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smith007
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Homework Statement
Trying to maximize the profit of a farmer using dynamic optimization. Each period the farmer has a stock of seeds. He can plant them at a cost c per seed or sell them for p. Every seed that is planted produces [itex]\gamma[/itex] seeds for next period. In period m there is no longer any demand for the seeds.
Profit function [itex]\pi[/itex] = p(1-[itex]\alpha[/itex]t)[itex]\gamma[/itex]xt - c[itex]\alpha[/itex]t[itex]\gamma[/itex]xt where [itex]\alpha[/itex]t is the proportion of seeds kept to sow at the end of the period.
We are trying to maximize [itex]\sum[/itex][[itex]\pi[/itex]t / (1+r)t] from t=0 to m-1
Initial stock of seeds is x0 = 1
[itex]\gamma[/itex] = 8
c = 3
p = 1
r = 0.1
m = 3
Homework Equations
Bellman Optimization.
The Attempt at a Solution
Define [itex]\beta[/itex] as 1/(1+r)t
Motion rule xt+1 = [itex]\gamma[/itex][itex]\alpha[/itex]txy
We know that there is no demand for the seeds in period 4 so x4 = 0 = [itex]\gamma[/itex][itex]\alpha[/itex]3x3
This means that x4 = 0
The value function V3 becomes:
V3 = p[itex]\gamma[/itex]x3
@ t = 2
V2 = max (p(1-[itex]\alpha[/itex]2)[itex]\gamma[/itex]x2 - c[itex]\alpha[/itex]2[itex]\gamma[/itex]x2 + [itex]\beta[/itex]V3
Which using motion rule gives
V2 = max (p(1-[itex]\alpha[/itex]2)[itex]\gamma[/itex]x2 - c[itex]\alpha[/itex]2[itex]\gamma[/itex]x2) + [itex]\beta[/itex](p[itex]\gamma[/itex][itex]\alpha[/itex]2[itex]\gamma[/itex]x2)
Normally at this point I would differentiate and to find the maximum and then recurse the answer back into t=1 but it is a linear function. So I am guessing I need to take some sort of corner soluition but I am not entirely clear how to proceed.
Any tips would be welcome. Thank you.