Question

The Georges Bank, a highly productive fishing area off New England, can be divided into two zones in terms of fish population. Zone 1 has the higher population per square mile but is subject to severe diminishing returns to fishing effort. The daily fish catch (in tons) in Zone 1 is \\[F_{1}=200\left(X_{1}\right)-2\left(X_{1}\right)^{2}\\] where \(X_{1}\) is the number of boats fishing there. Zone 2 has fewer fish per mile but is larger, and diminishing returns are less of a problem. Its daily fish catch is \\[F_{2}=100\left(X_{2}\right)-\left(X_{2}\right)^{2}\\] where \(X_{2}\) is the number of boats fishing in Zone \(2 .\) The marginal fish catch MFC in each zone can be represented as \\[\begin{array}{l} \mathrm{MFC}_{1}=200-4\left(\mathrm{X}_{1}\right) \\ \mathrm{MFC}_{2}=100-2\left(\mathrm{X}_{2}\right) \end{array}\\] There are 100 boats now licensed by the U.S. government to fish in these two zones. The fish are sold at \(\$ 100\) per ton. Total cost (capital and operating) per boat is constant at \(\$ 1000\) per day. Answer the following questions about this situation: a. If the boats are allowed to fish where they want, with no government restriction, how many will fish in each zone? What will be the gross value of the catch? b. If the U.S. government can restrict the number and distribution of the boats, how many should be allocated to each zone? What will be the gross value of the catch? Assume the total number of boats remains at 100. c. If additional fishermen want to buy boats and join the fishing fleet, should a government wishing to maximize the net value of the catch grant them licenses? Why or why not?

Short answer

a. Given unrestricted fishing, boats will distribute themselves in such a way that possibilities of returns in both zones are equal. The gross value of the catch can be calculated based on the catch from each zone and the selling rate. b. Under restricted conditions, the government maximizes the gross value of the catch by distribution of boats such that the marginal fish catch in each zone equals the cost of operating a boat. c. Granting additional licenses would be beneficial only if the marginal fish catch is greater than the cost of operating a boat.

Step-by-step solution

Part a: Determine the number of boats in each zone under unrestricted condition

First, the marginal fish catch (MFC) in each zone should be equal for equilibrium to occur. Set MFC1 equal to MFC2 and solve for the number of boats, resulting in the equation \(200 - 4X_1 = 100 - 2X_2\). Solve this equation to find the values of \(X_1\) and \(X_2\).

Part a: Calculating the gross value of the catch under unrestricted condition

Substitute the values of \(X_1\) and \(X_2\) found in the previous step into the equations for \(F_1\) and \(F_2\) to find the daily catch in each zone. Multiply the total catch by $100 per ton to find the gross value of the catch.

Part b: Determine the number of boats in each zone under restricted condition

To maximize the gross value of the catch, the boats should be distributed such that the marginal fish catch in each zone is equal to the cost of operating a boat. Set \(MFC1 = MFC2 = $1000\) (the cost of operating a boat) and solve for \(X_1\) and \(X_2\). Then make sure the total number of boats does not exceed 100.

Part b: Calculating the gross value of the catch under restricted condition

Substitute the values of \(X_1\) and \(X_2\) found in the previous step into the equations for \(F_1\) and \(F_2\) to find the daily catch in each zone. Multiply the total catch by $100 per ton to find the gross value of the catch.

Part c: Decision on giving additional licenses

If additional fishermen want to buy boats and join the fishing fleet, the government should grant them licenses only if the marginal fish catch in each zone is greater than the cost of operating a boat. Otherwise, the net value of the catch would decrease.

Key concepts

Marginal Fish Catch

Understanding the concept of marginal fish catch (MFC) is crucial in fishery economics. MFC refers to the additional amount of fish that can be caught by employing one more unit of effort, for example, adding another fishing boat in a specific zone. In the provided exercise, diminishing returns to fishing effort were highlighted, which essentially means that as more boats are deployed to fish, the additional catch per boat decreases. This is because fish stocks are limited and as effort increases (more boats), each additional boat catches less than the one before it.

In Zone 1, where the fish population per square mile is high but subject to severe diminishing returns, the MFC equation given was \(\mathrm{MFC}_{1}=200-4(\mathrm{X}_{1})\). Similarly, in Zone 2 with a lower population per mile but also lower diminishing returns, the MFC is given by \(\mathrm{MFC}_{2}=100-2(\mathrm{X}_{2})\). To optimize the number of boats, we set these two marginal catches equal, balancing the additional catch per boat across both zones and ensuring efficient utilization of boats.

Equilibrium in Fishery Economics

Equilibrium in fishery economics refers to a situation where resources are allocated optimally between different zones, such that neither zone can increase its catch without decreasing the catch in the other zone. In our exercise case, equilibrium is reached when the marginal fish catches (MFC) in both zones are equal. This ensures that each additional boat contributes the same amount to the total catch, regardless of the zone in which it fishes.

The pursuit of equilibrium helps in managing the fisheries sustainably by preventing overfishing in any particular zone. According to the solution steps, equating the MFCs from both zones allows us to determine an optimal distribution of 100 boats to maximize the total catch value, under both unrestricted and restricted scenarios laid out by government policies.

Resource Allocation

Resource allocation in the context of fisheries management concerns how fishing efforts, like the number of licensed boats, are distributed across different fishing zones to maximize catch, profit, or sustainability goals. The exercise demonstrates two different scenarios: unrestricted and restricted allocation by the government.

Under unrestricted conditions, boats will naturally distribute themselves based on where they can catch the most fish until the marginal fish catch equals out in each zone. When restrictions are enacted, the government controls the number of boats in each zone to achieve a desired objective, in this case, maximizing the gross value of the catch while ensuring sustainability. This controlled allocation ensures that the fishing effort does not exceed the ecological capacity of the fishery ecosystem while also considering economic efficiency.

Government Restrictions in Fisheries Management

Government restrictions play a pivotal role in fisheries management. These regulations are designed to maintain fish populations at sustainable levels and to ensure that the fishing industry remains viable in the long term. In the exercise, the government controls the number of boats that can fish in each zone.

Such restrictions can take various forms, including licensing a specific number of boats, limiting the total catch, or dictating how many boats can fish in certain areas. By allocating a fixed number of licenses and strategically distributing boats across zones as the solution suggests, the government aims to optimize resource use and prevent overfishing. They also have the authority to deny additional licenses if the current number of boats already maximizes the net value of the catch, protecting the fishery from exhaustion and securing the industry's future.