Question

It turns out that the effect of a change in the savings rate on the steady-state consumption is ambiguous. Let's try to show this with the power of algebra! Let A=2, L=100, and d=1/4 but do not pick a value for s. Solve for the steady-state values of K, Y and C by just leaving an "s" in there. This means that you will get functions of s as your answer! You should be able to look at K*(s) and Y*(s) and see easily that they will be increasing in s. C*(s), should be a parabola. Economists call the savings rate which maximizes this function the "golden rule" savings rate. You will learn how to find it explicitly in intermediate macro (you have to use calculus). Notice what the value of C* is if s=0 and what it is if s=1.

Answer 1

Let .

Per capita production,

or,

At steady state,

or,

or,

or,

or,

Thus,

Now,

So,

or,

So, steady state values of C, K, Y are functions of s.

To find golden rule of steady state, maximize with respect to .

or,

This is golden rule savings rate.

When ,

When ,