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Price Discrimination∗
Simon P. AndersonÜand RÈgis Renaultá
This version August 2008.
keywords arbitrage, nonlinear pricing, discriminatory
pricing, Ramsey-Boiteux pricing, marginal cost of
public funds
1 Introduction
Whenever we take a trip by train or by plane, we are often well aware that the price we paid was quite different from that paid by our fellow passengers with whom we are sharing the carriage or cabin. We can bemoan this situation, if we booked late and do not qualify for an age discount and our ticket was one of the more expensive ones or perhaps be pleased at having gotten a good price. The different prices are illustrations of what economists call discriminatory pric- ing. This seems to be a textbook case where a service which is identical (same journey, same date, same time, same comfort class) is sold at different prices 1. ∗We gratefully acknowledge travel funding from the CNRS and NSF under grants INT- 9815703 and GA10273, and research funding under grant SES 0452864 .Anderson, Alain BÈraud, Catherine de Fontenay, Robin Lindsey, AndrÈ de Palma, Emile We thank Anita Quinet, and Sarah Tulman for their comments and suggestions.Melbourne Business School and the Autoridade da Concorrencia in Lisbon for their hospitality. We would also like to thank
ÜDepartment of Economics, University of Virginia, PO Box 400182, Charlottesville VA 22904-4128, USA. [email protected] áThEMA, UniversitÈ de Cergy-Pontoise, 33 Bd. du Port, 95011, Cergy Cedex, FRANCE and Institut Universitaire de France. [email protected] (^1) Price discrimination is also common in other transportation services in addition to rail and air travel. Odlyzko (2004) provides examples from maritime transport, inland waterways,
Looking closer, it is a little oversimplified to claim that all travelers have actually received the same level of service. Less expensive tickets are often associated with numerous restrictions which clearly indicate a lower level of service. It is often necessary to buy the ticket a long time in advance, with restrictive conditions on cancellation and reimbursement. The traveler can then enjoy the same service as she would have had if she had paid full price ó unless, of course, her plans change at the last minute. However, to get the cheap fare she has to accept some risk, had she been obliged to change or cancel her ticket, or indeed she might have had to put up with some inconvenience due to having not changed her ticket in order to not lose money. There are also reduced fares for those satisfying certain ìdemographicî considerations. Senior citizens and children often pay lower fares. On May 5, for May 18, 2007 the prices for one-way second class travel be- tween Charleroi Sud (Belgium) and Paris Nord (France), proposed on the inter- net site www.thalys.com were as follows: Librys 59 E; Mezzo 44.5 E; Mezzo+ 32.5 E; and Smilys 20.5 E. The rates for Mezzo+, Mezzo, and Smilys can be reserved only if sufficient seats are still available, and only if one buys a round trip ticket. It is less expensive to buy a round-trip Smilys ticket than to buy a one-way Librys ticket. However, Smilys can only be bought when reserving two weeks before the departure date and it is the only fare which is non-exchangeable and completely non-reimbursable. Mezzo and Mezzo+ are reimbursable up to fifty percent of the price of the ticket, up to the departure and turnpikes, as well as railroads, to argue that price discrimination has been (and is still)prevalent in the development of these sectors.
have proposed definitions based on the comparison of price differences relative to cost differences. Stigler (1987) proposed comparing the ratio of the prices of two services with the ratio of their marginal production costs. By this criterion, a situation is discriminatory if the two ratios are unequal. Phlips (1983) on the other hand proposes comparing absolute differences. Then prices are discriminatory if the difference in marginal costs is not equal to the difference in prices. It is difficult to find a decisive argument for one definition over the other. 3 Both definitions indicate that prices can be discriminatory even if price differ- ences are small, just as it can be discriminatory if price differences are large. Suppose an airline brings passengers to a Parisian airport from which its in- ternational flights leave, and has everyone pay the same price for a flight to New York. This pricing discriminates against travelers living near Paris (see Tirole, 1988, 1993, for a similar example).^4 The definitions do not say whether such discrimination harms economic efficiency: the airlineís pricing scheme al- lows it to more effectively exploit its market power by bringing its transatlantic travelers to Paris. A firm with some market power and proposing different services can set its prices to get the greatest profit, and its prices will not bear any simple relation (absolute or relative) to marginal costs. Exercising market power is channelled through the ability to price above marginal cost. Offering diverse services can be seen as a way of discriminating insofar as it allows the firm to adjust (^3) See Clerides (2004) for a definition in which there is no discrimination if the pricing structure is not subject to arbitrage by buyers. (^4) For further discussion, see Section 3.5.
the service proposed and its pricing to a demand that differs from customer to customer.^5 Just as it is difficult to define discriminatory pricing, it is not easy to classify different discriminatory practices. The classic reference is Pigou (1938), who distinguishes between three possible degrees of discrimination, depending on the ability of the firm to distinguish between buyers who are prepared to pay a higher price and those inclined to pay less. Pigou defines first-degree discrimination as when consumers pay their maximal willingness to pay for each unit. This is also called perfect price discrimination. Pigou recognized that this first form of price discrimination might not have great practical relevance. He notes that the firm is better able to segment the market between different groups of buyers who have different demands. Ideally, the firm would like to segment the market into groups with similar willingness to pay; such segments could be ranked from highest to lowest willingness to pay. Such idealized segmentation constitutes second-degree price discrimination. As Pigou notes, however, in practice a firm can only imperfectly compartmen- talize consumers according to their willingness to pay. The firm must use characteristics which it can directly monitor, such as the type of good that is being transported (for example, livestock or pig iron^6 ) for a railroad that trans- ports freight, or the location of the buyer. This latter practice is third-degree price discrimination. Pigouís classification underlines the fact that discriminatory pricing is meant (^5) In this paper, we emphasize demand drivers and take marginal cost as constant. The proper attribution of costs is a complex problem in itself. (^6) See Leadbellyís ìRock Island Lineî.
The discussion above suggests that discriminatory pricing is tightly tied to the exercise of market power.^7 Under perfect competition, firms are constrained to sell their output at the price that is imposed by the market. It is then obviously impossible to sell different units of a good at different prices, or to try to affect prices by proposing a range of different services (at least as long as such services are sold in a competitive market). Even if the firm has some market power, its ability to discriminate between different buyers can be undone or mitigated by the buyersí ability to arbitrage between the different options proposed.^8 This arbitrage can take two forms, depending on whether one or several buyers are involved. If it is possible to transfer the good, buyers can exchange the good or service between themselves and the firm cannot charge different prices, because those buyers who benefit from the lowest price will be able to buy in order to resell to those who would otherwise have to pay more. For example, if someone has a membership card which allows her to obtain her tickets at a cheaper price, she could buy a large number of them and resell them to those without a membership card. Similarly, if a low fare is offered under the condition that the ticket should be bought sufficiently far in advance, then entrepreneurial individuals could buy a large quantity of these cheap tickets in order to sell them just before the date when the tickets are valid. Even though this type of arbitrage can be limited by transactions costs, it nonetheless (^7) Nonetheless, McAfee, Mialon, and Mialon (2006) propose a simple model in which they show that the extent of price discrimination has no theoretical connection to the extent ofmarket power. (^8) One other limit on the ability to price discriminate is that firms cannot effectively propose a price menu that is too complex. See Levinson and Odlyzko (2007) for a recent treatment.
represents a significant constraint on firmsí pricing strategies, so much so that firms often put in place several techniques to stop it. They often require one to present the membership card during the trip, or, for airlines, present a piece of identification which has on it the name matching that of the ticket holder. This type of rule also allows the firm to circumvent the second type of arbitrage, in which one customer with several options does not choose the one which was designed for her. For example, if a firm wants to discriminate on the basis of age, by presenting a driverís license the customer verifies that she is paying the price that she should. When buyers can practice such arbitrage between the different pricing options that are offered, Pigou (1938) says that demand is transferrable. In practice, firms can often discriminate without explicitly forbidding arbi- trage. The firm then has to explicitly worry about potential arbitrage when it is setting up a discriminatory tariff structure. While the firm cannot force its customers to not arbitrage, it has to set up the right incentives in its pricing plan. The analysis of arbitrage behavior across consumers is relatively compli- cated, and most of the literature on discriminatory pricing simply supposes that transactions costs are high enough to render it impossible. We, too, will implicitly invoke this assumption throughout the paper.^9 On the other hand, so-called personal arbitrageó by which a user can choose an option which is not intended for her purchaseó has been the subject of numerous studies, especially (^9) See Alger (1999) for an analysis of the constraints that are imposed by the potential of arbitrage involving several buyers.
pretation is to suppose that D describes the distribution of willingness to pay over different travelers. Each traveler only wants a single trip, and D (p) then indicates the number of travelers willing to pay at least p. It is also useful to define the inverse demand for each quantity q. This is the maximum price at which this number of trips can be sold, P (q). When each traveler only wants a single trip, P (q) is the willingness to pay of the marginal consumer: the individual who would not travel if the price were slightly higher. In order to simplify the analysis, suppose that the marginal cost of production is constant at rate c per unit, which is also therefore the average variable production cost (and is the cost generated by each traveler). Under perfect competition, the price would be given by this marginal cost, and the number of travelers would be D (c). We assume that services are provided by a private monopolistic firm, whose objective is to maximize its profit. We start with the case where demand is linear,
q = D (p) = a − bp, a > 0 , b > 0 , and ab > c.
The latter condition ensures that it is optimal to produce a strictly positive quantity. This situation is illustrated in Figure 1. The firm is constrained under uniform pricing to choose a single point on the demand curve. If it wants to carry q travelers, the highest uniform price that it could charge is:
p = P (q) = a^ −b q.
Figure 1: Uniform Monopoly Pricing The producer surplus, P S, which is profit gross of fixed costs, is represented by the rectangle above marginal cost (i.e. the mark-up per traveler), p − c, multiplied by the number of travelers q.^11 The optimal quantity must therefore maximize:
∙ (^) a − q b −^ c
q,
and the first order conditions for an optimal quantity can be written as:
a − 2 q b =^ c. The left hand side of this expression is marginal revenue, MR. This is a straight line with the same price intercept as inverse demand D, but its slope is twice as steep. It is easy to see from Figure 1 that the condition for equality between marginal cost and marginal revenue is satisfied for a quantity:
qm^ = D^2 (c )= a^ − 2 bc,
which is therefore half the quantity produced under perfect competition. The uniform price chosen is therefore:
pm^ = P (qm) = a^ + 2 b^ bc , 1 1 (^) See Anderson and Engers (2007b) for further discussion of the concept of producer surplus.
be measured in monetary terms in a way that can be compared with the extra profit extracted by the firm. An individualís consumer surplus is the difference between the maximum price that the consumer is willing to pay for a trip and the price that she actually pays. The inverse demand curve is constructed by ranking willingness to pay in decreasing order, so that the q trips are sold to the q travelers willing to pay most. Aggregate consumer surplus, CS, generated by the sale of quantity D (p) at price p, therefore corresponds to the area between inverse demand and the quantity from 0 to D (p).^13 For the solution described in Figure 1, this is represented by the triangle CS in Figure 2. The loss in consumer surplus resulting from a change from pricing at marginal cost to pricing at the monopoly level is given by the monopoly producer surplus plus the area DW L in Figure 2. This loss of consumer surplus can be interpreted as the sum of what travelers would have been willing to pay, collectively, in order to be able to access tickets priced at marginal cost as opposed to the monopoly price. Insofar as this total is larger than the monopoly producer surplus, there is a beneficial exchange possibility for all market participants which has not been realized (the firm would be ready to cut its price down to marginal cost if, in exchange, it could receive compensation that is at least as large as its producer surplus). Monopoly pricing therefore introduces an inefficiency which can be measured by that part of the loss of consumer surplus which is not offset by an increase in producer surplus. This deadweight loss is the area DW L in Figure 2. To 1 3 (^) See Anderson and Engers (2007a) for further discussion of the concept of consumer surplus.
understand this inefficiency more concisely, it is convenient to introduce the concept of social surplus. This is the sum of producer surplus and consumer surplus. Pricing at marginal cost enables the maximal social surplus to be attained. Deadweight loss is the reduction of social surplus caused by a higher price.^14 One reasonable objective for public policy could be to minimize deadweight loss. This objective could be obtained by a public firm, or one subject to regulation, pricing at marginal cost. Such a solution is not generally very satisfactory. When there are increasing returns to scale, such pricing will not cover production costs. For example, when marginal cost is constant, marginal cost pricing will generate zero producer surplus, so the firm will not cover its fixed costs. It follows that the firm must be partly financed by taxpayers. It may then be desirable to suffer some deadweight loss in order to avoid an overly large deficit. Pricing at marginal cost might then be replaced by pricing at average cost, so that the firm just covers its costs. Nevertheless, it seems rather arbitrary to impose the condition that the firm should not make losses. An alternative argument against pricing at marginal cost is that taxation induces inefficiency in the allocation of resources (see Meade, 1944). Optimal pricing must therefore strike a balance between maximizing social surplus in the markets served by the firm and the efficiency cost of raising tax revenue in the rest of the economy. The latter cost can be measured by the deadweight losses caused by the taxes in 1 4 (^) Any price below marginal cost causes a deadweight loss because it leads to the sale of some units for which consumers are willing to pay less than the extra social cost that theirproduction would engender.
p − c p =^
λ 1 + λ
|η (p)|. The price thus obtained is a special case of what are called Ramsey-BoÓteux prices, which ensure maximization of social surplus under the constraint that the firm returns a particular level of profits, for example to cover its fixed costs. Under this interpretation, λ therefore indicates the severity of the budget con- straint and is the marginal social surplus gain that could be obtained by reducing the profit level to be earned by one euro (see Ramsey, 1927, and BoÓteux, 1956). In the interpretations above, λ is necessarily positive, so that more weight is placed on producer surplus than on consumer surplus. A higher λ leads to a higher mark-up over marginal cost (with monopoly pricing resulting as λ goes to infinity).^18 This outcome is rather unsatisfying insofar as we might wish for public policy to respond not only to economic efficiency but also to redistribution. The current analysis does not need to take a stand on these issues. Thus, if a higher price (of a train ticket) allows the government to efficiently collect revenues, these could be used to give lump sum transfers to consumers. Nevertheless, a system of monetary transfers creates difficulties in itself because of the perverse incentives it may induce, as well as perhaps for reasons of polit- ical viability. If indeed transfers are not to be made directly from Government revenues because it is intrinsically costly to do so, then the pricing scheme may be used directly for transfers, and it may therefore be reasonable to put a larger 1 8 (^) This follows from applying the Implicit Function Theorem to the first order condition to the firmís problem above.
weight on consumer surplus than on producer surplus. This translates in our formal analysis to − 1 < λ < 0.^19 This allows us to understand why we might want to heavily subsidize certain services such as public transportation, even pricing below marginal cost. Finally, there may be other reasons for pricing below marginal cost, especially for transportation services. Up until now, we have not taken into account the possibility of positive or negative externalities such as pollution or congestion. This omission can easily be rectified by re- placing, in our analysis, marginal private cost with marginal social cost, which would be greater or smaller depending on whether the externality were negative or positive.
3 Discrimination and verifiable consumer char-
acteristics
3.1 Perfect discrimination
We first consider the case which is most advantageous to the seller. This arises when the seller has perfect information about the demand from each possible buyer. Perfect discrimination arises when the firm can use this information fully (first-degree discrimination, in Pigouís terminology). In order to use the information, the seller must be able to control the price and the characteristics of each unit sold to each buyer. For example, if an airline perfectly knew the needs and desires of all of its customers it could choose the price at which each 1 9 (^) A negative lambda might also be applied in a welfare analysis used to evaluate different market outcomes, and where thefrom) the public purse. Private fifirmsí surplus would usually be weighted (much) less thanrmsí losses do not contribute to (or need to be financed consumer surplus.antitrust circumstances, would put a weight on producer surplus of zero ( For example, a consumer surplus standard, as is arguably used in someλ = − 1 ).
Another method for getting to the same result would be to use a two-part tariff (see also Oi, 1971). Such a tariff would specify an entry, or membership, fee A that the consumer must pay in order to consume the good at all. If she joins, then she can buy as much as she wants at a price p. Setting p = c ensures that the consumer will therefore choose q∗, and she will therefore enjoy a surplus of V (q∗) − cq∗. She will join as long as the entry fee is not larger than this, and so the seller will set a fee as large as possible subject to this individual constraint; namely it will set an entry fee of A = V (q∗) − cq∗. The total price paid by the consumer is then A + cq∗^ = V (q∗), which means that her full surplus is extracted by the firm. The outcome is thus just the same as for the preceding pricing system. Even though a two-part tariff seems simpler, it still needs just as much information: while it may be easy to fix price at marginal cost, calculating the entry fee means knowing the consumerís surplus, and hence her full demand curve.^21 The pricing solution for a public firm under the similar assumption of per- fect knowledge and ability to discriminate is straightforward: it should choose exactly the same tariff structure. This is because all surplus is extracted from the consumer and earned by the firm for the public purse. 2 1 (^) No further complication is introduced for the preceding analysis when marginal costs are not constant. The optimal quantity,consumer surplus is extracted. A single (all-or-nothing) tari q∗, equates the demand price with marginal cost, and allff equal to gross consumer surplus at this quantity is optimal, and is equivalent to a two-part tarito the demand price P (q∗) and an entry fee of A = V (q∗) − P ff(q with a per-unit price equal∗) q∗ (^) (= CS (q∗)).
3.2 One good and several groups of buyers
We next consider a situation in which a firm can observe a characteristic or characteristics of buyersó such as age, job, or residential addressó and observing these characteristics allows the firm to infer something about demand. The firm can exploit some correlation between the observed variable and the individual demand in order to discriminate (perhaps such discrimination is not legal or socially acceptable, for example if it is based on gender or race). The firm can then choose a price that depends on the observed character- istic, and the individual who does not have this characteristic can be excluded from prices not meant for her. However, discrimination is imperfect insofar as consumers are bundled together onto the same characteristics (e.g. the same age group), and individuals still may differ by willingness to pay within the group (within a group, selling different quantities with nonlinear pricing or introduc- ing differentiation can be used to get potential buyers to reveal their tastes: see Section 4). The firm is therefore obliged to set a uniform price for each category (or group). Suppose, for example, there are two classes of buyers: youths under 26 and the rest of the population. Asking for identification (in the absence of fake IDs), the firm can know which category the buyer is in. It can therefore exclude arbitrage by which people in one group buy the good or service in or- der to sell it to people in the other group (for example, airline tickets with the travelerís name on them). In the absence of such arbitrage, demand from each group depends only on the price charged to members of the group. Consider the case of a public firm: the specialization to a private firm will
