Ars Mathematica

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.

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.