Dynamic programming and Bellman Equation

We consider the following optimization problem

subject to

where we assume

We convert it into the following Bellman equation

at time t

We guess .

Then we get

F.O.C. w.r.t. c_t is

We solve this for c.

Finally, we get the following policy function

in terms of A.

To identify A, we set for simplicity.

We substitute

back into Bellman equation.

since .

Comparing coefficients, we get

or

Finally we get the coefficient

Value function is

The policy function is

after replacing A by the resulting coefficient.