Dynamic programming and Bellman Equation

We consider the following maximization problem

where

for t=0,4,8,12,. . .

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

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

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

where we consider switch structure of leisure(or labor) quarterly.

The corresponding Bellman equations are

We guess

Then we have

FOC’s are

and

and

and

and

for each state.

for state 1 (*)

for state 2 (*)

for state 3 (*)

for state 4 (*)

On the other hand

for state 1 (**)

for state 2 (**)

for state 3 (**)

for state 4 (**)

from yesterday’s results with (*) and (**) above. For 1*’s

for state 1

for state 2

for state 3

for state 4

The Bellman equations are now going to be

They are

Arranging terms and comparing the coefficients,

So

that is the same as yesterday and before without labor.

The rest is

where we can solve it numerically by plug them in order as in 0518.

We solve out and we can say that the guess is right.

The policy functions are

for state 1

for state 2

for state 3

for state 4

Here, from

for state 1

for state 2

for state 3

for state 4

where they are constant.

Once we solve l^*’s together with other parameters, we can get k’ for each state.