Harvesting in a Fishery with Stochastic Growth and a Mean-Reverting Price

Abstract

We analyze a continuous, nonlinear bioeconomic model to demonstrate how stochasticity in the growth of fish stocks affects the optimal exploitation policy when prices are stochastic, mean-reverting and possibly harvest dependent. Optimal exploitation has nonlinear responses to the price signal and should be conservative at low levels of biological stochasticity and aggressive at high levels. Price stochasticity induces conservative exploitation with little or no biological uncertainty, but has no strong effect when the biological uncertainty is larger. We further observe that resource exploitation should be conservative when the price reverts slowly to the mean. Simulations show that, in the long run, both the stock level and the exploitation rate are lower than in the deterministic solution. With a harvest-dependent price, the long-run price is higher in the stochastic system. The price mean reversion rate has no influence on the long-run solutions.

Appendix

We expand the Hamilton–Jacobi–Bellman Eq. (5) and write as follows:

$$\begin{aligned} \delta V&= \mathop {\max }\limits _{h\ge 0} \bigg \{ \left( {y-\frac{c_1 }{x}-\gamma _1 h} \right) h+\left( {f\left( x \right) -h} \right) V_x +\left( {-\alpha y+\alpha (p_0 -\gamma _2 h)} \right) V_y\nonumber \\&\quad +\,\frac{1}{2}\,\sigma _x ^{2}\,x^{2}\,V_{xx} +\frac{1}{2}\,\sigma _y ^{2}\,y^{2}\,V_{yy} \bigg \} \end{aligned}$$

(6)

If we let $L$ denote the object of the maximum operator, (6) is written as $\delta V=\mathop {\max }\nolimits _{h\ge 0} L$. $L$ is concave in $h$ and the inner optimum is given by:

$$\begin{aligned} \frac{\partial L}{\partial h}=y-\frac{c_1 }{x}-2\gamma _1 h-V_x -\alpha \,\gamma _2 V_y =0 \end{aligned}$$

(7)

from which we have:

$$\begin{aligned} h=\max \left\{ {0,\frac{y-\frac{c_1 }{x}-V_x -\alpha \,\gamma _2 V_y }{2\,\gamma _2 }} \right\} \end{aligned}$$

(8)

From (8), we can derive an expression for a contour of the inner solution. In particular, the zero contour is characterized by setting the numerator equal to zero. We have:

$$\begin{aligned} y-\frac{c_1 }{x}-V_x -\alpha \,\gamma _2 V_y =0 \end{aligned}$$

(9)

Total differentiation with respect to $y$ yields an expression for $dx/dy$ along the contour; that is, the derivative of the x-component of the contour as a function of $y$ is:

$$\begin{aligned} \frac{dx}{dy}=-\frac{1-V_{xy} -\alpha \,\gamma _2 V_{yy} }{\frac{c_1 }{x^{2}}-V_{xx} -\alpha \,\gamma _2 V_{xy}} \end{aligned}$$

(10)

When the expression in (10) is positive we have backward folding.

Figure 8 shows the zero contour of the right-hand side of (10) (the black curve), where the expression is positive above (north of) the contour. The figure also shows the unsmoothed zero contour of the harvest rate (the grey curve; identical to the white curve in Fig. 2a). Note that (10) changes behavior outside the zero harvest contour because the harvest rate has a kink along its zero contour. While we should be careful to interpret (10) in the $h=0$ region, Fig. 8 demonstrates that the right-hand side of (10) is positive along the part of the zero harvest contour where we observe backward folding.

Fig. 8

Zero contours of the right-hand side of (9) (black curve) and the harvest rate (grey curve) for the deterministic model