Fishing in the Commons: In the text, we introduced the notion of the Tragedy of the Commons — and found its source in the emergence of externalities when property rights are not well established. This exercise demonstrates the same idea in a slightly different way.
A. Consider a self-contained lake which is home to fish that are sold on the market at p. Suppose the primary input into fishing this lake is nets that are rented at a weekly rate of r , and suppose the single input production frontier for fish has decreasing returns to scale.
(a) Draw a graph with fishing nets on the horizontal axis and fish on the vertical. Illustrate the marginal product of fishing nets.
(b) Recalling the relationship between “marginal” and “average” quantities, add the average product curve to your graph.
(c) If you own the lake, what is the relationship between the marginal product of fishing nets and prices (p,r ) assuming you maximize profit?
(d) Illustrate the profit maximizing quantity of nets n∗ on your graph. Then, on a graph below it that plots the production frontier for fish; illustrate the number of fish x∗ that are brought to market.
(e) Suppose you instead charge a weekly fee for every fishing net that fishermen bring to your lake. Does the number of fish produced and nets used change?
(f) Next consider a nearby lake that is identical in every way except that it is publicly owned — with no one controlling who can come onto the lake to fish. Assuming all nets are used with the same intensity, each fishing net that is brought onto the lake can then be expect to catch the average of the total weekly catch. Illustrate on your graphs how many nets  will be brought onto this lake—and how many fish  this implies will be brought to market each week.
(g)Which lake yields more fish per week? Which lake is being harvested for fish efficiently?
(h) Suppose that what matters is not just the current crop of fish but also its implication for the future fish population of the lake. Explain how the privately owned lake is likely to house a relatively constant population of fish over time while the publicly owned lake is likely to run out of fish as time passes.
(i) The trade in elephant trunks—or ivory—has decimated much of the elephant population in some parts of Africa but not in others, with hunters often slaughtering entire herds, removing the trunks and leaving the rest. In some parts of Africa, the land on which elephants roam is public property — in other parts it is privately owned with owners allowed to restrict access.
Can you guess from our lake example what is different about the parts of Africa where elephant herds are stable compared to those parts where they are nearing extinction?
(j) Why do you think that wild Buffalo in the American West are nearly extinct while domesticated cattle are plentiful in the same region?
B. Let n again denotes the fishing nets used in the lake and assume that r is the weekly rental cost per net. The number of fish brought out of the lake per week is x = f (n) = Anα where A > 0 and 0< α < 1, and fish sell on the market for p.
(a) Suppose you own the lake and you don’t let anyone other than yourself fish. How many fish will you pull out each week assuming you maximize profit?
(b) Suppose instead you allow others to fish for a fee per net — and you want to maximize your fees. Will more or fewer fish be pulled out each week?
(c) Next, consider the identical lake that has just been discovered near yours. This lake is publicly owned, and anyone who wishes to can fish there. How many fish per week will be pulled out from that lake?
(d) Suppose A = 100, α = 0.5, p = 10 and r = 20. How many fish are harvested per week in (a), (b) and (c)? How many nets are used in each case?
(e) What is the weekly rental value of the lake? If we count all your costs—including the opportunity cost of owning the lake, how much weekly profit do you make if you are the only one to fish on your lake?
(f) How much profit (including the opportunity cost of fishing on the lake yourself ) do you make if you allow others to fish on your lake for a per-net-fee? How much profit do the fishermen who pay the fee to fish on your lake make?
(g) How much profit do the fishermen who fish on the publicly owned nearby lake make?
(h) If the government auctioned off the nearby lake, what price do you think it would fetch if the weekly interest rate is 0.12% or 0.0012?
(i) If the government auctioned off the nearby lake with the condition that the same number of fish per week need to be brought to market as before, what price would the lake fetch?