Additional Problems on the Economics of Fisheries

Mathematical Economics Problems and Solutions - Section 7

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

Problem EF1. The population of magenta-spotted dodecahedral fish in a particular fishery follows the growth function F(x) = 987x - 0.45x² if the population is left undisturbed, were x is population. What is the equilibrium natural population? Assume that fractional fish are possible.  

Solution EF1. The equilibrium population level occurs when the growth rate of population is zero and the population is positive. That is, 0 = 987x - 0.45x² and

987x = 0.45x², so x = 987/0.45 = x = 2193.3333333333 fish

Problem EF2. For the fishery in Problem EF1, what is the maximum sustained yield, and the corresponding population?

Solution EF2. We seek to maximize F(x) = 987x - 0.45x².

FOC: We take F'(x) = 987 - 0.9x ≡ 0, so 0.9x = 987 and x = 987/0.9 =

x_MSY = 1096.666667.

SOC: F''(x) = - 0.9 < 0, so x_MSY is indeed a maximum.

MSY = F(1096.666667) = 987*1096.666667 - 0.45*1096.666667² = MSY = 541205 fish/unit of time. It will be possible to capture 541205 fish per unit of time while keeping the fishery population at 1096.666667 and then to capture another 541205 per unit of time indefinitely.

Problem EF3. Humans discover the magenta-spotted dodecahedral fish fishery and begin to harvest at a yield of h = 8Ex, where E is the amount of fishing effort and x is population. Find sustainable population x as a function of effort E.

Solution EF3. The net rate of fish population growth is now

F(x) - h = 987x - 0.45x² - 8Ex. At the sustainable population, the net rate of fish population growth is zero. Thus, (987 - 8E)x - 0.45x² = 0 and x = (987 - 8E)/0.45 =

x = 2193.33333333 - 17.7777777777E

Problem EF4. Using the information from Problems EF1-3, find sustainable yield as a function of E.

Solution EF4. Y(E) = qEx = 8Ex in this case. From Solution EF3, the sustainable population x is 2193.33333333 - 17.7777777777E. Thus, sustainable yield is

8E(2193.33333333 - 17.7777777777E) = Y(E) = 17546.666667E - 142.22222222E².

Problem EF5. Within the market for magenta-spotted dodecahedral fish, the price of fish is P = $50 per unit, and the total cost of fishing effort is TC(E) = 25000E. Find the Total Revenue Product Curve, expressed as a function of E. Use any relevant information from Problems EF1-4.

Solution EF5. TRP = P*Y(E). Y(E) = 17546.666667E - 142.22222222E² from Solution EF4, so TRP = 50(17546.666667E - 142.22222222E²) =

TRP = 877333.3333E - 7111.1111111E².

Problem EF6. Suppose that Multinational Corp. is the exclusive property owner of the magenta-spotted dodecahedral fish fishery and seeks to maximize its rents. Find the level of effort E that maximizes the rent from the fishery, and calculate the rent. Use any relevant information from Problems EF1-5.

Solution EF6. Rent = TRP - TC(E) = 877333.3333E - 7111.1111111E² - 25000E

Thus, Rent = 852333.3333E - 7111.1111111E² and

FOC: d[Rent]/dE = 852333.3333 - 14222.2222222E ≡ 0, so 14222.2222222E = 852333.3333, so E = 59.9296875

SOC: Rent'' = - 14222.2222222 < 0, so the E determined above is a maximum.

Problem EF7. Suppose that magenta-spotted dodecahedral fish "liberationists" take over the world and declare all fisheries to be open-access commons. Find the level of fishing effort that would occur under open access.

Solution EF7. Under open access, rents are dissipated to zero and thus

Rent = 852333.3333E - 7111.1111111E² = 0, so 852333.3333 = 7111.1111111E and

E = 119.859375

Problem EF8. Under open access, is the magenta-spotted dodecahedral fish fishery suffering from biological overfishing? Is it suffering from economic overfishing?

Solution EF8. The level of effort corresponding to maximum sustainable yield is the E that maximizes Y(E) = 17546.666667E - 142.22222222E².

Y'(E) = 17546.666667 - 284.44444444E ≡ 0, so E_MSY = 61.6875001. But

E_open-access = 119.859375 > E_MSY, so there exists biological overfishing.

Under open access, the fishery is suffering 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.

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