Euler Equation

We consider the following maximization problem

subject to

in aggregate variables with any depreciation

We have the same random process of Q and A as

and

where and are error terms with mean zero.

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 with labor L, it is going to be

At steady state, and

So

That is

Thus

for some L evaluated at some Q and A.

On the other hand, we have another FOC w.r.t. L. That is,

FOC w.r.t. L:

So we solve

and

for L* and K* given Q and A simultaneously.