Conventional wisdom states that targeting either stocks of capital or flows of production could have equivalent effects on externalities correction. However, this may not hold when externalities are generated from stocks that exhibit exogenous growth, such as CO$_2$ sequestration by trees and water purification by shellfish. I present a dynamic partial equilibrium model of privately-owned renewable resource extraction to explore how different incentive targets affect the provision of social benefits from natural resources over time. I find that while stock-targeted incentives can yield socially optimal levels of the natural resource in steady state, levels under flow-targeting (i.e. extraction or harvest) converge to those under no policy or business-as-usual (BAU). This is likely driven by revenue sources for the firm. Stock-targeting allows the firm to flexibly maximize profits over two sources of revenue: the natural resource stock and the private good. By contrast, flow-targeting solely changes the price of the private good, leading to similar profit maximization and extraction behavior as BAU. I then explore the importance of other policy design choices for dynamic environmental benefits provision by calibrating the model to scientific parameters from the Chesapeake Bay, where Eastern oyster (C. virginica) cultivation yields water filtration benefits. Both subsidies on stocks and taxes on harvest can increase environmental benefits, but harvest taxes at scales that produce noticeable increases yield substantial losses in producer surplus with minimal gains in consumer surplus, suggesting that the costs of such a policy may outweigh its benefits. The model provides a theoretical framework for program design aimed at minimizing environmental damages through the management of privately-owned living natural resources.