An Algebraic Theory of the Multiproduct Firm

The typical firm produces for sale a plural number of distinct product lines. This paper characterizes the composition of a firm’s optimal production vector as a function of cost and revenue function attributes. The approach taken applies mathematical group theory and revealed preference arguments to exploit controlled asymmetries in the production environment. Assuming some symmetry on the cost function, our central result shows that all optimal production vectors must satisfy a dominance relation on permutations of the firm’s revenue function. When the revenue function is linear in outputs, then the set of admissible output vectors has linear bounds up to transformations. If these transformations are also linear, then convex analysis can be applied to characterize the set of admissible solutions. When the group of symmetries decomposes into a direct product group with index K in N, then the characterization problem separates into κ problems of smaller dimension. The central result may be strengthened when the cost function is assumed to be quasiconvex.