Value function interation help

2020 11 11
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Hi,
I have a task to solve a utility optimization problem with dynamic programming using value function interation.
The task is quite similar to the classic cake eating problem, which there exist codes on, but I cant seem to work it out. Here are the information:
Utility function:
subject to:
You can choose the variables freely if you want, but in the problem Beta = 0.96, n=2 and alpha = 1/3
All help would be appriciated.

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