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Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Model
  5. Bertrand–Nash Equilibrium
  6. Feasibility of Complementary Alliances
  7. Conclusions
  8. Appendix
  9. References

This paper investigates the feasibility of full/partial complementary alliances in composite goods markets with network structure. There are multiple producers who each provide a complementary component of the composite good. In another related market, one single firm (a monopolist) produces another composite good which could be a substitute or complement of the composite good. The analysis shows that even a full alliance cannot be profitable when the number of producers is small and the two goods are close substitutes or complements. Moreover, none of the profitable alliances is stable, except the full alliance with two component producers.

Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Model
  5. Bertrand–Nash Equilibrium
  6. Feasibility of Complementary Alliances
  7. Conclusions
  8. Appendix
  9. References

Since the contribution by Cournot (1927), it is well known that the integration of two independent monopolists, each producing one complementary component, will reduce the sum of the two components' prices. The reason is that the two independent firms ignore the effects of the price-rising effect on each other's demand, while the integration internalizes this pricing externality. The integration benefits not only the integrated firms themselves but also the consumers (also see Tirole 1988). This property of integrations actually emerges in complementary alliances which are defined as alliances between complementary producers.1

Given this theoretical knowledge, it should be plausible to consider that most component producers would have an alliance relationship with their complementary partners. However, this is not always true in the real world. In particular, in network industries where multiple firms produce competing and/or complementary products, it can be observed that some firms ally with their complementary partners, while others do not do so. These network industries include the airline, multimodal transport, shipping, logistics, personal computers and software, domestic/international telecommunication services, ATMs and bankcards, as well as the market for vacations comprised of transportation and resort hotel industries.2

This observation raises the straightforward question of why, given clear complementary relationships among their products, some firms allied with the others, while others do not enter an alliance. From the viewpoint of economics, one considerable reason might be the costs for the formation of alliances. However, suppose that the alliance costs are limited (extremely close to zero), should these complementary partners form an alliance or act independently? This theoretical paper investigates the feasibility of complementary alliances and offers reasonable explanations for why complementary alliances may not occur in practice, even though the alliance could be beneficial for the allied firms and be desirable from the viewpoint of social welfare.

To achieve this purpose, this paper considers simple but common markets with a network structure. In one composite goods market, there are N firms who each provide a complementary component of composite good A. According to observations of network industries and previous network models, one representative related market is included in our model. In the related market, a monopolist (firm B) provides a differentiated good B, which could be either a substitute or a complement of composite good A. In addition, the market size of composite good A is relatively smaller than that of good B, due to some extent of incompatibility among the components. The complementary alliance takes the form of either a full alliance (i.e. all existing N complementary producers allied) or a partial alliance (some of the existing complementary producers allied, while the others, say outsiders, act independently).

Adopting a well-used linear demand for the two goods and assuming zero production and alliance costs, this study first shows that, in the cases of N = 2 and N = 3, even a full alliance is unprofitable for the allied firms when the degree of substitutability (complementarity) between goods A and B is large, whereas the full alliance in the case of N ≥ 4 is always profitable. This results from the fact that the full alliance benefits the allied firms due to the internalization of the pricing externality in the market for composite good A. However, the alliance forces firm B to lower (raise) its price when the two goods are substitutes (complements), which in turn causes losses for the allied firms themselves. The partial alliance analysis shows that an alliances size less than 80 per cent of all component producers is always unprofitable for any N ≥ 2 and any degree of substitutability (complementarity). Furthermore, the necessary alliance size for ensuring the profitability for the allied firms increases as these degrees become larger.

The intuitions for these results add new insights to the literature. In the partial merger/integration studies with one single homogenous market, one additional number of merger/integration firms benefits the outsiders. This well-known business stealing response works in our model. The additional contribution of our model is in considering the response by a firm (firm B) who acts on the related market. Throughout the paper, the response by firm B is called retaliation response in the substitutive goods case and business stealing response in the complementary goods case.

This paper also examines the stability of a complementary alliance by considering an open membership game. In the game, a stable alliance requires that no alliance member wants to unilaterally leave the alliance (internal stability) and that no outsider wants to join the alliance (external stability). The examination shows that all of the possible alliances are unstable except the full alliance case where N = 2 and the degree of substitutability (complementarity) is not so large. In other words, except for the above case, all allied members do not have an incentive to simultaneously break up the alliance contract, but each member has an incentive to unilaterally deviate from the alliance. This result offers another explanation for why some component producers do not form a complementary alliance in practice.

In the industrial organization literature, various studies have long argued that horizontal integrations may reduce the joint profits of the participating firms in non-cooperative oligopoly settings. Using a symmetric Cournot (quantity-setting) model with linear demand, Salant et al. (1983) show that an integration is not beneficial unless it involves over 80 per cent of all firms. Subsequently, Cheung (1992) shows that for demands satisfying that the marginal revenue of the industry is decreasing, the minimal size for a profitable integration is 50 per cent. Fauli-Oller (1997) shows that the minimal size for a profitable integration is increasing in the degree of concavity of demand. In contrast, using a Bertrand (price-setting) model with differentiated products, Deneckere and Davidson (1985) show that integrations of any size are beneficial because the reaction of non-integrated firms is the opposite of the quantity-setting case. This present paper is also a price-setting model. The essential difference from Deneckere and Davidson (1985) is that in their model with one single market, N products are imperfect substitutes and are in direct competition, whereas in our model with two related markets, N products sold in one market are complements with each other, and the N products are also substitutes or complements with the other product(s) sold in the related market.3

In contrast to the previous works with one single market, this paper considers a simple but commonly observable composite goods market with another related market. This present market structure could be viewed as a relatively general one in the literature of network markets. In the previous studies by Economides and Salop (1992), Beggs (1994), Park (1997), Lin (2005) and Bilotkach (2007), each of them essentially deals with one case of our market structures where N = 2 (with symmetric market size), and thus the issue of partial integrations/alliances and the corresponding argument of the stability of alliances were not included in their analysis. Under the market structure, our analysis clarifies the reactions of strategic pricing, not only for the allied firms and the outsiders but also including the reactions of a firm who supplies a good for another related market. Our findings of the profitability analysis offer a reasonable explanation for why some component producers do not ally with their complementary partners. The results of the stability analysis also offer another explanation for the concern.

This paper is organized as follows. Section 'The Model' presents the market structure. Section 'Bertrand–Nash Equilibrium' derives the Bertrand–Nash outcomes, which can be easily applied for the analysis of the full alliance and the partial alliance in Section 'Feasibility of Complementary Alliances'. Section 'Profitability of Full Alliances' details the conclusions.

The Model

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Model
  5. Bertrand–Nash Equilibrium
  6. Feasibility of Complementary Alliances
  7. Conclusions
  8. Appendix
  9. References

Let us consider a market for a composite good (composite good A) that consists of N (≥2) kinds of components. Each component i (= 1, 2, … , n) is provided by an independent firm (firm i) with a price pi. There is another market for a differentiated good (good B) provided by firm B. Firm B, a monopolist, sells good B with a single price p. The consumption relationship between composite good A and good B is either substitutive or complementary to some extent. Figure 1 illustrates the market structure of our model. Because our concern is on strategic alliances in the composite goods market, the markets for each individual component are not included in our model. This simplification can be justified by the fact that discriminated price strategies are widely used in network markets.

figure

Figure 1. Market Structures

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Suppose that component producers for composite good A are now considering an alliance with other component producers, so as to set their own component prices cooperatively. Let us denote the number of allying firms as M ∈ [1, N] and the set of allying firms as C. In the case where M = 1, no component producer allies and each component producer sets the price of its own component independently (i.e. C is the null set). In the case where M ∈ [2, N), a single decision-maker of M allied firms set the total price for their own components, while the remaining NM firms (i.e. firms not in C) and firm B continue to act independently. In the full alliance case (i.e. M = N), all component producers ally together and choose the single price of composite good A, taking the firm B's pricing behavior as given. The marginal production costs for each firm are constant and assumed to be zero.4

Consumers have preferences over composite good A and good B. In order to consume composite good A, they have to purchase each of the necessary components in fixed proportions (e.g. one unit of each). To illustrate attractive and suggestive results, a representative consumer with a quadratic utility function is considered. The utility function is written as U(q, Q) = a(γ · q + Q) − (1/2)(q2 + 2bqQ + Q2), where q and Q represent demands of composite good A and good B respectively. Parameter a > 0, and the substitutability (complementarity) between good A and good B will be measured by b ∈ (−1, 1), where positive (negative) b is associated with substitutes (complements).5 The degree of incompatibility among the components for composite good A is reflected by the exogenous parameter γ in the utility function, where 0 < γ ≤ 1. When γ = 1 (the full compatibility case), the marginal benefit from the consumption of composite good A is totally the same as that from good B. In the other case with some extent of incompatibility (γ < 1), the consumer receives less benefit if he/she purchases composite good A. The case of γ > 1 will be discussed in Section 'Effect of Compatibility'. Note that γa and a will become parameters in the intercepts of the derived inverse demand functions. Namely, the parameters represent the asymmetric market size for composite good A and good B respectively.6

The surplus of the representative consumer is measured by U(q, Q) − p · qP · Q. Note that composite good A is available at the total price p, which is the sum of the cooperative price offered by the allied firm (pC) and the prices of the NM outsiders (inline image). Thus, inline image. Given the prices, the consumer chooses the demands in order to maximize the consumer surplus. Then the demand functions for the two goods, A and B, can be derived and written as follows:7

  • display math(1)
  • display math(2)

To ensure the positive demands of both goods, it is necessary to impose γ > b.

Bertrand–Nash Equilibrium

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Model
  5. Bertrand–Nash Equilibrium
  6. Feasibility of Complementary Alliances
  7. Conclusions
  8. Appendix
  9. References

To investigate the effects of full alliances and partial alliances on the market performance, this section first solves the Bertrand–Nash outcomes for the three types of representative agents; an allied firm comprised of M firms of component producers, NM component producers outside the alliance (outsiders), and firm B. The joint profit function for the allied firm is defined as πC = pC·q.8 The profit function for firm iC (i.e. outsider i) is inline image. The profit function for firm B could be written as Π = P·Q. Each agent chooses its price to maximize its own profits/joint profits, taking the prices of the others as given.9 Profit maximization of each agent is respectively characterized by the following:

  • display math(3a)
  • display math(3b)
  • display math(3c)

Equations (3a), (3b), (3c) are actually the reaction functions for each agent, and those describe the strategically complementary and substitutive relationship among all firms.10 When the composite good A combined by N components is substitutive (complementary) with good B, the choices of pC as well as inline image and the choice of P are strategic complements (substitutes). Due to the complementary relationship, the choices of pC, inline image and inline image are strategic substitutes.

Now, applying the Bertrand–Nash supposition and solving the equation system (3a), (3b), (3c), the outcomes can be derived as listed in Table 1. It should be noted here that under the demand functions, outsiders i and j (ij and i, jC) are symmetric, and thus inline image in the equilibrium.

Table 1. Bertrand–Nash Equilibrium in Complementary Alliances
Prices inline image
inline image
inline image
Demands inline image
inline image
Profits inline image
inline image

Feasibility of Complementary Alliances

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Model
  5. Bertrand–Nash Equilibrium
  6. Feasibility of Complementary Alliances
  7. Conclusions
  8. Appendix
  9. References

Profitability of Full Alliances

This subsection analyzes the profitability of full alliances with various numbers of component producers. In the full alliance case, all N component producers for composite good A are allied. According to Table 1, the profitability can be examined by comparing the outcomes achieved under M = 1 (the no-alliance case) with those achieved under M = N (the full alliance case). The comparison result is summarized by the following proposition.

Proposition 1. In our model:

  • (a)
    A full alliance is always profitable for any N ≥ 2 if composite good A and good B are independent (i.e. b = 0).
  • (b)
    When N = 2 and N = 3, a full alliance is unprofitable if composite good A and good B are close substitutes or close complements, whereas it is always profitable when N ≥ 4.

Proof. See the Appendix.▪

Given the well-known result in the literature, that a full complementary alliance generally increases the allied firms' joint profits (Proposition 1-(a)), Proposition 1-(b) might be interesting. Forming a full alliance has a positive impact on the allied firms' joint profits through the internalization of the pricing externality in the market for composite good A. However, at the same time, the alliance forces firm B to reduce (raise) its price when the two goods are substitutes (complements). In other words, when b is positive, the formation of the alliance induces firm B to lower its price as a retaliation response to the decrease in p. This causes losses on the allied firms' joint profits. When b is negative, the formation of the alliance gives firm B room for raising its prices. This business stealing response also causes losses on the allied firms' joint profits. In markets where N = 2, N = 3 and, goods A and B are close substitutes or close complements (i.e. |b| is large), the losses due to firm B's reaction outweigh the gains due to the cooperative pricing, and thus, even a full alliance is unprofitable.11 However, in N ≥ 4 cases, the full alliance is always profitable for all b ∈ (−1, 1) because the gains resulting from the cooperation with a large number of firms outweigh the losses caused by the reaction by firm B.

Profitability of Partial Alliances

The profitability of partial alliances can be investigated by comparing the outcomes achieved under M = 1 (the no-alliance case) with those achieved under M ∈ [2, N) (the partial alliance case). Similar with the previous studies, the essential concern is on the relationship between the profitability and the ‘alliance size’. In other words, on the question of ‘how large an alliance size is needed for the alliance to be profitable for each allied member’.

Before considering the profitability of partial alliances, the following lemma can be derived.

Lemma.
  • display math
  • display math

Proof. See the Appendix.▪

Consider one outsider k entered the alliance formed by M ∈ [2, N) component producers. It reduces the total price of M + 1 components due to an additional internalization of the pricing externality.12 As the best response to this price decrease, each of the remaining outsiders raises the price of its own component (dpO/dM > 0), which corresponds to the famous business stealing effect.13 As a consequence, the total price of the composite good A falls (i.e. dp/dM < 0).14 In addition, in response to dp/dM < 0, firm B reduces (raises) its price when the two goods are substitutes (complements) because p and P are strategic complements (substitutes).

Now, the following proposition presents the profitability of partial alliances.

Proposition 2. In our model:

  • (a)
    An alliance size less than 80 per cent is always unprofitable irrespective of the consumption relationship between goods A and B.
  • (b)
    The minimum alliance size for ensuring the profitability is increasing in |b|.

Proof. See the Appendix.▪

According to (A3) in the Appendix, Fig. 2 can be drawn to demonstrate the results of Proposition 2 clearly. Given a particular value of b, the minimum alliance size for ensuring the profitability of the allied members (inline image) is shown by the curve. When b = 0 where composite good A and good B are independent, the formation of an alliance induces no reactions from firm B. However, as the lemma shows, the formation of a partial alliance gives the outsiders room to increase their prices (i.e. business stealing effects). If the alliance size is not large enough, the loss caused by the reactions of the outsiders will be dominant. Thus, a larger alliance is needed for ensuring the profitability. Interestingly, the inline image curve in the case of b = 0 is exactly the same as the minimum profitable merger size in the model of Salant et al. (1983).15 Although the model of Salant et al. (1983) is a Cournot (quantity-setting) model with substitutive goods and our model is a Bertrand (price-setting) model with complementary goods, the well-known duality between Cournot competition with substitutes and Bertrand competition with complements also holds regarding the minimum size of a profitable alliance (merger).16

figure

Figure 2. The Size of Profitable Alliances

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As a new insight to the literature, Proposition 2-(b) shows that the greater the value of |b|, the less likely for a complementary alliance to be profitable. This is because for a greater value of |b|, the formation of a complementary alliance, which reduces the total price of composite good A, induces greater undesirable reactions of firm B. These reactions (either retaliation or business stealing response) by firm B as well as the business stealing response by outsiders diminish the gains from forming complementary alliances.

Stability of Alliances

Let us examine the stability of a complementary alliance. Assume that all component producers simultaneously decide whether or not to join an alliance, and the alliance is formed by all component producers who decide to join it.17 A stable alliance is then defined as the Nash equilibrium outcome of the game.18 For an alliance comprised of M firms to be stable, the following two conditions need to be satisfied: (i) no alliance member wants to leave the alliance (condition of internal stability), and (ii) no outsider wants to join the alliance (condition of external stability). In our model, these conditions are:19

  • display math(4)
  • display math(5)

Using the results in Table 1 yields the following result.

Proposition 3. In our model, all possible forms of alliances are externally stable. However, all possible forms of alliances are internally unstable, except the full alliance case of N = 2 and |b| < 0.77. Thus, this is the only stable alliance in the market.

Proof. See the Appendix.▪

The intuition behind it is sensible. All possible alliances are externally stable because for any particular outsider, the benefit from the other producers forming a complementary alliance is greater than the benefit from entering the alliance and sharing the increased joint profits. On the other hand, even if an alliance itself would be profitable for all allied members, this alliance cannot be internally stable, except only in the case of N = 2 and |b| < 0.77. In a profitable alliance, all allied members do not have an incentive to simultaneously break up the alliance contract, but each member has an incentive to unilaterally deviate from the alliance. This is because if the other members will stay in the alliance, then it is preferable for a producer to act outside. The only exception emerges because deviating from the profitable full alliance with N = 2 means no alliance members remain, and thus no benefit can be obtained by leaving and acting as the outsider. These results, based on the argument of the stability of complementary alliances, also offer another reasonable explanation for why some component producers do not form a complementary alliance in practice.

It is worth noting that our argument of the stability also contributes to the literature by offering a plausible justification for the model assumption made by previous studies (Economides and Salop, 1992; Beggs, 1994; Park, 1997; Lin, 2005; Bilotkach, 2007). The important fact is that these studies only consider N = 2 case, not based on the results of stability analysis, but for the reason of simplification (e.g. see Economides and Salop, 1992, Section 'The Model'). The simplification can be supported by our theoretical finding that all possible forms of alliances are unstable except the full alliance case of N = 2.

Effect of Compatibility

In our study, it is assumed that the market size of composite good A is relatively smaller than that of good B (i.e. 0 < γ ≤ 1). The following proposition shows the relationship between the profitability/stability of alliances and the degree of compatibility (equivalently, the market size differential between composite good A and good B).

Proposition 4. In our model:

  • (a)
    Propositions 1 and 2 hold, independent of γ. In other words, the condition for ensuring the profitability of full alliances, as well as the minimum alliance size for ensuring the profitability of partial alliances, are independent of the degree of compatibility. However, the magnitude of the gains/losses from forming alliances is increasing in γ.
  • (b)
    The arguments on the stability of any possible forms of alliances (Proposition 3) are also independent of the degree of compatibility.

Proof. See the Appendix.▪

Consider one benchmark case of our study, b = 0, then Proposition 4-(a) is natural because the effects from the alliance (the internalization) and the business stealing by the outsiders occurred independently of the market size. This result can also be found in Salant et al. (1983), given the described duality between Salant et al. and this benchmark case of our study.20 However, the independence also holds for b ≠ 0 case including the reaction taken by firm B. Proposition 4 shows that even if there is another market for substitutive or complementary goods, the relative market size is un-restrictive for the critical value of size for whether or not the alliance is profitable and whether or not the alliance is stable. This is because the reaction taken by firm B becomes just proportionally smaller as γ becomes smaller, but note that the magnitude of the gains/losses from forming alliances is increasing in γ.

Throughout the paper, we assume 0 < γ ≤ 1 that might be natural in the context of compatibility/incompatibility among multiple components. However, the case of γ > 1 is possible when the demand for good A is relatively large. It should be emphasized that our results hold for the case of γ > 1, as long as γ < 1/b.21

Some Other Extensions

This subsection proposes some extensions of our model. These extensions may indicate that complementary alliances are more likely to be unprofitable.

Effect of Alliance Costs

Consider the existence of some costs for alliance formation C(M), which are assumed to be increasing with the number of the allied members (i.e. C′ > 0). Then, the alliance size for ensuring the profitability may not exist. This is because the alliance is not profitable, unless M is large, even without the alliance costs. Then, taking the increasing alliance costs into account, the gains from forming an alliance may be dominated by the alliance costs.

Other Related Markets Exist

Suppose there are some markets for substitutive or complementary goods of composite good A, other than the market for good B. It can be easily predicted that the formation of an alliance comprised of some component producers in market A induces negative reactions of all firms who produce substitutive or complementary goods, and thus, the complementary alliances are more likely to be unprofitable for allied firms. This is because all firms in the other markets take undesirable reactions for the allied members.

Conclusions

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Model
  5. Bertrand–Nash Equilibrium
  6. Feasibility of Complementary Alliances
  7. Conclusions
  8. Appendix
  9. References

This paper has argued the feasibility of a full/partial complementary alliance in composite goods markets, where some firms provide a complementary component that is necessary for forming one composite good, and one single firm provides its substitutive or complementary good. This market structure can be commonly observed in a number of network-oriented industries.

As described in the introduction, there are a number of previous studies that dealt with full or partial horizontal merger/alliances, based on a quantity setting-model with homogenous goods. On the other hand, there are also studies focused on full mergers/integrations. However, as far as we know, there is relatively little literature that addresses partial integrations/complementary alliances. This present study may provide some new insights to the literature. In addition, the analytical results are perhaps interesting, and may provide a reasonable explanation for why a number of complementary component producers do not form an alliance and act independently in practice.22 Furthermore, our results of the stability analysis contribute to the literature by providing a plausible justification for the previous studies which deal with the alliance between two complementary firms only.

There remain some restrictions of the model which are desirable to be extended in future research. In the present model, the relative market size of good A, which reflects the compatibility between N complementary components, has been supposed to be independent of the number of the allied firms. In practice, including the pricing strategy argued in the present study, the improvement of the compatibility might also be an important concern for the complementary firms as they are considering an alliance. In addition, it is plausible to consider that the compatibility of a composite good substantially depends on the number of the allied firms M. Thus, it is meaningful to extend the model to investigate the relevance between the degree of the compatibility and the profitable alliance, each of them endogenously decided by the firms' strategic behavior. In future work, these directions of extension as well as more general demand/cost functions should yield more generalized results and could provide meaningful policy implications.

Appendix

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Model
  5. Bertrand–Nash Equilibrium
  6. Feasibility of Complementary Alliances
  7. Conclusions
  8. Appendix
  9. References

Proof for Proposition 1

The joint profits of the fully allied firms, denoted by inline image, can be easily generated by setting M = N into πC in Table 1. Let inline image denote the sum of the profit of all component producers in the no-alliance case, where πI can be generated by setting M = 1 into πC.

Then, the profitability of full alliances can be shown as

  • display math(A1)

When b = 0, (A1) is positive for any N ≥ 2. (A1) is negative if inline image for N = 2, is negative if inline image for N = 3. However, (A1) is always positive for all b ∈ (−1, 1) when N ≥ 4. In addition, it is obvious that the sign of (A1) is independent of γ, but the magnitude of (A1) is increasing with γ. ▪

Proof. Proof for the lemma Using Table 1 yields the following results for all b ∈ (−1, 1):

  • display math
  • display math

where Γ ≡ (2 − b2)(NM + 1) + 2 > 0 and Δ ≡ (2 − b2)γb > 0.

In addition, for b > 0 (b < 0):

  • display math
  • display math

Proof for Proposition 2

The joint profit of the M allied firms is pC·q = πC (see Table 1). If the M firms do not ally, the sum of their profits are M·πI. Then the profitability of the M partial alliance can be shown as

  • display math(A2)

where

  • display math
  • display math
  • display math
  • display math

Thus, the condition for (A2) > 0 is NS < N < NL, where NS and NL are the smaller and the larger solution for η N2 + θ N + κ = 0. NS and NL are derived as follows:

  • display math
  • display math

Because NS(b, M) < N holds for any M < N and b ∈ (0, 1), the condition for (A2) > 0 reduces to N < NL(b, M). This condition can be rewritten as M > MS(b, N), or equivalently, inline image, where

  • display math(A3)

Then,

  • display math

Thus, inline image, which proves assertion (b).

Given inline image, the minimum value of inline image is obtained when b = 0. Then,

  • display math(A4)

From the above, the minimum value of inline image is 0.8 when N = 5. Thus, the profitable complementary alliance needs to involve at least 80 per cent of all component producers. This proves assertion (a). ▪

Proof for Proposition 3

From Table 1, the condition that is necessary for an internally stable alliance can be shown as

  • display math

The condition is satisfied when

  • display math(A5)

Because Ω is monotonically decreasing in (NM), the condition (A5) is most likely to hold when NM = 0. In this case, the following relationships hold:

  • display math

Thus, for any N = M ≥ 3, the condition (A5) cannot hold. The condition of (A5) holds (i.e. the alliance is internally stable) only when N = M = 2 and inline image. Next, consider the case of NM = 1.

  • display math

Thus, an internally stable alliance does not exist when N > M.

From Table 1, the condition that is necessary for an externally stable alliance can be written as

  • display math

The condition is satisfied when

  • display math(A6)

Because Φ is monotonically decreasing in (NM), the condition (A6) is most unlikely to hold when NM = 1. Then in this case,

  • display math

Thus, the condition (A6) always holds, which implies that all possible alliances are externally stable.

As a result, the internally and externally stable alliance is the full alliance of N = 2, M = 2 and |b| < 0.77. ▪

Proof for Proposition 4

Assertions (a) and (b) are immediately confirmed by the fact that the parenthetic term in (A3), Ω in (A5) and Φ in (A6) are independent of γ. ▪

Footnotes
  1. 1

    Recently, merger/integration activities have significantly decreased, while strategic alliance formation in network markets has increased since 2000. A strategic alliance might be viewed as a lesser form of merger, since alliance partners remain separate business entities and retain their decision-making autonomy (see Zhang and Zhang, 2006). Without loss of substance, this paper essentially considers the complementary alliance rather than mergers/integrations.

  2. 2

    For an overview of network industries, see Shy (2001).

  3. 3

    Recently, Davidson and Mukherjee (2007) consider the impact of horizontal mergers in the presence of free entry and exit, and show that quantity- and price-setting games yield similar predictions that even a small-sized merger is beneficial. Escrihuela-Villar (2008) considers partial horizontal mergers among quantity-setting firms in a repeated game setting. Méndez-Naya (2008) also investigates merger profitability in a mixed-oligopoly Cournot model.

  4. 4

    The results are identical for positive constant marginal costs with ‘price’ reinterpreted as differences between prices and marginal costs.

  5. 5

    This type of utility function was originally provided by Dixit (1979), and has been used in a number of differentiated duopoly studies (Singh and Vives, 1984; Vives 1985; Bárcena-Ruiz and Garzón, 2003).

  6. 6

    It is assumed that the degree of compatibility is independent of the number of component producers.

  7. 7

    Although the demand functions generated by this utility function are linear, it yields a number of suggestive conclusions that are likely to hold true in general (Varian, 1992, p. 294).

  8. 8

    This definition of the joint profit function is popular in the merger and alliance literature, such as Salant et al. (1983). It implies that a single decision-maker in an alliance chooses a total price of the M components, i.e. inline image. Notice that the profit maximization of an alliance choosing pC yields the same result as that choosing inline image.

  9. 9

    The costs for the alliance are ignored for simplicity. Section 'Some Other Extensions' discusses the existence of alliance costs.

  10. 10

    For the definition of strategic substitutes and complements, Bulow et al. (1985) is useful.

  11. 11

    To be precise, as shown in Appendix, a full alliance of N = 2 is unprofitable when inline image, and that of N = 3 is unprofitable when inline image.

  12. 12

    This can be confirmed by inline image in Table 1. Here, p(#) indicates the corresponding price in the case where # number of component producers are forming a complementary alliance.

  13. 13

    Although this paper considers Bertrand competition of symmetric complementary components for a composite good, whereas Salant et al. (1983) consider the Cournot competition of substitutive goods, the same business stealing reactions by the firms outside the alliance occurs here.

  14. 14

    This resulted from two price-reducing forces: (i) the single price of the M + 1 components decrease, as compared with the sum of the single price of M components and the price of firm k, and (ii) the price of each remaining outsider rises but their total number decreases.

  15. 15

    It can be confirmed that our equation (4) is exactly the same as α2 in footnote 6 (p. 193) of Salant et al. (1983).

  16. 16

    For this point, see Singh and Vives (1984) who show that Cournot (Bertrand) competition with substitutes is the dual of Bertrand (Cournot) competition with complements.

  17. 17

    For a more detailed explanation and survey of cartel or alliance group formation in industrial organization literature, see Bloch (2005).

  18. 18

    Note that this game necessarily admits a trivial equilibrium in which all component producers decide not to join the alliance. The focus of the analysis is on non-trivial equilibria for which an alliance of size M ≥ 1 emerges in the market.

  19. 19

    In the two conditions, π(#) indicates the corresponding profits in the case where # number of component producers are forming a complementary alliance.

  20. 20

    In the model of Salant et al. (1983) with one single market, the intercept (i.e. market size) and the slope of the linear demand function are unrestrictive for their results of merger profitability.

  21. 21

    See Hattori and Lin (2011, Section 'Feasibility of Complementary Alliances') for the importance of the relative size difference in demand on profitable alliance in markets with vertical and horizontal externalities.

  22. 22

    Given the well-known fact that forming complementary alliances improves social welfare, our results suggest the importance of encouraging complementary component producers to form the welfare-enhancing alliances.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. The Model
  5. Bertrand–Nash Equilibrium
  6. Feasibility of Complementary Alliances
  7. Conclusions
  8. Appendix
  9. References
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