On surplus-sharing in partnerships

For investment or professional service partnerships (in general, for partnerships where measures of the partners' contributions are available), we consider a family of partnership agreements commonly used in real life. They allocate a fixed fraction of the surplus equally and the remains, proportional to contributions; and they allow this fraction to depend on whether the surplus is positive or negative. We analyze the implications of such partnership agreements on (i) whether the partnership forms in the first place, and if it does, (ii) the partners' contributions as well as (iii) their welfare. We then inquire which partnership agreements are productively efficient (i.e. maximizes the partners' total contributions) and which are socially efficient, (i.e. maximizes the partners' social welfare as formulated by the two seminal measures of egalitarianism and utilitarianism).