Howardfs 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 
and it turns out to be
where
from FOC with respect to kf.
That is the same as yesterday and the day before yesterday.
Here, 
can be
determined by FOC with respect to 
.
Plugging the result of kf
above into this FOC, we get
where we have to solve it
for 
and that is
constant.
where
and
where
and
starting from 
and 
.