Appendix
Model of Regulated Open Access with Many Heterogeneous Fishers
Here we derive the aggregate cost function for the whole fishing community from individual fishers entry-exit decisions in response to profitability of fishing relative to outside opportunities. The model follows Quaas and Requate (2013), allowing for heterogeneous fishing skills and opportunity costs of effort.
Consider a continuum of fishers \(i\), where fisher \(i\) has fishing skill (idiosyncratic productivity parameter) \(\varphi \left( i \right)\). Assume that each fisher is endowed with one unit of effort. If the fisher uses \(e_{i} \in [0,1]\) units of effort in fishing, the catch \(h_{i}\) of that fisher is \(h_{i} = q{\kern 1pt} \varphi \left( i \right)e_{i} {\kern 1pt} X_{t}^{\chi } .\) As in Baland and Francois (2005), each fisher \(i\) has the choice between using effort in fishing or in a private project outside the fishery, which would yield a return \(\omega \left( i \right)(1 - e_{i} )\). Given these opportunity costs, and a fish price \(p\), fisher \(i\)’s net gain of fishing is:
$$\pi_{i} = p{\kern 1pt} q{\kern 1pt} \varphi \left( i \right){\kern 1pt} e_{i} {\kern 1pt} X_{t}^{\chi } - \omega \left( i \right){\kern 1pt} e_{i} = \left( {p{\kern 1pt} q{\kern 1pt} \varphi \left( i \right){\kern 1pt} X_{t}^{\chi } - \omega \left( i \right)} \right){\kern 1pt} e_{i} .$$
(20)
All individuals with positive net gain will spend the entire effort fishing, all others don’t fish at all. The marginal fisher i.* is the one where
$$p{\kern 1pt} q{\kern 1pt} \varphi \left( {i_{{}}^{*} } \right){\kern 1pt} X_{t}^{\chi } = \omega \left( {i_{{}}^{*} } \right).$$
(21)
Thus, all fishers with fishing skill higher than \(\varphi \left( {i_{{}}^{*} } \right) = \omega \left( {i_{{}}^{*} } \right)/\left( {p{\kern 1pt} q{\kern 1pt} X_{t}^{\chi } } \right)\), and all fishers with opportunity costs of fishing effort lower than \(\omega \left( {i_{{}}^{*} } \right) = p{\kern 1pt} q\varphi \left( {i_{{}}^{*} } \right){\kern 1pt} X_{t}^{\chi }\), will go fishing. We use \(E_{t}\) to denote the number of active fishers. As they are endowed with one unit of effort each, this is also aggregate effort.
We specifically assume a Pareto distribution of fishing skills, \(\phi (i) = \xi {\kern 1pt} i^{\xi - 1}\), as common also in other fields of economics where heterogeneous productivity is considered, such as international trade (Arkolakis et al. 2008, 2012). Thus, aggregate harvest by all active fishers is given by:
$$H_{t} = q{\kern 1pt} \left( {\int_{0}^{{E_{t} }} \varphi \left( i \right){\kern 1pt} {\kern 1pt} di} \right){\kern 1pt} X_{t}^{\chi } = q{\kern 1pt} E_{t}^{\xi } {\kern 1pt} X_{t}^{\chi } ,$$
(22)
which is the generalized Gordon-Schaefer harvest function specified in the main text.
Total harvesting costs are \(C_{t} = \int_{0}^{{E_{t} }} \omega (i){\kern 1pt} di.\) Relative to the marginal fisher with skill \(\omega \left( {i_{{}}^{*} } \right)\), individuals with lower opportunity costs are fishing and individuals with higher opportunity costs are using their effort outside the fishery. Thus, \(\omega^{\prime}\left( {E_{t}^{{}} } \right) > 0\), and we obtain that marginal costs of effort are positive and increasing. Finally, it follows that the zero-profit condition for the marginal fisher, equation (21), is equivalent to condition (5), as:
$$\frac{{dC_{t} }}{{dH_{t} }} = \omega (E_{t} ){\kern 1pt} \frac{{dE_{t} }}{{dH_{t} }} = \frac{{\omega (E_{t} )}}{{\frac{{dH_{t} }}{{dE_{t} }}}} = \frac{{\varpi (E_{t} )}}{{q{\kern 1pt} \xi {\kern 1pt} E_{t}^{\xi - 1} {\kern 1pt} X_{t}^{\chi } }} = \frac{1}{{q{\kern 1pt} X_{t}^{\chi } }}{\kern 1pt} \frac{{\omega (E_{t} )}}{{\varphi (E_{t} )}} = p,$$
(23)
as we have \(E_{t} = i_{{}}^{*}\).
Proof of Result 1
We first state some properties of the cost function (2). We get the following derivatives of the cost function (2)
$$C_{q} = - \frac{1}{{\xi {\kern 1pt} q}}{\kern 1pt} \left( {\frac{{H_{t}^{*} }}{{q{\kern 1pt} X_{t}^{\chi } }}} \right)^{{\frac{1}{\xi }}} {\kern 1pt} \hat{C}^{\prime}\left( {\left( {\frac{{H_{t}^{*} }}{{q{\kern 1pt} X_{t}^{\chi } }}} \right)^{{\frac{1}{\xi }}} } \right),$$
(24)
$$C_{{X_{t} }} = - \frac{\chi }{{\xi {\kern 1pt} X_{t} }}{\kern 1pt} \left( {\frac{{H_{t}^{*} }}{{q{\kern 1pt} X_{t}^{\chi } }}} \right)^{{\frac{1}{\xi }}} {\kern 1pt} \hat{C}^{\prime}\left( {\left( {\frac{{H_{t}^{*} }}{{q{\kern 1pt} X_{t}^{\chi } }}} \right)^{{\frac{1}{\xi }}} } \right),$$
(25)
and
$$C_{{H_{t} }} = \frac{1}{{\xi {\kern 1pt} H_{t}^{*} }}{\kern 1pt} \left( {\frac{{H_{t}^{*} }}{{q{\kern 1pt} X_{t}^{\chi } }}} \right)^{{\frac{1}{\xi }}} {\kern 1pt} \hat{C}^{\prime}\left( {\left( {\frac{{H_{t}^{*} }}{{q{\kern 1pt} X_{t}^{\chi } }}} \right)^{{\frac{1}{\xi }}} } \right).$$
(26)
Thus, we can also write \(H_{t}^{*} {\kern 1pt} C_{{H_{t} }} = - q{\kern 1pt} C_{q}\), and \(\chi {\kern 1pt} H_{t}^{*} {\kern 1pt} C_{{H_{t} }} = - X_{t} {\kern 1pt} C_{{X_{t} }}\). Differentiating the former equation with respect to \(H_{t}^{*}\) we get:
$$C_{{H_{t} H_{t} }} = - \frac{q}{{H_{t}^{*} }}{\kern 1pt} C_{{qH_{t} }} + \frac{q}{{\left( {H_{t}^{*} } \right)_{{}}^{2} }}{\kern 1pt} C_{q} \Leftrightarrow H_{t}^{*} {\kern 1pt} C_{{qH_{t} }} - C_{q} = - \frac{{\left( {H_{t}^{*} } \right)_{{}}^{2} }}{q}{\kern 1pt} C_{{H_{t} H_{t} }} .$$
(27)
Moreover, multiplying \(H_{t}^{*} {\kern 1pt} C_{{H_{t} }} = - q{\kern 1pt} C_{q}\) with \(C_{{H_{t} H_{t} }}\) and rearranging a bit, we get
$$C_{{H_{t} H_{t} }} {\kern 1pt} C_{q} = - C_{{H_{t} H_{t} }} {\kern 1pt} \frac{{H_{t}^{*} }}{q}{\kern 1pt} C_{{H_{t} }} .$$
(28)
Differentiating Eq. (11) in the main text with respect to \(q\), using the envelope theorem and doing small rearrangements, we find (omitting arguments):
$$\frac{{d\pi_{t}^{*} }}{dq} = C_{{H_{t} H_{t} }}^{{}} {\kern 1pt} H_{t}^{*} {\kern 1pt} \frac{{dH_{t}^{*} }}{dq} + C_{{H_{t} {\kern 1pt} q}}^{{}} {\kern 1pt} H_{t}^{*} - C_{q}^{{}}$$
(29)
$$\mathop = \limits^{{(8)}} \frac{{C_{{H_{t} H_{t} }}^{{}} {\kern 1pt} H_{t}^{*} {\kern 1pt} C_{{H_{t} {\kern 1pt} q}}^{{}} }}{{P^{\prime}(H_{t}^{*} ) - C_{{H_{t} H_{t} }}^{{}} }} + C_{{H_{t} {\kern 1pt} q}}^{{}} {\kern 1pt} H_{t}^{*} - C_{q}^{{}} = \frac{{P^{\prime}(H_{t}^{*} ){\kern 1pt} \left( {H_{t}^{*} {\kern 1pt} C_{{H_{t} {\kern 1pt} q}}^{{}} - C_{q}^{{}} } \right) + C_{{H_{t} H_{t} }}^{{}} C_{q}^{{}} }}{{P^{\prime}(H_{t}^{*} ) - C_{{H_{t} H_{t} }}^{{}} }}$$
(30)
$$\mathop = \limits^{(27)} \frac{{P^{\prime}(H_{t}^{*} ){\kern 1pt} \frac{{\left( {H_{t}^{*} } \right)_{{}}^{2} }}{q}{\kern 1pt} C_{{H_{t} H_{t} }} - C_{{H_{t} H_{t} }} {\kern 1pt} \frac{{H_{t}^{*} }}{q}{\kern 1pt} C_{{H_{t} }} }}{{P^{\prime}(H_{t}^{*} ) - C_{{H_{t} H_{t} }} }}$$
(31)
$$\mathop = \limits^{(28)} \frac{{C_{{H_{t} H_{t} }} {\kern 1pt} C_{q} }}{{P^{\prime}(H_{t}^{*} ) - C_{{H_{t} H_{t} }} }}\left( {1 - \left( { - \frac{{H_{t}^{*} {\kern 1pt} P^{\prime}(H_{t}^{*} )}}{{P(H_{t}^{*} )}}} \right)} \right){\kern 1pt} .$$
(32)
The result that consumer surplus increases with \(q\) directly follows from the assumption of a downward sloping inverse demand function according to which consumer surplus increases with and the result that catch increases with \(q\) as stated in Eq. (8).
Proof of Result 2
We have already proven the result for consumer surplus. Here we show that we obtain the same for producer surplus. Using Eqs. (16) in (18) from the mai text, we find:
$$\frac{{d\pi^{*} }}{dq} = \frac{1}{{(F^{\prime}(X^{*} ) - H_{X}^{*} )}}{\kern 1pt} \left( {\left( {C_{HX}^{{}} {\kern 1pt} H^{*} + C_{HH}^{{}} {\kern 1pt} H^{*} {\kern 1pt} H_{X}^{*} - C_{X}^{{}} } \right){\kern 1pt} H_{q}^{*} + \left( {C_{HH}^{{}} {\kern 1pt} H^{*} {\kern 1pt} H_{q}^{*} + C_{Hq}^{{}} {\kern 1pt} H^{*} - C_{q}^{{}} } \right){\kern 1pt} \left( {F^{\prime}(X^{*} ) - H_{X}^{*} } \right)} \right)$$
(33)
Differentiating condition (5), which determines the market equilibrium harvest, with respect to \(X\) and \(q\), we obtain:
$$\left( {P^{\prime}(H^{*} ) - C_{HH}^{{}} } \right){\kern 1pt} H_{X}^{*} = C_{HX}^{{}}$$
and
$$\left( {P^{\prime}(H^{*} ) - C_{HH}^{{}} } \right){\kern 1pt} H_{q}^{*} = C_{Hq}^{{}} .$$
Combining these two conditions, we have \(C_{HX}^{{}} {\kern 1pt} H_{q}^{*} = C_{Hq}^{{}} {\kern 1pt} H_{X}^{*}\). Note that the cost function (2) implies \(X{\kern 1pt} C_{X}^{{}} = \chi q{\kern 1pt} C_{q}^{{}}\) and \(X{\kern 1pt} C_{HX}^{{}} = \chi q{\kern 1pt} C_{Hq}^{{}}\) (see Appendix 2). We thus also have \(C_{X}^{{}} {\kern 1pt} H_{q}^{*} = C_{q}^{{}} {\kern 1pt} H_{X}^{*}\). Using these results, Eq. (C.1) simplifies to:
$$\frac{{d\pi^{*} }}{dq} = \frac{{F^{\prime}(X^{*} )}}{{F^{\prime}(X^{*} ) - H_{X}^{*} }}{\kern 1pt} \left( {C_{HH}^{{}} {\kern 1pt} H^{*} {\kern 1pt} H_{q}^{*} + C_{Hq}^{{}} {\kern 1pt} H^{*} - C_{q}^{{}} } \right).$$
(34)
The term in brackets is the short-term effect of more efficient fishing technology on profit, as given in Eq. (20). Under the assumptions stated in Result 1, this term is positive. Thus, the sign of \(d\pi^{*} /dq\) depends on the sign of the first factor. For a stable steady state, the denominator is negative. Thus, \(d\pi^{*} /dq > 0\) if and only if \(F^{\prime}(X^{*} ) < 0\), which is the case if and only if \(X^{*} > X^{{{\text{m}}sy}}\).
Generalized Model of Welfare from Myopic Explotation and Regulated Open-Access Fishery
To obtain some indications of how robust the above results are, we will look at a very general model of harvesting and welfare derived from the fishery. In general, aggregate instantaneous welfare derived from the fishery is described by a utility function:
$$U_{t} = U(H_{t} ,X_{t} ,q),$$
(35)
which is increasing in harvest, \(U_{{H_{t} }} > 0\), nondecreasing in stock size, \(U_{{X_{t} }} \ge 0\), and increasing in efficiency, \(U_{q} > 0\). We further assume that marginal utility of catch weakly increases with stock size, \(U_{{H_{t} X_{t} }} \ge 0\), and increases with efficiency, \(U_{{H_{t} q}} > 0\). As above, the assumption \(U_{q} > 0\) implies that the short-run effect of increasing efficiency—while keeping stock size constant—is positive. The instantaneous welfare function (35) can capture net economic surplus—the sum of consumer and producer surplus. The general formulation (35), applied to a commercial fishery, also allows to take into account that consumers’ willingness to pay for fish may include concerns for sustainability, for example in a way that the demand for fish positively depends on stock size.Footnote 2
The condition determining market equilibrium harvest in myopic exploitation is then simply:
$$U_{H} (H_{t}^{*} ,X_{t} ,q) = 0.$$
(36)
We assume that the net utility is separable as follows:
$$U(H_{t}^{{}} ,X_{t}^{{}} ,q) = \hat{U}(H_{t}^{{}} ,y(X_{t}^{{}} ,q))$$
(Assumption 1)
Harvesting profit with a Gordon-Schaefer harvest function is a special case, where \(y\left( {X_{t}^{{}} ,q} \right)\) can be interpreted as the catch per unit of effort. Formally, given the separability of utility, the condition (36) for harvest under restricted open access fixes the relationship between harvest and \(y(X,q).\) Thus, utility can be expressed as a function of harvest only.
More intuitively, Assumption 1 imposes a condition how harvesting efficiency affects instantaneous welfare; namely in a way such that an equivalent effect on welfare could be obtained by keeping harvesting efficiency constant, but changing the resource stock size in a specific way. This shows that Assumption 1 is related to the idea of labor-augmenting technical change in growth theory (Robinson 1938) where improving technology has the same effect on output as an expansion of labor force. Thus, we can interpret Assumption 1 as the property of ‘stock-augmenting’ harvesting efficiency.
Differentiating (36) with respect to \(X\) and \(q\) gives, adopting the assumption of stock-augmenting harvesting efficiency:
$$U_{HH} {\kern 1pt} H_{X}^{*} = - U_{HX} = - \hat{U}_{Hy} {\kern 1pt} y_{X}$$
(37)
and
$$U_{HH} {\kern 1pt} H_{q}^{*} = - U_{Hq} = - \hat{U}_{Hy} {\kern 1pt} y_{q} ,$$
(38)
respectively. Under Assumption 1, it follows that \(U_{X} {\kern 1pt} H_{q}^{*} = U_{q} {\kern 1pt} H_{X}^{*}\), where \(U_{x} = \hat{U}_{y} {\kern 1pt} y_{X}\) and \(U_{q} = \hat{U}_{y} {\kern 1pt} y_{q}\). From the assumptions on \(U\) it also follows that \(H_{q}^{*} > 0\).
Differentiating the biological equilibrium condition \(F(X^{*} ) = H^{*}\) with respect to q gives:
$$\left( {F^{\prime} - H_{X}^{*} } \right){\kern 1pt} X_{q}^{*} = H_{q}^{*} .$$
(39)
As stability requires \(F^{\prime} - H_{X}^{*} < 0\) (Eq. 15 main text), this equation implies, \(X_{q}^{*} < 0\). Now consider a steady state in which changes in \(q\) lead to changes in \(H^{*}\) giving repercussions on the steady state stock size \(X^{*}\). Taking these feedbacks into account, we obtain the following long-term relationship between \(U\) and q:
$$\frac{{dU^{*} }}{dq} = U_{q} + U_{X} {\kern 1pt} X_{q}^{*} = U_{q} + U_{X} {\kern 1pt} H_{q}^{*} {\kern 1pt} \frac{1}{{F^{\prime} - H_{X}^{*} }} = U_{q} {\kern 1pt} \frac{{F^{\prime}}}{{F^{\prime} - H_{X}^{*} }}.$$
(40)
The first equality holds by virtue of the envelope theorem while the second equality uses Eq. (39) that states how steady state stock size reacts to changes in \(q\) via changes in \(H\). In the last step we have used the Assumption 1 of stock-augmenting harvesting efficiency, which implies \(U_{X} {\kern 1pt} H_{q}^{*} = U_{q} {\kern 1pt} H_{X}^{*}\), as shown above. Thus, we can conclude that our main results hold for this generalized model of welfare as well. This is summarized as follows.
Theorem 1
The long-run equilibrium stock size is above the stock that generates the maximum sustainable yield if and only if the long-run equilibrium welfare increases with harvesting efficiency. Formally,
$${\text{sign}}\left( {X^{*} - X^{{{\text{msy}}}} } \right) = {\text{sign}}\left( {\frac{{dU^{*} }}{dq}} \right).$$
(41)
To illustrate this result a bit further, consider the Sengalesian case study again, but for the sake of the argument we assume a stock elasticity \(\chi = 0.65\)(Sect. 4), such that the market equilibrium harvest is a linear function of stock size.
Figure 3 illustrates the effect of increasing harvesting efficiency, \(q_{0} < q_{1} < q_{2}\). Stock-augmenting increase in harvesting efficiency compresses the welfare indifference curves to the left, i.e. at a given level of harvest welfare is the same, no matter what the level of harvesting efficiency is. Thus only the resulting harvest level affects the equilibrium level of welfare. For the increase in harvesting efficiency from \(q_{0}\) to \(q_{1}\), equilibrium harvest increases and thus the new equilibrium is reached at a higher welfare level. With the further increase from \(q_{1}\) to \(q_{2}\) the new equilibrium attains a lower welfare level.
Extension to Dynamic Setting
Here we extend the analysis to a dynamic setting by considering the problem to determine the time path of \(q\) that maximizes the present value of welfare, at a discount rate \(\rho\), subject to the biological dynamics and the condition for restricted open-access market equilibrium. Formally, the problem reads:
$$\mathop {\max }\limits_{{q_{t} }} \int_{0}^{\infty } e^{{ - \rho {\kern 1pt} t}} {\kern 1pt} U(H^{*} (X_{t} ,q_{t} ),X_{t} ,q_{t} ){\kern 1pt} dt$$
(42)
subject to Eq. (13) (main text) with \(X_{0}\) given and where \(H^{*} (X_{t} ,q_{t} )\) is implictly defined by the restricted open-access condition \(U_{{H_{t} }} (H^{*} (X_{t} ,q_{t} ),X_{t} ,q_{t} ) = 0\).
The current-value Hamiltonian for this problem is
$${\mathbb{H}} = U(H^{*} (X_{t} ,q_{t} ),X_{t} ,q_{t} ) + \mu_{t} {\kern 1pt} \left( {F(X_{t} ) - H^{*} (X_{t} ,q_{t} )} \right)$$
(43)
The first-order conditions for the optimal time path of \(q_{t}\) are, using \(U_{{H_{t} }} = 0\),
$$\frac{{\partial {\mathbb{H}}}}{{\partial q_{t} }} = U_{{q_{t} }} (H^{*} (X_{t} ,q_{t} ),X_{t} ,q_{t} ) - \mu_{t} {\kern 1pt} H_{{q_{t} }}^{*} (X_{t} ,q_{t} ) = 0$$
(44)
and
$$\frac{{\partial {\mathbb{H}}}}{{\partial X_{t} }} = U_{{X_{t} }} (H^{*} (X_{t} ,q_{t} ),X_{t} ,q_{t} ) + \mu_{t} {\kern 1pt} \left( {F^{\prime}(X_{t} ) - H_{{X_{t} }}^{*} (X_{t} ,q_{t} )} \right) = \rho {\kern 1pt} \mu_{t} - \dot{\mu }_{t}$$
(45)
Equation (44) states that the immediate marginal benefit of improved harvesting efficiency should equal the marginal opportunity costs in terms of reducing the stock. Condition (45) is closely related to the fundamental equation of resource economics (Clark 1990), but here it includes the effect of the stock size on the harvest quantity in regulated open access.
Given Assumption 1, it follows from Eq. (44) that \(U_{{X_{t} }} (H^{*} (X_{t} ,q_{t} ),X_{t} ,q_{t} ) - \mu_{t} {\kern 1pt} H_{{X_{t} }}^{*} (X_{t} ,q_{t} ) = 0\). Thus, the optimality conditions simplify to:
$$U_{{q_{t} }} (H^{*} (X_{t} ,q_{t}^{*} ),X_{t} ,q_{t}^{*} ) = \mu_{t} {\kern 1pt} H_{{q_{t} }}^{*} (X_{t} ,q_{t}^{*} )$$
(46)
and
$$F^{\prime}(X_{t} ) - \rho = - \frac{{\dot{\mu }_{t} }}{{\mu_{t} }}.$$
(47)
These equations define, for any given stock size \(X_{t}^{{}}\), a level \(q_{t}^{*} = q^{*} (X_{t} )\) of harvesting efficiency that maximizes the present value of welfare. Figure 4 shows the numerical result for the case of the Senegalese small-scale fishery on Sarinella Aurita detailed in Sect. 4, for a zero discount rate \(\rho = 0\). We use AMPL with Knitro to numerically solve the dynamic optimization problem; the code is available from the authors on request. In line with intuition, the figure shows that larger the current stock size, the larger is the current harvesting efficiency that maximizes the present value of welfare. The steady state equilibrium stock size and harvesting efficiency that optimize welfare is indicated by the intersection of the black vertical and horizontal lines.
The result corresponding to Theorem 1 is the following: Whenever the actual harvesting efficiency is above (below) \(q^{*} (X_{t} )\), an instantaneous increase of harvesting efficiency decreases (increases) the present value of welfare.