Proportional rule (bankruptcy)

The proportional rule is a division rule for solving bankruptcy problems.

According to this rule, each claimant should receive an amount proportional to their claim.

In the context of taxation, it corresponds to a proportional tax.

[1] There is a certain amount of money to divide, denoted by

(=Estate or Endowment).

Each claimant i has a claim denoted by

, that is, the estate is insufficient to satisfy all the claims.

The proportional rule says that each claimant i should receive

, where r is a constant chosen such that

In other words, each agent gets

Examples with two claimants: Examples with three claimants: The proportional rule has several characterizations.

It is the only rule satisfying the following sets of axioms: There is a variant called truncated-claims proportional rule, in which each claim larger than E is truncated to E, and then the proportional rule is activated.

That is, it equals

{\displaystyle PROP(c_{1}',\ldots ,c_{n}',E)}

:= min (

The results are the same for the two-claimant problems above, but for the three-claimant problems we get: The adjusted proportional rule[8] first gives, to each agent i, their minimal right, which is the amount not claimed by the other agents.

Formally,

Note that

implies

Then, it revises the claim of agent i to

, and the estate to

Note that that

Finally, it activates the truncated-claims proportional rule, that is, it returns

{\displaystyle TPROP(c_{1},\ldots ,c_{n},E')=PROP(c_{1}'',\ldots ,c_{n}'',E')}

:= min (

With two claimants, the revised claims are always equal, so the remainder is divided equally.

Examples: With three or more claimants, the revised claims may be different.

In all the above three-claimant examples, the minimal rights are

and thus the outcome is equal to TPROP, for example,

{\displaystyle APROP(100,200,300;200)=TPROP(100,200,300;200)=(20,40,40)}