Question

At which price p is the demand given by D(p) = e^(-p) neither elastic nor inelastic?

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

D'(p) = (-p). e^-(p+1)

ed = D'(p) *p/q

When ed=0

p^2/ e=0

p=0

TC=q^2 +4

AFC=TFC/q = 4/q

AVC=TVC/q = q

AC= AFC+AVC

AC= q+4/q

Minimization of AC implies

dAC/dq =0

1-4/q^2 =0

q= 2

Second order condition is AC"(q) >0

8/q^3 >0

Hence proved