Howard’s policy iteration(policy improvement algorithm)

We consider the following maximization problem with log utility function

subject to

where and

adding the utility from previous consumption with some weight.

The corresponding Bellman equation with expectation is

We firstly set a feasible policy function of K’ as

for .

Consequently,

Then, we can have

We also have

if we include all the terms related to in the constant.

The Bellman equation between the first two period is

FOC w.r.t. K’ is

Thus

that’s exactly the same as before in 0616. Parameter is not involved here.

Therefore, the policy function of C is

In the same argument as in 0616, we have

Here,

where

starting from and when t=1. The process of is exactly the same

as in 0616 because the policy function of K’ has the same form.