Marginal Cost Pricing of Recursive Lumpy Investments

Abstract

Problems involving optimal decentralization of decision-making in the presence of nonconvexities have a long, frustrating history. Solutions generally involve evaluation of a surplus term and economists have been unable to discover ways of revealing that term without the collection of hypothetical information. However, certain non-convex problems have additional structure which can be exploited in order partially to circumvent this difficulty. We will focus here on such a problem involving capacity construction. Suppose that we restrict attention to problems involving a single type of capacity (for example, electric power, telephone channels, highway through-put, bridges or dams), and for which the non-convexities are involved in the construction of capacity but not in the production of services from capacity. This class of problems seems fairly broad in scope and has a long history generally associated with the French economists (see Nelson (1964) for a survey). We begin by summarizing the French approach as discussed by Boiteux (Chapter 3 in Nelson, op. cit.). Boiteux discusses the non-convex replacement problem, using railway cars as the paradigm. First, he attempts to apply marginal analysis to the problem of pricing capacity in this situation. Under the usual rules of marginal cost pricing, the price of a ticket should be zero as long as the last car is not full, but should jump up to the marginal cost per person handled of adding an additional car as soon as it is full. Now, if at the zero price demand exceeds the capacity of this car, while at the second it falls short, there is clearly no equilibrium outcome; marginal analysis fails to tell us what to do. Considered in isolation, we are left with the classical non-convexity dilemma: to decide whether or not the extra car is justified, we must compare the cost against the full consumer surplus generated. However, Boiteux notices that the problem should not be considered in isolation.