<p>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.</p>

<p>Reference: <a href="http://nicolapersico.com/files/takings.pdf">Exchange Efficiency With Weak Property Rights.</a></p>

CC BY-SA 3.0
2015-04-26T09:33:14.970Z
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
CC BY-SA 3.0
2015-04-26T11:14:09.263Z
@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
CC BY-SA 3.0
2014-11-23T22:08:17.477Z
2014-11-23T22:08:17.477Z
Pburg
<p>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.</p>

<p>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.</p>

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

<p><strong>Necessary condition</strong> 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</p>

<p>$$x_1 \geq x_2 \Leftrightarrow u'(x_1) \leq v'(x_2) \quad (1)$$ </p>

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

<p>$$x_1 \geq x_2 \Leftrightarrow u'(x_1) \leq u'(x_2) \quad (2)$$ </p>

<p>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.</p>

<p><strong>Sufficient condition</strong></p>

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

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

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

<p>Alternatively, </p>

<p>$$ u(\epsilon) - u(0) &gt; v(x+\epsilon) - v(x) \, , \forall x&gt;0$$</p>

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

CC BY-SA 3.0
2014-11-23T21:17:11.007Z
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
CC BY-SA 3.0
2014-11-23T21:23:48.927Z
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
CC BY-SA 3.0
2014-11-23T21:24:56.733Z
Also, of note, JPE Vol 63, Iss 4 http://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
CC BY-SA 3.0
2014-11-23T21:27:02.727Z
@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
CC BY-SA 3.0
2014-11-23T21:28:09.503Z
@Pburg Right, I should put that somewhere in the introduction though, as the whole text bases on the assumption of positive marginal utility.
FooBar
CC BY-SA 3.0
2014-11-23T21:31:30.057Z
correction to previous comment, positive and finite marginal utility to eliminate the case of the first good being infinitely valuable.
Jason Nichols
CC BY-SA 3.0
2014-11-23T21:35:16.747Z
If you restrict marginal utility, I'm afraid I don't know any useful mathematical restriction, just the one that is somewhat obvious - which I added.
FooBar
CC BY-SA 3.0
2014-11-23T21:43:12.297Z
I'm thinking along the lines of if the monster's marginal utility function is A+B/n where n is the number of goods, and everyone else's function is C+D/n, where A>C what's the constraints on B and D to allow for a utility monster to appear? (i.e. to preclude, D must be > B+A/C or something along those lines)...
Jason Nichols
CC BY-SA 3.0
2014-11-23T21:06:20.447Z
2014-11-23T21:34:25.643Z
2014-11-23T21:34:25.643Z
FooBar
<p>There is a quantification of this "utility monstrousness" in the <a href="https://en.wikipedia.org/wiki/Fair_division" rel="nofollow">fair division</a> 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 <em>fair</em> (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).</p>

<p>Here, too, there is a tradeoff between fairness and social welfare. It is quantified by the <a href="https://en.wikipedia.org/wiki/Price_of_fairness" rel="nofollow">Price of Fairness</a> concept.</p>

CC BY-SA 3.0
2015-04-26T10:03:22.247Z
2015-04-26T10:03:22.247Z
Erel Segal-Halevi
<p>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:</p>

<blockquote>
  <p>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</p>
</blockquote>

<p>rules out <a href="http://en.wikipedia.org/wiki/Pareto_efficiency" rel="nofollow">pareto efficiency</a> -- 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. </p>

<p>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).  </p>

CC BY-SA 3.0
2014-11-23T21:12:06.613Z
2014-11-23T21:12:06.613Z
CompEcon
<p>Economics, especially in the modern school is broadly influenced by the <a href="http://en.wikipedia.org/wiki/Utilitarianism" rel="noreferrer">utilitarian</a> concept of utility. More so since the labor theory of value has been broadly replaced by the theory of marginal utility.</p>
<p>Additionally, <a href="http://en.wikipedia.org/wiki/Perverse_incentive" rel="noreferrer">perverse incentives</a> are commonly understood and well documented, and seem to be small scale imitations of <a href="http://en.wikipedia.org/wiki/Utility_monster" rel="noreferrer">Nozick's classical &quot;utility monster&quot;</a>.</p>
<p>Are there any observations of larger &quot;utility monsters&quot; (wherein consumption by an individual increases aggregate utility for a group while individual utility is reduced for all but the &quot;monster&quot; of the group)?</p>
<p>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.</p>
<p>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 &quot;economy&quot;, 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.</p>
<p>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?</p>
<p>I understand this could be a contentious question, but I'm asking not from a moral standpoint but strict aggregate utility.</p>
<h2>Updates</h2>
<p>The constraints regarding the system I'm modeling are (and looking for a general solution to):</p>
<ol>
<li>Marginal Utility for every unit of a good must be positive (or zero), finite, and declining (though never to below zero).</li>
<li>Finite Goods:
<ol>
<li>The available quantities of all goods must be finite, though they can be arbitrarily large.</li>
<li>There can be an arbitrarily large, though finite number of other goods in the system.</li>
</ol>
</li>
<li>Aggregate Utility must increase for all, while individual utility must decrease for all except one (the &quot;monster&quot;) when a good of a certain class (let's say &quot;cars&quot;) is transferred from any member of the group to the &quot;monster&quot;.</li>
<li>Condition 3 should be met for all transfers of &quot;cars&quot; from &quot;innocents&quot; (people not the monster) to the &quot;monster&quot;, to the exhaustion of &quot;cars&quot; from the system.</li>
</ol>
<p>Again, this is not a question about &quot;Can mutually beneficial trade exist under any circumstances?&quot; 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.</p>
<p><a href="http://existentialcomics.com/comic/8" rel="noreferrer">Inspiration for the question</a>:</p>
<p><img src="https://i.sstatic.net/uQ5UJ.jpg" alt="Inspiration" /></p>

CC BY-SA 3.0
2014-11-23T21:10:53.870Z
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
CC BY-SA 3.0
2014-11-23T21:14:07.983Z
@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
CC BY-SA 3.0
2014-11-23T21:16:35.257Z
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
CC BY-SA 3.0
2014-11-23T21:19:04.260Z
I know it's possible to make fair trades. That's not the question. :) The question is do these situations arise in serious studies, for example as a kind of Machiavellian justification for force, or conceivably as a justification for regulation or eminent domain. Or anywhere else for that matter.
Jason Nichols
CC BY-SA 3.0
2014-11-23T21:20:10.610Z
let's say I value car at 5, she values it at 3, and we both value cake at 1. If there are two cars and two cakes in the system, even if I have both cakes, I can't make a fair trade, but I could maximize the group's utility by trading, despite the fact that her utility decreased.
Jason Nichols
CC BY-SA 3.0
2014-11-23T21:32:59.450Z
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
CC BY-SA 3.0
2014-11-25T09:32:19.307Z
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
<utility><theory>
Are there Utility Monsters in Economics?
Jason Nichols
# Are there Utility Monsters in Economics?

(Asked by: username_0 on 2014-11-23)

Economics, especially in the modern school is broadly influenced by the [utilitarian](http://en.wikipedia.org/wiki/Utilitarianism) concept of utility. More so since the labor theory of value has been broadly replaced by the theory of marginal utility.

Additionally, [perverse incentives](http://en.wikipedia.org/wiki/Perverse_incentive) are commonly understood and well documented, and seem to be small scale imitations of [Nozick's classical "utility monster"](http://en.wikipedia.org/wiki/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.

## Updates

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.

[Inspiration for the question](http://existentialcomics.com/comic/8):

![Inspiration](https://i.sstatic.net/uQ5UJ.jpg)

**Tags:** utility, theory
> **Comments:**
> **username_1**: You are assuming there are no other goods? With complete markets, why wouldn't your daughter sell you the car (or exchange another good)?  
> **username_0**: @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.  
> **username_1**: 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.  
> **username_0**: I know it's possible to make fair trades. That's not the question. :) The question is do these situations arise in serious studies, for example as a kind of Machiavellian justification for force, or conceivably as a justification for regulation or eminent domain. Or anywhere else for that matter.  
> **username_0**: let's say I value car at 5, she values it at 3, and we both value cake at 1. If there are two cars and two cakes in the system, even if I have both cakes, I can't make a fair trade, but I could maximize the group's utility by trading, despite the fact that her utility decreased.  
> **username_1**: 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.  
> **username_2**: 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.  

---

## 4 answers

### Answer by username_1 on 2014-11-23. Score: 7

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.

Reference: [Exchange Efficiency With Weak Property Rights.](http://nicolapersico.com/files/takings.pdf)

> **Comments:**
> **username_3**: 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?  
> **username_1**: @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.  

---

### Answer by username_4 on 2014-11-23. Score: 6

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.

> **Comments:**
> **username_0**: this necessitates negative marginal utility, though. In the question, I specifically asked about cases without it, although otherwise this is a great answer.  
> **username_4**: 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 $v(x) = e^99 + u(x)$.  
> **username_0**: Also, of note, JPE Vol 63, Iss 4 http://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.  
> **username_0**: @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)  
> **username_4**: @Pburg Right, I should put that somewhere in the introduction though, as the whole text bases on the assumption of positive marginal utility.  
> **username_0**: correction to previous comment, positive and finite marginal utility to eliminate the case of the first good being infinitely valuable.  
> **username_4**: If you restrict marginal utility, I'm afraid I don't know any useful mathematical restriction, just the one that is somewhat obvious - which I added.  
> **username_0**: I'm thinking along the lines of if the monster's marginal utility function is A+B/n where n is the number of goods, and everyone else's function is C+D/n, where A>C what's the constraints on B and D to allow for a utility monster to appear? (i.e. to preclude, D must be > B+A/C or something along those lines)...  

---

### Answer by username_3 on 2015-04-26. Score: 4

There is a quantification of this "utility monstrousness" in the [fair division](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Price_of_fairness) concept.


---

### Answer by username_5 on 2014-11-23. Score: 3

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](http://en.wikipedia.org/wiki/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).


---

