## Debunking Economics, Part XIV: In Defence of Mathematics

Chapter 16 of Debunking Economics is a short comment on the use of mathematics in economics. Keen offers a defence of maths itself, suggesting that it is neoclassical economist’s misuse of the tool, rather than the tool itself that has caused the problems in economics today. He compares it to the story of a king who hears an awful tune played on the piano, and proceeds to shoot the piano.

Keen first recaps on some of the mathematical mistakes he has discussed throughout the book, such as the problems with demand and supply curves. I won’t go over these again here – that would be a summary of a summary – but will instead briefly note a couple of general problems with economist’s use of mathematics.

First, it seems economists are not ready to acknowledge the limits of mathematics: mathematicians have known for some time that some equations simply cannot be solved, or are incredibly difficult. Since economists are often dedicated to proving the existence of an equilibrium, they have to stick to overly simplistic analysis, where equations can definitely be solved. This causes them to rely overly on linear models.

Second, Keen makes a pithy mathematical observation about emergent properties and reductionism. Reductionism can be characterised as reducing something down to its component parts. However, if these component parts are multiplied together – rather than added – as you aggregate up, you will see a substantial change in behaviour at the aggregate level. Hence, reductionism has clear and obvious limitations.

Overall, I agree with Keen that mathematics is useful in economics. Jevons put it most accurately when he said “[economics] must be mathematical, simply because it deals with quantities.” However, this shouldn’t mean quantifying things with erroneous measures – such as capital – just for the sake of  mathematics. Equations have to have clearly defined parameters, can only be considered as good as their assumptions, and may not have clear implications. Such is the nature of modelling complex systems.

Update: I was going to leave this out for fear of digressing, but a couple of the comments reminded me of a quote Keen used to end the last chapter:

The real problem with my proposal for the future of economics departments is that current economics and finance students typically do not know enough mathematics to understand (a) what econophysicists are doing, or (b) to evaluate the neo-classical model (known in the trade as ‘The Citadel’) critically enough to see, as Alan Kirman [54] put it, that ‘No amount of attention to the walls will prevent The Citadel from being empty’. I therefore suggest that the economists revise their curriculum and require that the following topics be taught: calculus through the advanced level, ordinary differential equations (including advanced), partial differential equations (including Green functions), classical mechanics through modern nonlinear dynamics, statistical physics, stochastic processes (including solving Smoluchowski-Fokker-Planck equations), computer programming (C, Pascal, etc.) and, for complexity, cell biology. Time for such classes can be obtained in part by eliminating micro- and macro-economics classes from the curriculum. The students will then face a much harder curriculum, and those who survive will come out ahead. So might society as a whole.

This is from the (econo)physicist Joseph McCauley. It’s an interesting reversal of roles for economists, who often label critics as mathematically illiterate. Having said that, I think McCauley’s attitude shares some of the same characteristics that I hate to see in economists.

1. #1 by Donald Last on October 24, 2012 - 6:20 pm

What I find fallacious is the treatment and importance accorded to those well-known aggregates – C, G, I and GDP. These are basically accounting descriptions with strictly limited prescriptive value. For when manipulating them they in no way or at any time tell us what are the forces within those aggregates that are driving them and how they are working out; what actually happens within that accounting segment; what market forces are key – at any selected point in time – in driving them to fill up or to empty. And when one does descend to the shop floor to seek out the market forces and prices which households and businesses, large and small, are reacting to, well then it gets horribly complex. Which is why the generality of economists don’t like to go there being more comfortable with their ethereal differential equation models which pretentiously claim to know the forces at work and how they reach an equilibrium.

• #2 by Ben Brennan on October 24, 2012 - 7:28 pm

What I have found useful is to take the accounting identity, and use it in conjunction with a sum of incomes, the production function, and a wage and employment bargaining model. It gives you a simple but more robust model, even if it is fairly NC in nature.

• #3 by Unlearningecon on October 24, 2012 - 10:36 pm

Which production function? As I said to Roman below, many of the basic production functions just strikes me as bad science.

2. #4 by W. on October 24, 2012 - 7:57 pm

In so far as economists remained away of pure-math education (either at undergraduate or at graduate level), economics could rather hardly (if not just impossibly) improve it feedback with mathematics. It seems quite utopian to think about new, meaningful interactions between economics and mathematics, without having at the same time (or to be fair, a little bit earlier…), as just mentioned, economists educated within mathematics.

3. #5 by Roman P. on October 24, 2012 - 9:00 pm

Out of interest, Unlearningecon, do you study ODEs and dynamic modeling as a part of your curriculum?

• #6 by Unlearningecon on October 24, 2012 - 9:44 pm

I have studied basic ODEs but they are not within the core of the course. They are an area I am greatly interested in – I am considering doing a masters in applied maths or something similar if I can no longer stand economics.

• #7 by Roman P. on October 24, 2012 - 10:07 pm

Curious. In Russia, curriculum is semi-rigidly defined by the federal departments, and I believe that pretty much every Russian economist did at least a semester of ODEs. I did more (not that I remember analytical solutions to various equations now…). I’ve always thought that my education in MEPhI was substandard by the standards of the western universities, but it seems that I was spared high-level calculus and topology in favor of dynamic modeling, which is probably more useful for the real world.

• #8 by Unlearningecon on October 24, 2012 - 10:31 pm

No, I had to take a module from the maths school deliberately (nobody else in economics did) and my advisor commented to me that he was surprised they let me do it. Bear in mind that even my degree is pretty substandard by western standards, so I can’t imagine how rigid an education at, say, the University of Chicago must be.

ODEs are certainly a far more useful tool for economics than constrained optimisation of imaginary substances (capital, utility). I don’t really understand why economists can just posit relationships that beg about 1000 different questions – like Cobb-Douglas – and use simplistic mathematics on them in the hope of finding something out.

4. #9 by Aziz on October 24, 2012 - 10:51 pm

I’m working through this stuff slowly.

I think the thing that is most sorely missing from neoclassical economics is empirical demonstrability of assumptions. I think the way forward is to only use assumptions that are empirically demonstrable. Unfortunately, most of neoclassicism’s most cherished assumptions, e.g. equilibrium, utility maximisation, perfect competition, etc, etc, etc, are not supported by the empirical record.

Without empirically demonstrable assumptions, neoclassical economics is just a mathematical equivalent of praxeology.

• #10 by Unlearningecon on October 25, 2012 - 9:11 am

Yeah, absolutely. I don’t know why economists think they can be so cavalier about assumptions and it just shows how little they engage with the critical literature (there are many refutations of Friedman if you look for them).

5. #11 by Jon Finegold on October 24, 2012 - 11:20 pm

I think Jevons is wrong. Economists shouldn’t use math because economics deals with quantities. Economics isn’t just about quantification, but about explaining causal relationships. Mathematics is a logical “language,” and economists use it because it tends to be more precise than verbal logic.

• #12 by Unlearningecon on October 25, 2012 - 9:12 am

But causal mechanics can be expressed in terms of quantities, right? If we put X amount of good Y into machine W for Z period of time, we get N of good M.

• #13 by Jon Finegold on October 25, 2012 - 7:45 pm

Sure, but the task of mathematics isn’t to deal with quantities, but to explain the relationship between quantities.

• #14 by Unlearningecon on October 27, 2012 - 12:25 pm

How does that contradict that economics should use maths?

6. #15 by rjw on October 25, 2012 - 12:48 pm

I would make a few simple points about quantification.

Firstly, A lot of economic relationships, particularly aggregate relationships, are almost impossible to understand unless you can think conceptually about how things add up. National accounts, for example, are quantification, but hard to do without in any sensible form of macroeconomics. If you can’t see/understand that investment must, in accounting terms, be equal to the sum of domestic private, government, and foregin sector saving, then you will misunderstand a lot of macroeconomics.

Secondly, you do not need to use mathematical language to identify a causal relationship – for example, to identify the possible transmission channels for monetary policy, one can argue verbally. But to assess the relative weight/importance of how these channels work, at some point you need to think about how you measure relative influence. Hard to do this without some kind of quantitative thinking.

The problem is not maths, or quantitative thinking per se. These things are necessary. The problem is the association of complexity with insight. sometimes very simple models, mathematical or not, can be ful of insight. At other times, complicated models can be junk doe to underlying conceptual nonsense. The art of being a good economist is to judge these things on their merits.

7. #17 by Blue Aurora on October 25, 2012 - 6:18 pm

I’m glad to see that you reference the econophysicists again, Unlearningecon. However, not all of the econophysicists are as hostile to economics as Joseph L. McCauley himself is. I know that H. Eugene Stanley has reached out and cooperated with economists (namely Xavier Gabaix) and has published in standard economics journals.

• #18 by Unlearningecon on October 27, 2012 - 12:26 pm

That’s good to hear. My primary concern with econophysicists is that they might simply become a mirror image of economists (‘economists don’t understand us, everything they do is useless’ etc.)

8. #19 by Eric L on October 26, 2012 - 5:36 am

“require that the following topics be taught: calculus through the advanced level, ordinary differential equations (including advanced), partial differential equations (including Green functions)”

I enjoyed taking calculus in college, but I’m not so sure it was worth as much time as I spent on it. The concepts of calculus are very useful. The techniques, the variety of tips and tricks you learn to solve an integral or differential equations, not so much. In the real world, numerical methods will just about always suffice. But more importantly, the real world is full of integrals and differential equations that can’t be solved, and interesting phenomena that can only arise in the types of systems that can’t be solved; learning that manipulating equations to turn them into other equations is the way that math is done can be a bit of a handicap when it comes to modeling complex phenomena. The rest of Joseph’s suggestions are good; I would just stress that economists should be taught to express models as code and investigate their behavior empirically, and learn that this is not only a perfectly valid way of doing math but a far more powerful one that can reveal implications of your models that you would never have noticed by manipulating equations.

• #20 by Unlearningecon on October 27, 2012 - 12:25 pm

Excellent comment. Most applied sciences use approximations like Fourier or power series. Economists do not find themselves in need of these because they are using equations that are easy to solve elegantly.

9. #21 by Metatone on October 26, 2012 - 8:00 am

In line with Eric, I like the gist of Joseph’s thought – in reality most economists are much less good with maths than they think – if they were better they might understand the limitation of some of their models more. As someone who studied systems engineering, economists often seem to build epicycle models with maths, partly because they have a particular toolkit they are proficient with, but lack an overview of alternative methods.

However, I think there’s still a place for serious thinking about models and about macro and micro that isn’t “take a math degree” first. In the end, we need to educate economists about assumptions and models and the real economic world. And you won’t get any of that in maths classes.

We’ve seen this in finaphysics, where lots of interesting models are produced – they are assigned utility by how they profit, but all too often there’s a big gap where the discussion of the model (and how it relates to reality and how the causal factors relate to empirical mechanisms) should be.

• #22 by Unlearningecon on October 27, 2012 - 12:37 pm

Yes, pure mathematicians also often exhibit a preference for elegance over real world accuracy. Engineers are far better suited to the task.

And I think you are right that economic physicists sometimes seem to fall into similar traps, using their pet models for no apparent reason. I was puzzled by this, for example, which tries to model migration using electromagnetic forces. Why? I’ve no idea.

10. #23 by John Goodwin on October 28, 2012 - 6:01 pm

I’m a physicist by training. I too wish Keen and McCauley wouldn’t fight, though I’ll defend McCauley if I have to. McCauley, by the way, says that he is exiting the field of Econophysics and has sent his last book on the subject to the publisher, which I hope means it will be the 2012 Spring list, so we don’t have to wait until Fall 2012.

• #24 by Blue Aurora on October 28, 2012 - 7:44 pm

Joseph McCauley is exiting the field of econophysics? Do you have a source for this? Do you know why he is leaving? Cos’ I do think that he’s had an impact and he deserves to stay on…