Monday, January 30, 2012

A wise Austrian economist on mainstream understandings of preferences and the utility functions that represent them

"It should immediately be clear from this definition of “represent” that if one function, U(· ), represents the preference ordering, then any positive monotonic (i.e. rank order-respecting) transformation of U(· ) will also represent the ordering. The reason neoclassicals stress this seemingly trivial mathematical detail is that it (should) prevent any possible significance from being assigned to a “util.” For example, if U(a) = 20, while U(b) = 10, we can conclude is that a @ b, i.e. that the individual considers bundle a to be at least as good as bundle b [Bob used the symbol "@" to indicate the preference relation in this piece]. But we may not conclude that bundle a offers ten more “utils,” nor can we say that bundle a is “twice as good” as bundle b. Why not? Because our representation theorem tells us that the function V(· ), defined, say, as the square of the function U(· ), will also represent the preference relation. So if we used this monotonic transformation, we would have V(a) = 400 and V(b) = 100. It’s still true that the number assigned to the first bundle is higher, and thus we still conclude that bundle a is (ordinally) preferred by the individual. But now the utility assigned to the first bundle exceeds by three hundred that assigned to the second bundle, and it is no longer double the quantity."

- Bob Murphy, 2000


  1. This maybe a little just a tad off-topic, but it seems like has maybe misrepresented the way mainstream micro is taught, or at least in my experience. I actually started out on before going to "the darkside" (haha) of economic theory, and I was expecting to be confronted with these cardinal rankings in college. But both my intro and intermediate micro courses made it very clear that such measurements could not be made. We were told that any numerical values were arbitrary and used to illustrate ordinal preference. Maybe there is some more math-heavy stuff in the appendix that treats utility functions in this cardinal manner?

  2. Not saying that this proves that neoclassical economics consumer theory requires cardinal utility, but to give an example of the poor misunderstanding about cardinal utility, heres from my intermediate microeconomics textbook:

    "In this book we will refer to two types of rankings: ordinal and cardinal. Ordinal rankings give us information about the order in which a consumer ranks baskets. For example, for basket A in Figure 3.1 the consumer buys three times as much food and three times as much clothing as she does for basket D. We know that the consumer prefers basket A to D because more is better. However, an ordinal ranking would not tell us how much more she likes A than D.
    Cardinal rankings give us information about the intensity of a consumer's preferences. With a cardinal ranking, we not only know that she prefers basket A to basket D, but we can also measure the strength of her preference for A over D. We can make a quantitative statement, such as "The consumer likes basket A twice as much as basket D." A cardinal ranking therefore contains more information than an ordinal ranking." (p.72)

    Microeconomics 3rd edition. David Besanko and Ronald Braeutigam.

    1. As I said in the other post - I still think it's important to distinguish between what Besanko and Braeutigam actually think, and what they tell their intermediate undergraduates to drive home the difference between "ordinal" and "cardinal" preferences and what those labels imply. Those are two quite different things.

      Still, this is a fairly surprising passage. Usually these texts are good about pointing out how unable we are to talk cardinally - and they say this because the monotonic transformation point is so important.

  3. Again, I'm not denying that its incorrect to assume from this passage that this means that they MUST use cardinal utility, but its extremely misleading.

    1. I guess my point is that calling it "misleading" depends on where you want to lead them.

      If you want to lead students towards a basic grasp of the distinction between cardinal and ordinal so they keep ordinal in mind as they go through the rest of the semester using real-valued utility functions, it is leading them in the right direction.

      If you want to lead students towards being economists, it's probably going to lead them in the wrong direction. But only a small percent of students go on to do that, and the ones that do go on will have plenty of time to think through these issues.

    2. Let me put it this way: The claim "Using functional relations that represent ordinal binary preference relations is an important technique for economists" is not a fundamental concept that non-economists need to go through life understanding. What they need to go through life understanding is that a price ratio is a marginal utility ratio, that marginal utility is best thought of as decreasing, etc. etc.

    3. I don't see how you can say

      "If you want to lead students towards a basic grasp of the distinction between cardinal and ordinal so they keep ordinal in mind as they go through the rest of the semester using real-valued utility functions, it is leading them in the right direction."

      and then

      "The claim "Using functional relations that represent ordinal binary preference relations is an important technique for economists" is not a fundamental concept that non-economists need to go through life understanding."

      If your going to use mathematical functions that at appear to treat utility as a continuous cardinal variable and teach it to undergraduates, then you should make it a point that utility is ordinal and not cardinal under any circumstances. Writing passages like the above and then following with mathematical modeling of consumer behavior and demand curves etc, is misleading.

      The fact that passages above exist, there (at least in my experience) isn't nearly enough emphasis Neoclassical theory of consumer preference being ordinal, and the "behind the door" cardinal utility ideas you implicitly get in welfare economics, it isn't all that surprising that people think Neoclassical economics uses cardinal utility. Whether they actually do (not in consumer preference, but in other areas like WE), poor pedagogical devices on behalf of the teachers, or poor learning from the students seem to be the main possible reasons.

  4. Patch that quote from Besanko and Braeutigam shocks me. I can't believe they said that (given the context of our discussion here). Do they make any other points, or just move on?

  5. I haven't read the book in a while (its been quite some time since intermediate micro, and frankly I'd like to keep it that way), so I can't do a full exposition of the textbook. But its in my hand right now, and looking at my underlined notes, they do go on a talking about the two of them:

    "It is usually easy for consumers to answer a question about an ordinal ranking, such as "Would you prefer a basket with a hamburger and french fries or a basket with a hot dog and onion rings?" However, consumers often have more difficulty describing how much more they prefer one bundle to another because they have no natural measure of the amount of pleasure they derive from different baskets. Fortunately, as we develop the theory of consumer behavior, you will see that it is not important for us to measure the amount of pleasure a consumer receives from a basket. Although we often use a cardinal ranking to facilitate exposition, an ordinal ranking will normally give us enough information to explain a consumer's decisions." p.73

    Granted, they do reiterate that it "is not important for us" to use cardinal utility, but then later say, ordinal rankings will "normally" give us enough information to explain preferences. They don't definitively say "We will not be using cardinal utility in this textbook", from this they say "We will use cardinal rankings to represent ordinal relations for convenience, and this will apply most of the time". And then for the rest of the chapter they talk about levels of satisfaction, utility functions, total utility, and mapping a consumer's "utility" over the course of a week when he eats hamburgers. Again, I'm not saying that this means they use cardinal utility in preference functions, but frankly in books and the way its taught can be very ambiguous, if not downright misleading.

    1. I remember reading that textbook, and my interpretation was that they were treating cardinal utility as a "what if". Basically, much like an equilibrium analysis, they were using a model that assumed that one could measure utility as a tool in understanding preferences regarding certain ordinal baskets of goods as compared to each other. I don't think that they were saying that this is anything that explains real choices, but that it is helpful in understanding the ordinal structure with regard to the choices of an agent that subjectively rates basket A versus basket B (sort of like a 1-to-10 scale).

      I always read it as a hypothetical, like asking, "rate both Jessica Alba and Jessica Biel on a scale of 1-to-10, and then compare these two choices and place them into an ordinal ranking". It isn't like they're saying that they can actually use cardinal utility to explain certain choices, but that everybody does use their own unique rating system with regard to such choices, and that that is how they rank certain preferences on an ordinal scale. In the end, it ultimately comes down to ordinal preferences, because cardinal utility ultimately resolves to an ordinal ranking of preferences.

      In the practical sense, there certainly is nothing that can be drawn from this other than that actors prefer one action to the other. The use of units used to measure utility is nothing more than a mental exercise, a "what if" that in the real world is only present in the moment of choosing between one action vs. the other.


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