Marginal cost price rule for homogeneous cost functions

Mirman and Tauman (1982) show that axioms of cost sharing, additivity, rescaling invariance, monotonicity, and consistency uniquely determine a price rule on the class of continuously differentiable cost problems as the Aumann-Shapley price mechanism. Here we prove that standard versions of these axioms determine uniquely the marginal cost price rule on the class of homogeneous and convex cost functions, which are, in addition, continuously differentiable. This result persists even if the cost functons are not required to be convex.