Dynamic programming and Bellman Equation

We consider the following optimization problem

subject to

where we assume that and .

Here, is unit labor and its total time is normalize to one. That is,

is time for leisure. For simplicity, we use the same utility function u(1-l) as u(c).

It is weighted by compared to u(c).

We convert it into the following Bellman equation

at time t

We guess

that is the same as what we used without labor.

Then we get

Its first order conditions with respect to k_t+1 and respectively are

and

from the derivative of logarithm and the function of function rule.

So

(*)

So

(**)

The Bellman equation is now going to be

Arranging terms and comparing the coefficients,

So

The rest is

We have solved out. The guess is right. The value function is .

From FOC’s,

Thus

that does not depend on k.

Thus