What is a good resource for non-math managers to learn the calculus behind cost analysis?

Question

I purchased Economics of Strategy, 6th Edition by David Besanko which reviews were overwhelmingly positive and indicated it covers a lot of topipcs that interest me in broad terms. However, skimming through while I understand many of the chapters there are some segments that include I believe calculus for varius price, quantity, and production curves. For example:

Associated with the total cost function are two other cost functions: the average cost function, AC(Q), and the marginal cost function, MC(Q). The average cost function describes how the firm's average or per-unit-of-output costs vary with the amount of output it produces. It is given by the formula

$$ AC(Q) = \frac {TC(Q)}Q $$

If total costs were directly proportional to output---for example, if they were given by a formula, such as TC(Q) = 5Q or TC(Q) = 37,000Q, or more generally, by TC(Q) = cQ, where c is a constant---then average cost would be a constant. This is because

$$ AC(Q) = \frac {cQ}Q = C $$

Economics of Strategy, 6th Edition, David Besanko. p13

This math is well beyond my understanding and while I can certainly read the text and gain a lot without the mathematical skills I was wondering:

1. Is this relatively speaking a basic calculus formula that a "first year" calculus student might be able to learn and understand?
2. And if so is there a particular book that focuses on teaching pricing calculations like these for managers that don't have a math background?

Answer 1

The equation you give doesn't actually constitute calculus, it is simple "translation" and (pre-college) algebra. What they are saying is that you might have a "total cost function" given in terms of the variable $Q$, $TC(Q),$ that has the form $TC(Q) = cQ,$ where $c$ is some constant (real number, probably positive). If this were the case, then by using the formula for the "average cost function," $AC(Q)$ we would get $$AC(Q) = \frac{TC(Q)}{Q} = \frac{cQ}{Q} = c\frac{Q}{Q} = c \times 1 = c.$$ If, you were given some other total cost function, you would get a different result. Like if $TC(Q) = 5 Q^2$ we would get $$AC(Q) = \frac{5 Q^2}{Q} = 5\frac{Q^2}{Q} = 5 Q.$$

So, no calculus here yet. Although if you need a resource for calculus (just the math, not business), I can recommend the free textbooks at openstax. You could simply do the first few chapters, coming up with examples from the world of business and economics to help make the examples concrete. For instance in the example you have given us, the total cost function can be nearly anything (any function that changes with respect to time, or product quantity or price of components, etc.). If the function you dream up is with respect to time, and starts at zero, the $AC$ function is still the average cost function, given any total cost function dependent on time, $Q$, because you are dividing the "amount" of the total cost by the "amount" of total time. Just like how if you want to find the average speed of a car, you would divide the length of the trip by total time.

If you want a book that will more explicitly lay out the math in terms of business vocabulary, you should probably be looking at "Applied Calculus" books. I only know of one free one available at the opentextbookstore. I've never used it, but it looks like almost every other calculus book, and since undergraduate calculus hasn't changed for a couple hundred years, it should be fine :-)

Answer 2

That particular example is algebra. You're substituting $TC(Q)$ with $cQ$, and canceling the $Q$'s.

As for calculus, though, you may be able to get fairly far along by learning the integral and derivative of a polynomial function:

$$\int_a^b x^n dx = \frac{b^{n+1}-a^{n+1}}{n+1}$$

$$\frac{d}{dx}x^n = nx^{n-1}$$

as well as some basic concepts of dimensional analysis. (For example, if you have a plot of hourly rate (dollars per hour) and integrate with respect to time (hours), then the area under the curve has units of dollars (dollars per hour, times hours).