Math: Useful & Over-Used

Paul Krugman:

Noah Smith … on the role of math in economics … suggests that it’s mainly about doing hard stuff to prove that you’re smart. I share much of his cynicism about the profession, but I think he’s missing the main way (in my experience) that mathematical models are useful in economics: used properly, they help you think clearly, in a way that unaided words can’t. Take the centerpiece of my early career, the work on increasing returns and trade. The models … involved a fair bit of work to arrive at what sounds in retrospect like a fairly obvious point. … But this point was only obvious in retrospect. … I … went through a number of seminar experiences in which I had to bring an uncomprehending audience through until they saw the light.

Bryan Caplan:

I am convinced that most economath badly fails the cost-benefit test. … Out of the people interested in economics, 95% clearly have a comparative advantage in economic intuition, because they can’t understand mathematical economics at all. …. Even the 5% gain most of their economic understanding via intuition. .. Show a typical economist a theory article, and watch how he “reads” it: … If math is so enlightening, why do even the mathematically able routinely skip the math? .. When mathematical economics contradicts common sense, there’s almost always mathematical sleight of hand at work – a sneaky assumption, a stilted formalization, or bad back-translation from economath to English. … Paul[‘s] … seminar audiences needed the economath because their economic intuition was atrophied from disuse. I can explain Paul’s models to intelligent laymen in a matter of minutes.

Krugman replies:

Yes, there’s a lot of excessive and/or misused math in economics; plus the habit of thinking only in terms of what you can model creates blind spots. … So yes, let’s critique the excessive math, and fight the tendency to equate hard math with quality. But in the course of various projects, I’ve seen quite a lot of what economics without math and models looks like — and it’s not good.

For most questions, the right answer has a simple intuitive explanation. The problem is: so do many wrong answers. Yes we also have intuitions for resolving conflicting intuitions, but we find it relatively easy to self-deceive about such things. Intuitions help people who do not think or argue in good faith to hold to conclusions that fit their ideology, and to not admit they were wrong.

People who instead argue using math are more often forced to admit when they were wrong, or that the best arguments they can muster only support weaker claims than those they made. Similarly, students who enter a field with mistaken intuitions often just do not learn better intuitions unless they are forced to learn to express related views in math. Yes, this typically comes at a huge cost, but it does often work.

We wouldn’t need as much to pay this cost if we were part of communities who argued in good faith. And students (like maybe Bryan) who enter a field with good intuitions may not need as much math to learn more good intuitions from teachers who have them. So for the purpose of drawing accurate and useful conclusions on economics, we could use less math if academics had better incentives for accuracy, such as via prediction markets. Similarly, we could use less math in teaching economics if we better selected students and teachers for good intuitions.

But  in fact academia research and teaching put a low priority on accurate useful conclusions, relative to showing off, and math is very helpful for that purpose. So the math stays. In fact, I find it plausible, though hardly obvious, that moving to less math would increase useful accuracy even without better academic incentives or student selection. But groups who do this are likely to lose out in the contest to seem impressive.

A corollary is that if you personally just want to better understand some particular area of economics where you think your intuitions are roughly trustworthy, you are probably better off mostly skipping the math and instead reasoning intuitively. And that is exactly what I’ve found myself doing in my latest project to foresee the rough outlines of the social implications of brain emulations. But once you find your conclusions, then if you want to seem impressive, or to convince those with poor intuitions to accept your conclusions, you may need to put in more math.

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  • Robert Koslover

    Regarding questioning the value of using math to argue economics, doesn’t the mere existence of such a discussion among seriously-regarded economists suggest that “That Old SF Prejudice” ( may not be entirely without merit?

    • Not at all! Unless you think it is obvious that math models are always very useful to understand any topic. Which I don’t.

      • Robert Koslover

        Fair enough. Now, I note that at your own university, passing at least a few courses in mathematics is required to obtain an economics degree. (See .) Regardless of whether this helps students of economics actually understand economics any better, I’m willing to believe that making them study math is good for them!

      • The discussion of math models shows economics’ lack of scientific maturity. That it still needs to deal with problems prior to quantification speaks to the field’s level of development.

        (For a field (in psychology) that is more mature than economics, consider vision science.)

  • Is the intuition that “your intuitions are roughly trustworthy” itself trustworthy? That’s the sort of problem I’d expect Robin Hanson of all people to anticipate.


    “But in fact academia research and teaching put a low priority on accurate useful conclusions, relative to showing off, and math is very helpful for that purpose.”

    Yup, just make difficult-looking formulas out of simple concept and your paper suddenly looks a lot more impressive. But it would be useful to more often check against statistical errors and biases, that form of math is underused.

  • adrianratnapala

    Show a typical economist a theory article, and watch how he “reads” it: …
    If math is so enlightening, why do even the mathematically able
    routinely skip the math?

    I think you can replace “economist” with “physicist” and still have a true sentence. Physicists will pore over the details only for the few papers they have a specific reason to concentrate on. That is maybe 5% of the readers, but they are the most important 5% to write for.

    • blink0

      Yes, and in fact how experts “read” does not tell us much. All seem to agree that the math can be difficult, so the question is whether it is worth the time investment to go through it. Of course, then, we should read the results from several papers and then investigate the mathematics behind only those few that are most surprising, important, etc.

  • Vaniver

    I think your “Krugman replies” link should go here:

  • ben

    Yep, I completely agree. Someone (I can’t remeber who) said recently that the theoretical advances that tend to get the most traction within the discipline these days are those that allow young academics who are mathematically talented but not very creative to demonstrate how clever they are.

  • I agree that formal mathematical models help one think more clearly and forces one to state one’s assumptions up front, the problem is that, as George Box famously said, “most models are wrong …” and distinguishing the useful models from bad models is not worth the effort … As an aside, I would still rather read a chapter of The Prince, or Ancient Law, or The Theory of Moral Sentiments than 12 AER articles

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  • dmytryl

    I for one fail to see any, any what so ever interesting results arising from intuitive economics. None. Granted, the intuitive is how people actually decide, and the intuitive economics rules the world, but as far as transferable knowledge goes, it is literally zero.

    Meanwhile, mathematical economics produces very interesting results, for example, mathematical models which replicate Pareto distribution in a market game regardless of the distribution of “skill” as long as skill is sufficiently uniform. Or a wide variety of other simulations.

    • IMASBA

      It’s not always “intuitive” as in “common sense”. It’s often “intuitive” for someone who already studied mathematics (statistics in particular) and many concepts can be explained using simple analogies (so without complicated math), just like statistics 101 starts out with vases and dice instead of complicated formulas. And with economics there’s always the risk you get buried so deep in the math that you forget whether your model and the terms in it even make sense.

      • dmytryl

        Well, if it was the case then it would have been possible to also express formally, while the debate quoted by Hanson seems to concern things that can’t be so expressed. Here, Caplan is just a guy who’s trying to re-frame his inability to do mathematics well as something other than a product of lack of intelligence and perhaps laziness – he’s trying to present it as some sort of smart non participation in a silly game of status. That much is clear to most people that would read his passage, whereas a few would imagine that it is some sort of costly signalling that he can engage in because he’s a proven genius.

      • Caplan can do mathematics, he got his phd under Bernanke (although maybe he’s not good enough to compete with the top math jockeys for prestige). He’s just WAAAAAAAAAAYYYYYYYYYY too confident in his intuition.

      • dmytryl

        Well, I presume he’s not some sort of mystic or non-reductionist, so the intuitions in question got to be describable with mathematical processes. Such as, most likely, politically biased randomness uncorrelated to the truth value of statements.

      • Well, I presume he’s not some sort of mystic or non-reductionist

        I think he is non-reductionist—and something of a mystic (in that he believes in libertarian free will). While they’re not always or even usually mystics, I think all economists are non-reductionist. I’ve never seen someone propose, even in theory, to reduce economics to physics.

      • VV

        Stephen Diamond

        I think all economists are non-reductionist. I’ve never seen someone propose, even in theory, to reduce economics to physics.

        Nicholas Georgescu-Roegen attempted to do that, with somewhat questionable results:

        Steady-state economics by Herman Daly is an offshoot of that line of research, which is less bent towards theoretical formalization and more towards data.

        Physicist Tom Murphy endorses this theory on his blog:

        Mainstream economists tend to reject the theory.


        According to Wikipedia (parenthesis mine), Bryan Caplan is “a professor of Economics at George Mason University (Koch), esearch fellow at the Mercatus Center (more Koch), adjunct scholar at the Cato Institute (even more Koch)”. He is also an “anarcho-capitalist”.

        I suppose that math and extreme political ideology don’t mix well.

      • oldoddjobs

        You, on the other hand, view things from “nowhere”, right? My, how value-free you are! Nothing extreme here, move along.

      • VV

        @oldoddjobs:disqus Sure, I have Kim Jong-un, Fidel Castro and Lenin’s mummy pay my bills…

        @TGGP:disqus Specific mathematical models in economy are often criticized for including excessively strong, unrealistic assumptions.

        But that’s the good thing of doing math: you have to make all your assumptions explicit.

        By arguing in favour of “intuition” and against math, Caplan is trying to lower the standard of proof in economical discourse in order to get a free pass to peddle his favourite ideology without offering strong arguments or making his assumptions explicit.

      • dmytryl


        Yeah, exactly. Thing is, mathematical approach is more open to critique.

        Other thing is, the more formal is the reasoning the harder it is to fit it to particular ideology – I don’t doubt that even the Rand-inspired libertarianism these guys are spewing can be backed with mathematical models by very carefully making very specific but fairly plausible looking assumptions, but anyone capable of doing so has better things to do.

        There’s some very robust results – such as the distribution of wealth not being indicative of some highly superior skill of the very wealthy but being a general result of multiplicative game – these results are not very kind to libertarianism which needs it that the very rich got superpowers and should thus have the control. (Frankly, I don’t think bare bones libertarianism needs that, but the Randian variety, which exists to tell the rich how deserving they are, does)

      • oldoddjobs

        Which guys are spewing Rand-inspired libertarianism? Is that the same as Objectivism, or something else? When you say that the Randian variety “exists to tell the rich how deserving they are” do you mean that it was designed by Rand specifically to do that, with everything else being just a smokescreen? What about those made rich through government? Clearly they’re out. Which rich people are being “told” they deserve it? What end does it serve? (presumably they don’t need to be told anything). Arggggh you make so little sense it hurts.

      • VV

        Wikipedia is your friend:

        “According to Caplan, he was first introduced to libertarian capitalist political philosophy through the writings of Ayn Rand and that it was his interest in philosophy that drew him to study economics.”

        “What about those made rich through government? Clearly they’re out. ”

        “Which rich people are being “told” they deserve it?”
        I think the point is not that rich people need to be told, it’s the poor/middle class people who need to be told things such as “Inequality talk is about grabbing”

      • IMASBA

        “But that’s the good thing of doing math: you have to make all your assumptions explicit.”

        Not in economath: you can just pile model upon model until it becomes near impossible (for both the author and the reader) to figure out what all of this would actually mean in terms of basic assumptions. You can easily use parts of models that have conflicting assumptions and few people would notice, I think that is in part because people think less about the basic assumptions when math is involved because it’s hard and because math gives everything an aura of legitimacy. Economath is not like physics where most models are versions of other models with different basic assumptions (because the wrong assumptions get unmasked very quickly and decisively compared to the wrong assumptions in economics where models keep being used decades after the assumptions behind them have been trashed by psychology or statistics).

      • VV, I don’t know if there’s any correlation of math with ideology. I hear a lot of complaints about mathematical models in econ from folks on the left, whether Post-Keynesians or just Krugman fans who don’t like the ratex/neoclassical/RBC turn under Lucas, Sargent, Prescott etc. The more abstract complaint is that the math is so complicated the only models that are halfway tractable must closely resemble an overly functional view of markets with perhaps a few frictions. Of course, the “freshwater” folks aren’t necessarily right-wing either, Steve Williamson identifies as a liberal/progressive Democrat, Sargent & Sims are similar, even if they have a low opinion of Krugman and his arguments for fiscal stimulus.

      • I don’t know if there’s any correlation of math with ideology

        My prediction (“A taxonomy of political ideologies based on construal-level theory” — ): Managerialists and Monomaniacalists will embrace math: Demagogists and Utopianists will reject it. (This is basically a near-mode vs. far-mode conflict with respect to “means” or “method.”)

        (Caplan’s anarcho-capitalism probably counts as a form of Utopianism. Hanson is a Monomaniacalist. Krugman is a Managerialist.)

      • dmytryl

        I think the ones with views that are most of a far stretch will reject mathematics the most.

        Even simplified models can shed a lot of light on validity of intuitions. For example intuitively the richest must have some superpowers, which is why they are richest. But when there is a model exhibiting the same wealth distribution as the real world even when the “skills” are exactly identical, or when a monkey flinging poo at a chart performs as well as an economist, that really calls into the question our assumptions – now we know that mere shape of the distribution does not indicate corresponding distribution in skill.

      • Christian Kleineidam

        Looking at a distribution that you get by sampling the real world has nothing to do with math in the sense math is used in the discussion.

      • I think he is a non-reductionist, he insists that any emulation of him, even if it was in a body identical to his current one, would not “really” be him, but merely a copy. As Stephen Diamond references, he also believes in genuine free will.

        Stephen, is Robin Hanson not a reductionist? I think his objections to reducing economics to physics would be pragmatic rather than “in principle”.

      • The principle of charity inclines me to believe he’s not a reductionist. Robin seems to think something like economics governs all rational agents operating under principles of scarcity. But there seems general agreement among philosophers today that “rational agent” isn’t reducible to physics, as it would require an infinite disjunction to specify the corresponding physical states.

      • Gary Becker wrote “Irrational Behavior and Economic Theory” about how scarcity is enough, even with irrational agents, for much of basic economics to hold.

        Robin’s thinking seems inspired by evolutionary theorizing about selective pressures. Evolution/ecology is supposed to be reducible to genetics & biology, and in turn to chemistry and physics. I don’t know if the concept of “rationality” is needed to claim that entities behaving in a certain manner will displace those acting in a different one. Are single-celled organisms “rational”?

      • Do evolutionary biologists believe their field is reducible? Do geneticists hope to reduce evolutionary psychology? I don’t think so.

        I guess I’m not sure who is doing the supposing.

      • I was under the impression most scientists are reductionists, although I don’t have anything to back that up off the top of my head.

      • oldoddjobs

        Arrogant nonsense.

  • arch1

    I’m currently reading “The Trouble with Physics” by Lee Smolin. I find it a fascinating look at how the academic physics community works, from one person’s perspective. Some of his claims (my possibly-botched paraphrases) tie loosely to this discussion thread:

    -Physics is currently (2006) dominated by string theorists. In that subfield:
    -pulling off intricate technical calculations in one of the the 2-3 currently hot topics is the best way to advance one’s early career
    -there is a need for (and paucity of) revolutionary ‘seers’ (as opposed to consummate technicians) capable of reexamining foundational issues as did Einstein, Heisenberg, Bohr
    -string theory has as yet articulated no core principles of the kind that guided General Relativity
    -string theory so far has no connection with experiment
    -widely held assumptions (e.g. that someone-or-other has demonstrated string theory’s freedom from infinities) can persist for many years despite the fact that no one can produce the paper, because it does not exist.

  • Grant

    In software development we get lots of neat, intuitive ideas described in natural language. Then when we go to encode these informal ideas into a formal logic of code, we find many holes and problems with our intuitive understanding of the ideas. Over time we develop intuitions on how hard it will be to turn a natural idea into a formal one. These intuitions are never perfect because software development is usually about implementing unique ideas or mutating old ones, not re-doing something as it has been done before.

    In short, the act of encoding ideas into a formal language forces us to more thoroughly examine the topic than our intuitions allow. In the case of using a language with a strong type system, the compiler can actually do much of this forcing.

    When it is actually put into practice, is economics any different? You need math to create a prediction market, for example. This isn’t to say that math isn’t over-used for the purpose of seeming impressive, but I think it my be necessary to actually apply economics.

    • rrb

      in other words, you think it’s overused but useful?

  • Ely Spears

    For me it’s simple: the most trustworthy intuition that I have is that I should not trust my own intuitions or the intuitions of others. Rarely does the extra cost of verifying the intuition with rigor actually hurt me. Maybe that wasn’t true in less wealthy times, but it’s true for me now, so I can easily just pull a sort of General Thud kind of trick and say, ‘I won’t listen to you unless you use math.” Math is like unit testing for intuition. And companies that feel like the cost-to-benefit of writing unit tests is too low usually don’t continue being companies for very long. Programmers who feel they just have an intuition for writing correct code and so skip writing the tests usually don’t have a job for too much longer… So why not require rigor?

  • What seems largely missed in this discussion is that using mathematical models to express economic intuition (call it intuition1) itself depends on another intuition: the intuition that the model adequately expresses the first intuition. (Intuition2.)

    The principle properly governing use of math models seems to be something like: Use the mathematical formalization if and only if you think Intuition2 is more reliable (in the particular case) than intuition1.

    (Some here seem to be arguing that unless you can find a formalization for which Intuition2 is strong—or perhaps stronger than Intuition1–you’re not doing good economics. I don’t see how anyone besides a practicing economist can decide this: it depends on the maturity of the subfield.)

    Caplan maintains that Intuition1 is usually better than Intuition2. If this is true (or where it is true), it’s a serious criticism–not on the simple “cost-benefit” grounds Caplan raises but on the grounds that the economics is obscurantist.

    Caplan presents (there’s a link in the cited piece) an example of his own math modeling. He purports to show that educational level in a society reflects real economic development despite a countervailing trend for education increasingly to consist of unproductive signaling. I found his argument impossible to follow without study (which I didn’t give it)–which I consider bad even for a highly mathematical piece. (See “Overzealous Concision: Density” — and the Bourbaki quote) It was genuinely obscurantist: it left me wondering whether it was a spoof. (Conclude what you will.) But for those who claim that math in economics is always good, read Caplan’s piece on education and social decay for a possible counterexample. Could the issue not be addressed illuminatingly in a purely conceptual fashion?

    • IMASBA

      @Stephen Diamond

      “What seems largely missed in this discussion is that using mathematical models to express economic intuition (call it intuition1) itself depends on another intuition: the intuition that the model adequately expresses the first intuition. (Intuition2.)”


    • Eliezer Yudkowsky

      The intuition1-intuition2 distinction seems very important. The Simple Math of Economics is useful because there are cases where intuition2 (the inside-view belief in the premises) seems very strongly supported. Outside of those cases, an intuition1 is not much more convincing when expressed in math.

    • rrb

      tl;dr summary: use math iff your intuition about the appropriate model is stronger than your intuitions about appropriate conclusions

  • Peter St. Onge

    Half the price, double the benefit: formal logic instead of both intuition and economath.

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  • rrb

    missing from the last sentence: you may also need to put in more math if you suspect your clinging to an intuition because of ideology

  • Jayson Virissimo

    Even the ancients understood mathematics as a kind of filter to keep dumb people out of serious intellectual discussions. For instance, above the entrance to Plato’s Academy, there was purportedly a sign reading “let no one inapt to geometry come in.”

  • Weaver

    I find an unwillingness to go to maths is a good indicator of weak argument. “I’m not going to put a number on it” is often short for “I don’t want to formally structure my argument”.

    • IMASBA

      ” “I’m not going to put a number on it” is often short for “I don’t want to formally structure my argument”.”

      That’s not the kind of math Krugman and Caplan are referring to, if they were they’d be totally guilty of avoiding real argument like you say. They are talking about the (mostly unnecessary) use of bizarrely complicated calculus and such to show off, those articles sometimes make the average physics article look simple, but that’s all they do, it’s just about looks, not substance. If economists gave their lectures like that any half decent college would fire them on the spot because if they didn’t no student would ever pass a course.

      Economists who want to show off will find a way to use the description “the identity element of the abelian group of real numbers, except zero” to describe the number “1”.

      • Weaver

        That’s fair enough.

    • Christian Kleineidam

      Math is not about putting numbers on things. Math is about having axioms and proving theorems.

      • Alium

        The entire purpose of math, the reason why we use it is to quantify. An inordinate focus on the theorems and axioms without an ability to make predictions using real data makes for a truly useless framework.

        If your mathematical framework is not applied to make accurate, repeatable predictions; then your framework is not useful.

        Answer me this: can you explain why the Philips Curve has the statistical regression that it does? If you see oscillations in the relationship between inflation and unemployment rate, can your framework predict a-priori what the frequency of those oscillations are?

        Or is the usefulness of your mathematical framework limited to making generalizations, truisms and successfully being manipulated to confuse the public and cause the next recession?

      • Christian Kleineidam

        Mathematical logic is a subject that happens to be useful. It tells you whether certain things are true and others are false. Logic doesn’t need any numbers.

        Statistics is a subject on it’s own and a lot of mathematicians want to have nothing to do with it.

        I do care that the garbage collector of my android phone runs well. That has something to do with math, but it’s not about numbers.

  • Michael Wengler

    The formal econ I took was at Caltech so maybe me and my profs were a little happier with the math than the rest of econoworld. I realize Robin is Caltech also, but he has had an entire career to assimilate with the rest of econoworld, and my point here is that the math was very helpful in learning basic econ, for me and it seems for the others in my classes.

    Based on my non-economic work now, R&D for cell phones, I’d say you should always use at least enough math to run a sim, at least a trivial “sim” which is to show some numerical curves of one thing against another. Not only does this serve as a great check on your intuition, but you can look at what you must assume mathematically and ask yourself what each piece means, enhancing old intuitions even developing new intuitions.

    As to complex gobbledy-gook written out math, for me and apparently for most people that is sort of the 4th line of defense, there to check if you really get in to something but never the first step in figuring out what is going on.

  • IdPnSD

    The article says “But in fact academia research and teaching put a low priority on accurate useful conclusions, relative to showing off, and math is very helpful for that purpose.” – exactly correct.

    Both real numbers and money are false, because both are not objects of nature. Therefore, whatever you create using math or money, must also be false. Note that money is also a real number, and therefore must be false, since real numbers are false. Falsity of economics is obvious everywhere we look – we see wars, poverty, unemployment, pollution, migration etc. These are all happening because of money. Remove money, and create money-less economy (MLE) all these problems will vanish.

    Invalidity of math is very subtle. If you know engineering, it will be obvious for you. For example 3+4=5, is the common case in engineering. This is so because all variables in engineering, and even in economics, are bounded by lower and upper values. In this example 5 is the upper bound. No matter what the result of addition is, the output must be clipped to the upper bound 5. This makes all applications non-linear, but math requires linearity, a very false assumption for nature. The issue is more complex with infinity, another false object of math. Take a look at for more details. Under MLE even math will change and become more meaningful.