Howard’s policy iteration(policy improvement algorithm)

We consider the following maximization problem

subject to

where we assume that

and

We convert it into the following Bellman equation

We firstly guess the policy function as

for

We then calculate the value of k at time t in terms of the value of k at time 0.

For some labor

.

.

.

where c is some constant.

We secondly plug this relation into the following Value function

That is

Thus

that we calculate this beforehand.

We then consider two period problem in Bellman equation

That is

FOC w.r.t. k’ is

Thus

On the other hand, FOC w.r.t. is

Plugging the result of k’ above into this FOC, we get

Thus

Plugging it back to k’, we get

or

That’s all in one iteration.