Crop input response functions with stochastic plateaus.

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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