Adding stochasticity seems to have a small history.
Reed 1979 Reed, 1979 Uncertainty in next year’s stock, “growth uncertainty” at the time of determining harvest. Constant escapement policy, no chance of extinction.
Clark & Kirkwood 1986 Clark & Kirkwood, 1986 Uncertainty in this year’s stock, “measurement uncertainty” when determining harvest.
Roughgarden & Smith (1996) consider both, and add uncertainty in harvest. Sethi et al. 2005 (Sethi et. al. 2005) consider the economist’s optimization of this.
Statement of problem
\[ \max_{ \{ q_t \} \leq 0 } \mathbb{E} \left\{ \sum_0^{\infty} \alpha^t h_t \right\} \]
s.t. Fish population follows: $ x_t = z_t^g f( x_{t-1} - h_{t-1} ) $ and measured stock is given by: $ m_t = z_t^m x_t $ while harvest is: $ h_t = (x_t, z_t^i q_t) $
Where $ z^g $ denotes uncertainty in population growth, $ z^m $ denotes uncertainty in measurement for the stock assessment, and $ z^i $ denotes uncertainty in harvest implementation. Note we ignore the profit function for the moment and just maximize infinite-horizon harvest.
From this we must set up the stochastic transition matrix. Instead of defining this over the known current stock values $x_t $ as we did before, our decision state-space is defined over the measured values $ m_t $, which is given by:
\[ m_t = z_t^m z_t^g f\left( x_{t-1} - \min(x_{t-1}, z_{t-1}^i q_{t-1}) \right) \]
The trick here is assigning the probability distribution on $ m_t $. Otherwise the optimization proceeds as before with the modified stochastic transition matrix. Sethi et al., and Clark & Kirkwood choose uniform random variables over given intervals for each, which seems like a rather strange choice. Reed 1979 uses a normal distribution.
Have implemented a general simulation-based approach to generate the stochastic transition matrix for arbitrary distribution shapes for $ z_t^g $, $ z_t^m $, $ z_t^i $.
Further topics
(parameter uncertainty - more extensive literature)
(modified to be effort-based. Explore with an allee effect).
Misc
Completed review for THEE.
References
Sethi G, Costello C, Fisher A, Hanemann M and Karp L (2005). “Fishery Management Under Multiple Uncertainty.” Journal of Environmental Economics And Management, 50. ISSN 00950696, https://dx.doi.org/10.1016/j.jeem.2004.11.005.
- on Uncertain Renewable Resource Stocks: Optimal Harvest Policies And The Value of Stock Surveys, Colin W. Clark, Geoffrey P. Kirkwood, (1986) Journal of Environmental Economics And Management, 13 10.1016/0095-0696(86)90024-0
Optimal Escapement Levels in Stochastic And Deterministic Harvesting Models, William J Reed, (1979) Journal of Environmental Economics And Management, 6 10.1016/0095-0696(79)90014-7