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a Iowa Soybean Assoc., 4554 114th St., Urbandale, IA 50322
b Dep. of Agronomy, Iowa State Univ., Ames, IA 50010
c Dep. of Plant Science, Univ. of Connecticut, Storrs, CT 06269
* Corresponding author (pkyveryga{at}iasoybeans.com)
Received for publication November 26, 2006.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Abbreviations: EOR, economic optimum rate • EXP, exponential model • IRBE, incremental break-even rate • IRDP, incremental rate that gives the desired level of profit • IRNR, incremental rate that gives no yield response • LRP, linear response and plateau model • QRP, quadratic response and plateau model • QUAD, quadratic model; SR, square root model
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Spatial and temporal variability in crop responses to N fertilizer is a serious problem when calculating EORs (Blackmer et al., 1992; Mamo et al., 2003; Miao et al., 2006; Oberle and Keeney, 1990; Scharf et al., 2005). This variability is unavoidable because fertilizers are often applied before plants grow and because crop yield responses are greatly affected by weather and other factors that occur during plant growth. The problem of variability is exacerbated by the tendency of relationships between rates of N fertilization and yields to have only slight curvature at near-optimal rates of fertilization. Therefore, models used to quantitatively relate rates of N to yields often disagree substantially when calculating EORs (Black, 1993; Cerrato and Blackmer, 1990; Colwell, 1994; Frank et al., 1990; Heckman et al., 1996). The problems of model bias and variability in yield response interact because models inject significant systematic errors when curvatures are slight, because variability in yield response makes it difficult to identify the best model for any given site, and because the best model often varies due to differences observed in the nature of the response among sites.
The problem of calculating EORs amid variability in yield response is great enough that many researchers have concluded that producers cannot afford the economic risk associated with using EORs and should apply additional N (Babcock, 1992; Bullock and Bullock, 1994; Sheriff, 2005; Yadav et al., 1997). It is suggested that this extra N, often called insurance N, enables producers to benefit from situations where conditions are favorable for unusually large yield responses to N and, therefore, unusually large profits from fertilization. Despite the fact that insurance N is often applied, quantitative methods have not been described for calculating optimal amounts of insurance N.
A new multistep procedure has been developed to disaggregate the problems of model bias and variability in yield response when calculating EORs for N in corn (Kyveryga et al., 2007). This multistep procedure integrates established methods of calculating EORs and concepts of ex post and ex ante analyses. A sequence of steps and iteration of these steps makes it possible to minimize the problem of model bias in some steps while reducing unexplained variability in yield responses in other steps. Unexplained variability is reduced by forming categories that are based on information that is available to producers at the time of fertilization.
When compared with methods used in the past, the new procedure places much greater emphasis on the need for collecting and analyzing samples of yield responses that randomly distributed within the specific areas of interest. The results are called category-specific ex post EORs. In contrast to EORs calculated for individual trials, numerical values for category-specific ex post EORs vary with the amounts of information included in the analysis and vary only within a small range with normal year-to-year variability in weather or other factors. The overall procedure essentially involves a systematic search of all data collected in the past to identify the rate of fertilization that is most likely to maximize profits for any specified range of field conditions. Development of the new procedure was driven by the underlying assumptions that (i) the inability to collect a large number of observations of yield responses has been a primary barrier for calculating EORs in the past, and (ii) this problem would soon be alleviated by the use of precision farming technologies to collect data from hundreds of trials on producers' fields.
Observations made during development of the new procedure suggested there was a need to reevaluate the commonly accepted idea that EORs provide the best benchmark when discussing rates that are most likely to maximize profits for producers. A major reason is that EORs are best only in situations where producers have unlimited capital (i.e., no alternative place to invest their dollars). Although it has been recognized that this simplifying assumption of unlimited capital is not always appropriate (Voss, 1975), penalties for not considering this assumption seemed to be small relative to the uncertainty caused by interactions of model bias, variability in crop yield response, and small numbers of the yield response trials. Other useful benchmarks when evaluating rates of N fertilization to maximize profits, for example, the minimum rate, have been discussed in Waugh et al. (1973) and in Voss (1975).
Here we describe methods for calculating some alternative economic benchmarks for optimal rates of N fertilization. In addition, we explore the possible benefits of using these benchmarks when discussing optimal rates of N fertilization in situations where the problems of model bias and variability have been minimized (Kyveryga et al., 2007). Our approach uses relatively simple economic principles to integrate the concepts of alternative benchmarks for optimal rates as discussed by Voss (1975) with the use of discrete marginal analysis as discussed by Waugh et al. (1973).
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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The NONLIN procedure of SAS software (SAS Institute, 2002) was used to fit five yield response models: quadratic (QUAD), exponential (EXP), square root (SR), linear response and plateau (LRP), and quadratic response and plateau (QRP). Maximum yields for QUAD and SR models were calculated by equating the first derivatives of yield response functions to zero, solving for rates, substituting those rates into yield response function, and finally solving models for these rates.
The marginal products (MPs) of the yield response models were estimated by calculating the first derivatives of the models by using Mathcad 2000 Professional software (MathSoft, 2000). The MPs were plotted as continuous functions for the five models described above, which are commonly used to describe the relationships between yields and rates of N application. The MPs were plotted for all rates of N applied and for three selected rates within the near-optimal range. The methods used for discrete marginal analysis is found in introductory textbooks of microeconomics (Hyman, 1993; Samuelson et al., 1995) and agricultural economics (James and Eberle, 2000). Discrete marginal products (DMPs) were calculated by dividing a yield increase 
Yi (kg ha–1) resulting from an increase in rate of fertilization by the amount Ni (kg ha–1) of fertilizer N applied by the increment as indicated in the following equation.
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To enable clear enumeration of our ideas, we need to explicitly define some phrases and use them consistently throughout this manuscript. Accordingly, the phrase EOR of fertilization is used only to denote a rate that is calculated by using methods described by Heady et al. (1955) and Heady and Dillon (1961). As part of this definition, it must be recognized that numerical values described as EORs may contain some errors and, therefore, EORs have to be considered only as estimates of the rates that are truly optimal for conditions specified or assumed. We use the phrase economically optimal rate (without the acronym EOR for economic optimum rate) to denote the rate that is truly optimal and usually not known. Unless this distinction is clearly made, it is impossible to discuss errors and uncertainty in calculated EORs.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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The shape of MP curves from models used for describing relationships between yields and N rates is also shown in Fig. 3. For example, the QUAD model always imposes a bias so that the relationships between MPs and rates of N fertilization must be exactly linear across all rates of N. The EXP and SR models always impose a bias so that these relationships are curved, with the amount of curvature decreasing when reaching maximum yields. The subtle bias injected by the shape of each model used to describe relationships between N rates and MPs cannot be detected by analysis of the relationships between N rates and yields. Part of the problem is that trends across all N rates are subtly smoothed by using any given model. This hidden problem of model bias makes it difficult to objectively analyze which shape of the curvature that is, which model, best describes the relationships.
MPs from Discontinuous Models and Rate of Change in MPs
The use of the LRP and the QRP models, which are discontinuous models, to relate rates of N fertilization to yields resulted in abrupt changes in MPs at the rate of fertilization usually selected as optimal (Fig. 3). There is a need to question the value of such models for refining estimates of optimal rates because there are no theoretical reasons to expect abrupt changes in MPs at optimal rates of fertilization (Black, 1993; Sinclair and Park, 1993). The problem is less with the QRP model than with the LRP model, but the abrupt changes in MPs from positive values to zero substantially limit the ability to refine estimates of optimal rates in this range. Another problem with discontinuous models is that the N rate at which this abrupt change in MPs occurs (i.e., the rate at which two separate model segments are merged) is largely determined by data collected at rates that are substantially greater than or less than the optimal rate.
The second derivatives for the three continuous models, the QUAD, EXP, and SR, are shown in Fig. 4 . While the first derivatives show instantaneous slopes or the rate of change in yields when increasing rates of N fertilization, the second derivatives show the rate of change in MPs as fertilizer rate increases. The rate of change in MPs as shown in Fig. 4 depends on the model choice. For example, the rate of change in MPs is constant for the QUAD model but it is gradually decreasing for the EXP and SR models when increasing rates of N fertilization. For the QRP model, the second derivative is constant for the response portion (data not shown) and it is equal to zero for the plateau portion of the curve. The second derivative for the LRP model is equal to zero for both the response and the plateau portions (data not shown). These observations provide additional evidence that discontinuous models may have a limited value when refining EOR estimates in the near-optimal range.
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Effects of Change in Fertilizer-to-Grain Price Ratios
The MP curves in Fig. 3 provide a relatively simple way to assess the effects of changing the fertilizer-to-grain price ratio on disagreements among models used to calculate MPs or EORs. For example, if one considers an increase in the price ratio from 5 to 10, then the marginal cost increases to 10 kg grain kg–1 fertilizer N, and disagreements among the EXP, SR, and QRP models become greater. Although the effects of price ratios on disagreements in calculated values for EORs can be calculated from models that relate rates of fertilization to yields or relate EORs to fertilizer-to-grain price ratios as shown in Fig. 5
, these relationships are more difficult to interpret. Curves relating rates of N fertilization to MPs (Fig. 3) are easier to interpret because the key characteristic measured (i.e., the shape of the curve) is clearly distinguished from the price ratio, which is arbitrarily selected during the analyses. Plots of the MP curves are more desirable because they make it easier to recognize that the effects of the shape of the curve and the price ratio should be recognized as independent factors.
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Linearity of MPs and DMPs in the Near-Optimal Range
Inserts in Fig. 3 show MP curves of continuous models fitted only to rates in the near-optimal range for each crop category. A key point illustrated in the inserts is that MPs tend to be near linearly related to rates of N fertilization in the near-optimal range. These models, however, differ in the range of N fertilization over which these relationships are linear or near linear. The inserts in Fig. 3 do not show the effects of considering the near-optimal range for discontinuous models because discontinuous models are often difficult to fit to data with only small yield increase and when analyzing only three incremental increases in N rates.
The relationships between DMPs and rates of N fertilization also become near linear when considering the near-optimal range (Fig. 2). The linear relationships between DMPs and rates of N fertilization were statistically significant when all observations were analyzed as shown in Fig. 6 with r2 values: 0.05 (P = 0.008) for all trials (Fig. 6A), 0.05 (P = 0.05) for corn after corn (Fig. 6B), and 0.09 (P = 0.007) for corn after soybean (Fig. 6C). In contrast, relationships between yields and rates of N fertilization were statistically insignificant when fitting models to all observations in the near-optimal range for each crop category (data not shown). Higher r2 values for the relationships between DMPs and rates of N fertilization are expected because each DMP value is expressed as a yield response to N and, therefore, effects of factors other than N fertilizer tend to be reduced when analyzing DMPs compared with analyzing yield values. These observations also explain why many studies in the past (Anderson and Nelson, 1975; Blackmer, 1986; Cerrato and Blackmer, 1990) recommended applying a wide range of N rates when calculating EORs.
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Studying relationships between DMPs and rates of N fertilization across many sites offers three important advantages. First, this method eliminates the need to fit models and, therefore, avoids the problem of model bias. Second, optimal rates of N fertilization can be calculated by applying a smaller number of N rates that are restricted to the near-optimal range. This can help to increase the number of yield response trials within the area of interest. Third, the process of calculating EORs becomes easier and less computer intensive. The relationships shown in Fig. 7 provide a simple and practical way to calculate DMPs and EORs for various categories, estimate the amounts of uncertainty associated with these values, and assess the importance of differences among categories.
Alternative Benchmarks for Economically Optimal Rate
Although EORs are considered as the most appropriate benchmark to indicate economically optimal rates of N fertilization, these EORs may not be the best benchmark for all conditions (Colwell, 1994; Voss, 1975). Some possible benefits of using alternative benchmarks can be illustrated by defining three benchmarks that could be used as alternatives: the incremental rate that breaks even economically (IRBE), the incremental rate that gives the desired level of profit (IRDP), and the incremental rate that gives no yield response (IRNR). The word incremental is included in each alternative benchmark to denote that analyses were done by using discrete marginal analyses as described in this paper and that the discrete marginal analyses were based on the effects of the last 1 kg N ha–1 applied.
The IRBE is defined as the rate at which the marginal cost for the last 1 kg ha–1 applied equals the marginal value of the crop produced by the last incremental increase in rate of fertilization (Fig. 7). For all practical purposes, these rates are calculated based on the same economic principles as EORs; however, there are important differences in the methods used to calculate IRBEs and EORs. Predicted values from models are used to calculate EOR values, while IRBEs are calculated based on differences in treatments means without fitting curves to interpolate between rates of N actually applied. Calculated values for IRBEs and EORs should be expected to differ for any given dataset. Both IRBEs and EORs can differ because the effects of N rates outside the optimal range is minimized when calculating IRBEs from the raw yield data for the relationships between rates of N fertilization and yields. The calculated values for IRBEs were 186 kg N ha–1 for the category that includes all trials, 227 kg N ha–1 for corn after corn, and 141 kg N ha–1 for corn after soybean (Fig. 7). These IRBEs are within the same range as EORs that are calculated by using three continuous models with three rates of N fertilization analyzed in the near-optimal range (see inserts of Fig. 3). The IRBEs should be compared with the EORs calculated for the five models shown in Fig. 2 of the accompanying manuscript.
The IRDP (or the desired profit rate) is defined as the rate at which the marginal value of the crop produced by the last 1 kg ha–1 gives the minimal desired level of profit the producer expects from this last increment of N. Crop producers may decide, for example, not to apply the last kg of N if the marginal value of the crop produced is not 25% greater than the marginal costs of applying this last increment of N. The IRDP can be easily identified by using Fig. 7 and placing a horizontal line at 125% of marginal cost line as determined by the fertilizer-to-grain price ratio, identifying the point at which the DMP line intersects this line, and finding the rate of fertilization associated with this intersection. It would be reasonable for crop producers to expect a 25% return on fertilizer investments in situations when they can invest their money in other enterprises and receive a 25% return. Requiring this level of profit reduced optimal rates of N by 25 kg N ha–1 for the category that includes all trials, 27 kg N ha–1 for corn after corn, and 18 kg N ha–1 for corn after soybean.
The IRNR is defined as the lowest rate (to the nearest kg ha–1) needed to attain the highest yields that can be attained by adding fertilizer under the conditions studied. The highest yields have been widely described as maximum yields when using continuous models to describe relationships between rates of N fertilization and yields. The highest yields have been widely described as plateau yields when using discontinuous models. Both maximum and plateau yields are calculated by fitting the models. Determined values for maximum or plateau yields are not affected by the price of corn or the price of fertilizer. The IRNR, however, can be simply identified for each category in Fig. 2 as a rate where DMP equals zero.
Data presented in Fig. 2 and 7 illustrate how the method described here can be used to reduce problems associated with identifying economically optimal rates of fertilization. Differences between the incremental rates that give no yield response, the incremental rates that break even economically, and the incremental rates that give the desired level of profit can be clearly distinguished as long as enough observations of yield responses are made to calculate DMPs with a reasonable level of certainty and variability in DMPs is controlled by forming appropriate categories.
Data showing that DMPs of N fertilization were essentially linear in the near-optimal range suggest that application of insurance N was not a good investment. The economic risk associated with applying too little N was not greater than the economic risk associated with applying too much N within this range. If economic risk were the same, then applying insurance N would be expected to increase amounts of N lost to the environment. The problem of identifying the economic risk of applying too little or too much N can be avoided by using two alternative benchmarks that can be identified within the near-optimal range as shown in Fig. 7.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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More discussion is needed about alternative benchmarks for economically optimal rates of fertilization. Our analysis suggests that no single benchmark is appropriate for all situations. Deciding which benchmark to use is important when analyzing yield response data because calculated values for economically optimal rates of N fertilization largely depend on the amount of knowledge used during the analysis. Better estimates of economically optimal rates of N fertilization should enable crop producers to increase their profits while decreasing environmental problems associated with the use of N during crop production.
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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