Partnership's profit sharing: Linear and nonlinear contracts

Suppose that one party proposes to another a contract for sharing an uncertain profit which maximizes the former's expected utility, with respect to its beliefs, subject to a constraint on the latter's expected utility, with respect to the latter's beliefs. It turns out that the optimal contract, which we find, can be nonmonotone, as well as nonlinear, in the realized profit. To avoid the implausible lack of monotonicity, we formulate and solve a model constrained to have monotone increasing profits for both partners. If beliefs are identical, the (unconstrained) contract is shown to be monotone, and under certain conditions, linear. That might explain one famous contract from the history of jazz. If the other party can be assumed risk neutral, the linear contract reduces to the former receiving a constant amount, and the latter the residual net profit, as in the case of another famous contract from the history of jazz. Since in the type of partnerships, we have in mind the partners are always motivated to exert high effort due to other factors like reputation, our setting has no moral hazard or adverse selection, and the partnerships do not involve a large initial investment.