2. Conceptual and Econometric Framework

A risk-neutral producer’s objective is to maximize expected net returns from corn production:

(1) $${\rm{ma}}{{\rm{x}}_N}\;E\left( \pi \right) = {\rm{p}}E\left( y \right) - {\rm{r}}N$$

$${\rm{s}}.{\rm{t}}.\;\;\;\;\;\;\;y = {\rm{F}}\left( N \right),\;\;N \ge 0$$

where $E\left( \pi \right)$ is the expected net return ($/ha) for the producer, p is the corn price ($/kg), r is the nitrogen fertilizer price ($/kg), $N$ is the quantity of fertilizer applied (kg/ha), and $E\left( y \right)$ is the expected yield (kg/ha).

In the case of yield response to a single input, Tembo et al. (Reference Tembo, Brorsen, Epplin and Tostão2008) give the general form of the linear response stochastic plateau (LRSP) production function as:

(2) $${y_{it}} = F\left( {{N_{it}}} \right) + {\varepsilon _{it}} + {u_t}$$

where

(3) $$F\left( {{N_{it}}} \right) = \min \left\{ {{\beta _0} + {\beta _1}{N_{it}},\;\;{P_m} + {v_t}} \right\}$$

and t indexes over time, i is the plot or location, ${y_{it}}$ is the yield in plot i in year t, ${N_{it}}$ is the nitrogen applied to plot i in year t, ${\varepsilon _{it}} \sim N\left( {0,\sigma _\varepsilon ^2} \right)$ is a random error term, ${u_{t}} \sim N\left( {0,\sigma _u^2} \right)$ is an intercept year random effect, and ${v_t} \sim N\left( {0,\sigma _v^2} \right)$ is a plateau year random effect. In the Tembo et al. (Reference Tembo, Brorsen, Epplin and Tostão2008) specification, the error and random effects are assumed to be independent. The fixed-effect parameters are the intercept ${\beta _0}$ , the slope ${\beta _1}$ , and the average plateau yield ${P_m}$ .

Tumusiime et al. (Reference Tumusiime, Wade Brorsen, Mosali, Johnson, Locke and Biermacher2011) expanded the model of Tembo et al. (Reference Tembo, Brorsen, Epplin and Tostão2008) by allowing for additional randomness in the slope parameter and relaxing the assumption of independence in the variance parameters. $F\left( {{N_{it}}} \right)$ is then

(4) $$F\left( {{N_{it}}} \right) = \min \left\{ {{\beta _0} + \left( {{\omega _t} + {\beta _1}} \right){N_{it}}\;\;,\;\;{P_m} + {v_t}} \right\}$$

and ${\omega _t} \sim N\left( {0,\sigma _\omega ^2} \right)$ is the slope random effect. The random effect (variance) parameters ${\omega _t}$ , ${v_t}$ , and ${u_t}$ are normally distributed and allowed to be correlated.

The addition of these random effects and their correlation helps the production function capture the influence of stochastic events on the yield response. ^{Footnote 1} A closed-form for the likelihood function is not available for both model specifications, but it can be estimated using Gaussian quadrature approximations. Both model specifications can be estimated using frequentist techniques for the estimation of non-linear mixed-effects models. Ouedraogo and Brorsen (Reference Ouedraogo and Brorsen2017) detail an alternative Bayesian approach.

These stochastic plateau specifications are highly flexible and can also incorporate additional non-linearities in yield response. However, the use of the most flexible specifications is often limited in practice by the relatively short samples available from most agricultural experiments. In addition, estimating the expectation of ${y_{it}}$ requires the use of numerical integration. Using the estimates of the parameters of the production function in equation (4), we calculate $E\left( {{y_{it}}} \right)$ using standard methods of Monte Carlo integration.

2.1 An Alternative Parameterization

We propose and develop an SRFootnote ² type of plateau production function with random effects. This alternative parameterization of the linear stochastic plateau model is presented as a tool for the robustness check of the main results obtained with the LRSP. It also has the objective of helping researchers avoid local optima when estimating non-linear mixed-effects models. The SR has the direct interpretation of a stochastic level of nitrogen required to reach the yield plateau. It, therefore, represents a complementary specification to the LRSP.

Following Maddala and Nelson (Reference Maddala and Nelson1974), the SR specification for yield response to a single input can be expressed in the same fashion as equation (2). In this case, $F\left( N \right)$ has a threshold point ${\beta _2}{\rm{\;}}$ that includes a random effect such that

(5) $$F\left( N \right) = \left\{ {\matrix{ {\left( {{\beta _0} + {u_t}} \right) + {\beta _1}{N_{it}};\hskip 26pt{N_{it}} \lt \left( {{\beta _2} + {\nu _t}} \right)} \cr {\left( {{\beta _0} + {u_t}} \right) + {\beta _1}\left( {{\beta _2} + {\nu _t}} \right);\;{N_{it}} \ge \left( {{\beta _2} + {\nu _t}} \right)} \cr } } \right.$$

The threshold point ${\beta _2}$ is the minimum nitrogen level necessary to reach the plateau yield. As shown in equation (5), the threshold point is an unknown parameter determined by the data. ${v_t} \sim N\left( {0,\sigma _v^2} \right)$ represents the ${\beta _2}$ year random effect which shifts the minimum nitrogen level necessary to reach the plateau yield and consequently causes the plateau to shift up or down. Like in the LRSP, here ${u_{t}} \sim N\left( {0,\sigma _u^2} \right)$ is the intercept year random effect that allows the whole function to shift up or down. Without loss of generality, we can assume no correlation among ${u_{t\;}}\;{\rm{and}}\;{v_{t\;.}}$

Like the LRSP, the SR is an empirical embodiment of von Liebig’s Law of the Minimum and describes a production function where the corn yield increases in a linear fashion with the addition of nitrogen until the threshold point ${\beta _2}$ is reached. This threshold point ${\beta _2}$ defines the yield plateau; beyond the threshold point, additional nitrogen use will not increase the corn yield.

By including random effects for all of the parameters, equation (5) becomes:

(6) $$F\left( N \right) = \left\{ {\matrix{ {\left( {{\beta _o} + {u_t}} \right) + \left( {{\beta _1} + {\omega _t}} \right){N_{it}};\hskip 32pt{N_{it}} \lt \left( {{\beta _2} + {\nu _t}} \right)} \cr {\left( {{\beta _o} + {u_t}} \right) + \left( {{\beta _1} + {\omega _t}} \right)\left( {{\beta _2} + {\nu _t}} \right);\hskip 8pt{N_{it}} \ge \left( {{\beta _2} + {\nu _t}} \right)} \cr } } \right.$$

which is similar to the model specification presented in equation (4). Likewise, the variance parameters ${\omega _t}$ , ${v_t}$ , and ${u_t}$ are assumed to be correlated and normally distributed. Allowing for nonzero covariances permits a richer set of relationships between the random effects.

The key difference between the SR parameterization and the classic LRSP is the selection of parameters of interest. LRSP treats the plateau yield, ${P_m}$ , as a parameter and adds the random effect ${v_t}$ to ${P_m}$ to account for stochastic effects in the plateau itself. In doing so, LRSP captures most of the stochastic variation of the plateau yield but does not provide the researcher with information on how much of this variation can be attributed to nitrogen use. In contrast, SR treats the minimum nitrogen level to reach the plateau yield as a switching parameter. The yield plateau and its variation are then influenced and determined by this parameter and its random effect ${v_t}$ . This parameterization is an advantage if one wishes to explore how much of the yield plateau variation can be attributed to nitrogen use.

The impact of the random effects of this parameterization can be shown graphically. Figure 1(a) represents the shifts caused by random effect ${u_t}$ , which adjusts the production function across the entire domain of the input. The adjustment in terms of yield is constant across the level of nitrogen. Figure 1(b) shows the plateau shifts caused by the random effect ${v_t}$ and Figure 1(c) represents the plateau shifts caused by the random effect ${\omega _t}$ . As it is evident from the figures, the plateau can shift over time due to a number of factors, some of which are unobservable to producers and uncontrollable even in experimental settings. The random variation of the plateau—also viewed as the maximum yield one can obtain due to a limiting input—leaves the producer with the task of determining the best use and management of nitrogen. Recognizing this task as being left to the producer, we estimate the alternative SR parameterization where ${\beta _2}$ is modeled as a parameter instead of ${P_m}$ .

Figure 1.

Potential shifts induced by the random effects.

Similar to the LRSP, the estimation of $E\left( {{y_{it}}} \right)$ for the SR parameterization requires integration that must be solved numerically, and therefore, we use standard methods of Monte Carlo integration.

3. Data

We estimate and compare the nitrogen response models using experimental data collected from a long-term, no-tillage continuous corn study conducted at the Agricultural Research Development and Education Center (ARDEC), located in Fort Collins, Colorado. Nitrogen fertilization treatments included a control (no nitrogen) plus eight treatment levels (34, 68, 101, 134, 168, 202, 224, and 246 kg/ha).Footnote ³ All plots were irrigated with a linear-move sprinkler system. The study was initiated in 2001 on a clay loam soil field, with one to two percent slope, that had previously been continuously cropped to corn for 8 years. Plots are close together and assumed to have soil uniformity. Data include 16 years of crop production, from 2001 to 2017, excluding 2008 due to an unusual and severe hailstorm that decimated the corn crop.

4. Estimation

We estimated the LRSP and SR specifications using MATLAB NLMEFIT, a solver for non-linear mixed-effects models. To calculate expected net returns and profit-maximizing nitrogen rates for the different tillage systems, we use the parameter estimates in combination with Monte Carlo methods. Ten thousand draws from the jointly normally distributed vector $\left[ {{u_t},{\omega _t},{\nu _t}} \right] \sim N\left( {0,{\rm{\Omega \;}}} \right)$ are generated using the respective variance/covariance estimates of the LRSP and SR parameterizations. Nitrogen applications varying by one unit over the interval [0, 246] are also generated. Yield expectations are computed for each of the simulated nitrogen applications evaluated at the 10,000 draws of the vectors of random effects. Finally, profit-maximizing nitrogen rates are calculated for each model using a traditional grid search since the problem involves only one choice variable. ^{Footnote 4}

6. Summary and Discussion

We estimate irrigated no-tillage continuous-corn yield response functions to nitrogen using data from a long-term study of corn yields conducted at the Agricultural Research Development and Education Center located in Fort Collins, Colorado. For our estimation, we utilize the LRSP function of Tembo et al. (Reference Tembo, Brorsen, Epplin and Tostão2008), and to check the robustness of our results we also estimate an alternative parameterization using an SR plateau specification. Under equivalent assumptions on the random parameters and correlation of the random effects, this alternative parameterization performs as good as the classic LRSP. We suggest the SR specification as a useful robustness check for the frequentist estimation of plateau-type production functions with stochastic variations. The availability of various estimation approaches allows the researcher to check the stability and robustness of parameter estimates, and in the specific case of non-linear models, it helps avoid local optima.

Results show the expected profit-maximizing nitrogen rates range from 162 to 197 kg per hectare depending on the price scenario. When compared to the reported local use of nitrogen (269 kg/ha), our estimates and the prospective Colorado Extension recommendation suggest farmers overfertilize the corn crop in the study area. Overfertilization and excess nitrogen use not only cause economic losses to farmers, but they also exact environmental costs on society (Williamson, Reference Williamson2011).

Delgado and Bausch (Reference Delgado and Bausch2005) report nitrogen-leaching problems related to the excess use of nitrogen in northeastern Colorado. This excessive use of fertilizer can be attributed in part to the common use of yield-based methods for nitrogen recommendations. These are part of the principal algorithms of most decision tool software being sold to farmers for fertilizer management (Rodriguez et al., Reference Rodriguez, Bullock and Boerngen2019). The yield-based algorithms are based on achieving a specific corn yield goal such that if a farmer wants more corn bushels per acre, he needs to apply more nitrogen (Camberato, Reference Camberato2012). Such algorithms often disregard the existence of a yield plateau and von Liebig’s Law of the Minimum. Thus, avoiding excess nitrogen applications, and promoting alternative practices such as applying nitrogen in phase with crop demand should be encouraged in the area.

Naturally, our results are conditioned on the empirical situation at hand and may not hold for all data. But as a general rule, this study indicates that the SR stochastic plateau production function is a useful approach for measuring yield response. Future research could apply this alternative parameterization along the LRSP to other crops and locations to determine optimal input rates.