## 1. Introduction

Since Greenland obtained partial autonomy from Denmark in 2009,Footnote ^{1} an ongoing political issue has been securing increased economic independence, with fisheries being an important sector for the economy. For example, the primary fishing sector generated between 76 and 88 per cent of the total value of exports between 2013 and 2015 (Statistics Greenland, 2020).

Greenland halibut (hereafter Grl. halibut) has gained increasing importance for primary fisheries. In the period between 2013 and 2015, the Grl. halibut harvest constituted approximately 33 per cent of the total Greenlandic harvest of all fish species by weight and approximately 44 per cent by income, and the Grl. halibut price increased by approximately 47.5 per cent from 2010 to 2019 (Statistics Greenland, 2020). This has generated increased political pressure for larger Grl. halibut quotas, but concerns for overexploitation of the fish stock have been raised (Fiskerikommissionen, 2021). Thus, an important policy issue for Greenland is securing the long-run sustainable management of the Grl. halibut fish stock.

The largest amount of Grl. halibut is harvested on the west coast of Greenland. In this respect, at least three different fleet segments target Grl. halibut, namely, high sea vessels, coastal vessels above 6 m and coastal boats below 6 m.Footnote ^{2} A total allowable catch (TAC) is fixed each year based on biological recommendations, and this TAC is allocated to the high sea and coastal areas as total quotas (e.g., Ogmundsson and Bæk, Reference Ogmundsson and Bæk2021). The total high sea Grl. halibut quota is then divided between two fleet segments represented by vessels from Greenland and foreign vessels. The total coastal quota is also allocated to two fleet segments given by vessels above 6 m and boats below 6 m (e.g., Ogmundson and Haraldson, Reference Ogmundson and Haraldson2017Footnote ^{3}). Regarding the regulation of individual vessels, an individual transferrable quota system was introduced in 2012 for coastal vessels above 6 m targeting Grl. halibut in Disko Bay, Uummannaq and Upernavik, whereas time-limited licences without a maximum harvest are used for both high sea vessels from Greenland and coastal boats below 6 m (e.g., Ogmundsson, Reference Ogmundsson2019).Footnote ^{4}

At least two issues are relevant for optimal management of Grl. halibut. First, Grl. halibut spawn and grow in the high sea area (e.g., Boje, Reference Boje2002). However, according to Fiskerikommissionen (2021), the migration of young Grl. halibut from the high sea area constitute the main source of recruitment for the coastal stock size. Furthermore, Grl. halibut does not spawn in coastal areas (e.g., Boje, Reference Boje2002). Thus, the natural growth of Grl. halibut on the west coast of Greenland appears to differ between the high sea and coastal area. This implies that we shall operate with separate high sea and coastal fish stocks, but in current biological assessments, a common fish stock is assumed (e.g., Treble and Nogueira, Reference Treble and Nogueira2018). However, if this assumption can be questioned, it is relevant to investigate the implications for optimal management of Grl. halibut in terms of operating with two separate fish stocks rather than a common fish stock.

Second, from an economic perspective, it is not only primary fisheries but also secondary land-based processing industries that are important for Greenland (e.g., Ogmundson and Haraldson, Reference Ogmundson and Haraldson2017). For high sea vessels, the harvest of Grl. halibut is to a large extent processed on board, but to ensure land-based employment and profitability, a legal landing obligation is imposed. According to this obligation, high sea vessels shall deliver at least 25 per cent of the Grl. halibut for land-based processing (e.g., Ogmundson and Haraldson, Reference Ogmundson and Haraldson2017). In contrast, coastal vessels and boats normally deliver the entire harvest of Grl. halibut to land-based processing.Footnote ^{5} Thus, an important issue is the implications for optimal management of Grl. halibut in terms of considering the profit of land-based processing.

To our knowledge, only Fredenslund (Reference Fredenslund2022) has used a modelling approach to study the optimal management of Grl. halibut. By using a static, steady-state equilibrium model, Fredenslund (Reference Fredenslund2022) showed that the coastal Grl. halibut fish stock in Disko Bay, Uummannaq and Upernavik is overexploited. In reaching this conclusion, Fredenslund (Reference Fredenslund2022) used a model with two fleet segments (coastal vessels above 6 m and coastal boats below 6 m), but only one fish stock was considered, and land-based processing was not included. However, a modelling approach can also be used to investigate the effect of operating with separate high sea and coastal fish stocks and considering land-based processing. These topics constitutes important areas for future research.

The purpose of this paper is to address these research topics. We depart from a theoretical model at the industry level with a high sea and coastal fleet segment. The model is used to identify four scenarios for the optimal, long-run, steady-state equilibrium management of fishery resources that differ with respect to whether a common or two separate fish stocks exist and whether the land-based profit is considered. The model is parametrized using the high sea and coastal Grl. halibut fisheries on the west coast of Greenland as an empirical case. For Grl. halibut, we compare the outcome under four scenarios for optimal management. Furthermore, to investigate whether economic overexploitation of the Grl. halibut fish stock is occurring, the scenarios for optimal management are compared to the actual situation. Finally, when parameterizing the model for Grl. halibut, many estimated parameter values are highly uncertain, so to investigate the robustness of our results, we conduct sensitivity analyses by varying each parameter value separately with ±50 per cent.

With the analysis in this paper, we also contribute to at least two additional strands of related fishery economic literature. First, many empirical papers discuss long-run optimal management with several fleet segments and fish stocks. Examples of empirical papers operating with several fleet segments are Ulrich *et al*. (Reference Ulrich, Gallic, Dunn and Gascuel2002), Camber *et al*. (Reference Camber, Ourens, Vidal, Carabel and Freire2012) and Quinones *et al*. (Reference Quinones, Lopez-Rocha, Hernandez-Herrera and Torres-Irineo2021), while DuPont *et al*. (Reference Dupont, Fox, Gordon and Grafton2005), Danielsson *et al*. (Reference Danielsson, Stefansson, Baldursson and Thorarinsson1997) and Pope *et al*. (Reference Pope, Gislason, Rice and Daan2021) are examples of studies of actual fisheries that consider multiple fish stocks. However, the empirical literature on optimal management with multiple fleet segments and fish stocks does not consider the effect of land-based processing. By doing so, we make a novel contribution to the literature.

Second, several papers investigate the relationship between primary fisheries and land-based processing. A number of papers have shown that optimal management of primary fisheries will have a negative effect on the land-based profit (e.g., Casey *et al*., Reference Casey, Dewees, Turris and Wilen1995; Eythorsson, Reference Eythorsson2000; Norman-Lopez and Pascoe, Reference Norman-Lopez and Pascoe2011). More importantly, Grafton *et al*. (Reference Grafton, Kompas, Che, Chu and Hilborn2012) argue that the optimal harvest and stock size will be reduced when the land-based profit is taken into account, but only one fish stock and fleet segment is considered. In this paper, we investigate the implications for optimal management of including the land-based industry profit, and we operate with several fleet segments and fish stocks. Thereby, we make a novel contribution to the abovementioned literature.

## 2. Grl. halibut fishery on the west coast of Greenland

In this section, we will describe the Grl. halibut fishery on the west coast of Greenland, and descriptive statistics for this fishery are provided in table 1.Footnote ^{6}

*Sources:* Ogmundson and Haraldson (Reference Ogmundson and Haraldson2017), Fiskerikommissionen (2021).

From table 1, we see that the total Grl. halibut quota allocated to high sea vessels from Greenland and coastal vessels above 6 m increased between 2014 and 2015, while the quota allocated to coastal boats below 6 m increased during the entire period. A part of the high sea Grl. halibut quota is allocated to foreign vessels according to international fishing agreements covering a reasonably long period.Footnote ^{7} This implies that the high sea Grl. halibut quota allocated to foreign vessels was constant between 2013 and 2015 (table 1).

From Fiskerikommissionen (2021), we have information about the number of used licences for harvesting Grl. halibut for the period between 2013 and 2015, and we use this as a measure of the number of vessels participating in the fishery. From table 1, we see that only 4 production trawlers from Greenland participated in the high sea Grl. halibut fishery between 2013 and 2015. Furthermore, the number of coastal vessels above 6 m and boats below 6 m is approximately constant over the entire period. Compared to vessels above 6 m, a large number of boats below 6 m participate in the coastal Grl. halibut fishery.

Ogmundson and Haraldson (Reference Ogmundson and Haraldson2017) reported the total industry revenue and accounting costs for vessels targeting Grl. halibut in the high sea and coastal area, and this information can also be found in table 1.Footnote ^{8} The revenue and costs are identified for all vessels targeting Grl. halibut but covers the harvest of all fish species. From table 1, we see that the revenue and accounting costs for high sea vessels from Greenland and coastal boats below 6 m increased between 2013 and 2015. However, coastal vessels above 6 m experienced a considerable decrease in revenue and accounting costs between 2014 and 2015, although the Grl. halibut quota for this fleet segment has increased. The explanation for this fact is that the harvest of Grl. halibut by coastal vessels above 6 m decreased in 2015, indicating that the Grl. halibut quota was not fully utilized for this fleet segment (appendix A).

## 3. Theoretical model

In this paper, we use a theoretical model to construct four scenarios for optimal management of Grl. halibut on the west coast of Greenland. An overview of these scenarios is provided in table 2.

There are four important facts related to the theoretical model. First, we use a static model, which we solve for a long-run steady-state equilibrium.Footnote ^{9} Following Clark (Reference Clark1991), a static fisheries economic model implies that the undiscounted long-run economic yield for one period is maximized subject to the fish stocks being in a steady-state equilibrium.Footnote ^{10} Second, we operate with two fleet segments represented by high sea and coastal vessels. From table 1, the coastal fleet segment can be divided into vessels above 6 m and boats below 6 m. However, for simplicity, we have chosen only to operate with one coastal fleet segment. Third, we consider a model at the industry level, implying that we operate with revenue, costs and harvest for entire fleet segments. Fourth, we use a dual instead of primal formulation of the model. According to Neher (Reference Neher1990), a fishing technology is captured by a production function with a primal formulation, implying that fishing effort and stock size are control variables. With a dual approach, a cost function is introduced, and harvest and stock size are control variables. There is a close relation between a primal and dual formulation in the sense that a production function can be used to derive a cost function (e.g., Neher, Reference Neher1990).

### 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:

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:

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:

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:

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:

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:

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:

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. Functional forms and parameter estimation

Now we briefly summarize the parametrization of our model for the Grl. halibut fishery; more details can be found in appendix A. We have obtained a benchmark value for each parameter, but due to uncertainty we have generated an upper and lower bound by varying each parameter value by ± 50 per cent. An overview of the parametrization is provided in table 3.

### 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):

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:

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}

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}

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. Empirical results

To obtain empirical results for the Grl. halibut fishery on the west coast of Greenland, we used GAMS to solve the first-order conditions from section 3 numerically using the functional forms and parameter values from table 3. When solving the first-order conditions, we imposed nonnegativity constraints on all control variables, implying that the system of equations was solved using Kuhn–Tucker conditions. We have also used several different starting values for the control variables, and if various starting values lead to different solutions, we have selected the solution that provides the highest long-run economic yield.

Below, we will compare the results in the benchmark case with the actual situation for the period between 2013 and 2015, where averages over this period are used. The actual high sea harvest for vessels from Greenland and the actual coastal Grl. halibut harvest is found from observations in Ogmundson and Haraldson (Reference Ogmundson and Haraldson2017), while the high sea harvest allocated to vessels from other fishing nations is found by using *β*. To find the actual high sea and coastal profit, we use the difference between the revenue and costs from table 1, while the observations described in section 4.3 are used to find the actual land-based profit. In the scenarios with a common fish stock, we use the observations described in section 4.1 to obtain the actual Grl. halibut stock size, while the growth function is used to find the actual growth. With two separate fish stocks, we use the high sea and coastal Grl. halibut stock size described in section 4.1 as a measure for the actual stock sizes, while the high sea and coastal natural growth functions are used to find the actual high sea and coastal growth. The actual net migration is found using the migration function. Below, we will also summarize the results of our sensitivity analyses; details can be found in appendix C. In these analyses, we set each parameter value at the upper and lower bound, while the other parameters are fixed at the benchmark value. The results of sensitivity analyses are mainly used to investigate the robustness of our results.

### 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.

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.

## 6. Brief summary and main policy implications

In this paper, we consider four scenarios for optimal management of high sea and coastal vessels targeting the Grl. halibut fishery on the west coast of Greenland:

Scenario 1: common fish stock in the high sea and coastal area while disregarding the profit from land-based processing;

Scenario 2: separate fish stocks in the high sea and coastal area while disregarding the profit from land-based processing;

Scenario 3: common fish stock in the high sea and coastal area while including the profit from land-based processing; and

Scenario 4: separate fish stocks in the high sea and coastal area while including the profit from land-based processing.

In scenario 1, the optimal coastal harvest and profit are higher than the high sea harvest and profit, while the optimal high sea harvest and profit are higher than the coastal harvest and profit in scenario 2. For scenario 3, the optimal high sea harvest is higher than the coastal harvest, while the optimal coastal profit is higher than the high sea profit, and in scenario 4, we obtain approximately the same results as in scenario 2.

Thus, our empirical results are highly dependent on the chosen scenario. Provided that we believe in our results, a main policy implication is that a regulator (or manager) needs to be very explicit about the assumptions behind the policy decisions. Specifically, the regulator must explicitly consider: (1) whether common or separate fish stocks for various fleet segments exist; and (2) whether the profit of secondary fisheries-related activities will be considered. Since our results differ significantly depending on the choice of scenario, it will generate large mistakes in the harvest and profit allocated to various fleet segments if the policy decision is based on incorrect assumptions. This can be an argument for introducing a precautionary principle when making public decisions, since this principle minimizes the probability that wrong decisions will lead to catastrophic events.

For Grl. halibut, this issue is highly important because this fish species is important for the development of the economy in Greenland. Greenland is working towards full independence from Denmark, but a factor that limits the possibility of independence is the ability to generate income. Here Grl. halibut plays an important role since there has been a huge increase in the price of this species. However, an increase in the harvest of Grl. halibut puts pressure on the fish stock. Hence, the issue of long-run sustainable management of the Grl. halibut fish stocks, which we consider in this paper, is crucial to ensure economic and political independence.

## Supplementary material

The supplementary material for this article can be found at https://doi.org/10.1017/S1355770X23000128.

## Competing interests

The authors declare none.