Alternative Benchmarks for Economically Optimal ... - Semantic Scholar

Published online June 5, 2007

Alternative Benchmarks for Economically Optimal Rates of Nitrogen Fertilization for Corn Peter M. Kyveryga,* Alfred M. Blackmer, and Thomas F. Morris tors 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 in-

Reproduced from Agronomy Journal. Published by American Society of Agronomy. All copyrights reserved.

ABSTRACT Discussions of N fertilizer needs for corn (Zea mays L.) usually are based on the assumption that economic optimum rates (EORs) of N provide the best benchmark to indicate the rates most likely to maximize profits for producers. We describe methods for calculating some additional benchmarks and illustrate how efforts to improve N management could be advanced by recognizing several alternative benchmarks simultaneously. We analyzed data from 54 small-plot trials with a new procedure that disaggregates problems associated with systematic model bias and variability in crop yield responses to fertilizer N. We compared the commonly used method for calculating EORs to a method that is described as discrete marginal analysis at near-optimal rates of N fertilization. We described methods for calculating the incremental break-even rate, the incremental rate that gives no crop response, and the incremental rate of N that gives the desired level of profit. Discussions of the relative merits of each benchmark suggest that problems associated with calculating and using the concept of EORs can be avoided by presenting the alternative benchmarks as proposed in this paper.

E

CONOMIC OPTIMUM RATES of fertilization as defined by Heady et al. (1955) and Heady and Dillon (1961) provide an essential benchmark when discussing N fertilization practices for corn. Economic optimum rates explicitly denote rates of N fertilization expected to maximize net returns (i.e., profits) from fertilizer applied per unit of land area. This benchmark has been widely used in agronomic studies (Black, 1993; Bock and Hergert, 1991; Colwell, 1994; Voss, 1975). Use of such a benchmark is reasonable because modern corn production is a business and, therefore, maximization of profits is often a primary consideration when selecting rates of N fertilization. Economic optimum rates also provide an essential benchmark when discussing or calculating any loss of profit that producers would suffer if rates of fertilization needed to be reduced to address problems related to environmental pollution (Aldrich, 1980; Huang et al., 2001). 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 fac-

P.M. Kyveryga, Iowa Soybean Assoc., 4554 114th St., Urbandale, IA 50322; A.M. Blackmer, Dep. of Agronomy, Iowa State Univ., Ames, IA 50010; and T.F. Morris, Dep. of Plant Science, Univ. of Connecticut, Storrs, CT 06269. Received 26 Nov. 2006. *Corresponding author ([email protected]).

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.

Published in Agron. J. 99:1057–1065 (2007). Forum doi:10.2134/agronj2006.0340 ª American Society of Agronomy 677 S. Segoe Rd., Madison, WI 53711 USA

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AGRONOMY JOURNAL, VOL. 99, JULY–AUGUST 2007

cluded 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). MATERIALS AND METHODS We utilized datasets of yield response data presented elsewhere (Binford et al., 1990, 1992; Meese, 1993). The data analyzed were collected in 54 response trials with seven rates of N (0, 56, 112, 168, 224, 280, and 336 kg N ha21) that were applied just before planting to plots (12.2 by 4.6 m) of corn in a randomized complete block design with three replications. Yields were measured by hand-harvesting 7.6 m of row in the center of the plots and grain weights were corrected for moisture content. This collection of trials was used to emulate samples of trials conducted across many sites and years within a specified area and time. The crop preceding the corn trial was either corn or soybean [Glycine max (L.) Merr.], which provided three crop categories: corn grown after corn, corn grown after soybean, and all trials (grown after corn or soybean). Category-specific ex post EORs were calculated for each of three categories with the corn price set at $86.50 Mg21 and the fertilizer N price set at $0.44 kg21 for all calculations. Because the sample of yield responses in our dataset was not randomly

distributed within the area of interest, the EORs calculated were considered applicable only within an unspecified hypothetical area and time. 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 DYi (kg ha21) resulting from an increase in rate of fertilization by the amount DNi (kg ha21) of fertilizer N applied by the increment as indicated in the following equation.

DMPi 5

YN1 2 YN2 DYi 5 N1 2 N2 DNi

[1]

The DMPs were expressed as the amounts of yield response (kg of grain) received per unit of fertilizer applied (kg21 N). The values of DMPs are the same as the values for discrete marginal agronomic efficiency of N fertilization as described by Cassman et al. (2003). The resulting DMPs were used to represent the marginal products at midpoints of the incremental increases in rates of N applied. It should be noted that the DMPs do not rely on models to predict yields; the DMPs instead are based on the difference in yields between two successive rates of N rates. We use the word discrete to denote that we calculated marginal products for the increments of fertilizer N actually applied, which are different from calculation of marginal products from a model used to describe relationships between N rates and yields. Interpolations were made between the midpoints of the incremental increases in rates, but interpolations were made only after calculations of the DMPs. The marginal cost of fertilization was expressed as the amount of grain required to pay for the cost of 1 kg of N rather than the amount required to pay for the incremental increase in N rate. 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.

KYVERYGA ET AL.: ALTERNATIVE BENCHMARKS FOR OPTIMAL N

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RESULTS AND DISCUSSION Yield Responses and Discrete Marginal Products The observed relationships between rates of N fertilization and treatment means for yields of grain for three categories are shown in Fig. 1. The means are connected by straight lines to avoid injecting bias when fitting curves, a problem that is illustrated by Kyveryga et al. (2007). Although Fig. 1 gives an unbiased description of the effects of N on yields, this presentation of data has rather limited value because there are no quantitative methods for interpolating between the rates of N applied in the trials to refine estimates of rates that were economically optimal under these conditions. The observed relationships between N rates and discrete marginal products (DMPs) of N fertilization for each of the three categories are shown in Fig. 2. Discrete marginal products are expressed as kilograms of grain yield response per kilogram of N fertilizer applied. Data points are located midway between the rates of N applied in the trials because DMP points show the mean effects of incremental increases in rates of N fertilization rather than the yields obtained at a given application rate. The numerical values for DMPs are equal to the slopes of the corresponding line segments illustrated in Fig. 1. The mean values for DMPs are connected by straight lines to avoid any bias associated with fitting curves.

Marginal Products Calculated from Models Figure 3 shows relationships between rates of N fertilization and marginal products (MPs) as calculated from five models used to describe relationships between N rates and yields. The MP curves show the first derivatives or instantaneous slopes of the yield response curves. Disagreements among models indicate that one or more of the models have injected bias. A major problem illustrated in Fig. 3 is that the models used to describe relationships between N rates and yields cause errors when MP values are calculated. Cerrato and Blackmer (1990) also presented analyses showing that

Fig. 2. Discrete marginal products (DMPs) of N fertilization for the incremental increases in rates of N fertilization applied for three crop categories.

disagreements among models used to calculate EORs should be attributed to errors in calculated values of slopes for relationships between N rates and yields. Errors in calculated values for MPs result in errors in calculated values for EORs because EORs are calculated by solving models for the rates of N fertilization at which the MPs equal the marginal costs of fertilization. The horizontal dashed line in each graph of Fig. 3 shows the marginal cost of fertilization expressed as the amount of grain (i.e., 5.1 kg) required to pay for the cost of 1 kg of N, which is the fertilizer-to-grain price ratio. The EOR as indicated by any given model can be identified by merely extrapolating downward to the x axis from the point where the appropriate MP curve intersects the marginal cost line. Disagreements among models when calculating EORs can be assessed by extrapolating downward from the intersections of the appropriate MP curves with the marginal cost line. 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 Fig. 1. Relationships between corn grain yields and rates of N fertilization for three crop categories analyzed.

The use of the LRP and the QRP models, which are discontinuous models, to relate rates of N fertilization to


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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

Fig. 3. Marginal products (MPs) of N fertilization as calculated from five different models for all rates and for rates in the near-optimal range (inserts A, B, and C). EXP, exponential model; LRP, linear response and plateau model; QRP, quadratic response and plateau model; QUAD, quadratic model; SR, square root model.

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.,

Fig. 4. The second derivatives of continuous models used to describe the relationship between rates of N fertilization and grain yields. EXP, exponential model; QUAD, quadratic model; SR, square root model.



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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 nearoptimal range. An additional problem associated with the use of discontinuous models (i.e., LRP and QRP) is also illustrated by analyses already discussed. The relationships presented in Fig. 1, for example, suggest that the last incremental increase in rate of N fertilization decreased yields. Authors rarely state whether a statistical test was used to decide if yields are increasing, decreasing or not changing at higher rates of N before deciding on a model to describe the relationship. The selection of a model to describe the relationship between yields and N rates can be affected by personal judgments concerning the likelihood that the last increment of N did or did not decrease yields. The selection of a model should not be based on personal judgments because the estimate of the optimal rate of fertilization will be greatly affected by the choice of a model. Any measured decrease in yields at the highest rates of fertilization tends to have the effect of lowering the yield level identified as a plateau. Similar problems are encountered when the models used (i.e., EXP and SR) assume that yields should asymptotically approach a maximum when increasing rates of N fertilization. Although this problem can be reduced by applying several rates of N in the above-optimal range, this approach probably is not always the most practical way to refine estimates of optimal rates.

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-tograin 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 kg21 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. Within each category, the magnitude of disagreements in calculated values for EORs or MPs tended to change

Fig. 5. Effects of fertilizer-to-grain price ratios on economic optimum rates (EORs) as calculated from five different models used to describe relationships between rates of N fertilization and grain yields.

gradually with changes in rates of fertilization identified as being optimal (Fig. 3). When compared with MP values of other models, for example, the QUAD model gave (i) relatively low MP values if price ratios made optimal rates higher than 250 kg N ha21, (ii) relatively high MP values if price ratios made optimal rates of about 150 kg N ha21, and (iii) MP values nearly equal to the means of the other models if the price ratios made optimal rates about 50 kg N ha21. The gradual changes in magnitude of disagreements should be expected because some or all of the models are unable to flex at appropriate places to correctly describe the natural relationships observed between N rates and yields measured at the sites studied. This problem can be minimized by

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not fitting curve to yields collected at rates of fertilization above or below the near-optimal range (Kyveryga et al., 2007).


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 nearoptimal 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 r 2 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 nearoptimal range for each crop category (data not shown). Higher r 2 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. Analyzing the relationships in Fig. 6 is more practical than those analyzed in Fig. 3 because it is relatively easy to assess the amount of uncertainty associated with use of straight lines to interpolate between measurements when relationships are known to be essentially linear. However, it is much more difficult to assess the amount of uncertainty when curves are fit to interpolate between measurements made in situations where some curvature is expected, but there is no way to quantify the amount and the shape of curvature that accurately characterizes the yield responses observed, which is the situation shown in Fig. 3. Moreover, analyses presented in Table 1 show that MP values at midpoint rates for optimal ranges calculated from three continuous models as shown in inserts of Fig. 3 are almost identical to MP values at midpoint rates calculated by linear interpolations between the lower and the upper N rates in the near-optimal range. The exception is only a value for MP at a midpoint rate calculated by linear interpolation for the EXP model for corn after soybean crop category

Fig. 6. Relationships between discrete marginal products (DMPs) of N fertilization and N rates in the near-optimal range when all observations are included in the analysis.

(Table 1). These observations suggest that errors introduced by using linear interpolation of MPs are smaller than errors introduced by models selected to describe relationships between rates of N fertilization and yields in the near-optimal range. Figure 7 shows relationships between mean DMPs and rates of N fertilization restricted to the near-optimal range for each crop category. The practical importance of uncertainty associated with the assumption of linear relationships in Fig. 7 can be also assessed by making comparisons with the amounts of uncertainty associated with the measured mean DMPs for treatments. If the measured means were based on relatively few observations, then errors associated with the assumption of lin-


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Table 1. Marginal products (MPs) of N fertilization as calculated for various points within the near-optimal range from three continuous models used to describe relationships between N rates and MPs in the near-optimal range. MPs calculated by models


Model†

Midpoint of optimal Lower end Midpoint of Upper end of range calculated by of optimal optimal optimal linear interpolation range‡ range range between ends kg grain kg

All trials QUAD 9.7 EXP 10.8 SR 10.5 Corn after corn QUAD 7.9 EXP 8.4 SR 8.3 Corn after soybean QUAD 8.6 EXP 13.3 SR 9.7

21

N

5.7 5.3 5.4

1.7 2.6 2.3

5.7 6.7 6.0

5.2 5.0 5.1

2.6 3.0 2.9

5.2 5.7 5.6

3.5 2.0 3.0

21.6 0.3 20.9

3.5 6.8 4.4

† EXP, exponential model; QUAD, quadratic model; SR, square root model. 21 ‡ Rates included in the analysis: 112, 168, and 224 kg N ha for all trials; 168, 224, and 280 kg N ha21 for corn after corn; and 112, 168, and 224 kg N ha21 for corn after soybean.

earity often would be small relative to the amounts of uncertainty in the measured means. If the measured means combined data from categories that should be considered separately, then errors associated with the assumption of linearity often would be small relative to the amounts of uncertainty caused by the failure to consider the categories separately. For example, analyses revealed that classifying the sample of all trials into two categories with different N fertilizer needs based on previous crop (Fig. 2) increased profit about $15 ha21 (data not shown) because fertilizer was applied where it was needed the most. The profit gained is substantial because information about the previous crop is usually known and available without additional cost. When compared with normally accepted methods for calculating EORs, the method shown in Fig. 7 is likely to reveal situations in which the data collected are not adequate to provide the information desired. A common problem is that too few observations are collected within the near-optimal range. The use of category specific estimates of optimal rates makes it easier to address this problem because optimal rates for a given category tend to be relatively constant. The concept of category-specific estimates of optimal rates makes it possible to design trials to collect observations to identify the near-optimal range and design trials to collect follow-up observations to efficiently refine EORs estimates by focusing on observations in the near-optimal range of fertilization. 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

Fig. 7. Discrete marginal products (DMPs) of N fertilization calculated for three incremental increases in rates of N fertilization in the near-optimal range for each crop category.

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 bench-


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mark 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 ha21 applied. The IRBE is defined as the rate at which the marginal cost for the last 1 kg ha21 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 ha21 for the category that includes all trials, 227 kg N ha21 for corn after corn, and 141 kg N ha21 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 ha21 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 ha21 for the category that includes all trials, 27 kg N ha21 for corn after corn, and 18 kg N ha21 for corn after soybean. The IRNR is defined as the lowest rate (to the nearest kg ha21) 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.

CONCLUSIONS Problems associated with estimating economically optimal rates of N fertilization can be substantially reduced by calculating DMPs and by simultaneously using alternative benchmarks when discussing economically optimal rates of N fertilization. The key advantage of using discrete marginal analysis is that many hidden problems associated with model bias are avoided because calculations are based on differences in treatments means without fitting curves to interpolate between rates of N applied in the experiments. Although straight lines are used to interpolate between N rates applied in the near-optimal range in the final step of the analysis, the practical importance of any errors associated with this interpolation can be assessed by making comparisons with the amounts of uncertainty associated with measured mean DMPs. When this method is used, the ability to refine estimates of economically optimal rates depends largely on ability to define categories that have relatively little variability in yield response and collect samples that adequately characterize this variability. More discussion is needed about alternative benchmarks for economically optimal rates of fertilization. Our analysis suggests that no single benchmark is ap-



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