Talk:Marginal utility/Archive 3

Point 1: apparent confusion about the meaning of de Finetti’s conjecture

Back on Slam’s talk page where I appear to have been classified as an “orc” (mistakenly, I assure you), I said:

“...if ‘ordinal’ means a complete ordering, instead of a partial ordering, then indeed it would always imply a cardinal ordering too (just numbering by rank).”

and then I clarified

“When I talk about cardinal utility, I mean putting a number on each consumption basket, and ranking consumption baskets according to those numbers. As long as I put a number on every basket, the ordering becomes complete rather than partial... Thus, I think of cardinal utility as a special case of ordinal utility: it does the ranking by assigning numbers.”

Whether or not these are the definitions that are most correctly considered “neoclassical”, I hope I made clear what I meant. Therefore I think the following response from Slam does not actually address what I said:

“Your belief that any full ordering will correspond to a measure is false... I linked to a free, on-line copy of the article by Kraft et. al, which originally provided the mathematics that it is false...”

Notice that I never said that a full (complete) ordering was a measure; I said that any complete ordering could be used to define a numerical ranking. This is obviously true, as long as one bears in mind that in keeping with standard usage in contemporary microeconomics, I was talking about a ranking over ‘bundles’, not a ranking over individual goods.

Therefore, the truth or falsehood of the de Finetti conjecture is unrelated to my comment, because the dF conjecture did not address whether you can put a number on bundles; rather, it addressed whether a utility function over bundles is decomposable in a particular way. Let me explain what the conjecture said... I didn’t know what de Finetti’s conjecture was before I started reading this page, and I’m sure many of us didn’t, so just let me briefly explain what it says in nontechnical notation. You can check the details by reading the FIRST PAGE of Kraft, Pratt and Seidenberg, “Intuitive Probability on Finite Sets”, Ann. Math. Statist. Volume 30, Number 2 (1959), 408-419.) It’s really much clearer than you might expect!

De Finetti’s conjecture: Consider a utility function U over bundles of consumption goods. Suppose that utility function satisfies three properties called comparability, transitivity, additivity (explained in the paper). Then it is possible to decompose the utility function U by assigning a value u(x) to each individual consumption good x, and expressing the utility of a given basket as the sum over all the x in the basket of those values u(x). THIS VERSION IS SLIGHTLY INCORRECT, because the conjecture started from a ranking, not from a utility function. The correct version (though still stated in the most nontechnical way I can) is immediately below. --Rinconsoleao 07:16, 9 August 2007 (UTC)

De Finetti’s conjecture: Consider a ranking R over bundles of consumption goods. Suppose that ranking satisfies three properties called comparability, transitivity, additivity (explained in the paper). Then the ranking implies that we can assign a value u(x) to each individual consumption good x, such that the ranking of baskets by summing the u(x) over all the x in any given basket is equivalent to the original ranking R. --Rinconsoleao 07:16, 9 August 2007 (UTC)

The paper of Kraft et. al proves that de Finetti’s conjecture is FALSE. However, this observation is completely unrelated to standard treatments today of cardinal utility, and also ordinal utility. That’s because standard descriptions of preferences only rank (or give numbers to) bundles. They don’t attempt to do that for individual goods. See for example the discussion of ranking bundles on p.34-35 (Sections 3.1-3.2) of Varian (1987), ‘’Intermediate Microeconomics: A Modern Approach”, ISBN 0-393-95554-0. So the fact that you can’t break utilities of bundles into values of individual goods is in no way a criticism of today’s preference theories. And as I emphasized, it’s certainly not a disproof of my claim that an ordering of all bundles can be used to put numbers on those bundles (put 1 on the first bundle, 2 on the second, etc... or use reals if there is a continuum of bundles).

Any disagreements on these points? --Rinconsoleao 20:55, 8 August 2007 (UTC)

Absolutely. But I'll just note what is wrong with the central point of this section, rather than critiquing everything. Let “${\displaystyle B_{n}}$” be one of your bundles.

There is a very great difference between asserting that

${\displaystyle U\left(B_{1}\right)-U\left(B_{2}\right)}$

tells us nothing of use beyond its sign, and claiming that it is an impossible operation. Neoclassical economics doesn't deny the possibility of monotonic or of affine transformations of the utility functions; it declares the results meaningless. If, instead, it declared

${\displaystyle U\left(B_{1}\right)-U\left(B_{2}\right)}$

to be an impossible operations, then it would also be locking itself out of things such as

${\displaystyle {\frac {\partial U}{\partial x}}}$

and what goes with it. So let's imagine

${\displaystyle B_{1}=B_{2}\cup \left\{x_{n}\right\}}$

De Finetti's conjecture is, in effect, that there will always be a possible

${\displaystyle U\left(B_{2}\cup \left\{x_{n}\right\}\right)-U\left(B_{2}\right)}$.

—SlamDiego_←T 22:55, 8 August 2007 (UTC)

De Finetti's claim was much stronger than your last statement. It says that for any ranking over bundles (which satisfies three properties mentioned above), we can come up with a utility function U such that for all ${\displaystyle B_{2}}$,

${\displaystyle U\left(B_{2}\cup \left\{x_{n}\right\}\right)-U\left(B_{2}\right)=v(x_{n})}$.

In other words, we can uniquely define the value ${\displaystyle v(x_{n})}$ associated with a given good ${\displaystyle x_{n}}$, regardless of what other goods are in the bundle ${\displaystyle B_{2}}$. (Here ${\displaystyle v}$ is exactly the same notation used in paragraph 1 of the Kraft et al. paper.) Such a strong claim is never made in contemporary preference theory, which is why discussing de Finetti appears to me to be a red herring, just like discussing measure theory. Perhaps such claims were made early on in the theory of preferences; Slam obviously knows more history of utility theory than I do. If so, discussing de Finetti might be relevant at some point in the historical part of the article. --Rinconsoleao 08:00, 9 August 2007 (UTC)

I said “in effect” because the thing with which we should here concern ourselves is the marginal implications for the concept of marginal utility. You are still coonfusing allowing addivity is disallowing non-additivity.

I don't know where you think I equated allowing additivity with disallowing nonadditivity.

Explicitly in the next subsection, and implicitly here when claim “Such a strong claim is never made in contemporary preference theory, which is why discussing de Finetti appears to me to be a red herring”. Strength is about what is allowed or disallowed. In confusing the sort of strength to which we should object, you confuse the sort of disallowing to which we should object. —SlamDiego_←T 09:17, 9 August 2007 (UTC)

If DeFinetti's conjecture were correct, then neoclassical “ordinal” utility would allow additivity. (Exactly as I said: “in effect, that there will always be a possible ${\displaystyle U\left(B_{2}\cup \left\{x_{n}\right\}\right)-U\left(B_{2}\right)}$”.) Since he was wrong, neoclassical “ordinal” utility disallows additivity. That makes it less than fully general. —SlamDiego_←T 08:21, 9 August 2007 (UTC)

Some ordinal preferences (i.e. some orderings over bundles) allow additivity. The counterexample to the de Finetti conjecture simply showed that not all do. Thus ordinal preferences are sufficiently general to allow both cases. Whether some other conception of preferences is more general than this is another issue, about which I would be happy to be informed. --Rinconsoleao 08:56, 9 August 2007 (UTC)

The reason that I put “ordinal” in quotation marks when referring to the neoclassical conception so named is because the that conception always corresponds to a measure.

Thanks for making such a strong, specific clarification. I would appreciate it if you could clarify further. If U(x) is a neoclassical utility function (or a list of the numbers associated with a complete neoclassical ranking by order of preference), what is it a measure over? Are you asserting that it is a measure over the space of goods x? Or are you asserting that it is a measure over the space of 'utilities' u? Or are you asserting something else? --Rinconsoleao 09:54, 9 August 2007 (UTC)

Neoclassical economics defines a utility function as a mapping, with certain properties, over real numbers. It would be empty, in neoclassical economics, to call this a mapping over utilities, as utilities would be no more or less than that over which a utility function maps. —SlamDiego_←T 10:10, 9 August 2007 (UTC)

Fine. Now in what sense is that mapping a measure? --Rinconsoleao 10:42, 9 August 2007 (UTC)

Arithmetic performed upon it yields meaning full results. For example, as I stated elsewhere, the signs of differences are meaningful. That is why transformations must be at least monotonic (if not more rigidly restricted) to be accceptable, and why monotonic transformations are possible. I have already made these points. —SlamDiego_←T 10:47, 9 August 2007 (UTC)

OK, but if that's what you mean by measure, then it is not what is meant by measure in measure theory. Therefore, I would again suggest that we eliminate the link to measure theory, which will only confuse readers. A measure, in measure theory (as I assume you do know) is a function that maps sets to the real line, subject to certain regularity conditions. The main regularity condition is that the measure of the union of A and B is at least as large as the measure of A. One can perform meaningful arithmetic on functions which are not measures. --Rinconsoleao 11:07, 9 August 2007 (UTC)

For example, the utility function I constructed in example 3 below is not a measure, because the quantities of apples it evaluates can be regarded as sets of apples, but sometimes an expanded set leads to a smaller utility in this case (i.e. if we consider a set that contains two kilos of apples, and then a set of three kilos of apples that contains the previous set). Nonetheless, arithmetic performed on that utility function yields meaningful results. In particular, we can perform a monotonic transformation on that utility function (over its defined range [0,4]), by adding a constant to U, or by multiplying U by 100, or by taking the log of U, and in all cases, the ranking it implies over consumption bundles is preserved. --Rinconsoleao 11:12, 9 August 2007 (UTC)

No, and we already discussed this point at an earlier time. A utility function might also be a measure of apples, but that doesn't mean that a utility function of apples must be a measure of apples to correspond to a measure. A utility function, to be a utility function, must capture preferences. Thus, a utility function of apples can be a measure of the place that bundles occupy in a preference ranking. —SlamDiego_←T 11:43, 9 August 2007 (UTC)

OK, so we agree that a utility function is not a measure 'of' consumption goods, at least not in general. But now saying it is a 'measure of the place that bundles occupy in a preference ranking', is not a claim that it is a measure in the sense of measure theory. (If you disagree please explain why.) As you know, a measure in the sense of measure theory is a very specific type of function. If we are to have a link to measure theory, either you should explain why a utility function is a measure in the sense of measure theory, or you should explain whether there is some other link to measure theory that has not yet been clear from the discussion. --Rinconsoleao 12:26, 9 August 2007 (UTC)

Again, you're saying “we agree” when the truth is that you've wasted more of our time.

A quick-and-dirty answer is to note that probabilities can be assigned to bundles, which probabilities are measures, without necessarily increasing with the count of apples. I'm unsure what is wrong with your understanding of measure, but I'll leave you to sort it out. —SlamDiego_←T 12:40, 9 August 2007 (UTC)

Of course. Probability functions are measures (and any discussion of probability theory will tell you that). Utility functions are not measures (and if you can find a textbook which asserts that a utility function is a measure, I will be very interested. --Rinconsoleao 12:50, 9 August 2007 (UTC)

I spoke too quickly. Probabilities cannot be assigned to bundles, they are assigned to events. So your quick and dirty answer is based on a misconception. But probabilities are measures, because they are functions over events, which are treated as sets; and they obey the property that the union of A and B is at least as probable as A (i.e. the probability of 'A or B' is at least as great as the probability of A). Utility functions are not (in general) measures; they are defined over bundles (which can be treated as sets) but they do not (in general) have the property that the union of bundles A and B has at least as much utility as bundle A. Nor do they (in general) satisfy the other regularity conditions that define whether or not a function is a measure. --Rinconsoleao 13:12, 9 August 2007 (UTC)

No, now you've spoken too soon. Getting a bundle with ${\displaystyle n}$ apples is an event; getting one with ${\displaystyle n+1}$ is another. There is an isomorphism, in other words, between sets of bundles and sets of events. —SlamDiego_←T 13:21, 9 August 2007 (UTC)

No, this shows you don't know what an event means in probability theory. In a model where you receive bundles of apples randomly, 'receiving 6 apples' is not a subset of 'receiving 7 apples'. Instead, it is a subset of 'receiving 6 apples or receiving 7 apples'. Whereas when we treat groups of apples as consumption bundles, a particular set of 6 apples can indeed be a subset of a particular set of 7 apples. There is no isomorphism between these two uses. --Rinconsoleao 13:29, 9 August 2007 (UTC)

No, I didn't say that these events were subcases on of another. I said (correctly) that getting ${\displaystyle n}$ was an event (as illustrated in the remark below, about EU). And I said (correctly) that probabilities can max-up before the apples are exhausted, just as can utility. I was addressing your actual arguments, not whatever argument you may have intended to make. —SlamDiego_←T 13:37, 9 August 2007 (UTC)

Hence, for example, EU models the utility of a lottery as

${\displaystyle \sum _{i=1}^{n}\left[p\left(S_{i}\right)\cdot u\left(S_{i}\right)\right]}$

where ${\displaystyle S_{i}}$ is a bundle (state-of-the-world), ${\displaystyle p\left(S_{i}\right)}$ is the probability of that bundle, and ${\displaystyle u\left(S_{i}\right)}$ is its utility.

—SlamDiego_←T 13:37, 9 August 2007 (UTC)

Those equations in no way address whether utility is a measure, which is the question we are discussing. In that EU function, p(S) is indeed a measure, u(S) is not. --Rinconsoleao 13:43, 9 August 2007 (UTC)

Those equations weren't claimed to show that utility is a measure; they were offered to disprove your absurd claim that probabilities cannot be assigned to bundles. —SlamDiego_←T 13:45, 9 August 2007 (UTC)

Wow, now we are really talking in circles. Strictly speaking, a probability is assigned to the event of receiving a given bundle (or any and/or combination of events of this kind), rather than being assigned to the bundle itself. But we should stick to the issue at hand, which is: are utility functions measures? I have repeatedly asked for an argument to that effect. --Rinconsoleao 13:57, 9 August 2007 (UTC)

Since there is an isomorphism between the event of receiving a bundle and the bundle received, if there is a function that maps from the event to a probability, there is a function that maps from the bundle to a probability, so each bundle has a probability, strictly speaking. (And that is yet another point that I'd already made.)

What you've actually repeatedly done is offered arguments that don't quite work to the effect that utility is not a measure, and asked me to defend the claim that it is in the face of those arguments; so I've shown that they don't work. I'll think about a positive explanation (as opposed to negative arguments) when I get a chance.

—SlamDiego_←T 14:59, 9 August 2007 (UTC)

I will eagerly await any positive argument justifying the claim that utility is a measure. --Rinconsoleao 15:23, 9 August 2007 (UTC)

Many irons were dropped in my fire; they are mostly still in it, so I cannot respond as quickly as I would like.

From the perspective that I believe that you've been making your request, the quantification that neoclassical economics calls “marginal utility” is a measure of the set of uses associated with a bundle. Being careful here, one must either take the bundle to be the complete state-of-the-world, or take there to be a mapping from the bundle to that state, and then from that state to a set of uses, and then the measure to be on the set of uses. A bliss point exists when the inclusion of another unit of a good or services would cause a use of something to be lost. (The unit is then really a magrinal bad or disserivce.) That something could be easily seen, as when one is compelled to expend resources on disposal, but it could also take place at the neuronal level that you introduced in earlier discussion.

Again, responses may be delayed. —SlamDiego_←T 04:21, 19 August 2007 (UTC)

Hi again! (Eliminating indents.) No hurry, bc I am hoping at some point to read the McCulloch paper in more detail, which will take time. Just one comment on that comment: 'use' seems to be an inappropriate issue to bring into the discussion of the neoclassical theory. 'Use' is a fundamental element in the Austrian way of defining MU, but it simply never enters into the neoclassical definition (as far as I know). The neoclassical version just starts out by ranking bundles... don't you agree? --Rinconsoleao 10:17, 19 August 2007 (UTC)

While neoclassical theory at times may lose sight of that significance, it always resides implicitly. What, after all, are the real differences between saying that someone has a use for a good or service and that it has utility for her? (&c.)

The fact that a word doesn't appear doesn't mean that its concept is not present.

Indeed, the word “use” does not appear in most Austrian School definitions of “marginal utility”, even those written in English by Austrian School economists (including Mises). McCulloch uses words in a manner different both from the norm amongst neoclassical economists and from that amongst Austrian School economists. Even setting aside the problems of adopting peculiar nomenclature, I don't see his lexicon as ideal; I advocate that we pay attention to the conceptual content of his article.

—SlamDiego_←T 11:52, 19 August 2007 (UTC)

I was trying to be clear about the concepts, not the words. The only thing that explicitly appears in the neoclassical theory is a ranking. Doubtless there are many ways of interpreting that; some interpretations might involve 'use', but claiming that all do seems to be a very strong claim (so strong it sounds like original research). In contrast, Menger's theory, and McCulloch's development of it, are explicitly based on listing 'uses' (by whatever name) before going on to define marginal utility. That's a really interesting model which I had never encountered before I first read this page, and it's obviously worth pointing out. But I see no reason to claim that that model or theory or interpretation hovers at the background of every neoclassical economist's interpretation of the neoclassical theory. --Rinconsoleao 13:03, 19 August 2007 (UTC)

I understand that you were trying to be clear about the concepts. But explicitness is a matter of words. And the claim that use is in the notion implicitly is not particularly strong, nor “original research” as defined by Wikipedia policy.

I am not claiming that the Austrian School theory hovers in the background of the neoclassical theory; in fact, the neoclassical theory, descended from Benthamite origins, is very like the Austrian School theory turned on its head (which would then derive use from utility). The point, however, is that use is at least implicitly present in neoclassical theory (and explicitly so amongst the more utilitarian neoclassical economists), however ill-considered it may be fit in the model. —SlamDiego_←T 05:01, 20 August 2007 (UTC)

Let me explain explicitly, that the association of use with utility is a matetr of simple tautology (and hence not “original research” as defined internally by Wikipedia). Under the neoclassical conception or otherwise, we are talking about decisions as to which goods and services to use, and how to use them. Try to find a production of utility that cannot be said to result from some sort of use of a good or services. (Even gloating over possession qualifies as a use, and is certainly amongst the uses that the Austrian School theory must include.) —SlamDiego_←T 07:20, 24 August 2007 (UTC)

Backing up a bit, while I was keen to establish in discussion that the neoclassical conception is a measure, I am not insistent that the article link to that on measure theory if you can indeed propose an appropriately concise way within this article to clarify the notion of quantification. So long as people associate the word “quantification” with a rather strong conception, we need other words to make it plain that no quantification need be assumed. —SlamDiego_←T 11:52, 19 August 2007 (UTC)

Good, I agree. I think linking to measure theory is not the clearest way to define quantification. Of course, defining quantification does require explaining what it using other words! I'm still not ready to propose a rewording, but I will give it a try at some point. --Rinconsoleao 13:03, 19 August 2007 (UTC)

Well, you'd earlier declared such a term or term or terms to be available. :-/ In any case, right now the burden is carried by linkage, and I see no reason why in theory it could not be carried by a different linkage, so long as it was to an article that would clarify the general concept of quantification. —SlamDiego_←T 05:01, 20 August 2007 (UTC)

Another issue I was wondering about. You gave the Dewey decimal example as a ranking that is not 'quantified'. Seems like a great example, and maybe it would be helpful to have it in the article. My initial understanding of that example was that it showed that the difference between two ranked items is inherently meaningless, if the ranking is not 'quantified'. I still suspect that is what you mean, since many neoclassical treatments specifically point out that differences are meaningless in ordinal utility (I recently ran into a statement like that in the Kreps textbook). But then I realized that another interpretation of the example is that the Dewey decimal codes simply are not numbers (i.e. they have too many decimal points in them, and often include letters too). Which of these two points were you intending to make... or were you intending both? --Rinconsoleao 10:17, 19 August 2007 (UTC)

We can always map from even the most rococo expression of the Dewey Decimal system to one that contains only digits. (And a system that uses letters only to ordinate is plainly not in principle different from one that uses hexadecimal digits to ordinate.) The point is that ordinates can be numbers without being quantities.

I believe that I owe that example of the Dewey Decimal system to some paper by McCulloch. I suspect that it would be more appropriate to use that analogy in the article on utility than that on marginal utility. The article on marginal utility simply shouldn't bear responsibility for the full functionality of the article on utility.

Again, the neoclassical conception doesn't necessarily want to do much with the quantification, but it wants to do things such as play with signs (as in associating decreasing marginal utility with a negative second derivative). If no quantification can be fit to some rational orderings, then even this relatively modest use of quantification is unacceptable.

—SlamDiego_←T 11:52, 19 August 2007 (UTC)

And note that, in these cases, ${\displaystyle p\left(S_{i}\right)}$ is not ${\displaystyle n}$ or more apples, nor ${\displaystyle n}$ or fewer apples; is just exactly some ${\displaystyle n}$. —SlamDiego_←T 13:50, 9 August 2007 (UTC)

Your argument as to why utility functions are not measures would also preclude utility functions being measures — again, the probability of bundles of apples could increase to in apples to some point, and then decrease thereafter, analogously to a bliss point. While I don't know of the point being made explicitly in a textbook, it has certainly been made in the literature, as in McCulloch's article. —SlamDiego_←T 13:21, 9 August 2007 (UTC)

Again this is a mistaken interpretation of probability theory and measure theory. Measures increase when they go from subsets to supersets, and probabilities increase as they go from events which are subsets to events which are supersets. Example: the probability of 'receiving two apples' is less than the probability of 'receiving two apples or one apple', which is less than the probability of 'receiving two apples or one apple or seven apples'. That's the kind of monotonicity propertya mathematical description of probability must have. That's why measures are a good type of function to apply to probabilities. There is no analogous property in preference theory, which is why utility functions are not in general treated as measures. --Rinconsoleao 13:37, 9 August 2007 (UTC)

No, now you're effectively inserting words into what I said. I didn't refer to the event of getting ${\displaystyle n}$ or fewer apples, nor to the event of getting ${\displaystyle n}$ or more apples; I referred to the event of getting ${\displaystyle n}$ apples. —SlamDiego_←T 13:43, 9 August 2007 (UTC)

The monotonicity properties of measures only refer to events which are subsets or supersets. So your comment is irrelevant if you are talking about mutually exclusive events such as 'receiving one' or 'receiving two' or 'receiving three'. A monotonicity property is applicable to events like 'receiving one or less', 'receiving two or less', etc., because the former is a subset of the latter. This kind of monotonicity property has no role in utility analysis, which is why utility functions are not measures; probability functions are. --Rinconsoleao 13:48, 9 August 2007 (UTC)

My comment was relevant to refuting what you wrote, and (like every other comment) irrelevant to countless other things that might have been written. —SlamDiego_←T 13:55, 9 August 2007 (UTC)

In other words, it must be possible to decompose a utility function into two functions, one from states-of-the-world to nodes on a preference graph, and one from a preference graph to real numbers. Had you asked “In what sense is that mapping a measure of apples”, I would have answered to explain that it were not (necessarily) quite that. —SlamDiego_←T 12:03, 9 August 2007 (UTC)

Again, a mapping from positions on a preference graph to the real numbers is not a measure in the sense of measure theory. Let's delete the reference to measure theory. --Rinconsoleao 12:26, 9 August 2007 (UTC)

See above. —SlamDiego_←T 12:40, 9 August 2007 (UTC)

Even as they apply monotonic or affine transformations thereby showing that no one measure is particularly significant, they are performing transformations that wouldn't be possible if utility did not correspond to a measure.

(Moreover, I wouldn't have had to wrestle in the very first place with getting another editor to accept a definition without quantification of utility were the “ordinal” neoclassical conception not quantified.)

Now, indeed, you can find economists outside of the neoclassical mainstream whose use of the word “ordinal” is more general. (McCulloch for example tries to reclaim it.) But what we're arguing about here is not how to use the word “ordinal”; it's whether the neoclassical conception (however labelled) is fully general. Again: The disproof of De Finetti's conjecture showed that it is not. —SlamDiego_←T 09:17, 9 August 2007 (UTC)

Point 2: it’s hard to see why we would expect preferences to satisfy the properties in de Finetti’s conjecture

One of the properties assumed in de Finetti’s conjecture is ‘additivity’, which means that if bundle A is preferred to bundle B, and C is a bundle of goods disjoint from A and B, then the bundle of A together with C is preferred to the bundle of B together with C. In clearer English, this says that if two beers are preferred to one, then nine beers are preferred to eight. It is very hard to see why we would want to assume preferences must have that property! Standard treatments of utility usually make allowance for the possibility of bliss points, after which more consumption is bad. See Varian (1987) again, Figure 3.7, p. 43.

Any disagreements on these points? --Rinconsoleao 20:55, 8 August 2007 (UTC)

First, you've not quote got a handle on additivity. Additivity would, for example, also be violated if the tenth bed-sheet made the rope long enough for escape. (And this would be true even if yet more sheets remained desirable.)

Second, allowing for addivity is not disallowing for bliss points any more than allowing atheists to serve in Parliament disallows Christians.

Could you please give me an example of an additive utility function with a bliss point? I thought my beer example made this clear. --Rinconsoleao 09:47, 9 August 2007 (UTC)

Stop; pay attention to simple logic. Allowing additivity is not the same thing as requiring additivity.

You haven't (so far) proposed that people must have bliss points (and they're certainly not required for economic rationality); no one has proposed that utility must be additive.

Good, at least we agree on this. Additivity, and having a bliss point, are two properties we might want to consider as possibilities; a theory that considers preferences that can have one or both of these properties is more general than one which does not. --Rinconsoleao 10:09, 9 August 2007 (UTC)

Again, this is not so much agreement as your having wasted our time. —SlamDiego_←T 10:22, 9 August 2007 (UTC)

The point is that economically rational preferences can be additive, and therefore a conception of utility that disallows them is not as general as one that allows them.

—SlamDiego_←T 09:59, 9 August 2007 (UTC)

And my point is that ordinal preferences (meaning an ordering over consumption bundles) can have the additivity property. In that sense they are general. Disproving the dFC simply proves they don't always have that property. But that does not make them less general. --Rinconsoleao 10:09, 9 August 2007 (UTC)

You are abusively switching lexicons. We are not talking about what the Austrian School or some other group calls “ordinal” utility. We are talking about what the neoclassical school calls “ordinal” utility. And because that name is not a proper description — because the neoclassical conception is not fully general — I have consistently used quotation marks to sugegst that there is something off in that name. —SlamDiego_←T 10:22, 9 August 2007 (UTC)

I'm really not trying to gain rhetorical advantage by switching lexicons. I have consistently been trying to make clear which mathematical/econonomic concepts I am talking about, and trying to figure out whether they were the same ones you are talking about. That's why I explicitly said, many times, that I was talking about an ordering over bundles. Which historical schools they can be attached to is a question I have not tried to address. But it would be helpful to know whether 'neoclassical', in your usage, includes standard textbook treatments nowadays (such as the one you have requested that I not cite). --Rinconsoleao 10:53, 9 August 2007 (UTC)

Again, I am focussed on your actions, not on your intentions. You started making declarations of about “ordinal utility” as if these contradicted what I had said about “the neoclassical conception of ‘ordinal’ utility”.

Fine, so I may have made a wrong turn by failing to keep track of exactly which concept you were referring to. But therefore it would again be helpful to me to know whether 'neoclassical', to you, means standard contemporary usage such as Varian, or whether it refers to usages from the early 20th century, or to both, or to something else. --Rinconsoleao 11:32, 9 August 2007 (UTC)

Since I've not been using the past tense or words such as “former”, you naturally should take it as the present definition. (At the dawn of neoclassical economics, the definition would have been more Benthamite.) But I'd refer you to an author such as Kreps, who is far more careful than Varian. —SlamDiego_←T 11:54, 9 August 2007 (UTC)

And your wrong turn has wasted a huge amount of time and drowned this talk page in avoidable nonsense, as you attempted to argue that the issue of de Finetti's conjecture were a red herring. —SlamDiego_←T 11:58, 9 August 2007 (UTC)

That's certainly not the same thing as telling me what you mean by “ordinal utility”; rather, since the meaning of the one didn't match that of the other that's abusive lexicon switching. There is little excuse for it, since I plainly wasn't baldly saying “ordinal utility”, but always putting “ordinal” in quotation marks.

Citing Varian wouldn't at all change that. And implicitly citing Varian at me is not acceptable.

—SlamDiego_←T 11:21, 9 August 2007 (UTC)

Third, allowing for additivity allows for a class of economically rational preferences in the context of various economic circumstances.

Fourth, don't quote Varian at me. Especially don't quote his undergraduate text (or any other undergraduate text) at me.

—SlamDiego_←T 23:13, 8 August 2007 (UTC)

Slam, I am obviously not quoting Varian 'at you'. I am backing up my main points with page-specific references to widely available texts, which is what Wikipedia, wisely, asks us to do. I would hope our discussion will be accessible to many people, no only to the two of us. --Rinconsoleao 06:57, 9 August 2007 (UTC)

Wikipedia asks us to cite sources in articles, not to treat economists on talk pages as if they haven't taken undergraduate economics. I repeat: Do not quote Varian at me. —SlamDiego_←T 07:47, 9 August 2007 (UTC)

Slam, I really sincerely am not trying to talk down to you. And you are not the only person to whom I am directing my comments. My problem with the MU page is that it has never been sufficiently clear. If I quote Varian it is only to allow you (and others) to be sure you know what I mean with my arguments. For example, the page now talks about a 'highest quantitative utilization', three words which I simply could not understand until I clicked and found 'production possibilities frontier'. I did not feel that my intelligence had been insulted by that link. I was grateful for the clarification. --Rinconsoleao 09:01, 9 August 2007 (UTC)

I am less concerned with your intentions (which are not directly observable) than with your actions. You're expecting me to respond to you. Regardless for whose benefit you would do it, don't quote Varian at me. Don't quote any undergraduate text at me. Whatever you might think that you would want in my shoes, you're not in my shoes and I'm not you. —SlamDiego_←T 09:25, 9 August 2007 (UTC)

I am not talking only to you, because many people have been involved in this discussion. If you find it offensive to see that text mentioned, then bear in mind that someone else might want to see it. Where possible, I will always quote the least technical text I can find. --Rinconsoleao 09:31, 9 August 2007 (UTC)

Nonsense. The fact that we presumably have an audience doesn't change the fact that you are talking to me — indeed, if it did then “I am not talking only to you” would be absurd, as the “you” would refer exactly to the people to whom you were talking. And the fact that we have an audience doesn't legitimize quoting undergraduate textbooks at me. If you want to express yourself for their benefit, find a better, acceptable way. —SlamDiego_←T 09:39, 9 August 2007 (UTC)

Point 3: why all this fuss about measure theory?

I was surprised from the beginning when I saw ‘quantified’ utility defined as a ‘measure’, and in particular when that was linked to ‘measure theory’. I always thought of cardinal utility simply as putting numbers on consumption bundles as a way of ranking them. I don’t recall in my studies ever seeing measure theory mentioned in discussions of utility per se (though obviously it is mentioned when you get to expected utility, for example, since expected utility means weighting a bunch of utilities with respect to a probability measure).

It’s true that Stigler uses the word ‘measure’ a lot in the essay cited in this article.(Stigler, George Joseph; “The Development of Utility Theory”, I and II, Journal of Political Economy (1950), issues 3 and 4.) But my reading of that article is that he is using ‘measure’ in the intuitive, nontechnical sense, meaning actually being a quantity of something. In other words, he is asking whether utility has some natural units by which it might, in principle, be measured.

Let me give an example of what that might mean. Suppose we only consume apples, and we measure them in kilograms. That’s something you can actually measure. (The fact that there are other units of measurement that could be used in this context is irrelevant. That’s always true when we measure anything.) Suppose also that utility really is something that could be measured. For example, (JUST for the sake of argument!) suppose utility means milligrams of serotonin per liter of brain juice. Since that’s actually a quantity, we would represent it by a cardinal utility function, and likewise a cardinal marginal utility function. In other words, ${\displaystyle a}$ kilos of apple give ${\displaystyle U(a)}$ units of utility (milligrams of...), and likewise, at the margin, additional apples give ${\displaystyle u(a)={\frac {d}{da}}U(a)}$ units of utility per kilogram of apples.

For example, suppose ${\displaystyle u(a)}$ is defined for ${\displaystyle a\in [0,4]}$, because it is impossible to eat less than 0 or more than 4 kilograms of apples, and suppose the marginal utility function is ${\displaystyle u(a)=20-10a}$. Then by definition we must have

${\displaystyle U(a)=\int _{0}^{a}(20-10x)dx=20a-5a^{2}}$

which has a bliss point at ${\displaystyle a=2}$. Here, we are actually using the measure of apples in the sense of measure theory. Namely, we are integrating with respect to the measure of apples. (I.e. the dx represents a small change in the quantity of apples. And if we changed the units in which we measure apples, we would have to adjust the integral accordingly.) That’s what you do with measures in the sense of measure theory.

In this example, utility is not a measure over apples, because by definition a measure (in the sense of measure theory) gives a larger measure when the set it is measuring is increased to include more members (like a few extra apples). (See Thm. 10.2 in Aliprantis and Burkinshaw (1990), Principles of Real Analysis, 2nd ed., ISBN 0-12-050255-0, or any other real analysis textbook.) What utility ${\displaystyle U(a)}$ does measure in this example, both in the ordinary sense of the word, and in the measure theory sense of the word, is serotonin. Saying that “the utility of two kilos of apples is ${\displaystyle 20*2-5*2^{2}=20}$” in this context is a statement about the amount of serotonin in your brain juice.

It seems this is where the confusion about the relevance of de Finetti’s conjecture arose. De Finetti’s conjecture is trying (unsuccessfully) to divvy up the utility of a consumption bundle into statements about the utilities of individual goods. In other words, it was conjecturing the existence of a measure over goods. But that’s not what utility functions do. They don’t measure goods. They measure utility, as a quantity, under the assumption (whether it is a reasonable assumption or not) that utility can be quantified.

But my main point here is simply that mentioning measure theory at all is a red herring. It points to a lot of technical issues which there is no reason to go into, because standard treatments of preferences today only rank (or give numbers to) whole consumption bundles. (I can cite many more textbooks that talk about rankings and/or utility functions defined over bundles, without ever mentioning such a function over individual goods, if that would help.) They say nothing about the utility of particular goods. More precisely, they can say something about the utility of a particular good when it is at the margin, but they cannot and do not attempt to attribute utility to a particular good in general. In other words, a utility function is not, and is not intended as, a measure over goods.

Any disagreements on these points? --Rinconsoleao 21:52, 8 August 2007 (UTC)

The distinction between measure theory and measurement theory was already made in discussion with Thomasmeeks. You should not confuse the two, nor attempt to lecture me on either.

Here at least I think we are largely in agreement. In your discussions and in my example above we are both trying to distinguish the mathematical theory called measure theory from the day-to-day concept of measuring something with a ruler or a scale that it tries to formalize. ('Measurement theory' as a way of referring to the day-to-day concept is not standard wording. I would just call it 'measurement'.) --Rinconsoleao 07:35, 9 August 2007 (UTC)

It's not so much that we are in agreement as that you are just wasting my time and your own with misdirected integrals and lectures. —SlamDiego_←T 08:09, 9 August 2007 (UTC)

(There is, in fact, a field of measurement theory. See Basic Measurement Theory by Suppes and Zinnes.) —SlamDiego_←T 15:07, 9 August 2007 (UTC)

Thanks. That is a new one to me. I'll try to look that up. --Rinconsoleao 15:24, 9 August 2007 (UTC)

I have used the term “measure” because that is the mathematical term that applies, and I linked to the Wikipedia article on measure theory accordingly.

My suggestion, instead, is to drop references to measure theory because even though it is accurate and precise, most people don't understand it, and therefore mentioning it raises the possibility that there are deeper issues at stake. But when we ask whether utility could be 'measured' (again, see the Stigler papers you cite) I don't see any issues specifically related to measure theory per se. The question is simply whether there is some real meaning or not when we put a number on utility (for example, does it represent milliliters of serotonin, or any other definable quantitative meaning). If there is some issue where we specifically need techniques from measure theory to address these questions, you should point it out to us, please. --Rinconsoleao 07:35, 9 August 2007 (UTC)

We're not simply worried about numbers but about quantities. I would have been fine with the idea of just using the word “quantity” (and derivatives), quietly wiki-linked to the article on measure theory, except that you proved that the reader wouldn't get it when you barrelled ahead and switched the link to the article on “cardinal” utility. —SlamDiego_←T 08:09, 9 August 2007 (UTC)

The reason for using this term is that it gives accurate and precise mathematical meaning to “quantity”. Your own attempt, in one of your edits to the article to replace this reference with one to “cardinal” utility illustrates that such accuracy and precision is desirable.

Whether my reference to cardinal utility was justified is a separate issue. My usage of the term is coherent, as I explained in Point 1 above, though I'm not entirely sure whether it is the usage that should be called 'neoclassical'. But we can discuss that later. For now, I am advocating eliminating all reference to measure theory in this article, because it merely obfuscates things. --Rinconsoleao 07:43, 9 August 2007 (UTC)

It's not a separate issue. The issue is what keeps the reader from heading off in the wrong direction. You showed that using the word “quantity” wasn't enough; using the word “number” woudl be even worse. —SlamDiego_←T 08:09, 9 August 2007 (UTC)

I don't understand what you mean when you distinguish "quantity" from "number". Sure, I can imagine some distinctions, but I don't know which possible distinction you intend. Could you help me understand this? --Rinconsoleao 09:09, 9 August 2007 (UTC)

Consider the Dewey Decimal system. It assigns numbers to books; those numbers aren't quantities. —SlamDiego_←T 09:52, 9 August 2007 (UTC)

Good, I am getting closer to understanding you. Would it be fair to interpret 'those numbers aren't quantities' as 'those numbers do not represent amounts of anything; instead, they are merely labels'? --Rinconsoleao 10:21, 9 August 2007 (UTC)

Well, they may or may not be merely labels, but the point is that arthimetic performed on those numbers yields just nonsense. By contrast, with the neoclassical conception of “ordinal” utility, the sign of a difference (arithmetic subtraction) is meaningful. Indeed, the sign of the difference-of-differences might also be meaningful. —SlamDiego_←T 10:26, 9 August 2007 (UTC)

DeFinetti's focus was upon probability. The fact that the math applies to utility as well was apparently first recognized by McCulloch.

—SlamDiego_←T 23:29, 8 August 2007 (UTC)

The fact that the de Finetti conjecture is applicable more widely than probability theory was not first recognized by McCulloch (1977). The first page of the Kraft et al (1958) paper makes crystal clear that the conjecture is about rankings in general (such as preference orderings, and in fact the notation used was standard in preference theory in the 1950s) and that the application to probability is a special case (as explained in the third paragraph). --Rinconsoleao 07:22, 9 August 2007 (UTC)

You are again mistaken. The notation was not invented by economists nor for economics; it is a very standard notation for orderings in general, and could and can be found in works by mathematicians who were and are quite oblivious to what economists are doing. And the fact that Kraft &alii applied it to orderings in general does not meaning that they recognized any one specific application other than to probability orderings. (De Finetti's own writing didn't make explicit use of utility, and it took a degree of perspicuity for others to show how it was implicit therein.) —SlamDiego_←T 07:57, 9 August 2007 (UTC)

I did not assert that it was invented by or for economists; yes, you are right, it is mathematical notation; but yes, it was standard notation in economics at the time (Debreu's Theory of Value came out in 1959.) --Rinconsoleao 08:12, 9 August 2007 (UTC)

And I didn't say that you said that it was invented by economists. But what you did do was argue that the common notation illustrated that Kraft &alii had recognized the implications for utility, and I choked-off every argument that might be used to defend that illogical inference; doing so included noting that the authors needn't have got the notation (directly or even indirectly) from economists. —SlamDiego_←T 08:27, 9 August 2007 (UTC)

You are right that the fact that Kraft et al. wrote about orderings in general does not necessarily imply that they recognized that those orderings might be applicable to economics. But Pratt (one of the coauthors of Kraft) made a major contribution to preference theory in 1964 (J.W. Pratt (1964), 'Risk aversion in the small and in the large', Econometrica 32, pp. 122-136). Am I really supposed to believe that he was completely ignorant of the economics applications of orderings in 1958? --Rinconsoleao 08:18, 9 August 2007 (UTC)

Are we really supposed to believe that he did see it and then didn't turn around and say “Hey, guys, about this thing that you call ‘ordinal’ utility and think is fully general…”? Even brilliant people often over-look implications of their own work. I cannot guarantee that McCulloch was the first to see the implications for utility; he was the first to publish, though, which makes it apparent (exactly as I said) that he was the first. —SlamDiego_←T 08:27, 9 August 2007 (UTC)

Of course he would not have turned around and said that, because the disproof of the dFC does not show a lack of generality of 'ordinal' utility. It shows that some ordinal representations of utility satisfy the assumptions of the dFC, and some don't, which is not a lack of generality. --Rinconsoleao 09:30, 9 August 2007 (UTC)

Again: I advised have always referred to what neoclassical economics calls “ordinal” utility. I had Pratt likewise hypothetically use the phrase “this thing that you call ‘ordinal’ utility”. What the Austrian School calls “ordinal” utility is indeed fully general; but what the neoclassical school has been calling “ordinal” utility is not. So the question is: If Pratt was aware of the implications for that neoclassical conception, why didn't he say something? McCulloch did note the implications, and did say something. —SlamDiego_←T 09:46, 9 August 2007 (UTC)