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We maximize the present value by maximizing g(N) - SN over N; the condition for maximization is (d/dN){g(N)- SN} =0 so that (d/dN)g(N) = g'(N) = S. In fact, if you look back to the previous section, just above Exercise 6.7 and to Figure 6.7 you see that this is basically the same kind of condition that we had previously reached: the present value is maximized when the stock size is such that the tangent line of the biological growth curve has slope S (Figure 6.9a). Since we know that g(N) is a decreasing function of N, we recognize that this argument makes sense only if g(0) > S. But what if that is not true, as for example in the case of whales or rockfish, where g(0) ~ r may be 0.04-0.08 and the discount rate may be much higher (say even 12% or 15%)? Then the optimal behavior, in terms of present value, is to take everything as quickly as possible (drive the stock to extinction). This result was first noted by Colin Clark in 1973 (Clark 1973) using methods of optimal control theory. In his book on mathematical bio-economics (Clark 1990, but the first edition published in 1976) he uses calculus of variations and the Euler-Lagrange equations, and in his 1985 book on fishery modeling (Clark 1985), Colin uses the method of integration by parts that we have done here.

In a more general setting, we would be interested in discounting a stream of profits, not harvest, so our starting point would be

where p is the price received per unit harvest and c(N) is the cost of a unit of harvest when stock size is N. The same kind of calculation leads to a more elaborate condition (Clark 1990).

There is yet another way of thinking about this question, which I discovered while teaching this material in 1997, and which led to a paper with some of students from that class (Mangel et al. 1998) and which makes a good exercise.

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