Brad de Long, an economist, has a post up about the significance of how he dresses for specific audiences. In particular, the consequences of wearing ties:
With math-oriented students, however, a tie tells them that I spend too little time thinking about isomorphisms.
(This inspired n-category jokes in the comments.)
I was looking at an interview with Roger Myerson, one of the three winners of the 2007 Nobel Prize in Economics (technically the Sveriges Riksbank Prize). Myerson says
How would my career have been different without John Nash? What a wonderful question! My difficulty in answering it is an indication of how influential he has been in everything that I have done. I wanted to be a mathematical social scientist since I was 12 years old, when I read Isaac Asimov’s science-fiction novel Foundation. But my concept of what kinds of mathematical models should be studied was completely transformed when I read Nash’s and Harsanyi’s papers in college.
Cosma Shalizi has written a detailed posting on what’s wrong with econophysics. Econophysics is the application of certain ideas that have been influential in physics in recent years — power laws, phase transitions, nonlinear dynamics — to finance and economics. Econophysics articles are generally published in physics journals, and have not had much of an impact on economics as practiced in economics departments. If you love statistical mechanics, and you think it explains everything, then econophysics is the area for you.
Cosma also includes a section on what’s wrong with economics, because he intends on dying friendless and alone.
In a post on his weblog, Michael Greinecker mentioned some applications of homology to economics. While aimlessly websurfing for more information, I came across the homepage of Graciela Chichilnisky, a mathematical economist who has extensively applied topological techniques to economic questions. Chichilnisky has written nearly 200 articles, and includes PDFs on her site.
On a less happy note, Chichilnisky also links to a site dedicated to detailing her problems with Columbia University, where she is a tenured professor. Chichilnisky had successfully sued Columbia on the grounds of sex discrimination in the 90s. The two parties are now back in court over an alleged pattern of retaliation on the part of Columbia. An article on her experiences with the reviewing process also makes depressing reading.
Famously, the economist Gerard Debreu was close to the Bourbaki circle of mathematicians. This gives his book-length treatment of general equilibrium, Theory of Value, the reputation of being economics the way Bourbaki would write it.
I’ve been looking over Theory of Value, and while it is very abstract for an economics book, but anyone who thinks that the book could have been written by Bourbaki has never had the pleasure of the real thing. Debreu’s book has picture and everything. I would put it at the same abstraction level as Herstein’s Topic in Algebra.
I was doing some more reading on general equilibrium, when I came across On the Fundamental Theorems of General Equilibrium by Maskin and Roberts, which gives a succinct proof of the existence of general equilibrium, as well as the two subsidiary results known as the first and second welfare theorems. It’s particuarly good in spelling out the mathematical relationship between the different results. (Be warned, though. It’s completely unintelligible if you don’t have a separate description of the model handy, though.)
I found the paper via this post by Michael Greinecker on his weblog Yet Another Sheep.
I ran across an old article by Donald Saari from the Notices of the AMS,
Mathematical Complexity of Simple Economics, which explains how some simple models of the economy can have arbitrarily complicated dynamics.
The basic model in economics of the economy as a whole is that of general equilibrium (GE). General equilibrium is a model of the economy where goods are traded for money which are traded for goods. It’s assumed that all of the goods are used up, and no one has money left over; prices are assumed to take on a value such that both of these things occur. It’s also assumed that in equilibrium the supply and demand of each good are exactly equal: everyone who wants to buy or sell at the current price is able to. The implicit dynamical idea is that if demand exceeds supply, then prices will go up, and if supply exceeds demand prices will go down. In this model, markets as said to clear.
It’s not easy to show that such market-clearing prices even exist. Under certain convexity assumptions, they can be shown to exist using the Brouwer fixed point theorem. The model, as stated, has no dynamics: prices achieve their equilibrium values, and that’s where the story ends. But as I mentioned above, there is an implicit dynamic story that whenever prices are too high, they adjust downwards, and when they are too low, they adjust upwards.
Saari’s article is about the difficulties that arise when taking that model of dynamics seriously. Assuming that prices adjust in proportion to how far supply and demand are apart can lead to arbitrarily complicated dynamics, even chaotic dynamics.
The Annals of Improbable Research, perhaps the finest general-purpose scholarly journal today, has published a satire by Yoram Bauman of Greg Mankiw’s 10 principles of economics (from his introductory economics textbook). Most of it is hit-or-miss, but his summary of Principle #2 is a brilliant satire on the concept of opportunity cost.
Opportunity cost is easy enough to understand in simple descriptive situations, but I have never seen a precise mathematical definition of it that didn’t make it seem like a contentless idea.
I’m glad to see that finally someone is putting mathematics to some good use. I present to you The social norm of leaving the toilet seat down: A game theoretic analysis. (Via Crooked Timber.)
Ariel Rubinstein has made his book, Modeling Bounded Rationality, available online. Economic models tend to assume that humans are mistake-free calculating machines; economists have tried to introduce more realistic assumptions under the banner of bounded rationality. This is a far-from-settled problem, mainly because there are more ways to be wrong than there are to be right.