Dynamic programming and Bellman Equation

We consider the same optimization problem as yesterday’s

subject to

but we change the function h( ).

The schedule of capital

has exactly the same form as yesterday’s but l* is differently determined.

The optimal level of labor l* is derived from

that is the second one in FOC. Here,

that is also unchanged. That is

All we have to do is set the first derivative of h( ) and solve l* to satisfy the relation.

It is not difficult to find l* because the domain is limited. Use any tool or grid search.

(e.g.)　CRRA type function with weight

Then

where we find l* between 0 and . If we do not sleep and rest, .