The Economics of Fisheries: Practice Problems and Solutions

Mathematical Economics Problems and Solutions - Section 5

See Mr. Stolyarov's complete list of Mathematical Economics Problems and Solutions.

Note: Here, I will present solve problems typical of those offered in a mathematical economics or advanced microeconomics course. The problems were compiled by Dr. Charles N. Steele and are reprinted with his generous permission. The solutions to the problems are my own work and not necessarily the only way to solve the problems.

Problem 6. Logistic population growth: suppose the growth rate of a population of wild fish is given by F(x) = 10x - 0.01x² if the population is left undisturbed, were x is population.

a. What is the equilibrium natural population?

b. What is the maximum sustained yield, and the corresponding population?
c. Graph the population growth function. Which portions of this function correspond to biological overfishing?

Solution 6a. The equilibrium population level occurs when the growth rate of population is zero and the population is positive. That is, 10x = 0.01x² and 10 = 0.01x. Thus, x = 1000 fish

Solution 6b. The maximum sustained yield occurs at the maximum growth rate.

FOC: dF/dx = 10 - 0.02x ≡ 0, which means that x = 500 at MSY.

SOC: F" = -0.02 < 0, so x = 500 is a maximum. The growth rate at MSY =

10*500 - 0.01*500² = F(500) = 2500 fish/unit of time. It will be possible to capture 2500 fish per unit of time while keeping the fishery population at 500 and then to capture another 2500 per unit of time indefinitely. So MSY= 2500 fish.

Solution 6c. See included graph. Biological overfishing corresponds to all populations less than (left of) x = 500.

Problem 7. Schaefer model: suppose that the yield (harvest h) is given by h = 2Ex ,where E is the amount of fishing effort and x is population (here q, the productivity parameter, equals 2).

a. Find sustainable population x as a function of effort E.

b. Find sustainable yield as a function of E.

c. Graph the resulting Yield-Effort curve. Which portions of this curve correspond to biological overfishing?

Solution 7a. Now the net growth rate of fish is F(x) - h = 10x - 0.01x² - 2Ex. The equilibrium population level occurs when the net growth rate of population is zero and the population is positive. That is, (10 - 2E)x = 0.01x² and 10 - 2E = 0.01x. Thus,

x = 1000 - 200E

Solution 7b. The yield-effort curve is given by Y = qKE(1 - qE/r), where r =10, q = 2, K = 1000. Thus, Y = 2000E(1 - E/5) = Y = 2000E - 400E².

Solution 7c. See included graph. Biological overfishing corresponds to all levels of effort to the right of E_MSY.

8. Gordon model: Suppose price of fish P = 10, and total cost of fishing effort is TC(E) = 1000E.

a. Find the Total Revenue Product Curve.

b. Suppose the fishery is managed by an owner with exclusive property rights. Find the level of effort E that maximizes the rent from the fishery, and calculate the rent.

c. Find the level of effort that would occur under open access.

d. Under open access, is this fishery suffering from economic overfishing?
e. Under open access, is this fishery suffering from biological overfishing? Why or why not?

Solution 8a. TRP = P*Y(E). Y(E) = 2000E - 400E² from 7b. So

TRP = 20000E - 4000E²

Solution 8b. The rent-maximizing level of effort will be the level of effort that maximizes Rent = TRP - TC(E) = 20000E - 4000E² - 1000E = 19000E - 4000E².

d[Rent]/dE = 19000 - 8000E ≡ 0 and E = 19/8 = 2.375.

Rent will be 19000*2.375 - 4000*2.375² = Rent = $22,562.50

Solution 8c. Under open access, economic rent from the fishery would diminish to zero and 19000E - 4000E² = 0, so 19000E = 4000E² and the non-zero level of effort for this is

E = 19000/4000 = E = 19/4 = 4.75

Solution 8d. Under open access, the fishery issuffering from economic overfishing, because rents are being dissipated and the fishery is contributing no value on net to the economy. With less effort, rents could exist and the economic value of the fishery could be increased.

Solution 8e. Under open access, the fishery is suffering from biological overfishing. Biological overfishing occurs when E > r/2q. But here, r/2q = 2.5 = E, and 4.75 > 2.5, so the fishery is being fished at a level of effort greater than the one which corresponds to maximum sustainable yield.

See Mr. Stolyarov's complete list of Mathematical Economics Problems and Solutions.