The B.E. Journal of Economic Analysis & Policy

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1 The BE Journal of Economic Analysis & Policy Contributions Volume, Issue 0 Article 50 Cometitive Mixed Bundling of Vertically Differentiated Products Illtae Ahn Kiho Yoon Chung-Ang University, illtae@cauackr Korea University, kiho@koreaackr Recommended Citation Illtae Ahn and Kiho Yoon (0) Cometitive Mixed Bundling of Vertically Differentiated Products, The BE Journal of Economic Analysis & Policy: Vol : Iss (Contributions), Article 50 DOI: 055/ Coyright c 0 De Gruyter All rights reserved Download Date /6/ 5:8 AM

2 Cometitive Mixed Bundling of Vertically Differentiated Products Illtae Ahn and Kiho Yoon Abstract We examine mixed bundling in a cometitive environment that incororates vertical roduct differentiation We show that, comared to the equilibrium without bundling, (i) rices, rofits and social welfare are lower, whereas (ii) consumer surlus is higher in the equilibrium with mixed bundling In addition, the oulation of consumers who urchase both roducts from the same firm is larger in the equilibrium with mixed bundling These results are largely in line with those obtained in the revious literature on cometitive mixed bundling with horizontal differentiation Further, we conduct a comarative static analysis with resect to changes in quality differentiation arameters When the quality ga between brands narrows under no bundling and symmetric mixed bundling, rices and rofits decrease When quality differentiation is asymmetric across roducts, however, comlicated effects occur on rices and rofits due to strategic interdeendence that mixed bundling creates KEYWORD: mixed bundling, vertical differentiation, quality advantage, cometitive bundling We thank the editor and two anonymous referees for many invaluable suggestions The second author acknowledges that this work was suorted by a ational Research Foundation of Korea Grant funded by the Korean government (RF B00085) Download Date /6/ 5:8 AM

3 Ahn and Yoon: Mixed Bundling of Vertically Differentiated Products Introduction Mixed bundling, in which firms offer multile roducts for sale individually or at a discount if urchased together, is commonly observed in the real world Classic examles include stereo systems, software suites, fast-food restaurant meals, and concert tickets, which are tyically available at a discount as ackages alongside with individual roducts Moreover, the strategic use of mixed bundling in oligoolistic environments is gaining increasing imortance in the era of digital convergence For examle, many firms in the telecommunications and broadcasting industries offer the trile lay service of combining telehony, Internet access, and broadcasting with bundle discounts to attract new customers and to widen their business shere In this aer, we examine mixed bundling in a cometitive environment that incororates vertical roduct differentiation We comare the equilibrium rices, rofits, social welfare and consumer surlus with mixed bundling to those without bundling to see the effects of mixed bundling We also conduct a comarative static analysis with resect to changes in quality differentiation arameters There is an extensive literature on bundling Adams and Yellen (976), chmalensee (98), McAfee, McMillan, and Whinston (989), and Bakos and Brynjolfsson (999) establish that the monoolist may use mixed bundling as a rice discrimination device to increase rofits Other aers including Whinston (990) and alebuff (004) study how the monoolist makes strategic use of ure bundling or tying to foreclose cometitors These aers analyze the industry structure in which one firm holds a monooly ower or sufficient market ower The strand of literature that is closest to the current aer is those that study mixed bundling in a more cometitive and symmetric environment These works include Matutes and Regibeau (99), Anderson and Leruth (993), Economides (993), Reisinger (006), Gans and King (006), and Thanassoulis (007) 3 In these aers, each of two symmetric and horizontally differentiated firms roduces its own brand of two roducts 4 Matutes and Regibeau (99) study mixed bundling in the mix and match model of Matutes and Regibeau (988) and show that, for a wide range of arameters, firms choose to offer bundle discounts even though their rofits would be higher if they could agree not to be engaged in mixed bundling They also show that firms roensity to bundle is excessive from a social welfare ersective Anderson and Leruth (993) and Thanassoulis (0) contains relevant discussion on bundling in multimedia markets Excellent recent surveys include Kobayashi (005) and tole (007) 3 A recent aer by Armstrong and Vickers (00) is also relevant 4 As a matter of fact, Gans and King (006) first consider four indeendent firms and then integrated firms ee the next aragrah Published by De Gruyter, 0 Download Date /6/ 5:8 AM

4 The BE Journal of Economic Analysis & Policy, Vol [0], Iss (Contributions), Art 50 Economides (993) consider mixed bundling in somewhat different models but arrive at basically the same conclusion 5 Reisinger (006) is yet another aer that studies mixed bundling in a circular-city model with a discussion of location choice Thanassoulis (007) extends Matutes and Regibeau s location model by introducing small buyers who only desire one of the comonent roducts and by distinguishing between firm-secific and roduct-secific references This aer discovers that, comared to the equilibrium without bundling, rofits are higher but consumer surlus is lower with mixed bundling under firm-secific references The conclusions are reversed under roduct-secific references, with which the other aers are concerned In Gans and King (006), two unrelated roducts are roduced by four firms Pairs of firms may agree to offer bundle discounts when their roducts are urchased together In comarison to the aers above which assume that bundle discounts as well as stand-alone rices are set simultaneously, firms in this aer first decide on the level of bundle discounts and then simultaneously announce their stand-alone rices Gans and King show that both airs of firms offer bundle discounts such that all consumers urchase both roducts either from one air or from the other in equilibrium However, if both airs of firms are integrated, they choose not to offer bundle discounts in equilibrium, contrary to the findings above o aer, however, that the authors are aware has studied mixed bundling in a cometitive environment that incororates vertical roduct differentiation This is surrising since vertical differentiation in quality is widely observed in cometitive mixed bundling In fact, it would be an excetion rather than the rule that the roducts offered by different firms have the same quality Vertical differentiation may result from the roduct design or consumers ercetion due to installed base, incumbency, or firm s reutation As an examle, consider the broadcasting and telecommunications industry Both the hone comany and the cable comany offer the telehony service and the broadcasting service In many instances, the hone comany offers a high-quality telehony service, whereas the cable comany offers a high-quality broadcasting service The telecommunications industry itself in many countries, say in Korea, is another examle of a vertically differentiated oligooly engaged in mixed bundling Korea Telecom (KT) has market dominance in fixed-line telecommunications, whereas K Telecom (KT) holds the dominant osition in mobile telecommunications In recent years, each firm has exanded into the other s turf by merger and acquisitions, and started to offer bundles alongside with stand-alone telehony services Consumers general ercetion is such that KT s quality of fixed-line telehony service is suerior, but KT s mobile telehony service is suerior 5 Matutes and Reibeau (99) adots the Hotelling location model; Anderson and Leruth (993), the discrete choice model; and Economides (993), the linear demand model Download Date /6/ 5:8 AM

5 Ahn and Yoon: Mixed Bundling of Vertically Differentiated Products To analyze the effects of cometitive mixed bundling in vertically differentiated roduct markets, we consider a model in which two firms offer their own brands of two roducts One firm offers a high-quality brand of roduct and a lowquality brand of roduct, whereas the other offers a low-quality brand of roduct and a high-quality brand of roduct We assume that (i) there are no economies of scoe at the roduction level, ie, bundling does not induce cost savings, 6 and that (ii) consumers valuations for the two roducts are indeendent Hence, we rule out the rationale for bundling arising out of roduction and/or consumtion efficiency After introducing the model in ection, we consider the benchmark case in which the firms do not offer bundle discounts in ection 3 We then analyze the case of mixed bundling in ection 4: We start in ection 4 with the case when the firms quality advantages are symmetric In ection 4, we allow for asymmetric quality advantages That is, the magnitude of the quality ga between the brands of roduct may be different from that between the brands of roduct It is shown that, comared to the equilibrium without bundling, (i) rices, rofits and social welfare are lower, whereas (ii) consumer surlus is higher in the equilibrium with mixed bundling In addition, the oulation of consumers who urchase both roducts from the same firm is larger in the equilibrium with mixed bundling These results are in line with those obtained in the revious literature on cometitive mixed bundling with horizontally differentiated roducts Mixed bundling, both with horizontal differentiation and with vertical differentiation, intensifies cometition between the firms by extending it to the inter-roducts level, as the bundles become substitutes This lowers rices and rofits ocial welfare decreases in the horizontal differentiation models because some consumers who like one roduct more than the other and thus would be best served by urchasing from different firms are lured into buying bundles as a result of bundle discounts The decrease in social welfare in our model can be exlained in a similar way: some consumers who care about the qualities of both roducts switch from buying high-quality brands of both roducts to buying bundles However, mixed bundling in our model has additional effect It increases social welfare by shrinking or eliminating the consumer grou who urchase low-quality brands of both roducts ocial welfare decreases because the former effect dominates the latter We conduct a comarative static analysis with resect to changes in quality differentiation arameters Let A be the quality ga between the brands of one roduct and B be the quality ga between the brands of another roduct We 6 In fact, marginal roduction costs will be assumed to be identical for all quality brands and for all firms Published by De Gruyter, 0 3 Download Date /6/ 5:8 AM

6 The BE Journal of Economic Analysis & Policy, Vol [0], Iss (Contributions), Art 50 show in ection 3 that, when firms do not offer bundle discounts, equilibrium rices and rofits decrease as A or B gets smaller The reason is basically that firms comete more intensely when brands become more substitutable We show in ection 4 for symmetric mixed bundling that, with resect to a decrease in A B, the direction of changes in these equilibrium variables is the same as that in the revious section but the rate of changes is different In articular, rices and rofits decrease more slowly as A B decreases Hence, the rofit-reducing effect of mixed bundling gets exacerbated as the quality ga, A B, increases The asymmetric quality differentiation in ection 4 yields more interesting observations with regard to the comarative statics Let us assume without loss of generality that A B We first show that a decrease in A reduces both firms rofits, whereas a decrease in B increases one firm s rofits but reduces the other s rofits The reason is that a decrease in A narrows the quality ga between bundles and thus intensifies cometition In contrast, a decrease in B widens the quality ga which leads to lessened cometition, favoring the firm in a cometitively advantageous osition The comarative statics on the rofitreducing effect of mixed bundling become slightly different from the case of symmetric mixed bundling We also show that the effect of a decrease in A on rices is different from that of a decrease in B, due to rather comlicated strategic interdeendence between the roducts 7 We additionally establish that the equilibrium rices with mixed bundling converge to those without bundling when the quality ga in one roduct vanishes, that is, when B gets close to zero In ection 5, we extend the model by relaxing some of the assumtions and show that the main results continue to hold with these extensions ection 6 concludes The model Two firms, and, each offer their own brand of two roducts, and The roducts are differentiated in terms of their quality That is, firm offers a highquality brand of roduct and a low-quality brand of roduct, and vice versa We assume that the marginal cost of roduction is zero for both firms 8 Let k i denote the (stand-alone) rice charged by firm k, for roduct i, and k denote the bundle discount offered by firm k, when the two roducts k k are sold as a ackage We assume that i 0 and 0 The firms set their rices and bundle discounts simultaneously 7 Please refer to ection 4 for more details 8 The main results of the resent aer continue to hold even when we relax this assumtion of zero marginal costs ee ection 5 for more details 4 Download Date /6/ 5:8 AM

7 Ahn and Yoon: Mixed Bundling of Vertically Differentiated Products There is a oulation of consumers who urchase at most one unit of each of the two roducts We normalize the total oulation of consumers to be Consumers differ in their taste for quality A consumer s reference is denoted by ( x, y) [0,] [0,], where x reresents the valuation for roduct and y reresents that for roduct We assume that consumers valuations for the roducts are uniformly distributed on a unit square [0,] [0,] Hence, consumers references for the two roducts are indeendent A consumer with k the reference ( x, y ) enjoys a gross ayoff of v a x when she urchases one k unit of roduct from firm k,, and she enjoys a gross ayoff of v b y when she urchases one unit of roduct from firm k, Observe that this ayoff secification is equivalent to the one in the seminal work of haked and utton (983) and the subsequent works on vertical differentiation U to ection 4, we assume that v and v are sufficiently large such that every consumer urchases one unit of each of the two roducts (We relax this full market coverage assumtion in ection 5) Hence, a consumer may urchase (i) one unit of roduct from and one unit of roduct from ; (ii) one unit of each of the two roducts from ; (iii) one unit of each of the two roducts from ; or (iv) one unit of roduct from and one unit of roduct from Accordingly, a consumer with the reference ( x, y ) enjoys a net ayoff as follows: (i) v v a x b y ( ), (ii) v v a xb y( ), (iii) vv a xb y( ), or (iv) v v a xb y( ), where a and b are firm s quality arameters for roducts and, resectively, and a and b are firm s quality arameters for roducts and, resectively Here, a a, but b b, that is, firm has a quality advantage in roduct, whereas firm has a quality advantage in roduct For simlicity, we fix the qualities of the high-quality brands of both roducts to Thus, Assumtion a b and 0 a, b We make several notational conventions Firstly, we denote the arameters a and b as a and b, resectively econdly, we denote the bundle rice k k k as r k for k, Then, a consumer s net ayoff for the four Published by De Gruyter, 0 5 Download Date /6/ 5:8 AM

8 The BE Journal of Economic Analysis & Policy, Vol [0], Iss (Contributions), Art 50 urchase scenarios discussed above, as suressing the common comonent v v, can be rewritten as follows: (i) x y, (ii) x by r, (iii) ax y r, or (iv) ax by Let us denote the oulation of consumers who urchase roduct from firm and roduct from firm by D, the oulation of consumers who urchase both roducts from firm by D, and so on 3 Equilibrium without bundling As a benchmark, we first consider the case when no firm offers a bundle discount, that is, 0 Figure shows the consumer s choices without bundling Consumers with x ( )/( a) urchase roduct from firm, whereas consumers with x ( )/( a) urchase roduct from firm imilarly, consumers with y ( )/( b) urchase roduct from firm, whereas consumers with y ( )/( b) urchase roduct from firm Figure : The attern of consumer choice without bundling 6 Download Date /6/ 5:8 AM

9 Ahn and Yoon: Mixed Bundling of Vertically Differentiated Products Hence, the firms rofits are ( ) ; ( ) a b a b We have: Proosition The equilibrium is given by ( a) a b ( b),,, ; ab 5a4b D, D D, D ;, k k Proof The equilibrium can be easily obtained by setting / i 0 for i, and for k, QED Observe that the equilibrium rices i and i of roduct i deend only on the quality differentiation arameter for roduct i, not on that for the other roduct In this sense, cometition between the firms is comletely searated across the roducts, that is, there is no strategic interdeendence between the roducts This asect of cometition changes when the firms offer bundle discounts Observe also that when the quality differentiation between the firms diminishes (ie, as a or b increases toward ), both rices and rofits decrease The reason is that the firms comete more vigorously as the roducts become more substitutable ocial welfare without bundling is given by 6 a b axdx xdx bydy ydy 8 /3 /3 0 /3 0 /3 Consumer surlus, which is equal to social welfare net of the firms rofits, is (a b 4) /8 9 Both social welfare and consumer surlus increase as a or b increases 9 ote that we suress the common comonent ν ν Published by De Gruyter, 0 7 Download Date /6/ 5:8 AM

10 The BE Journal of Economic Analysis & Policy, Vol [0], Iss (Contributions), Art 50 4 Equilibrium with mixed bundling We now assume that the firms are engaged in mixed bundling Then the consumer s choice can be summarized by the following two diagrams Figure deicts the case when min{ r, r }, whereas Figure 3 deicts the case when min{ r, r } Figure shows that consumers with high valuations for both roducts urchase roduct from firm and roduct from firm Consumers with a high valuation for roduct but a low valuation for roduct urchase both roduct from, whereas those with a high valuation for roduct but a low valuation for roduct urchase both roduct from Finally, consumers with low valuations for both roducts urchase roduct from and roduct from Figure : The attern of consumer choice when min{ r, r } Figure 3 shows similar urchase atterns, excet that there do not exist consumers who urchase roduct from and roduct from Figure 3(a) shows the case when r r, and Figure 3(b) shows the case when r r If a consumer urchased roduct from and roduct from, then she would consume low-quality brands of both roducts When min{ r, r } r, 8 Download Date /6/ 5:8 AM

11 she can consume a high-quality brand of roduct (and a low-quality brand of roduct ) with a lower total rice by urchasing both roducts from imilarly, when min{, } r r r, she can consume a high-quality brand of roduct (and a low-quality brand of roduct ) with a lower total rice by urchasing both roducts from (a) (b) Figure 3: The attern of consumer choice when min{, } r r The firms rofits when min{, } r r r are, a r r a r b r a r b r r b r a r D r D π 9 Ahn and Yoon: Mixed Bundling of Vertically Differentiated Products Published by De Gruyter, 0 Download Date /6/ 5:8 AM

12 The BE Journal of Economic Analysis & Policy, Vol [0], Iss (Contributions), Art 50 π r D r D r a r b r a r b r a r r a r b, and their rofits when min{ r, r } r can be analogously given We can in fact establish that there cannot exist consumers who urchase low-quality brands of both roducts That is, the situation in Figure cannot haen in equilibrium Proosition We have min{ r, r } in equilibrium Proof ee Aendix QED The reason for this result is that, when min{ r, r } holds, either firm can rofitably increase to min{ r, r } or firm can rofitably increase to min{ r, r } such that there exist no consumers who urchase lowquality brands of both roducts By raising the rices of low-quality brands to make the otion of buying low-quality brands of both roducts unattractive, the firms induce consumers to urchase bundles 0 Observe that we have drawn Figure 3 under the resumtion that ( r )/( a) and ( r )/( b) If one (or both) of these inequalities does not hold, then there exist no consumers who urchase roduct from firm and roduct from firm That is, all consumers urchase either s bundle or s bundle However, we establish in Proosition 3 that this ure bundling outcome cannot haen in equilibrium In a situation where all consumers urchase either firm s bundle, we can show that one firm can rofitably reduce the standalone rice of its strong roduct to cature some consumers who have high valuations for both roducts but used to buy the bundle of the other firm 0 We note that Proosition holds under the full market coverage assumtion When the market is not fully covered, the deviation used in roving the roosition (raising the rices of low-quality brands) would result in reduced demand, with otentially ambiguous effects on rofits In fact, we show in ection 5 that D 0 in equilibrium if the full market coverage assumtion is droed The authors are indebted to an anonymous referee for ointing out this issue 0 Download Date /6/ 5:8 AM

13 Ahn and Yoon: Mixed Bundling of Vertically Differentiated Products r r Proosition 3 We have max, a b in equilibrium Proof ee Aendix QED We next establish that r r when a b That is, if firm s disadvantage in roduct is smaller than firm s disadvantage in roduct, so that the quality of firm s bundle is suerior to that of firm s bundle, then firm s bundle rice is higher in equilibrium Proosition 4 Assume that a b We have r r in equilibrium Proof ee Aendix QED 4 Mixed bundling under symmetric quality differentiation We now focus on the case in which the firms quality disadvantages are symmetric, that is, a b ection 4 below addresses a more general case in which a b The following roosition characterizes the equilibrium outcome Proosition 5 Assume that a b The equilibrium is given by 7( a) ( a), r r ; 3 3 9( a) D, D D, D 0; Proof ee Aendix QED Let us comare the equilibrium outcome with mixed bundling to that without bundling ince we are now focusing on the symmetric case for the moment, we need to set a b in Proosition Observe first that the oulation of consumers who urchase both roducts from the same firm, ie, D D, increases from 4/9 without bundling to 3/4 with mixed bundling This is because the firms comete to attract customers by offering bundle discounts Hence, the bundle rice r r ( a) / 3 is lower than the sum of stand-alone rices without bundling, which is equal to a ote also that the stand-alone rices of high-quality brands,, are lower with mixed bundling As a consequence, Published by De Gruyter, 0 Download Date /6/ 5:8 AM

14 The BE Journal of Economic Analysis & Policy, Vol [0], Iss (Contributions), Art 50 the firms rofits also decrease from 5( a) / 9 without bundling to 9( a) / 48 with mixed bundling Most of the aers on cometitive bundling with horizontal differentiated roducts, including Matutes and Regibeau (99), Anderson and Leruth (993), and Economides (993), show that the firms rofits decrease in the equilibrium with mixed bundling comared to the equilibrium without bundling We have established that this finding continues to hold when roducts are vertically differentiated As a matter of fact, basically the same intuition alies The reason why mixed bundling reduces rofits in our model is that it creates a new asect of cometition Recall from the discussion after Proosition that cometition between the firms is comletely searated across the roducts without bundling With mixed bundling, the cometition between firms is extended to the inter-roduct level as the bundles offered by the two firms become substitutes Mixed bundling in the horizontal differentiation models has the exact same role ocial welfare with mixed bundling is given by / / ( x y) dydx ( x by) dydx ( x by) dydx / / / / / / 5 a ( ) ( ) 0 ax y dydx ax y dydx / 0 x 6 Recall from the revious section that social welfare without bundling is (8 a) / 9 when a b ince (8 a) / 9 (5 a) / 6 ( a) /8 0, social welfare decreases with mixed bundling Mixed bundling has two countervailing effects on social welfare On the one hand, it increases social welfare by eliminating the consumer grou, reresented by D, who urchase low-quality brands of both roducts On the other hand, it decreases social welfare by shrinking the consumer grou, reresented by D, who urchase high-quality brands of both roducts As can be seen from Figures and 3 (with setting a b and r r ), the former effect increases social welfare by x We can show, from the first order conditions given in the aendix, that one firm s best resonse is to offer mixed bundling when the other does not For instance, when a b 0, firm s best resonse is to set r and if firm follows the equilibrium strategy without bundling, ie / 3, / 3, and r Thus, firms face a risoners dilemma situation, similar to the case of mixed bundling with horizontal differentiation ote that, though the individual roducts are vertically differentiated in our model, the bundles are horizontally differentiated, esecially when a b ome consumers refer firm s bundle while others refer firm s bundle, deending on their reference for the individual comonents of the bundles We areciate an anonymous referee for ointing this out Download Date /6/ 5:8 AM

15 Ahn and Yoon: Mixed Bundling of Vertically Differentiated Products /3 x /3 y [ x by ( ax by)] dydx [ ax y ( ax by)] dxdy /3 x ( a) ( a) x dydx, whereas the latter effect decreases social welfare by / / [ x y ( x by)] dydx [ x y ( x by)] dydx / /3 /3 /3 / / [ x y ( ax y)] dxdy [ x y ( ax y)] dxdy / /3 /3 /3 / / [ ( b) ydydx ( b) ydydx] / /3 /3 /3 3( a) 6 x x Thus, the latter effect dominates It is clear that firms offer bundle discounts mainly to attract the consumers in D who have high valuations for both roducts The extinction of consumer grou D may be viewed as a by-roduct of firms cometition at this front In other words, the latter effect is rimary whereas the former effect is auxiliary 3 That s why social welfare decreases with mixed bundling In existing models of cometitive mixed bundling with horizontal differentiation, such as Matutes and Regibeau (99), Gans and King (006), and Reisinger (006), social welfare decreases with mixed bundling due to the distributive inefficiency That is, social welfare decreases in these location models because some consumers who like one roduct more than the other and thus would be best served by urchasing from different firms are lured into buying bundles as a result of bundle discounts In this resect, the reason for the decrease in social welfare is similar to the case of vertical differentiation where some consumers who care about the qualities of both roducts switch from buying highquality brands of both roducts to buying bundles ote, however, that the equilibrium without bundling in horizontal differentiation models is socially y 3 Let us elaborate on this The dominance of the latter effect can be exlained by two reasons First of all, mixed bundling shrinks the size of D more than that of D Observe that the size of D used to be larger than that of D without bundling, due to high-quality brands quality advantages (and the assumtion of uniform distribution) Thus, mixed bundling affects the size of D more than that of D econdly, the consumers who used to be in D without bundling care much more about the quality than those in D Hence, a change in their consumtion has a bigger imact on social welfare Published by De Gruyter, 0 3 Download Date /6/ 5:8 AM

16 The BE Journal of Economic Analysis & Policy, Vol [0], Iss (Contributions), Art 50 otimal since the total distance travelled by consumers is minimized whereas the corresonding equilibrium in this vertical differentiation model is not The social otimum is achieved when all consumers urchase high-quality brands from both firms Thus, it is less obvious that social welfare decreases with mixed bundling in our model Finally, consumer surlus with mixed bundling is ( 3 a) / 4 Thus, consumer surlus increases with mixed bundling by the amount of 9( a) / 7 This is mainly due to rice decreases under mixed bundling As a matter of fact, it is straightforward to check that all consumers, including those who change their consumtion atterns, are better off under mixed bundling ummarizing the discussion, Proosition 6 Comared to the equilibrium without bundling, (i) rices, rofits and social welfare are lower, whereas (ii) consumer surlus as well as the oulation of consumers who urchase both roducts from the same firm is larger in the equilibrium with mixed bundling As for comarative statics, observe in Proosition 5 that rices as well as rofits decrease as a bincreases Observe also from the discussion above that both social welfare and consumer surlus increase as a bincreases A noteworthy observation with resect to the effect of a change in the arameter a b is that the rofit-reducing effect of mixed bundling gets exacerbated as the degree of vertical differentiation increases In articular, the magnitude of reduction in rofits as the firms offer bundle discounts, which is equal to 5( a)/99( a)/483( a)/44, increases with higher vertical differentiation, ie, with a decrease in a b As exlained before, cometition is intensified with mixed bundling since the bundles become substitutes This cometition-enhancing effect of mixed bundling, which certainly reduces rofits, is stronger when the cometition level without bundling is lower On the other hand, when the cometition level is already high even without bundling, as in the case when a b is sufficiently close to, firms do not have much room for bundle discounts Therefore, the rofit-reducing effect of mixed bundling becomes bigger when the quality ga increases 4 Mixed bundling under asymmetric quality differentiation Let us return to the general case in which a b Assume without loss of generality that a b We relegate the characterization of equilibrium to the aendix since it is technical as well as quite comlicated The technical lemma in Aendix shows that the equilibrium rices with mixed bundling deend on the 4 Download Date /6/ 5:8 AM

17 Ahn and Yoon: Mixed Bundling of Vertically Differentiated Products quality differentiation arameters a and b for both roducts This result contrasts with that for the case without bundling, where the equilibrium rices of roduct i deend only on the quality differentiation arameter for roduct i, not on that for the other roduct A detailed discussion on this strategic interdeendence will follow shortly Let us comare the equilibrium outcome with mixed bundling to that without bundling In short, the result is the same as that for the symmetric case We show in Proosition 7 that the equilibrium rices are lower and the oulation of consumers who urchase bundles is larger with mixed bundling Proosition 7 Assume that a b Comared to the equilibrium without bundling, (i) rices are lower, whereas (ii) the oulation of consumers who urchase both roducts from the same firm is larger in the equilibrium with mixed bundling Proof ee Aendix QED We can obtain the same results for rofits, social welfare, and consumer surlus as we have in Proosition 6 However, since the roof is tediously long albeit similar to that of Proosition 7, we only resent some numerical results 4 In articular, we resent the results for rofits in Tables and below, but relegate the results on social welfare and consumer surlus to Tables 5 and 6 in the aendix for exositional brevity Comaring these values with the corresonding equilibrium outcomes without bundling (obtained analytically at Proosition and whose numerical values are given in the arentheses at the tables below), we can easily see that rofits and social welfare are lower but consumer surlus is higher under mixed bundling 4 The roof is available uon request Published by De Gruyter, 0 5 Download Date /6/ 5:8 AM

18 The BE Journal of Economic Analysis & Policy, Vol [0], Iss (Contributions), Art 50 Download Date /6/ 5:8 AM b a Table under mixed bundling (05556) (05444) (05) (05333) (04889) (04444) (05) 036 (04778) 0389 (04333) (03889) (05) (04667) 037 (04) 0793 (03778) (03333) 04 (05) (04556) 0353 (04) 083 (03667) 0398 (03) (0778) 0473 (04889) (04444) 0397 (04) 086 (03556) 0430 (03) 0003 (0667) (0) 043 (04778) (04333) (03889) 090 (03444) 0473 (03) 0038 (0556) 0609 (0) (0667) 0496 (04667) (04) 0340 (03778) 090 (03333) 057 (0889) 0086 (0444) 0649 (0) 05 (0556) (0) (04556) 0393 (04) (03667) (03) 059 (0778) 048 (0333) 0705 (0889) 063 (0444) (0) (00556) 6

19 Ahn and Yoon: Mixed Bundling of Vertically Differentiated Products b a Table under mixed bundling (05556) (05) (05) 0339 (04667) 0380 (04556) (04444) 0306 (04) 0995 (04) 0884 (04) (03889) 089 (03778) 0709 (03667) 0599 (03556) 0488 (03444) (03333) 053 (03333) 04 (03) 03 (03) 003 (03) 009 (0889) (0778) 045 (0889) 035 (0778) 005 (0667) 096 (0556) 0806 (0444) 0696 (0333) (0) 0959 (0444) 0849 (0333) 0739 (0) 069 (0) 059 (0) 0409 (0889) 099 (0778) (0667) 0675 (0) 0564 (0889) 0454 (0778) 0343 (0667) 033 (0556) 03 (0444) 003 (0333) (0) (0) 039 (0556) 08 (0444) 070 (0333) 0059 (0) (0) (0) 0077 (00889) 0066 (00778) (00667) (00556) We now turn to the comarative static analysis Observe from Tables and that an increase in a reduces both firms rofits, whereas an increase in b increases firm s rofits but reduces firm s rofits This asymmetry is due to our assumtion of a b, or more accurately, it is due to the difference in the effect of changes in quality arameters on the firms cometitive advantage: An increase in a, which means a reduction in firm s quality disadvantage in roduct, diminishes firm s overall advantage On the other hand, an increase in b or a reduction in firm s quality disadvantage in roduct enhances firm s overall advantage In other words, an increase in a narrows the quality ga between the firms bundles, whereas an increase in b widens it This contrasts with the result in ection 3 that an increase in a or b reduces both firms rofits in the case without bundling, which is due to no strategic interdeendence across the roducts Observe also from Tables 5 and 6 in the aendix that both social welfare and consumer surlus increase in a and b As we have seen in the case of a b, the rofit-reducing effect of mixed bundling gets exacerbated as a b decreases When a b, we have a slightly different result When b decreases, we have the same result as in the case of Published by De Gruyter, 0 7 Download Date /6/ 5:8 AM

20 The BE Journal of Economic Analysis & Policy, Vol [0], Iss (Contributions), Art 50 a b We can verify from Tables, and Proosition that the reduction in both firms rofits gets larger as b decreases A decrease in a also makes the reduction in firm s rofits larger However, the reduction in firm s rofits turns out to be smaller for some arameter values as a decreases How can we exlain this difference? Recall from the case of a bthat firms do not have much room for bundle discounts when the quality ga is small This intuition also alies when a b When a or b (searately) decreases, both firms affordability to rovide bundle discounts increases as their rofits are higher without bundling Therefore, a decrease in a or b has an effect to exacerbate the rofit-reducing effect of mixed bundling However, there are additional asects to be taken into account when a b As exlained in the last aragrah, an increase in a narrows the quality ga between the firms bundles, whereas an increase in b widens it Thus, a decrease in b makes the rofit-reducing effect of mixed bundling larger, while a decrease in a tends to make it smaller Therefore, a decrease in a has an ambiguous overall effect on the rofitreducing effect of mixed bundling On the contrary, a decrease in b unambiguously makes the rofit-reducing effect of mixed bundling larger The effect of changes in quality arameters on the equilibrium rices is more intricate (We relegate the numerical results for rices to Tables 7, 8, 9 and 0 in the aendix) An increase in a lowers, r, and r, whereas its effect on is ambiguous 5 This is due to strategic interdeendences between the roducts that mixed bundling creates Without bundling, roduct and roduct are neither substitutes nor comlements The demand for roduct offered by firm (or ) is indeendent of the rice of roduct offered by firm (or ), and vice versa Mixed bundling changes this relationshi Figure 3(a) reveals the following strategic interdeendences: (i) the bundles offered by the two firms are substitutes; (ii) roduct by firm and the bundle by firm are substitutes, and so are roduct by firm and the bundle by firm ; (iii) roduct and the bundle by firm are substitutes, and so are roduct and the bundle by firm ; and (iv) roduct by firm and roduct by firm are comlements 6 Taking these interdeendences into account, suose that the quality of roduct by firm imroves, that is, the arameter a increases As the rival firm s disadvantage in roduct diminishes, firm will lower both the 5 ee, for instance, the case of b 08 or b 09 in Table 8 in the aendix 6 The bundles offered by the firms are substitutes in the sense that the demand for firm s bundle, D, is increasing in the rice of firm s bundle, r, and vice versa On the other hand, roduct by firm and roduct by firm are comlements, because the demand for both, which is equal to D, is decreasing in and 8 Download Date /6/ 5:8 AM

21 Ahn and Yoon: Mixed Bundling of Vertically Differentiated Products bundle rice r and stand-alone rice of roduct The bundle rice r also decreases by (i) and (ii) However, the effect on is ambiguous because a decrease in r and r lowers by (ii) and (iii) but a decrease in raises by (iv) The effect of an increase in b is slightly different An increase in b lowers both and r by a reason similar to the one given above On the other hand, Table 9 shows that the effect on r is ambiguous 7 This is because an increase in b, by imroving the quality of firm s bundle, has an effect of ushing u r Recall that an increase in a also has an effect of raising r But this effect is smaller than the effect on r of an increase in b since an increase in a narrows the quality ga between the firms bundles whereas an increase in b widens it Thus, an increase in a has a negative overall effect on r, whereas an increase b has an ambiguous overall effect on r Because of these differential effects on the quality ga, an increase in b unambiguously raises, which contrasts with the fact that an increase in a has an ambiguous effect on The following table summarizes the comarative static results for the mixed bundling case In the table, W denotes social welfare and C denotes k consumer surlus In addition, for k, denotes the difference in firm k s rofits under no bundling and under mixed bundling Table 3 Comarative statics for the mixed bundling case r r π π W C Δπ Δπ symmetric quality differentiation ( a b) a b asymmetric quality differentiation ( a b) a b + + We finally establish that, as b gets closer to, that is, as the quality ga in roduct vanishes, the equilibrium rices with mixed bundling converge to those without bundling 7 ee, for instance, the case of a 0 or a 0 in Table 9 in the aendix Published by De Gruyter, 0 9 Download Date /6/ 5:8 AM

22 The BE Journal of Economic Analysis & Policy, Vol [0], Iss (Contributions), Art 50 Proosition 8 Assume that a bwe have ( a) a lim lim r, lim 0, and lim r b b 3 b b 3 Proof ee Aendix QED Observe that the limits are identical to the rices given in Proosition for b When the quality ga in roduct vanishes, it does not matter for the gross ayoff whether to urchase firm s bundle or to urchase roduct from firm and roduct from firm This causes r to be equal to In addition, goes down to 0 since, otherwise, firm can increase its rofits by slightly lowering the rice This imlies that r is equal to and that the imlicit rice of roduct by firm is thus zero In short, as the two brands of one roduct become erfect substitutes, the firms rices of that roduct go down to 0 Therefore, there is no room for bundle discounts and mixed bundling yields the same outcome as the case in ection 3 without bundling 5 Extensions In this section, we extend the model in the revious sections to see whether the results are robust In articular, we relax the assumtion of full market coverage and the assumtion of zero marginal costs: The analysis shows that the main results continue to hold with these extensions 5 The assumtion of full market coverage The analysis so far has been based on the remise that all consumers urchase one unit of each of the two roducts One might wonder whether this assumtion of full market coverage lays a critical role in obtaining some of the results, in articular, the results that firms rofits and/or social welfare are lower with mixed bundling Indeed, without the full market coverage assumtion, more consumers may buy the roducts, and so the market may exand, under mixed bundling since both the bundle rices as well as the stand-alone rices become lower This ossibility of market exansion is a factor that could ossibly increase firms rofits and social welfare, but that could not have been taken into account with the full market coverage assumtion In this subsection, we relax the assumtion of full market coverage and 0 Download Date /6/ 5:8 AM

23 Ahn and Yoon: Mixed Bundling of Vertically Differentiated Products investigate the effect of mixed bundling on rices, rofits, consumer surlus and social welfare In doing so, we assume that v v 0 in the ayoff secification, instead of the revious assumtion that both v and v are sufficiently high 8 With this secification, a consumer has five more otions, in addition to the four otions mentioned in ection Her net ayoffs for these 5 otions are (v) x when she only urchases roduct from firm, (vi) ax when she only urchases roduct from firm, (vii) by when she only urchases roduct from firm, (viii) y when she only urchases roduct from firm, and (ix) 0 when she urchases nothing The net ayoffs for the other 4 otions are secified in ection Following the notation in ection, denote the oulation of consumers who urchase only roduct from firm by D X, the oulation of consumers who urchase only roduct from firm by D X, and so on Figure 4 shows the attern of consumer choice Figure 4: The attern of consumer choice without full market coverage 8 With ν and ν being nonzero but small enough, we would have qualitatively similar results However, the equilibrium values would be slightly different because the equilibrium rices are functions of ν and ν without full market coverage Published by De Gruyter, 0 Download Date /6/ 5:8 AM

24 The BE Journal of Economic Analysis & Policy, Vol [0], Iss (Contributions), Art 50 The firms rofits are given by ( D D ) ( D D ) r D, X X ( D D ) ( D D ) r D X X k k k When no firm offers a bundle discount, ie, r for k,, it is easy to verify from the first-order condition that the equilibrium rices, rofits, consumer surlus, and social welfare are given by ( a) a( a) b( b) ( b),,, ; 4a 4a 4 b 4 b 4( a) b( b) a( a) 4( b), ; (4 a) (4 b) (4 a) (4 b) 4(8 ab)(4 ab) ab[5( ab) 88] C ; (4 a) (4 b) 4(8 a b)( 5a 5 b) ab[5( a b) 4ab 56] W (4 a) (4 b) On the other hand, the equilibrium with mixed bundling is very comlicated to characterize, even for the symmetric case of a b We can only rovide the equilibrium outcomes numerically for given arameter values Table 4 shows numerical values of the equilibrium outcomes when a b 00, a b 05, and a b 099 ote that the first row in each cell, ie the w/o row, shows the equilibrium value without bundling, while the second row, the w/ row, shows the one with bundling a=b=00 a=b=05 a=b=099 Table 4 Equilibrium values with and without mixed bundling r r C W w/o w/ w/o w/ w/o w/ We can see that rofits and social welfare are lower whereas consumer surlus is higher under mixed bundling Therefore, the results on rofits, consumer surlus, and social welfare with the assumtion of full market coverage are still valid We can also see that, as in the case of full market coverage, mixed bundling Download Date /6/ 5:8 AM

25 Ahn and Yoon: Mixed Bundling of Vertically Differentiated Products lowers the bundle rices, r and r, and thus exands the oulation D and D of consumers who urchase both roducts from the same firm, but shrinks the oulation D of consumers who urchase high quality brands from each firm However, there are some differences in the effect on the stand-alone rices Recall that the stand-alone rices of high-quality brands are lower with mixed bundling in the case of full market coverage 9 When the market is not fully covered, on the other hand, we obtain three different outcomes in equilibrium: (i) when the ga between the high-quality brand and the low-quality brand is large, eg, a b 00, all individual rices,,, and, are higher with mixed bundling; (ii) when the quality ga is sufficiently small, eg, a b 099, all individual rices lower; (iii) when the quality ga is in the intermediate range, eg, a b 05, the rices of high-quality brands, and, are lower but the rices of low-quality brands, and, are higher The reason is as follows It is easy to see from Figures 3 and 4 that, when the market is not fully covered, X firm has an additional incentive to attract consumers in D and D X to exand D, and likewise for firm This exlains why the stand-alone rices of high-quality brands might increase with mixed bundling, esecially when the quality ga is large However, if the quality ga is small and thus cometition between the bundles becomes tougher, the rices of high-quality brands as well as the bundle rices have to be reduced, although the downward ressure on the former is less severe when the market is not fully covered This exlains the situations such as when a b 05 and a b 099 We are now ready to exlain why rofits and social welfare are lower with mixed bundling when the market is not fully covered First of all, mixed bundling decreases rofits and social welfare by shrinking the consumer grou D who urchase high-quality brands of both roducts This is because bundle rices are lower as well as the downward ressure on the stand-alone rices of high-quality brands is less severe, as exlained in the revious aragrah econdly, it increases rofits and social welfare by shrinking the consumer grous D X, X X X D, D, and D who urchase only one roduct Finally, it may decrease XX rofits and social welfare by exanding the consumer grou D who urchase neither roduct As for the rices of low-quality brands, firms have two countervailing incentives Firms have an incentive to sell more individual lowquality brands Firms also have an incentive to kee the consumers who urchase 9 The stand-alone rices of low-quality brands are irrelevant in the case of full market coverage since Proosition shows that no consumer urchases low-quality brands of both roducts in equilibrium Published by De Gruyter, 0 3 Download Date /6/ 5:8 AM

26 The BE Journal of Economic Analysis & Policy, Vol [0], Iss (Contributions), Art 50 the bundles If the latter incentive dominates the former incentive, the rices of XX low-quality brands would be higher, leading to an exansion of D This is what haens unless a b is very close to All in all, as the numerical results above shows, the first effect of mixed bundling turns out to be dominant, and we get that firms rofits as well as social welfare are lower with mixed bundling Table 4 also suggests that the comarative static results continue to hold That is, rofits decrease while both social welfare and consumer surlus increase as a b increases The stand-alone rices of high-quality brands as well as the bundle rices decrease as a b increases However, the effect on the standalone rices of low-quality brands is rather ambiguous: and increase as a b increases from 00 to 05, but they decrease as a b increases from 05 to 099 This is because an increases in a b has two oosing effects On the one hand, it enhances the quality of low-quality brands, thereby raising the demands for low-quality brands and so the stand-alone rices of low-quality brands On the other hand, it has an effect of reducing the rices by narrowing the quality gas and thus intensifying cometition The former effect dominates the latter when a b is small, but the situation is reversed when a b is large 5 The assumtion of zero marginal costs Let us now relax the assumtion of zero marginal costs The marginal costs of roducing a high-quality brand and a low-quality brand are denoted by c H and c L, resectively, with ch cl Then, with the assumtion of full market coverage and the symmetric quality differentiation of a b, we can obtain that the equilibrium outcomes without bundling are given by 0 ( a) ch cl ach cl, ; 3 3 [( a) ( ch cl)] [( a) ( ch cl)]( ach cl) D, D D, 9( a) 9( a) ( ach cl) D ; 9( a) 0 The quality ga must be larger than the difference in marginal costs, ie, ch cl a, for the equilibrium outcomes Otherwise, it would not be rofitable to roduce a high-quality brand 4 Download Date /6/ 5:8 AM