Optimal spatial management of renewable resources: matching policy scope to ecosystem scale

Authors: J. Sanchirico and J. Wilen, 2004.


This paper applies the formal apparatus of optimal control theory to a particular open access fisheries model. The optimization is to maximize rents, i.e. the problem is being solved from an economic viewpoint. The objective functional has the form:

$ \max_{\tau_i, \pi_i} \int_0^{\infty} e^{-\delta t} \sum_i R_i(E_i,x_i) dt$

where $R_i$ represents the rent from spatial patch $i$ with fish population $x_i$ and effort level $E_i$. The regulatory mechanisms are taxes, in this case: $\tau_i$ and $\pi_i$ represent landing and shipping taxes, respectively. The constraint equations have the form

$ x_i^{\prime} = f_i(x_i)x_i + ND_i(x_1, x_2, \cdots , x_n) - h_i(E_i,x_i)$

and

$ E_i^{\prime} = s_i R(E_i,x_i, \tau_i, \pi_i) + \sum s_{ij}[R_i(E_i, x_i, \tau_i, \pi_i) - R_j(E_j, x_j, \tau_j, \pi_j)]$,

where the rent function is

$ R(x_i, E_i, \tau_i, \pi_i) = [(p_i - \tau_i) h_i(x_i, E_i) - (c_i + \pi_i) E_i]$,

with $p_i$ the ex-vessel price and $c_i$ the per-unit effort cost. The problem is linear in the control variables, and thus leads to bang-bang control.

From here I get lost, unfortunately. There is a lot of text, a few sketches, a long discussion, but I miss the main point.