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

That is the most basic model.

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.

.

.

.

or

.

.

.

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

We have this policy function in one iteration.

If we do not get to any generalization in one iteration, we repeat this process.