Dynamic programming and Bellman Equation

We maximize discounted sum of utility functions across time t

subject to the budget constraint

where we sometimes denote c_t and k_t as c and k and k_t+1 as k’ by dropping t.

We consider the following Bellman equation

at time 0

and

at time t

subject to and respectively.

For simplicity, we can use log utility function and Cobb-Douglas production function.

That is

and

where we consider

In this case, Bellman equations are

at time 0

and

at time t