Fair and efficient allocation of "family goods"

Question

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. Has this theorem been generalized from individuals to families?

Answer 1

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.

Answer 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.

Answer 3

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.

Answer 4

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.