In Cost Functions and Marginal Cost you saw marginal cost cross average cost at the bottom of the average cost curve. The The Quotient Rule shows that this is not a coincidence of the pictures but a fact about every cost function.
Key idea
$$AC'(q) = \frac{MC(q) - AC(q)}{q}.$$ So average cost falls while $MC < AC$, rises while $MC > AC$, and is lowest where $MC = AC$. The same holds for average variable cost and $MC$.
Why it works
By the quotient rule, $$\frac{d}{dq}\left(\frac{C(q)}{q}\right) = \frac{qC'(q) - C(q)}{q^2} = \frac{C'(q) - C(q)/q}{q}.$$ The intuition is the grade-point average: one more unit costing more than the current average pulls the average up, and one costing less pulls it down. At the bottom the marginal unit costs exactly the average. Setting $MC = 0$ finds where total cost is flat, which is a different (and usually meaningless) point.
Worked example: Quadratic cost
$C(q) = 100 + q^2$. Where is average cost lowest, and what is it?
- $AC = \dfrac{100}{q} + q$, so $AC'(q) = -\dfrac{100}{q^2} + 1 = 0$ gives $q = 10$.
- $AC(10) = 10 + 10 = 20$.
- Check: $MC = 2q = 20$ at $q = 10$. Equal, as the rule says.
Answer: $q = 10$, $AC = 20$.
Worked example: Average variable cost of a cubic
$C(q) = 100 + 10q - q^2 + 0.1q^3$. Where is $AVC$ lowest?
- $AVC = 10 - q + 0.1q^2$, so $\dfrac{d\,AVC}{dq} = -1 + 0.2q = 0$ at $q = 5$.
- $AVC(5) = 10 - 5 + 2.5 = 7.5$, and $MC(5) = 10 - 10 + 7.5 = 7.5$.
- This minimum is the shutdown price of The Shutdown Rule: below \$7.50 the firm can't cover its variable cost.
Answer: $q = 5$, $AVC = 7.5$.
Worked example: With a constant marginal piece
$C(q) = 9 + 10q + \tfrac{1}{4}q^2$. Find the lowest average cost.
- $AC = \dfrac{9}{q} + 10 + \dfrac{q}{4}$, so $AC'(q) = -\dfrac{9}{q^2} + \dfrac{1}{4} = 0$ and $q = 6$.
- $AC(6) = 1.5 + 10 + 1.5 = 13$, and $MC(6) = 10 + \tfrac{1}{2}(6) = 13$.
- The constant 10 per unit shifts both curves up but does not move the minimum: only the trade-off between spreading the fixed cost (9) and the rising part ($q^2/4$) does.
Answer: $AC_{\min} = 13$ at $q = 6$.
The bottom of average cost matters twice later in the course: it is the break-even price for a competitive firm (at a lower price it loses money), and with free entry it is where the price settles in the long run (Long-Run Industry Equilibrium).
Common mistake
Setting marginal cost equal to zero to find the lowest average cost. For $C = 100 + q^2$, $MC = 2q = 0$ gives $q = 0$, where average cost is infinite. The minimum of a ratio comes from the derivative of the ratio: set $MC = AC$. Check: at your answer, compute both MC and AC. They must be equal.