Microeconomics,, 16th Canadian Edition

but note that and , where L is the

quantity of the variable factor used and w is its cost per unit.

Therefore,

Because w is a constant, it follows that AVC and AP vary inversely
with each other, and when AP is at its maximum value, AVC must
be at its minimum value.
18. A little elementary calculus will prove the point:

If w does not vary with output,

However, referring to note 15 (Equation 15.3 ), we see that

Thus,

TVC=L⋅w Q=AP⋅L

AVC= ALP⋅⋅wL = AwP

MC= ddTQC = dTdQVC = d(LdQ⋅w)

MC=ddQL ⋅w

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ddQL =M^1 P

MC= MwP