Question

In class we considered TC(q) = q^2 + 4. AFC(q) is always decreasing, and AVC(q) is always increasing. Thus, two forces affect average cost: AC(q) = AVC(q) + AFC(q). Is it true that AVC(q) = AFC(q) at the minimum of AC(q) (the two "balance each other")? either prove the result for an arbitrary TC(q) function, or find a counterexample.

Answer 1

AC = TC/q = q + 4/q

Min AC means derivating AC with respect to q and putting it equal to 0.

Thus,

1 - 4/q^{​​​​​​2}​​​​​ = 0

This makes q = 2

Now,

AFC = 4/q = 4/2 = 2

AVC = q2/q = q = 2

Thus, it is true that at minimum of AC, AVC = AFC