## Introduction

Many of the world's fisheries are in dire straits because of poor management and poorly delineated property rights with a third of these fisheries labelled unsustainable by the Food and Agriculture Organization of the United Nations (FAO 2018). This poor state of affairs has been with us for some time and the costs are substantial (Clark 2005). The estimated global cost of forgone rents for the year 2012 alone was for instance estimated to be US\$ 83 billion for that year alone (World Bank 2017).

The problem of overfishing is likely to be exacerbated by technology. Brander and Taylor (1997) demonstrated that improved productivity could lead to more aggressive harvesting of open access resources. This also applies to open access fisheries and fisheries subject to the tragedy of the commons (Whitmarsh 1990; Squires and Vestergaard 2013; Squires and Vestergaard 2018). There are several well documented examples where improved technology has led to the collapse of a fish stock (Hannesson et al. 2010; Gordon and Hannesson 2015). Kvamsdal et al. (2016) point out that even in managed fisheries, “…increased technical efficiency and progress, usually lead to overcapacity in national fishing fleets. Overcapacity creates national pressure for higher quotas…”. In a recent paper, Skonhoft and Quaas (2019) find that in a restricted open access or shared resource fishery if the long run steady state in a shared, but unmanaged, resource fishery is lower than the harvest rate associated with maximum sustainable yield, then improved technology may increase rents. However in larger, more accessible fisheries this condition is unlikely to hold.

The caveat in Skonhoft and Quaas (2019) not notwithstanding, it seems clear that improved technology worsens the market failure of a poorly managed fishery. This is unfortunate as improved technology has the potential to make the fishery more valuable and therefore makes it even more important that fish stocks are managed as well as possible and that the role of technology is understood. Remarkably little has been written on the role of technology in an optimally managed fishery. As stated in Squires and Vestergaard (2013): "… the normative literature has yet to formally analyze the impact of changes in disembodied and embodied technical progress and technical efficiency on optimum exploitation of common renewable resources."

The analysis in the literature on the effects of improved productivity is mostly concerned with steady state analysis and/or singular solutions resulting from bang-bang controls (Clark and Munro 1975; Caputo 1989; Squires and Vestergaard 2013, 2018). However steady state analysis is only locally relevant and policy advice for management of recovering stocks should not be based on steady state analysis. As we shall see, in fisheries such a strategy would be particularly ill advised as the effects of technology in the case of stock dependent costs go in completely different directions in steady state and in stock levels sufficiently below the optimal steady state. Squires and Vestergaard (2013, 2018) go some way in performing a formal analysis. Their contribution is contrasted with the results in the present paper in conjunction with the discussion of stock dependent cost functions.

The results are derived in general versions of the original canonical fisheries model (Clark and Munro 1975). The modelling of improved productivity is very basic and obtained by examining how an exogenous and unanticipated one-shot change in a productivity parameter affects optimal policy. This may seem restrictive. However, making the productivity change unanticipated allows us to study the pure effect of technology disentangled from the effect of preparing for an expected productivity increase. Also, even if a productivity increase is anticipated by the fishing industry it may still catch regulating authorities off guard. The present paper may thus inform regulators how to respond to such an event. It should also be noted that even if the interpretation of the change in the parameter in the present paper is as a productivity increase, other interpretations are also possible. E.g. an unanticipated permanent decrease in the price of an important normal input has the same effect on output as a productivity shock. See Bertoletti and Rampa (2013) for an analysis of what constitutes normal and inferior goods.

## The Canonical Schooling Fisheries Model

The following is the basic version of the fisheries model applied to a schooling fishery (Clark 1990, p 97). In theoretical models of pure schooling fisheries harvest costs are not dependent on stock size (Neher 1990, p 177). We assume that the net instantaneous benefits from harvesting is given by a continuous and strictly concave function of the harvest rate, h, and given by D(h) – C(h)α−1. Here D(h) is a benefit function, typically price multiplied by harvest rate, C(h)α−1 is a cost function and α is a parameter with higher α indicating better technology. It is assumed that D(0) = C(0) = 0. $$D^{\prime}(h)$$ and $$C^{\prime}(h)$$ are both assumed positive. $$D^{\prime } (h) - C^{\prime \prime } (h)\alpha^{ - 1}$$ is assumed negative and $$C^{\prime \prime } (h)$$ is assumed positive. We also make the important assumption that $$0 < D^{\prime}\left( 0 \right) - C^{\prime}\left( 0 \right)\alpha^{ - 1} < \infty$$. This assumption implies that we are not considering a subsistence fishery, but rather a commercial fishery or a fishery where capital and labour have alternative uses.

The fish stock is assumed to be governed by

$$\dot{x} = G\left( x \right) - h,\,\,\,x\left( 0 \right){\text{ given}}$$
(1)

The natural growth function G(x) is taken to be strictly concave, differentiable and satisfy G(0) = G(K) = 0 for some K > 0 and positive for all x ∈ (0, K). The specification of G(x) is in line with standard biological growth functions such as the logistic, which is used in phase portraits below, but the formal results do not require a parametric growth function. We assume there exists x ∈ (0, K) such that the derivative of the growth function equals the discount rate, $$G^{\prime}\left( x \right)$$ = ρ, which is reasonable for many commercially interesting fish species. The assumptions lead to the following optimization problem:

\begin{aligned} & V\left( {x\left( 0 \right),\alpha } \right) = \mathop {\max }\limits_{h\left( t \right) \ge 0} \int\limits_{0}^{\infty } {\left( {D\left( h \right) - C\left( h \right)\alpha^{ - 1} } \right)e^{ - \rho t} } dt \\ & {\text{subject}}\;{\text{to}}\;\dot{x} = G\left( x \right) - h\;{\text{and}}\;x\left( 0 \right) > 0\;{\text{given}} \\ \end{aligned}
(2)

The current value Hamiltonian for this problem is:

$$H = D\left( h \right) - C\left( h \right)\alpha^{ - 1} + \mu \left( {G\left( x \right) - h} \right)$$
(3)

The problem in (2) may be solved by using 4 sufficiency conditions for optimality found in e.g. Theorem 9.11.1 in Sydsæter et al. (2008). The first condition require that h, maximize the Hamiltonian at all t which implies:

$$\frac{\partial H}{{\partial h}} = D^{\prime}\left( h \right) - C^{\prime}\left( h \right)\alpha^{ - 1} - \mu \le 0 \, \left( { = 0{\text{ if }}h > 0} \right)$$
(4)

The second condition requires that the co-state variable satisfy:

$$\dot{\mu } = \left( {\rho - G^{\prime}\left( x \right)} \right)\mu$$
(5)

The third condition is that H is a concave function over (h, x) which is straightforward to confirm. The fourth condition requires that we establish that:

$$\mathop {\lim }\limits_{t \to \infty } \mu \left( t \right)e^{ - rt} \left( {\tilde{x}\left( t \right) - x\left( t \right)} \right) \ge 0$$
(6)

This must hold for all admissible functions $$\tilde{x}\left( t \right)$$ that satisfies (1). This condition can only be checked after an optimal solution has been found. However, from the phase diagram in Fig. 1 it is clear that µ(t) < ∞ as long as x(0) > 0. We further have that and 0 ≤ $$\tilde{x}\left( t \right)$$K for all possible $$\tilde{x}\left( t \right)$$ satisfying the differential equation, so the left hand side of (6) computes to zero for all possible $$\tilde{x}\left( t \right)$$.

Equation (4) defines h as a function of µ and α. This function is denotedFootnote 1:

$$h = \phi \left( {\mu ,\alpha } \right)$$
(7)

Note that (4) and Assumption 1 implies that $$D^{\prime}\left( 0 \right) - C^{\prime}\left( 0 \right)\alpha^{ - 1} < \mu$$ ⇒ ϕ(µ, α) = 0. For all x where (4) holds with equality it also holds that:

$$\frac{\partial h}{{\partial \mu }} = \phi_{\mu }^{\prime } = \frac{1}{{D^{\prime \prime } \left( h \right) - C^{\prime \prime } \left( h \right)\alpha^{ - 1} }} < 0$$
(8)
$$\frac{\partial h}{{\partial \alpha }} = \phi_{\alpha }^{\prime } = - \frac{{C^{\prime } \left( h \right)\alpha^{ - 2} }}{{D^{\prime \prime } \left( h \right) - C^{\prime \prime } \left( h \right)\alpha^{ - 1} }} > 0\;{\text{for}}\;h > 0$$
(9)
$$\left( {\frac{\partial \mu }{{\partial \alpha }}} \right)_{{h{\text{ is constant}}}} = - \frac{{\phi^{\prime}_{\alpha } }}{{\phi^{\prime}_{\mu } }} = C^{\prime}\left( h \right)\alpha^{ - 2} > 0{\text{ for }}h > 0$$
(10)

The inequality in (9) may lead us to infer that higher productivity implies larger harvest rates. As we shall see, the inequality (10) and how it affects the shape of the isocline for $$\dot{x} = 0$$ is part of the reason why this is wrong.

The model assumptions ensure that there exists a steady state (hss, xss, µss) = limt→∞(h(t), x(t), µ(t)) > 0 which if x(0) > 0 will be the long run equilibrium for an optimally managed fishery (Clark 1973; Cropper 1979). Convergence of (h(t), x(t), µ(t)) to the steady state (hss, xss, µss) will be along a stable saddle path towards a steady state characterized by $$\dot{x} = G\left( x \right) - \phi \left( {\mu ,\alpha } \right) = 0$$ and $$\dot{\mu } = \left( {\rho - G^{\prime}\left( x \right)} \right)\mu = 0$$. This stable path can be found in (x, µ) space by solving the following differential equation (Conrad and Clark 1987, Ch. 1.6.6; Judd 1998, Ch. 10.7):

$$\frac{{\dot{\mu }}}{{\dot{x}}} = \frac{{{\text{d}}\mu }}{{{\text{d}}x}} = \frac{{\left( {\rho - G^{\prime}\left( x \right)} \right)\mu }}{{G\left( x \right) - \phi \left( {\mu ,\alpha } \right)}},\,\,\mu \left( {x_{ss} ,\alpha } \right) = \mu_{ss}$$
(11)

The solution to (11) defines µ as a function of x along an optimal path. We denote the stable saddle path as µ(x, α). It is worthwhile to note that the stable saddle path µ(x, α) is in fact the derivative of the value function, V(x, α). As V(0, α) = 0 in this model, we can illustrate the value function in a phase portrait as the integral under the stable manifold. A phase portrait of the optimal solution is shown in Fig. 1.

Inserting µ(x, α) into (7) gives us optimal h as a feedback control:

$$h = \phi \left( {\mu \left( {x,\alpha } \right),\alpha } \right) = \Phi \left( {x,\alpha } \right)$$
(12)

The expression in (12) is a feedback control and gives us optimal harvest rates as a function of the stock and the productivity parameter and is usually derived in a dynamic programming framework. However, by using optimal control with its explicit focus on the shadow price we gain some additional structure to the problem that enables us to construct analytical results not directly accessible with dynamic programming where interior solutions are typically assumed.Footnote 2

One such result, crucial in the analysis below, may be found in Nævdal and Skonhoft (2018) who prove that if $$0 < D^{\prime}\left( 0 \right) - C^{\prime}\left( 0 \right)\alpha^{ - 1} < \infty$$ there exists a critical stock level xc > 0 such that h = $$\Phi \left( {x,\alpha } \right)$$ = 0 for all xxc. In other words; as long as the instantaneous marginal utility of harvesting is finite at h = 0, it is optimal for a well managed fishery to temporarily close down if fish stocks are sufficiently low.

## The Effect of Increased Productivity

Here we demonstrate that for a given stock size increased productivity reduces the optimal harvest rate if h is positive and the stock is below the steady state level, i.e. ф(x, α) is a decreasing function of α for all x∈(xc, xss). This result is built on 4 propositions that are given below. However a fairly rigorous explanation is given in Fig. 2.

In order to lay the ground for the propositions that follows, note that an increase in α will shift the $$\dot{x} = 0$$-isocline in Fig. 1 upwards for all x except x = 0 and x = K. For any given x in (0, K), then along the $$\dot{x} = 0$$-isocline we have that h = ϕ(µ, α) = G(x) is a positive constant. From (10) it follows that if α increases, then µ must increase if h is to remain constant except at x = 0 and x = K as $$C^{\prime}\left( 0 \right) = 0$$ implies that µ is unaffected by the change in α at these points.

### Proposition 1

Define Q(α) = $$\left\{ {\left( {x,\mu } \right)|0 < x < x_{ss} ,\dot{x} = G\left( x \right) - \phi \left( {\mu ,\alpha } \right) > 0,0 < \mu < D^{\prime}\left( 0 \right) - C^{\prime}\left( 0 \right)\alpha^{ - 1} } \right\}.$$ Let αhigh > αlow. As the $$\dot{x} = 0$$ isocline shifts upwards as α increases it follows that Q(αhigh) is a subset of Q(αlow). Then for any point (x, µ) ∈ Q(αhigh) it holds that $$\left( {{{{\text{d}}\mu } \mathord{\left/ {\vphantom {{{\text{d}}\mu } {{\text{d}}x}}} \right. \kern-\nulldelimiterspace} {{\text{d}}x}}} \right)_{{\alpha^{high} }}$$ < $$\left( {{{{\text{d}}\mu } \mathord{\left/ {\vphantom {{{\text{d}}\mu } {{\text{d}}x}}} \right. \kern-\nulldelimiterspace} {{\text{d}}x}}} \right)_{{\alpha^{low} }}$$ < 0.

### Proof

The slope of any solution through an arbitrary point in Q (αhigh) is given by:

$$\frac{{\dot{\mu }}}{{\dot{x}}} = \frac{{{\text{d}}\mu }}{{{\text{d}}x}} = \frac{{\left( {\rho - G^{\prime}\left( x \right)} \right)\mu }}{{G\left( x \right) - \phi \left( {\mu ,\alpha } \right)}}\,$$
(13)

This is true for all α ∈ (0,∞). From (10) we have that ϕ(µ, α) is an increasing function of α when (x, µ) ∈ Q(αhigh) and the conclusion follows.

Proposition 1 defines a set Q(α). This set is significant in the present context because the stable manifold, µ(x, α), is contained in Q(α) for all x ∈ (0, xss). What this proposition does is to take an arbitrary point in Q(αhigh), calculate the slope of any solution that passes through that point and show that the slope is steeper when α is higher. Proposition 1 can be used to prove the following proposition:

### Proposition 2

(a) µ(x, α) is an increasing function of α for all x ∈ (0, xss].

### Proof

As a productivity increase shifts the $$\dot{x} = 0$$ isocline upwards while the $$\dot{\mu } = 0$$ isocline is unchanged, it follows that µ(xss, αhigh) > µ(xss, αlow). If the proposition is false, then there must exist an x' ∈ (0, xss) such that µ(x', αhigh) = µ(x', αlow) and $$\left( {{{{\text{d}}\mu } \mathord{\left/ {\vphantom {{{\text{d}}\mu } {{\text{d}}x}}} \right. \kern-\nulldelimiterspace} {{\text{d}}x}}} \right)_{\begin{subarray}{l} \alpha = \alpha^{high} \\ x = x^{\prime} \end{subarray} } < \left( {{{{\text{d}}\mu } \mathord{\left/ {\vphantom {{{\text{d}}\mu } {{\text{d}}x}}} \right. \kern-\nulldelimiterspace} {{\text{d}}x}}} \right)_{\begin{subarray}{l} \alpha = \alpha^{low} \\ x = x^{\prime} \end{subarray} }$$. From Proposition 1 it follows that $$\left( {{{{\text{d}}\mu } \mathord{\left/ {\vphantom {{{\text{d}}\mu } {{\text{d}}x}}} \right. \kern-\nulldelimiterspace} {{\text{d}}x}}} \right)_{\begin{subarray}{l} \alpha = \alpha^{high} \\ x = x^{\prime} \end{subarray} } > \left( {{{{\text{d}}\mu } \mathord{\left/ {\vphantom {{{\text{d}}\mu } {{\text{d}}x}}} \right. \kern-\nulldelimiterspace} {{\text{d}}x}}} \right)_{\begin{subarray}{l} \alpha = \alpha^{low} \\ x = x^{\prime} \end{subarray} }$$ which is a contradiction.

Given the economic interpretation of the stable manifold as the derivative of the value function, this also makes intuitive sense; better technology increases the marginal value of the stock given the stock size.

The critical stock level xc at and below which optimal h is zero will also be affected by a productivity increase. We can prove that xc is in fact an increasing function of α.

### Proposition 3

The critical xc where h = 0 for all x < xc is an increasing function of α.

### Proof

Define the relationship between xc and α by xc = xc(α). From Proposition 2 we have that $$D^{\prime}\left( 0 \right) - C^{\prime}\left( 0 \right)\alpha^{-1}$$ = µ(xc(αlow), αlow) < µ(xc(αlow), αhigh). It follows that the value of xc(αhigh) that solves $$D^{\prime}\left( 0 \right)-C^{\prime}\left( 0 \right)\alpha^{-1}$$ = µ(xc(αhigh), αhigh) is larger than xc(αlow).

We can now prove the main result in this section.

### Proposition 4

Let αlow < αhigh. Then the feedback control ф(x, α) has the property that for all x < xss it holds that $$\Phi \left( {x,\alpha^{high} } \right) \le \Phi \left( {x,\alpha^{low} } \right)$$.

### Proof

Denote xc(αi) as $$x_{c}^{i}$$ for i = {high, low}. A consequence of the existence of xc and Proposition 3 is that there exists some $$x_{c}^{low}$$ such that $$\Phi \left( {x_{c}^{low} ,\alpha^{low} } \right)$$ = $$\Phi \left( {x_{c}^{high} ,\alpha^{high} } \right)$$ = 0 for all x ∈ [0, $$x_{c}^{low}$$] and that there is an interval ($$x_{c}^{low}$$, $$x_{c}^{high}$$] where $$\Phi \left( {x,\alpha^{low} } \right)$$ > $$\Phi \left( {x,\alpha^{high} } \right)$$ = 0. It follows from (5) that xss is determined and independent of α. It follows from $$\dot{x} = G\left( {x_{ss} } \right) - h = 0$$ in steady state that hss is also independent of α, thus $$\Phi \left( {x_{ss} ,\alpha^{high} } \right)$$ = $$\Phi \left( {x_{ss} ,\alpha^{low} } \right)$$. We must now account for the open interval ($$x_{c}^{high}$$, xss). If $$\Phi \left( {x,\alpha^{low} } \right)$$ < $$\Phi \left( {x,\alpha^{high} } \right)$$ for some subinterval of ($$x_{c}^{high}$$, xss) there must exist a pair $$\left( {\tilde{h},\tilde{x}} \right)$$ where $$\tilde{x}$$ ∈ ($$x_{c}^{high}$$, xss) such that $$\tilde{h} = \Phi \left( {\tilde{x},\alpha^{low} } \right)$$ = $$\Phi \left( {\tilde{x},\alpha^{high} } \right)$$ and $$\Phi^{\prime}_{x} \left( {\tilde{x},\alpha^{low} } \right)$$ < $$\Phi^{\prime}_{x} \left( {\tilde{x},\alpha^{high} } \right)$$. We can calculate $$\Phi^{\prime}_{x} \left( {x,\alpha } \right)$$ in a point $$\left( {\tilde{h},\tilde{x}} \right)$$ and when we insert from (8) and (11) we get that:

$$\Phi ^{\prime}_{x} \left( {\tilde{x},\alpha } \right) = \left( {\phi ^{\prime}_{\mu } \frac{{{\text{d}}\mu }}{{{\text{d}}x}}} \right)_{{\left( {h,x} \right) = \left( {\tilde{h},\tilde{x}} \right)}} = \underbrace {{\frac{1}{{D^{{\prime \prime }} \left( {\tilde{h}} \right) - C^{{\prime \prime }} (\tilde{h})\alpha ^{{ - 1}} }}}}_{{ < 0}}\underbrace {{\frac{{\left( {\rho - G^{\prime } \left( {\tilde{x}} \right)} \right)\mu \left( {\tilde{x},\alpha } \right)}}{{G\left( {\tilde{x}} \right) - \tilde{h}\left( {\mu \left( {\tilde{x},\alpha } \right),\alpha } \right)}}}}_{{ < 0}} > 0$$

We now compare $$\Phi^{\prime}_{x} \left( {\tilde{x},\alpha^{low} } \right)$$ and $$\Phi^{\prime}_{x} \left( {\tilde{x},\alpha^{high} } \right)$$ to see under what condition the same harvest rate at the same stock level yields $$\Phi^{\prime}_{x} \left( {\tilde{x},\alpha^{low} } \right)$$ < $$\Phi^{\prime}_{x} \left( {\tilde{x},\alpha^{high} } \right)$$. We calculate that

$$\begin{gathered} \Phi^{\prime}_{x} \left( {\tilde{x},\alpha^{low} } \right) < \Phi^{\prime}_{x} \left( {\tilde{x},\alpha^{high} } \right) \hfill \\ \Updownarrow \hfill \\ \frac{1}{{D^{\prime \prime } \left( {\tilde{h}} \right) - {{C^{\prime \prime } \left( {\tilde{h}} \right)} \mathord{\left/ {\vphantom {{C^{\prime \prime } \left( {\tilde{h}} \right)} {\alpha^{low} }}} \right. \kern-\nulldelimiterspace} {\alpha^{low} }}}}\frac{{\left( {\rho - G^{\prime}\left( {\tilde{x}} \right)} \right)\mu \left( {\tilde{x},\alpha_{low} } \right)}}{{G\left( {\tilde{x}} \right) - \tilde{h}}} < \frac{1}{{D^{\prime \prime } \left( {\tilde{h}} \right) - {{C^{\prime \prime } \left( {\tilde{h}} \right)} \mathord{\left/ {\vphantom {{C^{\prime \prime } \left( {\tilde{h}} \right)} {\alpha^{high} }}} \right. \kern-\nulldelimiterspace} {\alpha^{high} }}}}\frac{{\left( {\rho - G^{\prime}\left( {\tilde{x}} \right)} \right)\mu \left( {\tilde{x},\alpha^{high} } \right)}}{{G\left( {\tilde{x}} \right) - \tilde{h}}} \hfill \\ \Updownarrow \hfill \\ \frac{{\mu \left( {\tilde{x},\alpha_{low} } \right)}}{{D^{\prime \prime } \left( {\tilde{h}} \right) - {{C^{\prime \prime } \left( {\tilde{h}} \right)} \mathord{\left/ {\vphantom {{C^{\prime \prime } \left( {\tilde{h}} \right)} {\alpha^{low} }}} \right. \kern-\nulldelimiterspace} {\alpha^{low} }}}} > \frac{{\mu \left( {\tilde{x},\alpha^{high} } \right)}}{{D^{\prime \prime } \left( {\tilde{h}} \right) - {{C^{\prime \prime } \left( {\tilde{h}} \right)} \mathord{\left/ {\vphantom {{C^{\prime \prime } \left( {\tilde{h}} \right)} {\alpha^{high} }}} \right. \kern-\nulldelimiterspace} {\alpha^{high} }}}} \hfill \\ \Updownarrow \hfill \\ \frac{{\mu \left( {\tilde{x},\alpha_{low} } \right)}}{{D^{\prime \prime } \left( {\tilde{h}} \right) - {{C^{\prime \prime } \left( {\tilde{h}} \right)} \mathord{\left/ {\vphantom {{C^{\prime \prime } \left( {\tilde{h}} \right)} {\alpha^{low} }}} \right. \kern-\nulldelimiterspace} {\alpha^{low} }}}} < \frac{{\mu \left( {\tilde{x},\alpha^{high} } \right)}}{{D^{\prime \prime } \left( {\tilde{h}} \right) - {{C^{\prime \prime } \left( {\tilde{h}} \right)} \mathord{\left/ {\vphantom {{C^{\prime \prime } \left( {\tilde{h}} \right)} {\alpha^{high} }}} \right. \kern-\nulldelimiterspace} {\alpha^{high} }}}} \hfill \\ \end{gathered}$$

But the last inequality holds for all $$\left( {\tilde{h},\tilde{x}} \right)$$ including (hss, xss) as µ(x, α) is increasing in α and the denominator is decreasing in α.Footnote 3 This implies that there is at most one point $$\left( {\tilde{h},\tilde{x}} \right)$$ where $$\Phi \left( {x,\alpha^{low} } \right)$$ = $$\Phi \left( {x,\alpha^{high} } \right)$$ and we already know that $$\Phi \left( {x_{ss} ,\alpha^{low} } \right)$$ = $$\Phi \left( {x_{ss} ,\alpha^{high} } \right)$$. It follows that there are no values of x less than xss such that the corresponding harvest rate is larger with larger productivity.

Thus we have that in a schooling fishery a unanticipated technological shock will lead to lower catch for each given stock size, in turn ensuring a faster convergence to the steady state. To get some intuition about this result note that the canonical fisheries model is very similar to the well known Ramsey–Cass–Koopmans (RCK) model of economic growth. The only difference is that marginal productivity at zero use of input is assumed infinity in the RCK model, whereas the corresponding expression in the fisheries model, G'(0), is assumed finite. In the analysis of the RCK model the instantaneous elasticity of substitution, also known by its reciprocal, the intertemporal elasticity of substitution is an important explanatory factor. Roughly speaking, the higher the instantaneous elasticity of substitution, the more willing is the consumer to reduce consumption in the present in order to increase consumption in the future, see Blanchard and Fischer (1989, pp 39–43) for a discussion of the instantaneous elasticity of substitution and how it affects intertemporal tradeoffs in the RCK-model. If we can show that a productivity increase has the same effect as an increase in the instantaneous elasticity of substitution we have a way of understanding and generalising the above results to a more general cost function c(h, α). It is assumed that c(0, α) = 0 and that c(h, α) is increasing and concave with respect to h. It is also assumed that c(h, α) and $$c^{\prime}_{\alpha } \left( {h,\alpha } \right)$$ are decreasing with respect to α except when h = 0.

In the present model we can calculate the equivalent to the instantaneous elasticity of substitution for the integrand in (2). This expression is given by:

$$\sigma \left( {h,\alpha } \right) = - \frac{{D^{\prime}\left( h \right) - c^{\prime}_{h} \left( {h,\alpha } \right)}}{{h\left( {D^{\prime\prime}\left( h \right) - c^{\prime\prime}_{hh} \left( {h,\alpha } \right)} \right)}} > 0{\text{ for all }}h > 0$$
(14)

By using (4), (5) and the differential equation for x one can derive the following differential equation for h as a function of x which holds when (4) has an interior solution.

$$\frac{{\dot{h}}}{h} = \sigma \left( {h,\alpha } \right)\left( {\rho - G^{\prime}\left( x \right)} \right)$$
(15)

In macroeconomics this equation is known as the Keynes–Ramsey rule. From (15) it is clear that for any given x < xss, the higher the numerical value of σ(h, α) the faster is the rate of convergence towards the steady state value of hss, and therefore also towards xss. Differentiating σ(h, α) with respect to α yields:

$$\frac{\partial \sigma }{{\partial \alpha }} = \frac{{\overbrace {{c_{h\alpha }^{\prime \prime } (h,\alpha )}}^{ < 0}\overbrace {{\left( {D^{\prime \prime } (h) - c_{hh}^{\prime \prime } (h,\alpha )} \right)}}^{ < 0} + \overbrace {{\left( {c_{h}^{\prime } (h,\alpha ) - D^{\prime}(h)} \right)}}^{ < 0}\overbrace {{c_{hh\alpha }^{\prime \prime \prime } (h,\alpha )}}^{{{\text{sign }} = ?}}}}{{\underbrace {{h\left( {D^{\prime \prime } (h) - c_{hh}^{\prime \prime } (h,\alpha )} \right)^{2} }}_{ > 0}}}$$
(16)

With exception of $$c^{\prime\prime\prime}_{hh\alpha }$$, the sign of all terms in this expression follows from assumptions, first order conditions and second order conditions. It follows that $$c^{\prime\prime\prime}_{hh\alpha }$$ < 0 is a sufficient, but not a necessary condition for σ to be an increasing function of α and therefore for Propositions 3 and 4 to hold also with this more general cost function. Thus improved productivity has the same qualitative effect along the segment of the optimal path where h > 0 as an increase in the instantaneous elasticity of substitution has in the RCK model. Intuitively, this condition implies that the second derivative of costs with respect to h decreases in response to improved technology. As the steady state harvest and stocks level are fixed, this implies that net instantaneous utility increases more for high harvest levels close to the steady state than harvest levels further below steady state, which makes it optimal to sacrifice some instantaneous net utility in the near future in order to speed up the rate of convergence towards steady state. Remarkably as long as $$c^{\prime\prime\prime}_{hh\alpha }$$ < 0, this effect does not depend on the discount rate. It should be pointed out that although the canonical model of a schooling fishery and the RCK model are mathematically almost equivalent, they are of course models of very different things. Importantly, the RCK model has utility as the unit and ρ is the rate of time preference. In the fisheries model as long as we do not examine a subsistence fishery, but a commercial fishery the unit is money and the discount rate is the interest rate.

## Stock Dependent Costs

We now look at the case where the harvest costs depend on stock size, reflecting that for many fish-species it is easier to catch a unit of fish if the stock is abundant and if the stock of the fish goes to zero, then the marginal cost of harvesting a unit of fish goes to infinity. The canonical model is now:

$$\begin{gathered} V\left( {x\left( 0 \right),\alpha } \right) = \mathop {\max }\limits_{h \ge 0} \int\limits_{0}^{\infty } {\left( {D\left( h \right) - C\left( {h,x,\alpha } \right)} \right)e^{ - \rho t} dt} \hfill \\ {\text{subject to}}\,\dot{x} = G\left( x \right) - h{\text{ and}}\,x\left( 0 \right)\,\,\,{\text{given}}{.} \hfill \\ \end{gathered}$$
(17)

It is assumed that $$C^{\prime}_{h} \left( {h,x,\alpha } \right) > 0$$, $$C^{\prime}_{x} \left( {h,x,\alpha } \right) < 0$$, $$C^{\prime\prime}_{h\alpha } \left( {h,x,\alpha } \right) < 0$$ for all h and x > 0, $$C^{\prime\prime}_{h\alpha } \left( {0,x,\alpha } \right) = 0$$ and that $$C^{\prime\prime}_{hh} \left( {h,x,\alpha } \right) > 0$$. This is consistent with, but not restricted to, cost functions of the formFootnote 4

$$C\left( {h,x,\alpha } \right) = k\frac{{h^{a} }}{{\alpha x^{b} }}{\text{ where }}a > 1{\text{ and }}b > 0$$
(18)

Thus the technology is such that an increase in α lowers the marginal cost of harvesting. It follows from C(0, x, α) = 0 and $$C^{\prime\prime}_{h\alpha } \left( {h,x,\alpha } \right) < 0$$ that an increase in α also reduces costs for all positive h and x.

This model or models very similar to this one has been analysed in a large number of papers and is close to the analysis of the pure schooling fishery above. A typical phase diagram is shown in Fig. 3.

We keep the notation from the previous analysis so ϕ(µ, α) is the harvest rate that maximises the Hamiltonian, µ(x, α) is the stable manifold/derivative of the value function, ф(x, α) is optimal harvest rate as a feedback control and $$x_{c}^{j}$$ is the critical stock level below which one should not harvest when the productivity is of type j.

With stock dependent costs there is also a critical value xc determined by the lowest stock of x such that the first order condition for the maximised Hamiltonian is that its derivative with respect to h is zeroFootnote 5:

$$\frac{\partial H}{{\partial h}} = D^{\prime}\left( 0 \right) - C^{\prime}_{h} \left( {0,x_{c} ,\alpha } \right) - \mu = 0$$
(19)

We can use this to find conditions that ensure that xc is a increasing function of α.

### Proposition 5

$$C^{\prime\prime}_{hx} \left( {0,x,\alpha } \right) \le 0$$ is a sufficient, but not necessary condition for x c being an increasing function of α.

### Proof

Inserting µ(x, α) into (19) and implicitly differentiating gives the result:

$$\frac{{\partial x_{c} }}{\partial \alpha } = \frac{{\overbrace {{\frac{{\partial \mu \left( {x_{c} ,\alpha } \right)}}{\partial \alpha }}}^{ > 0}}}{{ - \left( {\underbrace {{C^{\prime\prime}_{hx} \left( {0,x_{c} ,\alpha } \right)}}_{{\text{sign = ?}}} + \underbrace {{\frac{{\partial \mu \left( {x_{c} ,\alpha } \right)}}{\partial x}}}_{ < 0}} \right)}}$$
(20)

If $$C^{\prime\prime}_{hx} \left( {0,x,\alpha } \right) \le 0$$ then the expression in (20) is positive.

Assuming that $$C^{\prime\prime}_{hx} \left( {0,x,\alpha } \right) \le 0$$ is in line with standard parameterisations of cost functions used in fisheries. It is also reasonable to expect that the higher the stock, the lower is the marginal cost with respect to h. However, one can not rule out that there are exceptions to this rule. One can e.g. envision changes in technology that improves productivity for some values of h and x and reduces it for others. If that is the case a more detailed numerical investigation is warranted to establish the sign of $${{\partial x_{c} } \mathord{\left/ {\vphantom {{\partial x_{c} } {\partial \alpha }}} \right. \kern-\nulldelimiterspace} {\partial \alpha }}$$.

### The Effect of Improved Productivity on Optimal Harvest Rate when x > xc

Clark and Munro (1975) demonstrate in a model that is linear in the harvest rate that an increase in productivity will lead to lowered steady state stocks. Caputo (1989) demonstrates in a fairly general non-linear model that the long-run effect of a productivity increase is also a lower stock level. Caputo (1989) also show and that a productivity increase may be commensurate with both a higher and a lower long term harvest rate hss. The initial response will however be an increase in the harvest rate. The results in Caputo (1989) are based on the response to a productivity shock if the initial stock is in steady state. However, even if the steady state response to a positive productivity shock implies that the steady state stock decreases when α increases it is still the case that ф(x, α) will be a decreasing function of α if x sufficiently small as proven in Proposition 6.

### Proposition 6

If Proposition 5holds, an increase productivity from αlow to αhigh implies that there is an non-empty interval $$\left[ {x_{c}^{low} ,\hat{x}} \right)$$ such that harvest levels for a given stock size is reduced, i.e. ф(x, αhigh) < ф(x, αlow) for all x in $$\left[ {x_{c}^{low} ,\hat{x}} \right)$$.

### Proof

It follows from the existence of xc > 0 for all α > 0 and Proposition 6 that there exists a non-empty intervals [0, $$x_{c}^{low}$$] such that ф(x, αlow) = ф(x, αhigh) = 0 and another non-empty interval [$$x_{c}^{low}$$, $$x_{c}^{high}$$] where ф(x, αlow) > 0 and ф(x, αhigh) = 0. Note that h = ф(x, α) is a continuous function of x and that ф($$x_{c}^{high}$$, αlow) > ф($$x_{c}^{high}$$, αhigh) = 0. Due to continuity, h cannot jump infinitely fast at $$x_{c}^{high}$$ and there is therefore an interval $$\left( {x_{c}^{high} ,\hat{x}} \right]$$ where ф(x, αlow) > ф(x, αhigh) > 0.

This effect remains regardless of the size of productivity increase. An important thing to note from Fig. 3 is that there is a lower bound to how low the steady state stock will be reduced by improved technology. Proposition 6 shows that below this lower bound, there will be an interval [$$x_{c}^{low}$$, $$\hat{x}$$) where the higher the productivity, the lower the harvest rate. The effect of increased productivity is summarized in Fig. 4.

Figure 4 shows that the results in the Clark and Munro (1975), Caputo (1989) and Squires and Vestergaard (2013) where increased productivity leads to lower steady state stocks are compatible with the results in the present paper. If increased productivity increases rents in steady state and the stock is low enough one should still decrease harvesting in order to increase stocks faster even if the desired steady state level has decreased. This is illustrated in Fig. 5.

A final technical remark is that the assumption that $$C^{\prime\prime}_{h\alpha } \left( {0,x,\alpha } \right) < 0$$ is crucial for the result in Proposition 5 that $${{\partial x_{c} } \mathord{\left/ {\vphantom {{\partial x_{c} } {\partial \alpha }}} \right. \kern-\nulldelimiterspace} {\partial \alpha }}$$ is positive for all α. It is however a sufficiency condition and not a necessary condition. If $$C^{\prime\prime}_{h\alpha } \left( {0,x,\alpha } \right) \ge 0$$ there may be some intervals of α values where $${{\partial x_{c} } \mathord{\left/ {\vphantom {{\partial x_{c} } {\partial \alpha }}} \right. \kern-\nulldelimiterspace} {\partial \alpha }}$$ < 0. However, as may be seen from Fig. 4, if $$\lim_{\alpha \to \infty } C\left( {h,x,\alpha } \right) = 0$$ then xc will increase, perhaps not monotonically, and eventually converge to the stock level that solves the equation $$G^{\prime}\left( x \right) = \rho$$. It follows that h = ф(x, α) will also be reduced as α increases, but perhaps not monotonically.
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