Policy function calculation from value function iterations

We consider the following maximization problem:

subject to .

Here, we use the following CRRA utility function:

We have the following switching parameters

for t=0,2,4,6,. . .

and

for t=1,3,5,7,. . .

We then consider the corresponding set of Bellman equation:

and

where value functions on the right hand side are switched. Here,

and .

Notice that grid method is valid only for a single value function.

We had better compute each policy function separately.