Value function iteration

We consider the following maximization problem

subject to

We introduce technology shock that is a random process. That is

where

for

We consider the following Bellman equation

We start from E[V0|theta]=0 and k’=0.

When j=0

Since and k’=0

When j=1

That is

Since

FOC w.r.t. k’ is

So

Plugging it into Bellman equation at optimum when j=1

Arranging the first term on the right hand side, we get

In term of k and theta, we can express it as

We can express it as

We can repeat this.

When j=2

That is

FOC w.r.t. k’ is

So

At least we can calculate k’ in the same way and we get

when j=n.

That is going to become

Thus

That’s what we got in 0511 by the method of guess and verify. That is not difficult.

However, the generalization of value function is a little bit complicated.