Debreu’s Theory of Value

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.

Maskin and Roberts on General Equilibrium

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.

Mathematical Complexity of Simple Economics

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.

Opportunity Cost Explained

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.

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Sunspot Equilibria

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.

Wanted: game theorists

The University of Liverpool (UK) has a vacancy for a post-doc researcher and for a PhD student, both in automated mechanism design. The expertise we are looking for includes game theory, mechanism design and auction theory, mathematical economics, and computational versions of same.  These posts are part of a major UK research project on market-based control of complex computational systems.

Details here and here.

Peer-review and its discontents

The latest issue of the Post-Autistic Economics Review is now out, available here.   It has an interesting article by philosopher Donald Gillies arguing against the centrally-organized reviews of university research activities which British academics have had to endure these last 20 years, and which now look likely to be adopted in Australia, NZ and elsewhere.  One argument he makes is that one’s peers are usually quite bad at judging the long-run impact and quality of one’s research, especially when the research is innovative, and Gillies gives the example of Frege’s Begriffsschrift, the first axiomatic treatment of propositional and predicate calculus.  When this was published in 1879, it was slammed by Frege’s contemporaries, and it was only recognized for the seminal work it is two decades later.  If Frege had been working in a British University a hundred years later, both he and his department may have faced termination by his university administration, given the hostility that his own peers felt towards his work; lots of departments have been closed, and academics made unemployed, as a result of the peer assessments of the British Research Assessment Exercise (RAE).

A longer version of Gillies’ paper is available on his web-site, here.  

Behavioral Economics

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.