22 May 2015

Cardinal vs. Ordinal Utility

David R. Henderson, Economics, Shameless Self-Promotion 41 Comments

Uh oh, it’s David R. Henderson and me vs. Tyler Cowen and David Friedman. How do you rank the gladiators? An excerpt:

There has been a flareup in free-market economist circles over the issue of “interpersonal utility comparisons.” First scrooge mcduckTyler Cowen wrote a post that took it for granted that a rich man got less utility from an extra dollar than a poor man did (although Cowen didn’t think that justified government redistribution by itself). In response, David R. Henderson expressed puzzlement that Cowen overlooked the fact that such an interpersonal utility comparison (IUC) was nonsense, because utility is an ordinal concept. Then a bunch of people,including David Friedman, jumped into the fray, claiming that Henderson was outdated because von Neumann and Morgenstern had shown how we could in fact construct cardinal, not ordinal, utility functions.

In the present blog post I’ll hit the key points in this dispute. To cut to the chase, I agree with David R. Henderson: The way economists use the term, “utility” is an ordinal concept, which expresses a subjective ranking, not an objective measurement. Therefore it makes no sense to say Jim gets more or fewer utils than Sally. Furthermore, the work of von Neumann and Morgenstern does not alter this basic fact: Whether we “believe in” cardinal utility has nothing to do with their demonstrations.

41 Responses to “Cardinal vs. Ordinal Utility”

  1. guest says:

    Notice that even though the “overall utility” of Lottery 2 is a larger number, it would make more sense, given the ranks of the fruit, to choose Lottery 1.

    In both lotteries, the 3rd-ranked fruit is the most likely to be won. But in Lottery 1, the 2nd-ranked fruit has a greater likelihood of being won than in #2.

  2. LK says:

    “Therefore it makes no sense to say Jim gets more or fewer utils than Sally … even if we thought it made sense to attribute units of utility to individuals, there is no reason to suppose we could compare them across individuals..”

    It is not possible to objectively measure the intensity of the emotions that attach to the concept of utility – yet.

    But it is absurd to say that a proposition like “Jim is happier than Sally” or “Jim gets more enjoyment from a unit of this same good than Sally” makes :no sense. You are saying that the subjective utilities of different people are totally incomparable even in principle: that would require that every single person’s emotions of happiness, satisfaction and pleasure (the emotions that subjective utility must be identified with) are utterly unique and in no way resemble anyone else’s. This is a bizarre, anti-scientific, and utterly unconvincing idea: it violates everything we know about human psychology, neuroscience and evolution.

    What’s more you have already conceded that one day science may be able to measure the intensity of the relevant emotions associated with utility:

    “It may be that one day neuroscientists come up with an objective way to quantify various degrees of happiness, such that they can coherently talk about Mary being ‘three times more satisfied’ than Bill.” (Murphy, Robert P. 2010. Lessons for the Young Economist. Ludwig von Mises Institute, Auburn, Ala. p. 41).

    • Tyler says:

      I think Bob is saying that it is impossible to make those comparisons *as an economist.*

      Personally, I’m fairly confident in claiming that Bill Gates receives less utility from a marginal dollar than does a homeless man. However, in making that claim, I am completely departing from the realm of economic science.

      • Ben B says:

        I’ve always thought that a malnourished, homeless person would be able to produce far fewer utils in any situation relative to a healthy, nourished billionaire.

    • Rick Hull says:

      LK,

      You seem to be deliberately obtuse here. Is it intentional to muddy the waters of utility theory with measuring intensity of emotion?

      Of course we may have an intuition about whether Jim is, in fact, happier than Sally. And particularly if Jim is held in a pleasant or comfortable state while Sallie is being tortured, then we can state with confidence that Jim is happier. But we cannot say by how much.

      I understand you find this view bizarre, anti-scientific, unconvincing, and in violation of everything we know about human psychology, neuroscience, and evolution. I can only state that I find this view to be the exact opposite in every category. Do you care to back up your findings?

      Bob’s (quoted) concession indeed leaves the door open for future philosophers and scientists to provide factual insight beyond mere intuition for making IPUC, and it’s baffling that you think this weakens his argument. His concession doesn’t change the fact that we don’t have the knowledge today and that there is zero hope for such knowledge in the forseeable future.

      Finally, regarding intuition, you should seriously try to grasp whether your confidence that the poor man gets more utils from a dollar than a rich man is based upon your imagining of yourself in both men’s shoes. That is, if you were poor, you would value the dollar more than if you were rich. It’s my belief that this imagining is what gives people the intuition that IPUC is possible. But please note that no IPUC has been done. There is only one person in this scenario.

      • LK says:

        “Is it intentional to muddy the waters of utility theory with measuring intensity of emotion?”

        If subjective utility is not in fact reducible to emotions like happiness, satisfaction and pleasure, then what is it?

        • Bala says:

          Utility is the subjective appraisement of the usefulness of a means towards end satisfaction. Nothing to do with emotions, happiness or satisfaction out there.

          • LK says:

            “Utility … Nothing to do with emotions, happiness or satisfaction out there.”

            Better break the sad news to Bob Murphy. Apparently he is such an idiot he thinks this:

            “marginal utility: The marginal utility of a good or service is the amount of satisfaction—utility—you get when consuming one unit of it.”
            Robert P. Murphy, The Politically Incorrect Guide to Capitalism, p. 18.

            “We can say that individuals rank outcomes in terms of happiness, utility, satisfaction, contentment, etc.
            Murphy, Robert P. Study Guide to Man, Economy, and State, p. 6.

            “In praxeology, happiness (or utility, or satisfaction) is a purely formal term, defined entirely by the subjective goals of the individual actor.”
            Murphy, Robert P., Study Guide to Human Action, p. 2.
            ————-
            Poor old Bob: doesn’t understand basic Austrian concepts.

            • Rick Hull says:

              That’s a cheap shot. Austrians hold no monopoly over utility theory. Secondly, your quotes are from Bob’s book addressed to (young) laymen. He is deliberately oversimplifying in an attempt to make things intelligible for his audience.

              It appears that your goal is to make Bob look foolish rather than engage with the substantive argument from this post. If you want to look for and point out contradictions — a generally noble effort — I suggest you should stay within the same realm, where the expected audience is peers rather than children.

              • Rick Hull says:

                Whoops, those quotes are not from LFTYE. My bad.

                Still, utils do not map to emotions, though satisfaction and happiness are imperfect analogs. These terms are useful for introducing utility theory, but utility is meant to be more formal and abstract, deliberately introduced to be used in place of the fuzzier happiness or satisfaction terms.

            • guest says:

              “Better break the sad news to Bob Murphy.”

              Doesn’t the last of your Murphy quotes get at the essense of what Bala was saying?

      • Harold says:

        We are looking for a category difference here – ordinal or cardinal. That category should not be affected by our ability to measure accurately. If accurate measurement is possible hypothetically (in the future say), then we are justified in treating them as cardinal, even if we cannot make the measurement with much accuracy today.

        We probably agree that distance is cardinal. If we had only a 1 yard stick, we could not measure accurately, but the property of distance would remain cardinal. It would make perfect sense to say 1 inch was half 2 inches, even though we could only measure to within 1 yard. The problem would not be that it makes no sense to talk thus, but that we couldn’t measure accurately enough to be sure we had twice one inch.

        “It may be that one day neuroscientists come up with an objective way to quantify various degrees of happiness, such that they can coherently talk about Mary being ‘three times more satisfied’ than Bill.” If utility is subject to such measurement in the future, then it makes perfect sense to talk of Mary being three times more satisfied than Bill right now. It is just that we have a problem measuring it.

    • guest says:

      “But it is absurd to say that a proposition like “Jim is happier than Sally” or “Jim gets more enjoyment from a unit of this same good than Sally” makes :no sense.”

      Jim is more happy than Sally … in what context that we can hold constant for both of them? (Since they would have to have the same preferences and the same opportunity costs for this to make sense.)

      Jim gets more enjoyment from a unit of this same good than Sally … given what end and given what opportunity costs that we would need to hold constant for both of them?

      Intensity of emotion is relative to subjectively ranked opportunities. So, without knowledge of the ranks of other opportunities – which change all the time, anyway – you won’t be able to compare the utility of something across two or more individuals.

      Two people would have to think exactly the same way, and have exactly the same opportunities for this to make sense. And even then, two people cannot occupy the same space, so some resources would necessarily be more readily available to one than the other, resulting in inequality of opportunity and making the comparison of utility impossible.

    • anon says:

      “But it is absurd to say that a proposition like “Jim is happier than Sally” or “Jim gets more enjoyment from a unit of this same good than Sally” makes :no sense. You are saying that the subjective utilities of different people are totally incomparable even in principle: that would require that every single person’s emotions of happiness, satisfaction and pleasure (the emotions that subjective utility must be identified with) are utterly unique and in no way resemble anyone else’s.”

      The claim is not that the happiness of A “in no way resembles” the happiness of B; presumably the function of the human brain means that my subjective experience of happiness is roughly analogous to that of another homo sapiens. It’s a massive inductive leap, but it’s one we make every day, mostly because it works out pretty well not to treat other people as though they’re unfeeling machines.

      The claim is a subtler one, namely that no objective means of studying, measuring, or otherwise charting the interaction of an aggregation of third-person Its–molecules, neurons, whatever–can ever give insight into the subjective sensation of the self, where I’m necessarily limited to “my” subjective experience. You can study the “substrate” or “correlate” of the brain all you like, but that gives no insight into the subjective experience of the subject. Even if you completely mapped and replicated the neural activity of a person, all you could do is use gross induction to guess at his interior state, which would be the world’s most obvious case of begging the question of whether you could compare intersubjective utility or happiness.

      It’s among the most obvious points that a philosopher could make and thus one of the most heatedly debated.

  3. Harold says:

    An aside on heat and temperature. “heat really is a cardinal, measurable quantity.” Heat certainly is, but heat is not the same thing as temperature. Heat is energy, measured in joules or calories. Temperature is the thing measured with the thermometer. The same amount of heat will cause a different temperature rise in different objects. Add a cardinal amount of energy (say 1 calorie) to 1g water and you get a 1°C temp rise. Add the same cardinal amount of heat to alcohol and you get about a 2°C rise. Put the two adjacent to each-other and they will equilibrate at the same temperature, but each will contain a different amount of heat energy.

    Since the Kelvin scale starts at absolute zero, it does make some sense to say 100K is twice 50K, which does not work with C or F.

    • Bob Murphy says:

      Harold, I had in mind your point about Kelvin and doubling, and so deliberately used F for that argument. And now that you mention it, I see what you mean about temperature vs. heat, but I’m not sure how I could have phrased things more accurately while retaining the basic point. I.e. I’m sure you agree that the reason we can only do positive affine transformations on our temperature scales is that heat is a cardinal, measurable thing, but then what? What would the next sentence be, to explain why even a “Kelvin Squared” scale wouldn’t work?

      • Harold says:

        Yes, I don’t think there was anything wrong with your basic point, just that heat vs temperature distinction is worth making.

    • Tel says:

      So did we just prove that people have cardinal utility, after all? The analogy proponents often invoke is temperature. It’s true, they will admit, any temperature scale is arbitrary. Three popular ones are Fahrenheit, Celsius, and Kelvin. It would be silly for someone to say that an object that’s 100 degrees F is twice as hot as an object that is only 50 degrees F, because the numbers wouldn’t work out like that in Celsius or Kelvin.

      I’d argue that C and F are not real temperature scales, the zero of Kelvin does mean zero, while the shifted scales C and F are just the result of people who knew they were measuring something, but didn’t fully understand what it was (i.e. they are first attempts to make a temperature scale, and wrong, but still used for historical reasons).

      https://en.wikipedia.org/wiki/Ideal_gas_law

      P V = n R T

      For a fixed volume and fixed gas, if you double the temperature you also double the pressure (which could for example be directly measured in a piston as doubling the force). If you halve the temperature you halve the pressure.

      So yeah, real temperature (Kelvin) is certainly a cardinal scale, with a real unit. It fully supports the multiplication operator.

    • Tel says:

      I’m not sure that this popular move really works. For one thing, I think the reason we reject a “Fahrenheit Squared” temperature scale is that we just know that heat really is a cardinal, measurable quantity. If all we wanted to do was rank objects in terms of their heat from hottest to coldest, then a “Fahrenheit Squared” scale would work.

      Be a little bit careful about that one too. Physical units are based on the multiplication operator, so the concept of squaring the unit must be possible… however it has a special meaning when you do that.

      Consider feet as a measure of length, if you square it, you end up measuring area, which is not the same thing as length (but area is a real thing you might want to measure). What happens when you square it once more? It’s weird, you end up with four dimensional hypervolume… no one uses that. The hypervolume is well defined as a concept, it has a physical meaning, but a completely different meaning to what you started out with which was length.

      So if you take a real temperature scale (Kelvin) and square it, you do get a well defined physical unit (the square Kelvin) but the meaning of this unit is weird. Temperature is based on kinetic energy divided by degrees of freedom put together with two background presumptions: first that Boltzmann statistics are in operation (e.g. bosons cannot have a genuine temperature, they don’t follow Boltzmann statistics, photons don’t bounce off one another, they pass through each other) and second that some sort of equilibrium has been reached where the same Boltzmann statistics describe the whole patch where you want to measure temperature (i.e. there is no hot spot off in one corner).

      So with a square Kelvin you have energy squared divided by degrees of freedom squared… both of those are a bit questionable as to how you would make a physical interpretation.

      https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law

      Just to prove that economics isn’t the only place where weird things happen, if you take the Kelvin unit and square it then square it again (i.e. fourth power) you now have something that is proportional to power emitted via radiation. Mind you there’s a magic constant in there, and if you look at the units of the Stefan–Boltzmann constant, you see Kelvin to the negative fourth power which just operates to cancel out your fourth power Kelvins and convert the units to something related to radiation. The wiki page shows the formula to create the Stefan–Boltzmann constant out of other more fundamental constants, I admit I haven’t studied up on how to demonstrate the derivation of that.

      To conclude, I accept your ordinal invariance if the unit is squared, but what I’m pointing out is that physical (cardinal) units have specially defined properties when multiplied by themselves or by each other. That goes to the heart of how physics works.

  4. Daniel Kuehn says:

    re: “The way economists use the term, “utility” is an ordinal concept, which expresses a subjective ranking, not an objective measurement. Therefore it makes no sense to say Jim gets more or fewer utils than Sally. Furthermore, the work of von Neumann and Morgenstern does not alter this basic fact: Whether we “believe in” cardinal utility has nothing to do with their demonstrations.”

    I think there are a couple issues to separate here. First, cardinal v. ordinal is a different question from IUC. Cardinality still does not necessarily get you to IUC.

    Second, utility and preferences. I don’t know how you’re using “expresses” here, but utility functions are not subjective rankings. Preferences are rankings, utility is mapped onto preferences. I’d have to think about what that implies for cardinality, but the whole point of distinguishing between preferences and utility is that different utility functions can map onto the same preference ordering (which is a big part of the intuition on why utility can’t be compared across individuals.

    I do want to echo LK’s clarification that IUCs can’t be made *yet*. It all boils down to brain chemistry, after all. We might be able to do IUCs one day. Who knows. Now as an ethical/welfare matter of course we may not care.

  5. Transformer says:

    ” The way economists use the term, “utility” is an ordinal concept, which expresses a subjective ranking, not an objective measurement. Therefore it makes no sense to say Jim gets more or fewer utils than Sally.

    Yes, if you define “utility” in such a way that it depends upon the cardinal ranking of people’s preferences then sure , it makes no sense to say Jim gets more or fewer utils than Sally. But that’s purely an artefact of your definition.

    Many non-economists would find it intuitively very likely that a starving person near a store would value a dollar more than a billionaire and would be a bit shocked to find that economics has nothing to say on the matter.

    If the only reason that economists cling to cardinality as the indicator of utility is that some of their theories depend upon it, then so much the worse for their theories!

    • Transformer says:

      I think I meant “ordinal” not “cardinal” (language is hard)

  6. Yancey Ward says:

    As usual, people confuse “should” with “is”.

  7. Harold says:

    “I think many people have no problem with their choices in the famous Allais paradox. It’s only a “paradox” if we assume that people ought to be maximizing expected utility.” The Alais paradox is to demonstrate the independence axiom may not be valid. If people are not maximising expected utility, what are they maximising?

    Trying to understand what utility actually is in the context of utility equations, I was once told that it is “that which is maximised” – which seems to mean that not being a utility maximiser is impossible.

    This definition means utility cannot be defined in terms of happiness, satisfaction and pleasure. We tend to think in these terms, and LK asks what utility is if not some combination of these emotions. It certainly is difficult to think what it is of not some combination of emotions. So is it “that combination of emotions that is maximised” by any particular choice? However, since we make mistakes, it would have to be “that combination of emotions that is expected to be maximised” by a choice.
    Sorry if this seems confused, but I do find it confusing.

    • Bob Murphy says:

      Harold wrote:

      Trying to understand what utility actually is in the context of utility equations, I was once told that it is “that which is maximised” – which seems to mean that not being a utility maximiser is impossible.

      Just be careful here: You can be a maximizer of utility (trivially) if we make sure we give a higher number of utils to a Lottery that is preferred to another Lottery. No problem.

      But there’s no way to explain people’s choices in the Allais paradox if we assume they get a certain number of utlls from the various monetary outcomes, and then assume the person is maximizing the mathematical expectation of the utils with the probability distribution of the lottery.

      In other words Harold, go back and look at my quote again. I didn’t say the person isn’t a utility maximizer, I said the person isn’t an *expected* utility maximizer. Those are different things.

      • Harold says:

        ” I didn’t say the person isn’t a utility maximizer, I said the person isn’t an *expected* utility maximizer. Those are different things.”

        As I see it, one explanation of the Allais paradox is that people are maximising their expected utility, but their utility is not only monetary outcome; it includes non-monetary aspects such as regret. If they get nothing and turned down a certain $1million, a person will feel really bad. If they get nothing, but changed the odds of $1million for 89% to 90%, then they won’t feel really bad. They understand this, and include the potential bad feeling in their utility calculation and opt for the safe option in one case and the risky option in the other.

        Now if we examine it closely, there is a very strong argument that going from 89% to90% really should give us the very same bad feelings as before, since after we have removed the bits of the choice that are the same in both cases, we are left with exactly the same choice. However, even after explanation, it does not feel that way. So we are correct to include the extra regret, even if that extra regret is actually based on an error – we know we will feel it anyway.

        The paradox does leave us with some difficult questions, but I don’t see that one of them needs to be whether we are expected utility maximisers.

        • Bob Murphy says:

          OK Harold sorry for treating you like you were in 3rd grade, I see that you have a sophisticated understanding of the Allais paradox. I get your point now.

  8. Bob Murphy says:

    Make sure you guys read the follow-up post if you’re confused about emotions vs. utility.

    • Bala says:

      And please make sure you read this as well. I thought that would be a better place to respond to LK and not throw Bob under the bus.

  9. Andrew_M_Garland says:

    Summary: “(Tyler Cowen assumed that a rich man gets less utility from an extra dollar than a poor man does, but granted that this alone does not justify government redistribution.)”

    Thank Gaia for small favors. Maybe a few more reasons can be found to justify redistribution. The government then can merrily redistribute, knowing that it is increasing happiness, making everyone better off on average, in a way which only force from mobs and guns can accomplish.

    Progressives dream of a world where they run everything and the populace hands to them all “excess” production with only a whimper. Their justification for this and for any regrettable violence along that path starts with dreams of global utility functions and average happiness.

  10. T. Dill says:

    Hi Bob,

    I explain on the most recent post on my blog (click my name) that IUCs aren’t impossible *because* utility is ordinal. Rather, IUCs are impossible for the same reason utility is ordinal.

  11. Grane Peer says:

    Wait, they both have extra dollars?

  12. Josiah says:

    Bob says:

    “Even if we retreat to the everyday usage of terms, it still doesn’t follow as a general rule that rich people get less happiness from a marginal dollar than a poor person. There are many people, especially in the financial sector, whose self-esteem is directly tied to their earnings.”

    I respond:

    Yes, but the question is not about how much a person cares about money in the abstract but how much utility they get from their marginal dollar. I submit that if you secretly took $100 out of a rich finance guy’s bank account he wouldn’t even notice, whereas if you did that for a poor person he’d not only notice but it could cause him severe difficulties.

    • Yancey Ward says:

      But that is exactly the heart of the matter, isn’t it- you can’t know that. You think it should be true, but you will never be able to prove it or quantify it- you simply are left with assuming it.

      • guest says:

        Right. It depends on what that marginal dollar means to each individual.

        The rich person may want to invest so heavily in the future that he is willing to drop a ton of cash on it, and will need to budget his food more carefully while his investment is maturing.

        The poor person may have found a warm place to stay next to a river that the government hasn’t yet tried to claim, and he’s good at hunting, so he doesn’t need to spend his marginal dollar right away.

    • Ben B says:

      The rich guy would definitely know that $100 was missing. He didn’t become rich by being careless with his marginal dollar. No, rich finance guys “care” a great deal about their marginal dollars; that’s why they’re rich finance guys.

  13. Josiah says:

    But that is exactly the heart of the matter, isn’t it- you can’t know that.

    Why can’t I know this?

    Can you estimate your net worth (or even your bank account balance) down to the last dollar? Within ten dollars?

    • guest says:

      No, because the real economy is measured in stuff, not dollars.

      Money is a means to an end, but money may not always buy the end.

      Your bank account does not contain your total worth, so knowing that number can only get you so far.

  14. Josiah says:

    No, because the real economy is measured in stuff, not dollars

    I think you may need to reread Bob’s article a little more closely.

  15. Keshav Srinivasan says:

    Bob, somewhat off-topic, but if people did obey the von Neumann-Morgenstern axioms, could that be used to solve the Sraffa multiple natural rates problem your PhD thesis was about? That is to say, if u is the vNM utility function, can’t the central bank set the interest rate equal to (u(x + 1 future dollars) – u(x future dollars)) / (u(x+1 present dollars) – u(x present dollars)), i.e. the exchange rate between present utils and future utils? That quantity is invariant under positive affine transformations.

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