s (weak) increasing returns to scale: Finally, cost complementarities imply that the multiproduct cost function exhibits the natural monopoly property,15 also referred to as subadditivity: C(Q)+C(Q")>C(Q'+Q") This means that if C(Q) is the cost function of a network firm, costs are lower than if the output were divided between two (or more) network firms using the same technology (Baumol et al. 1982). Because incremental costs sum to less than total cost, when there are cost complementarities, there is a temptation for regulators to think of dividing up total costs into incremental costs plus some cost allocation of the residual cost: Residual cost = However, any administrative allocation of residual cost is necessarily arbitrary. Suppose that all costs can be fully attributable to each service except for a joint and common fixed cost F. Thus the cost function would be C(Q)=F+Zd(Q)) 15 The concept of natural monopoly is generally attributed to John Stuart Mill (1961; originally published in 1848), who emphasizes the problem of wasteful duplication of transmission facilities that can occur in utility services. The connection between natural monopoly and regulation is developed by Leon Walras (1936) with reference to the construction and operation of railroads. The Design and Costs of Networks Regulators have long applied fully distributed cost (FDC) pricing rules to divide the fixed cost among a set of services. Typically, in network industries, regulatory commissions categorize services by customer classes and distribute joint costs "on the basis of some common physical measure of utilization, such as minutes, circuit- miles, message-minute-miles, gross ton-miles, MCF, or kwh" (Kahn 1971, p. 150). Among the rules of thumb used by regulators are the relative output rule, the relative attributable cost rule, and the relative revenue rule. The arbitrariness of such rules is easily demonstrated. By choosing different measures of output, regulators can easily generate substantially different cost allocations. Moreover