Dynamic programming and Bellman Equation

We consider the following optimization problem

subject to

where we assume that and .

We set a general function h( ) including weight.

This function is continuously differentiable, strictly increasing, and strictly concave.

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

Using yesterday’s results, we get

at optimum

and

using .

Suppose that we get the optimal level of labor l*. The Bellman equation is going to be

Comparing coefficients for the same kinds, we get

So

that is the same as what we got yesterday.

On the other hand,

Since

Since

Thus

in terms of l*.

We solved out and we can say that the guess is right.

Once the optimal labor l* is determined,

is also identified.

The optimal level of capital is determined at , so

Thus