Thursday, August 02, 2012
We Have Winners!
I've returned from backpacking in the Bob Marshall Wilderness (adventure report will follow at some point) to judge the UC Rothbard Challenge, as we're currently calling it. It was a little difficult to judge the two entries, in that neither specified which category they were entering. Both entries contained the correct calculation, and both addressed the "incommensurable units" argument. I've decided to award Mike McDonald first prize in category 1, Hulsmann Theory, since he makes more comments on the issue, and while expressing doubt at the outset ends up sufficiently certain of his answer that he puts his squared money where his mouth is. By default that leaves Chuck Grimmett as winner in category 2. Chuck also began by formally stating his starting assumption (as good mathematicians are wont to do) defining 100c=$1, which I appreciate. Well done to both! (Each of you should get your mailing address to me; some sort of prizes will be awarded. I believe you both know how to contact me.)
Both McDonald and Grimmett are exactly correct: 100c^2 = .01$^2. The supposed anomaly comes when someone simply squares the numerical amounts, leaves the units untouched, and ends up with $1 in the left pocket and 1 cent in the right pocket. (I regret to say I've seen a number of people with Ph.D.s make this error, to say nothing of a classroom of students.) It's crucial to square the units, just as it is with, say, inches and feet (12 in = 1 ft, 144 in^2 = 1 ft^2).
Both MM & CG suggest that squared cents and dollars make no sense (extra point to CG for the play on words, always appreciated here). But that's a red herring. Suppose instead I asked for entrants to take 12 inches and also 1 foot to the fifth power. I have no idea what ft^5 is and can't visualize it, but the answer is perfectly sensible: 248,832 in^5 = 1 ft^5. This is no more incoherent than, say E = mc^2. (What the heck are km^2/sec^2?) (Of course, this means that McDonald owes me 10,000 $^2. I guess I will have to send him my squared address.)
"Hulsmann theory" (again, pp. 8-9) completely boggles all these issues. I've pointed this out so much I feel like a stuck record (this is an allusion to an ancient technology), but I will make one more observation on Hulsmann's "Cardinality" argument. In making his "meaningless of the units" argument, Hulsmann says the following (p. 9):
The same problem appears on the side of price ratios. The common view that
sees no difficulty in the comparison of price ratios is unwarranted. The problem
becomes obvious once we recall that prices are themselves ratios. A price is not
just “3 dollars” but rather “3 dollars / 1 hamburger.” Now consider the ratios of this
price with two other prices, say, “1 dollar / 1 banana” and “2 dollars / 1 coke.” The
ratio of the hamburger and the banana prices would be “3 bananas / 1 hamburgers,” and the ratio of the hamburger and the coke prices would be “3 cokes / 2
It is clear that we encounter here exactly the same problems as above in the
case of ratios of preference ranks (see Hülsmann 1996, chap. 6). The first problem
is to interpret the meaning of the different units. What does (banana / hamburger)
and (coke / hamburger) actually mean?
Wow! Well, what does (dollar/hamburger) mean? The three textbooks I've used in teaching introductory microeconomics (Gwartney-Stroup, Steven Landsburg, John Taylor) all explain this within the first two or three chapters...this is the relative price of of a hamburger in terms of bananas. It's a concept that's ubiquitous in economics. Mises repeatedly makes use of it in Human Action, e.g. in his discussion of "autistic" action, in his discussion of the calculation argument, in discussing the price of money, and in an example of international trade and exchange rates.
Hulsmann and those who buy his line literally do not understand basic economics.