# Are there Utility Monsters in Economics?

Economics, especially in the modern school is broadly influenced by the utilitarian concept of utility. More so since the labor theory of value has been broadly replaced by the theory of marginal utility.

Additionally, perverse incentives are commonly understood and well documented, and seem to be small scale imitations of Nozick's classical "utility monster".

Are there any observations of larger "utility monsters" (wherein consumption by an individual increases aggregate utility for a group while individual utility is reduced for all but the "monster" of the group)?

Does a theory of declining marginal utility necessarily prevent such a thing if utility is said to remain non-negative? (i.e. simply having the ability to ignore excess goods). Obviously it prevents it if utility can become negative, unless the number of units of the good in question are fixed at less than the number required to reach negative total utility.

For a simplistic toy example, imagine a closed system consisting of myself, my five-year-old daughter, and two cars (full-sized). Allocating a car to her produces little marginal utility since she can't drive (or even reach the pedals), though presumably it's a non-zero amount. Consequently taking a car from her and giving it to me produces a net gain for the "economy", despite producing a presumable reduction in utility for her (assuming I'm not going to drive her around because I'm a terrible father in this example). Further, even assuming if she owned both cars, taking them both from her and giving them to me would produce an aggregate gain, as I could exploit the one car better than she could the two, and the second (or even third, etc), would not be an inconvenience, and in fact still offer me greater marginal utility as backup vehicles.

The question is, do such scenarios arise in practical economic settings where one group or individual is simply so much better able to make use of a good than another as to justify (in the sense of aggregate utility) taking it from the less able?

I understand this could be a contentious question, but I'm asking not from a moral standpoint but strict aggregate utility.

The constraints regarding the system I'm modeling are (and looking for a general solution to):

1. Marginal Utility for every unit of a good must be positive (or zero), finite, and declining (though never to below zero).
2. Finite Goods:
1. The available quantities of all goods must be finite, though they can be arbitrarily large.
2. There can be an arbitrarily large, though finite number of other goods in the system.
3. Aggregate Utility must increase for all, while individual utility must decrease for all except one (the "monster") when a good of a certain class (let's say "cars") is transferred from any member of the group to the "monster".
4. Condition 3 should be met for all transfers of "cars" from "innocents" (people not the monster) to the "monster", to the exhaustion of "cars" from the system.

Again, this is not a question about "Can mutually beneficial trade exist under any circumstances?" We've known that since before Ricardo. This is a question about the requirements for Aggregate Utility increase at the expense of most individuals when talking abut individual preference.

• You are assuming there are no other goods? With complete markets, why wouldn't your daughter sell you the car (or exchange another good)? – Pburg Nov 23 '14 at 21:10
• @Pburg additional goods don't necessarily matter. Let's assume we value cake the same amount, but both value cars more than cake. Giving her a piece of cake to "buy" the car, still lowers her total utility, raises mine, and the group's. – Jason Nichols Nov 23 '14 at 21:14
• If you value the car at 3, she at 2, and both value cake at 1, you could give her 2.5 units of cake? Linear utility. Maybe there isn't enough cake in this tiny economy for a trade to happen, but for a large it is more likely to work. – Pburg Nov 23 '14 at 21:16
• Depends what me mean by justify, but it seems like the second fundamental theorem of welfare economics addresses exactly these kinds of scenarios. We have a Pareto-optimal allocation that cannot be supported as a price equilibrium, but with transfers it is a quasi-equilibrium. – Pburg Nov 23 '14 at 21:32
• Seems a bit like game theory applied to a planned economy, I'm sure you can find subsets where there are greater benefits distributing resources where utility dictates.. but like in game theory exercises, the hardest part is actually getting everyone onboard. – NickW Nov 25 '14 at 9:32

There is a type of protection called a liability rule, where I, $A$, can take something from $B$, if I pay the damages $c$ which are court-ordered preemptively. Copyright law is all about liability rules. If the damages are correlated with $B$'s valuation appropriately, then efficiency holds. You are interested in the opposite case. If IP law doesn't get it right, this would be an example matching your hypothetical as I understand it. And some might say this is justified. A company might want to keep some research under wraps to prevent competitors from also using it and making it even more valuable to society through investment. So the expiration of a patent might fit this scenario in a roundabout way too if we start the model where the company already owns the patent and a court is deciding when to let it expire.

• IIUC, what you tell the OP is: you may take all cars from your daughter, but, you have to give her something else that will leave her with (at least) the same utility. Is this right? – Erel Segal-Halevi Apr 26 '15 at 9:33
• @ErelSegal-Halevi Under a liability rule, you'd definitely have to give something to your daughter. In principle, a lawmaker would want that to be of the same utility to ensure an efficient outcome. Otherwise, the daughter would reclaim those cars. – Pburg Apr 26 '15 at 11:14

Your toy example does not really happen in standard economic models. For one, because we assume identical preferences. Therefore, to the extent that we similarly care about different individuals, we need similar consumption (or whatever enters their utility) to maximize welfare.

That does not mean that all allocations are the same. It could be that one of the agents is especially lazy - he should optimally not work. Others will have to do extra shifts, but we will compensate them with more consumption: the utility per person over both consumption and working time will then be again relative to the pareto weights.

We keep focussing at classes of utility functions $u(x)$ that are finite and strictly increasing in $x$.

Necessary condition Strictly speaking, diminishing marginal utility is necessary but not sufficient to prevent the existence of such "utility monsters". Given two utility functions $u(x), v(x)$, you need that

$$x_1 \geq x_2 \Leftrightarrow u'(x_1) \leq v'(x_2) \quad (1)$$

and vice-versa for $v'$ and $u'$. Diminishing marginal utility only gives you

$$x_1 \geq x_2 \Leftrightarrow u'(x_1) \leq u'(x_2) \quad (2)$$

It is easy to see that when only (2) holds, given a sufficiently enough difference in the levels of $u, v$, we still find it optimal to give all of $x$ to one individual.

Sufficient condition

Sufficient condition for every individual having positive $x$ is that for every preference $u, v$:

$$\lim_{x\to 0}\,u(x) = -\infty \quad (3)$$

such that no matter the levels of $u, v$, marginal utility just blows up. This together with diminishing marginal utility should prevent utility monsters.

Alternatively,

$$u(\epsilon) - u(0) > v(x+\epsilon) - v(x) \, , \forall x>0$$

and vice-versa for $v, u$ and some $\epsilon > 0$ will ensure that you will never sacrifice all consumption of everyone else in order to give it to the other guy.

• this necessitates negative marginal utility, though. In the question, I specifically asked about cases without it, although otherwise this is a great answer. – Jason Nichols Nov 23 '14 at 21:17
• As far as I can tell the question only referred to questions without negative total utility (not marginal). Note that the level of utility being positive irrelevant: Assuming $u(x)$ corresponds to a preference relationship, so will any positive affine transformation of $u(x)$, including $v(x) = e^99 + u(x)$. – FooBar Nov 23 '14 at 21:23
• Also, of note, JPE Vol 63, Iss 4 darp.lse.ac.uk/papersdb/Harsanyi_%28JPolE_55%29.pdf re: the problems with summing individual utility functions to form aggregate social welfare functions. Postulate E deals specifically with individual preference incompatibility. Not a refutation or critique, just a note for expansion later. – Jason Nichols Nov 23 '14 at 21:24
• @FooBar Obviously it prevents it if utility can become negative was meant to imply (and will be corrected to state explicitly) that I get we can trend to zero if marginal utility can go negative, and that I'm asking after the mathematical constraints required assuming marginal utility cannot go negative (if additional units can simply be "ignored" without maintenance cost) – Jason Nichols Nov 23 '14 at 21:27
• @Pburg Right, I should put that somewhere in the introduction though, as the whole text bases on the assumption of positive marginal utility. – FooBar Nov 23 '14 at 21:28

The concept of the utility monster was intended by Nozick to criticize (1) interpersonal comparisons of utility where utility is a field subset, by revealing a certain absurdity in the scheme.

This is a thought experiment purely, because interpersonal utility comparisons are questionable empirically and theoretically, and this because utility is not a field subset (and even if it was, this would not imply by itself meaningfulness of interpersonal comparisons). As is well-known, there is no neurophysiological evidence for (1)-type utility. Indeed, two mind configuration states are isomorphic or not, and if not, that's it, nothing more can be said, assuming we go with the mainstream models of brain functioning.

Most importantly, as Gibbs and Fisher long ago argued, explaining and predicting behaviour does not require utility to be a measurable quantity, let alone such measurements for one person being comparable to measurements for another person. So in reality, there is no possibility of observing a utility monster. For they don't exist.

The utility monster is an additional, alternative argument against (1) being valid by revealing that it leads to a very strange prediction.

The references are the ones that started modern utility theory, where utilities are not elements of fields, but merely numerical labels covarying in restricted ways with transformations to preserve correspondence to ranked behaviours.

Utility here is postulated to and only to predict and explain that behaviour to which it corresponds. So it means nothing to say I like X more than I like Y both compared to my like of Z. Merely I prefer X to Y to Z, as is demonstrated by my observed behaviour. It means nothing to say that A and B like so much brandy more than so much whisky yet A gets more happiness from drinking brandy than B. Rather, A and B feel the need to take so much brandy and leave so much whiskey when offered one or the other, and are willing to forego the whiskey, nothing more or less is implied by the "utility" analysis, because U(B';B,W) > U(W';B,W), which means their mind is in a "happier" state, which is a different preferred configuration.

The utility numbers are distinguishing marks of ranks of continuously varying quantities, referring to lesser or more important wants that are eliminated by consumption of the quantities or remain where this consumption is absent. They don't add, but correspond to choices executed in a different order.

I give the references with historical priority:

Fisher, I. 1892. Mathematical Investigations in Theory of Value & Prices. Transactions of Connecticut Academy of Arts & Sciences 9:1-124. (It seems that the topic and novel approach and mathematics are given by Gibbs, the thesis instructor of Fisher, and Fisher worked it out into the modern neoclassical utility function and demand theory, to get his PhD.)

Cuhel, F. 1907. Zur Lehre von den Bedurfnissen. Innsbruck.

Bernardelli, H. 1938. End of Marginal Utility Theory? Economica 5:192 -212. 1952. Theory of Marginal Utility. Economica 19(3):254-268.

• Your covering a lot of ground here and making a lot of claims. Please expand. – Jason Nichols Nov 23 '14 at 21:53
• I was editing it while you posted, please check it now. I can add some more references, but they are the foundational works for modern utility functions, and not very recent (and very well know, e.g., I. Fisher). – user218 Nov 23 '14 at 21:56
• I still would not upvote this for a few reasons. 1) no links. 2) Overlooks the examples of utility monsters in simple small economies, and clearly demonstrated utility differential both presented in the question. 3) diverges from the main thrust of the question. I'm interested in building simulations that demonstrate concepts, not necessarily looking for research that criticizes those concepts. I'm not downvoting either, because I think this adds decent background to the question, but I think the answer could be made much better. – Jason Nichols Nov 23 '14 at 22:01
• the no links thing is a personal preference. When posters invoke names as authority without providing links to relevant studies, it makes me sad. – Jason Nichols Nov 23 '14 at 22:02
• I have the references in a paper nearby. A few moments... ;) I wasn't sure you would want rather old papers. I'm happy to hear that you do. – user218 Nov 23 '14 at 22:03

There is a quantification of this "utility monstrousness" in the fair division model. In that model, a resource has to be divided among $n$ partners, all of whom have the same rights to that resource but different preferences over parts of the resource. A division is called fair (according to one definition) if each person receives at least $1/n$ of the resource according to his own valuation (so there is no inter-personal utility comparisons).

Here, too, there is a tradeoff between fairness and social welfare. It is quantified by the Price of Fairness concept.

I haven't run into a "utility monster" in the literature. I can tell you one concept in which the utility monster is expressly ruled out. I believe this line:

wherein consumption by an individual increases aggregate utility for a group while individual utility is reduced for all but a the "monster" of the group

rules out pareto efficiency -- in your examples, clearly some person is being made better off while others are being made worse off. The first and second welfare theorems, for example, deal largely in pareto efficiency.

This doesn't exactly answer your question, but perhaps it is one reason for rarity of a "utility monster" in theory (at least as far as I've seen).