3.1. Scenario 1

Scenario 1 in table 2 covers optimal management with a common fish stock without taking the land-based profit into account. Initially, we derive a resource restriction in a steady-state equilibrium (see, e.g., Clark (Reference Clark1991) for an introduction to resource restrictions). We let x be the common Grl. halibut stock size for the high sea and coastal area and the natural growth function is denoted $F(x)$. In line with the convention, we assume an inverse U-shaped growth function in the sense that $F^{\prime} (x) > 0$ for $x < {x^{MSY}},\ F^{\prime} (x) < 0$ for $x > {x^{MSY}}$ and ${{F^{\prime\prime}}} (x) < 0$ for all x, where ${x^{MSY}}$ is the Grl. halibut stock size at maximum sustainable yield (MSY). The coastal industry Grl. halibut harvest is denoted ${h_C}$ (subscript C covers the coastal area), while ${h_H}$ is the high sea industry Grl. halibut harvest by vessels from Greenland (subscript H covers the high sea area). Furthermore, β denotes a constant scaling factor for the harvest by vessels from other fishing nations in the high sea area. Formally, we have that ${h_F} = \beta {h_H}$, where ${h_F}$ is the high sea industry Grl. halibut harvest by foreign vessels (subscript F covers foreign vessels). Now, the resource restriction in a steady-state equilibrium becomes:

(1)\begin{equation}F(x) - {h_H} - {h_C} - \beta {h_H} = 0.\end{equation}

From (1), the natural growth is equal to the aggregated Grl. halibut harvest in a steady-state equilibrium. The aggregated Grl. halibut harvest consists of the coastal harvest and the high sea harvest by vessels from Greenland and other fishing nations. Note that in the scenarios with a common Grl. halibut fish stock, interactions between high sea and coastal vessels arise due to the resource restriction (two fleet segments exploit a common fish stock).

Next, we derive an objective function and the coastal industry cost function for vessels targeting Grl. halibut is denoted ${C_C}({h_C},x)$. We assume that an increase in the Grl. halibut stock size generates a reduction in the costs since it is easier to harvest fish, implying that $(\partial {C_C}/\partial x) < 0$.Footnote ¹¹ Furthermore, we assume that $(\partial {C_C}/\partial {h_C}) > 0$ and $({\partial ^2}{C_C}/\partial {h_C}^2) > 0$, implying that the marginal coastal harvesting costs are positive and increasing (e.g., Clark, Reference Clark1991). The high sea industry cost function for vessels from Greenland is denoted ${C_H}({h_H},x)$, and for the derivatives of the high sea cost function, we adopt the same assumptions as for the coastal cost function $((\partial {C_H}/\partial x) < 0,\ (\partial {C_H}/\partial {h_H}) > 0\ \textrm{and}\ ({\partial ^2}{C_H}/\partial {h_H}^2) > 0)$. In addition, ${p_C}$ and ${p_H}$ denote constant coastal and high sea Grl. halibut prices, respectively.Footnote ¹² Now ${\pi _H}({h_H},x) ={p_H}{h_H} -{C_H}({h_H},x)$ is the total high sea industry profit for vessels from Greenland defined as revenue minus costs, while ${\pi _C}({h_C},x) ={p_C}{h_C} -{C_C}({h_C},x)$ is the total coastal industry profit from harvesting Grl. halibut. Note that in the high sea profit function, we have excluded the gain of providing foreign vessels with the opportunity to harvest Grl. halibut in the high sea area.Footnote ¹³ We can now define the long-run economic yield, $\pi ({h_H},{h_C},x)$, as the sum of the total high sea and costal profit:

(2)\begin{equation}\pi ({h_H},{h_C},x) = {\pi _H}({h_H},x) + {\pi _C}({h_C},x) = {p_H}{h_H} - {C_H}({h_H},x) + {p_C}{h_C} - {C_C}({h_C},x).\end{equation}

Note that in (2), we have disregarded discounting because we are interested in maximizing the long-run economic yield.

Now, the problem is to maximize (2) subject to (1), and for solving this problem, we set up the following Lagrange function:

(3)\begin{align} L & ={p_H}{h_H} - {C_H}({h_H},x) + {p_C}{h_C} - {C_C}({h_C},x)\nonumber\\ & \quad + \lambda (F(x) - {h_H} - {h_C} - \beta {h_H}), \end{align}

where $\lambda > 0$ is the shadow price, or the marginal user cost of the Grl. halibut fish stock (Anderson, Reference Anderson1977). By using ${h_H}$, ${h_C}$ and x as control variables, we obtain the following first-order conditions:

(4)\begin{gather}\frac{{\partial L}}{{\partial {h_H}}} = {p_H} - \frac{{\partial {C_H}}}{{\partial {h_H}}} - \lambda (1 + \beta ) = 0\end{gather}

(5)\begin{gather}\frac{{\partial L}}{{\partial {h_C}}} = {p_C} - \frac{{\partial {C_C}}}{{\partial {h_C}}} - \lambda = 0\end{gather}

(6)\begin{gather}\frac{{\partial L}}{{\partial x}} ={-} \frac{{\partial {C_H}}}{{\partial x}} - \frac{{\partial {C_C}}}{{\partial x}} + \lambda F^{\prime} (x) = 0\end{gather}

(7)\begin{gather}\frac{{\partial L}}{{\partial \lambda }} = F(x) - {h_H} - {h_C} - \beta {h_H} = 0.\end{gather}

According to (4), the optimal high sea harvest occurs where the marginal high sea profit is equal to the marginal user cost of the Grl. halibut fish stock corrected with the scaling factor for the harvest by vessels from other fishing nations. Equation (5) indicates that the optimal coastal harvest arises where the marginal coastal profit is equal to the marginal user cost of the fish stock. According to (6), the optimal stock size occurs where the sum of the high sea and coastal cost savings associated with an increase in the stock size is equal to the value of the marginal growth. Finally, (7) is identical to the resource restriction from (1). Equations (4)–(7) represent four equations with four unknowns (${h_H}$, ${h_C}$,$x$ and $\lambda$), and solving this equation system provides optimal values of the unknowns, which we denote ${h_H}^\ast$, ${h_C}^\ast$, $x^\ast$ and $\lambda^\ast$. By using these solutions, we can define the optimal high sea and coastal industry profit as ${\pi _H}({h_H}^\ast ,x^\ast ) ={p_H}{h_H}^\ast{-}{C_H}({h_H}^\ast ,x^\ast )$ and ${\pi _C}({h_C}^\ast ,x^\ast ) ={p_C}{h_C}^\ast{-}{C_C}({h_C}^\ast ,x^\ast )$, while the optimal long-run economic yield becomes $\pi ({h_H}^\ast ,{h_C}^\ast ,x^\ast ) = {\pi _H}({h_H}^\ast ,x^\ast ) + {\pi _C}({h_C}^\ast ,x^\ast )$. In this paper, we will identify the abovementioned indicators empirically for the Grl. halibut fishery on the west coast of Greenland.

3.2. Scenario 2

We now consider scenario 2, where we investigate optimal management with separate high sea and coastal fish stocks (table 2). Now, we must operate with two separate resource restrictions and we allow for migration from the high sea to the coastal area. To derive these restrictions, we let ${x_C}$ and ${x_H}$ be the coastal and high sea Grl. halibut stock size, respectively. The coastal and high sea natural growth functions of Grl. halibut are denoted ${G_C}({x_C})$ and ${G_H}({x_H})$, and as in section 3.1, we assume inverse U-shaped growth functions. Finally, the net migration of Grl. halibut is captured by $M({x_H},{x_C})$, and three facts are important in relation to this function: (i) we assume that $M({x_H},{x_C}) > 0$, implying a net migration from the high sea to the coastal area; (ii) the net migration from the high sea area is assumed to be identical to the net migration into the coastal area; and (iii) following Chapman et al. (Reference Chapman, Hulthen, Brodersen, Nilsson, Skov, Hansson and Bronmark2012), we assume that $(\partial M/\partial {x_C}) < 0$ and $(\partial M/\partial {x_H}) > 0,$ implying that the net migration decreases (increases) with an increase (decrease) in the coastal stock size and a decrease (increase) in the high sea stock size. Given this notation, the high sea and coastal resource restrictions become:

(8)\begin{gather}{G_H}({x_H}) - {h_H} - \beta {h_H} - M({x_H},{x_C}) = 0\end{gather}

(9)\begin{gather}{G_C}({x_C}) - {h_C} + M({x_H},{x_C}) = 0.\end{gather}

Equation (8) indicates that in a steady-state equilibrium, the high sea natural growth must be equal to the high sea Grl. halibut harvest by vessels from Greenland and other fishing nations plus the net migration of Grl. halibut to the coastal area. Similarly, (9) indicates that the coastal natural growth plus the net migration of Grl. halibut from the high sea area must be equal to the coastal Grl. halibut harvest in a steady-state equilibrium. Since we have two resource restrictions, we must introduce two shadow prices, and the marginal user costs of the coastal and high sea fish are denoted ${\varepsilon _C}$ and ${\varepsilon _H}$, respectively. Note that in the scenarios with two separate fish stocks, interactions between high sea and coastal vessels occur due to the net migration.

Regarding the objective function, the only difference compared to section 3.1 is that the separate stock sizes for the high sea and coastal areas must be considered in the cost functions. Thus, ${C_C}({h_C},{x_C})$ and ${C_H}({h_H},{x_H})$ are coastal and high sea industry cost functions for harvesting Grl. halibut and, for the derivatives of the cost functions, we adopt the same assumptions as in section 3.1. Now, the problem is to maximize the long-run economic yield from (2) (with the adjusted cost functions) subject to (8) and (9). To solve this problem, a Lagrange function is formulated (with ${\varepsilon _H} > 0$ and ${\varepsilon _C} > 0$ as Lagrange multipliers) and, since we have two separate fish stocks, the first-order conditions for ${x_H}$ and ${x_C}$ are:

(10)\begin{gather}\frac{{\partial L}}{{\partial {x_H}}} ={-} \frac{{\partial {C_H}}}{{\partial {x_H}}} + {\varepsilon _H}G_H^\mathrm{\prime }({x_H}) - {\varepsilon _H}\frac{{\partial M}}{{\partial {x_H}}} + {\varepsilon _C}\frac{{\partial M}}{{\partial {x_H}}} = 0\end{gather}

(11)\begin{gather}\frac{{\partial L}}{{\partial {x_C}}} ={-} \frac{{\partial {C_C}}}{{\partial {x_C}}} + {\varepsilon _C}G_C^\mathrm{\prime }({x_C}) - {\varepsilon _H}\frac{{\partial M}}{{\partial {x_C}}} + {\varepsilon _C}\frac{{\partial M}}{{\partial {x_C}}} = 0.\end{gather}

According to (10), the optimal high sea Grl. halibut stock size occurs where the marginal high sea cost savings from an increase in the stock size are equal to the value of the high sea marginal growth corrected for the net cost of the marginal migration to the coastal area. The optimality condition for the coastal fish stock in (11) can be interpreted in a similar way. Regarding the first-order conditions for ${h_H}$ and ${h_C}$, these are identical to (4) and (5) with two minor adjustments: (i) there are two distinct shadow prices, ${\varepsilon _H} > 0$ and ${\varepsilon _C} > 0$; and (ii) ${x_H}$ and ${x_C}$ are included in the cost functions. Now the adjusted versions of (4)–(5) and (8)–(11) represent six equations with six unknowns (${h_H}$, ${h_C}$, ${x_H}$, ${x_C}$, ${\varepsilon _H}$ and ${\varepsilon _C}$) and, as in section 3.1, the equation system can be solved for optimal values of the unknowns. As in section 3.1, we can also use the optimal values of the unknowns to find the optimal high sea and coastal industry profit and the optimal long-run economic yield.

3.3. Scenario 3

We now turn to scenario 3, where the land-based profit is considered in the case with a common fish stock (table 2). A legal landing obligation requires high sea vessels to deliver at least 25 per cent of their Grl. halibut harvest for land-based processing. However, we assume that high sea vessels have no incentive to deliver more than 25 per cent of their Grl. halibut harvest for land-based processing. This implies that $0.25\,{h_H}$ enters into the revenue and the cost function for the land-based processing industry. We also assume that coastal vessels deliver all Grl. halibut harvest for land-based processing, implying that ${h_C}$ enters into the relevant land-based functions. We let ${C_L}({h_C},0.25\,{h_H})$ be a land-based industry cost function (subscript L covers the land-based industry), and we assume that $(\partial {C_L}/\partial {h_C}) > 0,\ ({\partial ^2}{C_L}/\partial {h_C}^2) > 0,\ (\partial {C_L}/\partial {h_H}) > 0$ and $({\partial ^2}{C_L}/\partial {h_H}^2) > 0$. Furthermore, ${p_L}$ is a constant price for Grl. halibut delivered from land-based factories while α capture a constant land-based utilization rate. Specifically, α capture the share of the harvest that is utilized by the land-based industry.Footnote ¹⁴ Now we get that ${\pi _L}({h_C},{h_H}) = \alpha {p_L}({h_C} + 0.25{h_H}) - {C_L}({h_C},0.25{h_H})$ is the land-based industry profit, defined as the revenue minus the costs.Footnote ¹⁵ Compared to the scenario in section 3.1, additional interactions between high sea and coastal vessels now arise due to the land-based profit function. Now, the problem is to maximize the objective function from (2) with the land-based profit included subject to (1), and this problem can be solved by setting up a Lagrange function. We now let $\theta > 0$ denote the marginal user cost of the fish stock, and the first-order conditions for ${h_C}$ and ${h_H}$ become:

(12)\begin{gather}\frac{{\partial L}}{{\partial {h_H}}} = {p_H} - \frac{{\partial {C_H}}}{{\partial {h_H}}} + 0.25{p_L}\alpha - 0.25\frac{{\partial {C_L}}}{{\partial {h_H}}} - \theta (1 + \beta ) = 0\end{gather}

(13)\begin{gather}\frac{{\partial L}}{{\partial {h_C}}} = {p_C} - \frac{{\partial {C_C}}}{{\partial {h_C}}} + {p_L}\alpha - \frac{{\partial {C_L}}}{{\partial {h_C}}} - \theta = 0.\end{gather}

Compared to (4) and (5), the only difference is that the marginal land-based profit is considered when identifying the optimal high sea and coastal harvest in (12) and (13) while the first-order condition for the stock size is identical to (6) except that θ is included. Now the adjusted version of (6), (7) and (12)–(13) represent four equations with four unknowns (${h_H}$, ${h_C}$, x and $\theta$), and again, we can solve the equation system for the optimal values of the unknowns. The optimal values can also be used to find the optimal high sea and coastal industry profit, the optimal land-based profit and the optimal long-run economic yield.

3.4. Scenario 4

We now consider scenario 4, where we include the land-based profit function in the case described in section 3.2 (table 2). Now, interaction between high sea and coastal vessels also arises due to the land-based profit, and ${\gamma _H}$ and ${\gamma _C}$ denote the marginal user cost of the high sea and coastal fish stocks, respectively. We must incorporate the land-based profit into the objective function in the same way as in section 3.3, and the problem is to maximize the long-run economic yield in (2), including the land-based profit subject to the resource restrictions in (8) and (9). This problem can be solved by using a Lagrange function, and the first-order conditions for ${h_H}$ and ${h_C}$ are identical to (12) and (13), apart from the fact that the shadow prices are ${\gamma _C}$ and ${\gamma _H}$. Furthermore, the resource restrictions are given by (8) and (9), while the first-order conditions for ${x_H}$ and ${x_C}$ are identical to (10) and (11) with the adjustment that ${\gamma _C}$ and ${\gamma _H}$ is included. Now the adjusted versions of (8)–(13) represent six equations with six unknowns (${h_H}$, ${h_C}$, ${x_H}$, ${x_C}$, ${\gamma _H}$ and ${\gamma _C}$). The solution to this equation system represents optimal values of the unknowns, and these values can be used to find the optimal high sea and coastal industry profit, the optimal land-based profit and the optimal long-run economic yield.

4.1. The resource restrictions

First, we discuss how a growth function has been estimated for the scenarios with a common stock size of Grl. halibut. From section 3.1, the natural growth function must be inverse U-shaped, and to fulfil this requirement, we use a logistic specification (see, e.g., Clark (Reference Clark1991) for an introduction):

(14)\begin{equation}F(x) = rx\left( {1 - \frac{x}{K}} \right),\end{equation}

where $r$is the intrinsic growth rate and K is the carrying capacity. To estimate (14), we use time series for the stock size and harvest by all fishing nations of Grl. halibut on the west coast of Greenland for the period between 1997 and 2017 (MA Treble, personal communication, 2019, based on Treble and Nogueira (Reference Treble and Nogueira2018)).Footnote ¹⁶ We inserted the logistic growth function in (14) into the resource restriction from (1), and then the restriction was estimated with ordinary least squares (OLS) using harvest as the dependent variable (e.g., Elofsson and Svensson, Reference Elofsson and Svensson2019).Footnote ¹⁷ Thus, we obtain estimated parameter values for the intrinsic growth rate and carrying capacity in the scenarios with a common fish stock.

With two separate fish stocks, we chose to allocate the common Grl. halibut stock size to the high sea and coastal area. To do this, we use the relative distribution of one-year-old Grl. halibut in the high sea and coastal area in Disko Bay from 1997 and 2011 (OA Jørgensen, personal communication, 2019, based on Jørgensen (Reference Jørgensen2013)).Footnote ¹⁸ The total Grl. halibut harvest is allocated to the high sea and coastal areas by using the high sea quota share for 2015, which was calculated using the quotas reported in table 1. For the coastal natural growth, we again assume a logistic growth function, and ${r_C}$ is the coastal intrinsic growth rate, while ${K_C}$ is the coastal carrying capacity. Concerning the net migration function, we follow Chapman et al. (Reference Chapman, Hulthen, Brodersen, Nilsson, Skov, Hansson and Bronmark2012) and assume that the net migration depends on the relative density of the high sea and coastal Grl. halibut fish stock such that:

(15)\begin{equation}M({x_H},{x_C}) = m\frac{{{x_H}}}{{{x_C}}},\end{equation}

where $m$ is a net migration parameter. To estimate the coastal growth and migration functions, we use the same procedure as in the case with a common fish stock. Thus, we insert the coastal logistic growth and migration functions into (9) and then use OLS to estimate the coastal resource restriction with the coastal Grl. halibut harvest as the dependent variable. Hence, we obtain estimated parameter values for ${r_C}$, ${K_C}$ and m.

For the high sea natural growth function, we also assume a logistic growth function, with ${r_H}$ denoting the high sea intrinsic growth rate, while ${K_H}$ is the high sea carrying capacity. To estimate this growth function, we use the steady-state resource restriction for the high sea fish stock in (8), but we must consider that migration will be identical in the high sea and coastal areas. Thus, we use (15) to construct a time series for net migration using the observations for the high sea and coastal stock size and the estimated value of m (see Chapman et al. (Reference Chapman, Hulthen, Brodersen, Nilsson, Skov, Hansson and Bronmark2012) for a justification for this procedure). Now, the sum of the observations for the high sea harvest and net migration become the dependent variable when using (8) to estimate ${r_H}$ and ${K_H}$ with OLS.

In the resource restrictions in (1) and (8), β captures a constant scaling factor for high sea Grl. halibut harvest by vessels from other fishing nations. The amount of Grl. halibut harvest allocated to other fishing nations is determined by international fishing agreements. Thus, we can identify β as the high sea Grl. halibut quota share allocated to other fishing nations, and this can be calculated by using the information in table 1.Footnote ¹⁹ From table 3, four facts are important in relation to the estimated resource restrictions: (i) the carrying capacity with a common fish stock is higher than the coastal and high sea carrying capacity; (ii) the high sea carrying capacity is larger than the coastal carrying capacity because the stock size tends to be higher in the high sea area; (iii) the migration parameter is very low, implying that the interaction between the high sea and coastal area is lowFootnote ²⁰; and (iv) the share of the high sea harvest allocated to other nations represents an additional cost for high sea fisheries since Greenland does not obtain a payoff from international fishing agreements within our model.

4.2. High sea and coastal industry profit function

We assume the following high sea industry cost function, which fulfils the assumption about the derivatives from section 3:Footnote ²¹

(16)\begin{equation}{C_H}({h_H},x) = {c_H}\frac{h_{H}^2}{x},\end{equation}

where ${c_H}$ is a high sea cost parameter and ${C_H}({h_H},x)$ is the total high sea industry cost of harvesting Grl. halibut. To estimate the cost parameter in (16) in the case of a common fish stock, we use data on the total high sea accounting cost of harvesting for all fish species on the west coast of Greenland for the period between 2013 and 2015 from table 1.Footnote ²² From this, we obtain the total high sea cost of harvesting Grl. halibut for the period between 2013 and 2015 by using Grl. halibut quota shares defined as the high sea Grl. halibut quota divided by the high sea quotas on all species. To reduce the effect of random variations affecting the fisheries-related conditions in a given year, we take a simple average of the total cost observations over the period. From Ogmundson and Haraldson (Reference Ogmundson and Haraldson2017), we also have information about the high sea Grl. halibut harvest for 2013–2015, which is again averaged over the period. Furthermore, as described in section 4.1, we have information about the stock size for the period between 2013 and 2015, and we take an average of these observations. Now we have all the necessary information to calculate the high sea cost parameter by using (16).

For the high sea cost parameter with two separate fish stocks, the only difference compared to a common fish stock is that we use the average high sea fish stock for the period between 2013 and 2015 (instead of the average common fish stock) to calculate the high sea cost parameter using (16). For the coastal industry cost function, we assume the same cost function as in (16), and ${c_C}$ is a coastal cost parameter, while ${C_C}({h_C},x)$ is the total coastal industry cost of harvesting Grl. halibut. With both a common fish stock and two separate fish stocks, the only difference, compared with the high sea cost function, is that the data for the total coastal cost of harvesting all species and the coastal harvest of Grl. halibut are reported for vessels above 6 m and boats below 6 m (table 1). Thus, we must aggregate the costs and harvest for the two fleet segments to obtain the total coastal cost of harvesting all species and the total coastal Grl. halibut harvest.Footnote ²³

Concerning the output price, we assume a constant high sea and coastal price on the harvest of Grl. halibut (section 3.1).Footnote ²⁴ Furthermore, we can use the same high sea and coastal Grl. halibut price in the scenarios with a common and two separate fish stocks. In identifying the high sea price, we follow the recommendation by Flaaten et al. (Reference Flaaten, Heen and Matthiasson2017) and use the same data source and procedure to identify cost parameters and prices.Footnote ²⁵ Thus, we depart from the total high sea industry revenue of harvesting all species in table 1, and we use the same high sea Grl. halibut quota share as for the high sea cost function to calculate the total high sea industry revenue of harvesting Grl. halibut for the period between 2013 and 2015. As previously described, we average the revenue observations to reduce the effect of random price variations. From the estimation of the high sea cost functions, we also have information about the average high sea harvest of Grl. halibut. As an estimate for the high sea Grl. halibut price, we now use the average high sea revenue divided by the average high sea harvest. For the coastal Grl. halibut price, the only difference compared with the high sea price is that the total coastal industry revenue of harvesting all species and the harvest of Grl. halibut is reported for vessels above 6 m and boats below 6 m. Thus, we aggregate the revenue and harvest observations for these two fleet segments. Otherwise, we use the same procedure as for the high sea price. Three facts are important in relation to the parameters in the high sea and coastal profit functions (table 3). First, the high sea and coastal cost parameter is lower with two separate fish stocks than with a common fish stock.Footnote ²⁶ Second, the high sea cost parameter is higher than the coastal cost parameter with both a common and two separate fish stocks.Footnote ²⁷ Third, the coastal Grl. halibut price is lower than the high sea price.

4.3. Land-based industry profit function

A cost function that is consistent with the assumptions about the derivatives from section 3 is given by:Footnote ²⁸

(17)\begin{equation}{C_L}({h_C},{h_H}) = {c_L}{({h_C} + 0.25{h_H})^2},\end{equation}

where ${c_L}$ is a land-based cost parameter and ${C_L}({h_C},{h_H})$ is the total land-based industry cost of processing Grl. halibut. Three facts are important in relation to the land-based cost function in (17): (i) the land-based cost function depends on the total Grl. halibut harvest delivered to land-based processing; (ii) the land-based cost parameter, ${c_L}$, is assumed to be identical for the high sea and coastal Grl. halibut harvest; and (iii) the land-based cost function is identical in the scenarios with a common and two separate fish stocks.

For calculation of the land-based cost parameter, we use two main indicators: (i) the total industry costs for the primary fisheries industry of harvesting all fish species on both the east and west coasts (H Ogmundsson, personal communication, 2019; based on Ogmundsson (Reference Ogmundsson2019)); and (ii) the total industry costs for the whole fishing industry on both the east and west coasts (Ogmundson and Haraldson, Reference Ogmundson and Haraldson2017). The difference between these two cost measures represents a rough approximation for the total land-based processing industry costs covering the east and west coasts and all species. Next, we use relevant quota shares to obtain the total land-based costs of processing Grl. halibut on the west coast. These total land-based costs cover the period between 2013 and 2015 and, to reduce the impact of random variation, we take an average of these cost observations over the period. By using the information described in section 4.2, we can also calculate an average value of ${h_C} + 0.25{h_H}$, and now ${c_L}$ can be calculated by using (17).

The constant land-based Grl. halibut price is also identical for the scenarios with a common and two separate fish stocks and, to calculate this price, we use a similar procedure as for the land-based cost function. Thus, we obtained information about the total industry revenue generated by the primary fishing sector (H Ogmundsson, personal communication, 2019; based on Ogmundsson (Reference Ogmundsson2019)). Furthermore, we have data on the revenue earned by the entire fishing industry (Ogmundson and Haraldson, Reference Ogmundson and Haraldson2017). The difference between these two revenue numbers approximates the land-based processing revenue covering the east and west coasts and all fish species, and now we use relevant quota shares to obtain the revenue of processing Grl. halibut on the west coast. Information about this revenue is available for the period between 2013 and 2015, and to reduce the impact of random variation, we take a simple average of these numbers. We also obtained a measure for the average amount of Grl. halibut delivered from land-based processing factories (H Ogmundsson, personal communication, 2019; based on Ogmundsson (Reference Ogmundsson2019)). Now, the land-based price is found by dividing the average land-based revenue by the average quantity. The land-based utilization rate of Grl. halibut can be found by using two indicators: (i) the amount of Grl. halibut that goes into land-based processing which captures the demand (Ogmundson and Haraldson, Reference Ogmundson and Haraldson2017); and (ii) the amount of Grl. halibut that goes out of land-based processing factories which captures the supply (H Ogmundsson, personal communication, 2019; based on Ogmundsson (Reference Ogmundsson2019)). We have this information for the period 2013–2015; thus, as above, we take a simple average of these observations, and based on this, we can easily find a measure for α. From the land-based cost parameter, price and utilization rate summarized in table 3, it is clear that the marginal land-based profit is higher for high sea harvest than for coastal harvest.

5.1. Benchmark case

The results for the benchmark case are reported in table 4. To describe the results under each scenario, we focus on the distribution of the optimal industry harvest and profit between high sea and coastal vessels, and in the high sea harvest, we include the harvest by foreign vessels. As indicated in table 4, the optimal coastal harvest and profit are higher than the high sea harvest and profit in scenario 1 (a common fish stock without land-based profit). This result can be explained by the fact that the high sea cost parameter is higher than the coastal cost parameter (section 4.2). All other things equal, a higher cost parameter implies a lower harvest and profit.

Table 4.

Results for the Grl. halibut fishery on the west coast of Greenland

Turning to scenario 2 (two separate fish stocks without land-based profit), table 4 indicates that the optimal high sea harvest and profit are higher than the coastal harvest and profit. The main driver behind this result is that the high sea carrying capacity is higher than the coastal carrying capacity (section 4.1). A higher carrying capacity, all other things equal, implies a higher harvest and profit. The optimal high sea harvest is also higher than the coastal harvest in scenario 3 (a common fish stock with land-based profit), which can be explained by the fact that the marginal land-based profit is higher for high sea harvest than for coastal harvest (section 4.3). All other things equal, a higher marginal land-based profit implies a higher harvest. However, the optimal coastal profit is higher than the high sea profit in scenario 3, and the main explanation of this result is that the payoff from international fishing agreements with other nations is excluded in our model. The harvest by vessels from foreign nations represents an additional cost of high sea fishing (section 4.1). An additional cost, all other things equal, implies a lower harvest and profit. In scenario 4 (two separate fish stocks with land-based profit), the optimal high sea and coastal harvest and profit are approximately the same as those in scenario 2 (table 4), which is due to the low migration (section 4.1). All other things equal, a low net migration implies identical results in the scenarios with two separate fish stocks since the interaction between fleet segments is low. Turning to the land-based industry profit, table 4 indicates that the optimal land-based profit is lower than both the optimal high sea and coastal profit in both scenarios 3 and 4.Footnote ²⁹

Next, we compare the scenarios for optimal management with the actual situation. As indicated in table 4, the optimal high sea harvest and profit are lower than the actual harvest and profit in scenario 1, while the optimal coastal harvest and profit are higher than in the actual case. However, in scenarios 2 and 4, the optimal high sea harvest is close to the actual high sea harvest, while the optimal high sea profit is lower than the actual profit. Furthermore, the optimal coastal harvest and profit are lower than the actual coastal harvest and profit in scenarios 2 and 4. From table 4, we also observe that the optimal high sea harvest is higher than in the actual situation in scenario 3, while the optimal high sea profit, coastal harvest and coastal profit are lower than in the actual values. The optimal land-based profit is also lower than the actual land-based profit in scenario 3, while the optimal and actual land-based profits are almost identical in scenario 4 (table 4).

Finally, we discuss overexploitation of the Grl. halibut fish stock. In scenarios 1 and 3 (a common fish stock), table 4 indicates that the optimal stock size is higher than the actual stock size so, currently, the Grl. halibut fish stock is economically overexploited. Furthermore, in the actual situation, the total Grl. harvest is higher than the natural growth, so the Grl. halibut fish stock will decrease over time. With two separate fish stocks (scenarios 2 and 4), table 4 indicates that the optimal high sea stock size is higher than the actual stock size, implying that economic overexploitation of the high sea fish stock occurs. From table 4, we also see that the optimal coastal fish stock in scenarios 2 and 4 is close to the actual coastal fish stock. However, the actual coastal harvest is higher than the actual coastal natural growth, so the coastal fish stock will decrease over time.

5.2. Sensitivity analysis

When investigating the robustness of our results, we focus on the relative ranking of the optimal high sea and coastal harvest and profit.Footnote ³⁰ Our results are reasonably robust when excluding land-based processing (scenarios 1 and 2). Specifically, from section 5.1, we have that the optimal coastal harvest and profit are higher than the high sea harvest and profit in scenario 1. This result holds for all parameter variations apart from the lower bound of the coastal price. In scenario 2, the optimal high sea harvest and profit are higher than the coastal harvest and profit in the benchmark case (section 5.1). This ranking of the optimal harvest and profit only changes for the lower bound of the high sea price and the upper bound of the coastal price. However, in scenario 1, the optimal high sea harvest to vessels from Greenland and foreign vessels is zero for (i) the lower bound of the intrinsic growth rate; (ii) the upper bound for the scaling factor to other nations; (iii) the lower bound of the high sea price; (iv) the lower bound of the coastal cost parameter; and (v) the upper bound of the coastal price. An implication of this result is that it could be optimal to exclude high sea vessels from Greenland and other nations from the Grl. halibut fishery.Footnote ³¹

When taking land-based processing into account (scenarios 3 and 4), our results are reasonably sensitive to parameter variations. In scenario 3, we know from section 5.1 that the optimal high sea harvest is higher than the coastal harvest, while the optimal coastal profit is higher than the high sea profit. These results do not hold for (i) the lower bound for the carrying capacity; (ii) the lower bound for the intrinsic growth rate; (iii) the lower bound for the high sea cost parameter; (iv) the upper bound for the coastal price; (v) the upper bound for the land-based cost parameter; (vi) the lower bound for the land-based price; and (vii) the upper bound for the land-based utilization rate. In scenario 4, the optimal high sea harvest and profit are higher than the coastal harvest and profit in the benchmark case (section 5.1). This result changes for the lower bound for the high sea carrying capacity, the upper bound for the coastal carrying capacity, the lower bound for the high sea price and the upper bound for the coastal price.