Policy function calculation from Euler equation

We use Euler equation

with CRRA utility function

that leads to ln(x) as .

Then, we have

since

In steady state, we have

or

so the steady-state level of k will not change.

The paths to the steady-state level change.

We have to find so as to satisfy

given and . That is,

We repeat it from and .

Algorithm:

(1) Set Euler equation:

　and

(2) Find k* such that as steady-state level of k.

(3) Calculate given and from and up to t=n and save it.

(4) Change and find the best leading to .

(5) Change and repeat (3) and (4) to recover policy function of k.

Behaviors of policy functions of k according to

where the policy function also approaches that for ln( ) as .