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 ''$E$'' (=Estate or Endowment). There are ''n'' ''claimants''. Each claimant ''i'' has a ''claim'' denoted by ''$c_i$''. Usually, $\sum_^n c_i > E$, that is, the estate is insufficient to satisfy all the claims.

The proportional rule says that each claimant ''i'' should receive $r \cdot c_i$, where ''r'' is a constant chosen such that $\sum_^n r\cdot c_i = E$. In other words, each agent gets $\frac\cdot E$.

Examples

Examples with two claimants:
* $PROP(60,90; 100) = (40,60)$. That is: if the estate is worth 100 and the claims are 60 and 90, then $r = 2/3$, so the first claimant gets 40 and the second claimant gets 60.
* $PROP(50,100; 100) = (33.333,66.667)$, and similarly $PROP(40,80; 100) = (33.333,66.667)$.

Examples with three claimants:
* $PROP(100,200,300; 100) = (16.667, 33.333, 50)$.
* $PROP(100,200,300; 200) = (33.333, 66.667, 100)$.
* $PROP(100,200,300; 300) = (50, 100, 150)$.

Characterizations

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

* Self-duality and composition-up;
* Self-duality and composition-down;
* No advantageous transfer;
* Resource linearity;
*No advantageous merging and no advantageous splitting.

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 $PROP(c_1',\ldots,c_n',E)$, where $c'_i := \min(c_i, E)$. The results are the same for the two-claimant problems above, but for the three-claimant problems we get:

* $TPROP(100,200,300; 100) = (33.333, 33.333, 33.333)$, since all claims are truncated to 100;
* $TPROP(100,200,300; 200) = (40, 80, 80)$, since the claims vector is truncated to (100,200,200).
* $TPROP(100,200,300; 300) = (50, 100, 150)$, since here the claims are not truncated.

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, $m_i := \max(0, E-\sum_ c_j)$. Note that $\sum_^n c_i \geq E$ implies $m_i \leq c_i$.

Then, it revises the claim of agent ''i'' to $c'_i := c_i - m_i$, and the estate to $E' := E - \sum_i m_i$. Note that that $E' \geq 0$.

Finally, it activates the truncated-claims proportional rule, that is, it returns $TPROP(c_1,\ldots,c_n,E') = PROP(c_1'',\ldots,c_n'',E')$, where $c''_i := \min(c'_i, E')$.

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

* $APROP(60,90; 100) = (35,65)$. The minimal rights are $(m_1,m_2) = (10,40)$. The remaining claims are $(c_1',c_2') = (50,50)$ and the remaining estate is $E'=50$; it is divided equally among the claimants.
* $APROP(50,100; 100) = (25,75)$. The minimal rights are $(m_1,m_2) = (0,50)$. The remaining claims are $(c_1',c_2') = (50,50)$ and the remaining estate is $E'=50$.
* $APROP(40,80; 100) = (30,70)$. The minimal rights are $(m_1,m_2) = (20,60)$. The remaining claims are $(c_1',c_2') = (20,20)$ and the remaining estate is $E'=20$.

With three or more claimants, the revised claims may be different. In all the above three-claimant examples, the minimal rights are $(0,0,0)$ and thus the outcome is equal to TPROP, for example, $APROP(100,200,300; 200) = TPROP(100,200,300; 200) = (20, 40, 40)$.

See also

* Proportional division
* Proportional representation

References

{{Reflist

Bankruptcy theory