Howard’s policy iteration(policy improvement algorithm)

We consider the following optimization problem

subject to

where we assume

and

as in 0504. The corresponding Bellman equation is

We firstly set a feasible policy functions. That is, if we set

for

the other is automatically determined as

.

As in 0508, we have

if we regard c as some constant or irrelevant to . Then we can set

in the same way as what we did in 0608. Now the part of constant is partly a function

of but we ignore it.

That is

We consider the FOC with respect to k’ only. That is

Thus

That is exactly the same result as in 0608.

Once we get , the other policy function of c is

They are all we want. We do not have to consider the FOC w.r.t. c at all.

This argument will be applied in the same way to the model with labor and with both

labor and technology change.