Euler Equation

We consider the following optimization problem

subject to

where we assume that .　We now consider depreciation in capital.

We convert it into the following Bellman equation

where

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