Why does marginal cost cross average total cost at its minimum

Average total cost (ATC) is total cost divided by output, (ATC = $\frac{TC}{Q}$\). Marginal cost (MC) is the extra cost of producing one more unit, (MC = $\frac{$$\Delta$ TC}{$\Delta$ Q}\). If MC lies below ATC, each additional unit costs less than the current average, so the average falls; if MC lies above ATC, the average rises. The point where MC equals ATC is therefore the turning point of ATC, its minimum.

Mathematical proof

Deriving the crossing condition step‑by‑step

1. 1Write ATC as $ATC=TC/Q$.
2. 2Differentiate ATC with respect to Q: $\frac{d(ATC)}{dQ}=\frac{MC-ATC}{Q}$.
3. 3Set the derivative to zero for a minimum: $MC-ATC=0$ so $MC=ATC$.
4. 4Check the second derivative; it is positive when MC is increasing, confirming a minimum.

The derivative formula shows that the sign of $MC-ATC$ determines whether ATC is rising or falling. When $MC<ATC$, the derivative is negative, so ATC is decreasing. When $MC>ATC$, the derivative is positive, so ATC is increasing. Thus the equality $MC=ATC$ marks the exact point where ATC switches from decreasing to increasing, i.e., its lowest value.

Worked example: a firm produces output Q and incurs total cost TC as shown: Q=1, TC=10; Q=2, TC=18; Q=3, TC=27; Q=4, TC=38. MC between each unit is 8, 9, and 11 respectively. ATC values are 10, 9, 9, and 9.5. MC is below ATC at Q=2 (9<9), equal at Q=3 (9=9), and above ATC at Q=4 (11>9.5). The crossing at Q=3 coincides with the lowest ATC of 9, illustrating the rule with real numbers.

Key take‑aways

• MC below ATC ⇒ ATC falling
• MC above ATC ⇒ ATC rising
• MC = ATC ⇒ ATC at its minimum
• The crossing occurs because averages respond to marginal changes.

Sample cost schedule for a small plant