Dynamically Optimal Adoption of Farming Practices Which Degrade or Renew the Land

Many farming decisions are choices among discrete practices. This is particularly true for decisions that degrade or renew the land over time. Most models of soil erosion, acidity, or salinity, however, try to approximate discrete choices by a static model or by a dynamic model with continuous control. This study constructs a dynamic model for the optimal adoption of discrete practices. The model is an extension of free-time optimal control. Each discrete practice has its own optimal control problem and these control problems are linked over time to study optimal switching among practices. Land is often considered an exhaustible resource, but several practices renew rather than degrade the land. It is found that an optimal time path typically has an initial phase of degradation followed by a steady-state rotation between degrading and renewing practices. The initial phase could be one of renewal and the final phase one of abandoning or selling the land. The renewal of soil acidity, salinity and, perhaps, erosion makes agriculture sustainable into the indefinite future and shifts the focus of public policy from conserving an exhaustible land resource to attaining the optimal steadystate. It also shown that a discrete but static model or a dynamic but continuous model has no optimality properties whatever. This is unfortunate because the dynamic discrete-choice model is more difficult to solve. A few special cases may be easily solved but the general model requires large-scale mathematical programming with special gradient calculations.