Suppose you find yourself the sudden owner of a coastal fishery with the following characteristics:

Nt+1 = Nt + B - D

So Nt = B - D

b', the per-capita birth rate = b - aNt

d', the per-capita death rate = d + cNt

B = b'Nt

D = d'Nt

b = 1.4; d = 0.6; a=0.005, c=0.003

Nt = rNt[1-Nt/K] where Nt is the population of a species of fish that we are interested in

harvesting. You can think of the units of N and K being in thousands. Feel free to round your

answers to the nearest whole number. Also, assume that in any given year, fish first reproduce and then

you harvest them.

a)

Suppose you decide to maximize your sustainable yield by maintaining the fish stock at

the level corresponding to the maximum growth of the population size.

a. What is this level of fish stock?

b. What is the annual sustainable harvest at this level of stock?

c. What would be the annual percent return (a.k.a. the yield) of this harvesting scheme?

b)

Suppose that the market price for this particular species of fish is $10 (and each fish is

sold at this price regardless of its size, etc.). Suppose the cost of harvesting fish is $2 per

fish and your discount rate is zero.

a. What are your profits in the current period if you harvest according to the maximum sustainable yield criterion?

b. What is your combined profit for this period and the next, assuming the price does not change from year to year?

c)

Now suppose that the discount rate is 3% and you still harvest the maximum sustainable yield in both years. At the end of this period, you put your profits from into the bank.

a. What is your total profit at the end of period 2?

d)

Now suppose you face the following two choices: proceed as described in part e) OR harvest 30 fish now, put that profit into the bank, and harvest 1/3 of the stock

after fish reproduction [ (1/3)*(Nperiod 2+ Nperiod 2) ] in period 2. Assume the stock is at 50 at the start of period 1 (i.e. before the fish reproduce and before the first harvest).

a. Which option will you now choose? (Assume you are a profit maximizer)

b. What is the difference in profit between the two options?

e)

[Use Excel or a similar spreadsheet program (e.g., Google Docs, which is free)to solve

this problem.] Now suppose that you want to maximize your profits over a twenty year window, and that the price of fish remains constant over all 20 years. You can either

harvest the maximum sustainable yield each year OR harvest 30 fish in period 1 and 1/3 of the stock [ (1/3)*(Nt+ Nt) ] in each subsequent year. In both scenarios, you putyour

profits in the bank at the end of the year, to accumulate interest over the next 20 periods.

a. Which option should you choose, and what is the difference in profit between the two options at the end of the twentieth year?

b. What is the level of stock after the twentieth year's harvest in each case?

c. If the discount rate were significantly higher or lower, would you still choose the same option? Explain briefly.