Sunday, January 29, 2012

The grandest of generalizations

F.Y. Edgeworth, in defending the early marginalists' work on neoclassical microeconomics, talked about how the work was meant to "satisfy the soul of the philosopher with the grandest of generalizations". It's in this sense that I'm personally "satisfied" by the use of cardinal utility functions - something that's been discussed by stickman, the comment section of unlearningecon's blog, and Robert Vienneau.

My view is actually close to Mattheus's on what is actually true (preferences are ordinal, utility functions are made up, and interpersonal utility comparisons don't have a positive basis yet [that I know of - if a neuroscientist studying dopamine levels and human perception wants to challenge that I'm all ears]). But my views are quite far from Mattheus's on (1.) what it is mainstream economists think about these things, and (2.) how we should do microeconomics. With respect to those points, I can agree with Vienneau, stickman, and unlearningecon that a huge portion of Austrians have no clue what they are talking about.

So let's start at the beginning. On two separate points I think you really have to distinguish between two very different things:

1. You have to distinguish between "what mainstream economists think" and "what mainstream economists may or may not teach their undergraduate students"

2. You have to distinguish between "preferences" and "utility".

On point #1, I'm only going to concern myself with what mainstream economists think, because what they teach their students is a pedagogical issue and while I have my own thoughts on teaching undergrads, that doesn't really concern the discussion we're having.

Mainstream economists all agree that preferences are fundamentally ordinal relations. Indeed, that's how they're defined. It's a "preference relation" - it's not some preference scale. Everyone agrees that we can say "X is preferred to Y", or in some cases "X is indifferent to Y", but the sentence "X is twice as preferred as Y" doesn't make any sense because the preference relation is a binary relation. Now, we want to do something with that relation. So we usually assume certain things about preference relations. We say that agents are "rational". Now - a lot of critics of mainstream economics who don't know any better load a lot of junk onto that word "rational". But to say that someone has "rational preferences" is quite simple. It's just saying that (1.) the preference relations are complete, (2.) the preference relations are transitive. That's all.

The important question is - can rational agents settle on a maximal bundle given a set of constraints (typically a budget constraint) with these ordinal preferences.

Mainstream economists say "yes". I think different people learn a different answer to this question. I was taught the answer using Walker's Theorem (1977), because my professor thought it was particularly elegant. I understand there are other ways to show this, but learning Walker's Theorem was a bitch, so I've exerted exactly zero effort in learning any other method. This insight is the heart of mainstream micro. It's so important that we had to write the proof on our midterm and on our final. I think it was something like 1/8th of the points on our midterm and 1/5th of the points on our final. And since that was all of our grade except for 10% for homework, you can see how critical this point is. Ordinal preference relations can be maximized by a rational agent for any budget constraint. Please stop telling me mainstream economists don't think this.

OK, so this is settled. Now this is where Edgeworth's "grandest of generalizations" comes in.

We want to construct theories about the way the economy works motivated by action on the basis of these ordinal preference relations. But it's hard to do that with a collection of binary relations (X>Y, Y>Z, W>P, L>Q, etc.). So we ask ourselves: "can I think of a functional relation that represents these ordinal preference relations, which I can then analyze and use to theorize different things?". The key word here is "represents" - and this is also a point stressed by mainstream economists. What you want is a (cardinal) functional relation that will give you the same answer that you would get from a plausible set of ordinal preference relations. You want that because you can do calculus on a functional relation, but not on a set of binary relations.

Thus, we make up utility functions. We want utility functions that assign values to bundles that duplicate ordinal preference relations. If X is preferred to Y, we want U(X) > U(Y) for all X and Y under consideration. We also want to make sure we're talking about rational preference relations, so the function has to obey those rules as well. That's all that matters.

This gets to a point that stickman raised in some of the comment sections about monotonic transformations of utility functions. If U=x represents a preference relation, then U=ln(x) represents the same preference relation equally well. Indeed, there are lots of problems where you make that exact monotonic transformation because it's easier to solve U=ln(x) than U=x.

The point is, utility functions are meant to represent ordinal preference relations. They are not intended to give you information about the intensity of preferences. This is part of the reason that I'm somewhat skeptical of welfare analysis. It walks a dangerous line. Sometimes it's OK as a way of translating preferences into dollars. That doesn't cross the line of telling you how much more you enjoy something than another thing - it simply monetizes a preference relation. You don't know how much more you enjoy something unless you have a money-to-utils conversion, which of course is a conversion that doesn't exist (or at least that we don't have access to).


A lot of these discussions go wrong when people take what they think they were taught in undergraduate microeconomics and start assuming that's what mainstream economics is. Things are always simplified. The problems you spend a lot of time on as an undergraduate are very different from the problems you spend time on as a graduate student, and it's the latter that's somewhat closer to what economists actually spend time on. Undergraduates spend very little time on working with ordinal preference relations. Graduate students do. They work a lot with that because it's the underpinning of all the subsequent functional relations that we work with.

Critics need to stop and take a second to consider that maybe undergraduate economics doesn't spend much time articulating the step from ordinal preference relations to cardinal utility functions because that step - while necessary - is not important to get the fundamental insights of economics. If instructional time is scarce - and it is, because many don't continue on in economics - you disseminate more knowledge by expecting your students to trust you on the step between ordinal preference and cardinal utility. With scarce instructional time you are also probably better off finessing the difference between a utility function that is useful for theoretical applications and a preference relation that is what is actually motivating human action.

Anyway - trust me - this is not some incredible insight of the internet Austrians. We know this. We spend a considerable amount of time on it.

...and if you all are mean to me in the comment section my next post will be a proof of Walker's Theorem.


  1. Nobody tell Daniel about this. He'll think he won the argument.

  2. A large part of my concern about utility comes down to practical considerations. If the neoclassical model represented the information that Murphy's paper emphasized - and that was IT - then I would only have some minor quibbling about formalism, and certainly not enough to comment about on a blog. The larger beef I have is when I am instructed that we can divine from consumer surplus which consumer "got the better deal" or "lucked out" or "was made happier." This is nonsense. Simply because one customer was willing to pay 10 dollars but only had to pay 2, where another was willing to pay only 6 but had to pay 2 does not tell us anything about the height of either party's satisfaction, nor whether one party achieved more than the other.

    I nitpick on formalism sometimes, but it's the conclusions that really bother me. If the corpus of Austrian economics could be drawn in neoclassical terms, most of us would hardly tell the difference. But it's the mistaken monopoly theory, the mistaken welfare theory, the mistaken choice theory - which all in some way or other draw from utility theory - that really get me.

    1. But this is where I say that it walks a thin line, but it's fine. Consumer surplus is usually expressed in dollars, right? What's wrong with that. Other welfare analysis is - as I've said before - plausible and justifiable, but not necessarily scientific.

      To put it another way - have you ever heard consumer surplus discussed in utils?

      I don't think it usually is.

      re: "But it's the mistaken monopoly theory, the mistaken welfare theory, the mistaken choice theory - which all in some way or other draw from utility theory - that really get me."

      I get the issue with some welfare claims - but what's mistaken about choice theory or monopoly theory?

    2. Another question - would you feel better, knowing how economists actually think about this stuff - accepting this as reasonable as a pedagogical approach for undergraduates who aren't going to be economists?

      Would students get more value or less discussing preference relations and their relation with utility functions, if that would mean skipping more applied exercises with utility functions?

      I think there's a very strong pedagogical argument here.

    3. what's mistaken about choice theory or monopoly theory?

      I don't accept a lot of indifference analysis because it posits an impossible scenario where my entire income is spent on two goods irrespective of all sorts of other considerations. It reminds me of your favorite quote about remorseless logicians. Then there's literature on markets between perfectly substitutable goods and that doesn't even make sense.

      As for monopoly, there is no way to differentiate between a competitive and monopoly price. There is only the market price. Monopolists are just taking advantage of an inelastic demand curve like everyone else.

    4. I can understand and sympathize with it as a pedagogical device. Although every time I am taught something Austrian the undergrads say how much easier it is than the typical (neoclassical) way.

    5. - On indifference analysis - that argument makes no sense. What holds with two goods that doesn't hold with n goods? Kids learn addition with two numbers too. Should they question that it's a valuable concept because of that? I don't think this is a very serious objection.

      - On monopoly theory - I'm not sure what to say. It sounds like you accept monopoly theory. Monopoly prices are market prices. Monopolists are just taking advantage of a less elastic demand curve. Right. So? You say it's mistaken, but then you just repeat what I think everyone understands to be the way monopolies operate.

    6. Show me an indifference curve with n goods, Daniel. There are literally all sorts of things happening that the 2 dimensional graph doesn't take into account. It misses the fact that preference rankings change over time (so this castrates compensated welfare analysis), it misses the fact that a preference for one good can be caused by a preference of an unseen good, it misses that these rankings cannot be known by anyone except the person involved, and the dollars to util conversion is obviously nonsense. If you just want to draw a graph and say "Here are all the combinations of two goods where a fictional actor would be indifferent toward them," and then draw a budget line to show the most inexpensive - I wouldn't care. Again, a bit on formalism, but no problem. The problem is the compensating variation techniques and the welfare analysis.

      The label that monopolies are inherently undesirable is mistaken. The idea that the state should support public monopolies because of LRAC curve considerations, etc. is mistaken. The whole literature on perfect competition and its counterparts is wrong. One of my favorite chapters of MES is where Rothbard dismantles the neoclassical paradigm.

    7. dismantles the neoclassical paradigm... on monopoly theory.

    8. Mattheus, I must say that I also have no idea what you are talking about. For one thing, the entire point of consumer theory is to compare bundles of goods... not two-good trade-offs. (Though, again, such a simple example might be used for pedagogical purposes).

      What micro book are you using, if I may ask?

  3. re: "Consumer surplus is usually expressed in dollars, right? What's wrong with that."

    It's usually expressed in whatever the numeraire good is. And the main point is that equating dollars or the numeraire good to the unit(s) in demand in order to obtain consumer surplus is not a logically tight operation.

    If the eq'm price is 2 units of Y and Bob has a willingness to pay of 4 while Jon has a WTP of 3 we'd add up the surplus to get 3. We might also say that Bob got more surplus than Jon, which would be misleading for two reasons. Firstly, we only know that they prefer a unit of the good (say X) to whatever they're willing to pay. Secondly, any comparisons between the two require knowledge of a utility function - which, strictly speaking doesn't exist. Moreover, even to sum up the surplus across agents violates an ordinal preference relation by the first line of reasoning.

    All we know is that Bob prefers a unit of X to 4 units of Y, and that Jon prefers a unit of X to 3 units of Y. If they acquire a unit of X for 2 units of Y, strictly speaking, we can't subtract 2 from 4 for Bob, and 2 from 3 for Jon, and then add them together to get something meaningful. You have to make assumptions about indifference between a unit of X and 4 units of Y, and so on.....

    And the main problem with the pedagogical line of defense is that it's mildly dangerous. If we instill in a decent chunk of the populace that consumer surplus is a completely legitimate form of doing welfare analysis (which is implicitly true if you don't devote any significant amount of time to its shortcomings - again, because of the complexities involved), then you gradually get a cultural mindset that not only leaves the bulk of analysis that supports policy unquestioned (in that regard), but you also get a contemptuous group mindset with respect to people like those wacky Austrians telling us to take ordinal preferences seriously.

  4. I understand what Kuehn and other Neoclassicals say about cardinal utility functions.

    The main problem I have with the idea that Neoclassical economics is all ordinal is:

    1)Ask any general kid in a microeconomics or intermediate microeconomics class, or a cost benefit analysis class, etc. Ask them if they think utility is ordinal or cardinal. Dollars to donuts I'd be that a good amount of them would say cardinal, judging by the mass of equations they may see, and/or their experience with consumer surplus and welfare economics.

    For the sake of argument, while Neoclassical economics may have the best of intentions, and may truly be ordinal, IMO, I feel that this approach is not properly conveyed to students. And while students (or hopefuls, such as myself) will be taught otherwise in grad school, the majority of undergraduates will get a different impression. But as Kuehn properly puts it, this is a pedagogical issue and not really based on the economic science itself. But if Neoclassical economics is done ordinally, then they should stress it right from the getgo. And in all of my classes, and with my discussions with other students, this is not what is done.

    2)The more troubling problem is when we get to things like consumer surplus and cost benefit analysis. Expressed in dollars, consumer surplus implicitly tries to measure the benefit an individual will derive from buying goods at different prices, and then aggregating all these benefits to get a "social benefit". Or that some individuals will benefit "more" than others from a certain amount of money. With welfare economics you get things like the Compensation Principle or Kaldor Hicks, which was used alot in my cost benefit class. That the net winners in a welfare situation can give some of the benefit to the net losers (and everyone is no worse off than before), then (without even doing it-KH), social utility has increased. Then things seem to "bog down", and by "bog down", welfare economics becomes more ethical and less scientific, as Kuehn says above. And I think that is a big problem, especially because once we get to the "juicy stuff" (policy prescriptions/analysis), economics gets a little less scientific.

    1. The pedagogical point is a debatable one.

      What people have to stop saying is that mainstream economists think preferences are cardinal. That's simply wrong, and if you say that then you're ignorant of what mainstream economists think. Period.

      On welfare economics - I think it's when we get to policy prescriptions that this sort of thing is just fine. We're allowed to make judgement calls about standing and weighting individuals when we're discussing political questions. When we're talking about the science of human social behavior that's less tenable... which is exactly why mainstream economists hold to ordinal preferences represented by utility functions as a modeling technique.

  5. Bear in mind I'm not saying that certain Neoclassical "utility based conclusions", such as demand curves require cardinal utility. Things can be put into equations and still remain ordinal, you just have to be careful.

  6. "The pedagogical point is a debatable one."


    "What people have to stop saying is that mainstream economists think preferences are cardinal. That's simply wrong, and if you say that then you're ignorant of what mainstream economists think. Period."

    Agreed. (Again, I'm not saying that Neoclassical demand curves, etc can't be described ordinally). Even with that in mind, I'd still say Neoclassical versus Austrian price theory are radically different. But thats a whole different topic.

    The big question though, is that is the misconception due to poor research or bad pedagogical devices? I'd say its both, though the weighting would probably be different than yours.

    "On welfare economics - I think it's when we get to policy prescriptions that this sort of thing is just fine. We're allowed to make judgement calls about standing and weighting individuals when we're discussing political questions. When we're talking about the science of human social behavior that's less tenable... which is exactly why mainstream economists hold to ordinal preferences represented by utility functions as a modeling technique."

    When you get to policy prescriptions, then you are obviously allowed to make ethical judgements. Thats implicit in every policy judgement. The problem is if the economic science used to make those prescriptions (consumer surplus, welfare economics, etc), relies on some sort of implicit measurable utility, then you have a problem.

  7. 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. Right - but again - is this a "misunderstanding" or is it a deliberate pedagogical decision?

      This is the first point I said we have to be careful about. Do you think Besanko and Braeutigam actually think this, or do you think this is something they teach undergraduates to drive home the idea of ordinal vs. cardinal.

      Still, I have to say the passage is surprising. Usually textbooks are pretty up front the utils are meaningless, precisely because of the monotonic transformation point.

  8. I don't imply that this means they think this, but its very misleading, if not downright incorrect.

  9. This is me being mean so we get Walker's theorem.

    I agree that mainstream economics understands very well the fact that the cardinal utility functions only represent preference orderings. But I also think that some economists spend so much time in the land of cardinal utility that they learn to forget about preference orderings and think in terms of cardinal utility even when it no longer makes sense. So they may "know" that cardinal utility only represents cardinal utility, but they may not integrate that into their thinking in general. (Purely anecdote-based speculation on my part.)

    Also, a lot of the people who learn about cardinal utility in Econ 101 immediately forget the ordinal preferences point and go on to become policy makers and advocates who sincerely believe that people have utils and that if only we had a utils-meter they could make better policy decisions.

    PS: I can't login to comment. It shows me captcha and immediately redirects away...


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