Modifying Yield-Goal Based N and P Fertilizer Recommendations to Varying Input and Output Prices.

Many University fertilizer recommendations are based on yield goals and they do not explicitly account for crop and fertilizer prices.  For determining economic optimal fertilizer rates, a production function (yield response curve) is required.  Building upon a discussion of linear and curvilinear yield response to fertilizer, this research develops a framework where response is fundamentally linear for any particular site-year, but where expected response becomes curvilinear in the face of random weather across space and time.  We next posit several functional forms as candidates for generalizing expected yield response to N.  The functional forms were evaluated using historical N trial data from western and north-central Kansas involving wheat, corn, and grain sorghum.  The quadratic plateau functional form arose as the candidate of choice.  Given a defined functional form, we present the mathematics required to compute our suggested adjustments to current KSU N recommendations to accommodate changes in crop and fertilizer N prices.  Shortly after our N model development, and using KSU yield response to irrigation research, our model was expanded to incorporate this second yield factor, irrigation.  The two factors were assumed to drive yield in a quadratic plateau limiting factor framework.  Due to volatile P prices, the model was again expanded, this time to include price-based modifications to KSU P recommendations.  Lacking sufficient trial data to statistically select functional form, we assumed a quadratic yield response to P, as with N and water.  As with N (fertilizer and soil test N) and water (irrigation and rainfall), the sub-factors for P (fertilizer and soil test P) had to be integrated.  To mathematically accomplish that we assumed fertilizer P can fully compensate for soil test P.  Since this assumption is sometimes questioned, users of our model to guide P recommendations should carefully consider this critical assumption.