On Optimal Harvesting in Stochastic Environments: Optimal Policies in a Relaxed Model

TL;DR

This paper introduces a relaxed model for optimal harvesting in stochastic environments, proving the existence of optimal policies and deriving explicit bounds and formulas based on initial population size.

Contribution

It establishes a relaxed formulation that guarantees the existence of optimal harvesting policies and provides a closed-form expression for the value depending on initial conditions.

Findings

Existence of optimal relaxed harvesting policies proven.

Derived explicit bounds for the optimal value based on initial population.

Closed-form expression for the value function depending on initial population size.

Abstract

This paper examines the objective of optimally harvesting a single species in a stochastic environment. This problem has previously been analyzed in Alvarez (2000) using dynamic programming techniques and, due to the natural payoff structure of the price rate function (the price decreases as the population increases), no optimal harvesting policy exists. This paper establishes a relaxed formulation of the harvesting model in such a manner that existence of an optimal relaxed harvesting policy can not only be proven but also identified. The analysis embeds the harvesting problem in an infinite-dimensional linear program over a space of occupation measures in which the initial position enters as a parameter and then analyzes an auxiliary problem having fewer constraints. In this manner upper bounds are determined for the optimal value (with the given initial position); these bounds depend on the relation of the initial population size to a specific target size. The more interesting case occurs when the initial population exceeds this target size; a new argument is required to obtain a sharp upper bound. Though the initial population size only enters as a parameter, the value is determined in a closed-form functional expression of this parameter.