Monday, December 16, 2013

Math Arguments

The purpose of this post is to address many of the statements/arguments made by Austrians about math or their position on math.

One statement I have been hearing lately is along these lines, "Austrians are not against all math, just SOME math" or, "Austrians are not against math, just mathematical predictions" etc etc.  I find these claims to be rather telling of the person arguing on behalf of the Austrian position.  These statements tell me right away they do not even know the position of their own school of thought.  On their view, these 5 quotes need to be interpreted as only being against some math and/or mathematical predictions.  Here are the relevant quotations:

"The only economic problems that matter, defy any mathematical approach" - Ludwig Von Mises
"Now, the mathematical economist does not contribute anything to the elucidation of the market process"  - Ludwig Von Mises
"The equations formulated by mathematical economics remain a useless piece of mental gymnastics and would remain so even if they were to express much more than they really do" - Ludwig Von Mises

(These next two are my favorite)
"The mathematical method must be rejected not only on the account of its bareness.  It is an entirely vicious method, starting from false assumptions and leading to fallacious inferences.  Its syllogisms are not only sterile: they divert the mind from the study of real problems and distort the relations between various phenomena" - Ludwig Von Mises
"Mathematics cannot and does not enter into measuring ideas or values that determine human action.  There are no constants in these.  There is no equality in market transactions.  Therefore, mathematics does not apply.  The use of mathematics requires constants.  Mathematics cannot be used in economic theory" - Percy L. Greaves.

All of these quotes can be found in various articles on mises.org.

I am truly baffled how someone claiming to be an adherent of the Austrian School could read these, or any Austrian literature, and conclude that Austrians are only against the use of some math.  I have read a lot of Austrian literature, and I personally have never read anything that would support that claim.  Of course, quotations cannot be "proof" of anything, but I do think they provide rather strong evidence in favor of my argument.  Moreover, the Percy Greaves quotation is in response to the question, "is economics completely divorced from mathematics?"  Clearly, from his response he thinks it is.

Another statement I hear from Austrians is that neo-classicals do not give them any mathematical propositions they should accept.  This seems to be a rather silly statement.  For Austrians should accept all mathematical propositions that are true, from 1 + 1 = 2 to the propositions in set theory or algebraic topology etc.

However, to get specific I would like to point out two mathematical fields that have vast applications in economics.  First is game theory.  Game theory is a branch of mathematics first developed by Emile Borel, and then popularized by the works of Von Neumann, Morgenstern, Nash etc.  There is a plethora of economic questions game theory answers.  One example of such a question is - how do oligopolies decide on how much to produce given the production of the other firms?  Game theory provides the answer to this question.

Second, is functional analysis.  In general, functional analysis is the study of infinite dimensional vector spaces.  This field answers the question - how can a copper mining company extract Q tons of copper from a mine over T years and maximize its profit?  To find this function is one thing, and to prove it is the maximum of all functions is another.  I would like to ask an Austrian how to solve this problem without the use of mathematics?  It simply cannot be done.

Mathematics is vitally important to the study of economics, and to denounce it the way influential Austrian scholars have is exactly why I am not an Austrian economist.

3 comments:

  1. "Austrians should accept all mathematical propositions that are true, from 1 + 1 = 2 to the propositions in set theory or algebraic topology etc."

    And so we do.

    I consider game theory to be a branch of praxeology rather than of mathematics, though of course math is used to compute the optimal set of choices for each player. Which is not to say it uses math to predict the *actual* choices of each player, which is why it's of no use in economics. The (safe) assumption of game theorists is that each player will try to maximize his utility, his "payoff" in the face of imperfect information either about the details of the "game" or about the decisions other players would make that might ruin his strategy. Take the prisoners' dilemma, the classic example. Math is used to describe various possible strategies, to compute optimal outcomes etc., but it's of no use whatsoever in predicting the actual choices of each player - human action, the stuff economics is made of.

    "how do oligopolies decide on how much to produce given the production of the other firms? Game theory provides the answer to this question."

    So does economics, but only qualitatively. Neither branch of praxeology can predict the actual output of the firms - only that their owners will attempt to maximize their utility. Which is interesting because in this cartel scenario, the interests of individual producers are in conflict with those of the group, such that each producer has an incentive to overproduce to exploit the high prices established by cartelization and the other producers' underproduction. Not a single valid quantitative prediction emerges from this.

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  2. On functional analysis, copper miners aren't doing economics when they attempt to maximize or optimize output. Economists are interested in the fact that the miner will attempt to maximize profits, other things being equal, which is a more important caveat than you might think.

    Say I'm a copper miner. I keep close tabs on my mine, so I know that on average I can expect a ton of ore to yield a pound of pure copper, say. I've got some money in the bank and would like to increase my productive capacity, so on the back of a napkin I work out how much more I can produce in a month by hiring 5 more guys and buying $1M worth of gear. Maybe that would let me increase production by an even ton each month. Let's say we can simplify by holding all the variables constant - my mine's yield, the spot price of copper, factor prices, the weather, anything else that might affect my profit on a unit of product. Now I can math up a prediction that this project will pay for itself in one year, and increase my profits by 5% thereafter. None of this math has anything to do with economics - none of this has anything to do with human action. The economist is interested in whether, in the face of this prediction, made using assumptions which are unsafe in the real world, I will undertake the project. There are still other considerations - what is the market interest rate? Maybe I would be better off putting my money in savings than investing in this project. But in principle we can control for all those considerations, and still not predict what I will do, because we don't know my time preference - possibly I would rather buy some Lamborghinis to pick up chicks, than wait for the modest fruits of this investment.

    But even here, because we're looking at one guy and how I make my decisions in the face of a given set of circumstances, we're not doing economics - only praxeology. The economic question is, if I do make the investment and it does add a ton per month to the supply of copper, what will happen to the price of copper? All the economist can tell you is that it will be lower than it would otherwise have been. He can't say how much lower, and he certainly can't tell you what the price of copper will actually be, even if we stipulate that this is the only change worldwide to the supply of copper.

    Now back to the 30,000 foot view, everybody insists that you can legitimately use math in economics, and you're not the first to mention game theory as an alleged example of this, but nobody has ever actually offered up a valid economic mathematical proposition to be examined on its merits. Why is that?

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