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Renewable Resource Harvesting Under Correlated Biological and Economic Uncertainties: Implications for Optimal and Second-Best Management

Abstract

Biologists have long recognized environmental disturbances impact both the growth of renewable resources and the efficiency of harvest effort. However, models of renewable resource management under uncertainty have commonly assumed economic and biological uncertainties to be uncorrelated. We present examples of valuable fish species that experience correlated variation in biological growth and catchability, in response to a common environmental disturbance. Building correlation into a model of renewable resource management under uncertainty, we find correlation to alter the optimal response by managers to cost disturbances, and impact the value of retaining intra-period flexibility over harvest targets. Examining the performance of three harvest control mechanisms—harvest quotas, effort quotas, and taxes—reveals that positive correlation between costs and growth favors harvest quotas over effort quotas, and effort quotas over taxes (and vice versa). The model is then applied numerically to the Pacific bigeye tuna fishery, which experiences positively correlated shocks driven by the El Niño Southern Oscillation. Correlation qualitatively changes the optimal response to a cost shock, and harvest quotas are found to be strongly favored over both effort quotas and taxes.

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Notes

  1. 1.

    See Jensen (2008) for a review.

  2. 2.

    These observations were relayed to the authors by Georgia crabbers and scientists from the Georgia Department of Natural Resources (GDNR). Patterns in crab movement are confirmed in population survey data supplied by GDNR.

  3. 3.

    \(F\left( \theta \mid \eta \right) \) implies \(G\left( \eta \right) \) is conditional on \(\theta \), and that the two variables could be described by a joint CDF. This is not appropriate here, as we assume \(\eta \) is observed before \(\theta \). An alternative interpretation is that both \(\theta \) and \(\eta \) are driven by a common variable, for which \(\eta \) provides a more timely indication.

  4. 4.

    Certainty regarding stock levels implies that choosing escapement is equivalent to choosing harvest.

  5. 5.

    Or, after observing the environmental variable directly, which may occur before or after harvesting begins. This is equivalent to the case of sole ownership.

  6. 6.

    \(F\left( \theta \right| \eta _{1})\) is first-order stochastic dominant over \(F \left( \theta \right| \eta _{2}) \) i.f.f. \( F\left( \theta \right| \eta _{1}) \le F\left( \theta \right| \eta _{2})\).

  7. 7.

    If observing \(\eta _{t}\) reduces the variance of \(F\left( \theta _{t} \mid \eta _{t} \right) \) (second-order stochastic dominance), \(E_{t}\left[ g\left( e_{t},\theta _{t}\right) \right] \) will increase for the manager with target flexibility, and \(E_{t}\left[ g_{e}\left( e_{t},\theta _{t}\right) \right] \) will fall, with the combined effect lowering the expected marginal value of escapement and increasing harvests, or \(e_{t}^{f}\left( \bar{\eta }\right) < e_{t}^{q}\).

  8. 8.

    It is possible the opposing, indirect effect could be more than twice as large as the direct effect, thereby reversing the direction of optimal escapement and increasing the value of information.

  9. 9.

    Optimal escapement is state-independent; thus, if a mechanism is preferred to others in one period, it is preferred to others in all periods.

  10. 10.

    Twice-differentiating \(\int ^{\theta ^{h}}_{\theta ^{l}}\left( p\left( g\left( e, \theta \right) -\tilde{e}\right) -\int _{\tilde{e}}^{g\left( e, \theta \right) }c\left( s\right) ds\right) dF\left( \theta \mid \eta \right) \) with respect to \(e\) yields \(\delta \int ^{\theta ^{h}}_{\theta ^{l}}g_{ee}\left( p-c \left( g\left( .\right) \right) \right) - \left( g_{e}\right) ^{2}c\left( e\right) dF\left( \theta \right) \), which is strictly negative. The terms contained in the integral of the second term of (11) can be transformed with this expression and shown to be strictly negative.

  11. 11.

    The ITE will be binding if \(\eta _{t} c\left( e_{t}^{z}\left( z_{t},\eta _{t}\right) \right) <p \forall \eta \in \left[ \eta ^{l},\eta ^{h}\right] \).

  12. 12.

    Thank you to Simon Hoyle from WCPFC for help in identifying and accessing data.

  13. 13.

    This is a different objective than what we assume in our model, in which the manager is tasked with maximizing profits.

  14. 14.

    The changes in optimal harvest reported in scenarios 1 and 2 of Table 5 are so small in magnitude that it is unlikely a negative price elasticity would have discernible impacts.

  15. 15.

    Note, we have assumed only one random variable, which affects both functions.

  16. 16.

    Net revenue is concave in escapement; thus, concavity in growth implies the third term of (16) is concave.

  17. 17.

    We follow the procedure outlined by Costello et al. (2001).

  18. 18.

    This condition implies \(g\left( e_{t},\theta ^{l} \right) \ge \tilde{e}\left( \eta ^{h}\right) \) for all possible values of \(e_{t}\), which, in turn means \(g\left( e_{t},\theta ^{l} \right) \ge x_{t}\); if left unmolested, the stock will always experience net positive growth.

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Acknowledgments

We would like to thank Jason Shogren, John Tschirhart, and two anonymous referees for comments that have greatly improved this research. We also thank seminar attendees at the University of Wyoming, the NOAA Fisheries Science Center, Miami, FL, the University of Iceland, and conference participants at Camp Resources XVI in Asheville, NC, and IIFET 2010 in Montpellier, France. Kennedy appreciates support from a NOAA/NMFS - Sea Grant Fellowship in Marine Resource Economics. Any remaining errors are ours.

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Correspondence to Chris J. Kennedy.

A Mathematical Appendix

A Mathematical Appendix

Derivation of (2)

Rewriting (1) as a stochastic dynamic programming problem yields

$$\begin{aligned} V\left( x_{t}\right) =\max _{e_{t}} p\left( x_{t}-e_{t}\right) -E_{t}\int ^{x_{t}}_{e_{t}}\eta _{t}c \left( s\right) ds+\delta E_t [V\left( x_{t+1}\right) ] \end{aligned}$$
(16)

Recognizing that \(E_{t}\left[ \eta _{t} c\left( e_{t}\right) \right] =c\left( e_{t}\right) \), the following necessary condition must be met for an interior solution, \(e^{*}_{t}\), to exist

$$\begin{aligned} -p+c\left( e^{*}_{t}\right) +\delta E_{t}\left[ \frac{\partial V\left( x_{t+1}\right) }{\partial x_{t+1}} \frac{\partial x_{t+1}}{\partial e^{*}_{t}}\right] \end{aligned}$$
(17)

(16) is strictly concave in escapement,Footnote 16 a sufficient condition for (17) to characterize a maximum. The optimal control is state-independent, which allows for a closed form solution.Footnote 17 Define the maximum immediate harvest value as

$$\begin{aligned} \varphi \left( x_{t},\eta _{t}\right) \equiv p\left( x_{t}-\tilde{e} \left( \eta _{t}\right) \right) -\int \limits _{\tilde{e} \left( \eta _{t}\right) }^{x_{t}} \eta _{t}c\left( s\right) ds \end{aligned}$$
(18)

where \(\tilde{e} \left( \eta _{t}\right) \) is the escapement level at which price equals marginal harvest cost, \(p=\eta _{t} c\left( \tilde{e} \right) \). Using (18), (16) and (17) can be written as

$$\begin{aligned} V\left( x_{t}\right) =\max _{e_{t}} E_{t}\left[ \varphi \left( x_{t},\eta _{t}\right) -\varphi \left( e_{t},\eta _{t}\right) +\delta V\left( g\left( e_{t},\theta _{t}\right) \right) \right] \end{aligned}$$
(19)

and

$$\begin{aligned} \varphi '\left( e_{t}^{*},\eta _{t}\right) = \delta E_{t}\left[ \left( \frac{\partial V^{f}\left( x_{t+1}\right) }{\partial x_{t+1}}\right) g_{e}\left( e^{*}_{t},\theta _{t}\right) \right] \end{aligned}$$
(20)

Note that (20) is a function of \(e_{t}\), but not \(x_{t}\) or \(h_{t}\); i.e., when harvests are positive, the initial stock has no bearing on \(e_{t}^{*}\). The structure of the relevant optimization problem can be found via backward induction. In the terminal period \(T\), the value function is given by

$$\begin{aligned} V\left( x_{T}\right) =\max _{e_{T}} E_{T}\left[ \varphi \left( x_{T},\eta _{T}\right) -\varphi \left( e_{T},\eta _{T}\right) \right] \end{aligned}$$

Noting that at \(E_{T}\left[ \varphi \left( e_{T},\eta _{T}\right) \right] =E_{T}\left[ \varphi \left( \tilde{e}\left( \eta _{T}\right) ,\eta _{T}\right) \right] \) (there is no incentive to conserve in the final period, \(e^{*}_{T}=\tilde{e}\left( \eta _{T}\right) \)), the value function in \(T-1\) is given by

$$\begin{aligned} V\left( x_{T-1}\right)&=\max _{e_{T-1}} E_{T-1}\left[ \varphi \left( x_{T-1},\eta _{T-1}\right) - \varphi \left( e_{T-1},\eta _{T-1}\right) + \delta \varphi \left( g\left( e_{T-1},\theta _{T-1}\right) ,\eta _{T}\right) \right] \nonumber \\&= \max _{e_{T-1}} E_{T-1}\left[ \psi \left( e_{T-1},\eta _{T-1}\right) + \varphi \left( x_{T-1},\eta _{T-1}\right) \right] \end{aligned}$$
(21)

where \(\psi \left( e_{T-1},\eta _{T-1}\right) \equiv \delta \varphi \left( g\left( e_{T-1},\theta _{T-1}\right) ,\eta _{T} \right) -\varphi \left( e_{T-1},\eta _{T-1}\right) \) contains the terms from (21) that depend on \(e_{T-1}\), and \(E_{T-1}\left[ \psi \left( e_{T-1},\eta _{T-1}\right) \right] \) is the expected value (in period \(T\)) of harvesting the growth from \(e_{T-1}\) in period \(T\), net of forgone profit in period \(T-1\). \(E_{T-1}\left[ \psi \left( e_{T-1},\eta _{T-1}\right) \right] \) achieves an interior maximum at \(e^{*}_{T-1}\), so long as \(p>c\left( g\left( \tilde{e}\left( \eta _{T-1}\right) ,\theta _{T-1}\right) \right) \) for all combinations of \(\eta \) and \(\theta \).Footnote 18 Moving back one period

$$\begin{aligned} V\left( x_{T-2}\right)&= \max _{e_{T-2}} E_{T-2}\left[ \varphi \left( x_{T-2},\eta _{T-2}\right) -\varphi \left( e_{T-2},\eta _{T-2}\right) + \delta \left[ \psi \left( e_{T-1}^{*},\eta _{T-1}\right) \right. \right. \nonumber \\&\quad \left. \left. +\,\varphi \left( g \left( e_{T-1},\theta _{T-1}\right) ,\eta _{T}\right) \right] \right] \nonumber \\&= \max _{e_{T-2}} E_{T-2}\left[ \psi \left( e_{T-2},\eta _{T-2}\right) + \varphi \left( x_{T-2},\eta _{T-2}\right) + \delta \left[ \psi \left( e_{T-1}^{*},\eta _{T-1}\right) \right] \right] \end{aligned}$$
(22)

The choice of \(e^{*}_{T-1}\) is exogenous to the optimization problem in \(T-2\), and can be ignored in the determination of \(e^{*}_{T-1}\). Extending to the infinite-horizon case yields

$$\begin{aligned} V\left( x_{t}\right) =\max _{e_{t}} E_{t}\left[ \psi \left( e_{t},\eta _{t}\right) +\varphi \left( x_{t},\eta _{t}\right) + \sum \limits ^{\infty }_{\tau =1}\delta ^\tau \left[ \psi \left( e_{t+\tau }^{*},\eta _{t+\tau }\right) \right] \right] \end{aligned}$$
(23)

Thus, the problem can be broken down into a series of single-period optimizations in the form of (2).

Derivation of (4)

The numerator of (4) is equivalent to \(-\partial \zeta /\partial \eta _{t}\), where \(\zeta \) is the first order condition (3)

$$\begin{aligned} -c\left( e^{f}_{t}\right) -\delta \int \limits _{\theta ^{l}}^{\theta ^{h}}g_{e} \left( e^{f}_{t},\theta _{t}\right) \left( p-c\left( g \left( e^{f}_{t},\theta _{t}\right) \right) \right) \frac{dF_{\eta } \left( \theta _{t}\mid \eta _{t}\right) }{d\theta }d\theta \end{aligned}$$
(24)

The first term of (24) is the direct effect of \(\eta _{t}\) on the current cost; the second is the indirect effect on the shadow value of escapement. The second term is a Riemann–Stieltjes integral and can be evaluated via integration by parts

$$\begin{aligned}&-\delta \int \limits _{\theta ^{l}}^{\theta ^{h}}g_{e}\left( e^{f}_{t},\theta _{t} \right) \left( p-c\left( g\left( e^{f}_{t},\theta _{t}\right) \right) \right) dF_{\eta }\left( \theta _{t}\mid \eta _{t}\right) \nonumber \\&= \delta g_{e}\left( e^{f}_{t},\theta _{t}\right) \left( p-c \left( g\left( e^{f}_{t},\theta _{t}\right) \right) \right) F_{\eta } \left( \theta _{t}\mid \eta _{t}\right) |^{\theta ^{h}}_{\theta ^{l}}\nonumber \\&\quad \quad +\delta \int \limits _{\theta ^{l}}^{\theta ^{h}}g_{e\theta } \left( e^{f}_{t},\theta _{t}\right) \left( p-c\left( g \left( e^{f}_{t},\theta _{t}\right) \right) -g_{e}g_{\theta }c_{x} \right) F_{\eta }\left( \theta _{t}\mid \eta _{t}\right) d\theta \end{aligned}$$
(25)

The first term on the right hand side of (25) is zero, so long as the original supports on \(\theta \) hold. The second term, along with the direct effect, form the numerator of (4).

Proof of Proposition 2

The method we use is similar to Danielsson (2002). Throughout this derivation, we treat \(\eta _{t}=\hat{\eta }\) as deterministic, and drop time-scripts for succinctness. Consider the first order condition for the ITQ problem, given by (5).

$$\begin{aligned} -p+\int \limits ^{\eta ^{h}}_{\eta ^{l}}\eta c\left( e^{q}\right) +\delta \int \limits _{\theta ^{l}}^{\theta ^{h}}g_{e} \left( e^{q},\theta \right) \left( p-c\left( g\left( e^{q}, \theta \right) \right) \right) F\left( \theta \mid \eta \right) d\theta =0 \end{aligned}$$

Assuming an interior solution exists, and that all functions in (5) are continuous, there is a degenerate value \(\eta =\hat{\eta }=1\) such that \(e^{q}\) is also optimal in the deterministic case. (5) becomes

$$\begin{aligned} -p+\hat{\eta }c\left( e^{q}\right) +\delta \int \limits _{\theta ^{l}}^{\theta ^{h}}g_{e} \left( e^{q},\theta \right) \left( p-c\left( g\left( e^{q}, \theta \right) \right) \right) F\left( \theta \mid \hat{\eta }\right) d\theta =0 \end{aligned}$$
(26)

\(e_{t}^{q}\) maximizes the expected value of (2) given \(\hat{\eta }\). Similarly, under a tax, \(\tau _{t}^{*}\) maximizes the expected value of (2), and (9) becomes

$$\begin{aligned} \!-\!p\!+\!\hat{\eta }c\left( e^{\tau }\left( \hat{\tau },\eta _{t}\right) \right) \!+\! \delta \int \limits _{\theta ^{l}}^{\theta ^{h}}g_{e}\left( e^{\tau } \left( \hat{\tau },\eta _{t}\right) \right) \left( p\!-\!c\left( g \left( e^{\tau }\left( \hat{\tau },\eta _{t}\right) ,\theta \right) \right) \right) F\left( \theta \mid \hat{\eta }\right) d\theta =0\qquad \end{aligned}$$
(27)

When \(\eta _{t}=\hat{\eta }\), (10) becomes

$$\begin{aligned} \Delta ^{\tau q}&= p\left( x-e^{\tau }\left( \hat{\tau },\hat{\eta }\right) \right) -\int \limits _{e^{\tau }\left( \hat{\tau },\hat{\eta }\right) }^{x} \hat{\eta }c\left( s\right) ds -p\left( x-e^{q}\right) +\int \limits _{e^{q}}^{x} \hat{\eta }c\left( s\right) ds \nonumber \\&\quad +\delta \int \limits _{\theta ^{l}}^{\theta ^{h}}\left( p\left( g\left( e^{\tau } \left( \hat{\tau },\hat{\eta }\right) ,\theta \right) -\tilde{e} \right) -\int \limits _{\tilde{e}}^{g\left( e^{\tau }\left( \hat{\tau }, \hat{\eta }\right) ,\theta \right) }c\left( s \right) ds\right) F\left( \theta \mid \hat{\eta } \right) d\theta \nonumber \\&\quad -\delta \int \limits _{\theta ^{l}}^{\theta ^{h}}\left( p\left( g\left( e^{q}, \theta \right) -\tilde{e}\right) -\int \limits _{\tilde{e}}^{g\left( e^{q}, \theta \right) }c\left( s \right) ds\right) F\left( \theta \mid \hat{\eta } \right) d\theta \end{aligned}$$
(28)

The derivative of (28) with respect to the deterministic \(\hat{\eta }\) is

$$\begin{aligned} \frac{d\Delta ^{\tau q}}{d\eta }&= e^{\tau }_{\eta } \left( \hat{\eta }c\left( e^{\tau }\left( \hat{\tau },\hat{\eta } \right) \right) -p\right) -\int \limits _{e^{\tau }\left( \hat{\tau }, \hat{\eta }\right) }^{x}c\left( s\right) ds +\int \limits _{e^{q}}^{x} c\left( s\right) ds \nonumber \\&\quad +\delta \int \limits _{\theta ^{l}}^{\theta ^{h}}e^{\tau }_{\eta }g_{e} \left( e^{\tau }\left( \hat{\tau },\hat{\eta }\right) , \theta \right) \left( p-c\left( e^{\tau }\left( \hat{\tau }, \hat{\eta }\right) \right) \right) F\left( \theta \mid \hat{\eta } \right) d\theta \nonumber \\&\quad +\delta \int \limits _{\theta ^{l}}^{\theta ^{h}}\left( p\left( g\left( e^{\tau } \left( \hat{\tau },\hat{\eta }\right) ,\theta \right) - \tilde{e}\right) -\int \limits _{\tilde{e}}^{g\left( e^{\tau } \left( \hat{\tau },\hat{\eta }\right) ,\theta \right) }c \left( s \right) ds\right) F_{\eta }\left( \theta \mid \hat{\eta } \right) d\theta \nonumber \\&\quad -\delta \int \limits _{\theta ^{l}}^{\theta ^{h}}\left( p\left( g \left( e,\theta \right) -\tilde{e}\right) -\int \limits _{\tilde{e}}^{g\left( e,\theta \right) }c\left( s \right) ds\right) F_{\eta }\left( \theta \mid \hat{\eta } \right) d\theta \end{aligned}$$
(29)

By definition, (29) is equal to zero at \(\hat{\eta }\), and, given monotonicity in (26), (29) is also monotonic, implying that \(\Delta ^{\tau q}\left( e^{\tau }\left( \hat{\tau },\hat{\eta }\right) ,e^{q},\hat{\eta }\right) \) will represent either a maximum or a minimum on the bounded set \(\left[ \eta ^{l},\eta ^{h}\right] \); thus, examination of the second derivative of (28) with respect to \(\eta \) is sufficient to determine relative superiority of taxes or ITQs. If the second derivative is positive, (22) achieves a minimum at \(\hat{\eta }\) and taxes are deemed superior to ITQs (the expected payoff associated with an ITQ policy falls more rapidly with deviations in \(\eta \) than does the expected payoff associated with a tax). If the second derivative is negative, \(\hat{\eta }\) represents a maximum and ITQs are preferred. The second derivative is

$$\begin{aligned} \frac{d^{2}\Delta ^{\tau q}}{d\eta ^{2}}&= \left( e^{\tau }_{\eta }\right) ^{2}\hat{\eta } c_{x}+2e^{\tau }_{\eta }c\left( e^{\tau } \left( \hat{\tau },\hat{\eta }\right) \right) +e^{\tau }_{\eta \eta }\left( \hat{\eta }c\left( e^{\tau } \left( \hat{\tau },\hat{\eta }\right) \right) -p\right)&\nonumber \\&\quad +\delta \int \limits _{\theta ^{l}}^{\theta ^{h}} \left( \left( e_{\eta \eta }^{\tau }g_{e}+\left( e^{\tau }_{\eta }\right) ^{2}g_{ee} \right) \left( p-c\left( g\left( .\right) \right) \right) -c_{x}e^{\tau }_{\eta }g_{e}\right) dF\left( \theta \mid \hat{\eta } \right) \nonumber \\&\quad +\delta \int \limits _{\theta ^{l}}^{\theta ^{h}}e^{\tau }_{\eta }g_{e} \left( .\right) \left( p-c\left( g\left( .\right) \right) \right) F_{\eta }\left( \theta \mid \hat{\eta } \right) d\theta \end{aligned}$$
(30)

where \(e^{\tau }_{\eta }=\frac{de^{\tau }}{d\eta }=\frac{-c\left( e^{\tau } \left( .\right) \right) }{c_{x}\eta }>0\) and \(e^{\tau }_{\eta \eta }=\frac{d^{2}e^{\tau }}{d\eta ^{2}}=\frac{2c\left( e^{\tau } \left( .\right) \right) }{c_{x}\eta ^{2}}<0\). Recognizing that \(\frac{e^{\tau }_{\eta \eta }}{e^{\tau }_{\eta }}=-\frac{2}{\eta }\), and dividing any multiplying through by \(e^{\tau }_{\eta }\) and \(\eta _{t}\), respectively, yields

$$\begin{aligned}&\frac{d^{2}\Delta ^{\tau q}}{d\eta ^{2}} = 2p-\hat{\eta }c \left( e^{\tau }\right)&\nonumber \\&\quad +\delta \int \limits _{\theta ^{l}}^{\theta ^{h}} \left( \left( g_{e e}c\left( e^{\tau }\right) /c_{x}+2g_{e}\right) \left( c \left( g\left( .\right) \right) -p\right) + \left( g_{e}\right) ^{2}c\left( e^{\tau }\right) \right) dF\left( \theta \mid \hat{\eta } \right) \nonumber \\&\quad +2\delta \hat{\eta }\int \limits _{\theta ^{l}}^{\theta ^{h}}g_{e}\left( .\right) \left( p-c\left( g\left( .\right) \right) \right) F_{\eta }\left( \theta \mid \hat{\eta } \right) d\theta \end{aligned}$$
(31)

Integrating by parts the second integral in (31) yields (11).

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Kennedy, C.J., Barbier, E.B. Renewable Resource Harvesting Under Correlated Biological and Economic Uncertainties: Implications for Optimal and Second-Best Management. Environ Resource Econ 60, 371–393 (2015). https://doi.org/10.1007/s10640-014-9771-2

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Keywords

  • Renewable resource management
  • Environmental uncertainty
  • Prices vs. quantities
  • ITQs