Policy function calculation from Euler equation

We use Euler equation with labor

with CRRA utility function

that leads to ln(x) as .

Then, we have

and

from FOC w.r.t.

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

(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.

Effect of depreciation for behaviors of policy functions