Dynamic programming and Bellman Equation

We consider the following maximization problem

where

for t=0,2,4,6,. . .

and

for t=1,3,5,7,. . .

where we now switch utility function from leisure with weight.

The corresponding Bellman equations are

and

We guess

and

Then we have

and

FOC’s are

and

and

and

for each state.

for state 1 (*)

for state 2 (*)

For state 1

at optimum

For state 2

at optimum

from the results in 0508 with (*) and (**) above. For l¹* and l²*

for state 1 (*)

for state 2 (*)

The Bellman equations are now going to be

and

They are

and

Arranging terms and comparing the coefficients,

and

so

that is the same as yesterday.

The rest is

and

Thus

We have solved out and the guess is right.

The policy functions are

for state 1

for state 2

Here, from

for state 1

and

for state 2

where they are constant.

Once we solve l^1* and l^2* together with other parameters, we can get k’ for each state.