Optimal harvesting of fish stocks under a time-varying discount rate

Abstract

Optimal control theory has been extensively used to determine the optimal harvesting policy for renewable resources such as fish stocks. In such optimizations, it is common to maximise the discounted utility of harvesting over time, employing a constant time discount rate. However, evidence from human and animal behaviour suggests that we have evolved to employ discount rates which fall over time, often referred to as "hyperbolic discounting". This increases the weight on benefits in the distant future, which may appear to provide greater protection of resources for future generations, but also creates challenges of time-inconsistent plans. This paper examines harvesting plans when the discount rate declines over time. With a declining discount rate, the planner reduces stock levels in the early stages (when the discount rate is high) and intends to compensate by allowing the stock level to recover later (when the discount rate will be lower). Such a plan may be feasible and optimal, provided that the planner remains committed throughout. However, in practice there is a danger that such plans will be re-optimized and adjusted in the future. It is shown that repeatedly restarting the optimization can drive the stock level down to the point where the optimal policy is to harvest the stock to extinction. In short, a key contribution of this paper is to identify the surprising severity of the consequences flowing from incorporating a rather trivial, and widely prevalent, non-rational aspect of human behaviour into renewable resource management models. These ideas are related to the collapse of the Peruvian anchovy fishery in the 1970's.1

Behavioural Economics, Hyperbolic Discounting and Environmental Policy

“…Simulations employed the ode113 routine in MATLAB for solving non-stiff differential equations, and are explained in more detail by Duncan et al (2010).…”

“…The analysis can be facilitated by a change of variable from time t ∈ [0, ∞] to discount function D ∈ [1, 0], as presented in Duncan et al (2010). After specifying the Hamiltonian and solving for the unconstrained case (h > 0, x > 0), the necessary conditions for optimality give rise to the standard equations (transformed back into the t dimension)…”

“…(1) is equivalent to the expression given byClark (1990, p.23), which is F(x) = r x(x/K 0 − 1)(1 − x/K ), when A = r/(K 0 K ), K 0 = x and K = x. 18 SeeDuncan et al (2010) for a more detailed analysis.…”

Reconciling yield stability with international fisheries agencies precautionary preferences: The role of non constant discount factors in age structured models

“…In fisheries economics there are also some recent advances applying non constant discount factor in dynamic biomass management problems. Ducan et al (2011) examine harvesting plans when the discount factor increases over time. They find that the planner reduces stock levels in the early stages (when the discount factor is low) and intends to compensate by allowing the stock level to recover later (when the discount factor will be higher).…”

Optimal Control for Sustainable Consumption of Natural Resources

“…A common approach is the discounted utilitarianism model in which the utility gained by consumption of the resource is discounted over an infinite time horizon. The discount rate may be assumed to be fixed, time varying or uncertain (Farzin [1984], Duncan et al [2011]). However, discounting enforces a fundamental asymmetry between present and future generations, which may not agree with at least some definitions of sustainability (see for instance, Tietenberg and Lewis [2000] and Solow [1992]).…”