Howard’s policy iteration(policy improvement algorithm)

We continue to consider the following maximization problem

subject to

where we assume that

and

where

for

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 .

.

.

.

where

and

starting from and when t=1 for some .

The rest is almost the same as what we did yesterday.

That is

Thus

where

where

and

starting from and when t=1 for some

and

Here, can be calculated in the following way

.

.

.

As , it will become

that’s what we intend to set.