## Wednesday, November 25, 2015

### Math up

My view of why you should include math is best captured by Paul Krugman:
My point is that there seems to be a lot of implicit theorizing going on here — and at least at first glance, the implicit theorizing doesn’t make a lot of sense. I could be wrong, but that’s the whole point of simple models: to lay bare what you’re assuming, and make it clear what, specifically, is driving your conclusions.
Krugman says "models", and that's what I mean by math -- some sort of formal construct that organizes the various factors you are trying to explain. It doesn't have to be a precise DSGE model. Generic supply and demand diagrams are math. Working with orders of magnitude and asymptotic behavior qualify as math. Basically, math lets other people use your model.

Math is the basic universal language for talking about how things relate to each other. If you are saying X is related to Y, you are implicitly using math. If you don't want to use math, you're not allowed to say X is related to Y.

Of course, some people think math doesn't help illuminate anything. In its most dogmatic form, we can see the opposition to mathematics from Ludwig von Mises:
All of mathematical economics, with its beautiful curves and equations, is idle flirtation. The setting up of equations and the drawing of curves must be preceded by nonmathematical considerations; the setting up of equations does not broaden our understanding. Mechanical equations can be used to solve practical problems through the introduction of empirically acquired constants and data; but equations of mathematical catallactics cannot in the same way be of service to practical problems in the area of human action where constant relations do not exist.
This is of course completely disproved by the success of the theories of how auctions work. As well as the information equilibrium model ...

But in general the reason for mathematics is not that humans are mathematical, but that economists' models of the macroeconomy are mathematical -- whether they want them to be or not. When you start saying effect E has cause C, then you are assuming a functional relationship between E and C: E = f(C). From that we can say that if C is small, we have E ~ f(0) + a C ... and we're off to the races. I've just turned your statement of C causes E into a linear model with an equilibrium value E0 = f(0) and an "elasticity" a. If you say "but a and/or f(0) changes" ... then is it really C that is causing E? Note that linear function E ~ f(0) + a C has exactly the same form if E = f(a) where C is the elasticity. And f(0) changing means that something else is causing E [1].

The reason people don't use math is either a) they can't think clearly enough to use math or b) they want to pull the wool over your eyes. The stated defense is that math doesn't clarify (that's false; it does) or that human actions don't have constant relations (my question to von Mises would be "then why even write anything ... once written down as math or words, human actions will change"). If human actions don't have constant relations, then what does it matter? Even history is studied because people think certain things can cause e.g. wars in a fairly consistent way like competition for natural resources. People study WWI and WWII because they think the things that happened then could have benefit in future conflicts (preventing wars or fighting them). That is the entire point behind "history repeating itself". If history never repeated itself, then there wouldn't really be any reason to study it.

Not using mathematics for your model:

• Puts an upper bound on the complexity of your model (makes it easier to compete with smart people)
• Allows you to say what you're saying is novel even if it is equivalent to existing models (allows you to fool people into thinking you have new insight)
• Makes it impossible for anyone else to use your model (limits the use of your model to yourself)
• Allows your model to change in respose to new information without saying you are changing it in response to new information (so you can fudge any data)
• Allows you to get around stating clear predictions and precise conditions on those predictions (so you can fudge any prediction)
• Allows you to stay away from quantitative statements about the data (so you can fudge any data)
• Allows you to have effects of different magnitudes from causes of the same size (and vice versa) (so you can fudge any data)

A lack of math is not depth of argument or intuitive understanding, it is flim flam.

...

Update 7 September 2016

I should really have rephrased the part about "history repeating" in terms of "lessons of history" instead. People tend to take historical analogies too literally sometimes. An historical situation may enlighten you about the possibilities in a current situation (lessons), but there is almost never an exact analog of some historical event (repeating).

...

Footnotes:

[1] This is my opinion of those expectation operators in models. An E[u(c)] term may look like your model depends on consumption or utility, but really it just depends on E.

1. Your list of advantages to not using math is compelling... sign me up! ;^)

Your list made me think of my speculation about why no macro economist bloggers ever took you up on your model / forecasting challenge (assuming they were made aware of it):

What's the upside for them? Nobody who *matters* (notable macro types) takes challenges like that, so there's nothing there for them but negatives... especially when going up against an amateur / outsider. If they win, nobody who matters will care. If they lose, it's egg on the face... and people who will matter will probably just think it was a foolish thing to do in the first place.

Happy Thanksgiving Jason!

1. That sounds a lot like incentives ....

:)

Happy Thanksgiving.

2. Dang I really wish I had known this back at the end of high school when I was deciding what to major in during college

3. I only found this post today, thanks to a link in a recent post. For what it's worth, I agree with you 100% on this.

People who radically oppose maths in economics (like Mises, whom you quoted; or Joan Robinson, among Mises' more apparent than real opponents) remind me of Moliere's bourgeois gentilhomme, who had been speaking prose all his life without knowing it.

Either that, or they don't appreciate the need to be coherent which maths forces upon their users.

At any event, I don't agree with William Stanley Jevons' economics, but he was on to something when he wrote (and he, also thought of Moliere when he composed this):

"I contend that all economic writers must be mathematical so far as they are scientific at all, because they treat of economic quantities, and the relations of such quantities, and all quantities and relations of quantities come within the scope of the mathematics. Even those who have most strongly and clearly protested against the recognition of their own method, continually betray in their language the quantitative character of their reasonings. What, for instance, can be more clearly mathematical in matter than the following quotation from Cairnes's chief work[1]: 'We can have no difficulty in seeing how cost in its principal elements is to be computed. In the case of labour, the cost of producing a given commodity will be represented by the number of average labourers employed in its production--regard at the same time being had to the severity of the work and the degree of risk it involves--multiplied by the duration of their labours. In that of abstinence, the principle is analogous: the sacrifice will be measured by the quantity of wealth abstained from, taken in connection with the risk incurred, and multiplied by the duration of the abstinence.' Here we deal with computation, multiplication, degree of severity, degree of risk, quantity of wealth, duration, etc., all essentially mathematical things, ideas, or operations. Although my esteemed friend and predecessor has in his preliminary chapter expressly abjured my doctrines, he has unconsciously adopted the mathematical method in all but appearance." pp. xxi-xxii

"I hold, then, that to argue mathematically, whether correctly or incorrectly, constitutes no real differentia as regards writers on the theory of economics. But it is one thing to argue and another thing to understand and to recognise explicitly the method of the argument. As there are so many who talk prose without knowing it, or, again, who syllogise without having the least idea what a syllogism is, so economists have long been mathematicians without being aware of the fact. The unfortunate result is that they have generally been bad mathematicians, and their works must fall."

The Theory of Political Economy, 5th edition. Augustus M. Kelley, Bookseller. New York 1965.
Preface to the Second Edition (1879)

B.L. Zebub

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