William Baumol's Fairness Assumption

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Definitions

For convenience, the below definitions assume a world with two individuals, A and B, and two goods, X and Y

• Envy:

If A prefers to his own bundle, the bundle of goods held by B, A is said to envy B.

• Fairness:

The distribution of goods X and Y is fair if no envy is involved.

• Superfairness:

If the bundles held by both A and B can be reduced without giving rise to envy, then the distribution is superfair.

• Fairness Boundary:

The borderline between distributions which A considers to be unfair, and those that A considers to be more than fair.

Who cares about fairness?

• If preferences are strictly selfish, then why does fairness matter?
  

Selfish people are selfishly worried about others treating them unfairly. As a result, it is often times these innately selfish individuals who insist on rules of fairness. Fairness rules are accepted by selfish individuals as an insurance arrangement to make sure that they will not be mistreated. They pay for this insurance by guaranteeing others that they too will not be mistreated.

The Edgeworth Box

building and using an Edgeworth Box to map fair distributions

Superfairness

The concept of envy requires that each party evaluate the bundles received by others based on their own preferences. The bundles that occur at the intersection of an individual, A's, indifference curves and their mirror image (A's indifference curves that depict the valuation of bundles received by another individual, B) are said to be fair, as A is clearly indifferent between either bundle at that point. By tracing out an infinite number of these mirrored indifference curves, we can map A's fairness boundary based on these intersections. In exactly the same way we can construct the fairness boundary for B based on B's indifference curves and their mirror images.

By looking at the fairness boundaries using an edgeworth box it becomes very easy to see how superfair outcomes can arise. By comparing bundles that lie on A's fairness boundary with those that lie above it, it is easy to determine that while the bundle that lies on the fairness boundary is marginally fair to A, the bundle that lies above the fairness boundary is more than fair. Likewise, B would consider any point below his fairness boundary to be more than fair. Therefore, any point in the region that lies above A's fairness boundary and below B's represents a superfair distribution.

Multistage Superfairness

Although all points within a superfair region are in fact superfair, there exist some that are more superfair than others. If we consider a point tangent to A's fairness boundary on the highest of A's indifference curves within the region of superfair distributions, then any point just below and to the left of that point is superfair. This is also true for points just above and to the right of the point of tangency of B's fairness boundary with B's indifference curve. However, the first point offers A much more than he deems necessary to give him a fair share, while offering virtually no surplus to B. Clearly, this point represents a distribution that is “superfairer” to A. By choosing a point that is midway between these two “superfairer” points, the resulting distribution is “more superfair” to both A and B.

While it is very unlikely that their exists any one distribution which is the fairest of all, we can narrow the range of possible choices by eliminating those distributions which are in one some way unfair. This can be done by creating another Edgeworth Box on the interior of the region of superfairness, with the top right corner at A's most desired superfair distribution, and the bottom left corner at B's. By tracing out the superfair regions within this interior box, we can narrow the range of superfair solutions by eliminating those that are “less superfair.” This can be done again and again (hence the term multistage superfairness) in order to continually reduce the possible superfair distributions.