Dynamic programming and Bellman Equation

We consider the following optimization problem

subject to

We convert it into the following Bellman equation

at time t

We guess .

Then we get

Its first order condition with respect to k_t+1 is

(F.O.C)

because there is no term involved in k_t+1 on the left hand side.

From the function of function rule,

The derivative of with respect to k_t+1 is times –1 inside.

That is,

Substituting it back to Bellman equation in terms of guess form,

Arranging the same kinds of terms, we compare the coefficients and get

and

That is

and

where we have all solved out. That guess is right.

Value function is

The phase (k,k’) can be expressed as

from FOC

or simply