Question

This is my own question because I want to understand this as best as possible.

ATC = AFC + AVC

1) Can ATC increase if AFC decreases?

2) Can ATC decrease if AVC decreases?

3) If you could add anything else I would really appreciate it. I want to know how this average equation works.

Answer 1

(1)

ATC = AFC + AVC

AFC = TFC/Q where Q = quantity. TFC = Fixed cost is constant for all level of Output and hence d(AFC)/dQ = -TFC/Q² < 0, this implies that AFC is always decreasing.

In order for ATC to be increasing

d(ATC)/dQ = d(AFC)/dQ + d(AVC)/dQ  > 0 => -TFC/Q² + d(AVC)/dQ  > 0 => d(AVC)/dQ  > TFC/Q²

Hence ATC can increase If d(AVC)/dQ  > TFC/Q².

Example:

^{TC = 1 + Q2}

where TFC = 1 and TVC = Q2 here AFC =1/Q => d(AFC)/dQ = -1/Q² and d(AVC)/dQ = 2Q

Hence ATC will be increasing if 2Q > 1/Q² => Q > 2^1/3

Hence Yes ATC can increase if AFC decreases

(2)

ATC = AFC + AVC

AFC = TFC/Q where Q = quantity. TFC = Fixed cost is constant for all level of Output and hence d(AFC)/dQ = -TFC/Q² < 0, this implies that AFC is always decreasing.

Suppose AVC decreases => d(AVC)/dQ < 0

In order for ATC to be increasing

d(ATC)/dQ = d(AFC)/dQ + d(AVC)/dQ  > 0

But as discussed above, d(AFC)/dQ < 0 and d(AVC)/dQ is also lesser than => d(AFC)/dQ + d(AVC)/dQ < 0.

Hence d(ATC)/dQ < 0 and hence ATC is always decreasing in this case.

Hence No, ATC cannot increase if AVC decreases.

(3)

AFC is always decreasing as discussed above and because AFC is decreasing => ATC - AVC = AFC is also decreasing and hence As Q increases AVC and ATC gets closer to each other i.e. ATC - AVC is decreasing but ATC and AVC will never intersect because AFC = TFC/Q is not equal to 0 because TFC is not equal to 0