Optimal Growth, Renewable Resources and Sustainability

We study a growth model for a single resource-based economy, as an infinite-horizon op-timal control problem. The resource is assumed to be governed by the standard model of logistic growth, and is related to the output of the economy through a Cobb-Douglass type production function with an exogenously driven knowledge stock. The problem involves unbounded controls and the non-concave Hamiltonian. These preclude direct application of the standard existence results and Arrow’s sufficient conditions for optimality. We transform the original optimal control problem to an equivalent one with simplified dy-namics and prove the existence of an optimal admissible control. Then we characterize the optimal paths for all possible parameter values and initial states by applying the ap-propriate version of the Pontryagin maximum principle. Our main finding is that only two qualitatively different types of behavior of sustainable optimal paths are possible de-pending on whether the resource growth rate is higher than the social discount rate or not