Dynamic programming and Bellman Equation

We consider the same optimization problem as yesterday’s

subject to

but the form of production is now somewhat different.

We introduce technology shock that is a random process. That is

where

for

We can construct Bellman equation

at time t

where the former theta does not have expectation because it is known at time t.

We guess

Taking the derivative with respect to k_t+1, we get

So

(*)

The Bellman equation is now going to be

Comparing coefficients for the same kinds, we get

Therefore

Since

We have solved out so we can say that the guess is right.

From FOC,

Thus

where the schedule of (k,k’) shifts up or down depending on technology shock.