Price discrimination with correlated demands.

• Author: Eckel, Catherine C.

1. Introduction

In this paper we analyze customer-class price discrimination in the face of uncertain demands.(1) More precisely, we explore the pricing decision of a multiproduct monopoly facing random, correlated demands. We consider only a uniform price for each customer class; multi-part pricing is not investigated. We are interested in the extent to which uniform customer-class prices can exploit variations in demand characteristics.

If cost is a convex function of aggregate demand, then the expected cost of production is greater than the cost of the expectation of the rate of production.(2) Since the deviation of expected cost from the cost of the expectation is (to a second order approximation) proportional to the variance of demand, the monopolist can exploit the functional dependence of variance on prices to reduce expected costs.(3) This leads to optimal prices that discriminate on the basis of covariances among classes. In other words, prices are set to manipulate the variance of aggregate demand, much in the same way that investors can adjust the portfolio weights attached to individual securities in order to reduce the variance of their portfolio.

We observe correlation among consumers or consumer groups in many markets. Many situations in which price discrimination is practiced can be reinterpreted as covariance-based pricing. Think of demand for a facility by a church and a child-care center. The church has positive demand on weekends and the child care center has positive demand during the week. Observed demands are negatively correlated. Costs of providing services can be reduced by sharing the facility and pricing it on the basis of covariances.

A different example with a more explicit stochastic element, and so more directly related to our analysis, is the market for telephone calls. Businesses make calls predominantly nine-to-five, Monday-through-Friday. Individuals are more likely to call on evenings and weekends. The electricity market exhibits similar characteristics. In both we observe that demands are correlated, and that overall demand can be smoothed by covariance-based pricing. This pricing is similar in effect to peak-load pricing, but does not require time-of-day monitoring of demand.

In airline travel, price discrimination on the basis of customer group rather than time-of-day has a clear covariance component. The requirement that a passenger stay over a Saturday night to purchase a ticket at a discount price segments the market into business and non-business travellers. Non-business travellers have a lower covariance with peak demand than business travellers.

Finally, discounts to particular groups such as students or retired persons may also incorporate a covariance component. Retired persons are more likely to have off-peak demand at restaurants, hotels, or resorts. Students are more likely to have off-peak demand at movie theaters or lunch counters. Discounts to them may reflect their negative covariance with demand, in addition to the usual elasticity argument.

2. Stochastic Demands

   Consider a multiproduct monopoly facing several demand classes with random, correlated demands. Suppose that demand of each class is subject to a random disturbance that may be correlated with disturbances to other classes. Randomness may arise from fluctuations in income, temperature or other factors.(4) Demand of the ith class is thus:

   qi([p.sub.i], [u.sub.i] [is greater than or equal to] 0, i = 1,...,n. (2) where the [u.sub.i] are random variables with a nondiagonal covariance matrix. The function [q.sub.i] is twice continuously differentiable in [p.sub.i], with [q.sub.i]/ [p.sub.i] 0. This not only allows us to associate positive shocks with increases in demand, but also ensures the existence of the joint probability density function of the demands(5). The demands are also assumed to be non-negative,(6) so that, for all realization of [u.sub.i],

   [q.sub.i]([p.sub.i], [u.sub.i]) [is greater than or equal to] 0, i = 1, . . ., n. (2)

   We assume the first two moments of the demand functions are finite and given by:

   [Mathematical Expression Omitted]

   Given (3), (4) and (5), the mean and variance of aggregate demand are:

   where p = [[p.sub.], . . . [p.sub.n]] is the...