Cost Allocation and Pricing in a Supply Chain - An Application for Aumann-Shapley Prices

We consider the problem of choosing among different distribution channels for combinations of different products, and how to price, or share the costs under the various alternatives, in an efficient and fair way. The problem could also be interpreted in terms of producing different products in a joint production process, and choosing between technologies with different costs and cost structures. More specifically, we consider technologies with combinations of fixed and variable costs. The variable costs are assumed to be linear and separable in the products, i.e. for a given technology and product type, we have constant marginal costs. The optimal choice of distribution channel / production technology will depend on the total production plan, or demand. That is, both the level of the total quantities demanded, and also the relative shares of the demands for the different products influence what is the best solution. In a marginal cost pricing regime, this would lead to prices changing according to production level and product mix. The price changes would be abrupt, depending on the boundaries between the areas where the different production technologies dominate. As a function of output, the marginal cost prices may show large increments or decrements depending on which production or distribution technology is the best for the given product mix. In this setting we will consider cost sharing rules using game theoretic concepts. More specifically, we consider Aumann-Shapley prices, which can be interpreted as a natural extension of average cost prices to the case of joint production of several goods. Throughout, we illustrate the pricing rules in a small example, with two products, and several technologies to produce or distribute them.