A company is owned by two partners: partner $A$ holds a fraction $a$ and partner $B$ holds a fraction $b$ (with $a+b=1$). Partner $A$ wants to break the partnership. A common procedure for this is that $A$ should suggest a price $P$ for the entire company and let $B$ choose either of two options:
- Buy $A$'s share for $aP$, or -
- Sell his share to $A$ for $bP$.
This procedure incentivizes $A$ to suggest a fair price. Moreover, if $B$ thinks that $A$'s price is unfair, he can profit from this - if he thinks that the price is too low he can opt to buy and if he thinks that the price is too high he can opt to sell. The advantage of this procedure is that it guarantees fairness without requiring for an external arbitrator.
A possible problem with this procedure is that it requires $B$ to have enough cash for buying $A$'s share. If $B$ doesn't have enough cash, and cannot get credit, he might have to select the "sell" option even when the price is low. Moreover, if $A$ knows that $B$ is short in cash, he might exploit this fact and deliberately suggest a low price in order to force $B$ to sell his share.
MY QUESTION IS: is there an improved division mechanism that handles this problem?