Dynamic Optimization of Fertilizer Application with Carryover and Runoff

The Economics of fertilizer application has been an interesting issue in regard to crop production (Kennedy et al., 1973). Understanding the temporal dynamics complicates the issue along two dimensions: fertilizer carryover and fertilizer runoff. Fertilizer carryover measures the amount of fertilizer applied in the previous period that is available for crops in current growing period (Kennedy et al., 1973). Fertilizer runoff refers to fertilizer leaving the ground and flowing into water bodies. Thus, the optimal quantity of fertilizer to be applied in the current period depends on the amount of previous application, runoff, and carryover. Because fertilizer carryover and runoff makes optimization of application a complicated intertemporal problem, static optimization procedure that equate marginal benefit and marginal cost cannot be used to solve dynamic problems of a fertilizer application. This paper first builds an intertemporal net benefit maximization model for crop planting to solve the optimal applied quantity of fertilizer using dynamic programming method with emphasis on the comparative statics. Next, the net benefit maximization model and the optimal solution are used to predict the net benefit and the optimal application of nitrogen for production of switchgrass and corn planted in the University of Tennessee Research and Education Center at Milan, Tennessee. The goal of dynamic programming analysis is to determine an optimal resource application that achieves maximum net benefit. The optimal applied quantity occurs when marginal value of the initial net benefit equals the present value of all periods’ net benefits from managing the resource in addition to the final value of the resource at the end of the planning horizon. A quadratic yield response function is used to model corn and switchgrass response to nitrogen. The argument of the yield function is available amount of nitrogen for crop growth. The available amount of nitrogen is the sum of applied nitrogen and the nitrogen of carryover less the amount of nitrogen leaving the field. Switchgrass is a potentially important feedstock source to biofuel. It is a tall, hardy, and perennial grass native to North America (Rinehart, 2006). As a biomass feedstock, switchgrass can be used to produce ethanol. Switchgrass is usually produced for ten years or more. Thus, a ten year planning horizon is used in this study. For estimation of switchgrass yield response function, five years (2004-2008) of switchgrass yield data was obtained from field experiments. The land is characterized as a moderately well drained level upland. The corn and switchgrass yield response coefficients are used in the net benefit objective function. The other parameters of the net benefit model are initial amount of nitrogen in the soil, nitrogen price, output price, a discount rate, establishment budget, maintenance budget, and harvesting budget. Because carryover and runoff parameters are not available, the net benefit function is analyzed over series of parameter values. The optimal applied amount of nitrogen, the optimal available amount of nitrogen, and ten years of total net benefits are estimated for switchgrass and corn from the perspectives of carryover and runoff, with each rate fixed at 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9, respectively. The optimal applied amount of nitrogen, the optimal available amount of nitrogen, and ten years of total net benefits are estimated for switchgrass and corn, respectively. The optimal available amount of nitrogen will increase as the carryover rate increases. The optimal available amount of nitrogen and the optimal applied amount of nitrogen are concave functions of the runoff rate for switchgrass but not for corn because the yield function for switchgrass is a quadratic function of the optimal available amount of nitrogen but the yield function for corn is a quadratic plateau function of the optimal available amount of nitrogen. The total net benefits will increase as the carryover rate increases and decrease as the runoff rate increases. The ten year net benefit for switchgrass turns out to be equal to that for corn when the runoff rate is around 0.6. The ten year net benefit for switchgrass is less than that for corn as the runoff rate is less than 0.6 but more as the runoff rate is higher than 0.6. This paper provides a modeling framework for simulation or empirical analyses on estimating the fertilizer application and net benefit for production of bioenergy and conventional crops. Future research may be directed at using the predicted results for planting bioenergy and conventional crops to set a feasible fertilizer runoff constraint and nutrient management to reduce runoff flowing into water bodies to protect water resources.