Howardfs policy iteration(policy improvement algorithm)
We consider the following maximization problem with log utility function
subject to 
in aggregate variables as in 0515.
Suppose Lt=1 all the time for simplicity. In addition, we have
and
where 
and 
are error terms
with mean zero.
On the background, we assume
when 
The Bellman equation with expectation is
We firstly set a feasible policy function as
for 
.
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
Thus
that we we want and it is the same as what we got in 0515.
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.