Economics,
How do i find the time t for maximum present value of profitt
growth function given; w(t) = 4.8t^2 - 1.52t^3
w'(t) = 9.6t - 4.56t^2
Price function; p(w) 30 + 0.45 w(t)
w' = 0.45
Number of fish; R = 1
mortality rate for the fish m = 0.1
r = 0.05 per year / 0.0042 a month
Feed costs: Cf = 11 per kg, feed factor f t = 1.1
useage of food per fish F(t) = ft * w'
And given the time t what is then optimal harvesting point without rotation (only one harvest).
Just dont know how to proceed.
Optimizing porblem i have been using ;
(but i dont know if this is for rotation or not)
\(Max π(t)=V(t){e}^{-rt} - \int_{0}^{t}C{}_{f}F(u)R{e}^{-(M+r)u}du{}_{}\)
\(π'(t)=V'(t){e}^{-rt} - rV(t){e}^{-rt}-C{}_{f}F{}_{t}w'(t)Re{}^{-(M+r)t}=0\)
with V(t) = p(w) * B(t)
And B(t) = Re^-mt * w(t)
Apparently can be rewritten as
\(\frac{p'(w)}{p(w)}*w'(t*)+\frac{w'(t*)}{w(t*)}=r+m+\frac{{C}_{f}F(t*)}{p(w)w(t*)}\)
^but I have not been able to get anything useful from it.
