Policy function calculation from value function iterations

We consider the following switching model consisting of four states:

for t=1,5,9,13,. . .

for t=2,6,10,14,. . .

for t=3,7,11,15,. . .

for t=4,8,12,16,. . .

We then consider the corresponding set of Bellman equation:

where value functions on the right hand side are switched. Here,

In this case, the movement of capital soon gets into four-cycle process.

The points of four-cycle process are usually not steady-state points. For more states,

we can do the same thing.

Behavior of capital

The corresponding behavior of consumption

where the movement get into some cycle with four points after a few steps.

They are usually not stated-state levels.