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(1Xt) and the harvest of tuna Htis determined by Ht(Xt,Et) =qEtXt, where Etis 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.

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 ssfrom the fishery.

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

ii)Determine the level of effort Ethat maximizes steady-state profits ss. Be sure to explain how you found E.

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

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

iv)Explain the practical limitations of managing the tuna fishery using a harvest tax.