David R. Henderson  

Murphy on Interpersonal Utility Comparisons

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Bob Murphy has done a huge service by laying out clearly the economic reasoning behind my conclusions (here and here) that: (1) you can't make interpersonal utility comparisons and (2) utility is ordinal, not cardinal.

His piece is not long and I recommend it to anyone who is interested in the discussion.

This paragraph sums up his point:

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.

I will add one thing. In the various comments that people on my posts wrote, I got the impression that they think I'm making the point about interpersonal utility comparisons because I oppose government taking wealth from wealthy people and giving it to poor people. But that's not why. I see this as a purely technical issue in economics, an issue that one should get right regardless of one's views on forced distribution.

A little history here about the evolution of my views. I oppose government taking from the wealthy and giving to the poor unless the wealthy people first took it from the poor people. It's also true that when, as an undergrad, I first read the point about not being able to make interpersonal utility comparisons, I was excited because it helped me make the case against forced distribution. (I don't call it "redistribution" because to do so is to assume that someone "distributed" wealth in the first place. When Dwight Lee wrote the article "Redistribution" for my Encyclopedia, I persuaded him that therefore we needed to put the the word in quotes. You can see that in the print version but not in the on-line version.) But one of the goals of my professor Ben Klein at UCLA was to "knock my libertarianism out of me." (Those are my words, not his, but I think he would agree that that is what he was doing.) It wasn't that he necessarily wanted me to be less libertarian. It's that he wanted me always to distinguish between my libertarian views and technical economics and he wanted me, further, to enjoy technical economics separately from whatever conclusions it led to. He succeeded. That's why I hate bad arguments for my libertarian policy conclusions as much as I hate bad arguments for statist policy conclusions.

Bob Murphy also made that point well with this comment on my first post about Tyler Cowen's post:

If a certain physicist thought Einstein was wrong and that you could easily go faster than light, it would be weird if he said on his pop blog, "You can't sustain a solar system polity with rockets going five times faster than light, because eventually the fuel costs will be astronomical," without at least giving a nod to the fact that many of his colleagues would think he is speaking nonsense.

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CATEGORIES: Microeconomics

COMMENTS (24 to date)
Ben writes:

"Therefore it makes no sense to say Jim gets more or fewer utils than Sally". I have long been puzzled by this claim. It makes sense when you're comparing people who are on roughly an equal footing; I can see that the value I would get from a million dollars versus the value my neighbor would get from a million dollars are essentially not comparable. But when you're comparing people who are not on a roughly equal footing, it no longer makes any sense. It seems completely obvious to me that the value I would get from a million dollars is clearly greater than the value that Bill Gates would get from an additional million dollars. For him, it would be a rounding error; if you didn't tell him that you had put an extra million in his bank account, he probably wouldn't even notice. For me, on the other hand, it would be profoundly life-altering. To deny that a comparison can be made, and that my utils for that million are greater than Gates's utils for the million, seems like just burying one's head in the sand – missing the forest for the trees. I read Murphy's essay, and it didn't really answer this problem. I guess I would say: I can certainly accept that "utility" as economists use the term is ordinal and cannot be compared between individuals. If that is the case, however, then it seems to be a clearly flawed and severely limited concept that ought to be replaced by some new concept that *can* capture the fact (and I do claim it is a fact) that the marginal value of money for Bill Gates and for me *can* be compared, and is greater for me than it is for Gates. If "utility" is not up to the task, then economists ought to come up with something that is. Until then, they are missing a central part of economic reality.

Nick writes:

I understand the argument, but in your original post you said (emphases added)

One of the things we are most sure of in economics is that you can't compare utility, marginal or otherwise, across individuals. Utility is ordinal, not cardinal.
That gives the impression that this is a settled topic on which economists agree, which is not true. Your original post falsely implies that you are expressing a view on which economics as a profession has reached a consensus.

David R. Henderson writes:

Yup. I probably shouldn’t have said that. I probably should have said something like, “One of the things that many of us economists are most sure of."

D. F. Linton writes:

I doubt it would take a very large sheet of paper to list the full set of truly settled topics in economics

Pierre Lemieux writes:

Ben: What if I ask your neighbor and he does not agree? Or what if a third party believes that your neighbor would get more utility from refined pleasures (say, he would buy 40 cases of Krug Clos D'ambonnay champagne) than you from the vulgar pleasure of buying a top of the line Mercedez-Benz customized motorhome? (The "you" is purely rhetorical here, and follows from your own example.) One may agree with your neighbor or the third party above, or with you, but this is a normative (moral) value judgement (about distribution) that is outside economics.

david writes:
So did we just prove that people have cardinal utility, after all? ...

makes a curious philosophical claim; the idea behind neoclassical revealed preference is that one infers the underlying class of utility functions that is implied by those revealed choices. So people do not "have" cardinal or ordinal utils, either way, except inasmuch as their revealed choices can limit the uniqueness of a utility function.

they're getting a utility function pinned to them regardless, so to speak. The question is: what revealed choices does the agent have to make to contradict an existing utility function inferred from past choices? With cardinal utility, this is very easy (violate in the area that should be preserved up to an affine transform, i.e., Allais). With ordinal utility, this is much harder (prefer X ≿ Y ≿ Z, where X > Z > Y or somesuch).

vNM makes stronger claims on how choices should behave under risk, and "cardinalizes" in this respect, so to speak. Does this allow you to calculate a u(x) upon which to base non-subjective utils? No. Does this admit the methodological problems of cardinal utility, e.g., predictions of consistency in mental computation implied by affine transforms that then fail under experiment? Yes.

Does Murphy acknowledge as much? Yes.

Now - are there IUC implications? That's a trickier topic. If you took it as given that 'old' cardinality (i.e. prior to the emphasis on transforms) implied IUC, then you're doomed, because modern portfolio theory, composite good assumptions, etc are standard fare and these all provide units of wealth as a numeraire that is set against observable risk. That is, those ancillary modern theories all limit u(x). You might not be able to calculate how many apples to redistribute on a utilitarian basis, but that turns out not to matter, because apples are efficiently priced and you can just use their dollar value, and you have u(w) where w is a composite unit of Wealth set equal to $1 of portfolio returns, etc. Thence IUC.

Hazel Meade writes:

Sorry, but it does seem to me that both you and Bob Murphy are actually claiming that the supposed impossibility of making interpersonal utility meansurements is an argument against redistribution.

Bob Murphy:
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. And as the photo indicates, Scrooge McDuck really seems to enjoy money. Taking gold coins from Scrooge and giving them to a poor monk would not necessarily increase happiness, even in the everyday psychological sense.

David Henderson:

Tyler Cowen: 1. You cannot build and sustain a polity on the idea of redistributing wealth to take advantage of differences in the marginal utility of money across varying wealth classes.

He's right. You can't. But the reason you can't is that you can't measure differences in the marginal utility of money across people.

Yes, it does look like you're both arguing that it's impossible to tell if an extra dollar matters more to a millionaire than a poverty line worker, so therefore redistribution won't inprove utility.

Incidentally, try to stop and think for a few seconds about what Bob Murphey's argument above sounds like to someone in the lower three quintiles.

David R. Henderson writes:

Yes, it does look like you're both arguing that it's impossible to tell if an extra dollar matters more to a millionaire than a poverty line worker, so therefore redistribution won't inprove [sic] utility.
No. Because you can’t make interpersonal comparisons, you can't say that redistribution won’t improve utility. You can’t say it won’t hurt utility. You can’t say that it will leave utility unchanged. You can’t say anything about utility except that one person’s utility will be higher and one person’s utility will be lower.
You’re just adding more evidence, with your statement above, to my claim that people, in this case you, are drawing the wrong conclusion. This is purely a technical issue in economics.
Incidentally, try to stop and think for a few seconds about what Bob Murphey’s [sic] argument above sounds like to someone in the lower three quintiles.
I wouldn’t necessarily put it Bob’s way, because my claim is about utility, not happiness. The latter, though, is hard to measure also.
But, Hazel, do you really think I haven’t thought about this for more than a few seconds, given that I spent approximately 25 * 365 * 24 * 60 seconds (about 13 million seconds) in the bottom three quintiles? Moreover, it was during those years in the bottom three quintiles that I came to understand that you can’t make interpersonal utility comparisons.

Tiago writes:

Very well put.

And this brings about the problem of equivocation. "you can't distribute from the wealthy to the poor" - (first meaning) "because you can't make interpersonal utility comparisons" - (second meaning).

In the end, the utility that you and I are talking about, the one that matters - something like the amount of good a person can experience - can indeed be compared. In fact, making comparisons is unavoidable.

Jim Dow writes:

I really don’t think it’s a technical issue and this goes back to the comments made by David Friedman and others in the very first post. When I decide to give up my bus seat to an elderly person but not a young, fit person I’m making an interpersonal comparison of who among the three of us will value the seat more. So you cannot say that I cannot make interpersonal comparisons because I just did!

How do we do it? Mostly through introspection and empathy as Ben describes. And as Pierre points out, this is subjective. I apply my own estimates of the various utilities and you apply yours. If we all ended up with vastly different estimates it wouldn’t be very useful, but I would argue that in a number of important situations we probably end up in similar places.

Pierre writes “One may agree with your neighbor or the third party above, or with you, but this is a normative (moral) value judgement (about distribution) that is outside economics”

There’s certainly value judgments in there, which may be outside specific questions of economics but presumably not questions of policy.

Pajser writes:
"The way economists use the term, "utility" is an ordinal concept, which expresses a subjective ranking, not an objective measurement. "
If that is the case, this is just semantic issue. We need two terms - "utility in X's meaning", where X is the person who defined utility as "ordinal value"; and "utility in common sense meaning." In "common sense meaning", utility of one dose of penicillin for sick child is larger than utility of 1 inch of longer yacht for billionaire.
"I oppose government taking from the wealthy and giving to the poor unless the wealthy people first took it from the poor people. "
I see it this way: every material property is, by definition, restriction of the freedom of non-owners. As it is restriction of freedom, it requires justification. Only after we hear the justification, we can know whether redistribution is justified as well. If there is no justification of property at all, then redistribution from utilitarian reasons is justified.
Bob Murphy writes:

Last thing and I need to move on with my life: If someone wants to say, "It was good for me to give up my seat to the old lady on the best because she valued it more than I did," I don't have a problem with that, and I applaud the decision. I would do the same thing.

But that's not what Tyler said in his post. He talked about "differences in the marginal utility of money across varying wealth classes." That's not how a normal human being talks about old ladies valuing bus seats more than healthy young men, that's an economist using economist jargon and relying on a concept from utility theory.

Furthermore, I have seen people explicitly take the concept of diminishing marginal utility from consumer theory, and use it to justify redistributive tax policy. No, that doesn't make sense, for the reasons David and I have said.

To borrow an analogy from Walter Block: To physicists, holding a 20-pound weight at chest level for 10 minutes doesn't involve any "work." But to a layperson in everyday language, of course it involves a lot of work. It's fine to use the word in either sense, but it's not OK to smuggle in results from Newtonian mechanics into a discussion where everybody else is using the term in the everyday sense.

Hazel Meade writes:


Let me put it this way: Do you really think the ego boost that a Wall Street financier gets from earning an extra million dollars is not comparable to the benefits of giving a prosthetic limb to a disabled child? There is absolutely no measure we can use to tell which one of these things is "better"?

Charlie writes:

"Furthermore, I have seen people explicitly take the concept of diminishing marginal utility from consumer theory, and use it to justify redistributive tax policy."

If utility functions are purely ordinal, diminishing marginal utility as a concept doesn't exist, agreed?

The sign of the second derivative is not preserved under monotonic transforms.

So you think Tyler is wrong to even use the concept of diminishing marginal utility, which presupposes some cardinality in utility on his own economics blog?

Do you think it's wrong for any economist to invoke diminishing marginal utility ever?

Stephen Hicks writes:

Susan has 10 blankets in a closet in her heated home.

Jason is a homeless man on a cold night without a blanket.

Can we compare the utilities of one blanket in Susan’s possession and that same blanket in Jason’s possession?

Suppose we asked Susan, "Would you miss one blanket?" and she said, "Not really." And we asked Jason, "Would you like a blanket?" And he said, "Yes, indeed."

Can we then compare the utilities?

Jim Dow writes:

For anyone who’s left around, here’s why I disagree with what Bob Murphy said.

A number of commenters have expressed the commonsense notion that very rich people value additional money less than very poor people. Tyler’s “differences in the marginal utility of money across varying wealth classes” is just a fancypants way of saying the same thing. If we believe in one, we should believe in the other.

My point of view is that utility (and the maximization of it) is just a convenient simplification, it’s a useful construct, but it isn’t the truth. Writing down cardinal utility is especially convenient for capturing how income and substitution effects work which is why it is so commonly used, not because economists actually believe it. The shift to ordinal utility from cardinal utility reflected the belief that we could get the same outcomes but with less restrictive assumptions, which from a mathematical point of view is desirable. In the same way, the literature on aggregating utility functions is explicitly a literature on the mathematical properties of social welfare functions.

If we’re talking about actual policy then I’m not too worried about the mathematical properties of specific utility functions or social welfare functions since these are devices of convenience used to illustrate ideas, not real empirical things. Maybe particular assumptions about utility functions or social welfare functions could lead to paradoxes or undesirable characteristics, but from a policy perspective, what matters is how important those things are in practice. The math doesn’t give us any guidance there.

T. Dill writes:


I explain why your example of choosing who to give up a bus seat to is not an interpersonal utility comparison here. Basically, you are not comparing their utilities to each other. When you make a choice among alternatives, even a choice for the benefit of other people, you compare your utility among the alternatives and select the highest one. So it is a personal utility comparison, not an interpersonal utility comparison.

David Friedman writes:

I just want to point out, as I did once in my initial comment, that the issue of whether utility is ordinal or cardinal is not the same as whether you can make interpersonal comparisons. Von Neumann showed that you could treat utility as cardinal. Doing so makes the concept of utility a more useful tool, since it puts content into the idea of diminishing marginal utility and hence of risk aversion, providing a plausible and intuitive explanation of much observed behavior.

Utility could be cardinal but not comparable between people. My reason for not thinking that is the case was summarized by Jim Dow. We routinely take actions which imply that we have opinions about how much utility different people get from different things. Economists are supposed to believe in revealed preference, hence that those actions show that we, including economists, even, I suspect, including David H and Robert M., believe that we can compare the utility different people get from different things.

I am still waiting for either of them to claim to be completely agnostic as to whether the disutility I get from the prick of a pin is or is not greater than the disutility he would get from being tortured to death. If he is not willing to make that claim then he does not believe that utility is entirely incomparable between people.

T. Dill writes:


The pain, displeasure, and unhappiness you would experience from the prick of a pain are clearly less than what a normal person would get from being tortured to death. What does that have to do with interpersonal utility comparisons? If by utility we meant happiness, we would say people are happiness maximizers (observably false). If we thought disutility meant pain, we would say people are pain minimizers (again observably false). It is very hard to say what value people are maximizing (or values they are optimizing), and they are maximizing different things, so we use "utility" as a placeholder for our ignorance. Utility cannot be compared among people because nothing is being compared yet. You are not even at the starting line yet.

And insofar as utility is a placeholder for different things for different people, such comparisons are impossible anyway. How can one person have more happiness than another has satisfaction?

I go into some detail here.

D. F. Linton writes:

The fact that we can feel sufficient empathy to endure a pinprick to spare someone else death by torture, seem a complete red herring in a discussion of interpersonal utility comparisons in economics. First it is not universal, I for one would not have endured a pinprick to save Stalin from such a death. Second it is a simple rank ordering of preferences, not a ratio of cardinals.

Saying that it is better that M individuals suffer a single pinprick than that N individual suffer two pinpricks would be more relevant, but only if you could also say M+1 single pinpricks is not preferable to N double pinpricks. Even this kind of example ignores the fact that it matters who particularly is in which group.

Jim Dow writes:


I actually agree with you but just draw a different conclusion. On your blog, you write:

"First of all, on most comparisons of any particular value, the poor person gains more than the rich person from the marginal dollar. The poor person gains more happiness, pleasure, relief, etc., than the rich person in most circumstances. If you replace the placeholder of utility with something concrete, comparisons are valid. This isn't an interpersonal utility comparison, it is an interpersonal happiness//pleasure/relief comparison."

For me, utility is just a shorthand for happiness, avoidance of pain, etc. For some purposes the mathematical representation of utility does an OK job of capturing what we care about and for other purposes it doesn't. Use it when it's useful, but what we actually care about is the underlying person.

If we agree that the poor person gets more happiness/pleasure/relief from the marginal dollar then why do we care about the mathematical properties of some function that, at best, approximates this?

I also agree with you that there are different aspects to individual welfare. When I'm on the bus I expect that the elderly passenger will suffer more discomfort than the young passenger by standing. It's not just that I feel good doing a good dead, I'm making a comparison between the two individuals when I make my decision. This is probably easiest to do with physical aspects of utility. I wouldn't necessarily give up my ticket to either of them since it is hard for me to judge who values the trip more.

There are all sorts of practical issues in implementing policies from this, and talking about it in terms of happiness and pain is a good framework for it, which is why the discussion of cardinality was puzzling to me from the start.

Bob Murphy writes:

Let me just say one last thing because I realize it looks like I was ignoring David Friedman's question before. Here's David:

"I am still waiting for either of them to claim to be completely agnostic as to whether the disutility I get from the prick of a pin is or is not greater than the disutility he would get from being tortured to death."

No, I'm not agnostic about it David. I am quite certain that in the way I use the term disutility--and the way we use it in standard consumer theory--it is *nonsense* to say that. It's like asking me, "Is the color I get from running saltier than Jim's height?"

David, I agree that the *pain* you get from a pinprick is less than the pain I get from being tortured to death. Because pain is referring to a physical sensation that is clearly biological, and I assume that there is a vague way we could compare those things across people, which could be refined over time with advances in neuroscience etc.

But pain (or physical pleasure, etc.) is not the same thing as disutility / utility. A guy might choose to go to the gym and lift a lot of weights, over the alternative of sitting on the couch eating pizza. He gets more utility from the former, even though it gives him more pain / less pleasure.

If you want to say, "I am throwing out the way consumer theory has been done since at least Hicks up till Varian's current textbooks," at least notify your readers that you are doing that. You're acting like David (Henderson) and I are being obtuse. If you just said, "Yeah sure, the standard textbooks say that, but c'mon guys let's move beyond them..." then I wouldn't have even piped up. But I don't think that's the message the average reader is getting from our public disagreement on this issue.

Nathan W writes:

Just because we can't measure how many more utils are obtained from redistribution doesn't mean that it's not obvious that redistribution can increase the total number of utils enjoyed, and that solely from the principle of declining marginal utility. Sure there is a Scrooge and monk example, but the exceptions to declining marginal utility or the arbtrariness/impossibility of interpersonal utility comparisons would only be considered as an argument against redistribution is you were already predisposed to find argument against redistribution, imo.

Brian writes:

So just to be clear, folks like Bob Murphy and T. Dill say that utility cannot be compared among people because, well, utility doesn't refer to anything real. It's not pleasure or lack of pain or satisfaction, even though classical utilitarianism is based exactly on those things. To them it appears that utility is just a useful fiction, a conception that helps us think about people's preferences. In that light, whether utility is ordinal or cardinal is based solely on which approach is most useful in understanding the concept; utility is not actually either one, since it's not real.

I think that David Friedman, I, and others consider utility to be real. In making choices, people display well-defined preferences. Those revealed preferences are the result of physical and mental states in the human body, states that can be completely expressed in quantitative terms. Therefore, utility (1) is cardinal, and (2) can be compared between people because we all have a common biological basis for our decision making. Bob Murphy admits as much when he refers to a "vague way we could compare those things across people." I already referred to this "vague way" in my comment on David H's original post as utility plus uncertainty.

By the way, Bob Murphy is wrong about the weight lifter. A weight lifter may indeed be maximizing his pleasure by lifting weights. Sure, he endures short-term pain and soreness to build up his muscles, but presumably experiences much pleasure/satisfaction from the physical state he achieves. He is willing to suffer a short-term cost in exchange for a long-term gain certainly is no argument against equating utility with pleasure/satisfaction.

Finally, my question for David Henderson is this: do you also believe that utility is just a useful fiction? And if not, how do you continue to hold that utility cannot be cardinal when all physical states must be?

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