Euler Equation

We consider the following maximization problem

subject to

adding labor. In addition, we have

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

So far, we have almost the same form as what we did in 0627.

We define that F’(K,L) is the derivative of F(K,L) with respect to K.

When , it is going to be

At steady state, and

So

That is

Thus, for a certain level of L,

evaluated at some Q and A.

On the other hand,

FOC w.r.t. L:

So we solve

and

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

When L*=1, we have the same value of K* as in 0627.