Tuesday, March 1, 2011
The State-Contingent Model, Pages 163-166
Building upon the concept of option contracts in commodity trade, Arrow created a model to give application to the new math. Arrow took the idea of an options market and generalized it to include all that is in the economy, and every possible situation. The physical definition of any given commodity thus became dependent on the “state of the world.” This insight gave uncertainty to the theory of equilibrium, by allowing for the possibility that anything could happen in what Arrow called “the complete market.” A few years later, Debreu published the text Theory of Value: An Axiomatic Analysis of Economic Equilibrium, which came to essentially the same conclusions. The Arrow-Debreu model, when coded and put into computers, became what was essentially an early spreadsheet. A significant tool for those who followed, this "spreadsheet" was one of infinite dimensions, with “all the possibilities for all the markets for all the commodities in the world.”
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A for Jim.
ReplyDeleteIt's OK here to use the word "option", but it's a little careless because we use that word in finance to mean something very specific and different from this. So, "states of the world" is better: options are things that might happen that you have some control over, while states of the world are more like a situation beyond your control that you have to deal with.
The usefulness of Arrow's framework is not that "anything can happen" but that it disciplines modelers to stop using the excuse of surprises to avoid fixing their broken models. Metaphorically, it's like bad mystery novels, where the killer is dropped on you as a surprise in the last chapter. Arrow is pushing theorists to not be that way by giving them a framework where they have to include the possibility of that surprise in the last chapter.
FWIW: Jimmy Wales, who founded Wikipedia, was a student of mine in a graduate class in 1990. His biggest beef was with the Arrow framework, and how unlike reality it was. I tried to tell him that it wasn't about reality, but rather about constraining theorists to recognize that reality is full of surprises that they need to account for. He didn't buy it, left school early, moved to Chicago, traded options, made millions, and retired young. Good for him! But Arrow's modeling framework is more popular than ever.
BTW: the separating hyperplanes mentioned in this section are used in linear programming and constrained optimization in ECON 2500. They are the idea that on a graph, your constraints are coming from one direction, the objective you want to optimize from the opposite direction, and they will touch at a point of tangency (where a separating hyperplane keeps them from touching in other spots).