Why we use math in economics

It's the day after Labor Day, which means it is the first day of orientation for incoming MPAID students. Which also means that I have to give them a talk about the program and how it all fits together. In explaining our extensive and demanding curriculum, I emphasize that development is too important to be left to mushy thinking.

But having gone through an intense math camp for the last couple of weeks, taught by the inimitable Deb Hughes-Hallett, the students are a bit groggy at this point. Many wonder why they need to know about quasi-concavity and all that in order to be good at what they came here for--which is to improve the lives of the poor. 

So I tell them a story about Sir W. Arthur Lewis. When I was a master's student myself at Princeton, I once attended a lecture that he gave on real wages, the commodity terms of trade, and North-South income differentials.  The talk had no math in it. One of the younger faculty members of the economics department was sitting in the front row, and I could see him scratching his head in confusion throughout the talk.  A few minutes after Sir Arthur was done, this young professor jumped up in excitement and went up to the board. "Now I get it!" he exclaimed and began to scribble some equations on the board.  "This is the equation which relates to what you said in the first part of your talk, and this one expresses the other, and here is a third... and now finally we have three independent equations that determines your three endogenous variables..."  Sir Arthur kept on his bemused smile as his lecture was explained to him in mathematical terms.

The moral of the story is that if you are smart enough to be a Nobel-prize winning economist maybe you can do without the math, but the rest of us mere mortals cannot. We need the math to make sure that we think straight--to ensure that our conclusions follow from our premises and that we haven't left loose ends hanging in our argument.

In other words, we use math not because we are smart, but because we are not smart enough.

We are just smart enough to recognize that we are not smart enough. And this recognition, I tell our students, will set them apart from a lot of people out there with very strong opinions about what to do about poverty and underdevelopment. 

Comments

[robertdfeinman]

I got dressed down by Mark Thoma for my treatment of mathematics by economists. However, undaunted, I'll go at it again.

Economics deals with limited data. There are a number of standardized methods for trying to abstract this into some sort of mathematical equation. The most common are things like least squares or fitting to a polynomial function.

Economists do this, of course, but they omit things like error bars which are used in the physical sciences to show the degree of uncertainty. Then there is the truism that given a finite number of data points it is always possible to fit some curve to it to the desired degree of accuracy.

It is what happens next that is the problem. From this limited data set economists formulate generalized conclusions. To validate these conclusions requires additional data points which may not be forthcoming for a long time. So the predictive usefulness of the conclusions is not immediately apparent.

The other approach is to formulate a mathematical model and (usually) to overlay it with some explanation based upon one's understanding of human nature or behavior. This leads to such disparate groups as those who think people are altruistic vs those who think humans are selfish.

It appears that many model makers aren't interested in validating their models with data. This is sort of like what happens in physics (my field) the theorists make up the theories and the experimentalists test them.

I'm afraid that I feel that much of the more abstruse mathematical models used in economics are just academic window dressing. Cloistered fields can become quite introspective, one only has to look at English literature criticism to see the effect.

It may be that lesser minds need the mathematics, but if you can't explain your ideas without it then something is seriously wrong.

[Dani Rodrik]

"...if you can't explain your ideas without it then something is seriously wrong."

Absolutely. But distinguishing a good argument from a bad argument may require more.

[Arun Garg]

Krugman has written a brilliant essay on this topic also:

http://web.mit.edu/krugman/www/dishpan.html

[gabriel]

I understand the usefulness of math and modeling in Economics. It certainly provides some clarity. However, I am not supportive of economists who then forget that they have a "model", an abstraction of the world, and begin to believe that their model is the world. I find very few economists that are humble enough to admit that about their analysis. Dani, what are your thoughts on this matter? Didn't Ariel Rubenstein write a paper in Econometrica about modeling and stories he tells?

[terence]

Great - I'm halfway convinced. All you need to do now is to provide me with some sound empirical evidence (consistency, consistency) that there have indeed been fewer erroneous development prescriptions produced by mathed up economists than by the rest of us development folk.

I know this is a big ask, so perhaps you could just confine yourself to one decade. How about the 1980s?

[inthemachine]

While we are daring each other, how about a single example of a development initiative where the presence or absence of "math" was the root of its failure?

[Peter]

I think this math business is another example of intellectual displacement. More math or less is not really an issue, at least at this level of generality. There are two subtantive questions, however:

1. Is the math being used to elucidate (as Dani Rodrik would like) or to demonstrate technique for its own sake? My experience is that high tech math carries a cachet in itself across much of the profession. This leads to a sort of baroque over-ornamentation at best and, even worse, potentially serious imbalances in the attention given to different types of information and concepts.

2. Does the math, in order to be tractable, impose economically consequential assumptions? My favorite example is convex choice and possibility sets -- not so much by suppressing increasing returns (main diagonal) as interaction effects (off diagonal). Hence the status of economics as an "almost social science".

[Christian]

I certainly agree with the statement of "…if you can't explain your ideas without it then something is seriously wrong." However, I also agree with " … distinguishing a good argument from a bad argument may require more." But we should not forget that economics, in a regular speech, is defined as a social science that studies the production, distribution and consumption of good and services. The definition of economics has passed through many reviews and changes. Adams Smith (1776), for instance, regarded by some as the father of economics, first defines economics as "the science of wealth" or "the science relating to the laws of production, distribution and exchange."
However, the definition of economics varies according to human perceptions and evolved periodically to include in it human activity and welfare. Alfred Marshal (1890) , for instance, says that economics "is a study of mankind in the ordinary business of Life; it examines that part of the individual and social action which is most closely connected with the attainment and with the use of material requisites of well-being." Thereafter, as another example, we have a famous definition written by Lionel Robbins in his 1932 essay saying that "economics is a science which studies human behavior as a relationship between ends and scarce means which have alternative uses."
The introduction of human activity and furthermore, the introduction of human behavior in various aspects of the world economy, explicitly economics, brought several concerns not only to the definition of economics, but also to the outcomes of economics. Even though economics produces theories based on the correlation of empirical data with the observation of the society's behavior, its theories' outputs do not yield universal constants or follow natural laws. However and despite the countless cases of economic analysis where results are conclusive, the discomfort is basically due to the reliance on unrealistic or unobservable 'assumptions' due to the complexity of the human behavior and the world where we live.
Here perhaps is when economics and the use of math came together into debate. However, we cannot separate math from economic analysis, but certainly we should not forget that it is an instrument to understand, to some extent, the complex reality of our world and society and not by any means the ultimate conclusion or truth. And what students (any students or students of development/economics) should know is that "to be good at what they went there for—which is to improve the lives of the poor" they need to take into consideration that math in economics is a powerful tool, but when they create a model or an equation to explain human behavior in the economy, there are rules that might not apply even though it seems like they do. In addition, the one who pays the bill of any erroneous development prescription either produced by a mathed up economist or the rest of us development folks, at the end, are the poor that they are trying to help.

[gabriel]

This goes back to one of the issues addressed in an earlier post about what is appropriate to teach the students in Dani's program. If they spent two to three weeks studying how to separate hyperplanes or set theory in order to get through a grad level micro text, It is not obvious to me that this is an appropriate use of the limited time those students will be there. It seems to me a more appropriate approach would be to be more "practical" or "applied" approach to microeconomics. For example, if you are going to be involved in micro-credit programs, it is not obvious to me that one should be well versed in the models of Townsend vs the old Ohio State school of rural credit. A ph.d. student yes but maybe not a student who may be running a micro-lending organization. Well, that is my humble opinion.

[Jay]

What about the vast majority of people out there--the ones who are not smart enough to grasp the math? I guess they will never understand development. Every individual that hasn't had advanced level training in math should be automatically disqualified from having a strong opinion on poverty and underdevelopment. Well, that's just about most of the world, including nearly all political leaders in the developing world. Let's leave the strong opinions to the humble economists, the ones who realize that they're not smart enough.

[C. Ryan]

In defense of Peter's argument, an old friend from economic literature:

http://academic.reed.edu/economics/course_pages/354_s06/Siegfried_JPE_70.pdf

[kinglear]

The maths - and the economic arguments - are all very well. But somewhere along the line,the man on the Clapham Omnibus needs to understand it - and that can ONLY be in a very short sentence of short words.The greatest writers hone their sentences by shortening them, giving the words more punch, and making the meaning crystal clear. Would that Economists did the same! ( and no made up words)

[Per Kurowski]

Re: "The moral of the story is that if you are smart enough to be a Nobel-prize winning economist maybe you can do without the math"

Yes but also remember that life is a balancing act as so many highly sophisticated mathematical models are reminding so many sophisticated mathematical financial portfolio modeling engineers these days and who by the way included a couple of years ago a Nobel-prize winner that from what I have been told overdid it on math.
Even I who am not that great at math readily admit that math is great. But never forget that it is only an instrument to be used and not an instrument to obey blindly or worse, hide behind.

Actually at this moment particular moment, after having read some really senseless research papers that hide their mumbo jumbo behind equations I am not really sure what is more important promoting the undoubted usefulness of math or warning for the dangers of being overtaken by it. Nonetheless I respect and understand Dani Rodrik’s undivided call for math, after all it is the first days of the semester or term, and there will always be time for the nuances later.

By the way… Have yourselves all a very fun and productive semester or term.

[inthemachine]

Is it fair to mention that among those who do understand the math, it makes one heck of an efficient way to communicate? (and sometimes, gasp, its not important for *everyone* to understand what you mean while you are working out an argument). For all their complexity, equations also nicely set aside cultural connotations that wreak havoc on other forms of communication. X=5 means the same thing no matter what part of the world you are from.

[Matt Nolan]

I completely agree with what you have said here.

Even if maths is a more efficient way of understanding a set of economic concepts, being able to sensibly translate them into subjective, value laden words is a talent.

If we are not careful as economists, maths can become a set of smoke and shadows that we use to hide behind when we don't understand why something is happening. If you can describe your model without maths, then you truly understand it.

However, I struggle with the maths and the words, so I'm not sure I'm able to comment :)

[notsneaky]

"but they omit things like error bars"

Huh?

[Peter Boettke]

Dani,

Two points. First, Alfred Marshall had a wonderful saying for economists and it was "burn the mathematics." His basic idea was that mathematics is a very useful servant, but a horrible master. He did not argue that we shouldn't use mathematics to check our logic. Instead, he argued for the use of mathematics along the same lines you lay out. But then he turns around and says once you are tested your argument against mathematics you should "burn the mathematics" and commnicate the argument in the clearest use of language possible.

Second, there are some instances where the important questions defy mathematical rendering. Here I recommend Kenneth Boulding's 1947 review in the JPE of Samuelson's Foundations where he argues that the flawless precision of mathematical economics may prove impotent in addressing the complexity of the social world --- a world that was better analyzed with the literary vaguness of economic sociology.

Again, mathematics can be a good servant to thought, but not a very good master in the realm of economic discourse. Think of all the issues that were raised by non-mathematical scholars over the years that have proven essential to understanding development and social order: institutions (North), alternative legal systems in a positive transaction cost world (Coase), constitutional retraints on predation (Bucuahan), enterpreneurial activity (Schumpeter, Kirzner and Baumol), power and collective action (Olson), cooperation and coordination (Schelling), and spontaneous order (Hayek).

Especially in the field of development economics scholars such as Hirschmann and Streeten said a lot of valuable things without the use of mathematics.

So again I think it is a matter of having some perspective on the issue of mathematics. Is it a good signal of intelligence? It certainly can be. Does it guarantee good economics? Certainly not, we can prove much economic nonsense using higher mathematics. Being a good economist is about a lot of things, mathematical acuity may be on the list, but certainly doesn't exhaust it. History, philosophy, political theory, languages, demography, technology, etc. would be on my list of the skill set that usually is found in good economists.

[Eric H]

I agree with the conclusion of this post. However, that doesn't excuse overdoing it. This is a recurring theme in Econ Journal Watch. Examples:

Where would Adam Smith publish today?
http://www.econjournalwatch.org/pdf/SutterPjeskyEconomicsInPracticeMay2007.pdf

The Mathematical Romance: An Engineer's View of Mathematical Economics
http://www.econjournalwatch.org/pdf/GibsonCharacterIssuesApril2005.pdf

[pat toche]

I agree completely with Dani's conclusion. Yes, mathematics helps you to be smarter. So many times I have come up with ideas that seemed intuitively correct and brilliant, but once put in mathematical form turned out to be false leads.

I would add a cautionary note.

Some of the smartest ideas in economics can be expressed clearly with some mathematics. For instance, the matrices that summarize Ricardo's theory of comparative advantage is an (admittedly elementary) piece of mathematics. Smart ideas such as the market for lemmons or the theory of optimum taxation can be expressed clearly with mathematics.

In some cases, simple maths will do, in other cases the maths get more involved, and in a few cases the maths is really advanced.

The question that needs to be addressed is this: what is the optimal level of difficulty of the mathematics you should use to express an idea in economics? It must depend on your purpose. If your purpose is to illustrate or communicate an idea, you should try to use the simplest type of maths: take a hump-shaped function for your example. If your purpose is to run some simulations and attempt to replicate or estimate some dataset at some level of accuracy, then you may need a strictly concave function with a not-too-large number of parameters. But if your purpose is to show the generality of a certain property, then you may need to go for quasi-concavity, correspondences and other advanced stuff.

The problem I think, with the way things were taught when I was a student some 10 years ago or so, is that lecturers usually tried to prove the generality of certain theorems: they would immediately frame everything in terms of some little-known topology and try to prove that an equilibrium would exist even if the function was non-differentiable and had convex sections, as long as the convex sections were not-too convex according to some obtuse criterion.

And that's where it all goes very wrong, because the advanced mathematics should be used only for advanced research, best left to the McKenzies, Mas-Collell and Brocks of this world. The likes of Barro, Krugman or Mankiw, to name some famous mainstream economists, have never needed to venture into the world of Borel measures (or whatever), and have produced brilliant ideas.

So I'm afraid to say I do not approve of quasi-concavity being taught in most grad econ schools. I think strict-concavity should be more than enough.

P.S. As far as I know, Krugman and Mankiw have made mathematical mistakes in at least one of their published papers, the "History and Expectations" paper by Krugman, and the "Savers-Spenders" paper by Mankiw -- both papers are brilliant.

[Greg Ransom]

You need it to advance your careers and sustain your social status among your peers. That's the beginning and end of the story.

"The moral of the story is that if you are smart enough to be a Nobel-prize winning economist maybe you can do without the math, but the rest of us mere mortals cannot. We need the math to make sure that we think straight--to ensure that our conclusions follow from our premises and that we haven't left loose ends hanging in our argument."

[wjd123]

Where are the banal inconsistencies of life to be found in a mathematical formula?

[dearieme]

kinglear, are you proposalizing an endogeneous transitioning to an clarification proceduralization strategy initiative going forward?

[Joe S.]

It's way too late in the thread, but I'd like to add a comment anyway.

Anybody who has gone through the Feynman lectures has realized that physics is much harder to understand without the math. Is this really true for economics, or at least the parts of economics that are interesting and useful?

As far as I can see, math can be a useful test of an economic argument. If you can't translate it to math, and have the equations work, maybe you don't have the argument you thought you did.

But it is not like physics. You really don't (in my experience) need math to communicate substantial ideas in economics. Or if you do, it is hand-wavy math, like the idea of convexity.

[Darren McHugh]

My own feeling (I have a terminal MA from Queen's, one of the "best" econ schools in Canada) is that since economics deals with understanding quantitative phenomena of the economic world, tools of mathematical analysis (graphs, equations, etc) are certainly useful for building a framework in your mind with which to look at the world.

However, economics as taught at the graduate level combines an infantile infatuation with technical complexity with a criminal indifference to relevance and applicability. It is the worst form of boredom: both difficult AND pointless.

In my experience, algebra is essential in sound policy discussions but anything beyond that is overkill. Professional academics value the mathematical firepower the way an autistic savant would, but they are, unsurprisingly, universally oblivious to how irrelevant their work actually is.

[Darren McHugh]

Peter B: "Being a good economist is about a lot of things, mathematical acuity may be on the list, but certainly doesn't exhaust it. History, philosophy, political theory, languages, demography, technology, etc. would be on my list of the skill set that usually is found in good economists."

Absolutely. Sadly, graduate school in economics rewards ONLY mathematical acuity. In my experience, the students most favoured in my graduate program were the ones who were capable of quickly cranking out 40-page math-laden homework assignments, but knew nothing about anything and were capable of little more than cliched grunts when discussing any real-world policy question.