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 is expectation.

We guess

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

and

Thus

(*)

(**)

The Bellman equation is now going to be

where we have since when we take off the expectation.

Comparing coefficients for the same kinds, we get

Therefore

Since

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

From one of FOC’s,

Thus

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