Efficiency and fairness are two major goals of welfare economics.
The goal was first defined by David Schmeidler and Menahem Yaari.
We assume that each agent has a preference-relation on the set of all bundles of commodities.
Equivalently, each preference relation can be represented by a continuous utility function.
[2]: 79 Theorem 1 (Varian):[2]: 68 If the preferences of all agents are convex and strongly monotone, then PEEF allocations exist.
Assume that all resources in an economy are divided equally between the agents.
Since the preferences are convex, the Arrow–Debreu model implies that a competitive equilibrium exists.
All examples involve an economy with two goods, x and y, and two agents, Alice and Bob.
If Alice receives at least 3 units of x, then her utility is 6 and she does not envy Bob.
Similarly, if Bob receives at least 3 units of y, he does not envy Alice.
While there are other PEEF allocations, e.g. [(1.5,1);(2.5,1)], all have the same utility vector of (1,1), since it is not possible to give both agents more than 1.
There are several sufficient conditions that are related to the shape of the set of allocations corresponding to a specific efficient utility profile.
To see that it implies the condition of theorem 2, suppose there are two different allocations x,y with the same utility profile u.
Theorem 3 (Svensson):[4] If all agents' preferences are strongly monotone, and for every PE utility-profile u, the set A(u) is convex, then PEEF allocations exist.
Theorem 4 (Diamantaras):[5] If all agents' preferences are strongly monotone, and for every PE utility-profile u, the set A(u) is a contractible space (can be continuously shrunk to a point within that space), then PEEF allocations exist.
[6] Note: Every convex set is contractible, so Diamantaras' theorem is more general than the previous three.
Svensson proved another sufficient condition for the existence of PEEF allocations.
Moreover, all utility functions are continuously differentiable in the interior of the consumption space.
Suppose we create, for each agent, k copies with identical preferences.
Lemma:[7]: 528 An allocation is sigma-optimal, if-and-only-if it is a competitive equilibrium.
PEEF allocations might fail to exist even when all preferences are convex, if there is production and the technology has increasing-marginal-returns.
Proposition 6 (Vohra):[8] There exist economies in which all preferences are continuous strongly-monotone and convex, the only source of non-convexity in the technology is due to fixed costs, and there exists no PEEF allocation.
Thus, the presence of increasing returns introduces a fundamental conflict between efficiency and fairness.
Obviously, every EF allocation is EEF, since we can take Yi to be X for all i. Theorem 7 (Vohra):[8] Suppose all agents' preferences are strongly monotone, and represented by continuous utility functions.
PEEF allocations might fail to exist even without production, when the preferences are non-convex.
Proposition 8 (Maniquet):[9] There exist 2-good 3-agent division economies with strictly monotonic, continuous and even differentiable preferences, where there is domination at every Pareto efficient allocation.
For two agents, the adjusted winner procedure is a simple procedure that finds a PEEF allocation with two additional properties: the allocation is also equitable, and at most a single good is shared between the two agents.
For three or more agents with linear utilities, any Nash-optimal allocation is PEEF.
The fact that any Nash-optimal allocation is PEEF is true even in the more general setting of fair cake-cutting.
Therefore, the Nash-optimal rule gives each such piece Z to an agent j for which this expression is largest: