Policy function calculation from Euler equation

We use the same Euler equation with labor

with CRRA utility function

that leads to ln(x) as .

Then, we have

and

from FOC w.r.t.

with technology change parameter that is different from denoted as th in CRRA.

In steady state, we have

or

given

We have to find so as to satisfy

given and . That is,

We repeat it from and .

Algorithm:

(0) Calculate constant labor from

　 evaluated at some.

(1) Set Euler equation:

　

and

evaluated at some.

(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.

Behavior of policy function of k according to technology change