Dynamic programming and Bellman Equation

We consider the following maximization problem with log utility function

subject to

in aggregate variables. Suppose Lt=1 all the time for simplicity. In addition, we have

and

where and are error terms with mean zero.

On the background, we assume

when

When , it is unable or difficult to apply the method of guess and verify.

The Bellman equation with expectation is

at time t

We guess

Then we get

Now we take the derivative with respect to. That is

So

We plug this back into Bellman equation at optimum and we get

Here

and

Now we compare the coefficients in each kind.

Arranging the terms, we get

Thus, the value function is

From the first order condition

we have the following policy function

(*)

since