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Question 1

1: PV with coop

Answer 1

We assume that fishing occurs in periods and that the stock in t, R_t, depends on how much that was left in the previous periods (S_(t-1):

(1) R_t = S_(t-1) + G(S_(t-1))

a) Coop, every country leave behind eco optimal fish stock S^o, b) and fish from that optimal stock G(S^o).
c) optimal stock and its sus yield produce profit per period π^o(S^o)

The PV of profits for each country will be

(2) V^o = (π^0(1+r)) / (rN)

Question 2

2: One country abandon

Answer 2

Now, one country abandons. The country fishes down to S* instead of S^o. This is only found out when the period is over.
The best response from the other countries is to do the same.
Stock level S* will produce a profit of π(S)
The PV for each country is
(3) V* = (π(S))/(rN)

Question 3

3: When is it profitable to break the cooperation?

Answer 3

Breaking the cooperation is profitable if
V* + D > V^o
Where D is the profit the country gets in the period it deviates from the cooperative agreement. This is
D = [π^o(S^o))/N] + T (S^o + S*).

In the first part, the country gets its share of the cooperative profit that is π^o/N. In addition, it benefits T from fishing down the stock from S^o to S. This is solely for the deviating country. So, for it to be profitable:
T (S^o – S) > (π^o(S^o) – π(S))/rN
The one-time gain T must be greater than the PV of the difference between the cooperative an non-cooperative profits from the period after the deviation. This can specially happen if r or N is big enough.