Policy function calculation from value function iterations

Algorithm:

(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.

(4) Repeat (3) by setting v=v1 for many times or until some criterion is met.

(5) Find the final corresponding value of k as k’ according to the maximum value.

How to set U matters. Initial value of v is any value such as v=0.

We construct a maximum possible grid around steady-state value of capital.

We can roughly compute policy function of capital depending on the number of points

on the grid.

Today we consider the basic AK model and its Bellman equation:

where and

The case with

Convergence of value functions(from the bottom to the top)