# Posts Tagged Maths in Economics

### The Illusion of Mathematical Certainty

Nate Silver’s questionable foray into predicting World Cup results got me thinking about the limitations of maths in economics (and the social sciences in general). I generally stay out of this discussion because it’s completely overdone, but I’d like to rebut a popular defence of mathematics in economics that I don’t often see challenged. It goes something like this:

Everyone has assumptions implicit in the way they view the world. Mathematics allows economists to state our assumptions clearly and make sure our conclusions follow from our premises so we can avoid fuzzy thinking.

I do not believe this argument stands on its own terms. A fuzzy concept does not become any less fuzzy when you attach an algebraic label to it and stick it into an equation with other fuzzy concepts to which you’ve attached algebraic labels (a commenter on Noah Smith’s blog provided a great example of this by mathematising Freud’s Oedipus complex and pointing out it was still nonsense). Similarly, absurd assumptions do not become any less absurd when they are stated clearly and transparently, and especially not when any actual criticism of these assumptions is brushed off the grounds that “all models are simplifications“.

Furthermore, I’m not convinced that using mathematics actually brings implicit assumptions out into the open. I can’t count the amount of times that I’ve seen people invoke demand-supply without understanding that it is built on the assumption of perfect competition (and refusing to acknowledge this point when challenged). The social world is inescapably complex, so there are an overwhelming variety of assumptions built into any type of model, theory or argument that tries to understand it. These assumptions generally remain unstated until somebody who is thinking about an issue – with or without mathematics – comes along and points out their importance.

For example, consider Michael Sandel’s point that economic theory assumes the value or characteristics of commodities are independent of their price and sale, and once you realise this is unrealistic (for example with sex), you come to different conclusions about markets. Or Robert Prasch’s point that economic theory assumes there is a price at which all commodities will be preferred to one another, which implies that at some price you’d substitute beer for your dying sister’s healthcare*. Or William Lazonick’s point that economic theory presumes labour productivity to be innate and transferable, whereas many organisations these days benefit from moulding their employees’ skills to be organisation specific. I could go on, but the point is that economic theory remains full of implicit assumptions. Understanding and modifying these is a neverending battle that mathematics does not come close to solving.

Let me stress that I am not arguing against the use of mathematics; I’m arguing against using gratuitous, bad mathematics as a substitute for interesting and relevant thinking. If we wish to use mathematics properly, it is not enough to express properties algebraically; we have to define the units in which these properties are measured. No matter how logical mathematics makes your theory appear, if the properties of key parameters are poorly defined, they will not balance mathematically and the theory will be logical nonsense. Furthermore, it has to be demonstrated that the maths is used to come to new, falsifiable conclusions, rather than rationalising things we already know. Finally, it should never be presumed that stating a theory mathematically somehow guards that theory against fuzzy thinking, poor logic or unstated assumptions. There is no reason to believe it is a priori desirable to use mathematics to state a theory or explore an issue, as some economists seem to think.

*This has a name in economics: the axiom of gross substitution. However, it often goes unstated or at least underexplored: for example, these two popular microeconomics texts do not mention it all.

### 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.