Euler Equation

We consider the following maximization problem

subject to

in aggregate variables as in 0515 and 0616.

Suppose Lt=1 all the time for simplicity. In addition, we have

and

where and are error terms with mean zero.

On the background, we assume

for any

The Bellman equation with expectation is

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

When , it is going to be

At steady state, and

So

That is

Thus

evaluated at some Q and A.