The maths question in economics

Over at Noahpinion last week a post on the role of maths in economics generated plenty of comment.* This followed the award of the “Nobel Prize” in Economics to Shapley and Roth for work that is, in almost anyone’s book, highly mathematical. Noah Smith identified a number of reasons for using maths in economic analysis, each of which could be a good or a bad reason, depending on circumstance. His broad conclusion is that:

Math is not always the most appropriate tool in economics. But the more real successes economics achieves, the more good math it will use.

He argued that there are times when it would be appropriate to make less use of maths in economics. The argument here is summarised as:

The only time not to use math in econ is when we haven’t found the right math yet.

And in practice, I find that a few of the people calling for less math in economics … don’t seem to have any such goal in mind. There are a few people out there who would rather econ stay imprecise forever – so that nobody will ever be proved wrong or right, and we can let a million flowers bloom, and everyone’s scholarly opinion about the economy will be equally valid.

This is a debate that I spent some time thinking about a while ago. I have written a little about it in relation to the specific applied field of housing economics. It was interesting to revisit the topic for the first time in a while.

It strikes me, though, that this brief flurry of interest in the maths question is framing the discussion a bit too narrowly and missing some of the significant points.

One problem that economic analysis can be afflicted with is that, without care, methodology drives ontology. If the only tool you’ve got in your toolbox is a hammer then everything looks like a nail. When econometricians were restricted to working primarily with linear functions then all curves were linear, by assumption and as an approximation.  As the techniques of nonlinear dynamics became better understood and more widely used suddenly we were happy to accept all sorts of behaviours and possibilities – such as multiple equilibria or complex dynamics – that previous generations of economists couldn’t easily accommodate or actively sought to prove to be impossible.

That is just a variation on Smith’s point that ‘we haven’t found the right math yet’.  Seeking to contort the economy on the procrustean bed of an inappropriate mathematical technique can disguise more than it reveals.

But there is a more fundamental sense in which methodology – particularly mathematical formalism – can drive ontology.

Perhaps the most famous example is the case of Hicks’ 1937 interpretation of Keynes’ General Theory. Hicks seemed to have managed to tame Keynes’ approach into an economic framework that was intelligible to conventional economists of the time. However, in doing so he emptied Keynes’ approach of some of its most novel components – particularly the role of genuine uncertainty – because they cannot easily be incorporated into the mathematical frameworks used at the time. In his later years Hicks recanted. He felt he’d sent the debate off in an unhelpful direction. The risk/uncertainty problem is still with us. You can’t travel too far down the road of an economic discussion of uncertainty before it is operationalised as probabilistic risk, which is a completely different phenomenon.

That is why I would depart from Smith in his characterisation of the “less maths” brigade. I wouldn’t dispute that there are some with the motivation to insulate their theorising from any form of testing. But there have been some heterodox economists who eschew the use of much mathematics because they conceive of the economy is something that cannot be tamed and parameterised. They have an ontological stance which leads to a different methodological palette. If you conceive of the economy primarily as a discovery process involving agents operating in open systems making genuine choices under radical uncertainty, who use conventional decision rules and are subject to the double hermeneutic, then there is little to be gained from overly elaborate algebraic specification or heavy duty estimation. Structural stability is a chimera. The economy is an embodiment of Heraclitus’ famous aphorism: you cannot step into the same river twice.

Equally importantly, the maths question is about the allocation of scarce resources. Maths undoubtedly has a part in economic analysis, but does it justify the pre-eminence it currently has? That depends on your view of what economics is trying to achieve. Given disciplinary incentives it makes absolute sense for individuals to focus on signalling advanced mathematical ability, because they know that’s what gets published in prestigious journals and plays well at hiring time. It delivers clever models and analysis honoured as being “deep”.

But if the aim of economics is to advance our understanding of the economy then perhaps the allocation of effort to theory is less obviously justified. Twenty years ago Thomas Mayer wrote Truth versus precision in economics in which he argued that we can think of economic explanation as a chain. The economics profession seems intent on strengthening the links in the chain that are already the strongest – the models – to the detriment of improving the links in the chain that are weakest – the plausibility of assumptions, the behavioural foundations of the models, the operationalisation of concepts, the quality of the data used to test the models. And if a chain is only as strong as its weakest link then that isn’t a wise strategy.

Finally, there is the link between the mathematical models and the way in which they map on to the economy. One of the commenters on Noah Smith’s post cited Alfred Marshall’s famous quote:

(1) Use mathematics as a shorthand language, rather than as an engine of inquiry.

(2) Keep to them till you have done.

(3) Translate into English.

(4) Then illustrate by examples that are important in real life.

(5) Burn the mathematics.

(6) If you can’t succeed in (4), burn (3).

Smith is avowedly not a great fan of argumentum ad verecundiam, but this quote seems to me to have something useful to say. It can be interpreted in different ways. I tend to focus on point (4) and think of it as an anchor. It is a prescription for stopping economics drifting off into its own world of abstraction. It demands that the discipline is not engaged in mathematical pyrotechnics simply for the fun of it. The analysis has to be illustrated with examples from real life. And not by trivial examples or stylised facts but by examples that are of real world importance.

Some economists who prefer to work with serious mathematics never lose sight of what the discipline is ultimately trying to achieve. They are willing to anchor discussion in “examples that are important in real life”. But it is hard to dispute that some have drifted off into the ether, perceiving no great need or obligation to root what they are doing in advancing our understanding of the economy.

So students who come to economics to see if they can understand the world, address the pressing questions of the day, and maybe make the world a better place, end up having their heads stuffed full of maths which appears to have limited relevance to anything of significance. Master the technique; never mind the intuition, let alone the application. They may be mistaken in forming that impression, but it is understandable that they do. And that’s your problem right there.


* Apologies, being British I am programmed not to type “math” without the suffix, even though that is how it usually features in this discussion.

Image: © senoldo –


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4 replies »

  1. “So students who come to economics to see if they can understand the world, address the pressing questions of the day, and maybe make the world a better place, end up having their heads stuffed full of maths which appears to have limited relevance to anything of significance.”

    As a current graduate student in economics, I can relate to this very much–I find myself frustrated by this on a daily basis. Great article. I enjoyed it very much!

    • Hi Elizabeth, thanks for your comment. I don’t think you are alone in that frustration. It seems to me that quite a few people in and around the discipline have recognised the issue over a period of quite a few years.

      But it seems to me that the disciplinary incentives are such that the system is stuck in an equilibrium where relevance and application are not the most highly prized characteristics of analysis. And moving the system to another equilibrium, in which those characteristics are given priority, seems to face insurmountable co-ordination problems.

      I’m sure someone must have done an analysis of the history of the discipline as a path dependent process leading to a suboptimal equilibrium. But I don’t know of it. And if they haven’t then they should have done!

  2. Thank you for this great piece. I didn’t have time or energy to write a long comment on Noah’s blog and it felt like he took my short objections as glib rejoinders. Thank you for writing so cogently about the various angles on the problem.

    My own personal response came on 3 levels:

    1) The question of radical uncertainty that you referred to. Far too often in recent history it seems that once an economics theorem has been put into mathematical language all appreciation of the realms of uncertainty involved have been lost. Arguably it was this category of error that led to the misunderstanding about risk that brought us the financial crisis.

    2) We seem to have seen a lot of epicycle work in economics. The basic model (expressed in some simple curves) gives an intuition about what is going on, but subject to various limits. Lots of journal articles get published building vast edifices of epicycles onto the basic model, which improve it in a fragile way – that is to say they fit the dataset used to test it, but give no insight into the mechanisms at work. As such, some of them keep working as things change – and others don’t. But the crucial problem is the academic profession treats the mathematical edifice as evidence in itself.

    3) I was educated to a high level in systems engineering. I’d guess I’m about 0.5 to 0.75 the level of mathematical skill of Noah (for example) and I have physicist friends who are easily the equal of any economist mathematician. All of that knowledge and education informs my view that economists have a strange relationship with maths. Noah thinks of it as a language, but I’d say rather it needs to be thought of as a tool – and that’s not at all common in economics, because their curriculum doesn’t bring them ease with maths and so having fought mightily to absorb it in difficult circumstances they haven’t had time to develop other tools properly, so maths is their Swiss Army Knife.

    Now many other tools (simulation etc.) involve maths, but that’s not the same as the modelling economists’ journals tend to prize.

    4) For all the education in systems engineering, the basic lesson was the most important – the model isn’t reality and knowing that and appreciating the failure points is the key to success. It seems they don’t hammer that into you in economics in quite the same way…

    • Thanks for your comment. I agree with pretty much all you say. I think your point 4 is particularly important. It isn’t just that this point isn’t hammered in to economists – the mindset is almost the opposite at times. And simple models are often inappropriately used to generate policy prescriptions that are imported straight into the policy making process, with only limited attention to whether the model is actually a good representation of the social phenomenon in question. This can, of course, then have huge and damaging real world ramifications. The fallacy of misplaced concreteness is an ever-present danger.