Optimal Value Declaration in “Buy-Sell” Situations

Buy-Sell “Shotgun” clauses call for a partner who wishes to discontinue a partnership to declare a value for the business, and for the other partner to then buy her out or sell to her at this value. The resulting decision model for an expected utility maximizing individual, who is uncertain of the business' valuation by the partner, is analyzed. This model is also applicable to a “divide and choose” fair division method, as well as some historical tax/customs schemes, and is more general than comparable bidding/auctions models. After showing that the optimal declaration is always between the declarer's valuation and the fractile of the subjective distribution corresponding to the share owned, we show that the optimal declaration is always increasing in the valuation, and for a risk-neutral declarer also increasing in the share owned. We prove that the more risk-averse the declarer, the closer is the optimal declaration to the valuation, and the higher the probability that the partner who values the business more will end up owning it. Finally, we relate the optimal declaration to the degree of uncertainty concerning the partner's valuation.