Paul Krugman

responds to Noah Smith's

tale of disillusion with mathematical economics:

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.

I've heard such claims a thousand times. But after enduring four years of Princeton economath, and publishing two pure theory pieces (here and here), I am convinced that most economath badly fails the cost-benefit test.

Let's start with a truism: Some people have a comparative advantage in economic intuition; others have a comparative advantage in mathematical economics. The superior path to economic understanding varies from person to person and topic to topic.

Now let's move to an obvious fact: 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 professors in top undergraduate economics programs (MIT and Caltech aside) avoid economath because they know their students can't hack it.

What about the the remaining 5% who *do *understand some mathematical economics? An optimist would point out that this 5% does the lion's share of *original *economic thinking. Perhaps economath is vital scaffolding for intellectual progress despite its low pedagogical value.

Not really. Empirically, even the 5% gain most of their economic understanding via intuition. Don't believe me? Show a typical economist a theory article, and watch how he "reads" it: He reads the abstract and introduction, flips the theory pages, then reads the conclusion. If math is so enlightening, why do even the mathematically able routinely skip the math?

You could reply that mathematical economics shows that economic intuition is often wrong. I beg to differ. My experience: 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.

Krugman interestingly argues that economath helps economists grasp claims that seem intuitive after the fact:

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: even similar countries will end up specializing
in different products, and because there are increasing returns in many
sectors, this will produce gains from specialization and trade. But this
point was only obvious *in retrospect*. People in trade were not
saying anything like this until the New Trade Theory models came along
and clarified our thinking and language. Trust me, I was there, and went
through a number of seminar experiences in which I had to bring an
uncomprehending audience through until they saw the light.

I believe Paul. But I have a slightly different interpretation: His 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. Since they know no economath to blind them, they don't need economath to grasp the obvious-once-you-point-it-out.

Economath isn't utterly useless. Last year, for example, I used economath to convince my colleagues about a

subtle signaling fact. The main intellectual benefit of studying economath, though, is that
it allows you to detect the abuse of economath.

Example: During

my undergraduate independent study with Bill Dickens, I told him that compensating differentials and efficiency wages were
the same. Bill used math to show me I was wrong. The only reason I
erred, however, was that the textbook discussion of efficiency wages was incorrect! When I showed Bill the textbook, he admitted that the author had mistranslated the math of efficiency wages back into English. Kudos to Bill, but this incident made me respect economath less, not more as he hoped.

When I teach Ph.D. microeconomics, there's still a lot of math. I don't just teach my students economics; I also teach them what they need to know to succeed in the economics profession. Still, I strive to give them a better intellectual experience than I had. I only teach the math a typical econ Ph.D. might actually use one day. I use the time I save to interweave the subjects economists have to know to genuinely understand the world: psychology, history, political science, and philosophy for starters. I'm still not satisfied, but it's a start.

Don't put too much faith in "common sense". Just look at every time a politician says "the government should budget like a family does its budget."

This is obviously stupid to anyone who understands the difference between partial and general equilibrium, but it's "common sense" to a sadly large number of people.

The best thing I learned from studying physics is that common sense is always and everywhere wrong. It's served me well as I started moving beyond rational expectations and moved into behavioral economics.

"compensating differentials and efficiency wages were the same"

so true. what else could it be: work harder for more pay. Lazear showed time and again that incetive pay increases output and induces self-selection

I think your evidence is a bit weak. In particular it confuses the development process of the idea with the presentation process. Yes the economist flips over the theory pages and to the conclusion. But the same is often true with an empirical paper - one has to understand what the claim is in order to see if it's interesting/relevant/worthwhile to examine before one delves into the details. Yet without the work (whether the econometric model or theoretical model that fills those skipped pages) the author could not get to the point where they wrote the conclusion, and they certainly couldn't defend their work. Further, you haven't really demonstrated that Mr Krugman's point is wrong; it's one thing to describe a model in non-mathematical language - it's another to come up with that model from scratch.

As much as I really dislike the math I needed to learn in grad school, there have been times I've thought this argument has merit. Many people would probably vehemently disagree with this, but the verbal method does sometimes do things like "assume perfect competition."

An example of this was my discussions with people about what happens when a foreign central bank engages in expansionary monetary policy. I believe this may have an expansionary effect in the domestic economy (e.g. under free banking). The response I got to that was that it would be mediated by the exchange rate. But that assumes perfect competition. It isn't immediately that is true. I suppose that there are some verbal errors implicit in believing perfect competition isn't necessary, but they are non-obvious to professional economists I spoke with. So forcing yourself to say whether (for instance) the zero profit condition must hold for what you are saying to be true is useful, and econmath certainly does that.

I don't think "comparative advantage" means what you think it means.

This statement is ludicrous.

Speaking like a true Austrian!

It was about time this discussion finally comes out in the open.

Noah's post says it all, but then Noah is afraid to come to the obvious conclusion this post states so clearly: economath badly fails the cost-benefit test.

It is very difficult to disagree with Paul Krugman, but there is much to say about "economic intuition being atrophied from disuse".

Keep on debating, please.

Economists think they are capable of genuinely understanding the world?

The whole math-in-economics thing is really over-blown, in my opinion.

I agree with Krugman that math helps us think clearly when formalized logic is required. I would argue that mathematical logic succeeds in a way that verbal logic will never be able to do. The reason is that mathematical definitions are precise, while linguistic definitions are always a little vague and subject to interpretation.

But that's not what the real problem is. The real problem is that economists still insist on teaching students arcane mathematical concepts like LaGrangian equations. Take an optimization class from the university's

mathematicsdepartment and you learn to accomplish these economic tasks in very different ways.Mathematics has

moved onfrom a lot of the concepts that economists seem stuck on. That doesn't mean economists are doing wrong math or bad math, it just means that they haven't updated their theoretical logic in line with contemporary mathematics.That's a failure of economists to stay abreast of modern trends, not a problem inherent to using mathematics at all.

Bryan, I thought you were a fan of the signalling explanation for schooling. Surely this is a prime example? Economics majors are forced to do something difficult, just because it's hard, right? If it helps them in their official profession, that's a bonus, but the real value is screening out people so the signal of a degree is stronger.

I mean, I can agree that mathematical reasoning can get you places that intuition won't, but only if you have good measurements and preferably controlled experiments. Which is why it's more useful in science and engineering. Atoms don't lie to you.

I am inappropriately excited for the possibility that Paul Krugman will conquer his withering self-doubt and engage in a colloquy with Bryan Caplan.

You say that this 5% gains most of their original understanding from intuition, then illustrate it with an example where an economist gains understanding from reading someone else. I think they flip through because they trust that the author got the math right.

Any examples of this? I can think of so many counter-examples, and only a few examples. Comparative advantage? The importance of marginal incentives? All of these are blindingly obvious in a mathematical model but judging from the public discourse completely contradict common sense.

Well, I can explain models to intelligent laymen which are clearly ludicrous when expressed mathematically. I think what you describe as an 'atrophy' of intuition is in fact a learned unwillingness to take verbal arguments at face value, the result of encountering many verbal arguments which 'sound right' but have deep flaws.

It's no coincidence that people who reason verbally tend to have starkly differing viewpoints, whereas people who reason mathematically tend to broadly reach a consensus. Math provides a platform for identifying where, exactly, two individuals disagree, which is the most important step toward resolving disagreement.

Math is language for expressing logic. A shorthand for expressing intuition and a framework for seeing where that intuition might lead to. It is not a substitute for intuition. No one claims it is. Intuition and cursory observation told us that the sun revolves around the earth. Deeper observation and math disproved it. Now we have a better intuition about the solar system. So math has a place in how we understand reality. Presumably economics is a part of reality!

This is not to say that there are way too many papers that tweak some little mathematical property with little increase in human knowledge -- but that has more to do with the odd incentive structure of the professoriate than anything.

Econ math could just be in the building blocks phase of a much longer game. After all, it took 350 years for the benefits of classical mechanics and calculus to be fully realized. AI and insanely large databases could push economics into the engineering phase and out of the not even wrong/publishing papers/status whoring phase.

If we are living in a simulation then there is almost certainly "laws of economics" that could be described by mathematics , the application of which may look nothing like the attempts of the 20th century. :)

You may be right, but one may also skip over the math in a theory paper because one wants to save time and is not sufficiently interested in the details. When I buy a TV, I turn it on to watch it. I don't open it up and figure out in detail how it works. That doesn't mean the details don't matter. If the details were not done right, I wouldn't be able to watch.

Rigor is the cargo-cult of economists, that correlate with profound truths that, ergo, should be used to find profound truths. Thus Samuelson, after studying physics, created the new methodology via his Foundations book, thinking, that's how you advance a science like physics. Note that one of those key principles, stability, was soon discarded because no one knows how to model disequilibrium, but never mind, we have equilibrium derived from optimizing consumers, governments and firms.

Just as Post-Modern Art History has produced some interesting insights, any large literature will contain some gems, but that's just scattershot luck. Consider Krugman's increasing returns to scale argument: it was known before, being the basis of the centuries-old infant industry argument, and after Krugman it was no easier to apply. Think Detroit: what were the key conditions that allowed it enjoy increasing returns to scale in the early 20th century, but then decreasing returns to scale later in the century? Does Krugman's model help? No. Oops. It helps Krugman personally, in that he's now a certified genius, and so can expound more forcefully on immigration, Keynesian multipliers, and ethanol.

When primitive islanders saw the magic airplanes of WW2 airmen bringing cargo, they created bamboo airplanes to lure such cargo in. When Mao had everyone melt down their forks and hammers to present iron to the government, to modernize the economy, this was to create an industrial powerhouse. Similarly, rigor, from assumptions to conclusions, like a mathematical proof, is how to create a science about economies like physics.

So, what are we arguing about 70 years after Keynes? The same things. Misplaced rigor makes confabulation and rationalization easier, because there's always some model or complicated econometric technique that shows whatever one wants to show, and because it's complicated, you merely have to reference it when arguing policy, and so the key assumptions are never highlighted, and the debate stagnates. The science stagnates.

It's good to remember two quotes here:

Moderation in all things.

Never express yourself more clearly than you are able to think.

The fact that some economists skip the math part of papers says nothing about whether the math is useful. Good papers take the lessons from the math and turn them into words. A lot of researchers just trust that the math is done right, then harvest its fruits in the narrative section. That doesn't mean that the insights described in the narrative section could have been easily obtained without the math.

The idea that somehow the logic of mathematics and the logic of spoken debate are incompatible is fallacious. They are equivalent. A logically sound mathematical formulation will have an equivalently logically sound spoken statement

andit will be supported by empirical evidence. This is commutative.Our only hope for describing the world around us is through observing our surroundings. Much of the legerdemain we see in economath is a result of restricting the information with which the models are compared. This restricts the range of validity of the model while it is presented as having wider applicability. Call this

the fatal conceit of the pretense of knowledge.We need both forms of reasoning, however, we need to understand the limitations of both and ensure that we are not misapplying them either, the observations of

Das Kapitalcome to mind. As examples of sound logic, we have Mises' observations inSocialismand von Neumann and Morgenstern inTheory of Games and Economic Behavior.Because something is hard and difficult to understand does not mean that we should avoid it.Take it as a challenge to expand your understanding. If you don't train yourself in rigorous mathematics, it makes it impossible to understand when someone is blowing smoke.Bryan is wrong about this for many reasons, but the most obvious aspect is this--intuition and verbal logic are not useful for determining the actual size of an effect, or for hypothesis testing versus data with anything mildly subtle. Models allow for precise predictions and comparisons.

It's also the case that mathematics is the more efficient way to think about and express simple but fundamental ideas. Good luck actually convincing someone about comparative advantage using words alone. A simple illustrative numerical model can do the job much better and more convincingly.

Part of the emphasis on economath (or whatever you want to call it) is screening, that's for sure. As a practical matter, though, if you want to do serious statistical analysis (much less any serious projection or forecasting) with economic data, it's obviously indispensable.

Economath is also, as Donald McCloskey (as he then was) once pointed out, a useful rhetorical device against tyrants of any ideology who would otherwise be deaf to the advice of anyone suggesting that the tyrants are wrong, and have nasty habits of shooting messengers bringing bad news, quite literally much of the time.

PaceOrwell, few real tyrants have ever seriously tried to insist that two plus two literally makes anything but four. A wily researcher who realizes this can show the tyrant pages worth of data and equations and simply say, "Well, your majesty/my leader/Comrade Secretary/Mr. Department Chief, Imighthave made a mistake in my calculations. I'd be grateful for your guidance if you can see what I missed."Of course, even this doesn't always work, as Nikolai Kondratiev found out (when rewarded by Stalin for his statistic work with a long stay in jail before finally being executed). But it protected more economists from political interference than is appreciated. I'm told that between the war and the fall of communism much of the published work done in the field of "operations research" in Eastern Europe was thinly disguised mathematical economics, conducted in an environment relatively free from political interference.

To put it in Misesian terms (though I'll surely make him spin in his grave), mathematics is the ultimate

Wertfreidiscipline. Reasonable men (or people who want to be thought of as reasonable) can dispute how the world ought to work. Reasonable men can't dispute that two plus two equals four.(McCloskey's own opinion was that in a liberal democracy, such rhetorical tricks ought to be less necessary, and economists were better off finding other, better ways of persuading people of their beliefs. I'd like to believe that. As it is, regardless of how liberal or democratic you think modern North America really is, the world's tyrants and sociopaths aren't found only in presidential palaces, and rooting them all out isn't an option.

Even if it were, and our decision makers were all angels, they would surely always want their knowledge of the impact of a policy change to be as specific as possible---if not to the second significant digit, then at least the first. Economath isn't going anywhere.

The benefit-cost concept is key to this discussion. Can math be helpful in clarifying certain economic concepts? Yes, of course. But has the intellectual evolution of the discipline, in which mathematical formalism is now mandatory for publication in a "top tier" journal, benefited the advancement of economic understanding? I'd argue clearly not. It has essentially delegitimized all other paths to economic insight to the net detriment of the science.

RP Long,

You said, "I would argue that mathematical logic succeeds in a way that verbal logic will never be able to do. The reason is that mathematical definitions are precise, while linguistic definitions are always a little vague and subject to interpretation."

Aren't mathematical definitions always explained with verbiage? It seems that if someone were to carefully define their terms in a piece of verbal logic, the advantages of math go away. Therefore, the use of math would needlessly complicate things.

Bryan,

Suppose Adam Smith showed up at your doorstep one day and told you, "If a foreign country can supply us with a commodity cheaper than we ourselves can make it, better buy it of them with some part of the produce of our own industry, employed in a way in which we have some advantage." How would you convince him that even if the foreign country produced both commodities more cheaply that it may still be beneficial to engage in trade?

Using very basic economath, Ricardo convinced people of the then counter-intuitive theory of competitive advantage using numerical examples drawn from a 2 country 2 good mathematical model. The more advanced economaths of today have been able to extend his model to 2 country N good and N country 2 good cases.

Ricardo's brilliant economic intuition was likely responsible for leading him to this discovery. But if it wasn't for his ability explain his idea using mathematics then he would likely have failed to convince many people that his intuition was better than the consensus. In fact, it is extremely difficult to convince others that one's intuition is correct in any case because intuition is the ability to understand something immediately, without the need for conscious reasoning. Because intuition is so intangible we need to formulate some kind of reasoning to convey it to others and convince them that it's right.

Now because economics deals mostly with objects to which we can assign numerical values, the most convenient way to convey our intuition is through mathematical reasoning. This may be as mathematically simple as the Ricardian model or as nasty as the biggest General Equilbrium model. To me, both of these models utilize economath. If the former is considered intuitive to you then it seems that mathematical complexity is the issue you are deriding here.

I think that is both lazy and disrespectful. Lazy because it implies you are arguing that economath should be dismissed because it takes too long to sit down and reason through. And disrespectful because you are critisizing economists who may have excellent economic intuition, but are having trouble accurately translating their complex ideas into simple mathematics that seem "intuitive".

Thank you for the post and the argument.

I agree completely that economath very often fails the cost-benefit test. In fact, I think that your argument may even have stretched further than it needed to in order to support that position.

In common practice, economath has very real costs associated with it that outweigh the benefits that Krugman rightly points out. It is true that math can and does clarify relationships between key variables that can illuminate an argument.

However, economath also sets up many expectations for what economic results will look like, and those results are often much more specific than would be justified if not for the assumptions taken solely for the sake of doing economath. When those models fail, as financial risk models failed in the financial crisis, economists and practitioners do not go back and reevaluate their foundations. Instead, they use the old math models that "represent risk" as a standard by which new models will be compared. This argument is laid out in the article below, showing that costs are very real, regardless of some benefits.

http://distilledmagazine.com/economic-models-analysis/

Thanks again for the excellent post.

John B,

The process of carefully defining logical terms is precisely from whence mathematical logic originated in the first place.

I notice that no one here decided to make their argument with a mathematical model. Hmmm

There's also a very nice article on this very same topic by Simon Bishop in the recent issue of the

European Competition Journal: "Snake-Oil with Mathematics is Still Snakeoil".Maybe the advantage of maths is that you "know" (more strongly believe) that the reasoning is correct.

Bryan, how can you start out complaining about "too much theory"

and then follow up with more assertions than justifications?

I understand you are just expressing your point of view but I think the way you "argued" in this piece--by lining up your beliefs in a way that offers an alternative proposal to your opponent's--suffers from not justly explaining your pov. Of course economaths can brush aside questions ("Why did you assume particularly

that?" **waves hands vigorously**) but I think you've done the same in your argument in this piece.When you say "has a comparative advantage in economic intuition", you are doing two things I'm wary of. First you are hauling in the comparative-advantage framework, which you're going to hang all the ideas on. But how do I know how good the ideas of comparative advantage are here? Second, what is this "economic intuition" that 95% have a comparative advantage in? Does it come from years of business experience across all sectors? Does it come from squatting with the poor, being in the room with the FOMC, assessing land deals with RE moguls, counselling corporate boards all over the world, straining your back at copious startups across all industries, and reading more or less comprehensively through history books? No, and therefore we get paragraphs of speculation in economics research papers about "So what do you think normal people are actually like?". If you think the "economic intuition" that can be gained from economics classes amounts to more than "See a pattern (like 'MB=MC' or 'moral hazard') and look to apply that pattern in other situations" I'd like to know what that is.

Bryan, regarding the introduction of your econfaculty.gmu.edu/bcaplan/pdfs/standingtiebout.pdf. What justifies the claim that people are so mobile they can change local governments on a whim? According to "The Wall Street Journal's Complete Home Owner's Guidebook" staying in a mortgage less than 7 years is a money losing proposition in most cases.

Furthermore commuting costs may increase, and if an attractive government is geographically significantly far away (maybe the local government in Round Rock is not different enough from Pflugerville's) then you may have to get a new job, or sacrifice friend/professional/family connections.

I don't see why property tax needs to come into it. Bringing it back to economic intuition and why it may not be so great (especially for convincing people who don't trust you or don't share your priors): I assume it's because your economic intuition tells you "government" should be the bad guy.