Common Pool Politics and Inefficient Fishery Management

Abstract

Fisheries management often fails because total allowable catches (TACs) are set at inefficiently high levels. To study why decision-makers choose such high TACs, we model the annual negotiation on TACs as a dynamic game in discrete time. TACs are fixed by majority decision in a council consisting of decision-makers who are heterogeneous with respect to their discount rates. We show that the optimal feedback strategy for the less patient decision-makers will set inefficiently high TACs in Markov-perfect Nash equilibrium. A binding commitment to a long-term management plan could help solving this problem and lead to a more sustainable fishery management.

Appendix

1.1 Proof of Propositions 1–2

Assume a finite time horizon of $T$ periods. In the last period $T$, the optimal escapement levels are determined by the condition $\pi '(s_T^*)=0$, for both types of decision-maker. In period $T-1$, the Bellman equations thus read, using subscripts $T-1$ for the corresponding value functions,

$$\begin{aligned} v_{T-1}(x,i)&=\max \limits _{0\le s\le x}\left[ \pi (x)-\pi (s)+\rho _i\,E_z\left[ \pi (z\,g(s))-\pi (s_T^*)\right] \right] \\&=\pi (x)-\pi (S_i^*(x))+\rho _i\,E_z\left[ \pi (z\,g(S_i^*(x)))-\pi (s_T^*)\right] \\ V_{T-1}(x,p)&=\max \limits _{0\le s\le x}\left\{ \pi (x)-\pi (s)+\rho _p\,E\left[ \pi (z\,g(s))-\pi (s_T^*)\right] \right\} \\&=\pi (x)-\pi (S_p^*(x))+\rho _p\,E_z\left[ \pi (z\,g(S_p^*(x)))-\pi (s_T^*)\right] \\ v_{T-1}(x,p)&=\pi (x)-\pi (S^*_p(x))+\rho _i\,E_z\left[ \pi (z\,g(S^*_p(x)))-\pi (s_T^*)\right] \\ V_{T-1}(x,i)&=\pi (x)-\pi (S^*_i(x))+\rho _p\,E\left[ \pi (z\,g(S^*_i(x)))-\pi (s_T^*)\right] \end{aligned}$$

where we have used the guess (which is to be verified) that $S^*_j(x)=\min \{x,s_j^*\}$ is the optimal feedback policy for type $j\in \{i,p\}$.

In $T-2$, and in all periods before $T-2$, the Bellman equation for the case where type $p$ rules reads

$$\begin{aligned} V_{T-2}(x,p)\!&= \!\max \limits _{0\le s\le x}\left\{ \pi (x)\!-\!\pi (s)\!+\!\rho _p\,E_z\left[ q\,V_{T-1}(z\,g(s),i) +(1-q)\,V_{T-1}(z\,g(s),p)\right] \right\} \\&= \max \limits _{0\le s\le x}\Big \{\pi (x)-\pi (s)+\rho _p\,E_z \big [\pi (z\,g(s))\\&\quad +\,\,q\,\left( -\pi (S_i^*(z\,g(s)))+\rho _p\,E_{z'} \left[ \pi (z'\,g(S_i^*(z\,g(s))))\right] \right) \\&\quad +\,\,(1-q)\,\left( -\pi (S_p^*(z\,g(s)))+\rho _p\,E_{z'} \left[ \pi (z'\,g(S_p^*(z\,g(s))))\right] \right) \big ]-\rho _i\,\pi (s_T^*)\Big \} \end{aligned}$$

Using that $s_p^*$ is self-sustaining, i.e. that $g(s_p^*)>s_p^*$ with probability one, it follows for an interior solution for the optimization problem that $S_p^*(\hat{s}_p)=s_p^*$. Since $s_i^*<s_p^*$ this also implies $S_i^*(\hat{s}_p)=s_i^*$. Thus, the first-order condition for the optimization problem is

$$\begin{aligned} \pi '(s)=\rho _p\,E_z\big [\pi '(z\,g(s))\,z\,g'(z)\big ], \end{aligned}$$

which is identical to the first-order condition (5) for the optimal escapement level for type $p$. This proves Proposition 1.

In $T-2$, the Bellman equation for the case where type $i$ rules reads

$$\begin{aligned} v_{T-2}(x,i)&= \max \limits _{0\le s\le x}\left\{ \pi (x)-\pi (s)+\rho _i\,E_z\left[ q\,v_{T-1}(z\,g(s),i)+(1-q) \,v_{T-1}(z\,g(s),p)\right] \right\} \\&= \max \limits _{0\le s\le x}\big \{\pi (x)-\pi (s)+\rho _i\,E_z\big [\pi (z\,g(s)) +q\,\left( -\pi (S_i^*(z\,g(s)))\right. \\&\left. +\,\rho _i\,E_{z'}\left[ \pi (z'\,g(S_i^*(z\,g(s))))\right] \right) +(1-q)\,\left( -\pi (S_p^*(z\,g(s)))\right. \\&\left. +\,\rho _i\,E_{z'}\left[ \pi (z'\,g(S_p^*(z\,g(s)))) \right] \right) \big ]-\rho _i\,\pi (s_T^*) \big \} \end{aligned}$$

We start with the guess that the solution $\hat{s}_i$ to the optimization problem is self-sustaining, i.e. $z\,g(\hat{s}_i)>s_i^*$ with probability one, or equivalently $\underline{z}\,g(\hat{s}_i)>s_i^*$. This guess will be verified by the result that $\hat{s}_i\le s_i^*<\underline{s}$. Further, we use that $S^*_p(z\,g(s))=\max \{z\,g(s),s_p^*\}$. Thus, $S^*_p(z\,g(s))=z\,g(s)$ for $z<s_p^*/g(s)$ and $S^*_p(z\,g(s))=s_p^*$ else. Using these results, and using $\varphi (z)$ to denote the probability density function of $z$, we obtain the following Bellman equation

$$\begin{aligned} v_{T-2}(x,i)&= \max \limits _{0\le s\le x}\Bigg \{\pi (x)-\pi (s)+\rho _i\,E_z \bigg [\pi (z\,g(s))+q\,\left( -\pi (s_i^*)+\rho _i\,E_{z'} \left[ \pi (z'\,g(s_i^*))\right] \right) \\&+\,(1-q)\,\int _{\overline{z}}^{{s_p^*}/{g(s)}}\left( -\pi (z\,g(s)) +\rho _i\,E_{z'}\left[ \pi (z'\,g(z\,g(s)))\right] \right) \,\varphi (z)\,dz\bigg ]\\&+\,(1-q)\,\left( \int ^{\overline{z}}_{{s_p^*}/{g(s)}}\varphi (z)\,dz\right) \left( -\pi (s_p^*)+\rho _i\,E_{z'}\left[ \pi (z'\,g(s_p^*))\right] \right) -\rho _i\,\pi (s_T^*) \Bigg \} \end{aligned}$$

The first-order condition for the maximization problem reads

$$\begin{aligned} \pi '(s)&= \rho _i\,E_z\big [z\,\pi '(z\,g(s))\,g'(s)\big ] -(1-q)\\&\int _{\underline{z}}^{\frac{s_p^*}{g(s)}}\left( \pi '(z\,g(s))-\rho _i\,E_{z'}\left[ \pi '(z'\,g(z\,g(s))) \,z'\,\,g'(z\,g(s))\right] \right) \,z\,g'(s)\,\varphi (z)\,dz\nonumber \end{aligned}$$

(16)

This equation is solved by some $\hat{s}_i$ which is independent of $x$. Thus, the optimal policy for type $i$ is a constant escapement policy as well.

Comparing (16) to the first-order condition (5) for the optimal escapement level for type $i$, we find that the right-hand side of (16) is smaller than the right-hand side of (5), as the second term is negative: The term in brackets is monotone in $s$ (as $\pi$ and $g$ are concave). It is negative both at the lower and upper bound of integration: At the lower bound it is negative because $\underline{z}\,g(s)>s_i^*$, and

$$\begin{aligned}&\pi '(z\,g(s))-\rho _i\,E_{z'}\left[ \pi '(z'\,g(z\,g(s)))\,z'\,\,g'(z\,g(s))\right] \\&\quad >\pi '(s_i^*)-\rho _i\,E_{z'}\left[ \pi '(z'\,g(s_i^*))\,z'\,\,g'(s_i^*)\right] =0. \end{aligned}$$

At the upper bound it is negative because

$$\begin{aligned} \pi '(s_p^*)-\rho _i\,E_{z'}\left[ \pi '(z'\,g(s_p^*))\,z'\,\,g'(s_p^*)\right] >\pi '(s_p^*)-\rho _p\,E_{z'}\left[ \pi '(z'\,g(s_p^*))\,z'\,\,g'(s_p^*)\right] =0. \end{aligned}$$

This holds provided the upper bound of the integral is larger than the lower bound, i.e. if $s_p^*>\underline{z}\,g(\hat{s}_i)$. As $\hat{s}_i\ge s_i^*$, a sufficient condition for this to hold is $s_p^*>\underline{z}\,g(s_i^*)$.

Proposition 4 follows directly, as the (negative) second term on the right-hand side of condition (16) is decreasing in $s_p^\star$, which, in turn, is monotonically increasing in $\rho _p$.

1.2 Proof of Proposition 3

From Proposition 2:

$$\begin{aligned} v(x,1)&=\pi (x)-\pi (g(\hat{s}_i))+\rho _i(q(\pi (g(\hat{s}_i))+(1-q)C_3))\\ \pi '(\hat{s}_i)&=q \;\rho _i g'(\hat{s}_i)\pi '(g(\hat{s}_i))\\ q \; \rho _i&= \frac{\pi '(\hat{s}_i)}{g'(\hat{s}_i)\pi '(g(\hat{s}_i))} \end{aligned}$$

The RHS is increasing in $s$ (Reed 1979). Thus, if $q$ increases, $s$ has to increase, too.

1.3 Proof of Proposition 5

The optimal feedback-control rule $\bar{s}(x)$ is characterized by the Bellman equation

$$\begin{aligned}&\alpha \,\bar{V}(x)+(1-\alpha )\,\bar{v}(x)\\&\qquad =\max \limits _{s}\left\{ \pi (x)-\pi (s)+E\left[ \alpha \,\rho _p\,\bar{V}(z\,g(x))+(1-\alpha )\,\rho _i\,\bar{v}(z\,g(s))\right] \right\} \nonumber \end{aligned}$$

(17)

We guess the following value functions:

$$\begin{aligned} \bar{V}(x)&=\pi (x)+\bar{C} \end{aligned}$$

(18)

$$\begin{aligned} \bar{v}(x)&=\pi (x)+\bar{c} \end{aligned}$$

(19)

with constants $\bar{C}$ and $\bar{c}$. With this guess, the first-order condition for the right-hand side of (17) becomes

$$\begin{aligned} \pi '(s)=\left( \alpha \,\rho _p+(1-\alpha )\,\rho _i\right) \,E[\pi (z\,g(s))\,z\,g'(s)] \end{aligned}$$

(20)

This first-order condition is solved by a constant escapement level. This verifies the guess of the value function. Furthermore, the optimal escapement level that solves (20) is the same as the optimal escapement level for a hypothetical decision maker with discount factor $\alpha \,\rho _p+(1-\alpha )\,\rho _i$, which is a convex mixture of the discount factors of the two groups, and hence in between the two.