Howardfs policy iteration(policy improvement algorithm)
We consider the following maximization problem with log utility function
subject to 
where 
and 
.
The Bellman equation with expectation is
We firstly set a feasible policy function as
for 
.
Consequently, we have
as in what we set yesterday.
Then, we can have
We also have
due to the same kind of process mentioned before.
The Bellman equation between the first two period is
FOC w.r.t. Kf is
for some L.
Thus
On the other hand, FOC
w.r.t. L is
Substituting Kf into
this, we get
Thus
Thus
in terms
of 
.
We also have
The detailed calculation of initial value function is as follows:
.
.
.
where
starting from 
and 
when t=1.
Here,
where
starting from 
and 
when t=1.