Value function iteration

We consider the following optimization problem

subject to

where we assume that and as in 0507.

The corresponding Bellman equation is

We start with and k’=0 as in what we did the day before yesterday.

When j=0

Since and k’=0

On the other hand, FOC w.r.t. l is

When k’=0

So

and we express

in terms of l*. We set it as

When j=1

That is

FOC w.r.t. k’ is

So

Plugging it into Bellman equation at optimum when j=1

On the other hand, FOC w.r.t. l is

that is unchanged but k’ is changing. Plugging k’, we get

That is going to be

We can express as

in terms of l*. We can set

When j=2

That is

FOC w.r.t. k’ is

Plugging it into Bellman equation at optimum when j=2

On the other hand, FOC w.r.t. l is

Plugging k’, we get

That is going to be

We can express

in terms of l*. We can set

.

.

.

We repeat this and we get

and

by induction when j=n.

Since when 0<a<1

Thus

On the other hand,

Therefore

that is exactly the same as what we got in 0507.

In the same way we get

where we got the same things in the method of guess and verify.

The calculation of E is similar to yesterday’s.

When j=0

When j=1

When j=2

When j=3

.

.

.

When j=n+1

Since for 0<a<1 when we consider infinitely many terms

That is

Here, we can regard and it converges so we rewrite it as

Thus

where l* also approaches some constant value.