The creation of Stackelberg leadership through product bundling

Transcription

1 The creation of Stackelerg leadership through product undling Jeroen Hinloopen University of Amsterdam (and Tiergen Institute) and Economics Network for Competition and Regulation (ENCORE) NOT TO BE CONSIDERED FOR THE YEE AWARD March 005 Astract Product undling is interpreted as a comitment device for credily supplying the Stackelerg-leader output. Contrary to common wisdom this interpretation of product undling in most cases appears to e welfare enhancing, thus qualifying its negative ruling induced y price-discrimination and leverage theories. At the same time it does leave the market a multiproduct firm of formidale competitive strength. Key words: Stackelerg leader, undling. JEL Classification: L1, L41 Correspondence: University of Amsterdam, Faculty of Economics and Econometrics, Department of Economics, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands; J.Hinloopen@uva.nl; This paper is inspired y discussions with Stephen Martin aout operating systems and Internet rowsers. A preliminary version of the paper has een circulated under the title The incentives to undle for a Stackelerg leader. Responsaility for the papers content is solely mine. 1

2 1 Introduction The recent landmark antitrust case against Microsoft has put product undling centre stage in the deate of antitrust enforcement (Economides (001a, ), Fisher (001), and Ellig, 001). Bundling, throughout this paper treated as the sale of several distinct products as one non-separale undle, is generally perceived as a usiness practise that harms consumers; they are forced to uy additional products they might not want (Stigler (1968), Adams and Yellen (1976), McAfee, McMillan, and Whinston, 1989), and firms could transfer market power they enjoy in one market to another market (Bernheim (1984), Whinston (1990), Martin (1999), Naleuff, 004). On the other hand, marketing and distriution cost are lowered (Salinger, 1995), and rapid dissemination of new technologies could e enhanced, all possily leading to a socially desirale efficiency gain. Yet, at the end of the day product undling is viewed as a strategy to increase profits further while harming society at large. This paper offers another explanation for product undling; the aility of a firm to credily commit to supply the Stackelerg-leader output. Indeed, as the duopoly equilirium introduced y Heinrich von Stackelerg (1934) involves one firm, the Stackelerg leader, not to e on its reaction function, efforts aound to enrich Stackelerg s set up so that the ensuing equilirium is a (sugame perfect) Nash equilirium, including constraining capacity (Spence (1976), and Dixit, 1980), and allowing for unilateral export susidies (Brander and Spencer, 1980). For a multiproduct oligopolist that has the option to ecome the Stackelerg leader y undling its output to what it supplies in another, possily related market, a trade-off arises. On the one hand, Stackelerg-leader profits typically exceed those of a symmetric oligopolist, while on the other hand, the supply of the other commodity in the product undle ecomes a constrained optimization prolem. We show that the former effect always dominates the latter; independent of the type of demand sustitutaility of the two commodities (sustitutes or complements) the multiproduct oligopolist prefers to undle its two commodities in order to ecome the Stackelerg leader. We show further that product undling employed as a commitment device is in most cases welfare enhancing. To show our results we consider the extreme situation in which the multiproduct firm holds a monopoly over the supply of the commodity with which it undles another product. This way the effect that counters the profitaility of ecoming the Stackelerg leader through product undling is maximized. Asent product undling the multiproduct oligopolist does not necessarily prefers to e the Stackelerg leader in the market where it faces a

3 rival. This is more likely to e the case the more homogenous products are. As the Stackelerg equilirium involves higher total output compared to the Cournot alternative, enforcing the Stackelerg equilirium involves canialization of profits in the monopoly market. And the more homogeneous products are, the more this leads to a reduction in concomitant profits. At the same time, the likelihood that market performance deteriorates in the future increases if the multiproduct firm is the Stackelerg leader, independent of whether it undles or not. Indeed, eing a Stackelerg follower rather than a Cournot oligopolist yields lower profits. Accordingly, the range of fixed cost that allows the single-product firm to stay in the market shrinks if the multiproduct firm ecomes the Stackelerg leader. The model We consider two markets, 1 and, and two firms, A and B. FirmA, the multiproduct firm, has a monopoly over the production of good 1 (in market 1) while oth firms produce good (in market ). Inverse demand for the twogoodsisderivedfromthefollowingstandardquadraticutilityfunction: 1 U = q 0 + a X Q i 1 i=1 X Q i +θ i=1 Y Q i, (1) where q 0 is a numeraire good, where Q i = q A i + q B i is total quantity supplied of product i, andwhereθ [ 1, 1] captures the extent to which the demand for the two goods is related;θ>0 implies that the goods are sustitutes (eing perfect if θ =1) while θ<0 implies that the goods are complements (eing perfect if θ = 1);thetwogoodsareindependentindemandifθ =0. In order to guarantee stale equiliria we restrict the parameter space of θ to ( 1, 1). Note further that q B 1 =0and q A 1, q A, q B > 0. The inverse demand curves that (1) yields under appropriate udget restrictions are: i=1 p i = a (Q i + θq j ), () i, j =1,, i 6= j, wherep i is the price of product i. Firm A has the option to undle product 1 with product. The relation etween undles and total quantities supplied is given y 1 This utility function is quite common in the Industrial Organization literature (see e.g. Singh and Vives (1984), or Qiu, 1997). Martin (00) traces it ack to Bowley (194). 3

4 and Q 1 = A (3) Q = A + B, (4) where A and B are the numer of undles sold y firm A and B respectively. Using these relations etween undles and quantities allows us to derive the inverse demand functions in terms of undles. In particular, inserting (3) and (4) in (1) and performing standard optimizations yields: p A =a (1 + θ) A + B (5) p B = a (1 + θ) A + B. (6) As the undle of firm A includes product, which is identical to the single-product undle of firm B, firm A s undle always sustitutes to some extent for firm B s undle, and vice versa, depending on the extent to which products in markets 1 and are differentiated. Indeed, θ ( 1, 1) we have that p A / B, p B / p A < 0. Demand sustitutaility of product undles is less the more complementary are product 1 and product ; a undle containing oth products would then e perceived as more distinct from a undle containing one product only. Likewise, demand sustitutaility of product undles increases the more homogeneous are the products on markets 1 and ; a undle containing oth products would then e considered much less distinct from a undle containing one product only. 3 Market equiliria We solve the model in case firm A does not undle (Section 3.1) and in case it does (Section 3.). Throughout we assume that firm A acts as a Stackelerg leader in the market for good. The concomitant analysis in case market is a Cournot oligopoly is presented in the Appendix. 4

5 3.1 No undling Firm A maximizes its profits in the market for good 1, π A,1 =(p 1 c)q A 1, over q A 1, yielding as est response:,3 q1 A (q A SL,q B SF )= 1 (a c) θ(q A SL + q B SF ), (7) where c are constant marginal cost of production. Firm B s reaction function in the market for good equals 4 q B SF (q1 A,q A SL )= 1 (a c) q A SL θq1 A. (8) BeingtheStackelerg leaderfirm A anticipates firm B s ehaviour and incorporates it in its production decision in the market for good. In particular it is aware of (8) when determining its optimal output, yielding as reaction function: 5 q A SL (q1 A,q B SF )= 1 (a c) θq A 1. (9) Solving (7), (8) and (9) simultaneously yields: and q A 1 = (a c)(4 3θ) 8 3θ, (10) q J (a c)( θ) = 8 3θ, (11) J = A SL,B SF. The equilirium expressions for all relevant variales are summarized in Tale 1, oth in case firm A is the Stackelerg leader in market as when oth firms are Cournot oligopolists in that market. Superscript C refers to Cournot, superscript S refers to Stackelerg. That firm A does not undle is indicated with suscript ; suscript would indicate that firm A undles. SL stands for Stackelerg leader, SF stands for Stackelerg follower. 3 The second-order condition always holds, which is true for all optimizations carried out in the remainder of the paper.. 4 The Ruth-Horwitz condition for staility requires that π B SF q B >. π B SF q A qb, which translates into > 1. 5 The Ruth-Horwitz condition for staility again translates into > 1. 5

6 Cournot duopoly in market * Q C,1 λ(3 θ)/ Q C, λ( θ) π A C λ [(3 θ) +( θ) ] /4 π B C λ ( θ) /4 CS C λ 5 4θ 0θ +8θ 3 /8 TS C λ 59 44θ 8θ +8θ 3 /8 Stackelerg duopoly in market ** Q S,1 δ(4 3θ) Q S, 3δ( θ) π A SL δ [(4 3θ) +( θ) ] π B SF δ ( θ) CS S δ [6 6θ 1θ +9θ 3 ] TS S 3δ [18 14θ 3θ +3θ 3 ] Tale 1: Equilirium expressions for relevant variales in case firm A does not undle; CS = consumers surplus, TS = total surplus; * λ =(a c)/(3 θ ); ** δ =(a c)/(8 3θ ) 3. Bundling In order to ecome the Stackelerg leader in market firm A could undle its output in the two product markets. Profits of firm B in terms of its undle, π B SF = p B c B, imply the following reaction function: 6 B ( A )= 1 (a c) (1 + θ) A. (1) Firm A s profits, eing the Stackelerg leader, are then equal to π A SL = p A c A = 3 θ (a c) (1 + θ) A A. (13) Maximizing these over A give rise to the equilirium numer of undles 7 and, through (1), A = 1 (a c) (14) (1 + θ) B = 1 (a c). (15) 4 6 The Ruth-Horwitz staility condition requires that > (1 + θ), orθ<1. 7 The second-order condition for a maximum reads as (3 θ)(1 + θ) < 0, orθ> 1, while staility requires that 0 > (3 θ)(1 + θ), or θ > 1. 6

7 Cournot duopoly in market * Stackelerg duopoly in market ** A τ(3 θ)/(1 + θ) A ϕ/ B τ B ϕ(1 + θ)/4 Q C,1 τ(3 θ)/(1 + θ) Q S,1 ϕ/ Q C, τ(5 + θ)/(1 + θ) Q S, ϕ(3 + θ)/4 π A C τ (3 θ) /(1 + θ) π A SL ϕ (1 + θ)(3 θ)/8 π B C 4τ π B SF ϕ (1 + θ) /16 CS C τ (17 θ )/(1 + θ) CS S ϕ (13 + 5θ)(1 + θ)/3 TS C τ (39 8θ + θ )/(1 + θ) TS S 3ϕ (9 + θ)(1 + θ)/3 Tale : Equilirium expressions for relevant variales in case firm A undles; CS = consumers surplus, TS = total surplus; * τ =(a c)/(7 θ); ** ϕ =(a c)/(1 + θ). Oserve that the numer of undles supplied y firm B is independent of the extent to which products are differentiated; it always equals half the numer of undles that would e offered y an otherwise identical monopolist. This implies in particular that if market demand increases due to increased differentiation of products all additional demand will e supplied y firm A. 8 This has important consequences for firm A s incentives towards eing the Stackelerg leader in market, as explained in detail elow. Total quantities of the underlying goods are implied y (3) and (4): Q S,1 = 1 (a c) (16) (1 + θ) Q S, = 3+θ (a c). (17) 4(1 + θ) In Tale the equilirium expressions for the current scenario are summarized. 4 The incentives to e a Stackelerg leader Having characterized the different Cournot and Stackelerg equiliria we can address a numer of interesting issues. In this section we consider firm A s incentive to e the Stackelerg leader rather than a Cournot oligopolist in market. We also examine the concomitant welfare consequences. 8 That market demand increases if products ecome more differentiated corroorates economic intuition although it is not undisputed (see Martin, 00, p. 5-56). 7

8 4.1 Private incentives No undling In case firm A does not undle the firstresultofinterestis: 9 Proposition 1 Profits of firm A in market 1 compare as follows: π A SL,1 > π A C,1 θ ( 1, 0), πa SL,1 <πa C,1 θ (0, 1). Profits of firm A in market compare as follows: π A SL, >πa C, θ ( 1, 1). That firm A s profits in market increase if it is the Stackelerg leader rather then a Cournot oligopolist is standard in the literature. New in this context is the analysis of the concomitant impact on related markets. Oserve that if firm A is the Stackelerg leader in market total production in this market will e higher compared to the situation where market is a Cournot oligopoly. An increase in production in market puts upward pressure on the demand in market 1 if products are demand complements, yielding higher profits for firm A there. Likewise, an increase in production in market diminishes demand in market 1 if products are demand sustitutes, yielding lower profits for firm A there. Considering the sum of firm A s profits yields: Proposition θ ( 1, 1) such that π A SL π A SL >π A C θ ( 1,θ ) and <π A C θ (θ, 1). Proof. The difference etween profits of a non-undling Stackelerg leader and a non-undling Cournot oligopolist equals (see also Appendix A1): (4 π A SL π A C 3θ) =(a c) +( θ) (8 3θ (3 θ) +( θ) ) 4(3 θ =. ) (18) Oserve that is contineous in θ for θ ( 1, 1), andthat θ= 1 > 0, while θ=1 < 0. Hence, θ exists. Nummerical simulations of for θ ( 1, 1) show that θ is unique. If products are demand complements it is always profitale for firm A to e the Stackelerg leader in market, rather then a Cournot oligopolist. This follows also directly from Proposition 1. Numerical simulations indicate that θ equals Hence, also when products are mild sustitutes in demand it is profitale for firm A to e the Stackelerg leader in market ; for small ut positive θ the canailization of firm A s profits in market 1 due to the 9 The proof is availale upon request. 8

9 increase in production in market is more than off-set y the concomitant increase in its market- profits. This holds in particular for θ equalto0,the case considered y Martin (1999). Products are then independent in demand ut undles are nevertheless demand sustitutes (recall (5) and (6)). The novel part of Proposition is when firm A does not want to e the Stackelerg leader in market, that is, when θ (θ, 1]. The more homogeneous products are (the larger is θ), the less profits firm A can earn in market 1. For firm A to diminish the demand-stealing effectfrommarket it should rationalize production in that market. One way of doing that would e to act as a Cournot oligopolist rather than as a Stackelerg leader in that market. According to Proposition, for θ (θ, 1] the resulting drop in firm A s profits in market is more than compensated for y the concomitant increase in firm A s profits in market Bundling Comparing firm A s profits when it sells its outputs only as a undle yields: Proposition 3 With respect to firm A eing the Stackelerg leader in market ornot,firm A s profits when it undles compare as follows: π A SL,1 >π A C,1 θ ( 1, 1). Proof. The difference etween profits of a undling Stackelerg leader and a undling Cournot oligopolist equals (see also Appendix A1): π A SL π A C = (a c) (3 θ)(1 + θ) 8(7 θ), from which the proposition follows immediately. Proposition 3 implicitly indicates that the cannaalizing effect of market onmarket1isreducediffirm A sellsitstwocommoditiesasaundle, especially when products are relatively homogeneous. From comparing (17) and (9) in the Appendix it is clear that also in case firm A undles its two products for θ (θ, 1) total production in market increases if firm A ecomes the Stackelerg leader there. Yet, y eing the Stackelerg leader in that market firm A fixes the numer of undles sold y firm B at half he numer of undles that would e supplied y an otherwise identical monopolist, independent of the extent to which the two products are differentiated. In fact, from (14) and (15) follows that any increase in demand due to products ecoming more differentiated is supplied y firm A only. Indeed, in case the two firms would compete as Cournot oligopolist in market, the demand increase due to increased product differentiation would e supplied 9

10 y oth firms (see the Appendix, equations (6) and (7)). Whether or not undling alleviates firm A s need to rationalize production in market when products are relatively homogeneous depends on a comparison of all possile equilirium profits, which is dealt with in Section Welfare implications Considering consumers well-eing first, oserve that Proposition 4 With respect to firm A eing the Stackelerg leader in market or not, consumers surplus compares as follows: (i) CS S >CSC, and (ii) CS S >CSC. Proof. This follows from (i) min 1 θ 1 {CS S CSC } = {CSS CSC } θ=1 = 31(a c) /800 > 0,and(ii)min 1 θ 1 {CS S CSC } = {CSS CSC } θ=3/5 = 15(a c) /56 > 0. Proposition 4 extents the standard result that a Stackelerg leader-follower market yields higher consumers surplus than a Cournot oligopoly to the situation in which the Stackelerg leader is a monopolist on a related market. It mirrors the rationing argument that rules firm A s incentives for not eing a Stackelerg leader when it sells its two commidities not as a undle; only when products are differentiated up to θ do private incentives and consumers preference coincide in that firm A wants to e the Stackelerg leader in market, a desire that would also enhance consumers surplus. On the other hand, in case firm A does undle, its incentives for eing the Stackelerg leader in market coincide with those maximizing consumers surplus. Comparale oservations can e made for total surplus: Proposition 5 With respect to firm A eing the Stackelerg leader in market or not, total surplus compares as follows: (i) TS S >TSC, and (ii) TSS > TS C. Proof. This follows from (i) min 1 θ 1 TS S TSª C = TS S TSª θ=0.79 C ª (a c) /105 > 0, and(ii)min 1 θ 1 TS S TS C = TS S TS C = ª θ=1 7(a c) /88 > 0. Although firm A s total profits increase if it acts as a Cournot oligopolist in market rather than a Stackelerg leader whenever θ exceeds θ, it reduces total surplus as the resulting loss in consumers surplus exceeds firm A s profits gain. Accordingly, given that firm A sells its two commodities as a undle or not, total surplus always increases if it would act as a Stackelerg leader in market. 10

11 5 The incentives to undle for a Stackelerg leader In the previous section we have demonstrated that, contrary to common wisdom, for a sustantial part of the parameter range on θ firm A does not want to e the Stackelerg leader in market. Rationalizing total output in market y ehaving as a Cournot oligopolist dampens the cannializing of market 1 demand y market output, especially when the commodities of the two markets are relatively close demand sustitutes (i.e. 1 θ>θ 1). To conclude that for these situations it is of little value to consider firm A as pursueing a undling strategy in order to ecome the Stackelerg leader in market hinges on the comparison of a undling Stackelerg leader s profits with those of a non undling Cournot oligopolist. On the other hand, as shown in Proposition, for commodities that are complementary in demand or relatively mild demand sustitutes (i.e. 1 θ<θ 1) firm A does want to e the Stackelerg leader in market. For that firm A could e e considered to undle its outputs. In this section we thus turn to the complementary question of the previous section; suppose firm A is indeed the Stackelerg leader in market, does it then (still) have an incentive to sell its two products as a undle? 5.1 Private incentives The firstresultofinterestis: Proposition 6 With respect to undling or not, firm A s profits eing the Stackelerg leader in market compare as follows: π A SL < π A SL θ ( 1, 0) (0, 1). 0. Proof. This follows from min 1 θ 1 π A SL ª π A SL = π A SL π A SL = ª θ=0 Next, straightforward numerical simulations in comination with Propositions,3,6learnthatfirm A s profitscompareasfollows: π A C π A C π A C π A C <π A SL <π A C <π A C <π A SL <π A C <π A SL <π A SL <π A C <π A SL, θ ( 1,θ ), <π A SL, θ (θ, 0) (0,θ ), <π A SL, θ (θ,θ ), <π A SL, θ (θ, 1), where θ = 0.5, θ =0.1, andθ =0.39. From these profitrankingswe can draw the following conclusions. First, independent of the extent to which 11

12 products are differentiated and whether products are demand sustitutes or demand complements, firm A always wants to e the Stackelerg leader in market while selling its two commodities as a undle. Only in case the two products are independent in demand (θ =0) firm A would e indifferent etween undling and not undling provided that it is the Stackelerg leader in market. Bundling is therefore a profitale alternative for firm A to the need of rationalizing output in market when products are relatively homogeneous (i.e. θ (θ, 1)) inordertoprotectitsprofits in market 1. Second, for firm A to pursue Stackelerg leadership in market y undling its two commodities is a strategy that would make sense. In case the strategy succeeds the highest possile profits are otained; in case the strategy does not succeed (that is, firm A remains a Cournot oligopolist in market ) profits have increased due to undling (for θ ( 1,θ ))orprofits are ack at the original level as not undling is always a viale option if profitale (for θ (θ, 1)). 5. Welfare implications Consumer s stand towards a Stackelerg leader undling its two commodities can e taken from: Proposition 7 With respect to firm A undling its two commodities or not, eing the Stackelerg leader in market, consumers surplus compares as follows: CS S >CSS θ ( 1, 0), CSS <CSS θ (0, 1). Proof. Numerical simulation This highlights the positive effect of product undling when products are demand complements, while the opposite holds for demand sustitutes. Same applies in case of Cournot competition, as shown y numerical simulations plus the results of Propositions 4 and 7 yield: CS C <CSS <CSC <CS S θ ( 1, θ), CS C <CSC <CS S <CSS θ ( θ, 0), CS C <CS C <CSS <CSS θ (0, 1), where θ 0.4. Consumers always prefer a Stackelerg leader, undling or not, over a non-undling Cournot oligopolist. In case firm A is not a Stackelerg leader in market and provided that products are independent in demand, Martin (1999) shows that total surplus falls if firm A undles its two commodities and concludes (Martin, 1999, p. 375): Firm A s undling therefore reduces social welfare, compared to the 1

13 case in which firm A does not undle. This estalishes that an efficiency presumption for undling is unwarrendted. To generalize this result to any value of θ ( 1, 1), considerfirst: Proposition 8 With respect to firm A undling its two commodities or not, eing a Cournot oligopolist in market, total surplus compares as follows: TS C <TSC, θ ( 1, e θ), TS C >TSC, θ (e θ, 1). Proof. Numerical simulation Turning the the scenario considered here, that of a Stackelerg leader in market, gives us: Proposition 9 With respect to firm A undling its two commodities or not, eing a Stackelerg leader in market, total surplus compares as follows: Proof. Numerical simulations Summing up the ranking for total surplus yields: 6 Policy implications Moving on to firm B s profits note that sign{π Bc πb SF } = sign{(8 3θ ) 4(3 θ ) ( θ) }. Straightforward numerical simulations show that this sign is positive for θ ( 1, 1). Indeed, if firm A succeeds in ecoming the Stackelerg leader it not only can increases its profits, it also ecomes a more formidale competitor in that the range of fixed cost for which firm B can surveive in the market shrinks. From a welfare point of view the est situation that can arise is one in which firm A is the Stackelerg leader in market. One could then e inclined to view undling as an efficiency-enhancing strategy if it is used y firm A as a vehicle to ecome the Stackelerg leader in market. For the majority of product-types (θ ( 1,θ ))itisdirectlyprofitale for firm A to e the Stackelerg leader in market. Note that firm B s profits as a Stackelerg follower are always lower than as a Cournot oligopolist. For all product-types itimpliesalossinfirm B s profits. Indeed, even if ecoming the Stackelerg leader implies a reduction of firm A s profits, it always means an increase in its competitive strength towards firm B. However, the situation of firm A eing the Stackelerg leader in market is not on all accounts the most preferred situation. Note that all profits discussed so far are gross of fixed cost. As long as any firm s profits are sufficient to cover these cost it will find it profitale to stay in the market. 13

14 Martin [1999] consequently oserves that there is a range of fixed cost for which firm B would stay in the market if firm A does not undle, while leave the market if firm A does undle, since, in case of a Cournot duopoly in market, firm B s profits fall if firm A undles. We can make the same argument here, ut now in terms of firm A s aility to ecome the Stackelerg leader. Compared to the Cournot case firm B s profits always fall if firm A ecomes the Stackelerg leader in market. Hence, the range of fixed cost for which firm B would stay in the market shrinks if firm A succeeds in transforming itself from a Cournot oligopolist to the Stackelerg leader. Put differently, although a situation in which the multiproduct firm is the Stackelerg leader in market leads to the highest level of total surplus, it extends this firm s aility to drive its rival out of this market. 7 Conclusions We show that a Cournot oligopolist that has a monopoly over the production of another, related product has an incentive to ecome the Stackelerg leader in this related market. We also show that if this multiproduct firm succeeds in ecoming the Stackelerg leader it does not have an incentive to undle its output. This result differs from Martin s[1999]finding that a Cournot oligopolist that has a monopoly over the production of another, related product does have an incentive to undle its two products. It implies that if a monopolist undles its output with its production in a related market, it proaly is not the Stackelerg leader in this related market. It also means that uundling has no economic consequences if the monopolist meanwhile has ecome the Stackelerg leader in the related market. The desire for policy actions very much depends on the monopolist s aility to ecome the Stackelerg leader in related markets. If it succeeds, market performance improves ecause of undling. If it remains a Cournot oligopolist, market performance deteriorates ecause of undling. Recall however Martin s [1999] oservation: in case of a Cournot duopoly the range of fixedcostthatallowsthesingle-productfirm to stay in the market, shrinks if the multiproduct firm undles its output. Our analysis reveals a comparale mechanism: the range of fixed cost that allows the singleproduct firm to stay in the market shrinks if the multiproduct firm ecomes the Stackelerg leader. Indeed, although a situation in which the multiproduct firm is a Stackelerg leader yields to highest level of total surplus, it also leaves the market a multiproduct firm with formidale competitive strength. 14

15 8 Appendix Bundling in a Cournot duopoly In this appendix we replicate the analysis of Section 3 assuming that oth firms act as Cournot duopolists (note that this is a generalization of Martin [1999] who considers only the case of θ =0). 8.1 No undling Maximizing firm A s profits in market 1, (p 1 c)q A 1, over quantities yields: 10 q A 1 (q A,q B )= 1 (a c) θ(q A + q B ). (19) Doing likewise for oth firm A and B in market gives us: q A (q1 A,q B )= 1 (a c) θq A 1 q B (0) and q B (q1 A,q A )= 1 (a c) θq A 1 q A. (1) Solving (19), (0) and (1) simultaneously results the equilirium quantities: and q C A,1 = (a c)(3 θ) (3 θ ) () q C A, = q C B, = (a c)( θ) (3 θ. (3) ) 8. Bundling If firm A undles it maximizes (p A c) A, yielding: C (a c) A( B )= (1 + θ) B 4. (4) Firm B maximizes (p B c) B, which gives: C B( A )= (a c) (1 + θ) A. (5) 10 It is straightforward to check that for θ ( 1, 1) all second-order and staility conditions are met in case market is a Cournot oligopoly. 15

16 Solving (4) and (5) simultaneously results the equilirium numer of undles: and C A = (a c)(3 θ) (7 θ)(1 + θ) (6) which imply: C B = (a c) 7 θ, (7) and Q 1 = C A = (a c)(3 θ) (7 θ)(1 + θ) (8) Q = C A + C B = (a c)(5 θ) (7 θ)(1 + θ). (9) References [1] Adams, W. J. and Yellen, J. L., 1976, Commodity undling and the urden of monopoly, Quarterly Journal of Economics, 90: [] Bernheim, B. D., 1984, Strategic deterrence of sequential entry into an industry, RAND Journal of Economics, 40: [3] Bowley, 194, The mathematical groundwork of economics, OxfordUniversity Press. [4] Ellig, J. (editor), 001, Dynamic competition and pulic policy, Camridge University Press. [5] Economides, N., 001a, The Microsoft antitrust case, Journal of Industry, Competition and Trade,1, No. 1, pp [6] Economides, N., 001, The Microsoft antitrust case: rejoinder, Journal of Industry, Competition and Trade, 1, No. 1, pp [7] Martin, S., 1999, Strategic and welfare implications of undling, Economics Letters, 6, pp [8] Martin, S., 00, Advanced industrial economics (second edition), Blackwell Pulishers. 16

17 [9] McAfee, R. P., McMillan, J. and Whinston, M. D., 1989, Multiproduct monopoly, commodity undling, and correlation of values, Quarterly Journal of Economics, 104: [10] Naleuff, B., 004, Bundling as an entry arrier, Quarterly Journal of Economics, 119: [11] O Brien, D. P., 001, Comments on the Microsoft antitrust case, Journal of Industry, Competition and Trade, 1, No. 1, pp [1] O Toole, F., 001, Comment on the Microsoft antitrust case, Journal of Industry, Competition and Trade, 1, No. 1, pp [13] Qiu, L. D., 1997, On the dynamic efficiency of Bertrand and Cournot equiliria, Journal of Economic Theory, 75, No. 1, pp [14] Salinger, M. A., 1995, A graphical analysis of undling, Journal of Business, 68: [15] Seidmann, D. J., 1991, Bundling as a facilitating device: a reinterpretation of leverage theory, Economica, 58, No. 3, pp [16] Singh, N. and Vives, X., 1984, Price and quantity competition in a differentiated duopoly, RAND Journal of Economics, 15, pp [17] Stackelerg, H. von, 1934, Marktform und Gleichgewicht, Vienna: Springer. [18] Stigler, G. J., 1968, A note on lock ooking, in Stigler, G. J. (ed.), The Organization of Industries, Homewood, Il: Irwin, [19] Whinston, M. D., 1990, Tying, foreclosure, and exclusion, American Economic Review, 80: