Optimal Management of Groundwater under Uncertainty : A Unified Approach

Discrete-time stochastic models of management of groundwater resources have been extensively used for understanding a variety of issues in groundwater management for agriculture. Most models used suffer from two drawbacks: relatively simplistic treatment of the cost of water extraction (remarked in many papers in the literature) and lack of important structural results (monotonicity of extraction in stock, concavity of the value function etc), even in simple models. Lack of structural properties impede both practical policy simulation (due to the lack of robustness) and clear understanding of the resulting models and underlying economics. This paper provides a unifying framework for discrete-time stochastic groundwater models in two directions. First, the usual cost function is extended to encompass cases where marginal cost of pumping depends on the stock. Second, the analysis here dispenses with assumptions of concavity of the objective function and compactness of the state space, using instead lattice-theoretic methods. With these modifications, a comprehensive investigation of which structural properties can be proved in each of the resulting cases is carried out. It is shown that, for some of the richer models, more structural properties may be proved than for the simpler model used in the literature. In this set-up, convergence of the stock of resource under the optimal policy typically follows from monotonicity of extraction in stock. This paper introduces to the resource and agricultural economics literature, an important method of proving convergence to a stationary distribution which does not require monotonicity. This method is of interest in a variety of renewable resource settings