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Chapter 24: Problem 2

On the island of Pago Pago there are 2 lakes and 20 anglers. Each angler can fish on either lake and keep the average catch on his particular lake. On Lake Xthe total number offish caught is given by \\[ \mathbf{F}^{*}=\mathbf{1 0 L}_{\mathbf{x}^{-}}-L h \\] where \(L_{x}\) is the number of people fishing on the lake. For Lake Fthe relationship is \\[ F^{\gamma}=5 L_{y} \\] a. Under this organization of society, what will be the total number of fish caught? b. The chief of Pago Pago, having once read an economics book, believes it is possible to raise the total number of fish caught by restricting the number of people allowed to fish on Lake \(X .\) What number should be allowed to fish on Lake \(X\) to maximize the total catch of fish? What is the number of fish caught in this situation? c. Being basically opposed to coercion, the chief decides to require a fishing license for Lake X. If the licensing procedure is to bring about the optimal allocation of labor, what should the cost of a license be (in terms offish)? d. Explain how this example sheds light on the connection between property rights and externalities.

Short Answer

Expert verified
Answer: The maximum total catch of fish from both lakes is 106 fish, with either 2 or 3 anglers on Lake X and the remaining anglers on Lake Y. The cost of a fishing license for Lake X should be 54 fish to achieve this optimal allocation.

Step by step solution

01

a) Calculate the total number of fish caught on both lakes

First, we will add the total number of fish caught from Lake X and Lake Y. Since there are 20 anglers total, if \(L_x\) anglers are on Lake X, then there must be \(20 - L_x\) anglers on Lake Y. We can write this as: \\[ F = F^* + F^\gamma = (10L_x - L_x^2) + (5 (20 - L_x)) \\]
02

b) Determine the optimal number of anglers on Lake X to maximize the total catch

To maximize the total catch, we need to find the value of \(L_x\) that maximizes the function \(F\). This can be done by taking the first derivative with respect to \(L_x\) and setting it to zero: \\[ \frac{dF}{dL_x} = 10 - 2L_x - 5 = 0 \\] Solving for \(L_x\): \\[ L_x = \frac{10 - 5}{2} = \frac{5}{2} \\] However, since the number of anglers must be a whole number, we must choose between 2 and 3 anglers on Lake X. We need to test both cases to see which one yields the higher total catch. Case 1: \(L_x = 2\) \\[ F = (10(2) - 2^2) + (5 (20 - 2)) = 16 + 90 = 106 \\] Case 2: \(L_x = 3\) \\[ F = (10(3) - 3^2) + (5 (20 - 3)) = 21+ 85 = 106 \\] Either case results in the same maximum total catch of 106 fish. So, allowing either 2 or 3 anglers on Lake X would maximize the total catch of fish.
03

c) Calculate the cost of a fishing license for Lake X

Let \(C_f\) be the cost of a fishing license, measured in fish. To encourage the optimal allocation of anglers, the average catch per angler must be equal on both lakes. Given the case when \(L_x = 2\), we have: \\[ \frac{16}{2} = \frac{5(20-L_x) - C_f}{20-L_x} \\] Solving for \(C_f\): \\[ C_f = 5(20-L_x) - 8(20-L_x) = -3(18) = -54 \\] Therefore, the cost of a license should be 54 fish to bring about the optimal allocation of anglers on both lakes.
04

d) Property rights and externalities in the example

This example illustrates the connection between property rights and externalities because the chief's decision to require a fishing license for Lake X is an attempt to manage the common resource - the fish and the lakes. By assigning a cost or "property right" to the use of the lake through a fishing license, the chief is essentially internalizing the negative externality that occurs when too many anglers fish on Lake X, causing the total catch to decrease. By choosing the optimal license cost, the chief incentivizes the anglers to distribute themselves between the two lakes in a manner that maximizes the total catch, and thus leads to a more efficient use of the common resource.

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