Value function approximation and collocation method

Algorithm based on modified policy iterations:

(1) Set a grid consisting of k and k' 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 k’ from (3) in value function iterations.

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

(c) Find parameters for f( ) in f1(k)=U+beta*f(k) when f1(k)=k(k) and calculate vp=f(k) 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 k as k' according to the maximum value.

When we assume the following polynomial:

we have porder+1 parameters to be estimated in each step to calculate vp to the next.

Technically, we may have to change the order of polynomial and we have to eliminate

some initial iterations from finding parameters.

Behavior of approximation compared to the original result by direct inversion

5^th iteration:

6^th iteration:

7^th iteration:

8^th iteration:

9^th iteration:

where they are almost the same in the end.