# Application of Minimizing Average Cost

In my Business Calculus class we cover finding the production level that minimizes average cost. I get what average cost is measuring, but all the texts I've looked at don't really mention why we care to minimize it. Maximum profit relates to marginal cost, so points of minimum average cost are not even necessarily giving maximum profit (unless by coincidence (marginal cost)=(marginal revenue)=(average cost)). In what situations do we look at the problem and see that the answer is where the average cost is minimized?

The best I can come up with is this: A nonprofit is making medicine kits that it wants to sell at the lowest price. Since they don't want to make a profit, the price will be average cost. So finding the production level that minimizes average cost will allow them to sell at the lowest price.

This seems highly contrived. Any other good set-ups/questions that are answered by minimizing the average cost?

• Perhaps a company wants to declare that it has minimized its average production cost to sound good to shareholders or others, hoping they don't really think about whether that's a good thing. Oct 15 '14 at 17:19

Cost is fairly easy to calculate, so finding the minimum average cost is also easy.

Profit depends partly on revenue, which is much harder to predict. You can't know just how customers will react to each possible price that you charge. Your product may become a fad, or it may not. So how can you get a formula that gives you revenue, much less profit?

Minimum average cost is not the same as maximum profit, but it probably will not be terribly far from it. You can use it to easily approximate what you really want. The alternative is to get a worse approximation with more effort. At least, all that could happen fairly often.

Another consideration: a planner needs to take many things into account. The CEO may want to know both how to get minimum average cost and how to get maximum profit. A good CEO should know the proper weight to give each fact when making his decisions.

• Often average cost is minimized with infinite or unreasonably large quantities -- which is notably far from maximum profit.
– user173
Oct 16 '14 at 1:04

When prices can change quickly while it is difficult to adjust the output, then to produce at minimum average cost could be a good idea; especially, if avoiding to make a loss is important.

We are considering Total Cost $(C)$as a function of output, $q$, $C= C(q)$.

Average Cost is defined as

$$AC \equiv \frac {C(q)}{q}, q>0$$

Finding the minimum average cost production level $q^*$requires to set $AC$'s derivative equal to zero

$$\frac {dAC}{dq} = 0 \Rightarrow \frac{C'(q)q-C(q)}{q^2} = 0 \Rightarrow C'(q)q-C(q) = 0$$

$$\Rightarrow q^*: C'(q^*)=\frac{C(q^*)}{q^*}$$ (and you want the cost function to be convex, $C''>0 in order for this to be a minimum). But$C'(q^*)$is the marginal cost function calculated at$q^*$, so $$q^*: MC(q^*)=AC(q^*)$$ In other words, at$q^*$, average and marginal cost coincide. Now, we have to remember that in Economic Theory the concept of "profit" has nothing really to do with how the word is understood in every day business, or say, in Accounting. "Profit" is above and beyond some measure of "normal returns to capital", while Accounting profits, the "bottom line", have not subtracted from revenues capital returns fully (they have subtracted the returns to foreign-borrowed- capital, but not the returns to own capital). So roughly speaking, Economic Theory sees Accounting profits as "own capital returns" -and will characterize a portion of them as "profits" only if said portion exceeds the "normal rate of own capital return", however this is measured. Why this digression? Because in competitive markets the argument is that firms are more or less "price-takers" (they may "set the prices" but they have to eventually adjust them due to competition), meaning that the revenue function is linear in the price, and so marginal revenue does not depend on the quantity produced by a single company. Moreover, competition will drive the price down enough so that economic profits (as defined above) are extinguished: but this means that total revenue will equal total cost, and so marginal revenue (which equals average revenue here), will equal average cost. Since the company equates marginal revenue with marginal cost, this leads us to the conclusion/prediction that what we observe is companies producing at a point where all three are equal -and marginal cost is equal to average cost when the firms produce at$q^*\$ level.

On the practical side of things, producing at minimum average cost, maximizes your chances to make, after all, a profit, in the economics sense of the word. Note that the above imply that when doing cost calculations, in "costs" we should include the "opportunity cost of own capital", which is another way to talk about the "normal" returns to capital.

And yes, when market imperfections, market power, market asymmetries, market externalities, are taken into account, the above picture becomes an approximate benchmark case -and it is then when things get really interesting.

• Your "conclusion/prediction" does not follow from your argument on competitive markets. If in competitive markets, "economic profits are extinguished", then economic profits do not determine a production level at all.
– user173
Oct 16 '14 at 5:30
• @MattF. I don't understand your comment (and your downvote?): where in my answer do I say that economic profits "determine the production level"? Oct 16 '14 at 9:44
• If you're not determining the production level, you're not answering the question.
– user173
Oct 16 '14 at 12:20
• @MattF. Matt, I don't understand this comment either. My answer does determine the production level, through the assumptions on market structure. Market competition constantly pushes the price downwards. Why then it does not reach zero? Because below minimum average cost, the firms will make losses and production will be zero (this is a long-term equilibrium framework). So the downward movement of the price stops when we reach minimum average cost, and so the production level will be, in equilibrium, the one where average cost is minimum. Oct 16 '14 at 12:32
• I still don't see why "below minimum average cost, the firms will make losses". I hope you'll clarify in edits to your answer, but I won't comment further.
– user173
Oct 16 '14 at 12:58