The costate variable in natural resource optimal control problems (Q2745386)

We discuss the role of the costate variable (shadow value) for the resource stock in both nonrenewable and renewable resource problems. We separate the information in this variable into a scarcity and a cost effect. The scarcity effect is the portion of the shadow value that is due just to the scarcity of the resource relative to its demand, while the cost effect is a measure of the impact of the marginal unit upon future extraction costs. It is shown that in the nonrenewable resource, mining, problem both can exist simultaneously, but in the renewable resource, fisheries, model the two effects are mutually exclusive. In our analysis of the fisheries model we develop an expression for the time path of the marginal unit of fish stock. We do this using the theorem of Continuous Dependence on Initial Conditions. This result is then used to generate the conclusion that \(g(x)\) is the biological own rate of interest, where \(g(x)\) is the growth function for the resource stock, \(x\).