Ars Mathematica

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.

A sunspot equilibrium is a market equilibrium in which prices depend on otherwise irrelevant random variable. The name is inspired by a theory of nineteenth century economist William Jevons that sunspots affected the stock market. (The theory, while wrong, isn’t quite as rediculous as it sounds. Jevons thought, by looking at the data he had on sunspots and agricultural prices, that he detected a pattern that indicated that sunspots caused crops to fail, which in turn caused recessions.)

A sunspot equilibrium would then be a self-fulfilling prophecy. If everyone expected that sunspots caused recessions, that in principle could be sufficient to cause a recession, even if the cause-and-effect existed entirely in people’s heads. Note that this outcome, while not optimal, would still be individually rational: even if you knew that sunspots didn’t really cause recessions, you would know that everyone else was expecting a recession, so you would act accordingly.

Karl Shell, one of the inventors of the concept, has a list of links to his papers on sunspot equilibria. In particular, he links to a short survey article he co-authored with Bruce Smith.

I spotted a survey article, Behavioral Economics: Past, Present, and Future, which gives a guide to this fairly-new field of economics. The subject was born from a mathematical failure. Economists had given precise axioms as to how people would take into account time and uncertainty when making decisions. The axioms allowed precise predictions that (unlike most economics) could be tested in small-scale experiments with a few test subjects. The result was almost-total failure: nearly every prediction turned out to be wrong. Instead of this being the last word on the subject, this has inspired large amounts of research into finding empirical regularities in the discrepancies between the predictions and the experimental results, and formulating a new theory that is both precise and correct. It’s interesting because the original failure could have led to a turn away from mathematical modeling altogether, but it instead has led to research in improved mathematical modeling.