Policy function calculation from Euler equation

We use Euler equation

as in 0627 and 0708. We assume

that is CRRA utility function. As, it approaches ln(x).

We do not know policy function of K when.

In steady state, we have

or

Thus

　　for some Q and A that is not changed at all.

We have to find so as to satisfy

Powered by , we have

Thus

starting from and evaluated at some Q’s and A’s.

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 in CRRA utility function