Marginal Cost and Average Cost | Firm | Economics

In this article we will discuss about the relationship between marginal cost and average cost.

The three cost curves TC, AC and MC describe the same physical data and are, therefore, related mathematically. Let TC (q) be the total cost of output q, AC (q), is defined as the total cost divided by the amount produced, or

AC (q) = TC (q)/q …. (1)

Marginal cost, MC (q), is defined, precisely enough for our purposes, as the increase in total cost imposed by a unit increase in output.

Therefore:

MC (q) = TC (q + 1) – TC (q) … (2)

Making use of equation (1) we can also write:

MC (q) = (q + 1) AC (q + 1) – q AC (q)

= q AC (q + 1) – AC (q) + AC (q + 1) … (3)

Hence,

AC (q + 1) – AC (q) = MC(q) – AC(q + 1)/q … (4)

Suppose, now that output q lies in the range in which average costs are falling.

Then the left-hand side of this equation is negative, implying also:

MC (q) < AC (q + 1) < AC (q) … (5)

In words, if average costs are falling, then marginal costs are less than average costs.

By similar reasoning, if average costs are rising, making the left-hand side positive, MC (q) > AC (q + 1) > AC (q). Thus, average costs rise when and only when marginal costs are above average costs.

Finally, from the facts that average costs are above marginal costs when average costs are rising, it follows that the two curves must cross where the average- cost curve bottoms out—that is, where average costs are at their minimum.