Why we use mathematics in economics

Economics Nobel winner Jean Tirole put it succinctly as follows:

In a mostly auto-generated translation via Twitter’s web interface, this says “[we] use mathematics not because we’re smart but because we’re not smart”.

I agree wholeheartedly. Using mathematics in our work in economics (and in so many other areas of research) allows us to stand on the shoulders of giants and use their smarts. It’s on us to make good use of this powerful tool, honed over the centuries by so many brilliant people. Criticisms of using mathematics in economics are pointless; criticisms of using mathematics badly in economics are valuable.

On economics and evolutionary theory (wonky)

On my train ride to the office this morning, I saw this article by David Sloan Wilson, which is an update of Paul Krugman’s 1996 use of evolutionary biology to improve the methodology of economics. David Sloan Wilson’s post is an update on the developments in theory of evolution since then and their impact on Krugman’s assessment of how economics is, and should be, done. I highly recommend the Wilson article to anyone interested. In addition to being highly informative and well written, it has a good collection of citations worth following up.

Art and science, and some economics, too

The pursuit of beauty and economic theory: friends or enemies?

I read this post on Aeon by Frank Wilczek this morning. I posted about it on Facebook but doing so did not make me stop thinking about it. I want to put some of these thoughts down here, as they relate to economic theory, what draws me to it, and whether the draw of beauty in the mathematics used in economic theory detracts from the theory’s value to society. First of all, I invite you to follow the link above and come back to read the rest of my thoughts.

Wilczek has published a book recently, A Beautiful Question: Finding Nature’s Deep Design (Google Play link, Amazon link). I have not read it to the end yet, but I hope I will over the winter break. His discussion in the Aeon post immediately made me think about the art of mathematics and how it motivates me (and many others) to do economic theory and guides our attempts to do it.

Empires built on math

In my early years studying for my doctorate, I thought of math as a collection of empire-building tools for empires of the mind (castles in the sky might be another apt name). Economic theory, heavily math-laden, becomes in this view a galaxy far, far away where each theorist builds an empire (or at least a few starships) to impose order on the universe. Once you have laid down your assumptions, you then have a solid foundation for building and you are the emperor of your theoretical creations, as long as you can write down your mathematical arguments correctly.

This view is not far from Asimov’s galactic empire fiction. Indeed, Asimov himself thought that some sort of social physics exists, and if we can discover its laws, we could predict the unfolding of societies over vast, even galactic, scales. For economics, this perspective explains at least some of the psychological attraction theorists feel towards their castles in the sky. This attraction is particularly prominent for me when I am working on general equilibrium theory, which most of the profession has left behind.

Ethereal but irrelevant?

Good question! Let me first tell a story about logic and the demise of the Hilbert program in mathematics. Then I will talk about the fragmentation of economic theory, this one brought about by the confrontation of the theory with empirical tests, such as they are (famously much harder in the social sciences than in physics).

Hilbert was a prominent German mathematician. His famous and eponymous program was an attempt to make all mathematics utterly rigorous by putting it on a formal basis and to prove within that framework that mathematics is consistent. (Perhaps the best way to understand “formalization” in this sense, besides the obvious which is that it should be totally understandable to a computer, is by thinking about something von Neumann said: “Young man, in mathematics you don’t understand things. You just get used to them.” —source)

Hilbert’s program came undone when Kurt Gödel came up with his stunning, deep, and worldview shattering (for mathematicians) Incompleteness Theorems. The gist of these theorems is that if you build all of mathematics on a finite list of axioms, then you cannot prove by using this list of axioms all true theorems of mathematics and you cannot prove that the mathematics you have founded on these axioms is consistent (contains no contradictions).

Coming as it did in 1930, this devastating development seems to be a symptom of the fragmentation of European culture, which was soon to produce a horrible war. But what does it have to do with economic theory? Well, it tells us that the theoretical castles are indeed based on air, not stone foundations.

What’s worse, that’s not all that ails theoretical empire-building in economics. While there are plenty of valid criticisms to aim at empirical research in economics, its basic import is that grand, unified economic theories perform badly. This has led to the growth of “behavioral economics”. (Once again, let me point out how silly this name is: isn’t all economics behavioral? — I prefer psychological economics). However mainstream this field has become, it does have a shattering impact on beautiful theory-building that aims at wide applicability.

We contain multitudes and that can be beautiful too!

Well, it was good enough for Walt Whitman. How about we apply it to the entire society, whose functioning is the domain of study of economic theory, broadly conceived? Does the fragmentation I just talked about make further theory development ugly?

Many social scientists have written on this issue. To me, it seems that a fragmented field of theories can still be an object of beauty in the mind of the theorist. Think of it as a kaleidoscope (a word that literally means “a device for showing beauty”). This is how I attempt to keep my motivation up when I am struggling with long, messy arguments trying to prove something in complicated mathematical models of society that invariably leave my head spinning after only a couple of hours of effort.

Another good thing to say about the fragmented collection of models that constitutes modern economic theory is that it has produced some quite literally life-saving innovations (kidney exchanges, which came from a deeply mathematical field of economic theory called mechanism design theory). A good and easily readable account is Al Roth’s recent book Who Gets What — and Why.

Some others to read on related matters

In conclusion, here are some thoughtful works to read carefully that I constantly feel guilty for not having time to read carefully enough.

Daniel Little has a website portal and a blog about what it means to do social theory, even more widely construed than the widest view of economics.

Dani Rodrik has a recently published book in which he “argues that economics can be a powerful tool that improves the world―but only when economists abandon universal theories and focus on getting the context right”, as the book description on Amazon states. Another book I have started — so I really should wrap this post up now and get back to reading, even without having convinced myself that this gigantic post is as well written as possible! Of course it’s not. But I move on anyway. Ars Longa, Vita Brevis. Or, in the original language of Hippocrates, taken from the last link:

Ὁ βίος βραχύς, ἡ δὲ τέχνη μακρή, ὁ δὲ καιρὸς ὀξύς, ἡ δὲ πεῖρα σφαλερή, ἡ δὲ κρίσις χαλεπή. In English: Life is short, and art long, opportunity fleeting, experience perilous, and decision difficult.

Publication bias leads to error; psychology edition

Academic journals frequently do not publish papers that did not find an “interesting” effect. This comes under the “publication bias” moniker. This paper in PLOS One shows one specific ill effect this has on our perceptions of what research has shown: the papers that do make it to publication present an overly optimistic view of the effectiveness of psychological treatment of major depressive disorder.

More on “mathiness” in economics

Paul Romer has a new post about “mathiness”, this time in financial economics. Right at the top of the post, he includes two links to blog posts by Tim Johnson about how mathiness was used to obfuscate what the math says in finance, so that official investigations into the latest financial crisis, the one that started in earnest in 2007, would miss something. In Romer’s words, “People in finance used math to hide what they were doing.” The Johnson posts are rich in material from the beginnings of probability theory, incidentally, and surprised me with a connection to Aristotle and how thinking about money as a universal measure showed people the way to apply math to physics; Johnson also connects finance to a notion of justice. Fascinating stuff! I hope to carve some time out to delve in this more deeply.

H/T for the Paul Romer link: Mark Thoma, here.

Reacting to “Finding Equilibrium” by Till Düppe and E. Roy Weintraub

Another title for this post could be “Why people who talk about general equilibrium theory should always mention the Arrow-Debreu-McKenzie model, without omitting McKenzie”.

Prompted by a post by Joshua Gans, which I saw first on Google+, I recently bought the book Finding Equilibrium, as on the title of this post. The book contains an insightful discussion of the problem of assigning scientific credit in economics, as the book’s subtitle makes clear: Arrow, Debreu, McKenzie and the Problem of Scientific Credit. I finished the book today. It was published by Princeton University Press very recently, in 2014.

The authors continue an old project of the more senior of the two, Weintraub, that dates from the early 1980s. It has to do with why the Arrow-Debreu paper on the existence of general equilibrium gets more airtime in economic theory circles than McKenzie’s paper, even though the latter was submitted to Econometrica and published a bit earlier than the former.

I must disclose here that Lionel McKenzie was a member of my doctoral dissertation committee. I defended my dissertation at the University of Rochester in 1988 to earn my Ph.D. Furthermore, the professor responsible for my applying (and being accepted, I am sure) to the graduate program in economics at Rochester, Emmanuel Drandakis, was the second person to receive a Rochester Ph.D. in economics after McKenzie created the economics department and graduate program there in the late 1950s. Thus, I was immersed in the story of McKenzie’s unfair treatment in not receiving a Nobel Prize, unlike Arrow and Debreu. The story was “in the air” at Rochester, but I did not hear McKenzie himself talk about it, to the best of my recollection. This accords with my memory of McKenzie as a perfect gentleman; in the book I am talking about here, he is described as “classy” by one of the economists quoted there.

I enjoyed the book and read it in a couple of days, even as it came in the middle of a week that started with some worries in my personal life. It gave me the idea of writing a paper about the now ignored general equilibrium approach as it, I think, should be revived in conjunction with extensions to make it properly include collective goods and externalities (an approach that should proceed with techniques not only limited to the axiom-and-proof ones that were the hallmark of the emergence of general equilibrium in the 1950s). I have worked in this area before, myself, so I may have something new of interest to say by revisiting it. However, I did not want to wait to post here until I wrote a paper; that would mean quite a long wait!

Apart from the inclusion of some jargon from the sociology of science and (naturally) from economics, the book is very well written for a general audience. It flows well and is very instructive. The authors make the best of the fact that their topic allows them, by its very nature, to tell it as a story of people and their interactions. Readers always want stories (as we must remember every time we have to teach abstract ideas in the classroom).

As someone who is well versed in general equilibrium theory (although I’ve been neglecting it lately), I found the exposition of the theory’s inception in the book well done. Furthermore, it gave me a few perspectives and some context I did not already have. If I were to teach general equilibrium again, I would be sure to assign parts of this book to complement the rather dry and scary (for students) mathematics that dominate the theory.

Gans’s post that led me to this book concluded that, of the three main protagonists, Debreu emerges in the most unfavorable light. That is indeed my impression and it comes from documentary evidence of his behavior as a referee of McKenzie’s paper for Econometrica and also vis-a-vis his own co-author, Kenneth Arrow, from whom he kept secret his knowledge that McKenzie was already working on his own paper on pretty much the same topic. By contrast, McKenzie emerges clearly as the wronged party and Arrow as quite generous in giving credit to scholars who preceded him.

So, for the very few of you who say “Arrow-Debreu”, please do remember to say “Arrow-Debreu-McKenzie” when talking about general equilibrium theory.

[Edited 2014-09-29 to correct some infelicities in my use of English.]

RIP Gary Becker

Gary Becker, winner of the Nobel prize for economics in 1992, died on Saturday. There have appeared several obituaries and appreciations. I want to pay tribute to him by linking to three very good ones.

The first was written by Justin Wolfers in the Upshot. Wolfers explains well Becker’s influence on all the social sciences.

The second was written by Steven Levitt in the Freakonomics blog. The remarkable story of how Becker stepped in, unasked, to teach Levitt’s courses for some weeks when Levitt’s son died suddenly, stands out for me.

Finally, Kevin Bryan wrote a good appreciation of Becker’s influence in his blog, A Fine Theorem.

My own take is that, while Becker extended the homo economicus approach forcefully and with a recognition of the limitations that prevent any decision maker from being “perfectly rational”, he relied too much on methodological individualism, which may yet turn out to be the Achilles heel of the entire enterprise of economics. The way I see it, a very big step forward for economics has not yet been taken. This step would keep some aspect of methodological individualism while seriously, and tractably, incorporating the influence that people have on others’ preferences. De gustibus est disputandum. Until I make a contribution of some importance in this direction, however, I must simply tip my virtual hat to Becker’s fertile mind and his prodigious and influential output. If I ever make a contribution, then I will want to acknowledge that it was motivated, partly, by grappling with Becker’s ideas (on this, see the first link above, too).

Author thinks he has the key to revolutionizing economics

Blood circulation and economics?

The Economist’s latest issue has a review of Money, Blood and Revolution: How Darwin and the doctor of King Charles I could turn economics into a science, by George Cooper. The review intrigued me enough to take the morning of my last weekday of Spring Break and devote it to purchasing the Kindle version of the book and reading the crucial chapters. These are the chapters where Cooper discusses his view of economics and what ails it and then proposes a new conceptual framework for economics

I am disappointed. I spent my time investigating this book because I agree that economics needs change. I also liked that the author does not simply complain that economics needs changing, but has a proposal to make. However, the proposal was disappointingly vague. I started suspecting it would be so before I reached the proposal: the review does mention that economists would be unlikely to pay attention to it because it is not presented precisely enough, but, also, the description of economics that Cooper offers before coming to his proposal shows clearly that he does not know enough of the field he is criticizing.

To give an example of the gaps in Cooper’s knowledge, he says in section 7.3 that

The problem for mainstream economic theory is that the experimental evidence suggests that the way we choose to arrange our societies has enormous influence on how our economies actually work. However, there is simply no coherent way to integrate this observation into the neoclassical paradigm…

Really? Cooper does not seem to be aware of mechanism design theory, or its offspring that is making tremendous strides lately, market design. There is a lot riding on his usage of “coherent”, without which his ignorance of these fields would be utterly condemning of his diagnosis here.

A little later in the same section, Cooper complains that

Given the empirical evidence, it is unscientific not to at least consider whether democracy and government play a role in the promotion of economic growth.

Really? I have to exclaim again. Cooper has apparently not heard of the work of Acemoglu and Robinson, not to mention a legion of other mainstream economists who have examined exactly this question. Oh yes, and let’s throw in all of modern institutional economics, to boot. Cooper has a lot to learn, it appears.

Here is one more piece of evidence on the partial nature of Cooper’s knowledge of economics, as revealed in this book. In the entirety of section 7.5 he conflates all of mainstream economics with DSGE (dynamic stochastic general equilibrium) models. These models are indeed used in mainstream macroeconomics, but they are not the entirety of mainstream macroeconomics and of course macroeconomics is not ALL of economics.

And what about Cooper’s discussion in section 8.1 of competition, in the Darwinian sense, as opposed to individual maximization as in microeconomic theory. Methinks someone ought to show Robert Frank’s works to Cooper, not to mention the entire evolutionary game theory literature.

So, do I care for the proposal for reform that Cooper advances? I would, if he had told me how to formulate a model or two of the economy. He does not do so in this book. Instead, he gives a vague story about a flow that resembles (at least in Cooper’s mind) the circulation of blood in a body (hence the title of the book and the heading of this post). This flow is created by the social mobility that democracy enables, Cooper says. Yet, it is not clear what flows here, although I suspect it is money. Cooper does talk about income inequality in this connection. I suspect more serious thinkers, concerned with income inequality and the nature of contemporary economists (as am I), may be able yet to build on the vague suggestions of Cooper. Maybe Thomas Piketty, now that he has finished the labor of his upcoming (in English) magnum opus (which I am eager to read when it arrives in my Kindle in a few days). Maybe an economist well-versed in political economy, Acemoglu-style, can bring Cooper’s project to fruition. Maybe someone else. It seems certain to me, though, that, by giving us nothing precise to build on, Cooper has not advanced his self-professed goal to make economics more scientific.

Cooper is diagnosing the sickness of economics without having examined all parts of the patient, and it’s as though he’s showing us a bottle of colored liquid that supposedly has the needed medicine, but he does not explain the medicine’s formulation or how it is going to improve what he thinks is the entirety of the economic theory patient. If the medicine is ever made and administered and gets to improve some part of economics, I will be glad. But it would take someone with the theoretical chops needed to do the job that Cooper has only started. And the medicine may prove to be sugared water.