Dynamic programming and Bellman Equation

We consider the following maximization problem with CRRA utility function

subject to and

where is random return that is strictly positive.

We can construct Bellman equation

at time t

or

at time t

We guess

to find constant A for.

Plugging guess form into Bellman equation

or

where we can factor out non random variables from expectation operator.

FOC w.r.t. st:

So

Suppose

So

This way lead to some complicated calculation.

We use another relation called envelope condition.

That is

FOC of Bellman equation w.r.t. Wt is

From

we have

So

Define

for

Plugging st back to Bellman equation, we get

Comparing coefficients, we get

Thus

or

where we solve out and we can say that the guess is right. Value function is

Policy function of st has already been solved from envelope condition.

Once we factor out non random variables from expectation operator,

We can treat as a constant as it is.