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Product bundling and incentives for mergers and strategic alliances.

I. INTRODUCTION

This paper models firms' strategic choice of a merger or a strategic alliance in bundling their products with complementary products. Firms often practice bundling via a strategic alliance as observed in the bundling of iPhone and AT&T network services. (1) In some cases, however, firms may use a merger instead. In 2001, the European Commission (EC)'s decision to block a merger between GE and Honeywell was based mainly on a concern that the merger might facilitate foreclosure of competition with bundling. (2) Hence, the questions on issue are when and why firms choose an alliance over a merger for bundling, and vice versa.

We consider a model of four differentiated products in two complementary good markets in which a firm can improve profit only by bundling its product with a complementary product. We find that pure bundling is profitable, whereas mixed bundling is not. The profitability of pure bundling in this paper stems from the fact that consumers value the product differentiation of one product more than the other. By making the two products inseparable, pure bundling lowers the substitutability of the product with a lower value of differentiation. This enables firms to raise the price to as high as the other product that has a higher value of differentiation.

Whether firms merge or form strategic alliances affects the profitability of pure bundling differently. We first consider a simple case in which only one pair of firms decide whether to merge or to form an alliance prior to bundling. We find that firms benefit most from pure bundling if they form an alliance as the structure enables firms to maximally increase the price of the bundled products. In the case of a merger, however, a merged firm is always inclined to price more aggressively, internalizing the complementarities of two products (the Cournot effect). Thus, a merger entails intense price competition with rivals and reduces profits. In some cases, the merger may not even be profitable. Yet, the strategic advantage of a merger is that, if rivals are not merged, one of the rivals will exit as a result of a loss because of intense price competition. Thus, in the range of parameters where the merger is profitable and reduces the rival's profits enough to induce an withdrawal from the market, firms choose to merge to foreclose competition, whereas in other ranges, the optimal strategy is to form an alliance.

However, in the full game where all firms decide simultaneously whether to merge and what type of bundling to offer, we find that bundling occurs only through strategic alliances in equilibrium. When mergers do occur, they do not entail bundling, as that would only result in severe price competition and no gain. In the present framework, bundling reduces welfare. Firms' increased profits are purely a transfer from consumer surplus, while consumers incur an additional welfare loss because some are unable to choose their optimal pair of mix-and-match system.

This paper extends the literature on bundling by analyzing firms' incentives to choose a strategic alliance or a merger in practicing bundling and highlighting the differential effects of organizational forms on bundle pricing and competition.

The effect of a merger on the profitability of bundling has been discussed in several papers. Using a framework of linear demands for bundling two complementary products, Economides (1993), Beggs (1994), Choi (2008), and Flores-Fillol and Moner-Colonques (2011) show that a merger can improve the profit from mixed or pure bundling as a result of the Cournot effect. These papers highlight two offsetting aspects of the Cournot effect that a merger triggers: although the resulting competitive pricing can enhance merging firms' profits by expanding their market share, it can also reduce their profits by reducing the prices that they charge. Hence, the profitability of bundling and merger in these papers essentially depends on the size of the own price effect, which is measured by the increase in demand for a product due to a decrease in its price, in comparison to the size of the cross price effect, which is measured by the increase in the demand due to an increase in the prices of competing products. If the two are very close, the effect of increased price competition dominates the effect of market share expansion, and the merger reduces profits.

As the size of the two effects are the same in the case of Hotelling framework with full market coverage, a merger always reduces profits, other things being equal (Matutes and R6gibeau 1992 and Armstrong 2006). To explain how the profitability of mixed bundling can arise in this framework, Gans and King (2006) introduce sequential pricing for bundled products and stand-alone products. They show that a pair of firms can improve profits by "cobranding" (alliance) or by bundling after integration, because the pricing structure allows the firms to price-discriminate "loyal consumers" through a bundled discount. However, the profitability disappears in a symmetric equilibrium where both pairs of firms merge or offer co-branding.

In contrast, this paper highlights the importance of asymmetry in consumer valuations of product differentiation in two markets. This difference in tastes (measured in transportation costs) implies that the own price effect in one market is lower than in the other, which critically differentiates this paper from the others in which demand parameters are assumed to be the same in both markets before and after bundling. This difference generates the profitability of pure bundling in the current framework because pure bundling makes both products' own price effect equally low. By making the two bundled goods inseparable, pure bundling makes a lower valued product, good 1, as valuable as a higher valued product, good 2. This makes the own price effect of good 1 as low as that of good 2, which increases the price of good 1. Because the increase in profits requires that consumers be unable to purchase the lower valued product separately from the higher valued product, there is no profitability in mixed bundling.

This paper is organized as follows. Section II outlines the benchmark case prior to bundling. In Section III, we compare two different ways to bundle, namely, a strategic alliance and a merger. In Section IV, we characterize equilibrium bundling strategies and show when firms merge and when they choose strategic alliances in order to bundle. In Section V, we discuss the effects of having more competitors, unsaturated market demand, and imperfect complementarity on the profitability of pure bundling. Section VI provides a summary of the results, a discussion of the limitations of the results, and a suggestion for future work.

II. THE MODEL AND A BENCHMARK

The model extends the standard differentiated products model used in Matutes and Regibeau (1992). Consider two markets, Market 1 and Market 2. Consumers purchase at most one unit of each of the two complementary products. We assume that consumer valuations of the two goods, [v.sub.1] and [v.sub.2], are high enough to guarantee that the two markets are fully covered. Moreover, the two goods are complementary, that is, [v.sub.12] [much greater than] [v.sub.1] + [v.sub.2], where [v.sub.12] is the value of consuming both goods. In each market, a continuum of consumers of a unit mass are uniformly distributed on an interval [0, 1]. A typical consumer is characterized by her location on the unit square, [x.sub.12] = ([x.sub.1], [x.sub.2]) [member of] [0, 1] x [0, 1]. In each market i, i = 1, 2, there are two firms, [A.sub.i] and Bi, competing h la Hotelling. Assume that [A.sub.i] (Bi, respectively) is located at [x.sub.i] = 0 ([x.sub.i] = 1), for i = 1, 2. A consumer with [x.sub.i] in market i incurs a disutility of [t.sub.i][x.sub.i.sup.2] [([t.sub.i](1 - [x.sub.i]).sup.2], respectively) to reach a firm [A.sub.i] ([B.sub.i]). Without loss of generality, assume that [t.sub.2] [greater than or equal to] [t.sub.1] > 0. Let [p.sub.ij] be the price of firm j in market i, for i = 1, 2, and j = A, B prior to bundling. Then, in market i, for a given [p.sub.ij], for a consumer who is indifferent between [A.sub.i] and [B.sub.i], [v.sub.i] - [p.sub.iA] - [t.sub.i][([x.sub.i]).sup.2] = [v.sub.i] - [p.sub.iB] - [t.sub.i][(1 - [x.sub.i]).sup.2]. Thus, firms' demands are [D.sub.ij]([P.sub.ij], [p.sub.ik]) = ([p.sub.ik] - [p.sub.ij])/(2[t.sub.i]) + 1/2, for i = 1, 2, j, k = A, B, j [not equal to] k. For simplicity, we assume that the two firms in each market i have identical constant marginal costs, that is, [C.sub.iA] = [C.sub.iB] = [c.sub.i], for i = 1, 2. Each firm must incur a fixed cost of [f.sub.i] to produce. Prior to bundling, the optimal prices and the profit for firms are

(1) [p.sup.*.sub.iA] = [p.sup.*.sub.iB] = [t.sub.i] + [c.sub.i], and

(2) [[pi].sup.*.sub.iA] = [[pi].sup.*.sub.iB] = ([t.sub.i]/2) - [f.sub.i], for i = 1, 2.

III. STRATEGIC ALLIANCE VERSUS MERGER

Suppose that firms are deciding whether to merge or to form an alliance with a complementary good producer in order to bundle their products. As there is no synergy from a merger, a merger does not alter market conditions as long as it does not affect the taste parameter [t.sub.i] in the two markets. This parameter [t.sub.i] represents how much consumers value product differentiation of the product i. Thus, it is specific to each product i. For a consumer who buys good 1 from [A.sub.1], the key factor is how she would feel if she were to buy the good from [B.sub.1] instead. It does not matter whether good 1 is produced by a new firm created by a merger of [A.sub.1] and [A.sub.2] or by [A.sub.1] alone. Hence, the merger cannot alter [t.sub.i], i = 1, 2. Similarly, bundling the two products does not alter the taste parameters. (3)

Throughout this paper, the timing of the game is as follows. At Stage 0, firms decide whether to merge with a complementary good producer to sell their product in a bundle. At Stage 1, firms decide whether to offer mixed, pure, or no bundling. Deciding to practice bundling without a merger implies that firms have decided to form an alliance. At Stage 2, firms set their prices. At Stage 3, consumption occurs.

We have three cases to consider--a pair of firms merge, both pairs of firms merge, and no firms merge but offer bundling via strategic alliances. In this section, we first analyze the case in which only one pair of firms, A1 and [A.sub.2], decides to merge and bundle. We characterize the outcomes when they offer either mixed or pure bundling by a merger or a strategic alliance. In Section IV, we describe the equilibrium merger or alliance decisions by all firms considering all possible scenarios.

A. Unprofitable Mixed Bundling

Let [D.sub.iA] and [p.sub.iA] be the demand and price of a stand-alone product [A.sub.i] and [D.sub.bundle] be the demand for a bundle. As the bundled products compete with the stand-alone products, if there is no discount for a bundle, consumers have no reason to buy a bundle instead of two stand-alone products. We consider a type of alliance under which firms are required only to consider offering a "voluntary" discount for their component of a bundle. Suppose Ai offers a discounted price [[delta].sub.i][p.sub.iA] for its component in a bundle, where [[delta].sub.i] [member of] [0, 1], i = 1, 2. Firm [A.sub.i]'s profit from a strategic alliance is [[pi].sup.s.sub.iA] = ([p.sub.iA] - [c.sub.i])[D.sub.iA] + ([[delta].sub.i] [p.sub.iA] - [c.sub.i]) [D.sub.bundle]. (4) On the other hand, if the firms are merged, the profit from mixed bundling is [[pi].sup.m.sub.M] = [SIGMA] ([p.sub.iA] - [c.sub.i])[D.sub.iA] +([p.sub.b] - [SIGMA][c.sub.i])Dbundle, where Pb is the merged firm's bundle price. The difference between an alliance and a merger is that allied firms may not care about the joint profit, whereas a merged firm does. In either case, we find unilateral mixed bundling to be unprofitable.

PROPOSITION 1. (1) Suppose that [A.sub.1] and [A.sub.2] form an alliance to offer mixed bundling. They do not offer a discount for a bundle. As a result, the allied firms do not gain from mixed bundling.

(2) Suppose that [A.sub.1] and [A.sub.2] merge. The profits from mixed bundling are lower than the profits without bundling.

Proof. All proofs are provided in the Appendix.

Mixed bundling is unprofitable for a merged firm because it toughens competition. As the two products are complementary, increasing the price of product l reduces the profit from product 2 as well as the profit from product 1. A merged firm sets its prices internalizing the complementarity, whereas allied firms do not. Naturally, prices are more competitive in a merger. Expanding market coverage with a bundled discount will improve the merged firm's profit only if the firm can recoup the profits by charging more on the stand-alone products. However, as rivals respond to the discount by cutting their stand-alone prices, the merged firm cannot increase its stand-alone prices by much, and thus, loses profit by mixed bundling.

In a multiproduct duopoly model with [t.sub.1] = [t.sub.2], Matutes and R4gibeau (1992) and Armstrong (2006) show that a multiproduct firm does not gain from unilateral mixed bundling. The second result in Proposition 1 states that the same is true even when [t.sub.2] > [t.sub.1] and when the rivals are not merged.

In the proof of Proposition 1 in the Appendix, we also show that the losses from mixed bundling are greater when [t.sub.2] > [t.sub.1]. This is because, when [t.sub.2] > [t.sub.1], the merged firm must offer a greater bundle discount to make consumers switch to a bundle because consumers are less inclined to shop around for the second product. A greater bundle discount will create more intense price competition for the stand-alone products, making it more difficult for the merged firm to profit from mixed bundling.

The advantage of a strategic alliance for mixed bundling is that it does not intensify competition as much as a merger would. However, in expanding their market share by offering a discount for loyal consumers, each firm tries to free-ride on the other allied firm's bundled discount. Thus, in the end, no discount is offered and there is no gain from bundling.

This result complements that of Gans and King (2006) and explains why allied firms must pre-commit to a bundled discount before setting stand-alone prices to make co-branding profitable in their framework. However, since the profitability disappears if both pairs of firms offer bundled discounts, we explore a different channel of profits to explain the incentives for firms to form an alliance or to merge to bundle in this framework. In the following section, we show that as long as [t.sub.2] [not equal to] [t.sub.1], firms can profit from pure bundling through an alliance by reducing consumers' choices.

B. A Strategic Alliance for Pure Bundling

Suppose firms [A.sub.1] and [A.sub.2] agree to sell their products only in bundles. Since sales of individual components are no longer available, consumers have only two choices, namely, to purchase the two products either from [A.sub.1] and [A.sub.2] or from [B.sub.1] and [B.sub.2]. (5) Allied firms set their prices independently. Let [p.sub.AA] be the price of a bundle offered by [A.sub.1] and [A.sub.2]. Then, [p.sub.AA] = [p.sup.sp.sub.1A] + [p.sup.sp.sub.2A], where [p.sup.sp.sub.1A] and [p.sup.sp.sub.2A] are the optimal prices chosen by [A.sub.1] and [A.sub.2], respectively. Firms claim their share of the revenues from joint sales according to a pre-negotiated sharing rule. Suppose that each firm keeps [phi] [member of] [0, 1] fraction of their own contribution to the profits and shares 1 - [phi] fraction with the other allied firm. Then, each firm's profit from a strategic alliance for pure bundling is

(3) [[pi].sup.sp.sub.iA] = [phi][D.sub.AA] ([p.sub.iA] - [c.sub.i]) + (1 - [phi])[D.sub.AA](P-[i.sub.A] - [c.sub.-i]) - [f.sub.i]

and [[pi].sup.sp.sub.iB] = [D.sup.sp.sub.iB] - [c.sub.i]) - [f.sub.i] for i = 1, 2. If [phi] = 1, firms claim their own contribution only while they share the profits equally if [phi] = 1/2. (6) For any predetermined [phi], the two firms agree to sell their products only in a bundle if they both find that bundling is at least as profitable to them as before the alliance.

PROPOSITION 2. Suppose that [A.sub.1] and [A.sub.2] form an alliance to bundle their products. At the optimum, [phi] = 1, [p.sub.AA] = 2[t.sub.2] + [c.sub.1] + [c.sub.2] and [p.sup.sp.sub.iB] = [t.sub.2] + [c.sub.i], for i = 1, 2. The market shares are [D.sub.AA] = [D.sup.sp.sub.iB] = (1/2) and the profits are

(4) [[pi].sup.sp.sub.iA] = [[pi].sup.sp.sub.iB] = ([t.sub.2]/2) - [f.sub.i],

for i = 1, 2. Thus, pure bundling is always profitable if [t.sub.2] > [t.sub.1].

As a result of pure bundling, prices, demand, and profits no longer depend on [t.sub.1]. This is because as pure bundling makes the two goods inseparable, the firms in market 1 can act like the firms in market 2 and charge for their product in terms of [t.sub.2]. Before bundling, the price of product 2 was higher than that of product 1 because consumers have a strong preference for the product differentiation of the second product, [t.sub.1] < [t.sub.2]. Because consumers can no longer purchase [A.sub.2] ([B.sub.2], respectively) without purchasing [A.sub.1] ([B.sub.1]), product 1 becomes as valuable as product 2 to consumers. Thus, consumers become less responsive to product l's price changes. As a result, firms in market 1 can charge as much as firms in market 2 do, which increases their profits.

Consider the example of tying the iPhone to AT&T network services. Suppose that consumers have a strong preference for particular smart phones, although they are not particular about the providers of network services. The results in Proposition 2 imply that, if the network services are not bundled with smart phones, the prices of the network services would be lower because consumers are able to select any network service provider or smart phone manufacturer. However, if the network services are sold only in bundles with smart phones, the firms expect consumers to choose one bundle over another on the basis of the smart phone that is included. If a consumer buys a bundle with AT&T instead of a bundle with Verizon, it is probably because she likes the iPhone more than another phone. Then, even if AT&T charged a little more for its network services, it would not lose much demand because consumers who prefer the iPhone would not switch to the Verizon network. Hence, by exclusively bundling network services with smart phones, firms are able to charge more for their network services. As a result, the price of a bundle is higher than the sum of the stand-alone prices prior to bundling.

Note that even the non-allied firms are at least weakly better off as a result of pure bundling. Because the profitability comes from making separate purchases of A2 and A I impossible, when one pair of complementary goods are tied, the other pair of products become naturally tied in the current framework of two firms in each market. Thus, non-allied firms receive the same increase in profits as the allied firms. (7)

However, the profitability depends greatly on whether the firms are merged or not. In the next section, we show that mergers reduce the profitability of pure bundling by initiating aggressive competition.

C. A Merger for Pure Bundling

Suppose now that [A.sub.1] and [A.sub.2] merge and that the merged firm M practices pure bundling for the two products at [p.sub.M]. Let [p.sup.p.sub.1B] and [P.sup.p.sub.2B] be the prices of rivals, [B.sub.1] and [B.sub.2], respectively, when M practices pure bundling. Consumers can buy both goods from M, or they can buy good 1 and good 2 from [B.sub.1] and [B.sub.2] separately.

PROPOSITION 3. Suppose that [A.sub.1] and [A.sub.2] are merged. The optimal prices are [p.sub.M] = (5/4)[t.sub.2] + [c.sub.1] + [c.sub.2], [p.sup.p.sub.iB] = (3/4)[t.sub.2] + [c.sub.i], for i = 1, 2. The market shares are [D.sup.p.sub.M] = (5/8), [D.sup.p.sub.iB] = (3/8), and the profits are

(5) [[pi].sup.p.sub.M] = (25/32)[t.sub.2] - [SIGMA] [f.sub.i] and [[??].sup.p.sub.iB] = (9/32)[t.sub.2] - [f.sub.i].

for i = 1, 2. Thus, if [t.sub.1] < (9/16)[t.sub.2], the merged firm's profit increases with pure bundling.

COROLLARY 1. Pure bundling is more profitable by means of a strategic alliance than by a merger.

A merger lowers the profits of pure bundling with aggressive competition by internalizing the complementarity of the two products (the Cournot effect). By making it cheaper to purchase a bundle of two products from the merged firm than purchasing them separately from rivals, [p.sub.M] < [p.sup.p.sub.1B] + PPB, the merged firm gains market share. However, as this comes at the expense of reducing the price to (5/4)[t.sub.2], if this price is much below the pre-bundling prices [t.sub.1] + [t.sub.2], the merger can become unprofitable. Thus, the profitability of the merger depends on the size of [t.sub.1] and [t.sub.2]. If [t.sub.2] is not too large, that is, [t.sub.1] > (9/16)[t.sub.2], the decrease in profit because of a price cut is larger than the gain from a greater market share. Thus, the merger is unprofitable. Merger is profitable only if [t.sub.2] is sufficiently larger than [t.sub.1] ([t.sub.2] > (16/9)q).

If they form an alliance instead, the firms would charge a higher price for product 1 while maintaining the market share at the pre-bundling level. Consequently, pure bundling is more profitable by means of a strategic alliance than by a merger.

This result contrasts with that of Zhang and Zhang (2006) where a strategic alliance or a merger (for pure bundling) provides the same outcome. They define a strategic alliance as partial ownership, whereas in this paper it means only profit sharing and cooperation. In their paper, each allied firm i holds an [alpha] fraction of firm j's profit. As [alpha] grows, firm i is more concerned about how its pricing affects firm j's profit. When [alpha] = 1, it is equivalent to the case of a merger. Thus, [alpha] represents the degree of a partial merger between two allied firms. As their demand structure is the same before or after bundling, if [alpha] = 0, there is no change by an alliance. The only way to improve profit is by internalizing complementarity. Thus, both pairs of firms form a full alliance (merger, [alpha] = 1) in equilibrium.

In contrast, in this paper, two markets have different values of product differentiation, which makes it optimal for the allied firms to maintain complete autonomy ([phi] = l, or [alpha] = 0). In this paper, if two products are tied, the substitutability of product 1 decreases, thereby increasing the price. In such a situation, any anticipated fraction of shared profits only increases firms' incentives to price competitively, internalizing the complementarity with the other product. Thus, prices are higher only if allied firms completely ignore the complementarity. This is possible only if the allied firms are completely autonomous.

In response to the aggressive price competition that the merged firm initiates, rivals lower their prices. In particular, this results in [B.sub.2]'s price being lower than the price before bundling, [p.sup.p.sub.2B] = (3/4)[t.sub.2] + [c.sub.2] < [p.sup.*.sub.2B] = [t.sub.2] + [c.sub.2]. As the rivals' market share is also lower after the merger, this implies that B2 incurs a loss. This is because only the firms in market 1 improve their profits from pure bundling while the merger causes the prices to fall in both markets. From (5), if [t.sub.2] < (32/9)[t.sub.2], [B.sub.2] exists because its revenue cannot cover the fixed cost. In contrast, [B.sub.1] may still earn a higher profit than before bundling if [t.sub.1] < (9/16)[t.sub.2]. This happens when the increase in the price of product 1 is larger than the decrease in market share.

While a merger may not be as profitable as a strategic alliance for [A.sub.1] and [A.sub.2] at the time, the fact that one of the rival incurs a loss because of the merger provides an important motivation to merge, when the loss may lead to the rival's exit. In the subsequent section, we discuss the incentive for a merger in this context.

D. The Incentive for A Merger

Now we analyze the merger incentives for A 1 and [A.sub.2] at Stage 0. The possibility of a counter-merger by [B.sub.1] and [B.sub.2] will be discussed in the next section. By combining the results from Sections A through C, we obtain the following predictions.

PROPOSITION 4. Consider a game in which Ai and A2 decide to merge.

(1) If (16/9)[t.sub.1] < [t.sub.2] < (32/9)[f.sub.2], [A.sub.1] and [A.sub.2] merge and offer pure bundling to foreclose competition.

(2) Otherwise, [A.sub.1] and [A.sub.2] do not merge, and firms offer pure bundling through strategic alliances.

If [A.sub.1] and [A.sub.2] merge, the merged firm offers pure bundling, as pure bundling is a dominant strategy for the merged firm irrespective of rivals' strategies. When (16/9)[t.sub.1] < [t.sub.2] < (32/9)[f.sub.2], a merger is profitable [(16/9)[t.sub.1] < [t.sub.2]] and [B.sub.2]'s loss is large enough to induce the firm to exit ([t.sub.2] < (32/9)[f.sub.2]). As B2 exits market 2, [B.sub.1] also becomes nonviable, as [B.sub.1] alone cannot compete against the merged firm's bundle. Consequently, the merge-and-bundle strategy induces foreclosure in both markets. (8) The merged firm earns [[pi].sup.p.sub.M] and enjoys a monopoly profit after foreclosure. Thus, at Stage 0, [A.sub.1] and [A.sub.2] merge. In other ranges of parameters, however, either a merger is unprofitable, or foreclosure is not possible even though a merger is profitable. Thus, no merger takes place and [A.sub.1] and [A.sub.2] choose a strategic alliance.

Choi (2008) also discusses the possibility that bundling combined with a merger may lead to a foreclosure. However, Choi (2008) does not analyze why firms choose a merger for bundling given that there are alternative means of bundling such as an alliance. Moreover, in Choi (2008), foreclosure is one of the possible outcomes of pure bundling followed by a merger. In contrast, foreclosure is a certain outcome in this paper if a unilateral merger occurs, given that the merger is purposefully chosen by firms with the single intention of foreclosing competition, because an alternative and more profitable means of bundling, i.e., strategic alliance, cannot induce a foreclosure. This type of merger would be of more concern to antitrust authorities. (9)

IV. EQUILIBRIUM MERGER AND STRATEGIC ALLIANCE

Now suppose that a possible merger is discussed between [B.sub.1] and [B.sub.2] as well as between [A.sub.1] and [A.sub.2]. At Stage 0, two pairs of firms, ([A.sub.1], [A.sub.2]) and ([B.sub.1], [B.sub.2]), simultaneously decide whether to merge, and at Stage 1, each pair of firms decides whether to bundle and how to bundle. Proposition 5 summarizes the equilibrium of the game.

PROPOSITION 5. (1) If (16/9)[t.sub.1] < [t.sub.2] < (32/9)[f.sub.2], the unique subgame perfect Nash equilibrium of the game is that both pairs of firms merge and never practice bundling.

(2) Otherwise, the unique extensive-form trembling hand perfect Nash equilibrium (THPNE) of the game is that both pairs of firms practice bundling only by strategic alliances.

A merged firm can realize the advantage of utilizing the Cournot effect only if its rivals are not merged. When a counter-merger occurs, the profitability disappears as a result of too much competition because the merged rival firms also price aggressively, internalizing the complementarity of two products. In this situation, if bundling is offered, both merged firms incur great losses.

As the incentives for exclusionary bundling by means of a merger exist only if the merger enhances competition just enough to reduce a rival's profit without reducing the merged firm's profit too much, a counter-merger eliminates the incentive as well. Thus, given the symmetry of the problem, in the range where a unilateral merger can induce the foreclosure of unmerged rivals, (16/9)[t.sub.1] < [t.sub.2] < (32/9)[f.sub.2], both pairs of firms merge with an intention to foreclose. However, after observing that both mergers have occurred, they decide not to bundle as it would only result in a severe price war between the two merged firms. In all other ranges of parameters, a merger is never a dominant strategy, because it is either unprofitable or unable to induce foreclosure. Hence, no merger occurs, and firms offer bundling through strategic alliances. (10)

In equilibrium where bundling occurs, although firms are better off, consumers are much worse off. Consumers pay higher prices for the two products than before and some are unable to choose their preferred combination of products. The increase in profits for the firms is a welfare transfer from consumer surplus. Hence, pure bundling reduces welfare even if foreclosure does not occur in equilibrium.

V. DISCUSSION

The profitability of pure bundling in this framework requires that (a) product differentiation is valued differently in the two markets and that (b) tying the two products enhances the value of the lower-valued product. As both conditions are not difficult to satisfy, the result of this paper is expected to be robust. Yet, the second condition requires further stipulation because how much tying can enhance the value of tied products depends on specific market conditions. Among many conditions that matter, this paper focuses on the role of firms' organizational structure. A strategic alliance utilizes the benefit of pure bundling most because unlike a merger, it lacks the ability to internalize the complementarity between the products and thus, does not intensify competition.

Other nontrivial factors for determining the level of profitability in the current framework are (1) duopoly competition structure, (2) perfect complementarity of the two products, and (3) fully saturated markets. In this section, we discuss the effects expected if these conditions are relaxed.

The current framework of Hotelling competition is not suitable for incorporating these changes. Thus, for the discussion below, we consider a general framework of differentiated products. As foreclosure incentives and the need for antitrust scrutiny are relevant issues only in highly concentrated markets, we focus on the case of bundling by dominant firms.

In addition, to minimize the departure from the current framework, we assume in Sections A and B that the market coverage is nearly full in the sense that market expansion due to competitive pricing is expected to be minimal. We explain the importance of full market coverage in Section C.

A. The Effect of N > 2 Firms

Having more than two firms in each market will not cause the profitability of pure bundling to disappear because the profitability arises from making the purchases of two products inseparable so that firms can increase the price for a bundle. However, the effectiveness of bundling might be less. The firms may need stronger dominance or stronger consumer loyalty for their bundled products because when a pair of firms tie their products, they gain the ability to raise the price only to those consumers who are very loyal to the products (direct effect).

An interesting aspect of this case is that pure bundling would soften competition indirectly by limiting the number of available standalone products for consumers (indirect effect). While consumers are able to keep their option to mix and match, there will be fewer options to choose, and this enhances the market power of the firms who do not practice bundling. Consequently, pure bundling would enhance the market power of both bundling and unbundling firms. If the indirect effect is stronger than the direct effect, having more firms would not necessarily reduce the profitability of pure bundling. Extending the model to analyze the direct/indirect effects would be an interesting venue for future research.

Foreclosure by a merger may become less of an issue in a market with N > 2 firms for a couple of reasons. First, with more competitors, obviously, it is more difficult to foreclose competition. Second, pure bundling may, in fact, enhance unbundling firms' market power as a result of strong indirect effect, in which case foreclosure would be more difficult to achieve.

B. The Effect of Partial Complementarity

When two products are not perfectly complementary as some consumers purchase only one of the two products, having such consumers for single-product demand will affect the profitability of mixed bundling and pure bundling differently.

The profitability of pure bundling may be enhanced as a result of partial complementarity, if the substitutability of the single product is low. Tying two goods together will prevent consumers from purchasing a single product. When the substitutability is low, firms will profit by charging a price for two to consumers who consume only one product of a bundle. For example, suppose some consumers are very attached to a specific type of coffee filter, which is now tied to a mug. Although they don't need the mug, they must purchase a bundle and pay for the two products if they want to have the filter. However, if the substitutability is high, with pure bundling, firms lose the single-product demand to their competitors who offer standalone sales of close substitutes.

In the case of mixed bundling, single-product demand is not directly affected by the bundling as stand-alone purchases are possible. The demand is affected only indirectly by the effect on stand-alone prices. With single-product demand, an increase in the stand-alone price has a greater effect on the revenues from stand-alone sales, and thus, firms will be less inclined to raise the price. However, as they need to offer a discount to consumers who purchase a bundle, without increasing the stand-alone prices, they are likely to incur losses. Thus, as the single-product demand increases, a bundled discount become less profitable.

Flores-Fillol and Moner-Colonques (2011) consider how merged firms' profits from mixed bundling are affected by single-product demand. In their framework, however, each firm is a monopoly for single-product demand in the sense that the single demand for product i depends only on [p.sub.i] . As a result, an increase in the demand gives more latitude to firms to make up for their losses from a bundled discount, contrary to what we anticipated above. This is because the model lacks competition for the demand. If firms face competition from a close substitute by another firm for the single demand, we expect the profitability of mixed bundling to be lower. Moreover, the model in Flores-Fillol and Moner-Colonques (2011) does not allow single-product consumers to switch to a bundle instead if the stand-alone price increases. In such a case, it will be more difficult to recoup the profit from standalone revenues, making mixed bundling less profitable.

C. The Effect of Unsaturated Markets

In this paper, the model assumes fully saturated demand in both markets. As a result, firms do not gain from competitive pricing as it reduces their revenues without affecting their market share. This is why mixed bundling is not profitable and a counter-merger leads to a decision to not bundle in this paper. Mixed bundling typically creates a discount for loyal consumers who purchase a bundle. Such a discount is profitable if it results in a greater market share for the firms. When markets are saturated, an increase in market share can only be obtained by stealing business from rivals. Thus, a bundled discount is strategically profitable only when some rivals do not offer it. However, as the incentives are symmetric for all firms, all firms offer a discount and no firm gains.

Instead, firms turn to pure bundling, which improve profits by increasing prices without affecting the demand for the products. The increase in the prices and few choices for consumers result in welfare losses.

However, if the demand can expand as a result of price decrease as in Beggs (1994), Choi (2008), Economides (1993), and Flores-Fillol and Moner-Colonques (2011), the results would be quite different. In these papers, when the own price's effect is much larger, a bundled discount creates new customers who would not have purchased otherwise. This expansion in market demand is the key factor in making mixed bundling profitable and possibly more profitable than pure bundling. As the profitability arises from lower prices and increased market coverage, if there is no possibility of foreclosure, bundling is unlikely to raise antitrust concerns.

Hence, the findings of this paper must be understood in the context that bundling is likely to be welfare-reducing if its effect on demand is minimal. In both the case of the projected tying of an aircraft engine and avionics technology after the merger of GE and Honeywell and the case of tying the iPhone to network service, the effect on demand was expected to be minimal. This paper shows that there is a greater need for antitrust scrutiny in the case of bundling in a saturated market.

VI. CONCLUSIONS

This paper investigates firms' incentives for mergers and strategic alliances in bundling. There is a range of parameters in which firms choose to merge with an intention to foreclose competition. However, in most cases, firms prefer strategic alliance for bundling. Moreover, in equilibrium, a merger never leads to exclusionary bundling as a counter-merger removes the possibility of foreclosure. Thus, firms are more likely to choose a strategic alliance in bundling. Strategic alliances improve all firms' profit as they soften competition.

In this paper, pure bundling improves firms' profits by making two products with nonidentical values inseparable and thus equally valuable. As a result, firms can charge equally high prices for both. This result is likely to hold as long as the two products are complementary, the markets are highly concentrated, and consumers value the product differentiation of one product more than the other.

To check the robustness of the result, we discuss the anticipated effects of relaxing three main conditions of this framework: (1) duopoly, (2) full coverage, and (3) perfect complementarity.

Having more competitors will not eliminate the profitability of pure bundling since the profitability arises from reducing the number of consumer choices. The effectiveness of pure bundling will depend on other factors such as how dominant the allied firms are. The profitability may be enhanced by softening competition for stand-alone products. Similarly, partial complementarity may enhance the profitability of pure bundling if consumer loyalty to the bundled products is strong enough.

We find that the possibility of market expansion (unsaturated market) is crucial in determining the antitrust implication of bundling. Firms are likely to choose mixed bundling to expand market coverage by using a bundle discount if a large increase in demand is possible (Choi 2008; Economides 1993; Flores-Fillol and Moner-Colonques 2011). However, if demand increase is minimal, firms are likely to choose pure bundling to soften competition and an internal organization that maximizes the profits from softened competition, a strategic alliance. It will be interesting to investigate the antitrust implications in a model that shows how the extent of market expansion influences firms' choices of mixed bundling or pure bundling in future work.

In order to focus on the effect of firms' organizational structure on the profitability of bundling, we abstract away from many other reasons that motivate bundling. Bundling can improve efficiency in many ways. It can reduce transaction costs, create economies of scope, generate guided investment incentives, make it easier for firms to enter a new market, and mitigate agency problems. (11) Hence, for the purpose of antitrust scrutiny, a balanced consideration of all these aspects is necessary.

ABBREVIATIONS

FOC: First-Order Condition

THPNE: Trembling Hand Perfect Nash Equilibrium

doi: 10.1111/ecin.12047

APPENDIX

1. PROOF OF PROPOSITION 1

For simplicity, assume that [c.sub.1] = [c.sub.2] = [f.sub.1] = [f.sub.2] = 0 for this proof. Suppose [p.sub.iA], [p.sup.s.sub.b], and [p.sup.m.sub.b] are the price of stand-alone product i, i = 1,2, and the prices of bundled products when firms form an alliance and when firms are merged, respectively. Let [[lambda].sub.0], 0 = {s, m}, be the total bundled discount offered by the firms when they form a strategic alliance s, or use a merger m to practice bundling.

Suppose [x.sup.0.sub.1] and [x.sup.0.sub.2] are the marginal consumer who is indifferent between a bundle and ([B.sub.1], [A.sub.2]), and the consumer indifferent between a bundle and ([A.sub.1], [B.sub.2]), respectively. Then, [x.sup.0.sub.1] = (1/2) + ([p.sub.1B] - [p.sub.1A] + [[lambda].sub.0])/[2t.sub.1], and [x.sup.0.sub.2] = (1/2) + ([p.sub.2B] - [p.sub.2A] + [[lambda].sub.0])/[2t.sub.2]. Similarly, there exists a consumer [x.sup.1.sub.1] = (1/2) + ([p.sub.1B] - [p.sub.1A])/[2t.sub.1] who is indifferent between ([A.sub.1], [B.sub.2]) and ([B.sub.1], [B.sub.2]), and a consumer [x.sup.1.sub.2] = (1/2) + ([p.sub.2B] - [p.sub.2A])/[2t.sub.2] indifferent between ([B.sub.1], [A.sub.2]) and ([B.sub.1], [B.sub.2]). The market demands for a bundle and stand-alone products are [D.sup.0.sub.bundle] = {[x.sup.0.sub.1][x.sup.0.sub.2] - ([[lambda].sup.2.sub.0])/[8t.sub.1][t.sub.2]}, [D.sup.0.sub.1A] = [x.sup.1.sub.1] (1 - [x.sup.0.sub.2]), and [D.sup.0.sub.2A] = [x.sup.1.sub.2] (1 - [x.sup.0.sub.1], respectively, for 0 = {s, m}.

A. Unilateral Mixed Bundling Under Strategic Alliance

Let [[delta].sub.i][p.sub.iA] be the discounted price of [A.sub.i]'s product in a bundle when firms form an alliance, where [[delta].sub.i] [member of] [0, 1], i = 1,2. Then, [p.sup.s.sub.b] = [SIGMA] [[delta].sub.i][p.sub.iA], and the total bundle discount offered by the allied firms is [[lambda].sub.s] = [SIGMA](1 - [[delta].sub.i])[p.sub.iA] [greater than or equal to] 0. If [[delta].sub.i] = 1, [p.sup.s.sub.b] = [p.sub.1A] + [p.sub.2A], there is no bundle discount. Then firm [A.sub.i]'s profit is written as [[pi].sup.s.sub.iA] = ([p.sub.iA] - [c.sub.i]) [D.sup.s.sub.iA] + ([[delta].sub.i] [p.sub.iA] - [c.sub.i]) [D.sup.s.sub.bundle].

Using [x.sup.0.sub.i] - [x.sup.1.sub.i] = ([[lambda].sub.s]) /2[t.sub.i], the first-order condition (FOC) with respect to [p.sup.S.sub.iA] and [[delta].sub.i] can be written as

(A1) ([partial derivative][[pi].sup.s.sub.iA])/[partial derivative][p.sup.s.sub.iA] = [[LAMBDA].sub.1] + [[delta].sub.i] [[LAMBDA].sub.2] = 0, and

(A2) ([partial derivative][[pi].sup.s.sub.iA])/[partial derivative][[delta].sub.i] = ([p.sup.s.sub.iA] - [c.sub.i]) [[gamma].sup.1.sub.i] + [[LAMBDA].sub.2],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. At the optimum, [[delta].sub.i] > 0. Then, from (A1), [[LAMBDA].sub.2] = [[LAMBDA].sub.1] /[[delta].sub.i]. Plugging this into (A2), we get ([partial derivative][[pi].sup.s.sub.iA]/[partial derivative][[delta].sub.i])|[[delta].sub.i] = 1 = (1 - [x.sup.0.sub.-i]) [([p.sup.s.sub.iA] - [c.sub.i])/([2t.sub.i]) - [x.sup.1.sub.i]] = 0 because from (A1), at [[delta].sub.i] = 1, ([p.sup.s.sub.iA] - [c.sub.i])/([2t.sub.i]) - [x.sup.1.sub.i] = ([x.sup.0.sub.i] - [x.sup.1.sub.i])([x.sup.0.sub.-i] + [x.sup.1.sub.-i]) /2=0 given that [x.sup.0.sub.i] - [x.sup.1.sub.i] = [[lambda].sub.s] /([2t.sub.i]) = 0. Thus, at the optimum [[delta].sub.i] = 1, i = 1, 2.

B. Unilateral Mixed Bundling Under Merger

Under a merger, [[lambda].sub.m] = [p.sub.1A] + [p.sub.2A] - [p.sup.m.sub.b] [greater than or equal to] O. The merged firm's profit is [[pi].sup.m.sub.M] = [p.sub.1A] [[GAMMA].sub.1] + [p.sub.2A] [[GAMMA].sub.2] - [[lambda].sub.m] [[GAMMA].sub.3], and firm [B.sub.i]'s profits are [[pi].sup.m.sub.iB] = [p.sub.iB] {1 - [[GAMMA].sub.i]}, i = 1, 2, where [[GAMMA].sub.1] [equivalent to] [x.sup.1.sub.1] + ([lambda]/([2t.sub.1]))([x.sup.0.sub.2]) - ([[lambda].sup.2.sub.m]/([8t.sub.1][t.sub.2])), [[GAMMA].sub.2] [equivalent to] [x.sup.1.sub.2] + ([[lambda].sub.m]/([2t.sub.2])) ([x.sup.0.sub.2]) - ([[lambda].sup.2.sub.m] /([8t.sub.1][t.sub.2])), and [[GAMMA].sub.3] [equivalent to] [x.sup.0.sub.1] [x.sup.0.sub.2] - ([[lambda].sup.2.sub.m] /(8[t.sub.1][t.sub.2])).

From the FOCs with respect to [p.sub.iA], [[lambda].sub.m], and [p.sub.iB], we obtain

(A3) [[GAMMA].sub.i] + [[[lambda].sub.m]/2[t.sub.i]] ([x.sup.0.sub.-i]) - [p.sup.m.sub.iA]/2[t.sub.i] - [p.sup.m.sub.- iA] [[lambda].sub.m] / 4[t.sub.1][t.sub.2] = 0

(A4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(A5) 1 - [[GAMMA].sub.i] = [p.sub.iB]/2[t.sub.i],

for i = 1, 2. Combining the two equations (A3) and (A5), we get

(A6) 2[p.sub.iA][[lambda].sub.m] = [[lambda].sup.2.sub.m] + 12[t.sub.i]([t.sub.-i] - [P.sub.-iB]).

(1) If mixed bundling is offered, it has to be that [[lambda].sub.m] > 0.

Suppose [[lambda].sub.m] =0. In this case, [p.sub.iA] = [t.sub.i]. Then, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, [[lambda].sub.m] > 0.

(2) For [[lambda].sub.m] > 0, the merged firm's profits are highest when [t.sub.1] = [t.sub.2].

Combining the equations (A3) through (A5), we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Among all the roots that satisfy [PSI]([lambda]*, [t.sub.1], [t.sub.2]) = 0, let [lambda]* ([t.sub.1], [t.sub.2]) > 0 be the solution that gives the highest profit [[pi].sup.m*.sub.M]. Assume that such a [[lambda].sup.*.sub.m] ([t.sub.1],[t.sub.2]) exists. By construction, [[lambda].sup.*.sub.m]([t.sub.1],[t.sub.2]) is unique. Let [p.sup.m*.sub.iA] = [p.sub.ij] ([[lambda].sup.*.sub.m]([t.sub.1], [t.sub.2]), [t.sub.1], [t.sub.2]) be the optimal stand-alone price at [[lambda].sup.*.sub.m] ([t.sub.1],[t.sub.2]), for i = 1,2. Plugging the equations (A3) through (A6) into [[pi].sup.m.sub.M], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [t.sub.2] = [t.sub.1][mu], where [mu] [greater than or equal to] 1. Then, the profit function [[pi].sup.m*.sub.M] ([[lambda].sup.*.sub.m] ([t.sub.1], [t.sub.2]), ([t.sub.1], [t.sub.2])) can be rewritten as [[pi].sup.m*.sub.M] ([[lambda].sup.*.sub.m]([t.sub.1], [mu]), ([t.sub.1], [mu])). By Envelope Theorem,

(A7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [C.sub.1] = 1/2 + ([p.sup.m*.sub.2B] - [p.sup.m*.sub.2A] + [[lambda].sup.*.sub.m] ([t.sub.1], [t.sub.2])) /(2[t.sub.1][mu]) = [x.sup.0.sub.2] > 0. [C.sub.2] has to be positive. If not, [t.sub.2] [less than or equal to] [p.sup.*.sub.2B], and combined with (A5), the equation (A3) reduces to (3/2[t.sub.2])([t.sub.2] - [p.sup.*.sub.2B] - ([p.sup.m*.sub.1A][[lambda].sup.*.sub.m]/4[t.sub.1][t.sub.2]) - [[lambda].sup.*2.sub.m]/(8[t.sub.1][t.sub.2]) < 0, which contradicts. Thus, [t.sub.2] > [p.sup.*.sub.2B] at the optimum. If C3 < 0, [partial derivative][[pi].sup.m*.sub.M] ([[lambda].sup.*.sub.m] ([t.sub.1],[mu]), ([t.sub.1], [mu]))/[partial derivative][mu] < 0. If [C.sub.3] > 0, combining terms in [C.sub.1], [C.sub.2], and [C.sub.3], we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Therefore, [partial derivative][[pi].sup.m*.sub.M] ([[lambda].sup.*.sub.m] ([t.sub.1],[mu]), ([t.sub.1], [mu]))/[partial derivative][mu] < 0 for all [t.sub.1] > 0 and [mu] > 1, and thus, [[pi].sup.m*.sub.M] ([[lambda].sup.*.sub.m] ([t.sub.1], [t.sub.2]), ([t.sub.1], [t.sub.2])) is highest when [mu] = 1, that is, [t.sub.1] = [t.sub.2].

(3) If [t.sub.1] = [t.sub.2], [t.sub.1] > [[pi].sup.m*.sub.M] ([[lambda].sup.*.sub.m] ([t.sub.1],1), ([t.sub.1],1)).

When [t.sub.1] = [t.sub.2], without bundling, the merged firm earns [[pi].sub.M] =(1/2)([t.sub.1] + [t.sub.1]) = [t.sub.1]. Mixed bundling is not profitable when [t.sub.1] = [t.sub.2], that is, [t.sub.1] > [[pi].sup.m*.sub.M]([[lambda].sup.*.sub.m]([t.sub.1], 1), ([t.sub.1], 1)). See Proposition 1 in Matutes and R6gibeau (1992) for the proof.

(4) Mixed bundling is never profitable for all [t.sub.2] [greater than or equal to] [t.sub.1] ([mu] [greater than or equal to] 1).

Without bundling, the merged firm earns [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Combining these results, we get [[SIGMA].sub.i] [[pi].sup.*.sub.iA] = (1/2) ([t.sub.1] + [t.sub.1][mu]) [greater than or equal to] [t.sub.1] > [[pi].sup.m*.sub.M] ([[lambda].sup.*.sub.m] ([t.sub.1], 1), ([t.sub.1], 1)) > [[pi].sup.m*.sub.M] ([[lambda].sup.*.sub.m] ([t.sub.1],[mu]), ([t.sub.1], [mu])). Therefore. mixed bundling is never profitable for all [mu] [greater than or equal to] 1.

2. PROOF OF PROPOSITION 2

A consumer with [x.sub.12] buys a bundle from [A.sub.1] and [A.sub.2] if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Note that depending on the level of [[alpha].sup.s] + [beta], the demand functions are different. If [gamma] < [[alpha].sup.s] + [beta] < 1, the demand for a bundle is [D.sub.AA] = [D.sup.sp.sub.iA] = [[alpha].sup.s] + [beta] - [gamma]/2 = [[alpha].sup.s] + 1/2 and the demand for [B.sub.i]'s product is [D.sup.sp.sub.iB] = 1/2 - [[alpha].sup.s], for i= 1,2. Case (b) in Figure Al depicts this. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Similarly, if 0 < [[alpha].sup.s] + [beta] < [gamma], [D.sub.AA] = [D.sup.sp.sub.iA] = [[t.sub.1] + [[t.sub.2] + [DELTA]].sup.2] / (8[t.sub.1][t.sub.2]) and [D.sup.sp.sub.iB] = 1 - [D.sub.AA].

(1) First, we show that the optimal prices lie in the range where [gamma] < [[alpha].sup.s] + [beta] < 1.

Suppose [[alpha].sup.s] + [beta] > 1, that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This occurs when [??] are much higher than [p.sub.AA]. However, for [B.sub.i], setting the price in the range where [[alpha].sup.s] + [beta] > 1 is never a best response for any given [p.sub.AA]. We show this below.

When [[alpha].sup.s] + [beta] > 1, [[alpha].sup.s] + [beta] - [gamma] > 0 since [gamma] < 1. For a positive market share for [B.sub.i], it must be that [[alpha].sup.s] + [beta] - [gamma] < 1, that is, [??] < [t.sub.2] + [t.sub.1]. Thus, 0 < [t.sub.2] - [t.sub.1] < [??] < [t.sub.1] + [t.sub.2]. For any given [p.sub.AA], from the FOC for [B.sub.i], [p.sup.sp.sub.i]B must satisfy ([??] - [c.sub.i]) = ([t.sub.1] + [t.sub.2] - [??])/2. This implies that [??] [[[t.sub.1] + [t.sub.2] - [??]].sup.3] /(16[t.sub.1][t.sub.2]) - [f.sub.i] in this range. As [t.sub.2] - [t.sub.1] < [??] < [t.sub.1] + [t.sub.2], [[pi].sup.sp.sub.iB] = [[[t.sub.1] + [t.sub.2] - [??]].sup.3] /(16[t.sub.1][t.sub.2]) - [f.sub.i] < ([t.sub.1]/2) [gamma] - [f.sub.i]. In contrast, if [[alpha].sup.s] + [beta] < 1, [DELTA] = [SIGMA] [p.sup.sp.sub.iB] - [p.sub.AA] < [t.sub.2] - [t.sub.1], and [p.sup.sp.sub.iB] satisfies ([p.sup.sp.sub.iB] - [c.sub.i]) = ([t.sub.2] - [DELTA]). Then, in this range, [p.sup.sp.sub.iB] = [([t.sub.2] - [DELTA]).sup.2] /(2[t.sub.2]) - [f.sub.i] > ([t.sub.1/2]) [gamma] - [f.sub.i] since [DELTA] < [t.sub.2] - [t.sub.1]. Therefore, [B.sub.i] never has an incentive to set the price so that [[alpha].sup.s] + [beta] > 1.

By symmetry, [A.sub.i] has no incentive to set its price so high that [[alpha].sup.s] + [beta] < [gamma] (i.e., [??] = [SIGMA] [p.sup.sp.sub.iB] - [??] < [t.sub.1] - [t.sub.2] < 0). Thus, the optimum is in the range where [gamma] < [[alpha].sup.s] + [beta] < 1.

(2) When [gamma] < [[alpha].sup.s] + [beta] < 1, from the FOCs for [A.sub.i], we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, [p.sub.AA] - [c.sub.1] - [c.sub.2] = 4[t.sub.2][phi][D.sub.AA]. Combined with the FOCs for [B.sub.i], [DELTA] = 2[t.sub.2](1 - [phi] - 2[[alpha].sup.s] (1 + [phi])). Thus, [[alpha].sup.s] = [DELTA]/2[t.sub.2] = (1 - [phi])/(3 + 2[phi]) and [D.sub.AA] = (1 - [phi])/(3 + 2[phi]) + 1/2. Then, [[pi].sup.sp.sub.iA] = ([t.sub.2]/2)[phi][[(4 + [phi])/ (3 + 2[phi])].sup.2] is an increasing function of [phi]. Hence, when [phi] = 1, the allied firms maximize their profits. In this case, [p.sub.AA] = 2[t.sub.2] + [c.sub.1] + [c.sub.2], [p.sup.sp.sub.iA] = [p.sup.sp.sub.iB] = [t.sub.2] + [c.sub.i], [D.sub.AA] = [D.sup.sp.sub.iB] = 1/2, and the profits are [[pi].sup.sp.sub.iA] = [[pi].sup.sp.sub.iB] = ([t.sub.2]/2) - [f.sub.i], i = 1.2. [B.sub.1] is strictly better off and [B.sub.2] is indifferent. (12) Note that the demand functions depend only on [t.sub.2] in this range. Thus, the optimal prices only depend on [t.sub.2]. That is, firms find it optimal to compete in terms of [t.sub.2] because [t.sub.2] > [t.sub.1] ([gamma] < 1).

3. PROOF OF PROPOSITION 3

A consumer with [x.sub.12] buys both goods from M if and only if [x.sub.2] [less than or equal to] [[alpha].sup.M] + [beta] - [gamma][x.sub.1], where [[alpha].sup.M] := ([p.sup.p.sub.1B] + [p.sup.p.sub.2B] - [p.sub.M])/(2[t.sub.2]) = d/(2[t.sub.2]).

(1) Suppose [[alpha].sup.M] + [beta] < 1 [??] 0 < d < [t.sub.2] - [t.sub.1].

The demand for a bundle is [D.sup.P.sub.M] ([p.sub.M], [p.sup.p.sub.1B], [p.sup.p.sub.2B]) = [[alpha].sup.M] + 1/2 and the demand for an individual product t is [D.sup.p.sub.iB]([p.sub.M], [p.sup.p.sub.1B], [p.sup.p.sub.2B]) = 1/2 - [[alpha].sup.M], for i = 1, 2. The profits are [[pi].sup.p.sub.M] = {[[alpha].sup.M] + 1/2} {[p.sub.M] - ([c.sub.1] + [c.sub.2])) - [SIGMA] [f.sub.i] and [[??].sup.p.sub.iB] = {1/2 - [[alpha].sup.M]} ([p.sup.p.sub.iB] - [c.sub.i]) - [f.sub.i] for i = 1,2. Solving the FOCs, we obtain [[alpha].sup.M] + 1/2 = 5/8, and thus, [D.sup.p.sub.M] = 5/8 and [D.sup.p.sub.iB] = 3/8. The optimal prices are [p.sub.M] = (5/4)[t.sub.2] + [c.sub.1] + [c.sub.2] < [p.sup.p.sub.1B] + P[p.sup.p.sub.2B], and [p.sup.p.sub.iB] = (3/4)[t.sub.2] + [c.sub.i], for i= 1,2. These prices satisfy the condition d < [t.sub.2] - [t.sub.1] only if [t.sub.1] < (3/4)[t.sub.2]. The merged firm earns [[pi].sup.p.sub.M] = (25/32)[t.sub.2] - [SIGMA] [f.sub.i] from pure bundling. Thus, pure bundling is profitable if [t.sub.1] < (9/16)[t.sub.2].

(2) Pure bundling is unprofitable if [[alpha].sup.M] + [beta] > 1.

Suppose [[alpha].sup.M] + [beta] > 1. In equilibrium, it must be that 0 < [[alpha].sup.M] + [beta] - [gamma] < 1 to guarantee [D.sup.p.sub.iB] >0. That is, [t.sub.2] - [t.sub.1] < [p.sup.p.sub.1B] + [p.sup.p.sub.2B] - [p.sub.M] < [t.sub.1] + [t.sub.2]. Let [s.sub.iB] := [{[t.sub.1] + [t.sub.2] - d}.sup.2] / (8[t.sub.1][t.sub.2]) be the market share of [B.sub.i] in market i, where d := [P.sup.p.sub.1B] + [P.sup.p.sub.2B] - [p.sub.M]. Then, the demands for a stand-alone product from [B.sub.i] and a bundle are [D.sup.p.sub.iB] ([p.sub.M], [p.sup.p.sub.1B], [p.sup.p.sub.2B]) = [s.sub.iB] and [D.sup.P.sub.M]([p.sub.M], [p.sup.p.sub.1B], [p.sup.p.sub.2B]) = 1 - [s.sub.iB], respectively. The FOCs are given by [p.sub.iB] - [c.sub.i] = (1/2)Y, and

(A8) [p.sub.M] - ([c.sub.1] + [c.sub.2]) = 2Y - ([t.sub.1] + [t.sub.2]) = [8[t.sub.1][t.sub.2] - [Y.sup.2]]/2Y,

where Y := ([t.sub.1] + [t.sub.2] - d). In the range where [t.sub.2] - [t.sub.1] < d < [t.sub.1] + [t.sub.2], 0 < Y < 2[t.sub.1]. The unique solution satisfies

(A9) Y = 1/5 {([t.sub.1] + [t.sub.2]) + [square root of ([([t.sub.1] + [t.sub.2]).sup.2] + 40[t.sub.1][t.sub.2])]}.

implying that d = ([p.sup.p.sub.1B] + [p.sup.p.sub.2B] - [p.sub.M]) = [{4([t.sub.1] + [t.sub.2]) - [square root of ([([t.sub.1] + [t.sub.2]).sup.2] + 40[t.sub.1][t.sub.2])]}/5]. For the prices that satisfy (A9), the condition d > [t.sub.2] - [t.sub.1] holds only if [t.sub.1] > (3/4)[t.sub.2]. In the subgame, the merged firm earns [[pi].sup.p.sub.M] = (1 - [s.sub.iB])([p.sub.M] - ([c.sub.1] + [c.sub.2])) - [SIGMA] [f.sub.i] = [2Y/(8[t.sub.1][t.sub.2])](2Y - [([t.sub.1] + [t.sub.2])).sup.2] - [SIGMA] [f.sub.i] with pure bundling and [[pi].sub.M] = (1/2)([t.sub.1] + [t.sub.2]) - [SIGMA] [f.sub.i] without bundling. Then, [[pi].sup.p.sub.M] < [[pi].sub.M], i.e.,

(A10) ([t.sub.1] + [t.sub.2]) ([t.sup.2.sub.1] + [t.sup.2.sub.2] - 568[t.sub.1][t.sub.2]) + ([t.sup.2.sub.1] + [t.sup.2.sub.2] + 162[t.sub.1][t.sub.2]) [square root of R] < 0,

where R = [t.sup.2.sub.1] + [t.sup.2.sub.2] + 42[t.sub.1][t.subp.2]. This is because (i) the first term in (A10) is negative since from (A8), 8[t.sub.1][t.sub.2] - [Y.sup.2] > 0 iff Y > ([t.sub.1] + [t.sub.2])/2 [??] [([t.sub.1] + [t.sub.2]).sup.2] <32[t.sub.1][t.sub.2] and (ii) [square root of R] < (23/7)([t.sub.1] + [t.sub.2]), and thus, ([t.sub.1] + [t.sub.2])([t.sup.2.sub.1] + [t.sup.2.sub.2] - 568[t.sub.1][t.sub.2]) + ([t.sup.2.sub.1] + [t.sup.2.sub.2] + 162[t.sub.1][t.sub.2]) [square root of R] < ([t.sub.1] + [t.sub.2])([t.sup.2.sub.1] + [t.sup.2.sub.1] - 568[t.sub.1][t.sub.2]) + ([t.sup.2.sub.1] [t.sup.2.sub.1] + 162[t.sub.1][t.sub.2])(23/7)([t.sub.1] + [t.sub.2]) = [10([t.sub.1] + [t.sub.2])/7](3[t.sup.2.sub.1]- 25[t.sub.1] [t.sub.2] + 3[t.sup.2.sub.1]) < 0 in the range where [t.sub.2] > [t.sub.1] > (3/4)[t.sub.2] given that from (i) and (A9), Y < 6([t.sub.1] + [t.sub.2])/7. Therefore, pure bundling is unprofitable when [[alpha].sup.M] + [beta] > 1.

4. PROOF OF PROPOSITION 4

Figure 2 provides the payoff matrices in three subgames: when [A.sub.1] and [A.sub.2] merge, when no pair of firms merge, and when both pairs of firms merge. In the first two subgames in Figure 2, there are multiple equilibria. However, all of the equilibria give identical payoffs to the firms. The subgame perfect Nash equilibrium follows from backward induction. (13) The proofs of (1) and (2) are straightforward.

5. PROOF OF PROPOSITION 5

Let [[pi].sub.m.sup.MA] and [SIGMA][[pi].sup.m.sub.iA] be the post-merger profit for [A.sub.1] and [A.sub.2] when both merged firms offer mixed bundling and the post-merger profit when the other merged firm ([B.sub.1] and [B.sub.2]) alone offers mixed bundling. Similarly, we can define [[pi].sup.m.sub.MB] and [SIGMA][[pi].sup.m.sub.iB].

(1) First, we show that if both pairs of firms merge, neither mixed bundling nor pure bundling is profitable. If both pairs of firms merge and offer mixed bundling, none of the merged firms gains. See Armstrong (2006; pp. 123-124) for the proof. If both merged firms offer pure bundling, both incur losses. Let [P.sub.MA] and [P.sub.M8] be the prices of bundled products offered by the two merged firms, respectively. A consumer with [x.sub.12] buys both goods from the merged firm of [A.sub.1] and [A.sub.2] iff [x.sub.2] [less than or equal to] [[alpha].sub.MM] + [beta] - [gamma][x.sub.1], where [[alpha].sub.MM] := ([P.sub.MB] - [P.sub.MA])/(2[t.sub.2]). The optimal prices are [p.sup.*.sub.MA] = [p.sup.*.sub.MB] = [t.sub.2] + [C.sub.1] + [C.sub.2] and the profits are [[pi].sup.*.sub.MA] = [[pi].sup.*.sub.MB] = (1/2)[t.sub.2] - ([f.sub.1] + [f.sub.2]) < [SIGMA][[pi].sup.*.sub.iA] = [SIGMA][[pi].sup.*.sub.iB]. The out-come is identical even if one of the two merged firms do not offer pure bundling. The last case in Figure 2 shows the payoffs in the subgame when both pairs of firms are merged.

(2) Next, we find that when both firms are merged, "'not bundle" is a dominant strategy. (a) From the results in (1), Proposition 2, and Armstrong (2006), we get [[SIGMA].sub.i][[pi].sup.*.sub.ij] [[pi].sup.m.sub.ij] and [[SIGMA].sub.i][[pi].sup.*.sub.ij] > [[SIGMA].sub.i][[pi].sup.m.sub.ij] i = 1,2, j - A, B. (b) Armstrong and Vickers (2010) show that if both merged firms offer mixed bundling, in the symmetric equilibrium, the bundle discount [[lambda].sub.m] is less than [t.sub.1] = min {[t.sub.1], [t.sub.2]}. Thus. the bundle price is higher under mixed bundling than under pure bundling. Sum of the stand-alone prices under mixed bundling is even higher than the bundle price. On the other hand, in the symmetric equilibrium with pure bundling, each merged firm gets the same market share (the half) as it would under mixed bundling. Hence, with the same market share at higher prices, merged firms earn higher profits under mixed bundling. That [[pi].sup.m.sub.Mj] > [[pi].sup.p.sub.Mj] for j = A, B. (c) >From Matute and Regibeau (1992; Proposition I) and Armstrong (2006), if the rival firms are also merged and offer mixed bundling, the merged firm earns higher profits by not mixed-bundling that is [[SIGMA].sub.i][[pi].sup.m.sub.ij] > [[pi].sup.m.sub.Mj] i = 1,2, j = A, B. >From (a) through (c), "not bundle" is a dominant strategy.

(3) Now we derive the equilibrium of the game in each parameter range.

(a) When (16/9)h < [t.sub.2] < (32/9)[f.sub.2]

Eliminating a weakly dominated strategy, we get (not bundle, not bundle) as a unique prediction in the subgame when both pairs of firms are merged. The unique subgame perfect Nash equilibrium is that both pairs of firms merge, but no bundling occurs in equilibrium.

(b) When foreclosure is not possible.

The parameter space is divided into two by whether (16/9)[t.sub.1] < [t.sub.2] or not. Suppose (16/9)[t.sub.1] < [t.sub.2], that is, a unilateral merger is profitable. The payoffs for firms in this case are shown in the first reduced form extensive game in Figure 3. There are two subgame perfect Nash equilibria: (Merge, Merge) and (Not, Not). However, only (Not, Not) is an extensive-form THPNE.

Suppose [B.sub.1] and [B.sub.2] are playing a mixed strategy of choosing to merge with probability [epsilon] for 0 < [epsilon] < 1. Each pair of firms decides not to merge if and only if the expected profits from merging are lower than the expected joint profits from not merging. Then, for a small enough [epsilon], that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] choosing not to merge is a best response for [A.sub.1] and [A.sub.2]. By symmetry, [B.sub.1] and [B.sub.2] also place a minimal weight on "Merge" for a small enough [epsilon] when [A.sub.1] and [A.sub.2] are playing a mixed strategy of choosing to merge with probability [epsilon]. Thus, (Not, Not) is the unique extensive-form THPNE.

If (16/9)[t.sub.1] > [t.sub.2], a unilateral merger is unprofitable. The payoffs are given in the second extensive-form game in Figure 3. There are two subgame perfect Nash equilibria in this game: (Merge, Merge) and (Not, Not). Only (Not, Not) is the unique extensive-form THPNE of this game. Suppose [B.sub.1] and [B.sub.1] are playing a mixed strategy of choosing to merge with probability e for 0 < [epsilon] < 1. For all [epsilon] < 1, choosing not to merge is a best response for [A.sub.1] and [A.sub.2], and the same holds for Biand Bi- Thus, when foreclosure is not possible, in equilibrium, no firms merge, but firms offer pure bundling through strategic alliances.

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(1.) Upon the release of the iPhone in 2007, Apple entered into an exclusive arrangement with AT&T that prevented consumers from buying the iPhone without subscribing to the AT&T network service. This arrangement ended in 2011 as a result of an antitrust suit ("Verizon Finally Lands the IPhone" http://online.wsj.com/article/SB10001424052748704739504576068170230339348.html.)

For details of class action suits, see Smith et al. v. Apple. Inc. et al. (2007) and Holman et al. v. Apple, Inc. et al. (2007).

(2.) Hewitt (2002), "Portfolio Effects in Conglomerate Mergers," OECD. Best Practice Roundtables in Competition Policy No. 37.

(3.) If the transportation costs are the "actual" cost of travel, they depend only on the travel distance and not on the type of the product. That is, [t.sub.1] = [t.sub.2] = t. In this case, mergers do not affect the travel cost as long as the merged firms continue to shelve their products independently as before. Alternatively, if the merged firms shelve the two products together in a bundle, the cost of travel is reduced by half. (See Armstrong and Vickers 2010 for this type of bundling that reduces shopping costs.) We interpret [t.sub.i] as a taste parameter rather than the actual travel cost because the former represents more general cases.

(4.) This profit function implies that each allied firm's share of the profits from bundle sales is only proportional to its contribution, that is, ([[delta].sub.i][p.sub.iA] - [c.sub.i]). This type of strategic alliance represents the least integrated form of strategic relationship between two allied firms, granting the maximal level of independence to the firms. This contrasts with a merger, which is the most integrated form of relationship.

We present this form of strategic alliance for mixed bundling in order to make it consistent with the optimal form of strategic alliance for pure bundling in Section B. However, considering an alternative form of strategic alliance does not affect the result. For example, suppose allied firms prenegotiate how to divide joint profits. Let [[phi].sub.i] be [A.sub.i]'s share of the joint profit from bundle sales, [SIGMA][[phi].sub.i] = 1. Then, [[pi].sup.sps.sub.iA] = ([p.sub.iA] - [c.sub.i])[D.sub.iA] + [[phi].sub.i] [[SIGMA].sub.i] ([[delta].sub.i][p.sub.iA] - [c.sub.i])[D.sub.bundle]. We show that mixed bundling is not profitable in this case either. The proof is available upon request.

(5.) The market outcomes are the same even if [B.sub.1] and [B.sub.2] offer mixed or pure bundling through a strategic alliance. Hence, the analysis applies to all the three cases, when [B.sub.1] and [B.sub.2] offer mixed or pure bundling via a strategic alliance, or when they don't.

(6.) The relationship between allied firms can be defined in many different ways. In Section C, we briefly discuss an alternative form of strategic alliance in Zhang and Zhang (2006) and the effects on the profitability of bundling.

(7.) In Section V. we briefly discuss the effects of more firms in the market on the profitability of pure bundling.

(8.) This resembles the result of vertical foreclosure in Ordover, Saloner, and Salop (1990) although there is no role of bundling in their paper. In this paper, bundling is required for foreclosure. That is, without bundling, a merger alone cannot induce foreclosure.

(9.) Several papers consider the possibility of using bundling to foreclose competition or to deter entry. See Choi and Stefanadis (2001); Nalebuff (2004); and Whinston (1990) for examples. See also United States v. Microsoft, 253 F.3d 34, 87 (D.C. Cir. 2001) and Carlton and Waldman (2002) for more discussion on the use of bundling as a method of strategic foreclosure.

On the other hand, many other studies show bundling is likely to benefit consumers through lower prices and cost savings. For an example, see Armstrong and Vickers (2010).

Bundling is often used as a device for price discrimination (as in DeGraba 1994; McAfee, Mcmillan, and Whinston 1989; and McCain 1987) or product differentiation (as in Chen 1997), in which case the welfare implications are mostly arabiguous.

(10.) In this paper, mergers and strategic alliances result in different pricing strategies for firms because firms become completely integrated following a merger and coordinate their prices for bundled products, whereas allied firms set their own prices independently. However, because a merged firm can also decide how to reorganize its production structure, one might wonder whether a merged firm would have an incentive to fully integrate the merging partners in the current framework. A few studies have shown that firms may have strategic incentives to maintain competition among their operating units. For examples, see Baye, Crocker, and Ju (1996) and Mialon (2008). In the present model, if a merged firm chooses to maintain competition within the merging partners, the resulting structure resembles that of a strategic alliance. Given that a merger is only motivated by the possibility of foreclosure, when a merger does occur (i.e., when (16/9)[t.sub.i] < [t.sub.2] < (32/9)[f.sub.2]) it would not be optimal for the merged firm to operate autonomous divisions as it cannot induce foreclosure without full integration.

(11.) See Kobayashi (2005) for details about reasons for bundling.

(12.) In the current framework of optimal distribution rule [phi], the incentive compatibility condition for a merger or an alliance is satisfied as long as bundling increases the size of the pie to divide. Given that [B.sub.2] is indifferent, in order to make sure that [B.sub.2] agrees on bundling, there can be an arrangement of a lump-sum profit transfer. For example, [B.sub.2] will strictly prefer an alliance if [epsilon] > 0 amount of a lump-sum transfer of the increased profit from [B.sub.1] to [B.sub.2] occurs. As long as 0 < [epsilon] < ([t.sub.2] - [t.sub.1])/2, both firms profit from the pure bundling under a strategic alliance.

(13.) Figure 2 does not include the payoff of the merged firm after foreclosure, as it would require considering a new game between the merged firm and new rivals (potential entrants) in a later period, which involves completely different market parameters such as entry costs [E.sub.i] of a potential entrant [b.sub.i] in market i, i = 1, 2. The current reduced form is justified as long as the merged firm's profit after foreclosure is higher than the profits from bundling through a strategic alliance. While it is straightforward, we provide the proof that the profit from foreclosure after merger is higher than bundling via strategic alliance in a supplementary appendix, which is available upon request.

SUE H. MIALON, The author is grateful to Mark Armstrong, R. Preston McAfee, Kaz Miyagiwa, Russell Pittman, and participants in the 2011 American Law and Economics Association meetings at Columbia Law School for their helpful comments. This paper was previously circulated under the title of "Exclusionary Bundling: The Motive for Mergers."

Mialon: Assistant Professor, Department of Economics, Emory University, Atlanta, GA 30322. Phone 404-7128169, Fax 404-712-4639, E-mail smialon@emory.edu
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