Dynamic programming and Bellman Equation

We consider the following maximization problem with log utility function

subject to and without labor as in 0522.

Today we generalize yesterday’s model and consider n states. That is

for state1

for state 2

.

.

.

for state n

The corresponding n Bellman equations are

.

.

.

where

.

.

.

and each probability lies between 0 and 1.

We just guess

.

.

.

On the analogy of yesterday’s results, we have

for state 1

for state 2

for state 3

.

.

.

for state n

while coefficients in value functions are given by

and

as in ME=b. That is

and