The Value of Mathematical Knowledge | James Howard The Value of Mathematical Knowledge | James Howard

James Howard A Mathematician, a Different Kind of Mathematician, and a Statistician

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The Value of Mathematical Knowledge

A few years ago, I was teaching public finance. My students, generally, while grad students, probably had not had microeconomics. And if they have, it really didn’t stick. So I spend the first few weeks “reviewing” microeconomics, which means I teach all the bits of micro they will need to get through public finance. So far, so good.

So, I gave a 20-minutes introduction to pricing. This was the whole marginal cost = variable cost = long-run average cost = price spiel. As a general rule, I don’t give proofs in class. Proofs are for journal articles and lesser professors. But I do want the student to have an intuitive understanding of why something is true. For those baked in the formal sciences, a proof does that. For those not, well, a walk-through is usually sufficient.

So I give them \(AC = VC + FC / n\), where n is the number of units. In a proof, I would write \(\lim_{n \rightarrow \infty}{AC} = \lim_{n \rightarrow \infty}{VC + FC / n} = VC + 0 = VC\), LaTeX it, and move on. But that’s not going to help, like, normal people. So instead, I go back to \(AC = VC + FC / n\) and I asked, “What happens to this as we sell more units?” Blank stares, so I say, “I give you,” and point, “a dollar. How much do you have?” And the student responds, “a dollar.” So then I ask, “Okay, I am going to split a dollar between you and you; then how much do you each get?” “Fifty cents.” We continue this to a hypothetical million students, and they are starting to get the gist of the limit, if not the specifics. Then we kind of realize the FC can just disappear.

At this point, I’ve fairly well convinced the class that \(AC = VC = MC\) and am starting on the price when I hear a student note that they didn’t expect the math to be so difficult. This is like, eighth-grade algebra, setting aside the limit I kept hidden under the table anyway. And it wasn’t the limit that scared them. It was the algebra.

And that’s frustrating because it shouldn’t be that hard.