Tuesday, October 2, 2012

Further thoughts on mathematics and economics

After almost 3 years of blogging to a reduced audience, I observed a sudden spike in my stats after my post about the UMKC conference got re-blogged (or re--tweeted, who knows) in the legendarily active econo blogosphere. So I thought I should ride the wave and elaborate on my remarks. 

First of all, as pointed out by a commenter, I only attended the last day of the conference, which according to the unwritten laws of scheduling was 
bound to be the least exciting (especially on a Saturday!). I guess I should expect that - after all that was the spot on the schedule reserved for me... But still, 
after being blown away by the vibrance and intellectual rigour of the likes of Stephanie Kelton and Scott Fulwiller when I met them at Fields a few months ago, I had great 
expectations about the event, and was disappointed to find out that most of the people I wanted to meet had already left (his lordship notwithstanding). 

But on a more substantive note, I see from the comments (here and elsewhere), that the role of mathematics in economics is a bit of a raw nerve, hence the urge to elaborate.

Despite being a mathematician, I do not think that mathematical modelling is the most important part of economics, but I do think that it is somewhat essential. Here I'm reminded of the famous 
saying that "logic is to mathematics what hygiene is to life: it's clearly essential, but not what it's all about". The same goes for mathematics and economics: historical awareness, acute observations, and empirical plausibility come first in economic reasoning, but I don't see how much progress can be made without mathematics. Notice that this is not about pedagogy, but about being able to even formulate crucial statements. 

To borrow from other fields, it would be nearly impossible to even conceptualize something like the basic reproductive number  in epidemiology without a mathematical model, and this means the difference between being able to handle a pandemic or not. For another example, no amount of analogy or logical thinking can pinpoint the phenomenon of bifurcations. The fact that smooth changes in some underlying parameter can cause a system to completely change its qualitative behaviour is not something that is predicated by logic and hard thinking alone - it needs mathematics. Needless to say, the list could go on and on.

Finally, just in case you associate mathematics too heavily with neoclassical economics, remember that the final blow to general equilibrium was dealt by a mathematical result: the SMD theorems essentially tell us that the whole framework (and much of neoclassical economics with it) is only guaranteed to work for the trivial case of one agent and one good. 


  1. Actually, to address your last point Mathaeus, Neoclassical was dealt the REAL blow in the capital debates a few decades before SMD, and that was done through logic first and foremost, but was illustrated using input-output tables by Pierro Sraffa.

    But I think I can speak for heterodox economists in general that we almost fully agree with the rest of your statements.

  2. "but I don't see how much progress can be made without mathematics. "

    Mathematics always underlies everything, but raw mathematics isn't necessarily the best modelling tool.

    The model is less important than the communication of the model. People have to be able to understand it and what it means.

    Which is why in computer system modelling we dropped formal methods years ago for a more pragmatic and higher level modelling approach. You get less domain level errors and more understanding, but it is still type theory underneath.

    The case in point is variable names. I would use 'turnover_of_wage_share_per_annum' rather than 'w'.

    1. Formal methods are very much still alive and kicking and are absolutely essential when it comes to developing 'high integrity' software such as safety-critical and security-critical software e.g. in flight-control or encryption systems.

      Similarly if we want to 'fly' the complex, unstable and potentially dangerous system we know as the economy safely, it makes no sense at all not to reason about it via mathematically rigorous models. Of course it goes without saying that mainstream neo-classical economics has given this approach a bad name, but that does not mean to say that Steve Keen are Mathaeus wrong to try.

    2. "It makes no sense at all not to reason about it via mathematically rigorous models."

      It makes no sense to reason about it *exclusively* with rigorous models only two men and ET can understand.

      The economy is people interacting and the system will be more stable the more people understand how it works and what simple steps are in place to dampen the oscillations.

      You have to communicate models to others in terms *they* can understand.

      That is why front facing system do not use high-order mathematics in the modelling.

    3. In arguing for formal methods I wasn't arguing against heuristics, empirical observation and logic. Similarly it's perfectly possible that an exceptional practitioner can excel both as a theoretician and a communicator. Richard Feynman comes immediately to mind.
      Steve Keen is more than capable of expressing his ideas in lay terms, but it is his underlying mathematics that is producing real results inside the profession.
      For example he has used mathematics with devastating effect to demolish the superficial mathematical models espoused by the mainstream - fighting them on their own ground so to speak. And it’s having an effect: witness his online jousts with Krugman etc. over past few months (first they ignore you, and then they laugh at you....).
      Whilst more constructively, with help from Matheus, Steve has used mathematics to unify the MCT and MMT formalisms.

  3. I think that if you want to substantiate this you need to give an example: what in economics must be expressed in mathematical terms? Even the best models that I've seen are just didactic, really, and you can understand them pretty well without mathematics.

    Of course, I'm of the opinion that mathematics actually DAMAGES economics in that it focuses the attention on the formal properties of the model and detracts attention from the dubious assumptions laid out. This, I think, was why Joan Robinson was so good at picking neoclassical economics apart.

    1. The entire subject of decision making under uncertainty is impossible without mathematics, since it requires axiomatic probability theory to avoid well known paradoxes. Even notions like Knightian uncertainty are best expressed in terms of robustness with respect to specification of probabilities and sample spaces.

      But to give you an example outside my own field, game theory has been used as a replacement for naive utility optimization to serve as a basis for modelling economic interaction, and it is fundamentally a mathematical theory.

      As for your last remark, every mathematical result worth its name needs to spell out the precise assumptions on which it is based. Anyone who tries to obscure this is not doing either mathematics or economics.

    2. "game theory has been used as a replacement for naive utility optimization to serve as a basis for modelling economic interaction, and it is fundamentally a mathematical theory."

      And it is an inaccurate model - hence why the Spectrum auctions in the UK went mad.

      It's not a good idea for an Autistic to model human social interaction.

      Mathematical models are useful approximations and abstractions to help you make sure that the logic is consistent.

      But you always have to check that your abstraction is appropriate.

      For example are you absolutely sure that relying on a Philips Curve in your models is appropriate and empirically justified?

  4. "The entire subject of decision making under uncertainty is impossible without mathematics, since it requires axiomatic probability theory to avoid well known paradoxes. "

    Impossible . . . yet human beings make decisions under uncertainty every day, whether they know any mathematics or not.

    The mathematical rigor and elegance of axiomatic choice theory made it an uphill battle for behavioral economics to be taken seriously. I doubt psychologists needed much convincing, however, that humans aren't expected utility maximizers.

    1. Oh come on Winslow, that's a cheap shot...

      Fish don't solve differential equations, clouds don't run computer simulations, Amazonian tribes don't read anthropology, etc.

      There is a difference between the object of study in a model and the tools used to study them.

      And I didn't mean axiomatic choice theory, which is far too rigid as you point out. I meant that any other alternative, including behavioural economics, needs to take mathematics (and axiomatic probability theory) into account.

  5. I agree Mathematics is essential to building a good alternative to mainstream economics. However, I think there are issues where non-mathematical modes of presentation are warranted.

    My paper comparing the late 18th/ early 19th "Dollarzone" to the Eurozone is primarily a historical and institutional analysis with no formal mathematical model. However I think it was worthwhile and contained useful information. On the other hand, that doesn't mean I'm opposed to a mathematical model of the Eurozone.

    1. Indeed Nathan! For example, I still think that Panics, Manias, and Crashes is the best account of financial crisis (and macroeconomics in general) ever written. Any model attempting to describe these phenomena needs to pay close attention to the historical and institutional setting in which they occur. I'm simply objecting to the notion that math is at best irrelevant and at worst down right dangerous for economics.

    2. True. To be fair to Lord Skidelsky, I just re-watched a video a friend took of the Q and A session. He didn't say mathematics is irrelevant, only that before a mathematical model is presented, the modeler needs to ask (and answer) this question: why is this model necessary?

      To me he was taking Einstein's dictum to heart that "the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience".

      If you had started out with a methodology slide explaining why you thought the mathematics was necessary I bet he would have been satisfied.

  6. I agree with the main post that mathematics should be used to check the hygiene of our theories, but why is an exception made of axiomatic probability theory?

    Lord S. is a fan of Keynes, whose Treatise on Probability challenges the standard theory's assumptions. He later applies this to explain crashes. So isn't there scope for a renewed mathematical challenge here?

  7. There is a reference to the 'madness' of the UK spectrum auctions. Actually, the team were tasked with raising the maximum revenue directly from the auction, and to take no account of any broader impact. I think we should all be pleased that it wasn't more successful. To me, the madness was that lessons weren’t learnt from this mathematically-driven demonstration of the flaws in the then economic dogma. Mathematics is dangerous if you misuse it.

  8. Matheus,

    Can't post at Steve Keen's blog - have some login issues. So here are two posts on the talk/paper you have with Keen: