Tuesday, October 9, 2012

Math in economics papers: An explanation of an interesting trend in the data

Justin Wolfers links to a new paper presenting the upward trend in the number of equations per paper in economics. Here are two figures from that paper:

Source: Espinosa et al., 2012

What's your reaction?

Wow - big growth in the number of equations!

Yes, definitely (and I think that's almost certainly a good thing although of course how its done needs to be approached with care).

But I was also struck by the decline in equations after 1980 even though the trend in econometric output kept going up during that period. What is causing that?

My theory: that may be the impact of New Classical and Real Business Cycle economics.

It's just a thought. If you think about pre-New Classical stuff as being big systems of equations it seems to make sense. That would taper off with New Classical work. My sense is that a lot of the bigger DSGE type stuff is done in central banks and not necessarily put in articles, although to the extent that it is that happened later and may explain the trend picking up again.

Or I could be completely wrong. But I found that dip interesting. I *quickly* searched the article and did not see an explanation for the break in 1980.


  1. Ah Daniel, but what kind of mathematics is being used? Linear and additive mathematical equations, or non-linear and non-additive mathematical equations?

    According to Dr. Michael Emmett Brady, Keynes endorsed the use of non-linear and non-additive mathematical equations when they were soundly applied with good logical foundations. Keynes opposed the application of *linear and additive* mathematical equations to data that was non-homogeneous over time, like that of financial markets or long-term fixed capital investment. See the following article by Dr. Michael Emmett Brady for Keynes's approach to the use of mathematics in the social sciences for more information!


    1. Perhaps I am missing your point, but I seem to remember quite a few linear equations in the General Theory itself, Blue Aurora.

      Use the math that does the job, definitely.

      Otherwise I'm not sure what you're getting at. What is the problem with linear equations exactly? I don't have time to read a Brady article - perhaps you could provide a synopsis. I think a lot of equations in economics - particularly constraints that we optimize against and various accounting identities - are quite naturally linear. Where am I going wrong Blue Aurora?

    2. Yes, in The General Theory, there are equations that Keynes uses that can be described as linear and additive. But the point of Brady's 1988 article in Synthese is that it demonstrates that Keynes wasn't opposed to the use of mathematics, it's just how they were used. Linear and additive equations don't work when it comes to data that constantly shifts - hence Keynes's criticisms of Jan Tinbergen and the econometricians in the late 1930ies and early 1940ies. This is why you could make the argument that the Lucas Critique is a footnote to Keynes's criticisms of the field of econometrics.

      Have you re-read Book V of The General Theory yet?

      Keynes uses differential and integral calculus to model multiple unemployment equilibria. He is using those forms of mathematics as best as he can. Keynes was playing by Pigou's rules of the game for a purely theoretical argument, and he bested Pigou by demonstrating that there are multiple equilibria in addition to that of a full employment equilibrium. I know this isn't exactly the best explanation in the case of The General Theory, so I'm going to have to defer to Dr. Michael Emmett Brady for more information.

  2. This scares me — not only does it smack of faux-rigor — mathematicization lineates what may be nonlinear phenomena, and simplifies what may be more multivariate phenomena — but it puts up a wall around much economics research to make it accessible only to mathematically-minded readers.

    I basically agree with Keynes — simple models with as few assumptions as possible. But also I think it is wise to not write in mathematics what could otherwise be written in English — to use mathematics as a supplement rather than as a default language (and also a means to exclude non-mathematicians).

  3. Aziz: You may want to read this paper in addition to Dr. Michael Emmett Brady's 1988 paper in Synthese for more information on Keynes's attitude toward mathematics. According to Dr. Brady, Keynes was capable of doing difficult maths, and he would fit comfortable in the realm of "Chaos Theory".


  4. I think the growth in equations has some upsides and some down-sides. On the one hand, math offers a very precise language which can make communication easier. Similarly, when talking with fellow programmers, we will frequently just write down code because it expresses things more precisely and allow us to cut out hours of explanation.

    However, the problem with math-based models is that it can be easy to lose sight of the underlying reality. When you've spent enough time manipulating an equation, the variables become just variables which can be manipulated at will. I recall in college building a simple but very nifty game theory model for the sale of indulgences. It appeared to work quite well and I was running forward with all sorts of simulations and permutations on the equations until I explained it to my roommate in English who pointed out I had just predicted the end of Catholicism. It's not that the model was bad, it's just that it had some limitations that were obvious in English, but not in the Math.

  5. If only the up-and-to-the-right chart included a corresponding increase in the rate of re-translation into English. Oh well, maybe that's what the blogosphere is for.

    "1) Use mathematics as 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. This I do often." - Alfred Marshall.


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