You are hired by the Ministry of Natural Resources of your favorite country and are entrusted with the management of the yellowstripedbigmouthcod, henceforth called \"the sh.\" Let mt denote the sh stock at the beginning of period t. Given this sh stock, 56?, a E (0, 1), new sh will be born, and only a fraction 1 a E (0, 1) of the beginningofperiod sh stock will survive into the next period. Let yt denote the quantity of sh caught in period t. 1. (15 points) Describe the transition equation which describes the evolution of the sh stock over time. 2. (15 points) Suppose you impose a constant quota for the catch each year, Q'. Derive an expression for the maximal sustainable (i.e.; steady state) catch. You are an economist and know that catching sh cost resources. Let C(zt, yt) denote the cost of catching 3;, sh when the beginning of period sh stock is mt. The function C(mt, gt) is increasing in yt, innitely dif ferentiable and jointly convex in (1.9,, gt). Assume that C(z, O) = 0 and lim,,_,00(z, y) = 00. The price of sh is normalized to 1. Suppose that you are employed for one year only, and that your salary increases with the prot the shery makes in that year. Greedy economist that you are, you want to maximize your salary. 3. (15 points) What is the prot maximizing policy? Will you ruin the shery (that is, catch all available sh)? Now you get a longterm contract, and you are told to maximize the discounted present value of current and future prots of the shery. The time discount factor 5 6 (0,1) is constant, the price of sh is constant at 1, and the cost of catching sh is given by the function 0 dened above. 4. (15 points) Dene the dynamic optimization sequence problem. 5. (15 points) Show the functional equation associated with this optimization problem. 6. (15 points) Characterize the steady state of the solution to the dynamic optimization problem. Suppose you are no longer the single manager of the shery, but there are two eet operators catching sh. Let ya denote the catch of the i-th operator, 2'. = 1, 2. Assume that the cost function for the -th operator is C(mt) ya, that is, there is a constant marginal cost to her own catch, but the cost depends on the total sh stock. Assume that C(m) -'y is increasing in y, decreasing in x, and jointly convex in (my). 7. (10 points) Suppose the -th operator takes the current and future actions of the other operator as given (for example, operator 1 takes {y2,t}':0 as given). Characterize the optimal policy of the ith operator. EXTRA CREDIT (10 points) Consider a Nash equilibrium Where each operator takes the cur- rent and future actions of the operator as given. Characterize the steady state of this Nash equilibrium