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. Formal definition. There is a certain amount of money to divide, denoted by "formula_1" (=Estate or Endowment). There are "n" "claimants". Each claimant "i" has a "claim" denoted by "formula_2". Usually, formula_3, that is, the estate is insufficient to satisfy all the claims. The proportional rule says that each claimant "i" should receive formula_4, where "r" is a constant chosen  such that formula_5. In other words, each agent gets formula_6. Examples. Examples with two claimants: Examples with three claimants: Characterizations. The proportional rule has several characterizations. It is the only rule satisfying the following sets of axioms: Truncated-proportional rule. There is a variant called truncated-claims proportional rule, in which each claim larger than "E" is truncted to "E", and then the proportional rule is activated. That is, it equals formula_14, where formula_15. The results are the same for the two-claimant problems above, but for the three-claimant problems we get: Adjusted-proportional rule. The adjusted proportional rule  first gives, to each agent "i", their "minimal right", which is the amount not claimed by the other agents. Formally, formula_19. Note that formula_20 implies formula_21. Then, it revises the claim of agent "i" to formula_22, and the estate to formula_23. Note that that formula_24. Finally, it activates the truncated-claims proportional rule, that is, it returns formula_25, where formula_26. 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 formula_39 and  thus the outcome is equal to TPROP, for example, formula_40.