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.