Dynamic programming and Bellman Equation

We consider the following maximization problem with log utility function

subject to and for simplicity.

There is no expectation but we have two kinds of states alternately. That is

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

and

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

We can construct the following two Bellman equations

and

where value functions appear alternately.

We just guess

and

That is

Now we get FOC w.r.t. respectively. They are

(*)

and

(**)

Plugging (*) and (**) into Bellman equation at optimum, we get

They are

Comparing coefficients in each kind, we get

So

Similarly

The rest is

and

So

We have solved out and we can say that the guess is right.

From (*) and (**), we have

and