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There is no utility function

Alex Tabarrok and Matthew Yglesias comment on “the marginal utility of money income.” I’ll have to write something longer about this some day, but for now let me just reiterate my current understanding that there is no such thing as a utility function. Rather than people arguing over the shape of the utility function, I hope they can move forward to thinking more directly about what people will do with their money.

From my earlier blog entry:

Like Dave Krantz, I’m down on the decision-theoretic concept of “utility” because it doesn’t really exist.

The utility function doesn’t exist

You cannot, in general, measure utility directly, and attempts to derive it based on preferences (based on the Neumann-Morgenstern theory) won’t always work either because:

1. Actual preferences aren’t necessarily coherent, meaning that there is no utility function that can produce all these preferences.

2. Preferences themselves don’t in general exist until you ask people (or, to be even more rigorous, place them in a decision setting).

So, yeah, utility theory is cool, but I don’t see utility as something that’s Platonically “out there” in the sense that I can talk about Joe’s utility function for money, or whatever.

Call it value, not utility

The above is commonplace (although perhaps not as well known as it should be). But my point here is something different, a point about terminology. I would prefer to follow the lead of some decision analysis books and switch from talking about “utility” to talking about “value.” To the extent the utility function has any meaning, it’s about preferences, or how you value things. I don’t think it’s about utility, or how useful things are. (Yes, I understand the idea of utility in social choice theory, where you’re talking about what’s useful to society in general, but even there I’d say you’re really talking about what society values, or what you value for society.)

Just play around with the words for a minute. Instead of “my utility function for money” or “my utility for a washer and a dryer, compared to my utility for two washers or two dryers” (to take a standard example of a nonadditive utility function) or “my utility for a Picasso or for an SUV,” try out “my value function for money” or “the value I assign to a washer and a dryer, compared to the value I assign to two washers or two dryers” or “the value I assign to a Picasso or to an SUV.” This terminology sounds much better to me.

P.S. See Dave’s comments here.

Also see the comment thread.

11 Comments

  1. Manoel Galdino says:

    I saw no references to the historical use of the word utility. In the past utility was used for, well, people who developed "utilitarism" (Bentham and so on).

    Than, the concept was transformed, primary into a function and represented only ordinal valuations (not cardinal ones). The reveled preference theory is, well, a theory, ie., it is falseable! So, it can be wrong! Why this would bother anyone?

    When there is uncertainty, then we have to work with cardinal utilities, and here we have a problem to interpret what utility is. Economist seem to not realize that we are using utility in a quite diferent meaning.

    So, I think we should keep the word utility for ordinal utility, and maybe use another word for cardinal utility.

  2. Daniel says:

    So… it's basically just a word choice issue?

    Surely people shouldn't make any Platonic overstatements about utility functions, but your dispute seems to be less with substance and more with style here.

  3. John says:

    I haven't seen anything cause more petty arguments than utility (value) functions. Anybody can find fault with any proposal and come up with their own (often minor) variation. I remember when I wanted to use the word "utility" in a paper and someone warned me not to, saying that doing so would be like throwing meat into a pack of dogs.

    This is a shame. When a group can't agree on a value function, they revert to a conventional approach based on mathematical convenience that may not reflect anyone's values.

  4. bill says:

    When I'm helping companies make decisions, we often split their preferences into a value function and a u-curve.

    The value function takes all their inputs and outputs a value, usually NPV. Using this deterministic function, we can assign distributions to the inputs and we get a distribution on NPV.

    In order to choose between two different distributions on NPV, we use the u-curve. The u-curve takes a distribution on NPV and outputs a certain equivalent, one number that represents the whole distribution. We then choose the alternative with the highest certain equivalent.

    For example, if the company is risk neutral over the range of NPVs represented in the decision, then the certain equivalent is simply the expected value of the distribution.

    The value function lets us calculate value when there is no uncertainty. The u-curve is only used when there is uncertainty.

    "Utility" tends to mash these two meanings together. I think the value function is useful in making decisions, and I think the u-curve is useful in making decisions.

  5. Dave says:

    There's also no way to conduct interpersonal comparisons of utility, even if we accept the present construct of utility functions.

  6. Daniel Lakeland says:

    I've got a couple of questions about this. First off, in a Decision Theory course that I'm in now, the term "Value Function" is used in the context of certainty, whereas "Utility Function" is used in the context of uncertain outcomes. I suspect this distinction is at the heart of the separation of these two terms. However, it's not at all clear to me that there's a useful distinction here. Why should a utility function (incorporating risk aversion) reveal different preferences than the value function, since a "certain" gamble is just the limit of a gamble between two things as the probability of getting one of them goes to 1.

    Secondly, I imagine that deciders "inconsistency" is due to their own uncertainty about how they will feel about getting different options. in other words, if there is a utility function, it is only meaningful as an average concept, and the decider will reveal to you not a true utility based preference, but a utility preference plus some random "error".

    perhaps as a reasonable model, the "utility function" is really a sub/super/martingale from which at each decision the decider samples a single outcome and makes their decision on that basis.

    From the standpoint of decision analysis, where the decider is willing to spend time to discover their preferences, it seems reasonable to assume the existence of a utility, at least in this stochastic sense.

  7. Daniel Lakeland says:

    Also, your point about the jargon is that utility seems to imply some "platonic" usefulness. But people who actually use this concept already understand that the utility function is to model the "usefulness to a single person" or "desirability by person x" there is no "platonic" utility certainly! This seems to be a minor point of jargon.

    I'm much more interested in questioning whether utility functions as a model for choice making are actually useful than questioning the use of "utility" the word to describe the concept.

  8. Frank says:

    I'm all for nixing the casual use of "utility" seen in Yglesias' post. I don't really know what macroeconomists mean by it, but…
    1. I hope you're okay with "preferences." For example, the revealed preference relation is a very convenient tool/term.
    2. As an Anon pointed out on your last post, economists don't trust survey data, so "asking" for preferences is a turnoff.
    3. As you know, a social choice problem (as used in Arrow's impossibility theorem) has preferences over outcomes, not utility functions. And the objective is a method of making group decisions that is not only efficient (gets "what's useful to society in general"), but also sensible and non-dictatorial. Because no such method exists (Arrow's possibility theorem), we need to put some restrictions on preferences. Utility functions over outcomes X that are quasilinear in money transfers T
    U_i(x,t)=v(x)+t_i
    are a convenient way of getting around this problem.
    **The punchline**: There's no reason why v can't be called person i's value of outcome x, as you propose. Then the marginal value of money can be talked about in terms of what you get in x.

  9. We'll abandon the notion of utility when statisticians abandon the notion of probability. Both are formalisations of purely subjective notions. And pretty handy ones at that.

  10. michael roberts says:

    Yech. Those are fighting words to many economists.

    Oddly, and sadly I think, most economists probably have the same notion of value (utility?) that you do. Formal notions of utility and welfare trace to the concepts of "equivalent variation" and "compensating variation," which are really about what people do with their money. And this is the way economists think about it.

    It is true that economic terminology often makes for many meaningless food fights. Economists are largely to blame for that.

    But I think it's much more about awkward terminology than it is about substance.

  11. Robert Goldfarb says:

    I do not find Gelman's argument helpful, for the following reason. A utility function is a modeling device. Asking whether it is
    "real/true" is approximately like asking whether a "vacuum" is real/true, and then abandoning the concept because no one could ever
    observe/create one on Earth. One appropriate "test" of the "merit" of
    each concept is whether it is helpful in constructing theories/models
    that are themselves useful. A deeper test is whether substituting another modeling device produces richer/better/etc results. So in this
    view one modeling concept gets dumped (or, at least, should get dumped) when another comes along that is SHOWN to be more productive. Gelman might be able to argue in favor of
    "value" that way, but that is not how I read his argument.

    Note the previous comment (I'm paraphrasing)about "we'll abandon 'utility' when you statisticians abandon 'probability'" is consistent with my argument, though I doubt the commenter had exactly my argument in mind.