The pedagogy of parameters

A difficult thing that I ask my students to do is to parameterize everything.

The ideas that my game theorists come up with for applied theory projects are uniformly great. The puzzles they want to study are rich with potential, and many are easily original enough to be of publishable grade.

Compared to coming up with ideas, kicking it up a notch into something that looks like an economics paper is much harder. There are two main hurdles:

  1. Shrink the scale of the problem until it’s manageable enough to solve, but still tells you something useful.
  2. Work with parameters, not numbers.

The first one is the hard part about modeling in general, and I’m not sure if there’s any other way to get the knack for that other than experience and imitation. It’s also, of course, the reason why Realism Cops get themselves worked up about theory in general, but that’s a problem for another day.

The second one I think really gets at a key challenge of economics pedagogy in general: how do we shift from numbers to parameters?

For example, introductory texts typically use tables of numbers to demonstrate different types of cost, matrix games with numerical payoffs to illustrate game theory, and so on. At some point we have to move to parameterization if we want to show how to do the hard work of counterfactual analysis and comparative statics that economics is good at.

Part of this is just about the cultural connotations of math and algebra. Algebra has a Big Deal problem. Everyone’s conditioned to see math as otherworldly math skill as innate, so that equations feel like more of a big deal than numbers. As a side note, by the way, another Big Deal that I think is special to the U.S. is so-called “cursive”—it feels like a big deal because it’s made out to be a big deal with a fancy name. The cursive mystique and debate always confuses me because when I was at school in the UK it was just, I don’t know, writing?

Anyway, getting over the algebra and calculus hump is definitely part of what’s going on here. But I don’t think it can be the whole story, not least because my upper-level students have long since cleared the math hurdle. At this level it’s the difference between assuming what numbers go in the game tree or the matrix, and specifying a parameterized payoff function or distribution of types. We’re at a higher order of problem here.

I want to make the case, then, that there is more to the act of switching from numbers to parameters than being comfortable with algebra. Abandoning numbers is itself a Big Deal. The cultural meaning of numbers is special, connoting precision, certitude, and fact. [I mean this in the sense of the Mary Poovey argument.] Moving to parameterized analysis may be dislocating for a student if they perceive parameterization as relatively ambiguous and imprecise.

All that has truly changed is the moment at which the analysis changes from general to specific: instead of presupposing a particular case, parameterization leaves the particular to the end. The input to the machine becomes flexible rather than fixed, but the machine itself does not change.

But I don’t think that’s how it feels. It feels like we have shifted from a world of certitude to a world of radical uncertainty. It feels more comfortable to resolve uncertainty before we feed the machine than to embrace uncertainty in the analysis, even though the numbers in the matrix are just as arbitrary as the value of a parameter. The numbers come from the same place as the parameterization: educated guesses. It’s not truthier because it says “10” instead of “p”.

I try to sell it as the lazy person’s solution. It’s like writing a script to automate some task, but not telling anyone. You can go off to “run the new numbers” but just take the afternoon off, because you ran all the numbers at once when you parameterized a model instead of using numerical payoffs.

Less cynically, you can trace uncertainty over preferences, probabilities, or what have you through your analysis, and avoid contributing to the number mystique as well as avoiding its siren call. It’s like this: you don’t have all the answers, because you didn’t pretend you could pin down numbers. But you do have all the answers, because you know why which numbers matter and how. Magic!

I think this is all related to the economism problem too. The wily replacement of “good” with “efficient” in all too many introductory economics courses shifts us from the discomfort of moral dilemmas to the comfort of quantification. Abstract things become measurable in dollars, and if numbers are facts, what are bigger numbers but bigger, better facts? Small numbers become wrong, and the unquantifiable becomes positively obscene.

And so we perpetuate existing power structures. We feel comfortable using the devil’s metric of efficiency to play his advocate in favor of ticket scalping, price gouging, or abolishing the minimum wage. We tack on “… and we must balance that against concerns for equity”, but fail to see the racism and classism that’s inherent to equating ability to pay with value to society. We follow the lead of Marshall in thinking we are studying “man… in the ordinary business of life“, under-counting and under-valuing both women and the non-pecuniary.

The tone and tenor becomes a Google memo that believes that “objectivity” is the same as “neutrality”. Students who are rich, white, and powerful will find it easy to nod along with the lessons of “efficiency”. Do the Fundamental Theorems of Welfare Economics not teach us, after all, that efficiency and equity can be separated? [Spoiler: they do not.] Small wonder that “woke students are choosing not to pursue careers in Econ“.

It would be better for us to engage with the unquantifiable. It would permit us to be actively feminist and anti-racist in our pedagogy instead of hiding behind the false idol of efficiency. But we can really help ourselves out by embracing at the introductory level these same challenges and approaches that we already routinely teach and use at the upper levels. Numbers and quantification can be de-emphasized in favor of parameterization and abstraction, with a net gain in rigor as well as in professional ethics.

In sum then, in general, when we try to teach parameterization, I think the challenge is double-sided. We should try to demystify math and algebra as counter-programming to their cultural connotations. But we should also work to remystify numbers, as counter-programming to their cultural connotations. Algebra should be raised up a peg or two, but numbers should be brought down to meet it half-way.

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