The Costate Variable in a Stochastic Renewable Resource Model

In this paper we discuss the costate variable in a stochastic optimal control model of a renewable natural resource, which we call a fishery. The role of the costate variable in deterministic control models has been discussed extensively in the literature. See, for example, Lyon (1999), Clark (1990, pp. 102-107), and Arrow and Kurz (1970, pp. 35-37); however, there is little discussion of this variable for stochastic models, even though the costate variable has similar roles in the two models. In both models the costate variable is a shadow value of the associated state variable, and as such has the role of rationing the use of the state variable. In addition, as has been shown in Lyon (1999), in natural resource problems the costate variable can be partitioned into a scarcity effect and a cost effect. We show that this same partitioning can be done in the stochastic renewable resource problem. We discuss and contrast the similarities and differences in these concepts for deterministic and stochastic models. In addition, we present a numerical example help solidify the results.