Considering Economic Efficiency in Ecosystem-Based Management: The Case of Horseshoe Crabs in Delaware Bay

Abstract

The welfare gains from incorporating ecosystem considerations into fisheries management are unclear and can vary widely between systems. Additionally, welfare gains depend on how ecosystem considerations are adopted. This paper uses an empirically parameterized bioeconomic model to explore the welfare implications of two definitions of ecosystem-based fisheries management (EBFM). We first define EBFM as a fishery management plan that maximizes the net present value of ecosystem services. We then explore an alternative definition that adds ecosystem considerations to a fishery managed with regulated open access. Our biological model reflects horseshoe crabs in Delaware Bay, which are harvested in a commercial fishery and are ecologically linked to migrating shorebirds populations, e.g. the endangered red knot. We find that introducing ecosystem considerations to a regulated open access fishery generates welfare gains on par with gains from addressing the commons problem even when fishery rents are completely dissipated. Additionally, solving the commons problem within an EBFM approach can provide substantial welfare gains above those from solving the commons problem in a single-species management framework.

Appendix

In this appendix we derive the first-order necessary optimality conditions for the fishery manager’s delayed optimal control problem under ECON-EBFM defined in Eqs. 8–12. We assume all conditions required by Theorem 4.2 in Göllmann et al. (2009) are satisfied so that the theorem applies.³⁶ We first work with the present-value Hamiltonian and then turn to the current-value Hamiltonian.

Construct the present-value Hamiltonian as

$$\begin{aligned}&\mathcal {H}^\text {p }(t, C_t, C_{t-\tau }, R_t, E_t, \lambda ^\text {p }_t, \xi ^\text {p }_t, \zeta ^\text {p }_{e,t}, \zeta ^\text {p }_{c,t}, \zeta ^\text {p }_{r,t}) \\&\quad = {} e ^ {- \rho t} \left[ p q C_t E_t - \delta E_t^2 + w (R_t - R_\text {m }) ^ \alpha \right] \\&\qquad {} + \lambda ^\text {p }_t \left[ g_c C_{t-\tau } \exp (- C_{t-\tau } / K_c^*) - \eta _c C_t - q C_t E_t \right] \\&\qquad {} + \xi ^\text {p }_t g_r R_t \left\{ 1 - R_t / K_r / a \cdot [1 + \exp (b_0 + b_1 C_t)] \right\} \\&\qquad {} + \zeta ^\text {p }_{e,t} E_t + \zeta ^\text {p }_{c,t} C_t + \zeta ^\text {p }_{r,t} R_t, \end{aligned}$$

where a superscript p indicates association with the present-value Hamiltonian, $\lambda ^\text {p }_t$ and $\xi ^\text {p }_t$ are the two costate variables associated respectively with Eqs. 9 and 10, and $\zeta ^\text {p }_{e,t}$, $\zeta ^\text {p }_{c,t}$, and $\zeta ^\text {p }_{r,t}$ are multipliers associated with the nonnegativity constraints Eq. 11.

The first-order necessary optimality conditions are then given by

$$\begin{aligned} 0 = \frac{\partial \mathcal {H}^\text {p }}{\partial E_t} = e ^ {- \rho t} (p q C_t - 2 \delta E_t) - \lambda ^\text {p }_t q C_t + \zeta ^\text {p }_{e,t}, \quad 0 \le t \le T, \end{aligned}$$

(23)

$$\begin{aligned} \begin{aligned} - \dot{\lambda }^\text {p }_t&= \frac{\partial \mathcal {H}^\text {p }}{\partial C_t} + \left. \frac{\partial \mathcal {H}^\text {p }}{\partial C_{t-\tau }} \right| _{t+\tau } = e ^ {- \rho t} p q E_t - \lambda ^\text {p }_t (\eta _c + q E_t) \\&\quad - \xi ^\text {p }_t g_r R_t ^ 2 / K_r / a \cdot b_1 \exp (b_0 + b_1 C_t) + \zeta ^\text {p }_{c,t} \\&\quad + \lambda ^\text {p }_{t+\tau } g_c (1 - C_t / K_c^*) \exp (- C_t / K_c^*), \quad 0 \le t < T - \tau , \end{aligned} \end{aligned}$$

(24)

$$\begin{aligned} \begin{aligned} - \dot{\lambda }^\text {p }_t = \frac{\partial \mathcal {H}^\text {p }}{\partial C_t}&= e ^ {- \rho t} p q E_t - \lambda ^\text {p }_t (\eta _c + q E_t) - \xi ^\text {p }_t g_r R_t ^ 2 / K_r / a \cdot b_1 \exp (b_0 + b_1 C_t) \\&\quad + \zeta ^\text {p }_{c,t}, \quad T - \tau \le t \le T, \end{aligned} \end{aligned}$$

(25)

and

$$\begin{aligned} \begin{aligned} - \dot{\xi }^\text {p }_t = \frac{\partial \mathcal {H}^\text {p }}{\partial R_t}&= e ^ {- \rho t} w \alpha (R_t - R_\text {m }) ^ {\alpha - 1} \\&\quad + \xi ^\text {p }_t g_r \left\{ 1 - 2 R_t / K_r / a \cdot [1 + \exp (b_0 + b_1 C_t)] \right\} \\&\quad + \zeta ^\text {p }_{r,t}, \quad 0 \le t \le T. \end{aligned} \end{aligned}$$

(26)

Also, the optimal solution should maximize the Hamiltonian among all admissible control and state trajectories that satisfy the nonnegativity constraints Eq. 11. The transversality condition is simply

$$\begin{aligned} \lambda ^\text {p }_T = 0. \end{aligned}$$

(27)

Nonnegativity of multipliers and the complementarity condition guarantee

$$\begin{aligned} \zeta ^\text {p }_{e,t}, \zeta ^\text {p }_{c,t}, \zeta ^\text {p }_{r,t} \ge 0 \quad \text {and} \quad \zeta ^\text {p }_{e,t} E_t = \zeta ^\text {p }_{c,t} C_t = \zeta ^\text {p }_{r,t} R_t = 0. \end{aligned}$$

(28)

We now turn to the current-value Hamiltonian, defined simply as $\mathcal {H} = e ^ {\rho t} \mathcal {H}^\text {p }$. The current-value costate variables and current-value multipliers are defined accordingly as

$$\begin{aligned} \lambda _t = e ^ {\rho t} \lambda ^\text {p }_t, \quad \xi _t = e ^ {\rho t} \xi ^\text {p }_t, \end{aligned}$$

(29)

and

$$\begin{aligned} \zeta _{e,t} = e ^ {\rho t} \zeta ^\text {p }_{e,t}, \quad \zeta _{c,t} = e ^ {\rho t} \zeta ^\text {p }_{c,t}, \quad \zeta _{r,t} = e ^ {\rho t} \zeta ^\text {p }_{r,t}. \end{aligned}$$

(30)

Differentiation with respect to time in Eq. 29 yields

$$\begin{aligned} \dot{\lambda }_t = \rho \lambda _t + e ^ {\rho t} \dot{\lambda }^\text {p }_t \quad \text {and} \quad \dot{\xi }_t = \rho \xi _t + e ^ {\rho t} \dot{\xi }^\text {p }_t. \end{aligned}$$

(31)

Substitute Eqs. 29–31 into Eqs. 23–28 and then we obtain the conditions in current-value terms. Eqs. 23–26 become

$$\begin{aligned}\begin{gathered} E_t = [q C_t (p - \lambda _t) + \zeta _{e,t}] / (2 \delta ), \quad 0 \le t \le T, \\ \begin{aligned} - \dot{\lambda }_t + \rho \lambda _t&= p q E_t - \lambda _t (\eta _c + q E_t) - \xi _t g_r R_t^2 / K_r / a \cdot b_1 \exp (b_0 + b_1 C_t) + \zeta _{c,t} \\*&\quad + e ^ {- \rho \tau } \lambda _{t+\tau } g_c (1 - C_t / K_c^*) \exp (- C_t / K_c^*), \quad 0 \le t < T - \tau , \end{aligned} \\ \begin{aligned} - \dot{\lambda }_t + \rho \lambda _t&= p q E_t - \lambda _t (\eta _c + q E_t) - \xi _t g_r R_t^2 / K_r / a \cdot b_1 \exp (b_0 + b_1 C_t) \\*&\quad + \zeta _{c,t}, \quad T - \tau \le t \le T, \quad \text {and} \end{aligned} \\ \begin{aligned} - \dot{\xi }_t + \rho \xi _t&= w \alpha (R_t - R_\text {m }) ^ {\alpha - 1} \\*&\quad + \xi _t g_r \left\{ 1 - 2 R_t / K_r / a \cdot [1 + \exp (b_0 + b_1 C_t)] \right\} + \zeta _{r,t}, \quad 0 \le t \le T. \end{aligned} \end{gathered}\end{aligned}$$

Equation 27 becomes $\lambda _T = 0$. Equation 28 becomes

$$\begin{aligned} \zeta _{e,t}, \zeta _{c,t}, \zeta _{r,t} \ge 0 \quad \text {and} \quad \zeta _{e,t} E_t = \zeta _{c,t} C_t = \zeta _{r,t} R_t = 0. \end{aligned}$$