Problem 3: Regulating the Fishery [3 points]

As above, let's assume that a tuna stock at the beginning of time t, denoted by the variable Xt, grows according to the difference equation Xt+1 = Xt + F(Xt) Ht, where the net-growth function is F(Xt) = Xt(1 Xt) and the harvest of tuna Ht is determined by Ht(Xt, Et) = qEtXt, where Et is the amount of industry fishing effort. Also assume that fishermen are price-takers in the market for tuna, and thus receive the market price p per unit of tuna. The cost of harvesting tuna is C(Et) = cEt.

A. Suppose that you are the sole owner of the tuna stock and your goal is to choose a level of effort that maximizes steady-state profits ss from the fishery.

i) [0.5 points] Write down the objective function for the sole-owner's steady-state profit maximization problem.

ii) [0.75 points] Determine the level of effort E that maximizes steady-state profits ss. Be sure to explain how you found E .

iii) [0.5 points] How does the optimal amount of effort E compare to the steady-state open access level of effort EOA that you found in Problem 2? Why are they different?

B. Now suppose you are the manager of an open access tuna fishery and you have the option to impose a harvest tax on tuna.

i) [0.75 points] Suppose fishermen have to pay a tax of t for every unit of tuna that they catch. How would you choose the tax level t to maximize the tuna fishery rents? Find a value for t if c = q = 0.01 and p = 3.

ii) [0.5 points] Explain the practical limitations of managing the tuna fishery using a harvest tax.