Family Plans: Market Segmentation with Nonlinear Pricing. Bo Zhou. Business Administration Duke University. Date: Approved: Preyas S.

Transcription

1 amily Plans: Market Segmentation with Nonlinear Pricing by Bo Zhou Business Administration Duke University Date: Aroved: Preyas S. Desai, Co chair Devavrat Purohit, Co chair Wilfred Amaldoss Peng Sun Dissertation submitted in artial fulfillment of the reuirements for the degree of Doctor of Philosohy in Business Administration in the Graduate School of Duke University 014

2 ABSTRACT amily Plans: Market Segmentation with Nonlinear Pricing by Bo Zhou Business Administration Duke University Date: Aroved: Preyas S. Desai, Co chair Devavrat Purohit, Co chair Wilfred Amaldoss Peng Sun An abstract of a dissertation submitted in artial fulfillment of the reuirements for the degree of Doctor of Philosohy in Business Administration in the Graduate School of Duke University 014

3 Coyright by Bo Zhou 014

4 Abstract In the telecommunications market, firms often give consumers the otion of urchasing an individual lan or a family lan. An individual lan gives a certain allowance of usage (e.g., minutes, data) for a single consumer, whereas a family lan allows multile consumers to share a secific level of usage. The theoretical challenge is to understand how the firm stands to benefit from allowing family lans. In this aer, we use a game theoretic framework to exlore the role of family lans. An obvious way that family lans can be rofitable is if it draws in very low valuation consumers whom the firm would choose not to serve in the absence of a family lan. Interestingly, we find that even when a family lan does not draw any new consumers into the market, a firm can still benefit from offering it. This finding occurs rimarily because of the strategic imact of the family lan on the firm s entire roduct line. By allowing high and lowvaluation consumers to share joint allowance in the family lan, the firm is able to raise the rice to extract more surlus from the individual high valuation consumers by reducing the cannibalization roblem. urthermore, a family obtains a higher allowance comared to the urchase of several individual lans and therefore contributes more rofits to the firm. We also observe different tyes of uantity discounts in the firm s roduct line. inally, we identify conditions under which the firm offers a ay as you go lan. iv

5 Dedication To my wonderful and loving wife, Yue. v

6 Contents Abstract... iv ist of Tables... ix ist of igures... x Acknowledgements... xi 1. Introduction Model irm Consumers Game Structure Benchmark: Analysis of Individual Plans Analysis of Individual and amily Plans Model with Individual and amily Plans Otimal Individual and amily Plans Profitability of amily Plans Non Aversion to Overage Usage Analysis of Individual Plans Analysis of Individual and amily Plans Extensions ull Model with Positive Marginal Costs and Mindless Consumtion Non Overlaing Consumer Segments vi

7 6..1 Individual Plans Individual and amily Plans Alternative Model for amily Utility Other amily Comositions amily amily Multile amily Segments Correlated Consumtion Reuirements Deterministic Consumtion Individual Plans Individual and amily Plans Endogenizing Overage Aversion Model Analysis of Individual Plans Analysis of Individual and amily Plans Profitability of amily Plans Discussion and Conclusion Aendix A: Serving the Segment Only A.1 Individual Plan A. Individual Plan and a amily Plan A.3 Euilibrium Conditions Aendix B: Proof of Proositions vii

8 Aendix C: Examles of Wireless Plans Aendix D: Notation Tables References Biograhy viii

9 ist of Tables Table 1: Notation Table of the Main Model Table : Notation Table for Endogenizing Overage Aversion (Section 7) ix

10 ist of igures igure 1: Consumers Usage Distributions igure : Individual and amily s Usage Distributions... 6 igure 3: Illustration of a Plan igure 4: AT&T Individual Plans igure 5: AT&T amily Plans x

11 Acknowledgements I am deely indebted to my advisers Preyas Desai and Debu Purohit, for their hel and suort during the ast five years. This dissertation would not have been ossible without their hel. Their mentorshi has been incredible, and has shaed my develoment in so many dimensions. I am going to miss the time sent with them in the conference room on the third floor. I would like to thank Wilfred Amaldoss for his suort, for challenging my thoughts and ideas, for generously sharing his time and feedback with me throughout the rogram. I would also like to thank Jim Bettman for taking the time to meet with me and offer his insights on my work. I am very grateful to Peng Sun for his invaluable comments throughout this rocess. I would also like to thank Carl Mela for always being suortive and encouraging, Rick Staelin for his encouragement and feedback across various stages of the PhD rogram. I feel very grateful to the members of Duke Marketing aculty for their helful feedback on this dissertation. They have made this rogram as nurturing as ossible. I am also grateful for the suort and hel from the doctoral students. Most of all, I would like to thank my wife, Yue Qin, for her continuous love and encouragement throughout the doctoral rogram. inally, I would like to thank Theodore, who has been my most loyal friend and the best listener for the ast two years. xi

12 To those listed above and to those who I inevitably neglected, thank you all. xii

13 1. Introduction In the U.S. telecommunications market, firms give consumers the otion of urchasing either an individual lan or a family (or shared) lan. An individual lan gives a certain allowance of usage (e.g., minutes, data, etc.) for a single consumer, whereas a family lan allows multile consumers to share a secific level of usage. or examle, AT&T offers a 450 minute individual lan for $39.99 and a 700 minute family lan for $ rom a researcher s ersective, the theoretical challenge is to understand how a firm stands to benefit from allowing family lans. In this aer, we offer a novel and heretofore unexlored rationale for family lans and also demonstrate that when a firm allows consumers to share joint allowance by choosing a family lan, consumers and firms are both better off. Extensive research has examined bundles of roducts or services offered by firms, but this is the first aer to secifically consider bundling of consumers. The roblem we ose alies to situations in which consumers are uncertain about their needs for the services rovided by the firm. or examle, one may know that one needs a cellhone, but still have uncertainty about just how many minutes would be needed over the course of a month. Moreover, the level of need may vary from month to month, which makes estimating usage even more comlicated. Knowing that consumers will face these and other tyes of uncertainty, firms must devise contracts that anticiate consumers exected usage. One articular resonse made by firms is to offer 1

14 consumers contracts in which they ay a single fee for a block of service (e.g., 600 minutes of usage, etc.). If consumers usage exceeds the re determined allotment, they ay an additional er unit fee for the additional units of service they consume (e.g., 50 cents/minute). irms have also resonded by offering family or shared lans that allow multile users to share a single block of service. Although one may intuit that this aroach would in some cases reduce consumers level of uncertainty, this result is not enough to exlain why a family lan makes sense from the ersective of the firm. In this aer, we show that introduction of family lans can reduce the cannibalization roblem in the firm s roduct line design, and we identify the conditions under which the firm can maximize rofits by offering both individual and family lans. Our aer contributes to the literature on vertical differentiation and second degree rice discrimination. Starting with the canonical model of Mussa and Rosen (1978), there has been a rich stream of research on vertical differentiation and consumers self selection. Biyalogorsky and Koenigsberg (013) analyze the design and introduction of a roduct line when the firm is uncertain about consumer valuations for the roducts. Desai (001) studies whether the cannibalization roblem affects a firmʹs rice and uality decisions with both horizontal and vertical differentiation, in both monooly and duooly settings. Dobson and Kalish (1988) suggest a heuristic algorithm to solve the monoolist s roblem of ositioning and ricing a roduct line. Johnson and Myatt (003) analyze multiroduct monooly and duooly, and rovide an

15 exlanation for the common strategies of roduct line runing. Kim and Chhajed (00) consider a monoolist s roduct line design roblem with multile attributes. Maskin and Riley (1984) show that under a searability assumtion, strong conclusions can be made about the nature of otimal incentive schemes. Moorthy (1984) studies the roblem of imlementing market segmentation through consumer self selection, and shows that cannibalization comes into lay so a monoolist needs to simultaneously determine the whole roduct line. Moorthy and Png (199) show that seuential introduction is better than simultaneous introduction when cannibalization is a roblem. Rochet and Stole (00) introduce indeendent randomness into the agents outside otions and find the existence of no distortion at the bottom. Our aer extends this line of research by incororating consumers uncertainty about their usage needs. Unlike the revious research, in our model even consumers within the same segment end u consuming different uantities in euilibrium. To the best of our knowledge, by modeling both individual lans and family lans, our aer is the first to simultaneously consider second degree and third degree rice discrimination. We show how the firm can benefit from emloying both discrimination schemes comared to only using second degree rice discrimination. Our aer is also related to the growing literature on non linear ricing in service industries. Ascarza, ambrecht and Vilcassim (013) analyze the effect of tariff structure under two and three art tariffs and find that consumers who switch to three 3

16 art tariffs from two art tariffs significantly overuse minutes. Bagh and Bhargava (013) examine the firm s choice of a ricing scheme when tariff management costs are significant enough to imose a constraint on tariff size. Goalakrishnan, Iyengar and Meyer (01) study the behavioral effects of multi art tariffs in a lab setting and find consistent evidence for diminishing returns of exerience with regard to reducing consumers sub otimal decisions. Grubb (009) shows the otimality of the three art tariff ricing scheme in the resence of overconfident consumers. Grubb (01) develos a model of inattentive consumtion and finds that when inattentive consumers are heterogeneous and unbiased, bill shock regulation reduces social welfare in fairly cometitive markets. By contrast, Jiang (01) uses billing data to estimate the welfare effects of billshock regulation in mobile telecommunication markets and redicts an increase in consumer surlus. Iyengar et al. (011) use data from a field exeriment to show that consumers derive lower utility from consumtion under a two art tariff than ay eruse ricing, which results in lower retention of customers and lower usage of the service. Kolay and Shaffer (003) show that offering a menu of rice uantity bundles is more rofitable than offering a menu of two art tariffs absent cost considerations. ambrecht, Seim and Skiera (007) show that demand uncertainty is a key driver of choice among three art tariffs and that this uncertainty both decreases consumer surlus and increases rovider revenue. Narayanan, Chintagunta and Miravete (007) 4

17 develo a model for the choice and usage of local telehone service and find that the value of information to consumers is modest, and also that a major roortion of this value is in information about consumers tyes. Sundararajan (004) analyzes the nonlinear ricing of information goods and finds that offering fixed fee ricing in addition to a nonlinear usage based ricing scheme is always rofit imroving in the resence of nonzero transaction costs, and also that there may be markets in which a ure fixed fee is otimal. In general, research in this area has demonstrated the advantages of three art tariffs for the service roviders. We build on this stream of research by incororating shared consumtion, which allows family members to join the same three art tariff contract, and study the imact of such bundling on a service rovider s ricing and rofitability. Our aer is also related to the rich literature on bundling. The traditional exlanation for bundling that economists have given is that it makes rice discrimination strategies more owerful by reducing the role of unredictable idiosyncratic comonents of valuations (see, for examle, Adams and Yellen 1976, Bakos and Brynjolfsson 1999, 00, itt and Chen 005, McAfee, McMillan, and Whinston 1989, and Schmalensee 198, 1984). urthermore, mixed bundling (i.e., when a consumer may buy any individual comonent or the bundle) has been shown to be generally more rofitable than ure bundling and ure comonent sales (Banciu, Gal Or, and Mirchandani 010, Basu and Vitharana 009, Ibragimov and Walden 010, Prasad, 5

18 Venkatesh, and Mahajan 010, Venkatesh and Mahajan 1993, etc.). Our aer differs from the revious research by focusing on bundling consumers instead of roducts; in addition, we show the otimality of this aroach under certain conditions. The rest of the aer is organized as follows. Section resents the model. Section 3 obtains the otimal menu of individual lans. Section 4 analyzes the otimal menu of individual and family lans, and comares the roduct line strategies with those in Section 3. Section 5 considers a case in which consumers are not averse to overage usage. Section 6 lays out the extensions of the base model. Section 7 resents a model that endogenizes overage aversion. Section 8 resents our conclusions. 6

19 . Model In this section, we lay out the assumtions related to the firm, consumers and the structure of the game between them. We start with the assumtions on the firm..1 irm A monooly firm rovides telecommunication service to consumers. or ease of exosition, throughout this aer we denote the firm as a service rovider and the roduct as minutes of airtime. The firm knows the distribution of different tyes of consumers (in terms of their valuation of a minute of airtime) but cannot directly identify each individual s tye. or each tye of consumer, the firm also knows the distribution of the need for minutes of communication. The firm offers consumers a menu of individual lans from which to choose. Each lan is a three art tariff contract, denoted by (,, O ), where is the included allotment of minutes for which the marginal rice is zero (hereafter, is referred to as uantity), is the rice consumers ay for regardless of their actual consumtion, and O is the overage rice er unit for additional usage beyond the lan s uantity. Note that the overage charge can be different across lans. We assume that the service 7

20 rovider s marginal cost er unit is zero, which is a reasonable aroximation for information services. 1 In addition to the individual lans, the firm may decide to offer a family lan, which allows two individuals to articiate in a single contract. In this family lan, as long as the sum of both members consumtion stays within the uantity, these consumers do not incur overage charges. On the other hand, if their total consumtion goes over the family lan s allotment, then they need to ay overage charges.. Consumers There are several key characteristics of consumers in telecommunication markets. irst, they differ in their valuation for telecommunication services, i.e., how much they value a minute of airtime. Second, they have different communication needs (e.g., some consumers reuire more minutes than others during a given consumtion eriod). Third, they have uncertainty about their need for communication at the time of urchase (e.g., it is hard to redict the exact number of minutes one will need in the consumtion stage). ourth, they dislike being in the overage region because they incur additional costs of monitoring their usage, aying overage fees (i.e., the additional charges imosed by the service rovider after consumers have exceeded their uantity ermitted under their lan); imortantly, these costs also ensure that consumers cut 1 The analysis in which the marginal cost er unit is not zero is resented in the aendix. Most of our results are ualitatively the same. 8

21 back on consumtion, something that they would not have done in the absence of these costs. To model consumers heterogeneity of valuation for telecommunication services, we assume that there are two consumer segments in the market: a high valuation segment denoted by and a low valuation segment denoted by. The sizes of the and segments are (1 ) and. Consumers know their own tyes, and their utility function is given by V( i, ) i, where i {, },. This utility function is increasing and weakly concave in uantities, and decreasing in rice. Consumers valuation arameter means that an tye consumer obtains a higher utility than an tye consumer for a given uantity, as well as a higher marginal utility. This utility function satisfies the Sence Mirrlees single crossing condition: the cross artial derivative V (, ) 0 1, for all. To cature consumers heterogeneity in terms of their communication reuirements and their uncertainty about how many minutes they will need in the consumtion stage, we assume consumers consumtion needs are stochastic, come from a known distribution and are indeendently distributed. In articular, the segment s consumtion reuirement is uniformly distributed between [,1 ] ( 0 ), i and that of the segment is uniformly distributed between [0, 1]. In other words, the 9

22 exact amount of minutes consumers need is random, and the realization of the uantity needed is generated by two indeendent distributions, where one distribution ( ) stochastically dominates the other ( ). Note that tye consumers need at least minutes, below which they do not obtain sufficient utility to initiate the urchase decision. thus measures the heterogeneity of consumers usage needs. Ex ante, consumers do not exect to gain additional utility by continuing to use the service once their reuirement has been met. Therefore, if the reuirement realization, ˆ, is within a lan s uantity, consumers will sto consuming at ˆ. Both tyes of consumers gain zero utility beyond the suorts of their reuirement distributions; in addition, their outside otion gives them zero utility. Segment Segment igure 1: Consumers Usage Distributions Extensive behavioral research has demonstrated consumers ain of aying, where a difference in the transarency of ayments caused by different ayment mechanisms changes the ain of making a ayment, and conseuently influences sending and consumtion behavior (Prelec and oewenstein 1998, Soman and Gourville 001, Soman 003). To cature the feature that consumers dislike being in the 10

23 overage region, we introduce a arameter k ( 0 k 1) to measure this degree of aversion. In articular, consumers start curtailing their consumtion once they reach their lan s uantity such that they only use k roortion of the difference between the realization of their reuirement and their lan s uantity. Secifically, when the draw from the distribution, ˆ, is beyond a lan s uantity, consumers will use [ k( ˆ )] minutes. When k 0, consumers are extremely averse to overage consumtion and sto at their lan s uantity, even if their realized reuirement is beyond that uantity. When k 1, consumers are not at all averse to overage consumtion and they kee consuming minutes u to their reuirement s realization, the same way they did before reaching their lan s uantity. In reality, however, consumers may start curtailing consumtion even before reaching their lan s uantity. In this case, our framework rovides a first order aroximation to the intentional usage reduction near the lan s uantity and the resulting utility reduction. Note that consumers are aware of their behavior on curtailing overage, artly because they dislike aying a higher fee comared to the unit rice included in the lan. There are lenty of examles in other contexts that share the same behavior attern, such as in the auto leasing scenario where consumers actively avoid driving the car once the mileage restriction has been exceeded. To simlify our analysis and sharen our focus on Consumers dislike aying overage fees more comared to the rice of the lan. Overage fees are tyically higher than the unit rice included in the lan, thereby causing consumers dislikes. 11

24 bundling consumers with non linear ricing, we assume that the arameter for curtailing overage consumtion, k, is the same across both consumer segments. Now we introduce our model of a family. amilies may comrise different tyes of consumers. To cature heterogeneity within a family in a arsimonious way, we assume that a family consists of one tye consumer and one tye consumer. 3 We also assume that the total number of families in the market is f ( f 0 ). It follows that the sizes of the single and segment are (1 f ) and ( f ), resectively. To ensure the existence of single individual consumers (consumers who cannot buy family lans), we assume that (1 ) f and f. A family s valuation for uantities,, is given by (subscrit denotes family). Because both individuals in a family curb their overage consumtion by the roortion of k, a family s aversion to overage usage is also characterized by k. Consumers are risk neutral and make urchase decisions based on their exected utilities from different lans. Their exected utility consists of four comonents: (1) utility rior to reaching a lan s uantity, () utility after reaching that uantity, (3) the rice of the lan, and (4) the overage ayment. Conditional on choosing the lan (,, O), an tye consumer s exected utility is given as: 3 The analysis in which a family comrises two tye or two tye consumers is resented in Section 6. 1

25 EU(,,, ) EV ( (, ) ) Pr ( ) EV ( (, ), kpr ) ( ) O E(, k) Pr( ) O 1 1 v(, ) f ( ) d v(, k( )) f ( ) d k( ) f ( ) d. O (1) The first term in the above exression, v (, ) f ( ) d, is the tye consumer s exected utility conditional on her reuirement realization being smaller than the lan s uantity. The second term, 1 v(, k( )) f ( ) d, is her exected utility, conditional on her reuirement realization exceeding the lan s uantity and her starting to curtail her overage consumtion by k. The next term,, is the lan s rice, and the last term, 1 k ( ) f ( d ), reresents the consumer s O exected overage ayment. Similarly, if an tye consumer chooses the lan (,, O), her exected utility would be: EU(,,, ) E( V(, ) ) Pr( ) E( V(, ), k) Pr( ) O E(, k) Pr( ) O v(, ) f ( ) d v(, k( )) f ( ) d k( ) f ( ) d. O () 13

26 .3 Game Structure The game between the service rovider and consumers contains two stages. In the first stage, the service rovider resents two individual lans, (,, O) and (,, O), and consumers self select the best lans for themselves. In addition, the service rovider may decide to offer a family lan, (,, O), in which case, the consumers who comrise this family will comare this joint lan with the two individual lans that the individual members would choose. Two dimensions of uncertainty exist in this stage. irst, the service rovider does not know each individual consumer s tye but takes this information asymmetry into account when designing the lans to maximize its exected rofit. Second, neither the service rovider nor the consumers know exactly how many minutes each individual will need in the next stage; they only know the and segments usage distributions. Consumers choose which lan to urchase based on their exected utilities and ay the lan s rice, or, accordingly. When the choice is a family lan, consumers with family members take the entire family s exected utility into account and ay the rice. In the second stage, the communication reuirement is realized for each consumer. At the end of this stage, those whose reuirement realizations are beyond their lans uantities curtail their overage consumtion by the roortion k and make the overage ayment. 14

27 or a more interesting analysis, in the main body of this aer we consider the situation in which it is more rofitable for the service rovider to serve both the and segments than to serve only the segment. The condition for this situation to arise in euilibrium is given by: 1 k 1 (1 )(1 k) [(1 k)(1 ) (1 ) (1 k) (1 ) 1 (1 k) ] (1 k)(1 ) (1 ) 0. We assume the above condition to hold in the main text, and discuss the situation in which it is more rofitable to serve only the segment in the aendix. 15

28 3. Benchmark: Analysis of Individual Plans In this section, we analyze the case in which the service rovider attemts to maximize its rofits by offering only two individual lans to consumers. The service rovider chooses two sets of uantities, rices, and overage fees in these two individual lans, (,, O) and (,, O), to target the high valuation consumers (the segment) and the low valuation consumers (the segment), resectively. After observing the two three art tariff lans, consumers self select the best lans for themselves. As is well known from the classic literature on self selection (Mussa and Rosen 1978), the service rovider must ensure that the segment will not choose the lan, (,, O), targeted at the segment, and vice versa. In addition, the service rovider must ensure that both segments of consumers obtain nonnegative utilities from urchasing the lans targeted at them. Thus the new elements in our framework are the three art tariff lans and uncertainty on the reuirement for uantities. The service rovider s rofit,, is given by (1 )[ E(, k) Pr( )] [ E(, k) Pr( )]. O O The first term in this rofit function, (1 )[ E(, k) Pr( )], O gives the rofit from the segment. is the rice of the lan, and the term, E (, kpr ) ( ), catures the segment s exected overage O ayment after choosing the lan. Similarly, the second term in the rofit function gives the rofit from the segment, while the term, E (, kpr ) ( ), 16 O

29 catures the segment s exected overage ayment after choosing the lan. ormally, the service rovider sets its menu of lans to maximize its exected rofit: E[ ] Max (1 )[ E(, k) Pr( )] [ E(, k) Pr( )] O O 1 1 O O (1 )[ k( ) f ( ) d] [ k( ) f ( ) d], subject to EU(,,, ) EU(,,, ), (3) O O EU(,,, ) EU(,,, ), (4) O O EU (,,, ) 0, (5) O EU(,,, ) 0. (6) O The left hand side of Constraint (3), EU (,,, ), is an tye O consumer s exected utility of urchasing the lan. She ays the rice uantity of, and faces the overage charge er unit O, gets a. The right hand side of Constraint (3), EU(,,, ), is the tye consumer s exected utility if she O urchases the lan. In this case, she ays the rice, gets a uantity of, and faces the overage charge er unit O. Similarly, the left and the right sides of Constraint (4) cature an tye consumer s exected utilities of buying the and the lans, resectively. Constraints (3) and (4) ensure that both consumer segments voluntarily choose the lan directed to them. Constraints (5) and (6) ensure that each segment will buy the lan directed to it rather than not buy anything at all. In the main body of this aer, we focus on the more interesting situation where the incentive comatibility (IC) constraint 17

30 for the segment, Constraint (3), is not trivially satisfied. 1 In other words, the binding constraints are this incentive comatibility constraint for the segment, Constraint (3), and the individual rationality (IR) constraint for the segment, Constraint (6). In this case, tye consumers best outside otion is the lan, (,, O). To aid intuition, we exand the binding IC constraint for the segment as follows: EU(,,, ) E( V(, ) ) Pr( ) E( V(, ), k) Pr( ) O E(, k) Pr( ) O EU(,,, ) EV ( (, ) ) Pr ( ) EV ( (, ), kpr ) ( ) O E(, k) Pr( ). O (7) Two oints are worth highlighting in Constraint (7). irst, if an tye consumer selects the lan, she is more likely to run into an overage region because. Second, with the lan, she starts curtailing the overage consumtion earlier, at the uantity, which leads to a reduced utility in the overage region. The advantages for an tye consumer of selecting the lan are a lower rice and a otentially lower overage charge er unit O (confirmed ex ost). As we solve the firm s constrained otimization roblem, we obtain the following otimal menu of individual lans. (We use suerscrit to denote the otimal rices, uantities and overage fees in this case.) 1 A sufficient condition for this situation to occur is that is lower than a threshold. We discuss the other situation, in which the incentive comatibility constraint for the segment is trivially satisfied and thus the only binding constraints are the two individual rationality constraints, in the aendix. 18

31 The uantities of the and lans are: 1, (1 )((1 k) k ) 1. (8) (1 k)( (1 ) ) The rices for two individual lans, and, are given as: ( (1 k(1 (1 ))) (1 k)(1 ) (1 )) (1 k) ( (1 )) ( (1 kkk) (1 k)(1 ) (1 )), (9) ( k (1 k) )(1 k) (1 k) ( (1 )). (10) The overage fees are O and 0. O irst, note that the service rovider can charge tye consumers full valuation for uantities,, as the overage rice. This result occurs because as soon as tye consumers continue to use minutes in the overage region, the firm knows their marginal utility and can fully extract it. By contrast, the firm offers free extra minutes to tye consumers, 0. This is because the uantity for the lan, O 1, is the uer bound of the segment s reuirement distribution. Given that no one would consume more than 1, we can interret this result, 1, as that tye consumers obtain an unlimited lan. By contrast, tye consumers never obtain their maximum reuirement in their lan, 1. Intuitively, consumers are willing to ay a higher fixed rice when their lans included uantities are higher. In this context, the service rovider faces two major 19

32 trade offs. On the one hand, it wants to decrease the included uantity so that consumers are more likely to ay overage fees, which can be shown to be higher than the average rice er unit within a lan. On the other hand, if the firm decreases a lan s uantity, it forgoes some rofits with certainty because consumers are willing to ay a lower fixed rice. In addition, because consumers will curtail their overage consumtion if their reuirement realizations are beyond a lan s uantity, the firm cannot fully rea the benefits of a large reuirement realization. The curtailing behavior occurs more often with a lower uantity, which leads to another loss of revenues. Because the latter factor dominates in the offering of the lan, given a higher valuation for uantities, the service rovider does not ut any restriction on the uantities of the lan. This result is consistent with the no distortion at the to result from the literature on vertical differentiation. Now we characterize the imact of the curtailing arameter, k, on the two individual lans. Proosition 1. The lan s uantity,, and its rice, 0, both decrease when k increases (i.e., when consumers curtail their overage consumtion to a lesser extent). The lan s uantity, increases., is indeendent of k. The lan s rice,, may increase or decrease as k All the roofs are given in the aendix. Proosition 1 shows that the imact of k on the two lans is different. We discuss the intuition seuentially. The reason for the

33 first art of the roosition is that when consumers curb their overage consumtion less ( k is larger), the service rovider has incentives to reduce the lan s uantity to cature the increased benefits from the overage charges. Once the uantity,, is reduced, the service rovider must also reduce the lan s rice,, to comensate for an tye consumer s lowered willingness to ay. The comarative statics, k, can be either ositive or negative, deending on the heterogeneity of consumers valuations, ( ). It is interesting to note that although the lan s uantity, 1, is indeendent of the curtailing arameter k, its rice is deendent uon k. The imact of the extent of curtailing on the lan s rice is through the binding IC constraint for the segment. When tye consumers valuation,, is above a threshold, an increase in k will lead to an increase in the lan s rice, i.e., k 0. The intuition is that a higher k leads to a lower uantity for the lan, which makes this best outside otion for the segment less aealing, even after accounting for its rice reduction. Therefore, the service rovider can raise the lan s rice to benefit from consumers smaller aversion toward overage consumtion. Concerning the rice er unit within the lan, it is easy to show that. This ineuality means that the rice er unit within the lan is indeed lower than its 1

34 overage rice, a circumstance that is consistent with industry ractice. In other words, tye consumers are given some uantity discounts from the lan as comared to their maximal willingness to ay er unit,. Interestingly, another form of uantity discount also exists when comaring across the individual lans. Proosition. When (1 k) (1 (1 k) ) 0, the er unit rice in the lan is lower than that of the lan, i.e.,. Otherwise, the reverse is true. Similar to many roduct categories in which uantity discounts are observed, the high valuation consumer segment urchases a roduct with a higher uantity ( ), and enjoys a lower rice er unit when (1 k) (1 (1 k) ) 0. This condition is more likely to hold when or k increases. A higher imlies an lan with higher uantities ( 0 ), which increases the attractiveness of the segment s best outside otion. Thus the service rovider is more likely to offer uantity discounts with resect to the included uantity in the lan, in order to revent the segment from selecting the less exensive lan. owever, although a higher k decreases both the lan s rice and uantity, the overall effect of k on its rice er unit is ositive, k 0. This effect increases the likelihood of uantity discounts offered to the segment.

35 To summarize, in this section we have characterized the otimal individual lans and how overage aversion affects the terms in these lans. Next we move on to examine the situation when the service rovider offers family lans in addition to the individual lans. 3

36 4. Analysis of Individual and amily Plans In the revious section, we analyzed the situation when the service rovider offered two individual lans. In this section, we analyze the case when the service rovider offers two individual lans, (,, O) and (,, O), and a family lan, (,, O). 4.1 Model with Individual and amily Plans The service rovider uses two individual lans to target single high valuation consumers and single low valuation consumers, whereas it uses the family lan to target consumers who have families. Note that a family lan is jointly urchased and consumed by two family members; single, individual buyers cannot urchase a family lan. In other words, self selection of family lans goes in one direction, and the service rovider can identify family buyers tyes when a family lan is sold to them. Therefore, by offering three different lans, the firm utilizes both second degree and third degree rice discrimination. Recall that a family consists of one tye consumer and one tye consumer, and the size of the family segment in the market is f. It therefore follows that the size of the single segment is (1 f ) and the size of the single segment is ( f ). urthermore, a family s valuation arameter,, is given by. 4

37 Before we resent our analysis, we first establish a family s reuirement distribution below. Note that the density function of the sum of two indeendent random variables, X and Y, is the convolution of the density functions of the oerant distributions, f X ( x) and fy ( y ). The random variable, Q X Y, has density function: ( f f )( ) f ( y) f ( y) dy f ( x) f ( x) dx. X Y X Y Y X 1997.) (See Grinstead and Snell emma 1. Assume random variable Q is the sum of two indeendent random variables: Q X Y, where X is uniformly distributed between (0, 1) and Y is uniformly distributed between (,1). Then Q s robability density and cumulative distribution functions are given below: f Q 0,,, 1, ( ) ( ), 1, 0,. 0,, ( ), 1, ( ) Q 1,. (( ) ), 1, (11) Thus, the family s reuirement distribution is a triangular distribution with the density eak at 1. Its variance is the same as the sum of variances of the and segments distributions due to indeendence. Note that both consumers and the firm know this reuirement distribution in the first stage of the game. 5

38 Segment Segment Segment igure : Individual and amily s Usage Distributions Similar to the analysis in Section 3, the service rovider must make sure that the two single consumer segments will choose individual lans targeted to them. In addition, in order to induce eligible consumers to urchase the family lan, the service rovider needs to set the lan so that the exected joint utility from urchasing it is weakly greater than the sum of exected utilities obtained from two individual lans (which the two family members could have chosen). The service rovider s rofit,, is given by (1 f )[ E(, k) Pr( )] ( f)[ E(, k) O O Pr( )] f[ E(, k) Pr( )]. The first term in this rofit function, O (1 f )[ E(, k) Pr( )], gives the exected rofit from the O single segment where is the rice of the lan, and the term, E (, kpr ) ( ), catures the segment s exected overage O ayment after choosing the lan. Similarly, the second and third terms in the rofit 6

39 function give the rofits from the single segment and the family segment, resectively. The exression within the third term, E (, kpr ) ( ), catures the O family segment s exected overage ayment after choosing the family lan. ormally, the service rovider chooses three lans to maximize its exected rofit: E[ ] Max (1 f)[ OE(, k) Pr( )] ( f)[ OE(, k) Pr( )] f [ OE(, k) Pr( )], subject to EU (,,, ) EU (,,, ), (1) O O EU(,,, ) EU(,,, ), (13) O O EU (,,, ) EU (,,, ) EU (,,, ), (14) O O O EU (,,, ) 0, (15) O EU(,,, ) 0, (16) O EU (,,, ) 0. (17) O The left hand side of Constraint (1), EU(,,, ), is an tye O consumer s exected utility of urchasing the lan. She ays the rice uantity of, and faces the overage charge er unit O, gets a. The right hand side of Constraint (1), EU(,,, ), is the tye consumer s exected utility if she O urchases the lan. In this case, she ays the rice, gets a uantity of, and faces the overage charge er unit O. Similarly, the two sides of Constraint (13) cature an tye consumer s exected utilities of buying the and the lans, resectively. 7

40 Introduction of a family lan brings two new constraints, Constraints (14) and (17), into the service rovider s otimization roblem. The left hand side of Constraints (14) and (17), EU (,,, ), is a family s joint exected utility of urchasing the O family lan. This family ays the rice, gets a total shared uantity of, and faces the overage charge er unit O if the sum of two family members consumtion exceeds the lan s uantity. By contrast, the right hand side of Constraint (14), EU (,,, ) EU (,,, ), gives the sum of exected utilities for an O O tye consumer who is buying the lan and an tye consumer who is buying the lan. Constraints (1) and (13) imly that both the and segments voluntarily choose the lan directed to them. By contrast, Constraint (14) imlies that families will buy the family lan instead of two individual lans. Constraints (15) through (17) ensure, resectively, that the,, and family segments will buy the lan intended for them rather than not buy anything at all. Next, we examine the service rovider s otimal menu of lans. As in Section 3, we focus on the more interesting situation in which the incentive comatibility constraint for the segment, Constraint (1), is not trivially satisfied. 4. Otimal Individual and amily Plans When given the otion of choosing a family lan, a single tye consumer s best outside otion is still the lan, (,, O), whereas a single tye consumer s 8

41 best outside otion is no consumtion. A family s best outside otion is for each member to searately choose his or her best individual lan: (,, ) for the tye consumer and (,, O) for the tye consumer. In other words, the binding O constraints are the incentive comatibility constraints for the segment (Constraint (1)) and the family segment (Constraint (14)), as well as the individual rationality constraint for the segment, Constraint (16). By solving the firm s constrained otimization roblem, we obtain the following otimal menu of lans. (We use to denote the otimal rices, uantities and overage fees in the resence of family lans.) The uantities for the three lans are: 1,, (1 )((1 k) k) (18) 1. (1 k)((1 f) (1 ) ) Both the lan and the family lan obtain their maximum reuirements, 1 and. It can be interreted that the single segment and the family segment both obtain unlimited lans. Similar to the scenario in Section 3, the lan s uantity, decreases in the curtailing arameter k. The rices of these three lans, which all deend on k and f, are given as:, ( ((1 f)(1 k) k(1 )) (1 k)(1 ) (1 )) (1 k) ((1 f) (1 )) ( ((1 f)(1 k) k(1 )) (1 k)(1 ) (1 )), (19) 9

42 ( (1 ) (1 (1 ) ), k k ) f k k (1 k) ((1 f) (1 )) (0) 1 ((1 k) 1 ( (1 ) ) ( )) (1 ) ] {( ) k) [ ( f f k k f }. (1 k)((1 f) (1 )) (1) The overage fees are 0, O 0, and O O, all of which are indeendent of k and f. Note that the overage fee structure is the same as it would be without a family lan. When a lan (the lan) is caed in uantities, the service rovider charges this segment s full valuation ( ) as the overage rice. In the case of lans that rovide unlimited uantities, the firm does not charge anything to those buyers ( 0, 0 ) because no one would consume more than the uer suorts of O O their reuirement distributions. Recall that f is the total number of families in the market. Next, we discuss the imact of f on the uantities and rices of the three lans. Proosition 3. The size of the family segment, f, affects the otimal lans in the following way: (i) The uantity of the lan, and, are indeendent of f., decreases in f. The uantities of the and the family lans, (ii) The rice of the lan, and, increase in f., decreases in f. The rices of the and the family lans, Recall that when designing the roduct line for two individual lans, the service rovider must account for the otential cannibalization roblem. In articular, the firm 30

43 must ensure that the high valuation segment urchases the more rofitable lan by obtaining at least a utility of EU(,,, ). In other words, the firm must balance O reducing the attractiveness of the lan, so that the segment will not switch and thus can be further exloited, with a lowered revenue stream from the segment due to a reduction in the lan s uantity,. With the introduction of a family lan, the size of the lan buyers decreases, i.e., f for f 0, which means that the single segment s imortance also decreases. As the size of the family segment, f, increases, the firm can reduce the uantity of the individual lan less aealing to the single segment. Once the uantity to make this outside otion is reduced, however, the rice for the lan must also be decreased to account for a lower willingness to ay from the single tye consumers. Still, when the size of the family segment, f, increases, the service rovider can raise both and because the reduction in decreases the attractiveness of the best outside otion for both the single tye consumers and the tye consumers with families (recall the IC in Constraint (1)). In the absence of a family lan, we have already observed uantity discounts when comaring two individual lans (Proosition ). An interesting uestion is whether the uantity discounts of the lan comared to the lan still exists in the resence of a family lan. It is also interesting to assess the rice er unit in the family 31

44 lan and comare this with those of the two individual lans. The next roosition discusses rice uantity ratios across lans and resolves these two issues. Proosition 4. When f is below a threshold: (i) The rice er unit in the lan is lower than that of the lan, i.e., (ii) The rice er unit in the family lan is lower than that of the lan, i.e.,.. The first ineuality on uantity discounts of the lan comared to the lan holds to ensure that the single segment would select the more exensive lan instead of the lan, with which they could have benefited from both a lower rice and ositive utility ( ) in the overage region. The second ineuality results from the comosition of a family. Due to the inclusion of a lower valuation consumer in the family lan, the service rovider needs to lower the rice er unit in the family lan comared to the lan. inally, it should be noted that all of these rices er unit within a lan are lower than the lan s overage rice er unit,, which dislays another form of uantity discount and is also consistent with industry ractice. The condition on f is tediously comlicated and is given in the aendix. As is often the case, here we discuss the more intuitive sufficient condition. When or k increases, the condition on f is more likely to be satisfied. Concerning the first ineuality in Proosition 4, a higher imlies an lan with higher uantities 3

45 ( 0 ), which increases the attractiveness of the segment s best outside otion. Thus the service rovider is more likely to offer uantity discounts with resect to the included uantity in the lan to revent the segment from selecting the lan, which is less exensive. Moreover, although a higher k decreases the lan s rice as well as its uantity, the overall effect of k on its rice er unit is ositive, k 0. This effect increases the likelihood of uantity discounts offered to the segment. Concerning the second ineuality in Proosition 4, a higher or k imly a higher willingness to ay on the art of an tye consumer. Therefore, the service rovider no longer needs to offer as much of a uantity discount to a family in order for them to bundle an tye consumer with an tye consumer into the joint lan. Next, we comare the two individual lans in the resence of the family lan with the two individual lans in the absence of the family lan (as in Section 3). This comarison rovides further insights into how the introduction of the family lan changes the terms of the individual three art tariffs. Proosition 5. The changes in two individual lans after the introduction of the family lan are as follows: (i) Both the lan s uantity,, and its rice,, are lower, i.e., and. 33

46 (ii) The lan s uantity,, stays the same, but its rice,, is higher, i.e., and. In determining a lan s uantity, the service rovider faces two main trade offs. It wants to decrease the included uantity in order to induce consumers to ay overage fees with a greater robability; these fees are higher than the average rice er unit within a lan (excet for unlimited lans). owever, it also wants to increase a lan s uantity so that it can charge buyers a higher rice. Remember that consumers curtail their overage consumtion if their reuirement realizations are beyond a lan s uantity, so that the service rovider cannot fully rea the benefits of a large reuirement realization. or this reason, a higher uantity results in a smaller loss from consumers overage consumtion due to their delayed curtailing behavior. The first factor, couled with the consideration for the IC constraint for the segment, dominates the second factor in determining the uantity and rice for the lan. Because the detailed intuition for the resulting observations, and, is resented in the discussion of Proosition 3, it is not reeated here. By contrast, given the relatively high valuations and ( ), the firm s incentive in raising the included uantities is stronger than the oosite. Therefore, the service rovider offers its maximum reuirements for communication for 34

47 the segment and the family segment, resectively. Again, this result is consistent with the no distortion at the to result from the literature on vertical differentiation. By bundling some tye consumers with some tye consumers in the family lans, the service rovider is able to better extract surlus from the single segment by charging these consumers a higher rice,. In other words, the segment with families oses some negative externalities to the single segment. Recall that from Proosition 3, the firm utilizes the family lan to effectively reduce the number of the single lan s buyers and thus their imortance as well. Then the firm can lower the lan s uantity to make it less attractive to the single tye consumers as their best outside otion. As a result, the firm can adjust its lan to better exloit the single tye buyers. Because the uantity in the lan, 1(the overage fee, 0, O does not matter in this case), has attained its maximal value, the only instrument left to change in the three art tariff is the rice,. The service rovider thereby raises this rice to better rice discriminate the single segment, which leads to. Now that we have analyzed how the introduction of the family lan changes the two individual lans, in the next roosition we discuss how the introduction of the family lan affects the likelihood of observing uantity discounts in the lan as comared to the lan. 35

48 Proosition 6. Comared to the case without the family lan, uantity discounts between the and the lan,, is less likely. The intuition behind this roosition stems from the results in Proosition 5. While the introduction of the family decreases the lan s uantity and rice, it also increases the lan s rice and kees the lan s uantity unchanged. The latter result leads to a higher rice uantity ratio for the lan comared to that in Section 3,. Therefore, it is less likely for the service rovider to offer uantity discounts to the lan comared to the lan after the introduction of the family lan. 4.3 Profitability of amily Plans Proositions 4 and 5 have shown the oosing imact of the family lan on the rices of the lan and the lan, as well as the uantity discounts offered to the family lan users. Two natural uestions concern whether introducing the family lan would be more rofitable overall and when this strategy emerges in the euilibrium. We comare the service rovider s rofits with and without the family lan, E, and rovide answers to these uestions in the roosition below. Proosition 7. When ((1 k) k ) (1 ) (1 k)( )( (1 f) (1 )) 0, E and it is more rofitable for the service rovider to offer a family lan and two individual lans, i.e., E E. Otherwise, offering only two individual lans is more rofitable. 36