Euler Equation

We consider the following optimization problem

subject to

where we assume that and .

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

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

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

since .

Recall that the derivative of logarithm is reciprocal of inside the logarithm times

the derivative of the term inside.

Thus

evaluated at some .

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