Relationship Between MC, AVC and ATC

The way Marginal Cost interacts with the two average-cost curves follows a general rule that applies to any "marginal versus average" pair of magnitudes, not just cost: as long as the marginal value is LESS than the average, the average is being pulled DOWN; as soon as the marginal value is MORE than the average, the average is being pulled UP; and the average is neither rising nor falling — that is, it is at its minimum (or maximum) — exactly at the point where marginal equals average.

A simple analogy makes the logic clear: a cricketer's batting average falls whenever her latest innings score is below her existing average, rises whenever it is above the average, and stays unchanged only when the latest score exactly equals the average. Cost behaves in exactly the same arithmetic way, with MC playing the role of the "latest score" and AVC/ATC playing the role of the "average."

Applying this rule: so long as $MC < AVC$, each additional unit costs less (at the margin) than the existing average variable cost, so AVC keeps falling; once $MC > AVC$, each additional unit costs more than the existing average, so AVC starts rising. AVC is therefore at its minimum exactly at the output where $MC = AVC$ — the MC curve cuts the AVC curve exactly at AVC's lowest point, always from below. The identical logic applies to ATC: the MC curve cuts the ATC curve exactly at ATC's minimum point, again from below. …