Mathematical Complexity of Simple Economics
June 1st, 2007 by WaltI 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.
June 3rd, 2007 at 1:03 pm
I wonder how much relevance this has to economic theory. It seems to be symptomatic of the larger failure of economics to construct a plausible theory of the economy in disequilibrium and how it moves to equilibrium. Most economists seem to be pretty uninterested in that failure but I have the feeling that they should pay more attention.
June 3rd, 2007 at 9:20 pm
economist say if 100 ppl take $100 worth of goods to a swap meet and trade among themselves then everyone will go home with more than $100 worth of goods - and i say if the same ppl take the same goods back each week for a year they will all become millionaires in no time
June 3rd, 2007 at 10:11 pm
mortimore: that’s really an amazingly facile misstatement. The biggest problem is that you’re completely disregarding the fact that value is an extremely local phenomenon. I value a given collection of goods differently than you do.
If I value some collection of your goods more than a collection of mine, and similarly you value that same collection of your goods less than you do that same collection of mine, then we both benefit (in our own estimation) by trading. Nothing has been created or destroyed, but we both feel better off, since we assigned different values to the goods involved in the trade.
June 4th, 2007 at 10:10 pm
I agree with John. The key question asked in a proper college Economics course is this: “What is one dollar worth?”
June 5th, 2007 at 4:16 am
Johan Richter said:
“I wonder how much relevance this has to economic theory. It seems to be symptomatic of the larger failure of economics to construct a plausible theory of the economy in disequilibrium and how it moves to equilibrium. Most economists seem to be pretty uninterested in that failure but I have the feeling that they should pay more attention.”
Correction: (mainly U.S.) economists dreaming of the Nobel price who still suffer from the illusion that economics is on par with physics seem to take this “general equilibrium” stuff seriously.
June 6th, 2007 at 5:20 am
Well, yes GE theory is still taught (which I think is a good gthing since I think it helps you understand th economy) and researched (which I am not so sure is a good thing) but General equilibrium is an equilibrium theory.
And so is almost all other mainstream economic theories. I think this intellectual straightjacket is unfortunate.
June 6th, 2007 at 7:35 am
I quite enjoyed studying GE theory. The mathematics is nice and not too difficult.
Sure enough, computable general equilibrium (CGE) modeling is still a very active field of research, but I fail to see the relevance of those models (too many restrictive and unrealistic assumptions for one thing).
As for a plausible theory of the economy in disequilibrium and how it moves to equilibrium, I was actually refering to a (limited) number of “economists” employing mathematical techniques borrowed from quantum mechanics or some other field of advanced physics trying to resolve that problem. I suppose they deserve some credit for trying, but whether e.g. the so-called isomorphism of “expected value” and “density matrix” is going to save the day, I doubt it …
June 7th, 2007 at 8:21 am
“What is one dollar worth?”
Answer: whatever someone will give you in exchange for it, in goods or services.
Different people may arrive at different values. You are likely to accept those goods or services which you consider to be worth less than a dollar, and accept those goods or services which you consider to be worth more than a dollar.
The point is, different people will evaluate the dollar differently. Dollars are fungible; one is the same as another (as currency, one might be collectable to a numismatist).
John Armstrong is right, that two people can have “… assigned different values to the goods involved in the trade.”
June 7th, 2007 at 9:29 am
JVP: It reminds me of the hypothetical involving four dollar bills. One is a real dollar bill signed by Andy Warhol, one is a perfect counterfeit signed by Andy Warhol, one is a real dollar bill with a perfect forgery of Andy Warhol’s signature, and one is a perfect counterfeit with a perfect forgery of Andy Warhol’s signature. Discuss.
June 7th, 2007 at 11:45 am
Question:
How is “… a universal price mechanism ..” [from p 226] different from a grand unified theory or theory of everything?
D Saari, “Mathematical Complexity of Simple Economics” discusses the dynamics of markets. Could this also be applied to the dynamics of energy econmics in general realtivity and quantum mechanics?
I have read Tamer Basar and Geert Jan Olsder, “Dynamic Noncooperative Game Theory (Classics in Applied Mathematics)”, [SCIAM]. I was very impressed by how they eventually came to use Pursuit-Evasion Games [chapter 8] to demonstrate this complicated mathematics for physical applications in various types of engineering.
I have read Bernd Heidergott, Geert Jan Olsder and Jacob van der Woude, “Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications (Princeton Series in Applied Mathematics) “. I was astonished to see how simplified game theory algebras can be using this form of dioid, k-algebra. There are various other forms such as Min-Plus [or Tropical] that heve even been extended from real to complex numbers.
Petri nets are used for graphing these powerful algebras. Nodes are used in lieu of vertices and arcs rather than edges. Form my perpsective these could be called “Node Operator Algebras” since they appear be so similar to Vertex Operator Algebras.
Petri nets also use transitions and tokens. The latter might be able to represent energy quanta.
Max-plus algebra, Wiki review:
http://en.wikipedia.org/wiki/Max-plus_algebra
Max-plus algebra:
http://www-rocq.inria.fr/MaxplusOrg
or
http://amadeus.inria.fr/gaubert/maxplus.html
Petri Nets:
http://www.informatik.uni-hamburg.de/TGI/PetriNets/
Algebra over a field:
http://en.wikipedia.org/wiki/K-algebra
June 9th, 2007 at 9:06 am
John Armstrong: that’s a mere lemma in this:
Umberto Eco, “The Original and the Copy”, in Francesco J. Varela and Jean-Pierre Dupuy (eds), Understanding Origins, Kluwer Academic Publishers, 1992, pp.273-303.
Let us imagine the following:
In 1921 Picasso asserts that he has painted a portrait of Honorio Bustos Domeq. Fernando Pessoa writes that he has seen the portrait and praises it as the greatest masterpiece ever produced by Picasso. Many critics look for the portrait, but Picasso says that it has been stolen.
In 1945, Salvador Dali announces that he has rediscovered this portrait in Perpignan. Picasso formally recognizes the portrait as his original work. The painting is sold to the Museum of Modern Art, under the title ‘Pablo Picasso: Portrait of Bustos Domeq, 1921.’
In 1950, Jorge Luis Borges writes an essay (’El Omega de Pablo’) in which he maintains that:
1. Picasso and Pessoa lied because nobody in 1921 painted any portrait of Domeq.
2. In any case, no Domeq could have been portrayed in 1921 because such a character was invented by Borges and Bioy Casares during the forties.
3. Picasso actually painted the portrait in 1945 and flasely dated 1921.
4. Dali stole the painting and copied it (masterfully). Immediately afterwards, he destroyed the original.
5. Obviously the 1945 Picasso was perfectly imitating the style of the early Picasso and Dali’s copy was indistinguishable from the original. Both Picasso and Dali used canvas and colors produced in 1921.
6. Therefore the work in New York is the deliberate authorial forgery of a deliberate forgery of a historical forgery (which mendaciously portrayed a non-existent person).
In 1986 there is found an unpublished text of Raymond Queneau asserting that:
1. Bustos Domeq really existed, except his real name was Schmidt. Alice Toklas in 1921 maliciously introduced him to Braque as Domeq and Braque portrayed him under this name (in good faith) imitating the style of Picasso (in bad faith).
2. Domeq-Schmidt died during the blanket bombing of Dresden and his identity papers were destroyed in those circumstances.
3. Dali really rediscovered the portrait in 1945 and copied it. Later, he destroyed the original. A week later Picasso made a cop of Dali’s copy, then the copy by Dali was destroyed. The portrait sold to the MOMA is a fake painted by Picasso imitating a fake painted by Dali imitating a fake painted by Braque.
4. He (Queneau) has learnt all this for sure from the discoverer of Hitler’s diaries.
All the individuals involved in this story are now dead. The only object we have at our disposal is that hanging in the MOMA.
June 13th, 2007 at 1:40 pm
I might point out that the Kakutani fixed point theorem is more general than Brouwer’s and thus frequently used instead since it gives equilibrium in more general circumstances.
Another fun fact about Brouwer’s theorem is that it can be shown to be equivalent to the nonexistence of draws in the game of Hex.