Value function iteration

We consider the following maximization problem

subject to

where

and

for

We consider the following Bellman equation

We start from E[V0|theta]=0 and k’=0 that are the same as before.

We have a set of FOC’s with respect to k’ and l as well.

When j=0

We set k’=0 initially.

FOC w.r.t. l is

that is always the same while k’ is changing inside. That is

where we have to solve nonlinear equation for l*.

Since and k’=0

We can express it as

When j=1

That is

Since

FOC w.r.t. k’ is

that is similar to the case without labor except . So we have

FOC w.r.t. l is

that is always the same while k’ is changing. In this case, plugging k’ into this

where we have to solve nonlinear equation for l*.

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

that is the same as the case without labor.

When j=2

That is

Since

FOC w.r.t. k’ is

that is similar to the case without labor except . So we have

FOC w.r.t. l is

that is always the same while k’ is changing. In this case, plugging k’ into this

where we have to solve nonlinear equation for l*.

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

that is exactly the same form as that without labor.

.

.

.

We go on and we can generalize in the limit.

Policy functions of k’ and l are

That is going to be a constant to solve the following

that does not depend on k.

when j=n.

That is going to be

Value function would be where

Here, we can regard in the limit so we rewrite it as

where c2 and c3 are the same as what we got yesterday for log utility for leisure.