<p>Suppose there are two families: Family U has $n_u$ members, and family V has $n_v$ members. Utility function of member $i$ of family U is:
\begin{eqnarray*} u_i(x_u, y_u) = a_ix_u + y_u \end{eqnarray*}
where all $a_i$s are positive for all $i\in\{1,2,\ldots, n_u\}$,</p>

<p>and utility function of member $j$ of family V is:
\begin{eqnarray*} v_j(x_v, y_v) = b_jx_v + y_v \end{eqnarray*}
where all $b_j$s are positive for all $j\in\{1,2,\ldots, n_v\}$.</p>

<p>Also, suppose $\min_i a_i \geq \max_j b_j$.</p>

<p>Suppose the total endowment vector of $X$ and $Y$ is $(\omega_X, \omega_Y)$. </p>

<p>For any $\theta \in [\max_j b_j, \min_i a_i]$, define $m :=  \displaystyle\frac{\theta\omega_X}{2} + \frac{\omega_Y}{2} $.</p>

<p>Check that if $\displaystyle\frac{m}{\theta} \leq \omega_X$, then $\displaystyle (x_u, y_u) = \left(\frac{m}{\theta}, 0\right)$ and $\displaystyle (x_v, y_v) = \left(\omega_X - \frac{m}{\theta}, \omega_Y\right)$ is Pareto efficient envy free allocation, and on the other hand if $\displaystyle\frac{m}{\theta} &gt; \omega_X$, then $\displaystyle (x_u, y_u) = \left(\omega_X, m-\theta\omega_X\right)$ and $\displaystyle (x_v, y_v) = \left(0, m\right)$ is Pareto efficient envy free allocation.</p>

CC BY-SA 3.0
2017-02-16T13:41:09.997Z
What is the meaning of the requirement $\min_i a_i \geq \max_j b_j$.?
Erel Segal-Halevi
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2017-02-16T14:18:25.360Z
All the members of family U have higher MRS then all the members of family V.
Amit
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2017-02-16T14:54:22.573Z
I think for 2 families and linear preferences, this requirement can be removed. I have to work on the details  yet.
Erel Segal-Halevi
CC BY-SA 3.0
2017-02-16T15:09:37.690Z
I think it will be difficult to remove this requirement because we want allocation to be envy free. The conditions may not appear neat even if somehow it is relaxed. But this result holds for a larger class of utility functions. It will be a good idea to extend the result to include preferences of other type. For example: A version of it can also be proved for Cobb Douglas preferences.
Amit
CC BY-SA 3.0
2017-02-16T13:03:03.820Z
2017-02-16T13:03:03.820Z
Amit
<blockquote>
  <p><em>Right now I'm not sure about the equivalence of the relabeling, and therefore the usefulness of this anwer -- see comments below.</em></p>
</blockquote>

<p>This is the beginning of an answer and an attempt to demonstrate how strong the necessary assumptions would have to be to guarantee existence.</p>

<p>Let's transform the problem into one that's equivalent but a bit easier to work with. Instead of indexing over families, let's instead index over the agents (members of families). The key to this relabeling is understanding that families can be written as constraints: If agents $i$ and $j$ belong to the same family, then $x_i=x_j$ and $y_i = y_j$.</p>

<p>Now we're back in the standard environment with individual agents (not families) but with these familial constraints. Recall the proof of Varian's theorem, which you link in the question. It uses the existence of a competitive equilibrium from equal incomes. In this context, we would need the existence of a competitive equilibrium from equal incomes in which the familial constraints were also met. This is going to be very difficult to do. For instance, consider $i$ and $j$ are in a family, and 
$$
u_i=x_i + \varepsilon y_i \:\: \text{ and } \:\: u_j = \varepsilon x_j + y_j
$$
where $\varepsilon&gt;0$ is tiny. These preferences are monotonic and convex. Basically, one family member cares about $x$ and the other cares about $y$. If each of the two agents is purchasing $x$ and $y$ to maximize his or her utility, you would not expect $x_i^* = x_j^*$ or $y_i^* = y_j^*$ in the competitive equilibrium (see <strong>addendum</strong> at end).</p>

<p>This is why you certainly need some assumption on preference similarities within families (at least to use a version of Varian's proof). My sense is that if you give me any arbitrarily small difference in preferences between family members, I can construct an example around it where there exists no CEEI in which they choose the same allocation. And then, at the very least, you can't use Varian's proof.</p>

<p><strong>Two questions:</strong></p>

<ol>
<li>Do you agree that my reformulation of the problem is formally equivalent to yous?</li>
<li>Can you think of any assumption weaker than assuming preference homogeneity within the family that I can try to invalidate with a counter-example?</li>
</ol>

<p><strong>Addendum:</strong> Remember that in a competitive equilibrium, each agent's marginal rate of substitution (MRS) equals the price ratio. Here, my agents have constant and different MRS's, so there can exist no competitive equilibrium with a price ratio that equals both of their MRS's. If each agent has an MRS that varies, then perhaps they could happen to be equal at the equilibrium price ratio. So maybe you could get away with some notion of local homogeneity of familial preferences. But you need to have them be locally homogenous at the competitive equilibrium, which is exactly what you're trying to prove exists, so it would be a bit circular.</p>

<p><strong>Important note:</strong> As mentioned previously, I'm assuming that the only way to prove existence is how Varian did it, via CEEI. There may be other proof techniques that skirt these issues, but I suspect not.</p>

<p><strong>Beyond CEEI:</strong> As the OP points out in the comments, proving existence of PEEFs through CEEI as Varian does is somewhat restrictive. I do not have a lot to say about proving existence of PEEFs directly, but the following is readily apparent: For any allocation satisfying your Pareto efficiency condition (ignore envy-freeness for the moment), for any $i,j$ such that $x_i, x_j, y_i, y_j &gt; 0$,
$$MRS_i = MRS_j$$
If this weren't true, there would be a Pareto improvement. Competitive equilibrium essentially equates the MRS's through the price ratio, but you still need to equate these MRS's just to find a Pareto efficient allocation. I think the familial constraints will make this very difficult -- it's not hard to come up with an environment and familial constraints such that there exists no Pareto efficient equilibrium satisfying those constraints. In any case, this could be another partial step towards an answer: Forget about envy-freeness. First try to come up with an assumption on preferences (and maybe on familial constraints) that guarantees the existence of a Pareto efficient allocation that satisfies familial constraints. Then worry about envy.</p>

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2015-12-29T10:16:23.947Z
I share your intuition that a CEEI with familial constraints usually does not exist. But, there are many more PEEF allocations than CEEI allocations. In many cases, a CEEI is essentially unique while there are many different PEEFs. As an example (without familial constraints), take $u_1 = 2 x_1 + y_1$ and $u_2 = x_2 + 2 y_2$ and the total endowments are (4,4). The only CEEI allocation is [(4,0) ;  (0,4)]. However, the range of PEEF allocations goes from [(3,0); (1,4)]  to [(4,1); (0,3)]. CEEI is only a single point in that range.
Erel Segal-Halevi
CC BY-SA 3.0
2015-12-29T10:19:06.363Z
I found in Varian's original paper: http://www.sciencedirect.com/science/article/pii/0022053174900751 proofs of existence of PEEF allocations, which do not rely on CEEI and so are valid even in situations in which a CEEI does not exist (the preferences are not strictly convex). So far, I haven't managed to understand these proofs, but they may be relevant.
Erel Segal-Halevi
CC BY-SA 3.0
2015-12-29T16:23:43.543Z
@ErelSegal-Halevi In your example, any allocation in which both agents get strictly positive quantities of both goods is Pareto inefficient, no? I'm struggling to understand your ranges. More generally, though, I agree with you. I've added an additional section on proving PEEFs directly (without CEEI). I don't think you'll find it particularly satisfying, but it's about all that's obvious to me right now.
Shane
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2015-12-30T07:02:37.617Z
you are right. The ranges of PEEF allocations in my example are: $[(x_1,0),(4-x_1,4)]$ where $x_1\in [3,4]$, and $[(4,4-y_2),(0,y_2)]$  where $y_2\in [3,4]$
Erel Segal-Halevi
CC BY-SA 3.0
2015-12-31T07:37:19.630Z
Regarding your addition: what exactly do you mean by $MRS_i$? Is $i$ the index of a family or of a family-member? And how exactly do you calculate the MRS of a family with different members?
Erel Segal-Halevi
CC BY-SA 3.0
2015-12-31T13:15:15.430Z
@ErelSegal-Halevi It's the MRS of an individual member. I'm still working with the relabeling in which I index over individuals, not families. I don't see how you could define an MRS for a family with diverse preferences. Well, I do, but you'd have to use an arbitrary SWF to aggregate across the individuals within the family...
Shane
CC BY-SA 3.0
2015-12-31T15:30:04.390Z
I got confused because you used the indexes $i,j$ also for the allocations $x_i,x_j,y_i,y_j$, but the allocations are for families. So IIUC, your MRS condition is that, if agent $i$ is in family 1 and agent $j$ is in family 2, and if $x_1,x_2,y_1,y_2>0$, then $MRS_i=MRS_j$, right?
Erel Segal-Halevi
CC BY-SA 3.0
2015-12-31T17:49:46.400Z
Hmm... _All_ of my indexing is over individuals, even with $x_i,x_j,y_i,y_j$. So I was just saying: "take any two individuals, same family or not, ... " But now I'm wondering whether this relabeling actually does work, because I'm not directly accounting for the fact that if $i$ and $j$ are in the same family, and $x_i = x_j = 1$, then that only uses 1 unit of good $x$, not 2. Now I'm questioning the equivalence of the relabeling. Families are not just a constraint (in that people must share the same goods), they're also a benefit, in that goods are public/shared within the family.
Shane
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2015-12-31T17:54:39.707Z
If you want to continue over email, see my user profile.
Shane
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2015-12-28T21:15:18.193Z
2015-12-31T17:55:34.847Z
2015-12-31T17:55:34.847Z
Shane
<p>Suppose the preferences of all agents in all families are monotone and convex (the standard assumptions of consumer theory). </p>

<p>Then, a Pareto-efficient envy-free allocation always exists when there are two families. However, it might not exist when there ar three or more families.</p>

<p>Proofs and examples can be found in <a href="https://arxiv.org/abs/1811.06684" rel="nofollow noreferrer">this working paper</a>.</p>

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2018-11-19T10:11:04.160Z
2018-11-19T10:11:04.160Z
Erel Segal-Halevi
<p>The problem statement seems to imply that X and Y cannot be substitutes (an electrical device cannot be used as home furniture).</p>

<p>A Pareto-efficient envy-free allocation exists when:</p>

<p>For at least one agent, at least some goods have negative utility or are complements, and agents can chose to not consume. </p>

<p>Example:</p>

<ol>
<li>Agents A and B are in family F1. </li>
<li>The utility function of agent A is:</li>
</ol>

<p>Ua = -X1-X2-Y1-Y2</p>

<ol start="3">
<li>The utility function of agent B is:</li>
</ol>

<p>Ub = X1-X2+Y1-Y2 </p>

<ol start="4">
<li>Agents C and D are in family 2.</li>
<li>Agent C has a utility function:</li>
</ol>

<p>Uc = -X1-X2-Y1-Y2</p>

<ol start="6">
<li>Agent D has utility function:</li>
</ol>

<p>Ud = -X1+X2-Y1+Y2</p>

<p>Solution:</p>

<p>F1 prefers (X1, Y1) and agent A will chose to not consume any good.</p>

<p>F2 prefers (X2, Y2) and agent C chosen to not consume any good.</p>

<p>These are really semantic arguments and there is no meaningful equilibrium without assuming shared preferences.</p>

CC BY-SA 3.0
2016-01-05T16:14:30.203Z
Could you perhaps make your statements more precise? For instance, what are "negative complements"? And please offer at least a heuristic argument supporting the claims, if not a full proof, so that we can understand your reasoning.
Shane
CC BY-SA 3.0
2016-01-05T18:32:25.403Z
It looks from your utility functions like Agents A and B care about the other family's consumption? And I don't follow the idea of "choosing not to consume." Are you supposing that a member of family 1 can choose to consume anywhere in $[0,x_1]$?
Shane
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2016-01-06T04:04:12.537Z
Edited the answer. Youre correct on the second point. If agents are required to consume then the argument doesn't apply.
D J Sims
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2016-01-05T05:43:43.867Z
2016-01-06T04:02:40.297Z
2016-01-06T04:02:40.297Z
D J Sims
<p>Consider an exchange economy with two goods, e.g. home furniture (x) and electrical equipment (y). The interesting thing about these goods is that, when a family owns a bundle, all members of the family enjoy the same bundle (it is like a "club good" but only for the family).</p>

<p>There are two families. In each family, there are different members with different preferences over bundles. Assume that all preferences are monotonically-increasing and strictly convex.</p>

<p>An <em>allocation</em> is a pair of bundles, $(x_1,y_1)$ for family 1 and $(x_2,y_2)$ for family 2.</p>

<p>An allocation is called <em>envy-free</em> if:</p>

<ul>
<li>All members of family 1 believe that $(x_1,y_1)$ is at least as good as $(x_2,y_2)$;</li>
<li>All members of family 2 believe that $(x_2,y_2)$ is at least as good as $(x_1,y_1)$.</li>
</ul>

<p>An allocation is called <em>Pareto-efficient</em> if there is no other allocation of bundles to families such that all members of all families weakly prefer and at least one member of one family strictly prefers.</p>

<p><strong>Under what conditions does a Pareto-efficient envy-free allocation exist?</strong></p>

<p>If each family has a single member, then a Pareto-efficient envy-free allocation exists; this is a famous <a href="https://en.wikipedia.org/wiki/Varian%27s_theorems" rel="nofollow">theorem of Varian</a>. Has this theorem been generalized from individuals to families?</p>

CC BY-SA 3.0
2015-12-23T16:36:54.197Z
Very strong definition of envy-freeness. One would guess you would somehow aggregate the preferences first and then claim that there is no envy according to the aggregated preferences.
Giskard
CC BY-SA 3.0
2015-12-23T18:16:53.137Z
@denesp indeed, I thought about aggregating preferences, e.g. using a social welfare function. But, every selection of such a function would be arbitrary and not sufficiently motivated.
Erel Segal-Halevi
CC BY-SA 3.0
2015-12-27T19:23:17.593Z
@ErelSegal-Halevi Do you want us to also assume that the utility of each member of each family is weakly increasing in the quantity of $x$ and $y$ their family receives? If so, I have a very unsatisfying condition for you under which a Pareto-efficient, envy-free allocation exists: Suppose that, for each family, each member of that family has the same preferences... :P
Shane
CC BY-SA 3.0
2015-12-28T19:16:19.693Z
@Shane weak monotonicity seems like a reasonable assumption. If, in each family, all members have the same preferences, then each family is actually like a single agent, so we are back at the standard setting...
Erel Segal-Halevi
CC BY-SA 3.0
2015-12-28T20:06:47.737Z
What about the case where $x_1 = x_2$ and $y_1 = y_2$? Assuming weak monotonicity, then this must be Pareto and envy-free. From there, we could maybe make some small epsilon changes?
Kitsune Cavalry
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2015-12-28T20:25:33.823Z
@KitsuneCavalry While envy-free, that allocation isn't necessarily Pareto efficient.
Shane
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2015-12-28T20:26:51.557Z
You are correct.
Kitsune Cavalry
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2015-12-28T20:35:54.890Z
@ErelSegal-Halevi Indeed, that would put us back in the standard setting. That was kind of my point. My suspicion is that you will need to make a very strong assumption on preferences to guarantee existence. In fact, there may be no weaker assumption that guarantees existence than assuming the standard setting. It's a very good question -- I'm still thinking about it.
Shane
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2017-01-26T14:27:20.010Z
Interesting problem. Any progress?
Amit
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2017-01-26T18:44:57.897Z
@Amit Not much.  I only have a solution for two families and linear preferences.
Erel Segal-Halevi
CC BY-SA 3.0
2017-02-16T13:33:36.097Z
I have posted a proof of existence of Pareto efficient envy free allocation. It is also with linear preferences and two families.
Amit
<reference-request><pareto-efficiency><pure-exchange-economy><fairness>
Fair and efficient allocation of "family goods"
Erel Segal-Halevi
# Fair and efficient allocation of "family goods"

(Asked by: username_0 on 2015-12-23)

Consider an exchange economy with two goods, e.g. home furniture (x) and electrical equipment (y). The interesting thing about these goods is that, when a family owns a bundle, all members of the family enjoy the same bundle (it is like a "club good" but only for the family).

There are two families. In each family, there are different members with different preferences over bundles. Assume that all preferences are monotonically-increasing and strictly convex.

An *allocation* is a pair of bundles, $(x\_1,y\_1)$ for family 1 and $(x\_2,y\_2)$ for family 2.

An allocation is called *envy-free* if:

* All members of family 1 believe that $(x\_1,y\_1)$ is at least as good as $(x\_2,y\_2)$;
* All members of family 2 believe that $(x\_2,y\_2)$ is at least as good as $(x\_1,y\_1)$.

An allocation is called *Pareto-efficient* if there is no other allocation of bundles to families such that all members of all families weakly prefer and at least one member of one family strictly prefers.

**Under what conditions does a Pareto-efficient envy-free allocation exist?**

If each family has a single member, then a Pareto-efficient envy-free allocation exists; this is a famous [theorem of Varian](https://en.wikipedia.org/wiki/Varian%27s_theorems). Has this theorem been generalized from individuals to families?

**Tags:** reference-request, pareto-efficiency, pure-exchange-economy, fairness

> **Comments:**
> **username_1**: Very strong definition of envy-freeness. One would guess you would somehow aggregate the preferences first and then claim that there is no envy according to the aggregated preferences.  
> **username_0**: @denesp indeed, I thought about aggregating preferences, e.g. using a social welfare function. But, every selection of such a function would be arbitrary and not sufficiently motivated.  
> **username_2**: @username_0 Do you want us to also assume that the utility of each member of each family is weakly increasing in the quantity of $x$ and $y$ their family receives? If so, I have a very unsatisfying condition for you under which a Pareto-efficient, envy-free allocation exists: Suppose that, for each family, each member of that family has the same preferences... :P  
> **username_0**: @username_2 weak monotonicity seems like a reasonable assumption. If, in each family, all members have the same preferences, then each family is actually like a single agent, so we are back at the standard setting...  
> **username_3**: What about the case where $x_1 = x_2$ and $y_1 = y_2$? Assuming weak monotonicity, then this must be Pareto and envy-free. From there, we could maybe make some small epsilon changes?  
> **username_2**: @username_3 While envy-free, that allocation isn't necessarily Pareto efficient.  
> **username_3**: You are correct.  
> **username_2**: @username_0 Indeed, that would put us back in the standard setting. That was kind of my point. My suspicion is that you will need to make a very strong assumption on preferences to guarantee existence. In fact, there may be no weaker assumption that guarantees existence than assuming the standard setting. It's a very good question -- I'm still thinking about it.  
> **username_4**: Interesting problem. Any progress?  
> **username_0**: @username_4 Not much.  I only have a solution for two families and linear preferences.  
> **username_4**: I have posted a proof of existence of Pareto efficient envy free allocation. It is also with linear preferences and two families.  

---

## 4 answers

### Answer by username_4 on 2017-02-16. Score: 2

Suppose there are two families: Family U has $n\_u$ members, and family V has $n\_v$ members. Utility function of member $i$ of family U is:
\begin{eqnarray\*} u\_i(x\_u, y\_u) = a\_ix\_u + y\_u \end{eqnarray\*}
where all $a\_i$s are positive for all $i\in\{1,2,\ldots, n\_u\}$,

and utility function of member $j$ of family V is:
\begin{eqnarray\*} v\_j(x\_v, y\_v) = b\_jx\_v + y\_v \end{eqnarray\*}
where all $b\_j$s are positive for all $j\in\{1,2,\ldots, n\_v\}$.

Also, suppose $\min\_i a\_i \geq \max\_j b\_j$.

Suppose the total endowment vector of $X$ and $Y$ is $(\omega\_X, \omega\_Y)$.

For any $\theta \in [\max\_j b\_j, \min\_i a\_i]$, define $m := \displaystyle\frac{\theta\omega\_X}{2} + \frac{\omega\_Y}{2} $.

Check that if $\displaystyle\frac{m}{\theta} \leq \omega\_X$, then $\displaystyle (x\_u, y\_u) = \left(\frac{m}{\theta}, 0\right)$ and $\displaystyle (x\_v, y\_v) = \left(\omega\_X - \frac{m}{\theta}, \omega\_Y\right)$ is Pareto efficient envy free allocation, and on the other hand if $\displaystyle\frac{m}{\theta} > \omega\_X$, then $\displaystyle (x\_u, y\_u) = \left(\omega\_X, m-\theta\omega\_X\right)$ and $\displaystyle (x\_v, y\_v) = \left(0, m\right)$ is Pareto efficient envy free allocation.

> **Comments:**
> **username_0**: What is the meaning of the requirement $\min_i a_i \geq \max_j b_j$.?  
> **username_4**: All the members of family U have higher MRS then all the members of family V.  
> **username_0**: I think for 2 families and linear preferences, this requirement can be removed. I have to work on the details  yet.  
> **username_4**: I think it will be difficult to remove this requirement because we want allocation to be envy free. The conditions may not appear neat even if somehow it is relaxed. But this result holds for a larger class of utility functions. It will be a good idea to extend the result to include preferences of other type. For example: A version of it can also be proved for Cobb Douglas preferences.  

---

### Answer by username_2 on 2015-12-28. Score: 2

> *Right now I'm not sure about the equivalence of the relabeling, and therefore the usefulness of this anwer -- see comments below.*

This is the beginning of an answer and an attempt to demonstrate how strong the necessary assumptions would have to be to guarantee existence.

Let's transform the problem into one that's equivalent but a bit easier to work with. Instead of indexing over families, let's instead index over the agents (members of families). The key to this relabeling is understanding that families can be written as constraints: If agents $i$ and $j$ belong to the same family, then $x\_i=x\_j$ and $y\_i = y\_j$.

Now we're back in the standard environment with individual agents (not families) but with these familial constraints. Recall the proof of Varian's theorem, which you link in the question. It uses the existence of a competitive equilibrium from equal incomes. In this context, we would need the existence of a competitive equilibrium from equal incomes in which the familial constraints were also met. This is going to be very difficult to do. For instance, consider $i$ and $j$ are in a family, and
$$
u\_i=x\_i + \varepsilon y\_i \:\: \text{ and } \:\: u\_j = \varepsilon x\_j + y\_j
$$
where $\varepsilon>0$ is tiny. These preferences are monotonic and convex. Basically, one family member cares about $x$ and the other cares about $y$. If each of the two agents is purchasing $x$ and $y$ to maximize his or her utility, you would not expect $x\_i^\* = x\_j^\*$ or $y\_i^\* = y\_j^\*$ in the competitive equilibrium (see **addendum** at end).

This is why you certainly need some assumption on preference similarities within families (at least to use a version of Varian's proof). My sense is that if you give me any arbitrarily small difference in preferences between family members, I can construct an example around it where there exists no CEEI in which they choose the same allocation. And then, at the very least, you can't use Varian's proof.

**Two questions:**

1. Do you agree that my reformulation of the problem is formally equivalent to yous?
2. Can you think of any assumption weaker than assuming preference homogeneity within the family that I can try to invalidate with a counter-example?

**Addendum:** Remember that in a competitive equilibrium, each agent's marginal rate of substitution (MRS) equals the price ratio. Here, my agents have constant and different MRS's, so there can exist no competitive equilibrium with a price ratio that equals both of their MRS's. If each agent has an MRS that varies, then perhaps they could happen to be equal at the equilibrium price ratio. So maybe you could get away with some notion of local homogeneity of familial preferences. But you need to have them be locally homogenous at the competitive equilibrium, which is exactly what you're trying to prove exists, so it would be a bit circular.

**Important note:** As mentioned previously, I'm assuming that the only way to prove existence is how Varian did it, via CEEI. There may be other proof techniques that skirt these issues, but I suspect not.

**Beyond CEEI:** As the OP points out in the comments, proving existence of PEEFs through CEEI as Varian does is somewhat restrictive. I do not have a lot to say about proving existence of PEEFs directly, but the following is readily apparent: For any allocation satisfying your Pareto efficiency condition (ignore envy-freeness for the moment), for any $i,j$ such that $x\_i, x\_j, y\_i, y\_j > 0$,
$$MRS\_i = MRS\_j$$
If this weren't true, there would be a Pareto improvement. Competitive equilibrium essentially equates the MRS's through the price ratio, but you still need to equate these MRS's just to find a Pareto efficient allocation. I think the familial constraints will make this very difficult -- it's not hard to come up with an environment and familial constraints such that there exists no Pareto efficient equilibrium satisfying those constraints. In any case, this could be another partial step towards an answer: Forget about envy-freeness. First try to come up with an assumption on preferences (and maybe on familial constraints) that guarantees the existence of a Pareto efficient allocation that satisfies familial constraints. Then worry about envy.

> **Comments:**
> **username_0**: I share your intuition that a CEEI with familial constraints usually does not exist. But, there are many more PEEF allocations than CEEI allocations. In many cases, a CEEI is essentially unique while there are many different PEEFs. As an example (without familial constraints), take $u_1 = 2 x_1 + y_1$ and $u_2 = x_2 + 2 y_2$ and the total endowments are (4,4). The only CEEI allocation is [(4,0) ;  (0,4)]. However, the range of PEEF allocations goes from [(3,0); (1,4)]  to [(4,1); (0,3)]. CEEI is only a single point in that range.  
> **username_0**: I found in Varian's original paper: http://www.sciencedirect.com/science/article/pii/0022053174900751 proofs of existence of PEEF allocations, which do not rely on CEEI and so are valid even in situations in which a CEEI does not exist (the preferences are not strictly convex). So far, I haven't managed to understand these proofs, but they may be relevant.  
> **username_2**: @username_0 In your example, any allocation in which both agents get strictly positive quantities of both goods is Pareto inefficient, no? I'm struggling to understand your ranges. More generally, though, I agree with you. I've added an additional section on proving PEEFs directly (without CEEI). I don't think you'll find it particularly satisfying, but it's about all that's obvious to me right now.  
> **username_0**: you are right. The ranges of PEEF allocations in my example are: $[(x_1,0),(4-x_1,4)]$ where $x_1\in [3,4]$, and $[(4,4-y_2),(0,y_2)]$  where $y_2\in [3,4]$  
> **username_0**: Regarding your addition: what exactly do you mean by $MRS_i$? Is $i$ the index of a family or of a family-member? And how exactly do you calculate the MRS of a family with different members?  
> **username_2**: @username_0 It's the MRS of an individual member. I'm still working with the relabeling in which I index over individuals, not families. I don't see how you could define an MRS for a family with diverse preferences. Well, I do, but you'd have to use an arbitrary SWF to aggregate across the individuals within the family...  
> **username_0**: I got confused because you used the indexes $i,j$ also for the allocations $x_i,x_j,y_i,y_j$, but the allocations are for families. So IIUC, your MRS condition is that, if agent $i$ is in family 1 and agent $j$ is in family 2, and if $x_1,x_2,y_1,y_2>0$, then $MRS_i=MRS_j$, right?  
> **username_2**: Hmm... _All_ of my indexing is over individuals, even with $x_i,x_j,y_i,y_j$. So I was just saying: "take any two individuals, same family or not, ... " But now I'm wondering whether this relabeling actually does work, because I'm not directly accounting for the fact that if $i$ and $j$ are in the same family, and $x_i = x_j = 1$, then that only uses 1 unit of good $x$, not 2. Now I'm questioning the equivalence of the relabeling. Families are not just a constraint (in that people must share the same goods), they're also a benefit, in that goods are public/shared within the family.  
> **username_2**: If you want to continue over email, see my user profile.  

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### Answer by username_0 on 2018-11-19. Score: 1

Suppose the preferences of all agents in all families are monotone and convex (the standard assumptions of consumer theory).

Then, a Pareto-efficient envy-free allocation always exists when there are two families. However, it might not exist when there ar three or more families.

Proofs and examples can be found in [this working paper](https://arxiv.org/abs/1811.06684).


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### Answer by username_5 on 2016-01-05. Score: -2

The problem statement seems to imply that X and Y cannot be substitutes (an electrical device cannot be used as home furniture).

A Pareto-efficient envy-free allocation exists when:

For at least one agent, at least some goods have negative utility or are complements, and agents can chose to not consume.

Example:

1. Agents A and B are in family F1.
2. The utility function of agent A is:

Ua = -X1-X2-Y1-Y2

3. The utility function of agent B is:

Ub = X1-X2+Y1-Y2

4. Agents C and D are in family 2.
5. Agent C has a utility function:

Uc = -X1-X2-Y1-Y2

6. Agent D has utility function:

Ud = -X1+X2-Y1+Y2

Solution:

F1 prefers (X1, Y1) and agent A will chose to not consume any good.

F2 prefers (X2, Y2) and agent C chosen to not consume any good.

These are really semantic arguments and there is no meaningful equilibrium without assuming shared preferences.

> **Comments:**
> **username_2**: Could you perhaps make your statements more precise? For instance, what are "negative complements"? And please offer at least a heuristic argument supporting the claims, if not a full proof, so that we can understand your reasoning.  
> **username_2**: It looks from your utility functions like Agents A and B care about the other family's consumption? And I don't follow the idea of "choosing not to consume." Are you supposing that a member of family 1 can choose to consume anywhere in $[0,x_1]$?  
> **username_5**: Edited the answer. Youre correct on the second point. If agents are required to consume then the argument doesn't apply.  

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