how to price discriminate when tariff size matters

Articles in Advance, pp. 1–16ISSN 0732-2399 (print) � ISSN 1526-548X (online) http://dx.doi.org/10.1287/mksc.1120.0720

© 2012 INFORMS

How to Price Discriminate When Tariff Size Matters

Adib BaghDepartment of Economics, University of Kentucky, Lexington, Kentucky 40506,

[email protected]

Hemant K. BhargavaGraduate School of Management, University of California, Davis, Davis, California 95616,

[email protected]

Firms that serve a large market with many diverse consumer types use discriminatory or nonlinear pricingto extract higher revenue, inducing consumers to separate by self-selecting from a large number of tariff

options. But the extent of price discrimination must often be tempered by the high costs of devising and man-aging discriminatory tariffs, including costs of supporting consumers in understanding and making selectionfrom a complex menu of choices. These tariff design trade-offs occur in many industries where firms face manyconsumer types and each consumer picks the number of units to consume over time. Examples include wirelesscommunication services, other telecom and information technology products, legal plans, fitness clubs, automo-bile clubs, parking, healthcare plans, and many services and utilities. This paper evaluates alternative ways toprice discriminate while accounting for both revenues and tariff management costs. The revenue-maximizingmenu of quantity-price bundles can be very (or infinitely) large and hence not practical. Instead, two-part tariffs(2PTs), which charge a fixed entry fee and a per-unit fee, can extract a large fraction of the optimal revenue witha small menu of choices, and they become more attractive once the costs of tariff management are factored in.We show that three-part tariffs (3PTs), which use an additional instrument, the “free allowance,” are an evenmore efficient way to price discriminate. A relatively small menu of 3PTs can be more profitable than a menuof 2PTs of any size. This 3PT menu can be designed with less information about consumer preferences relativeto the menu of two-part tariffs, which, in order to segment customers optimally, needs fine-grained informationabout preferences. Our analysis reveals a counterintuitive insight that more-complex tariffs need not always bemore profitable; it matters whether the complexity is from many choices or more pricing instruments.

Key words : price discrimination; multipart tariffs; nonlinear pricing; tariff costsHistory : Received: August 22, 2011; accepted: February 29, 2012; Preyas Desai served as the editor-in-chief and

Duncan Simester served as associate editor for this article. Published online in Articles in Advance.

In late 2001, senior executives of the newly formedVirgin Mobile USA grappled with their pricing strat-egy. Should they offer complex price schedules thatincreased revenue through price discrimination butalso led to high customer acquisition costs? Or shouldthey offer a highly simple plan that sacrificed oppor-tunities for price discrimination but enabled Virgin toacquire customers at lower cost?

(see McGovern 2003)

1. IntroductionThis paper evaluates and compares alternative pric-ing schemes in their ability to price discriminateefficiently, i.e., increase the firm’s revenue while hold-ing the administrative costs of implementing the pric-ing scheme low. Price discrimination is a powerfulweapon in the managerial toolkit for raising firms’profits when they face heterogeneous consumers. Thefirm offers a common nonlinear price schedule (map-ping quantity to price) to all consumers who thenself-select and separate as they maximize their ownsurplus (Maskin and Riley 1984, Goldman et al. 1984,

Sharkey and Sibley 1993). Firms employ a variety ofpricing schemes in order to implement price discrim-ination (see, e.g., Iyengar and Gupta 2009), many ofwhich are illustrated in Table 1. We refer to these pric-ing schemes as tariffs to represent any nonlinear pric-ing schedule, following Wilson (1993), who applies itto both regulated and unregulated pricing.

How much should a firm price discriminate? Whynot simply pick the revenue-maximizing schedule?1

There are several reasons to limit the extent of pricediscrimination. First, the price specification gets morecomplex or lengthy as the firm continues to increaserevenue through more discriminatory pricing. Thisraises the cost of implementing the pricing schemewhile the corresponding revenue gains get increas-ingly smaller; hence at some point, the higher tariffcosts offset the revenue gain (Murphy 1977, Dolan 1984,

1 Here, and henceforth, revenue is defined after accounting for pro-duction and distribution costs; tariff administration costs are dis-cussed separately.

1

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Published online ahead of print July 12, 2012

Bagh and Bhargava: How to Price Discriminate When Tariff Size Matters2 Marketing Science, Articles in Advance, pp. 1–16, © 2012 INFORMS

Table 1 Examples of Nonlinear Pricing Structures

Tariff structure Product Example Size

Flat rate (unlimited use) Internet access SBC: $17.99/mo. 1Music download Rhapsody: $9.99/mo. 1Online retailing Amazon Prime shipping: $79/yr. 1

Two-part tariff Golf club Del Monte: $285/yr. + $42/day of play 2Three-part tariff Data center RimuHosting: $20/mo., 30 GB allowance, $1/GB 3Menu of two two-part tariffs International calls (United States

to India)AT&T: $2.95/mo. plus $0.39/min. or $3.99/mo. plus $0.28/min. 4

Menu of two three-part tariffs Wireless service Verizon: $40/mo., 450 min., $0.45/min.; or $80/mo., 1,350 min.,$0.35/min.

6

Menu of quantity-price bundles Web hosting GoDaddy: $3.99/mo. for 10 GB, $5.99/mo. for 150 GB, or $7.99/mo.for unlimited space

6

Freight delivery FedEx (also DHL, UPS) > 300Block-declining tariff (taper) Online data storage Amazon S3 storage: 15 c/GB for 0–50 TB, 14 c/GB next 50 TB,

13 c/GB next 400 TB, 12 c/GB thereafter7

Notes. Tariff size is measured as the number of parameters in the tariff specification. c/GB, cents per gigabyte.

Wilson 1993, Miravete 2007). Second, designing alengthier tariff generally requires more fine-grainedinformation about customer preferences, raising thefirm’s information and decision costs (Chen and Iyer2002, Acquisti and Varian 2005) and infringing oncustomers’ privacy considerations (Odlyzko 2003).Third, consumers are often overwhelmed—and failto choose anything—when given too many choices(Iyengar and Lepper 2000). This raises the firm’scosts of explaining the choices and guiding cus-tomers’ self-selection toward a purchase. The Har-vard Business School case on Virgin Mobile’s entryinto the U.S. market notes that Virgin Mobile’scompetitors—who offered fairly complex price dis-crimination schedules—spent about $100 per cus-tomer on sales staff hired to help customers with themyriad of menu options (McGovern 2003). Such hightariff management costs make it imperative for thefirm to price discriminate efficiently, i.e., extract morerevenue while avoiding a lengthy or complex pricespecification. In an empirical study of the U.S. cel-lular telephone industry, Miravete (2007, p. 1) con-cludes that firms “should only offer [a] few tariffoptions” because of the costs of tariff management.In a similar vein, Villas-Boas (2004) emphasizes thatadvertising and communications costs, which increasewith the number of offers, exert downward pressureon the number of choices (i.e., menu size) offered toconsumers.

Because nonlinear pricing involves indirect seg-mentation of the market (through self-selection byconsumers), the firm must distort the options offeredto certain segments in order to ensure that each seg-ment picks the intended option. Different nonlinearpricing schemes vary in the revenue they generaterelative to the ideal case of perfect price discrimina-tion. But they also differ on the tariff management

costs they impose. Our objective is to compare differ-ent pricing schemes on their efficiency at price discrim-ination; i.e., we care about both revenues and tariffmanagement costs. The magnitude of tariff costs rela-tive to revenue depends on the specific context, henceit is not practical to combine costs and revenue intoa single objective function. However, tariff size (thenumber of parameters used in specifying the priceschedule) serves as a useful proxy for evaluating howefficient a pricing scheme is at price discrimination,because, generally speaking, it is positively correlatedwith both higher revenue and higher tariff cost.2

We write tariff size as S = n × m, where n is thenumber of parameters in the pricing scheme and mis the menu size or the number of choices in themenu. The rightmost column of Table 1 clarifies theseterms. For instance, a single two-part tariff 4F 1 p5 has2 × 1 = 2 parameters (F is a fixed entry fee and pis marginal price per unit). A single quantity-pricebundle 4q1T 5 also is of size 2 (T is total price forq units). A single three-part tariff has three parame-ters (F 1 p, and free allowance Q).3 An m-item menuof two-part tariffs (or bundles) has tariff size 2m,and an m-item menu of three-part tariffs has size 3m.The firm can increase price discrimination by eitherproviding more options (increasing menu size m)or adding more parameters to the pricing scheme(increasing n). Either approach involves an increase

2 Dolan (1984) suggests tariff complexity as an alternative proxyfor tariff efficiency. Tariff complexity is the number of break pointsin the tariff. For example, a menu of m two-part tariffs has m− 1breaks, as does a m-block tariff; a single three-part tariff on theother hand, has one break point. Our results remain valid underthis alternative framing.3 Some authors refer to three-part tariffs as “fixed-up-to” tariffs, ortwo-part tariffs with inclusive consumption. Our usage is consis-tent with more recent marketing publications, including Lambrechtet al. (2007) and Iyengar and Gupta (2009).

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Bagh and Bhargava: How to Price Discriminate When Tariff Size MattersMarketing Science, Articles in Advance, pp. 1–16, © 2012 INFORMS 3

in S. For example, starting with a single 4q1T 5 bun-dle, the firm could either move to a menu of bundles(S = 4) or offer a three-part tariff (S = 3).

This paper demonstrates that three-part tariffs(3PTs) are highly efficient at price discriminating alarge population of heterogeneous consumers. Theargument is as follows. First, ignoring the effectof tariff management costs, the revenue-maximizingscheme is a menu of quantity-price bundles, but themenu is typically very large and often nonenumer-able. Second, however, when menu size is restrictedto be less than the revenue-maximizing menu size,then two-part tariffs can yield higher revenue thanquantity-price bundles, keeping the overall tariff sizethe same across the two regimes. These two claimscan be gleaned from the literature and are covered in§4. Third, we demonstrate that 3PT menus are evenmore efficient when tariff size is restricted. We showthat under fairly plausible conditions, even a single3PT (i.e., a specification of size 3) can outperform asubstantially larger menu of two-part tariffs (2PTs).We also show that, in general, a size-constrained spec-ification composed of 3PTs can produce higher rev-enue than an equal-size specification composed of2PT menu or bundles. Another salient point is thatthis 3PT design can extract higher revenue with lessor incomplete information.

We emphasize that our result is not merely that a3PT menu outperforms an equal-sized menu of 2PTs(or an equal-sized menu of bundles): that outcome istrivial and the comparison unfair, because a 3PT sub-sumes both a 2PT and a bundle (set Q = 0 to get the2PT and p = � to get the bundle), and the 3PT menuwould have more parameters than the 2PT menu.Instead, we focus on a combination of higher rev-enue, lower tariff size, and less accurate informationneeded to design it. We caution that the 3PT menuemployed in our constructive proofs is not the optimal3PT menu; the latter would, of course, be even moreprofitable, but it would also require more informationabout demand preferences.

2. Examples and IntuitionWe start with a series of examples to demonstrate thekey results and the intuition behind them. Becauseprior literature already demonstrates that a menu oftwo-part tariffs is superior to a menu of quantity-price bundles when tariff size matters (see §4), ourexamples compare two-part tariffs with three-part tar-iffs having equal or fewer parameters. Example 1below considers the most simple stylized setting withjust two consumer types. We use this simple exam-ple only to convey the intuition behind the result.Example 2 provides a six-segment setting in whichour result is more meaningful: we show that a single

3PT (3 parameters) beats the fully separating optimaldesign within the space of two-part tariffs (12 param-eters); hence it is cheaper to implement and it gener-ates higher revenue.

Example 1. The market has two segments, i = 1, 2,and the proportion of Segment 1 is �1 = 008.The demand curves of consumers in i = 1, 2are, respectively, D14p5 = 4100 − p5 and D24p5 =

250 − 1025p. The optimal menu of two-part tariffs(fully separating, with four parameters) is 4F11 p15 =

421093043135035 and 4F21 p25= 410113804105 with a rev-enue of $5,529.41. The single three-part tariff (threeparameters), 4F 1Q1p5 = 44137609616407174015, gener-ates higher revenue: $5,750.50.

The single three-part tariff produces 4% higherrevenue than the optimal menu of two-part tariffs,despite having lower tariff size. The result holds aslong as �1 is large enough. The intuition is that with a2PT menu—which relies on self-selection—the firm’sdesign of 4F21 p25 is constrained by the fear that high-value customers would pick 4F11 p15 meant for thelower segment. The large �1 amplifies this constraintand forces the firm to increase the consumption (andextractable surplus) of the low-value segment ratherthan distort their allocation in order to extract extrarevenue from the high-value customers. In contrast,the single 3PT constructed above allots Segment 1 cus-tomers only the free allowance Q; hence the choice ofthe linear fee parameter p is constrained only by thepreference of Segment 2. With this design (in whichSegment 1 generates the same revenue as the 2PTmenu), the firm extracts incrementally greater rev-enue from the high-value segment. This intuition isexplained in more detail in §6.1.

Example 2 (Single Three-Part Tariff Beats Menuof Six Two-Part Tariffs). The market has six seg-ments, with proportion �i of the customers and(inverse) demands vi4q5 (see Table 2). On one hand,

Table 2 Tariff Design and Outcomes for Example 2

Market descriptionSegment i 1 2 3 4 5 6Proportion �i 007 0015 0008 0004 0002 0001Inverse 3 − q 505 − q 605 − q 705 − q 803 − q 9 − q

demand vi 4q5

Optimal 2PT menuFixed fee Fi 108597 201786 208739 407336 606898 907786Variable fee pi 100714 100000 008750 006000 003500 000000Quantity 109286 405000 506250 609000 709500 900000

consumed qiRevenue �21 i 309260 606786 707958 808736 904723 907786

Single 3PTRevenue �̂31 i 309260 701148 809005 1006862 1201148 1303648

Note. The optimal two-part tariff has tariff size 12, whereas a single three-part tariff, with just three parameters, has a higher revenue and lower tariffcost.

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Figure 1 Revenue from Each Segment Under the Optimal Six-Item Menu of Two-Part Tariffs and a Single Three-Part Tariff

0 2 4 6 8 10

0

5

10

15

Quantity consumed (q )

Tot

al p

rice

paid

T(q

)(q, T (q)) forSegment 1Segment 2Segment 3Segment 4Segment 5Segment 6

Single

three

-par

t tariff

(3pa

ramete

rs)

Optimal menu of two-part ta

riffs (12 parameters)

1 2 3 4 5 6

0.0

0.5

1.0

1.5

2.0

2.5

Consumer segment

Ave

rage

pric

e pa

id

Single three-part tariff

Optimal menu of two-part tariffs

Note. The left panel shows the total revenue from each segment, and the right panel shows the average price per unit consumed.

we have the optimal menu of two-part tariffs, speci-fied in the table. It provides a unique option for eachsegment and yields a total revenue �∗

2 =∑

i �i�21 i =

500158. On the other hand, we have a single three-parttariff 4F̂ 1 Q̂1 p̂5= 4309260110928611078575, derived fromthe 2PT menu solution by fixing Q̂ = q∗

1 and F =�211,and then setting p̂ to maximize with respect to theresidual demand of Segment 2. The single 3PT has�̂3 = 5033089.

The single 3PT produces 6.2810% higher revenuethan the optimal fully separating 2PT menu, and itgenerates at least as much revenue from each segmentas does the 2PT menu (see Figure 1). And with only3 tariff parameters, it has lower tariff managementcosts than the 2PT menu, which has 6 tariff choicesand 12 parameters. Designing this single 3PT requiresprecise knowledge only of v1, v2, and �1. Finally,note that both pricing schemes implement quantitydiscounts (decreasing average price from 2.0357 to1.8525, as depicted in Figure 1), even though the out-lay in the single 3PT (as in any 3PT) is nonconcave.

The density of consumer types in these two exam-ples is monotonically decreasing in their valuations,but this is not necessary for the result to hold. Forexample, suppose the Segment 1 customers in Exam-ple 2 are split into Segments 1a, 1b and 1c (compris-ing 10%, 40%, and 20% of the market, respectively),with 1a having the lowest valuation and 1c the sameas original Segment 1. Then, depending on the valu-ations of Segments 1a and 1b, the optimal 2PT menuwould have six, seven, or eight items. And underanalogous conditions, it would still be possible toexceed this revenue with a one-, two-, or three-itemmenu of three-part tariffs. This idea is covered in §6,and Theorem 6.4 describes when a K-item three-part

menu can outperform a fully separating I-item menuof 2PTs (with 3K < 2I). Finally, Example 5 demon-strates that the 3PT design can extract higher revenuewith less (or less accurate) information than neededto design the optimal 2PT menu.

3. Problem StatementFor each pricing scheme � , we distinguish between(i) the outlay it specifies, a mapping from a subset Qof possible consumption levels Q into R+ (Sharkeyand Sibley 1993); and (ii) the specification of this out-lay, i.e., whether the mapping is announced explicitlyor implicitly. A menu of quantity-price bundles of theform 4qi1Ti5 is an explicit specification of the outlay:every quantity-price combination available to the useris stated explicitly.4 For implicit tariffs, the generalformulation is a piecewise linear (or “block”) func-tion that specifies marginal usage fees p11 p21 0 0 0 1 pk,and k−1 break points q11 0 0 0 1 qk−1. Whereas an explicitprice specification must explicitly state exactly howmany offers the firm intends to make available to con-sumers, an implicit specification embeds an infiniteseries of offers in a compact declaration. This dis-tinction is vital to understanding the relative perfor-mance of alternative pricing schemes under differentcircumstances.

The central question examined in this paper is,which pricing scheme is most efficient at price discrimi-nation when tariff size matters? Suppose a firm facesa market of heterogeneous consumers indexed by a

4 Technically, this menu contains implicit offers because one couldconsume an unspecified quantity between two explicitly specifiedlevels and pay the higher of the two prices. However, this is ared herring, because such a consumption choice would never beoptimal for the consumer.

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type parameter i ∈ I (with distribution G). Let vi4q5be type i consumer’s marginal valuation for the qthunit of consumption (with q ∈ Q, and vi4q5 decreas-ing in q). Type i consumer’s demand curve Di4p5is the inverse function of her marginal valuations.5

Let D represent the collection of these disaggre-gated demand functions along with the distributionG. Faced with D, the firm needs to announce a pricemenu, a mapping �2 Q 7→ R+ that specifies a tariffT = �4q5 for each consumption level q. It is aware thatgiven a specification � , each type i consumer will pickthe option that maximizes her net surplus Vi4q5−�4q5(if positive), where Vi4q5 =

∫ q

0 vi4q5 dq is the total val-uation for q units. Let ç4�3D5 denote the firm’s netrevenues if it chooses a specification � . Our interest isin the following question: What specification optimallysatisfies the firm’s profit from price discrimination subjectto a constraint S̄ on tariff size? That is, we solve theproblem

�∗= arg max

�

ç4�3D5 subject to S4�5≤ S̄1

subject to consumers’ IC and IR constraints, where ICis the incentive compatibility and IR is the individualrationality.

Following prior literature on nonlinear pricing, wechoose to compare three major players—quantity-price bundles, two-part tariffs, and three-part tariffs—in identifying the optimal tariff specification undertariff size constraints. The candidacy of the explicitprice specification, a menu of quantity-price bundles,is obvious: the profit-maximizing price schedule isa mapping from Q to R+; hence the optimal profitç∗4D5 can always be achieved by a menu of quantity-price bundles (though this menu may not always becomputable). Among the piecewise linear price spec-ifications, every concave outlay6 can be representedwith a menu of two-part tariffs, whereas any noncon-cave outlay can be covered with a menu of three-parttariffs (§6.4 of Wilson 1993). Hence our analysis willdirectly cover these three alternative schemes.

4. Bundles vs. Two-Part TariffsA majority of the literature on nonlinear pricingfocuses on quantity-price bundles and two-part tariffs

5 Some authors have considered valuation functions that reflectadditional factors such as consumers’ uncertainty about theirdemand (Clay et al. 1992, Lambrecht et al. 2007, Rochet and Stole2002, Png and Wang 2010) and demand externalities caused by net-work effects (Oren et al. 1982, Masuda and Whang 2006).6 Concavity represents that marginal price is always declining inquantity. An outlay could offer quantity discounts everywhere (i.e.,average price is always declining in quantity) or impose quantitypremia somewhere. Concavity is not synonymous with quantitydiscounts. An outlay can be nonconcave yet have quantity dis-counts everywhere (illustrated in Example 2).

(see, e.g., Shy 2008, Iyengar and Gupta 2009). Maskinand Riley (1984) show that when the distribution oftypes can be approximated with a continuous dis-tribution F that is strictly increasing (positive den-sity everywhere), continuous, and differentiable, therevenue-maximizing menu of bundles almost alwaysimplements a concave outlay. The key condition forthis result, that the distribution of types has a non-decreasing hazard rate, is satisfied by nearly all con-tinuous distributions and is essentially equivalent toa log-concavity requirement on the distribution func-tion (Bagnoli and Bergstrom 2005). The interest intwo-part tariffs stems from the fact that they arenearly optimal (when the market has a large numberof types) and widely employed in practice. Faulhaberand Panzar (1977) show that the revenue under amenu of two-part tariffs increases monotonically inmenu size and, in the limit, equals the revenue fromthe fully nonlinear tariff when this outlay is con-cave. In addition, a menu of two-part tariffs is nearlyoptimal even when the revenue-maximizing outlayis nonconcave (see §6.4 in Wilson 1993). The near-optimality of two-part tariffs also holds in the pres-ence of competition (Armstrong and Vickers 2001).Hence, when tariff costs are inconsequential, then amenu that is optimal in the class of 2PT menus willtypically also be optimal or near-optimal in the classof all nonlinear pricing (NLP) mechanisms.

Now consider the role of tariff costs. Theprofit-maximizing bundle is of infinite size andnonenumerable when the distribution of types is con-tinuous; with a noncontinuous function, it would stillneed to explicitly specify as many options as thenumber of consumer types. With a restriction on tar-iff size, this bundle solution is no longer feasible,and size-constrained two-part tariffs gain an advan-tage over size-constrained bundles. The argument fol-lows from Wilson’s (1993) demonstration that a smallmenu of two-part tariffs can capture a bulk of themaximum revenue possible from price discriminationunder self-selection constraints. Miravete (2007) esti-mates that a two-part tariff menu with around fiveitems would capture over 90% of the revenue from thefully nonlinear tariff. Moreover, both Wilson (1993)and Miravete (2007) provide efficient procedures tocompute the size-constrained optimal two-part tariffmenu. In contrast, the size-constrained bundle solu-tion requires a brute-force numerical procedure overa combinatorial search space, and it is likely to beinferior to a two-part tariff menu. This is because ak-item bundle menu makes only k offers to the largeconsumer population—defeating the key goal of non-linear pricing—whereas a two-part tariff menu withthe same tariff size makes an infinite series of offers,thereby extracting different levels of revenue from

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consumer types who pick the same item from themenu. The following example further illustrates thattwo-part tariffs are superior over bundles at price dis-criminating between heterogeneous consumers whenthe size of the menu is constrained.

Example 3. Suppose that there is a large num-ber of agents (say, I > 10), with agent i’s demandfunction being Di4p5 = ai · 41 − p5, and the a′

ishave a discrete uniform distribution on the inter-val (0, 1). Suppose that the tariff size is restrictedto S ≤ 4. Then the optimal bundle menu (with twoitems) is approximately 4q11T 15= 400222221001481475and 4q21T 25 = 4006666671002962975 with revenue of00148148, whereas the optimal menu of two two-part tariffs is approximately 4F11 p15= 40001610065 and4F21 p25= 4001610025 with a revenue of 0016.7

In Example 3, the tariff size restriction S ≤ 4 impliesthat the menu of either bundles or two-part tariffscontains only two items. Because of this, the menuof quantity-price bundles can fetch only two lev-els of revenue, T 1 or T 2. With a two-part tariff,the firm gains additional flexibility because the lin-ear price component allows it to embed many moreoffers. Despite stating just two tariff options, thefirm can extract different levels of revenue from eventhose consumers who pick the same option from themenu. The example is robust because the same logicapplies to alternative demand specifications. Wilson(1993, Example 6.3) demonstrates the argument usingdemand functions of the form D4p5= ai − p.

We emphasize that two-part tariffs are not alwayssuperior to bundles. This is trivially the case whentariff size is irrelevant. For certain products, two-parttariffs may not even be feasible. Contrast, for example,natural gas to bagged charcoal: the former can be soldvia a piecewise linear tariff, but the latter requiresbundles of specific quantities. In addition, with dis-crete consumer types, two-part tariffs have the weak-ness of making too many offers (even a single two-parttariff implies a continuous outlay function with aninfinite series of offers). This makes it difficult for thefirm to entice each consumer to pick the offer thatextracts the best revenue from them, and the profitis strictly lower than the bundle profit (Kolay andShaffer 2003). But the extent of disadvantage reducesas the number of customer types increases. There-fore, two-part tariffs play a prominent role in price

7 Computing the exact size-restricted optimal menu (of either twobundles or two 2PTs) poses a combinatorial challenge and is cum-bersome. The tight approximation stated above is obtained byapproximating the discrete distribution with a continuum of agentsthat are uniformly distributed on 60117 and then applying the tech-nique of (Wilson 1993; see Example 6.2). As the number of agentsI increases, these approximations become exact.

discrimination for products that are targeted to largepopulations. They are particularly attractive whenproducts are consumed over time because they allowthe customer to choose consumption quantity (andtotal price) after having chosen the tariff; in contrast,a quantity-price bundle requires customers to makeboth commitments at the same time. Moreover, com-pared with linear pricing, the fixed-fee componentof 2PTs is an attractive way to offset fixed costs invertical channels (Raju and Zhang 2005) or the per-customer transaction costs that occur even when cus-tomers have little or no use (Sundararajan 2004).

Apart from these theoretical results, the superior-ity of two-part tariffs in the presence of tariff costs isfurther validated by firms’ choice of pricing schemesin practice. Firms generally do not use quantity-pricebundles when either Q or I is large. Miravete (2007,p. 1) notes that “fully nonlinear tariffs are rarelyimplemented as such but rather through a menu ofself-selecting tariff options generally consisting of afixed fee plus a constant charge per unit of usage.”Examples include water, gas, or electric utilities; wire-less phone service plans; software as a service; legalservices; and healthcare services. When bundles mustbe used (e.g., bags of charcoal), then the firm typi-cally provides only a limited set of options, and thefew exceptions that do exist help explain why bundlesare not attractive in such settings. For instance, par-cel transportation firms such as Federal Express, UPS,and DHL display their price schedules as a menuof quantity-price bundles. Such menus contain over150 items and run several pages long, even for a spe-cific pair of zones and a specific speed of delivery.The following observation summarizes the findingsfrom existing literature about two-part tariffs versusbundles.

Observation. Quantity-price bundles are the optimalway to price discriminate when tariff size is inconsequen-tial or the market has a handful of distinct consumer types.Otherwise, two-part tariffs are usually superior to bundleswhen tariff size is limited.

5. Three-Part Tariffs vs.Two-Part Tariffs

Given the apparent superiority of two-part tariffs inprice discriminating efficiently in the presence of tariffcosts, we inquire whether there is an even more par-simonious way to price discriminate. Wilson (1993)indicates an evolution over time from linear to two-part to three-part tariffs (see the historical note at theend of his book). In this section we ask whether three-part tariffs can dominate two-part tariffs after control-ling for tariff size.

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We adopt the classical framework of indirect seg-mentation (Stole 2007), where the monopoly firmcannot separate consumers ex ante but can offer amenu of choices to induce customers to self-selectinto separate segments. Without loss of generality,we assume that the firm’s marginal cost of pro-duction, c, equals 0. We model consumer hetero-geneity via a single-type parameter i ∈ 811 0 0 0 1 I9 (afinite, enumerable support, with arbitrary density)that captures differences in marginal valuation. Thisdiscrete framing of customer heterogeneity is asymp-totically equivalent to a continuous distribution ofsegments; however, this discrete framing is mostuseful when the across-types differences are muchsharper than within-types differences. Recall that foreach i ∈ I , vi4q5 and Vi4q5 represent Segment i cus-tomer’s marginal and total valuations, respectively.Segment i has �i proportion of customers. All cus-tomers have diminishing marginal value for the ser-vice, and we rank the customer segments in increas-ing order of marginal valuations (so, for each q, aSegment i + 1 customer has higher marginal valua-tion than a Segment i customer). We do not imposeany additional restrictions on the hazard rate, demandelasticities, or other requirements for concave or uni-modal payoff function.8

Assumption 1. For every i, the demand Di is finite at0, and the marginal valuation function vi4q5 is (1) differ-entiable (and, in particular, v′

i is finite) and decreasing inq, with (2) vi+14q5 > vi4q5 for all q.

Customer i’s demand function can be written asqi = Di4p5, where vi4qi5 = p. Let p̄i be Segment i’smarginal valuation for the first unit of the prod-uct (so that D4p5 = 0 for p > p̄i). The firm offers amenu of two-part (or three-part) tariffs from whicha Segment i customer will choose the tariff and con-sumption quantity qi that maximizes her net surplus(if positive, see Appendix A.1 for details). The opti-mal, fully separating, I-item menu of two-part tariffssolves the following revenue maximization problem(after substituting in all the binding constraints):

maxi2 4Fi1 pi5

ç2PT =

I∑

i=1

�i4piDi4pi5+ Fi5

=

I∑

i=1

�i

(

piDi4pi5+i∑

j=1

∫ pj−1

pj

Dj4�5 d�

)

0

8 Strictly speaking, we assume �is such that the optimal 2PT menuis fully separating: there is a distinct tariff for every segment. How-ever, if it were not and the optimal 2PT menu implements somepooling of segments, then we just redefine I as the number ofunique items in the menu (fewer than the original I) and redefinethe affected �is as the sum of consumers that pick the same tariffitem i from the optimal menu. At any rate, the ability of a small-sized 3PT menu to outperform the optimal 2PT menu increaseswhen the 2PT menu has some pooling.

The optimal tariff design follows directly from thefirst-order conditions for the optimal p′

is and then sub-stituting the optimal p′

is into the F ′i s.

Lemma 1. Under Assumption 1, the optimal, fully sep-arating menu of two-part tariffs 64F ∗

i 1 p∗i 57 satisfies

p∗

i =

(1 −∑i

j=1 �j

�i

)

4Di+14p∗i 5−Di4p

∗i 55

−D′i4p

∗i 5

1 (1)

F ∗

i =

∫ p̄1

p∗1

D14�5 d� +

∫ p∗1

p∗2

D24�5 d�

+ · · · +

∫ p∗i−1

p∗i

Di4�5 d�1 (2)

with p∗1 > p∗

2 > · · ·> p∗I = 0.

Note that p∗I = 0 because we assumed that c = 0.

Lemma 1 can be deployed to produce exact solutionsfor the optimal 2PT menu after a specific demandfunction is identified (the solution for linear demandis illustrated in §6.3).

6. ResultsBefore diving into the results, we provide a map ofthe results and their positioning with respect to theliterature and design of price discrimination schemesin practice. Our first result, Theorem 1, formalizesthe conclusion of Example 2 by deriving a sufficientcondition for the existence of a single three-part tar-iff that generates higher revenue than a 2PT menuof any size. Section 6.3 illustrates the general resultsfor the special case that each individual customer haslinear demand. Theorem 2 in §6.4 provides the gen-eral result: conditions for a small-K three-part tariff tosurpass the optimal 2PT menu. Finally, §6.5 explainswhy the design of the superior 3PT menu is less sen-sitive to information error and hence imposes lowerinformation requirements and tariff design costs. Allproofs are in Appendix A.2.

From §4, we recall that two-part tariffs are opti-mal or near-optimal when the distribution of con-sumer types I is (i) continuous and (ii) satisfies certainsmoothness assumptions (essentially, log-concavity)that lead to the optimal schedule being concave.When the optimal schedule is not fully concave, then3PTs have an inherent advantage. The striking fea-ture of Theorem 1 is that a single 3PT (or in its gen-eral counterpart, Theorem 2, a small-K 3PT menu)outperforms the large and possibly infinite 2PT menu.The superiority of 3PT does not require imposition ofquantity premia, despite nonconcavity in the outlay(see Example 2). Our results also apply when condi-tions (i) or (ii) fail. Log-concavity, which implies thatpreferences not be too diverse (Cowan 2007), is vio-lated when adjacent types are distinct, leaving pointsof zero mass in the density function. One example

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is when the market has distinct segments such ashome and business users, students and professorsand labs and corporate users, or professional andamateur users. In such markets, cross-segment dif-ferences may be much sharper than within-segmentvariations (Wedel and Kamakura 2000), leading tomultiple modes in the (continuous) demand distribu-tion (Johnson and Myatt 2003). Then the apparatus ofMaskin and Riley (1984) becomes inapplicable; how-ever, the (more general) formulation of §5 still applies,and the optimal 2PT menu is still given by Lemma 1.The 3PT design outperforms this 2PT menu whilekeeping tariff size small. Finally, when the size of themenu is constrained, then our results are meaningfuleven if the unconstrained-optimal schedule is concaveeverywhere.

6.1. Single Three-Part Tariff OutperformsOptimal 2PT Menu

We develop the result by constructing a particular sin-gle three-part tariff 4F̂ 1 Q̂1 p̂5 using the optimal 2PTmenu. Set the free allowance Q̂ to q∗

1 , the optimal con-sumption level of Segment 1 under the optimal menuof two-part tariffs. Similarly, set F̂ to the total amountpaid by Segment 1 (sum of the fixed fee and usagefee). Therefore, by construction, the single 3PT andthe 2PT menu both extract identical profit from Seg-ment 1 customers. Now, set the third parameter p̂ inthe 3PT to optimize the profit from Segment 2 cus-tomers, p4D24p5− Q̂5.

Quasi-optimal three-part tariff:

Q̂ = q∗1 =D14p

∗15

F̂ = F ∗1 + p∗

1Q̂

p̂ =4D24p̂5− Q̂5

−D′24p̂5

0 (3)

Let ç̂3PT denote the profit from this design. Note thatthis is not an optimal tariff even within the subspaceof single three-part tariffs. In fact, because p̂ is opti-mized using Segment 2 demand only (rather thanthe types higher than 2), it is equivalent to solvinga problem where types 31 0 0 0 1 I collapse into type 2.Therefore this quasi-optimal tariff can be viewed as a“least optimistic” 3PT design. Yet even this subopti-mal single three-part tariff can outperform the opti-mal menu of two-part tariffs.

Theorem 1 (Single Three-Part Tariff Outper-forms the Optimal 2PT Menu). Suppose there areI customer segments, with demand functions Di4p5 andweights �i, such that the optimal menu of two-part tariffsis fully separating and has I items of the form 4F ∗

i 1 p∗i 5.

Then there always exists a threshold �̂1 ∈ 40115 such that if�1 ∈ 4�̂1115, then the quasi-optimal single three-part tariff

defined in Equation (3) generates strictly higher revenuethan the optimal 2PT menu.

The result is striking because the two-part tariffdesign carries a tariff size 2I while the three-part tariffgenerates higher revenue with only three parameters(recall Example 1, where 2I = 12). Clearly, the latterbecomes even more advantageous once tariff manage-ment costs are factored in. The intuition behind theresult is that when the market is tilted toward thelow-value segment, the 2PT design must set a rela-tively low p∗

1 in order to minimize the deadweightloss for this segment. Yet this low p∗

1 reduces thefirm’s price discrimination ability by making it harderto induce higher-type customers to pick a differentitem from the menu (i.e., successively, an item withhigher F and lower p). In contrast, the three-part tariffdesign decouples the unit variable fee for the low-value segment and the remaining higher-value seg-ments. It serves as a quantity-price bundle for thelow-value segment (i.e., they consume no more thantheir allowance Q), whereas the higher segments con-sume fewer units than under the 2PT menu (becausep̂ > p∗

2 = 0). Thus, the single three-part tariff functionsas a hybrid that combines the best features of linearpricing and quantity-price bundle.

To see the logic and intuition behind the result moreclearly, consider the special case of I = 2. In Figure 2,T 1 represents the revenue realized from Segment 1via the 4F ∗

1 1 p∗15 from the optimal 2PT menu, and q∗

1is the corresponding consumption level. In the single3PT, we set F̂1 to T 1 and Q̂ = q∗

1 . Now, we just needto compare the two pricing schemes on the revenuethey generate from Segment 2. For the 2PT menu,each Segment 2 customer generates an additional rev-enue R2 (i.e., each Segment 2 customer pays T 1+R2).With the quasi-optimal 3PT, each Segment 2 customergenerates a revenue R3 in excess of T 1. Figure 2(a)depicts these two regions. Note that the difference inprofits ç̂3PT −ç2PT is the difference R3 −R2. Cancel-ing the common region in R2 and R3, we can writeç̂3PT − ç2PT = R̃3 − R̃2, as depicted in Figure 2(b). Inpanel 2(a), the triangular area marked DWL1 is thedeadweight loss for Segment 1 customers. When �1is large, the optimal 2PT menu has a smaller p∗

1 inorder to reduce this deadweight loss (see Figure 2(c)).However, the reduction in p∗

1 limits the firm’s abilityto price discriminate and extract excess revenue fromSegment 2 customers, thereby reducing the region R̃2.Because of this, the single three-part tariff starts dom-inating the optimal 2PT menu for large enough �1.

Finally, we note that the incremental gain fromusing a single three-part tariff is not monoton-ically increasing in �1. That is, if ç2PT 4�15 andç̂3PT 4�15 denote the profits from optimal 2PT and thequasi-optimal 3PT, respectively, at a given �1, then

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Figure 2 Revenues from Each Segment 2 Customer, in Excess of Revenue from Segment 1 Customers, for the Optimal 2PT Menu (Region R2) andthe Quasi-Optimal Single 3PT (Region R3)

Quantity

T1

q1* q2

*

p2*

p1*

p̂

Pric

e

Dt

D2

R2

R3

q1* q2

*

p2*

p1*

p̂

Pric

e

Quantity

T1

R3~

R2~

q1* q2

*

p2*

p1*

p̂

Pric

e

Quantity

R3~

R2~

T1

(a) Revenue in excess of revenue fromSegment 1 customers (for low λ1)

(b) Revenue comparison after eliminatingregion common to R2 and R3

(c) Revenue comparison when�1 is high (implying low p1)*

DWL1

ã4�15=ç3PT 4�15−ç2PT 4�15 is not monotone in �1.This can be understood by examining the behaviorof ã4�15 starting at �1 = 0. For very small values of�1, the low type is ignored, and hence ã4�15 = 0 asboth the two-part tariff and three-part tariff extractfull surplus from Segment 2. As �1 increases, bothtypes get served, and the 2PT menu discriminatesbetter than the single 3PT; hence ã initially becomesnegative. But as �1 continues to increase, the logic ofTheorem 1 kicks in: both types continue to be served,and ã turns positive as the single quasi-optimal 3PTdominates. But for very high �1, the market collapsesinto a single customer type, and ã becomes 0 (takethe extreme case �1 = 1). Therefore, Theorem 1 is mostuseful when �1 is larger than �̂ but not too close to 1.

6.2. Determining When to EmployThree-Part Tariffs

Whereas Theorem 1 merely asserts an existence con-dition for the superiority of a single three-part tariff,the actual values of the threshold �̂1 can easily be cal-culated for any given demand functions. For instance,if the demand functions of individual consumers areof the form Di4p5 = ai − bi4p5, then an upper boundfor �̂1 is 0.75 (see Corollary 1). That is, �̂1 = 0075 issufficient (but not necessary) for a single three-parttariff to dominate. The parameter conditions for theresult cover many practical applications. The upperbound of 0075 simply implies that there are threetimes as many “low-value” consumers as there arehigher-value consumers. This distribution occurs inseveral markets, for instance in information technol-ogy (IT), telecommunications, healthcare, and otherservices where a small segment of “heavy” usershas vastly different consumption patterns than theremaining majority of “light” users. For IT services,the ratio of light to heavy users routinely exceeds 4:1(20% of users account for over 80% of usage).

We can think of a few examples where the firmemploys only a single three-part tariff even thoughmarket conditions suggest intensive price discrim-ination (because the firm faces large numbers ofcustomers who are substantially heterogeneous intheir demand preferences). For example, the Uni-versity of California (UC) offers a single legal plan(to tens of thousands of employees) that bundles afree allowance of eight hours’ legal consultation intothe annual premium, with additional hours billed ata specified rate. The pricing strategy fits our find-ings because (based on discussions with the bene-fits managers at UC) most employees have very lowconsumption while a small fraction of heavy usersaccounts for the majority of consumption. Similarly,many fitness clubs include one or two sessions witha physical trainer in the initiation fee and charge(about $60/session) for additional sessions, consumedby only 10%–20% of members. Another example isroadside assistance services where the annual mem-bership fee includes a towing allowance up to somedistance (e.g., in Kentucky, AAA covers 100 miles forthe $37 annual fee and charges $3 per mile abovethat). Arguably, a majority of members (both in expec-tation and ex post) utilize only this free allowance,and a small number goes over.9 Other examples thatfit the pattern (though we do not have empirical datato confirm the claim) include website design compa-nies that offer up to N pages with the service planand charge a fee for additional pages and Web hostingcompanies that offer similar plans using some compu-tational metric such as the number of gigabytes trans-ferred, the number of FTP accounts, or the number ofindividual email accounts.

9 We thank one of the reviewers for pointing out this example andfor inspiring us to verify consumption patterns with the Universityof California and fitness clubs.

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6.3. Illustration: Linear DemandsThe computations described thus far can be mademore precise given a specific functional form ofthe demand functions for individual consumers.We demonstrate using demand functions of the formDi = ai − bip, where bi ≥ bi+1 and ai+1/bi+1 ≥ ai/bi forall i. This specification implies that for each Seg-ment i, the marginal valuations (vi4q5= 4ai −q5/bi) aredecreasing at a constant rate 1/bi, and the total val-uation V 4q5 is quadratic in q. This functional formis commonly employed to model the demand pref-erences when consumers demand multiple units ofa product (Lambrecht and Skiera 2006). Moreover,it is not very restrictive because it applies to indi-vidual demand rather than market demand, and theaggregate market demand faced by the firm remainsnonlinear.

Corollary 1. For demand functions Di4p5= ai − bip,

p∗

1 =41 −�154a2 − a15

�1b1 + 41 −�154b2 − b151 p̂ =

a2 − a1 + b1p∗1

2b21

and the quasi-optimal three-part tariff 4F̂ 1 Q̂1 p̂5 generatesstrictly higher revenue than a 2PT menu of any size when-ever �1 ∈ 4�̂1115 (where �̂1 = 43b25/4b1 + 3b25) or, equiva-lently, �1/41 −�15 > 43b25/b1 and �1 < 10

For the special case where b1 = b2, this yields athreshold �̂1 = 0075 as stated above. Moreover, oncethe firm knows the parameter values of the demandfunction, it can use a more refined computational pro-cedure to compute a tighter threshold �̂1. For exam-ple, this procedure would yield �̂1 =

√005 ≈ 00707

for the case where bi = 1 for all i. Finally, if �1 > �̂1does not hold, then it is still possible that, for some1 <K < I , a menu of K 3PTs (size 3K) generates moreprofits that the optimal menu with m2 2PTs (size 2m2),even when 3K ≤ 2m2. This is demonstrated in the nextsection.

6.4. K-Item Menu of Three-Part TariffsTheorem 1 states an extreme result, that a single three-part tariff can outperform an arbitrarily large optimal2PT menu when there is a sufficiently high concentra-tion of low-type (Segment 1) customers. The thresh-old �̂1 depends on the demand functions Di and theweights �i. However, if the demand distribution issuch that �1 is not larger than the threshold �̂1, thena single three-part tariff may fail to beat the optimal2PT menu. Might there still be some value in the logicof price discriminating via a small menu of three-parttariffs?

We identify conditions under which a K-item menuof three-part tariffs (with small K such that K <42m25/3, where m2 is the number of options in theoptimal 2PT menu) produces higher revenue than a

2PT menu of any size. As a first step, we extend thelogic of Theorem 1 to K = 2, defining a two-item,three-part tariff as follows. The first item in this solu-tion is a “package deal” (a quantity-price bundle)identical to the first item in the 2PT menu: Q̂1 =D14p

∗15

and F̂1 = F ∗1 + p∗

1Q̂1, with p̂1 set sufficiently high todeter additional consumption by Segment 1. The sec-ond item is also derived from the second item in the2PT menu—Q̂2 = D24p

∗25 and F̂2 = F ∗

2 + p∗2Q̂2—and the

p̂2 in this offer is chosen to optimize the 3PT residualprofit from Segment K + 1 = 3. This item serves as apackage deal for Segment K = 2, whereas customersin Segments 31 0 0 0 1 I consume additional units at p̂2.This two-item 3PT menu beats the optimal 2PT menuso long as �2/41 − �15 is large enough. If this fails,repeat the computation for K = 3. The formal result isstated by relating the threshold level at the Kth typeto the discretized hazard rate HR4K5.10

Theorem 2 (K-Item Three-Part Tariff Outper-forms the Optimal 2PT Menu). In the setting of The-orem 1, there exists for every K = 11 0 0 0 1 I − 1 a corre-sponding threshold �K < 1 such that a K-item menu ofthree-part tariffs outperforms the optimal 2PT menu ifHR4K5 is finite and HR4K5 > �K . The 3PT menu has Q̂i =

Di4p∗i 5 and F̂i = �∗

21 i for all i ≤ K and p̂K = arg maxp4p ·

4Dk+14p5− Q̂K5 (while the remaining p̂is can be set suffi-ciently high).

An example of price discrimination via a shortmenu of three-part tariffs is the wireless telecommu-nications industry. Most cell phone service providersin the U.S. market their wireless voice service to tensof millions of consumers with a handful of three-part tariffs. To determine such a menu, a firm wouldlook for the smallest K that satisfies Theorem 2. Theresult demonstrates that a short 3PT menu can dom-inate two-part tariffs on both revenue and tariff costeven when the hazard rate satisfies the usual “mono-tone increasing” property (whereupon the optimalschedule implies a concave tariff outlay and quantitydiscounts everywhere). The exact form of the thresh-old (for each K) can be specified algebraically andmore tightly for any given functional form of Di’s. Forinstance, if we assume the linear demand specifica-tion used in §6.3, then a K-item 3PT menu generatesmore revenue than the optimal 2PT menu if HR4K5 isfinite and

HR4K5 > �K =3bK

bK + 3bK+10 (4)

Note that the above condition involves the distribu-tion of types and is distinct from the monotonicity of

10 For demand massed at points z1 < · · · < zn, the discrete hazardrate at zi ≤ zn is HR4zi5 = Pr4Z = zi5/41 −

∑

j<i Pr4Z = zj 55 (Barlowet al. 1963, Shaked et al. 1995).

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the hazard rate (a distribution with an increasing haz-ard rate can be skewed to the left or to the right; forexample, consider the Burr distribution with variousparameters). The condition is independent of whetherthe optimal NLP mechanism offers quantity premiaor quantity discounts.

To illustrate the mechanics of Theorem 2, sup-pose I = 4, with 4�11 0 0 0 1�45 = 400351004510015100055,v14q5 = 4 − q, v24q5 = 5 − q, v34q5 = 6 − q, and v44q5 =

7 − q. The optimal 2PT menu is fully separating, withrevenue of 8.66. To design the 3PT menu, we firsttest Equation (4) at K = 1: this test fails (because itrequires 0035 > 3/4), and it may be verified that thequasi-optimal single three-part tariff generates lowerrevenue (8.17). Next we test the condition at K = 2;i.e., 0045/0055 > 3/4. The test succeeds; the two-item3PT menu produces higher profit.

Theorem 2 can be interpreted as stating that three-part tariffs are a more efficient way to price discrimi-nate when consumption is dominated by a small groupof heavy users. As to the applicability of the condi-tion, we note that (1) it needs to hold only for somesmall K (rather than just K = 1), and (2) it can besatisfied either by the exogenous distribution of seg-ments or by the “postpooling” distribution createdafter customers pick their surplus-maximizing tar-iff from the 2PT menu. These points are graphicallydemonstrated in Figure 3 and illustrated with the fol-lowing example.

Example 4. Suppose that there are six types of cus-tomers with demand functions Di4p5 = ai − p, withai+j = ai + j�:

1. If �1 = 005 and �2 = 004, then the condition fails atK = 1 but is satisfied at K = 2 (see Figure 3(a)). A two-item 3PT menu outperforms the six-item optimal 2PTmenu.

2. If �1 = 0031, �6 = 0029 (and the remaining seg-ments are each 10% of the market), the demanddistribution is not “sufficiently right-skewed” (see Fig-ure 3(b)), and the condition fails at every K < I . How-ever, because the optimal menu of two-part tariffspools segments 1 through 5, we compute the post-pooling �s (thus the revised �1 = 0071; see Figure 3(c))and find that the condition is satisfied at K = 1. Thus,a single 3PT outperforms the optimal 2PT menu.

6.5. Sensitivity to Information ErrorTheorems 1 and 2 have demonstrated that when thefirm faces a large heterogeneous market, a singlethree-part tariff (or, more generally, a relatively smallmenu of such tariffs) produces higher revenue than amenu of two-part tariffs of any size. There is a sec-ond benefit: designing the three-part tariff requiresprecise information about segments 11 0 0 0 1 k+ 1 only,less than what is needed to design the optimal 2PT

Figure 3 Distribution of Customer Types

0.6

0.4

0.2

02 3 4 5 61

0.4

0.2

02 3 4 5 61

(a) Exogenous distribution: Right-skewedat K = 2

(b) Exogenous distribution: Not right-skewedat any K

Customer types

Customer types

0.8

0.6

0.4

0.2

061

(c) Postpooling: Right-skewed atK = 1

Customer types

Notes. The exogenous distribution of segments is not right-skewed. How-ever, the optimal 2PT menu in Example 4 pools types 1–5, and the resultingdistribution (of customers that pick the same item from the menu) is skewedto the right.

menu. It does not require information about higher-value customer segments, making it unnecessary forthe firm to undertake costly information-gatheringactions that might infringe on the privacy preferencesof its highest-value customers (which it would, whendesigning the optimal 2PT menu). We start with anexample and then provide a formal proof.

Example 5 (Information Requirements). Assumethat I = 3, with Di4p5 = ai − p and �1 = 008, �2 =

0018, and �3 = 0002. The firm knows, correctly, thata1 = 1 and a2 = 2, and it believes that a3 = 3. With

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this information, the optimal 2PT menu is 4F ∗1 1 p

∗15 =

40046875100255, 4F ∗2 1 p

∗25 = 4005339511001111115, and

4F ∗3 1 p

∗35= 400861111105. Using 4F ∗

1 1 p∗15, the correspond-

ing quasi-optimal single 3PT is 4F̂ = 00468751 Q̂ =

00751 p̂ = 006255.

Denote the true (or realized) value of a3 as aR,and consider the ex post performance of the twotariff designs for various values of aR ∈ 6a2 = 2147.Let ç2PT 4aR5 and ç3PT 4aR5 denote, respectively, the2PT menu and single 3PT ex post profits when aR isrealized:

1. 2PT menu: For aR ∈ 63147, customer 3 will pay F ∗3

(because p∗3 = 0) and consume aR, yielding a constant

ç2PT 4aR5 = 00526111. Incorrect information hurts thefirm because it should have set a higher F ∗

3 . For aR ∈

62137, customer 3 will choose to buy 4F ∗2 1 p

∗25 and will

pay F ∗2 + p∗

24aR − p∗25. The quantity ç2PT 4aR5 increases

monotonically, and incorrect information still hurtsthe firm because it should have set a lower F ∗

3 andinduced this customer to separate.

2. Single 3PT: Recall that this is a least optimisticdesign obtained by collapsing all higher types intotype 2 (hence equivalent to assuming a3 = 2). There-fore, the 3PT design will be identical for all values ofaR, and the ex post profit under incorrect information(even though it increases with aR) will be identical tothe perfect-information profit. Hence incorrect infor-mation about Segment 3 does not hurt the firm in thiscase.We write the percentage superiority of the 3PT overthe 2PT menu as a function of the true realization aR:

PctSup4aR3a35 =ç3PT 4aR5−ç2PT 4aR5

ç2PT 4aR5

=

000026 + 000103aR005219 + 000125aR

if aR ∈ 621371

−000042 + 000125aR005219 + 000125aR

if aR ∈ 631470

Computing the respective ex post profits, we see thatas aR increases from 2 to 4, PctSup4aR3a35 increasesin an almost linear fashion from 6.2% to approxi-mately 9% (there is a kink at aR = 3). This meansthat, with probability 1, the single 3PT will outper-form the 2PT menu for any possible realization ofa3. The advantage of a single 3PT will be more pro-nounced if obtaining the information pertaining to thebelieved value of a3 (which is required for design-ing the 2PT menu) involves nonzero costs. The 3PTdesign requires knowing only that a3 is in a certaininterval.

Now we prove the information claim formally, firstfor k = 1 and then for higher k. To prove for k = 1,we must show that the firm can both determine thedesirability of a single 3PT and design it with less

information than needed to compute the 2PT menu.First, consider the determination of desirability, i.e.,the threshold �̂1 in Theorem 1. The result is eas-ily seen for the special case where the individualdemand functions are linear (Corollary 1) because�̂1 = 43b25/4b1 + 3b25 is independent of preferences oftype 3 and higher. The claim for the general demandcase is proved in the following result.

Corollary 2. The threshold that ensures superiorityof the quasi-optimal three-part tariff (�̂1 of Theorem 1)depends only on the demands of Segment 1 and 2 cus-tomers.

Next, consider the design of the single 3PT. FromEquation (3), this design only depends on the 2PTdesign for Segment 1 (i.e., F11 p1) and the demandfunction for Segment 2. From Lemma 1, F1 and p1,in turn, also depend only on the demand parame-ters for Segments 1 and 2. The rest of the modelparameters (D31 0 0 0 1DI , �21 0 0 0 1�I ) are not needed. Wenote that one could increase revenue further by fullyoptimizing the single three-part tariff, but that wouldrequire precise information about the demand prefer-ences and distribution of all the segments. Hence wecan propose the following managerial heuristic.

When the firm is generally aware that the mar-ket is dominated by the low-type segments (i.e., highenough �1 as required by Theorem 1), then it is betterto (a) invest in learning accurately the preferences ofSegments 1 and 2 and then construct a single three-part tariff as specified above, rather than (b) incurgreater investments to learn the preferences of all con-sumer segments and design an optimal 2PT menu.

Finally, we note that the claim about informationrequirements also applies when the firm must employa small K-item 3PT. To determine whether Equa-tion (4) is satisfied at some specific k, the firm needsdetailed information only for types 11 0 0 0 1 k. Often,firms can easily identify factors that might signal theneed for a short menu of three-part tariffs rather thanrelying on self-selection from a very large menu. Thisapproach is particularly useful when the consump-tion pattern (the Ds in our model) can be predictedby demographic factors (such as ethnicity, race, gen-der, income level) for which the aggregate distribution(the �s in our model) is known at the local or regionallevel but which cannot be used as the basis for pricediscrimination.

7. Concluding RemarksThis paper has examined the firm’s choice of a pric-ing scheme when tariff management costs are signif-icant enough to impose a constraint on tariff size.A quantity-price bundle works well when the markethas a small number of distinct consumer segments

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and when tariff size is inconsequential. Otherwise,two-part tariffs have previously been considered tobe highly effective and more appropriate than bun-dles. We show that when it is costly to devise andmanage discriminatory tariffs, firms that face a largemarket with many diverse consumer types are bet-ter off using three-part tariffs, a pricing scheme thatprice discriminates more efficiently. We derived thisresult using a formulation of discrete consumer types,which can approximate a continuous distributionwhen the number of types is arbitrarily large.

We designed a quasi-optimal menu of three-parttariffs that outperforms the optimal menu of two-part tariffs (and hence outperforms a 2PT menu ofany size) when the distribution of types is sufficientlyskewed to the right (more customers in the low-type than in the high-type segments). This conditionis observed in many markets—for instance, when amajority of customers are light users whereas a smallfraction are heavy users. When a single three-part tar-iff does not dominate, we show that a short menuof three-part tariffs will generally still be more effi-cient at price discrimination than a much larger menuof two-part tariffs. The novel insight of our analysisis that the goals of price discrimination can be bet-ter achieved by employing additional pricing instru-ments (i.e., increasing the number of parameters inthe tariff) than by inducing more fine-grained self-selection (i.e., increasing the number of items in themenu). For instance, in the last decade, telecommuni-cations firms have introduced many instruments suchas rollover minutes, contract periods, off-peak dis-counts, and friends and family specials.

Our results apply in many business settings thatemploy three-part tariffs. In addition to the exam-ples discussed in §6.2 (e.g., legal plans, fitness clubs,automobile clubs), applications include telecommu-nications and IT industries (see, e.g., Mitchell andVogelsang 1991, Masuda and Whang 2006), park-ing, healthcare plans, and many services and util-ities (Brown and Sibley 1986). For instance, phonecompanies charge a fixed fee for the line and addi-tional service fees, credit card firms charge merchants(and consumers) an annual fee plus per-transactionfees, bars and nightclubs have cover and per-drinkfees, and managed hosting plans charge monthly feeplus fees linked to quantity. Multipart tariffs alsoplay a pricing and coordination role in intra- andinterbusiness contracts such as manufacturer–retailercontracts, which often feature advance commitmentplus an option to buy additional units (Kuksov andPazgal 2007, Cachon 2004), and bonus-based compen-sation packages for top executives or sales personnel(Chen 2005).

Our results should be useful to marketing managersconcerned about the cost of price discrimination and

about the information necessary to price discriminate.Finally, our findings are also relevant when firms’fears about tariff costs and complexity lead them tooffer a uniform price to all customers (either a sin-gle per-unit fee or a single fee for unlimited con-sumption). In such cases, the firm sacrifices profitsby opting for too much simplicity. Instead, a singlethree-part tariff could raise profit substantially with-out compromising simplicity. Moreover, because thefixed-fee component of a 3PT would deter the lowest-value segments, the firm could extend the pricingscheme to a small menu consisting of one three-parttariff (with fairly low F and Q) and an unlimited con-sumption price with high F . The evolution of VirginMobile’s pricing plans is a good illustration. Concernsabout tariff simplicity and costs motivated a launchwith extremely simple and uniform pricing, but Vir-gin now employs a small menu of three-part tariffs toget the dual benefits of simplicity and higher revenue.

There are multiple reasons why firms prefer three-part tariffs, and the advantage of 3PTs over 2PTs canbe amplified under any of these characteristics. 3PTsare preferable when (1) in the presence of segmentdevelopment and data collection costs that increasewith the length of the menu, (2) there are negativeexternalities as a result of congestion effects (Masudaand Whang 2006), (3) there are some time-inconsistent(naïve) consumers with uncertain demand who areunaware of the fact that they are time inconsis-tent (Della Vigna and Malmendier 2006, Eliaz andSpiegler 2006), and (4) consumers exhibit a biastoward tariffs with high allowance, preferring tariffswith high Q̂ independent of their consumption pat-terns (Lambrecht and Skiera 2006). And finally, 3PTscan be used to target overconfident consumers whounderestimate the variance of their future demands(Grubb 2009).

Appendix

A.1. Customer Choice Under a Multipart TariffGiven a multipart tariff, a segment i customer’s optimalconsumption quantity qi depends only on Di and p. Specif-ically, qi = Di4p5 because this consumption level producesmaximum surplus:

Si4p5=

∫ Di4p5

04v4q5− p5dq =

∫ p̄i

pDi4�5 d�0 (5)

The remaining components of the tariff will influencewhich, if any, tariff to accept. Since a two-part tar-iff 4F 1 p5 implies an outlay T 4q5 = F + pq, the cus-tomer will accept the tariff if and only if Si4p5 ≥ F .Similarly, the customer would accept a three-part tar-iff 4F 1Q1p5—which implies an outlay T 4q5= F + p4q −Q5+

but also provides Q “free” units—if and only ifSi4p5+ p · min8Di4p51Q9≥ F + p · max8Di4p5−Q109. Finally,given a menu of choices, the customer will accept the itemthat maximizes Si4p5− F .

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A.2. Formal Proofs of Results

Proof of Lemma 1. The seller’s problem is

max4F11 p151 0001 4FI 1 pI 5

I∑

i=1

�i4Fi + piDi4pi55

subject to F1 ≤

∫ v1405

p1

D14�5 d�1

F2 ≤ F1 +

∫ p1

p2

D24�5 d�1

000

FI ≤ FI−1 +

∫ pI−1

pI

DI 4�5 d�0

The above constraints are all binding and can be substitutedinto the objective function. Then the problem becomes

maxp11 0001 pI

�1

[

p1D14p15+∫ a1

p1

D14�5 d�

]

+�2

[

p2D24p25+∫ a1

p1

D14�5 d� +

∫ p1

p2

D24�5 d�

]

+ · · · 0

The first-order condition (FOC) for p1 yields

p1 =�2 + · · · +�I

�1

D24p15−D14p15

−D′14p15

0

Similarly, for 1 ≤ i ≤ I , the FOCs yield

pi =�i+1 + · · · +�I

�i

Di+14pi5−Di4pi5

−D′i4pi5

0 �

Proof of Theorem 1. Both the optimal 2PT menu andthe quasi-optimal single 3PT of §6.1 extract the same sur-plus from Segment 1 customers. Therefore, we only need toshow that the superior single 3PT of extracts more surplusthan the optimal 2PT from customers in Segments 21 0 0 0 1 I .Let �∗

21 i be the surplus extracted from a consumer of type iusing the optimal 2PT menu. The total profit under the opti-mal 2PT is then given by

ç2PT =

I∑

i=1

�i ·�∗

21 i0 (6)

For i = 21 0 0 0 1 I , the Segment i profit in excess of the profitfrom the adjacent segment i− 1 is

4�∗

21 i −�∗

21 i−15

= p∗

i 4Di4p∗

i 5−Di−14p∗

i−155+∫ p∗

i−1

p∗i

4Di4�5−Di−14�55 d�

≤ p∗

i−14Di4p∗

i 5−Di−14p∗

i−1550 (7)

This follows from the fact that the area representing theexcess profit is a parallelogram contained in a rectangle withheight p∗

i−1 and base Di4p∗i 5 − Di−14p

∗i−15. Now, let �̃2PT 1 i =

4�∗21 i − �∗

2115 denote the residual profit from Segment i,defined as the Segment i profit in excess of the profit from

Segment 1. Therefore, we can establish an upper bound onthis residual as follows:

�̃2PT 1 i =

i∑

j=2

4�∗

21 j −�∗

21 j−15

≤

i∑

j=2

p∗

j−14Dj4p∗

j 5−Dj−14p∗

j−155

≤ p∗

1

i∑

j=2

4Dj4p∗

j 5−Dj−14p∗

j−155

= p∗

14Di4p∗

i 5−D14p∗

1550 (8)

For comparison, under the single 3PT, the residual profitfrom Segment i customer is

�̃3PT 1 i = p̂4Di4p̂5− Q̂50 (9)

With the above construction of the residual profits underthe two pricing schemes, and as a result of the direction ofthe inequality in Equation (8),

ç̂3PT −ç2PT ≥

I∑

i=2

�i4�̃3PT 1 i − �̃2PT 1 i50 (10)

Now consider what happens when �1 is relatively large(close to 1)—equivalently, when �1/4

∑Ii=2 �i5 is large. From

Equation (1) it can be inferred that p∗1 gets very small (since

v′i is finite, D′

i is never 0, and the other term—the ratio4D24p5−D14p55/−D′

i4p5—in Equation (1) is bounded for allp). Therefore, at the extreme, as �1 → 1, we know that each�̃2PT 1 i → 0. And, in general, the residual profits �̃2PT 1 i in the2PT menu become very small as �1 gets large. By contrast,each of the residuals in the quasi-optimal 3PT is boundedfrom below: the right-hand side of Equation (9) remainsstrictly positive because p̂ remains bounded from below byarg maxp8pD24p5−D14059, which is strictly positive, and byAssumption 1, implying Di4p̂5 > Q̂ for i > 1. Putting every-thing together leads to the conclusion that as �1 gets large,the quasi-optimal 3PT generates higher revenue than theoptimal menu of two-part tariffs. �

Proof of Theorem 2. The proof works along the linesof the proof for Theorem 1. For the 2PT menu, the rev-enue from Segment i (ç∗

21 i) and the total revenue ç2PT isas in Equation (6). Now, for Segments K + 11 0 0 0 1 I , definethe “excess revenue over Segment K” for Segment i =

K + 11 0 0 0 1 I , analogous to Equation (7) but replacing 2 withK + 1. For the 3PT, replace p̂ and Q̂ with p̂K and Q̂K .Now, corresponding to Equations (8) and (9), we get theinequalities

�̃2PT 1 i ≤ p∗

K4Di4p∗

i 5−DK4p∗

K551

�̃3PT 1 i = p̂K4Di4p̂K5− Q̂K50

From Lemma 1, p∗K becomes arbitrarily small as

the ratio 4∑I

i=K+1 �i5/�K decreases, or equivalently, if4∑I

i=K �i5 < 41 + zK5�K (for some zK > 0). Rearrangingthis term yields the condition HR4K5 > �K , where �K =

1/41 + zK5 < 1. �

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Proof of Corollary 1. Let the optimal 2PT be4F ∗

1 1 p∗151 0 0 0 1 4F

∗n 1 p

∗n5, and let Q̂ = D14p

∗15. Equation (18) can

be reduced to

p̂ =D24p̂5− Q̂

b20 (11)

Now using Equations (8) and (9) (and the notation intro-duced in the proof of Theorem 1), we have

�̃3PT 12 − �̃2PT 12 > 4p̂− p∗

156D24p̂5− Q̂7− p∗

16D24p∗

25−D24p̂57

= 4p̂− p∗

156D24p̂5− Q̂7− p∗

1 · b2 · p̂

= 4p̂− p∗

156D24p̂5− Q̂7− p∗

16D24p̂5− Q̂71 (12)

where the second equality holds because b2p̂ =D24p̂5− Q̂ byEquation (11). For i > 2,

p̂6Di4p̂5−Di−14p̂57− p∗

i 6Di4p∗

i 5−Di−14p∗

i 57

> 4p̂− p∗

i 56Di4p̂5−Di−14p̂571

where the second inequality holds because Di+1 −Di is non-decreasing in p (recall bi ≥ bi+1) and 6Di4p̂5 − Di−14p̂57 >6Di4p

∗i 5−Di−14p

∗i 570

The above inequalities imply that if p̂ > 2p∗1 , then �̃3PT 1 i >

�̃2PT 1 i for all i ≥ 2, and therefore

p̂ > 2p∗

1 ⇒ ç̂3PT >ç2PT 0 (13)

More specifically, for linear demand functions satisfyingthe assumptions in §6.3, we have

p̂ =a2 − a1 + b1p

∗1

2b21 (14)

p∗

1 =41 −�154a2 − a15

�1b1 + 41 −�154b2 − b150 (15)

Substituting Equation (15) into Equation (14) and simplify-ing, we obtain

p̂ =4a2 − a15

2b2

(

�1b1 + 41 −�15b2

�1b1 + 41 −�154b2 − b15

)

0 (16)

Now using Equations (16) and (15), the first inequality in(13) becomes simply

�1 >3b2

b1 + 3b21 (17)

and whenever the above inequality holds, a single 3PT out-performs any menu of 2PTs. �

Proof of Corollary 2. We will show that �̂1 can becomputed by solving an inequality of the form f 4g4�155 >h4g4�155, where g, f (given in Equations (18) and (19)), andh are functions of �1 whose forms depends only on D1

and D2.Using an argument similar to the one used in the proof

of Corollary 1 (Equations (12) and (13)), there is a function hthat depends only on D1 and D2 such that if p̂ > h4p∗

15, then�̃3PT 12 − �̃2PT 12 > 0. Moreover, for i > 2, we have �̃3PT 1 i −

�̃2PT 1 i > 0. Therefore, if p̂ > h4p∗15, then a single 3PT beats

the optimal 2PT.

Now, consider the design of the 3PT. The F̂ and Q̂ compo-nents of the quasi-optimal 3PT depend only on the p∗

1 com-ponent of the 2PT menu (see Equation (3)). From Lemma 1,we apply Equation (1) to i = 1, which yields

p∗

1 =Sol ·

[

p=

(

1−�1

�1

)

1−D′

14p54D24p5−D14p55

]

=g4�151 (18)

where g is some function that depends on D1 and D2 only,not on the demand specification for the remaining Segmentsi = 31 0 0 0 1 I . Knowledge of these parameters immediatelyyields F̂ and Q̂. To compute the p̂ component of the single3PT, we use Equation (3) to get

p̂ = Sol ·

[

p =D24p5−D14p

∗15

−D′24p5

]

= f 4p∗

15 (19)

for some function f . Therefore p̂, like p∗1 , depends only on

D1, D2, and �1, and the firm need not know the detaileddistribution or preferences of Segments 31 0 0 0 1 I . �

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how to price discriminate when tariff size matters

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