Dynamic programming and Bellman Equation

We consider the following maximization problem with log utility function

subject to and .

We have the following switching parameters

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

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

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

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

for each quarter.

We can construct the following two Bellman equations

where value functions appear in order and go back to the first one.

We just guess

That is

Now we get FOC w.r.t. respectively. They are

So

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

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

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

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

Plugging these four into Bellman equations at optimum, we get

They are

Comparing coefficients in each kind, we get

So

where the slope coefficients in value functions are the same.

The rest is

They are

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

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

From the FOC’s, we have the following policy functions

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

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

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

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

as in yesterday’s.