Appendix to Chapter 7 – Computation and evaluation of optimal price schedules

Abstract

Introduction The optimal nonuniform price schedule P* (Q) generates more consumer surplus for a given revenue requirement than any other self-selecting tariff. In operational terms however, it is important to know just how much better it does over arguably simpler Ramsey and Fully Distributed Cost pricing rules. To answer this question, we need to be able to compute these alternative pricing rules and evaluate the consumer surplus and revenues generated for a variety of assumptions about demand and cost conditions. These computations represent a potentially difficult numerical problem where the willingness to pay reflects differences in tastes across a population of individuals. The consumer surplus and revenue integrals have themselves to be integrated over consumer types. In the case of the optimal nonuniform pricing rule, there are functions to be twice integrated that involve an optimal price schedule P*(Q) that is only implicitly defined. This implicit definition of the price schedule P*(Q) arises from the fact that the optimal price is derived from the first order conditions specific to a given customer type θ. The optimal price schedule is then defined in terms of the maximum quantity consumer type 6 will purchase. In other words, given a willingness to pay function p(Q,θ) decreasing in Q and increasing in consumer type θ, the maximum type θ will consume is given as the Q for which P = p(Q,θ) . Thus, provided the self-selection constraint is satisfied it is possible to define (implicitly) the unique function P = P*(Q) for all consumer types θ.