Read this article now - Try Goliath Business News - FREE!   
You can view this article PLUS...

• Over 5 million business articles
• Hundreds of the most trusted magazines, newswires, and journals (see list)
• Premium business information that is timely and relevant
• Unlimited Access

Tell Me More   Terms and Conditions

Purchase this article for $4.95

...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])= and cov([[omega].sub.it], [[omega].sub.i't]) = [for...

NOTE: All illustrations and photos have been removed from this article.



More articles from American Journal of Agricultural Economics
Differentiation and synergies in rural tourism: estimation and simulat..., May 01, 2008
Vernon W. Ruttan. Social Science Knowledge and Economic Development: A..., May 01, 2008

Looking for additional articles?
Search our database of over 3 million articles.

Looking for more in-depth information on this industry?
Search our complete database of Industry & Market reports by text, subject, publication name or publication date.

About Goliath
Whether you're looking for sales prospects, competitive information, company analysis or best practices in managing your organization, Goliath can help you meet your business needs.

Our extensive business information databases empower business professionals with both the breadth and depth of credible, authoritative information they need to support their business goals. Whether it be strategic planning, sales prospecting, company research or defining management best practices - Goliath is your leading source for accurate information.