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,. . .

that is an application as in 0516.

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

so

for state 1 (**)

Similarly

for state 2 (**)

The Bellman equations are now going to be

and

They are

and

Arranging terms and comparing the coefficients,

and

so

The rest is

Thus

G is also solved after we get E. We have solved out and the guess is right.

The policy functions are

for state 1

for state 2

Here

so

for state 1

Similarly

for state 2

Finally we get

for state 1

for state 2