c) For the switching regression model, the optimal quantity of nitrogen is 216.9 lbs [acre.sup.-1], 102.4 lbs [acre.sup.-1], and 0.0 lbs/acre when r is equal to $0.01 [lb.sup.-1], $0.2 [lb.sup.-1], and $0.6 [lb.sup.-1]. This translates into an expected yield, E(y | x), of 71.22 bu/acre, 46.58 bu/acre, and 24.54 bu/acre. (d) For the perfect information case the average optimal quantity of nitrogen at all prices is [x.sub.m] = 38.32 lbs/acre, and the average plateau yield of [u.sub.m] = 41.78 bu/acre is obtained.