The Externality when fishing in the Commons: In exercise 21.6, we showed that free access to a fishing lake causes overfishing because fishermen will continue to fish until the cost of inputs (i.e. fishing nets, in our example) equals average rather than marginal revenue product.

A. Suppose that the lake in exercise 21.6 is publicly owned.

(a) What is the externality that fishermen impose on one another in this lake?

(b) Seeing the problems one involving this externality, how would you go about setting a Pigouvian tax on fishing nets to remedy the problem? What information would you have to have to calculate this?

(c) Suppose instead that the lake is auctioned off to someone who then charges per-net fees to fishermen who would like to fish on the lake (as in A(e) of exercise 21.6). How do you think the fees charged by a profit maximizing lake-owner compare to the optimal Pigouvian tax?

(d) Do you think it is easier for the government to collect the information necessary to impose a Pigouvian tax in part (b) or for a lake-owner to collect the information necessary to impose the per-net fees in part (c). Who has the stronger incentive to get the correct information?

(e) How would the price of the lake that the government collects in (c) compare to the tax revenues it raises in (b)?

(f) Suppose instead that the government tries to solve the externality problem by simply setting a limit on per-net fishing licenses that fishermen are now required to use when fishing on the public lake. If the government sets the optimal cap on licenses and auctions these off, what will be the price per license?

(g)What does each of the above solutions to the Tragedy of the Commons share in common?

(h) Legislators who represent political districts (such as Congressmen in the U.S. House of Representatives) can be modeled as competing for pork barrel projects to be paid for by the government budget. Could you draw an analogy between this and the problem faced by fishermen competing for fish in a public lake? (This is explored in more detail in end-of-chapter exercise 28.2 in Chapter 28.)

B. Let N denote the total number of fishing nets used by everyone and X = f (N) = AN^α the total catch per week. As in exercise 21.6, let r be the weekly rental cost per net, let p be the market price for fish and let A > 0 and 0 < α < 1.

(a) The lake is freely accessible to anyone who wants to fish. How much revenue does each individual fisherman make when he uses one net?

(b)What is the loss in revenue for everyone else who is fishing the lake when one fisherman uses one more net?

(c) Suppose that each fisherman took the loss of revenue to others into account in his own profit maximization problem when choosing how many nets n to bring. Write down this optimization problem. Would this solve the externality problem?

(d) A Pigouvian tax is optimally set to be equal to the marginal social damage an action causes when evaluated at the optimal market level of that action. Evaluate your answer to (b) at the optimal level of N to derive the optimal Pigouvian tax on nets.

(e) Suppose that all fishermen just consider their own profit but that the government has imposed the Pigouvian per-net tax you derived in (d). Write down the fisherman’s optimization problem and illustrate its implications for the overall level of N. Does the Pigouvian tax achieve the efficient outcome?

(f) Suppose the government privatized the lake and allowed the owners to charge per-net fees. The owner might do the following: First, calculate the maximum profit (not counting the rental value of the lake) he would be able to make by simply fishing the lake himself with the optimal number of nets—then set the fee per net at this profit divided by the number of nets he himself would have used. What per-net fee does this imply?

(g) Compare your answer to (f ) to your answer to (d). Can you explain why the two are the same?

(h) Suppose A = 100, α = 0.5, p = 10 and r = 20. What is the optimal Pigouvian (per-net) tax and the profit maximizing per-net fee that an owner of the lake would charge?