Euler Equation

We consider the following optimization problem

subject to

where we assume that and .

Now follows a certain random process.

We convert it into the following Bellman equation

where

where is also a state variable.

Now we calculate Euler equation.

(1) FOC w.r.t. k’:

or

(2) Envelope condition(FOC w.r.t. k):

(3) Shift the envelope condition ahead by one period:

(4) Substitute the first one into this:

Thus

So far, we have almost the same form as yesterday’s.

When , it is going to be

At steady state, and and .

So

That is

Thus

evaluated at a certain.

On the other hand, from FOC w.r.t. ,

since .

Thus

evaluated at a certain

where we have to solve it for together with above simultaneously.

If we assume

for

as in 0603, we have

and

because in the limit and therefore in the limit

Recall that

shown in 0611.