Constrained stochastic cost allocation

This paper presents a model of a multi-divisional firm to share the joint yet uncertain and fixed cost of running a central operational unit. A firm aims at allocating this cost ex ante, subject to constraints imposed by the asymmetric and limited liabilities of the different divisions. We study solutions that are made up of a vector of ex ante payments which are allocated in absence of costs, and a remaining solution that is contingent on the cost. Under a mild continuity condition we find different classes of egalitarian solutions. The class of egalitarian proportional solutions is characterized by dependency on the disutility of the total cost instead of details of the distribution. In this class, there is a unique proportional solution which systematically minimizes the maximal transfer. A fundamentally different egalitarian solution is the stochastic egalitarian constrained equal costs solution. It is characterized using a local symmetry property which states that incremental costs should be distributed equally among those divisions with sufficient liability. This egalitarian solution has a smaller largest transfer than any egalitarian proportional solution. We conclude by showing how our results generalize when egalitarianism is replaced by a more general fairness property.