If you have spent an egregious portion of your life in economics classrooms, you have doubtless heard that the utility functions used therein are representations of purely ordinal preference rankings. As your professors likely told you, although utility functions assign numbers to various bundles of goods, the numbers themselves have no meaning beyond their order. If two apples, one orange takes the value 25, and one apple, two oranges takes the value 5, all that tells us is that the former bundle is preferred to the latter. The utility function is unique up to a monotonic transformation.
What that means is that we can change the function in any way we like so long as the order remains unchanged. We could add a constant to our utility function and our theory would be unchanged. We could cube our function, and, since cubing is a monotonic (i.e. order-preserving) transformation, that would also have no impact. Thus, although the utility function appears cardinal, it merely expresses ordinal preferences.
Knowing this, you can imagine my surprise when I found myself teaching a classroom full of undergraduates that utilities are, in fact, cardinal! The particular problem involved deriving “expected utility.” The neoclassical theory of how people form preferences when facing known uncertainty, for instance when evaluating a coin toss that could give them $15 for heads or $5 for tails, slips into cardinal utilities so subtly that few undergraduate students notice the shift.
A demonstration will help to illustrate the point: Suppose that Sally is wondering how much she would pay for a fair coin toss that would yield $15 for heads or $5 for tails. Sally prefers more money to less, so it would seem that we can choose any utility function that yields u($15)>u($5). Let’s test that.
Suppose Sally has a utility function of u(x)=x, where x is her wealth. Then, according to the theory of expected utility, Sally should value the coin toss at 0.5u($15)+0.5u($5)=10=u($10). So far so good. Now suppose we apply a monotonic transformation to that utility function, make it u(x)=x^3. Now 0.5u($15)+0.5u($5)=1,750=u($12.05). Uh oh! Now Sally would pay $2.05 more for the exact same coin toss, so our monotonic transformation clearly did more than just change some arbitrary utility numbers. [1] Before we had a preference ranking that had $11 preferred to a $15/$5 coin toss, and now we have the $15/$5 coin toss preferred to $11. What happened?
The answer, as I suggested above, is that the von Neumann-Morgenstern utility functions used in expected utility theory are different from the purely ordinal utility functions of your intermediate micro course. The misguided attempt to derive people’s valuations of lotteries (that is, events with random payoffs drawn from known distributions) from the probabilities and valuations of all the possible outcomes depends on those valuations being cardinal. In the case where u(x)=x, the utility function doesn’t just express that $15 is preferred to $10 is preferred to $5; it expresses that the intensity of preference for $15 over $10 is the same as the intensity of preference for $10 over $5. In the case where u(x)=x^3, $15 is more intensely preferred over $10 than $10 is over $5.
Not only does this theory depend on the impossible notion of cardinal utilities, it is demonstrably false. The Allais Paradox is one such demonstration.[2] Allais illustrated the “paradox” with two mathematically equivalent choices: The first (hypothetical) choice was between (1) a 100% chance of getting $1 million, and (2) an 89% chance of getting $1 million, a 10% chance of getting $5 million, and a 1% chance of getting nothing. The second (hypothetical) choice was between (A) an 11% chance of $1 million and an 89% chance of getting nothing, and (B) a 10% chance of $5 million and a 90% chance of getting nothing.
Both choices, that between (1) and (2) and that between (A) and (B), are a choice of sacrificing an 11% chance at $1 million for a 10% chance at $5 million. The other 89% should be irrelevant. Yet many people choose (1) over (2) and (B) over (A), demonstrating a set of preferences that cannot possibly be consistent with expected utility theory. The apparent cause of this inconsistency is that people value certainty, so the jump from a 99% chance of getting money to a 100% chance is valued higher than the jump from 10% to 11%.
In addition to this value placed on certainty, a great many things can influence a person’s subjective valuation of a lottery, just as a great many things can influence a person’s subjective valuation of a ham sandwich or a new shirt. A person with a strong religious objection to gambling might value the $15/$5 coin toss described above at less than $0, which would certainly be ruled out by the theory. Economists should have the humility to realize that they will never discover invariant laws governing how people form their subjective values.
This is a case where Mises’ distinction between praxeology and psychology is useful. Praxeology, the study of human action, examines purposive action given the actors’ subjective valuations and their beliefs. How people form their valuations is ultimately psychological and subject to many changing factors; it lies outside the realm of praxeology. The ordinal utility functions of neoclassical consumer theory, those that are unique up to a monotonic transformation, are ultimately praxeological. Von Neumann-Morgenstern utility functions, despite their notational similarity to the former ordinal functions, are merely the basis of flawed psychological theorizing about the formation of subjective valuations under a very specific type of uncertainty. They have no place in sound economic theory.
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[1] Strictly speaking, one can transform a von Neumann-Morgenstern utility function by adding a constant or multiplying by a positive scalar without altering the agent’s preferences, just as one can convert from Fahrenheit to Celsius without changing the temperature. All other transformations, monotonic or otherwise, will alter agents’ preferences.
[2] M. Allais. (1953). Econometrica. Vol. 21, No. 4, pp. 503-546
Article Stable URL: http://www.jstor.org/stable/1907921
Tags: austrian economics, economcis, Monotonic Transformations, Praxeology, psychology, von Neumann-Morgenstern