Economics
Q 1) Find the time t for maximum present value of the profitt
Q 2) what is then the optimal harvesting time (without rotatation, max profit).
given info;
growth function for the fish, with t as number of years:
\(w(t) = 4.8t{}^{2}-1.52t{}^{3}\)
Price function:
\(p(w)= 30 + 0.45 w(t)\)
Number of fish; R = 1
mortality rate for the fish m = 0.1
Bank intrest r = 0.05 per year or 0.0042 a month
Feed costs: Cf = 11 per kg, feed factor ft = 1.1
useage of food per fish F(t) = ft * w'
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Optimizing porblem I have been using ;
(but i dont know if this is for rotation or not)
\(Max π(t)=V(t){e}^{-rt} - \int_{0}^{t}C{}_{f}F(u)R{e}^{-(M+r)u}du{}_{}\)
with
\(V(t)= p(w)*B(t)\)
\(B(t)=Re{}^{-mt}*w(t)\)
\(π'(t)=V'(t){e}^{-rt} - rV(t){e}^{-rt}-C{}_{f}F{}_{t}w'(t)Re{}^{-(M+r)t}=0\)
rewritten as
\(\frac{p'(w)}{p(w)}*w'(t*)+\frac{w'(t*)}{w(t*)}=r+m+\frac{{C}_{f}F(t*)}{p(w)w(t*)} \)
