Characterizing Distributive Principles in Hierarchies with an Application to Wage Design

We study the allocation of surplus in hierarchical organizations when marginalism, hierarchical marginalism, and egalitarianism conflict. We represent these principles by the Shapley, Permission, and Equal Division values and introduce their convex combinations, which we call Egalitarian Shapley-Permission values. We provide an axiomatic characterization of this family by first characterizing a broader class whose weights may vary across coalitions, and then identify the conditions under which these weights must be common across unanimity components. Applying the family to wage design in a hierarchical firm, we show that the manager-to-worker wage ratio is bounded above by the manager's span of control, and that a manager is paid at least as much as each worker for every degree of input substitutability if and only if the Shapley component receives zero weight. In large organizations with nearly complementary worker inputs, the managerial premium depends jointly on span of control and input substitutability.