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]

Formal definition

There is a certain amount of money to divide, denoted by (=Estate or Endowment). There are n claimants. Each claimant i has a claim denoted by . Usually, , 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

Examples with two claimants:

  • . That is: if the estate is worth 100 and the claims are 60 and 90, then , so the first claimant gets 40 and the second claimant gets 60.
  • , and similarly .

Examples with three claimants:

  • .
  • .
  • .

Characterizations

The proportional rule has several characterizations. It is the only rule satisfying the following sets of axioms:

  • Self-duality and composition-up;[2]
  • Self-duality and composition-down;
  • No advantageous transfer;[3][4][5]
  • Resource linearity;[5]
  • No advantageous merging and no advantageous splitting.[5][6][7]

Truncated-proportional rule

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 , where . The results are the same for the two-claimant problems above, but for the three-claimant problems we get:

  • , since all claims are truncated to 100;
  • , since the claims vector is truncated to (100,200,200).
  • , since here the claims are not truncated.

Adjusted-proportional rule

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 , where .

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

  • . The minimal rights are . The remaining claims are and the remaining estate is ; it is divided equally among the claimants.
  • . The minimal rights are . The remaining claims are and the remaining estate is .
  • . The minimal rights are . The remaining claims are and the remaining estate is .

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, .

See also

References

  1. William, Thomson (2003-07-01). "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey". Mathematical Social Sciences. 45 (3): 249–297. doi:10.1016/S0165-4896(02)00070-7. ISSN 0165-4896.
  2. Young, H. P (1988-04-01). "Distributive justice in taxation". Journal of Economic Theory. 44 (2): 321–335. doi:10.1016/0022-0531(88)90007-5. ISSN 0022-0531.
  3. Moulin, Hervé (1985). "Egalitarianism and Utilitarianism in Quasi-Linear Bargaining". Econometrica. 53 (1): 49–67. doi:10.2307/1911723. ISSN 0012-9682. JSTOR 1911723.
  4. Moulin, Hervé (1985-06-01). "The separability axiom and equal-sharing methods". Journal of Economic Theory. 36 (1): 120–148. doi:10.1016/0022-0531(85)90082-1. ISSN 0022-0531.
  5. 1 2 3 Chun, Youngsub (1988-06-01). "The proportional solution for rights problems". Mathematical Social Sciences. 15 (3): 231–246. doi:10.1016/0165-4896(88)90009-1. ISSN 0165-4896.
  6. O'Neill, Barry (1982-06-01). "A problem of rights arbitration from the Talmud". Mathematical Social Sciences. 2 (4): 345–371. doi:10.1016/0165-4896(82)90029-4. ISSN 0165-4896.
  7. de Frutos, M. Angeles (1999-09-01). "Coalitional manipulations in a bankruptcy problem". Review of Economic Design. 4 (3): 255–272. doi:10.1007/s100580050037. hdl:10016/4282. ISSN 1434-4750. S2CID 195240195.
  8. Curiel, I. J.; Maschler, M.; Tijs, S. H. (1987-09-01). "Bankruptcy games". Zeitschrift für Operations Research. 31 (5): A143–A159. doi:10.1007/BF02109593. ISSN 1432-5217. S2CID 206811949.
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