Price discrimination with correlated demands.

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.

II. 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. In addition, demand is assumed to be monotonically increasing and at least once differentiable in the disturbance. That is, if [q.sub.i][p.sub.i], [u.sub.i]) is the value of [q.sub.i] given a realization [u.sub.i], then [q.sub.i]/ [u.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 n-dimensional vector of prices.

III. Pricing with Stochastic Cost Effects

To abstract from the issue of capacity choice, assume that the firm can instantaneously and costlessly vary its capital stock to match any realization of aggregate demand. The cost function c(Q) exhibits positive marginal costs, c' > 0, and may be either convex or concave, c" = [ .sup.]c/ [Q.sup.2] >< 0. Strict concavity Corresponds to the traditional notion of natural monopoly.

Jensen's inequality implies that:

[Mathematical Expression Omitted]

that is, the expectation of aggregate operating costs is greater or less than the cost of the expectation of aggregate production depending upon whether the cost function is concave or convex.

Suppose that the risk neutral firm sets prices prior to the realization of uncertain demands in order to maximize ex ante social welfare, which is the expected sum of consumer and producer surpluses:

[Mathematical Expression Omitted]

where u = ([u.sub.i] . . . [u.sub.n]) is the vector of disturbances.(7) The optimal pricing rule is:(8)

[Mathematical Expression Omitted]

where:

Markup depends on the covariance between [q.sub.i]/ [p.sub.i] and c', and on [epsilon.sub.i], the elasticity of expected demand by the ith class with respect to [p.sub.i]. Elasticity enters because the monopolist can only lower aggregate variance indirectly, by manipulating prices. The covariance determines the direction of the price manipulation. If this covariance is positive, for example, the firm will want to stimulate demand of the ith class by lowering its price. The familiar marginal cost pricing rule is optimal only if the covariance is zero, which will occur if the cost function is linear.

This covariance is difficult to interpret. If cov( [q.sub.i]/ [p.sub.i], c') < 0, for example, this means that demand tends to be flatter at levels associated with higher marginal cost. If c (Q) is convex, and if the derivative of demand is larger in magnitude the larger is [u.sub.i], then the sign of cov ( [q.sub.i]/ [p.sub.i] c') is opposite the sign of cov([u.sub.i], Q) and thus of cov([q.sub.i], Q). This relationship is examined in more detail below, and is illustrated by the example in Section 4, where uncertainty is multiplicative.(9) Returning to the pricing rule, we can see that the monopolist will typically want to increase price (over expected marginal cost) when demand of class i covaries positively with Q and when cost is convex. By doing so, it reduces the variance of aggregate demand, and so E(c).

The advantage of reducing variance when cost is convex can be seen in Figure 1. Suppose that Q = E([Q.sup.*]), that [Q.sup.*] = [Q.sup.*.sub.1] with probability .5 and [Q.sup.*] = [Q.sup.*.sub.2] with probability .5. The expected cost of production is E[c([Q.sup.*]) > c(Q). Suppose the monopolist manipulates prices to lower variance: for example he effects a mean-preserving spread such that Q = [Q.sub.1] < [Q.sup.*.sub.1] with probability .5 and Q = [Q.sub.2] < [Q.sup.*.sub.2] with probability .5. (Of course, price would affect Q as well, but we ignore this effect to focus on the importance of a change in variance.) Clearly, his expected cost is lowered to E[c(Q)] < E[c (Q.sup.*)]. The opposite is true for concave cost functions, where there are traditional economies of scale.

These "stochastic cost effects" are more obvious in a special case of the general pricing rule (10). For any well-behaved c(Q), the deviation of the expected cost of production from the cost of expected production is proportionate, to a second-order approximation, to the variance of aggregate demand:

[Mathematical Expression Omitted]

Ignoring products of second and third-order terms, with this approximation the pricing rule (10) becomes:

[Mathematical Expression Omitted]

This markup has an interesting interpretation. Notice that:

[Mathematical Expression Omitted]

By definition,

[Mathematical Expression Omitted]

Hence,

[Mathematical Expression Omitted]

Similarly, since

[Mathematical Expression Omitted]

it follows that:

[Mathematical Expression Omitted]

Substituting (16) and (18) into (14) yields:

[Mathematical Expression Omitted]

We now define the beta coefficient of the ith class as [beta.sub.i], the ratio of the covariance of the slope of the ith demand with aggregate demand to the variance of aggregate demand,

[Mathematical Expression Omitted]

[beta.sub.i] measures the change in the variance of aggregate demand from a change in price of the ith class. It is akin to the beta coefficient of the Capital Asset Pricing Model (CAPM), and possesses much the same intuition: just as an investor can manipulate portfolio shares of alternative assets to determine the variance of his portfolio, so too can the monopolist manipulate the mean and variance of customer-class demands in order to smooth aggregate demand. The difference is that the monopolist cannot directly choose the contribution of each class to aggregate demand, but can only influence variance indirectly, through price.

The numerator of [beta.sub.i] is the covariance of the slope of class demand with aggregate demand. Since Q = [EPSILON.sub.i] [q.sub.i] cov( [q.sub.i]/ [p.sub.i]Q can be decomposed into cov( [q.sub.i] / [p.sub.i], [q.sub.i] + cov( [q.sub.i]/ [p.sub.i], [q.sub.i] + cov( [q.sub.i]/ [p.sub.i], [EPSILON.sub.j] [is not equal to] [q.sub.j]. If the distribution of demand "spreads out", a high realization of [u.sub.i] causes demand to flatten, so that ( [q.sub.i]/ [p.sub.i], [q.sub.i]) is negative.(10) The second component can be positive or negative, and its sign will depend upon the [sigma.sub.ij]'s. If cov( [q.sub.i]/ [p.sub.i], [EPSILON.sub.j] [is not equal to]i [q.sub.j] < 0 [is not equal to] j, [beta.sub.i] is unambiguously negative. If [sigma.sub.ij] < 0, i [is not equal to] j, [beta.sub.i] generally will be smaller in magnitude and may even be positive.

Substituting [beta.sub.i] into pricing rule (13) gives:

[Mathematical Expression Omitted]

where:

[Mathematical Expression Omitted]

Note that the sign of the weight [delta.sub.i] attached to [beta..sub.i] varies directly with the sign of the second derivative of the cost function, and the sign of [beta.sub.i] is typically opposite the sign of cov([q.sub.i], Q). Thus, if cost is convex, the monopolist sets prices higher for classes that have a strong positive covariance with aggregate demand at the margin, thereby hedging against aggregate risk and lowering expected operating cost. If [beta.sub.i] is positive, price will be below marginal cost, reflecting its value to the firm as a variance-smoothing component of its demand portfolio.

IV. An Example

The meaning of the pricing rule is transparent in the following example: Suppose that there are two classes of consumers, 1 and 2. Agents in both classes have homothetic preferences defined over a numeraire good and the good produced by the monopolist. The demand by the ith class for the monopolist's product is then linear in the income [y.sub.i] of the ith class:

[q.sub.i]([p.sub.i], [y.sub.i]) = [g.sub.i]([p.sub.i])[y.sub.i] (23) where [dg.sub.i]/[dp.sub.i] < 0. Each consumer knows his own income, but the monopolist knows only the joint density function of the two incomes.

Denote the second moments of this distribution by var([y.sub.i]) = [sigma.sup.2.sub.i], i = 1, 2, and cov([y.sub.i], [y.sub.j]) = [sigma.sub.ij], i [is not equal to] j. Suppose finally that the cost function is quadratic in total output:

[Mathematical Expression Omitted]

The covariance between the slope of the ith demand and marginal cost is then

[Mathematical Expression Omitted]

The slope of demand covaries with marginal cost because the demands themselves covary. In turn, the markup over expected marginal cost for the ith class is determined by the covariance of the ith demand with aggregate demand

[Mathematical Expression Omitted]

Mark-up is positive if demand covaries positively with aggregate demand.

V. Extensions

In this section we apply the model of section III to two traditional problems in public enterprise pricing: profit maximization and second-best or break-even pricing.

Profit Maximization

The firm now maximizes expected profits:

[Mathematical Expression Omitted]

The optimal price [p.sup.pi.sub.i] obeys:

[Mathematical Expression Omitted]

The notation is the same as for the welfare maximization problem. The optimal price contains the usual inverse-elasticity monopoly pricing term, modified by the covariance term in equation (10). The profit-maximizing firm exploits the covariance of demands in exactly the same way as the welfare-maximizing firm. As usual, the profit maximizing price exceeds the welfare maximizing price.

Second-Best or Ramsey Pricing

Suppose that a welfare-maximizing firm is required to meet a minimum-profit constraint, E([pi]) [is greater than or equal to] T. T can be positive or negative, but exceeds the unconstrained welfare-maximizing profit level. The optimal price [p.sup.s.sub.i] is then:

[Mathematical Expression Omitted]

The Lagrange multiplier on the profit constraint, [gamma] [is greater than of equal to] 0, modifies the monopoly elasticity effect. If is [gamma] zero, (25) is the same as (10), the welfare-maximizing rule. If [gamma] is large, (25) converges to (24), the profit-maximizing rule. This follows the pattern found in non-stochastic pricing formulations.

VI. Conclusion

Correlation among customer class demands alters optimal prices in predictable ways. If cost is convex, then the firm can reduce expected cost by reducing variance. Its optimal prices reflect this desire to reduce variance. Prices incorporate the covariance between marginal cost and the slope of the demand curve, which we argue is closely related to the covariance between class demand and aggregate demand. In an example, we show that class demands that covary positively with aggregate demand are charged higher prices, and vice versa. Covariance enters these pricing rules through "betas" that play a role akin to that of stock betas in determining the market value of a share of stock. Aggregate demand is the analog of the stock portfolio. The firm can alter the share of each class in revenue and the variance of each class by altering prices. The desire of the firm to manipulate expected sales and variance depends on the value of smoothing demand, which is determined in part by demand characteristics such as elasticity and covariance.

Profit-maximizing prices contain the usual monopoly elasticity term, absent from the welfare prices, which implies lower prices in the welfare case. Second-best or Ramsey prices contain a modified elasticity term, and so fall predictably between welfare- and profit-maximizing prices. The covariance component of price is identical in all three cases.

(1) Eckel[11 analyzes the problem of customer-class pricing without uncertainty. We extend her analysis here and in Eckel and Smith[21] to the case of uncertain demands. (2.) Rothschild and Stiglitz[6] and McCall[4] describe stochastic cost effects in the context of a single-product monopolist. (3.) Alternatively, if the firm must commit to a fixed level of capacity prior to the realization of demand, it may suffer a loss of revenue if realized demand exceeds capacity. In this case the firm again has a further incentive to smooth demand by manipulating prices, since by reducing the variance of aggregate demand it can reduce the probability of demand exceeding capacity. We discuss this case in Eckel and Smith[2]. (4.) While demand is uncertain from the perspective of the firm, consumers act with complete information. Each consumer knows all prices and his income when he makes his decision. (5.) The joint p.d.f. of the [q.sub.i] is f[u(p,q)]//J/, where f[.] is the joint p.d.f of the [u.sub.i] and /J/ is the Jacobean of the vector of demands with respect to the disturbances. /J/ is diagonal since [q.sub.i]/ [u.sub.j] = 0, i [is not equal to] j; it is strictly positive since [q.sub.i]/ [u.sub.i] > 0. (6.) This assumption rules out the usual specification of additive uncertainty, [q.sub.i] = [q.sub.i][p.sub.i] + [u.sub.i], where [Eu.sub.i] = 0. Nonnegativity is maintained by imposing a lower bound [-[q.sub.i]([p.sub.i] on [u.sub.i]. As noted by several authors (Meyer[5], Sherman and Visscher[7] or Tschirhart and Jen[8]) this is vexatious since the lower bound is a function of price. Our formulation avoids this complication. However, as an observant referee has pointed out, additive uncertainty can still be salvaged with some clever assumptions about the distribution of [u.sub.i]: If [Eu.sub.i] > 0 and [u.sub.i] has a bounded support sufficiently high to make the lowest price non-negative, then additive uncertainty still satisfies our assumption (2). (7.) We assume that the firm is risk-neutral. To ensure a unique maximum, we also assume that the welfare function is strictly concave. As usual, this imposes restrictions on the degree of convexity of the cost function. Risk aversion would provide an additional source of concavity, and strengthen our results. The presence of risk aversion would increase the impact of the covariance of (10) on the markup. (8.) To derive pricing rule (10), consider the first-order condition with respect to [p.sub.i]:

[Mathematical Expression Omitted]

Use the fact that cov( [q.sub.i]/ [p.sub.i], c') = E [q.sub.i]/ [p.sub.i]Ec' - E(c' [q.sub.i]/ [p.sub.i] and invoke the definition of the elasticity of expected demand in equation (11) to arrive at equation (10). (9.) Although this relationship may not hold in unusual or perverse cases, we cannot construct a counter example. (10.) According to Theorem 236 of Hardy, Littlewood and Polya[3], the covariance of an increasing and a decreasing function of a random variable is negative.

References

[1.] Eckel, Catherine, "A General Model of Customer Class Pricing." Economics Letters, 17, 1985, 285-89. [2.] _____ and William T. Smith. "Multiproduct Pricing and Capacity Choice with Correlated Demands." Department of Economics, VPI&SU, Working Paper 89-06-01, Revised January 1992. [3.] Hardy, Godfrey Harold, John E. Littlewood, and George Polya. Inequalities, Second edition. Cambridge: Cambridge University Press, 1952. [4.] McCall, John J., "Probabilistic Microeconomics." Bell Journal of Economics and Management Science, Autumn 1971, 403-33. [5.] Meyer, Robert, "Monopoly Pricing and Capacity Choice Under Uncertainty." American Economic Review, June 1975, 326-37. [6.] Rothschild, Michael, and Joseph Stiglitz, "Increasing Risk II: Its Economic Consequences." Journal of Economic Theory, March 1971, 66-84. [7.] Sherman, Roger, and Michael Visscher, "Second-Best Pricing with Stochastic Demand." American Economic Review, March 1978, 41-53. [8.] Tschirhart, John, and Frank Jen, "The Behavior of a Monopoly Offering Interruptible Service." Bell Journal of Economics, Spring 1979, 244-58.