Value function approximation and collocation method

We generalize the algorithm in terms of x in the following:

(1) Set a grid consisting of x and x' columnwise and rowwise respectively.

(2) Calculate utility for consumption as U using the grid matrix above.

(3) Starting from a certain v, update v1=U+beta*v' so as to maximize v1.

(a) Find the corresponding x’ from (3) in value function iterations.

(b) Calculate utility from x and x’ as r.

(c) Find parameters for f( ) in f1(x)=U+beta*f(x) when f1(x)=k(x) and calculate vp=f(x) with the parameters we get.

(d) Repeat the whole thing by setting vp=vp1 until some criterion is met.

(4) Repeat (3) by setting v=v1 for many times.

(5) Find the final corresponding value of x as x' according to the maximum value.

Where we change all the variables related to k to x.

Behavior of capital on a single policy function to a steady state level