Mergers of complements, endogenous product differentiation and welfare

The static analysis shows that a merger among complementary input suppliers or complementary patent holders benefits the consumers and the society by reducing the input prices. We show that the effects of a merger of complements are not so straightforward in a dynamic set up with endogenous product differentiation in the final goods market. The merger of complements reduces the total input prices and increases product differentiation. However, whether it increases or decreases consumer surplus and welfare depends on the market expansion following product differentiation, the number of merged input suppliers and the intensity of competition. Hence, in a dynamic setup with endogenous product differentiation, the antitrust authorities may need to be more careful about mergers of complements. Our analysis has also relevance for vertical mergers.


Introduction
As discussed by Cournot (1927), mergers among complementary producers benefit the consumers by reducing prices of the complementary products, suggesting that the antitrust authorities may not need to be concerned about mergers among complementary producers.Similarly, complementary patent pools help to reduce the royalties charged by the patent holders and are often preferred by the antitrust authorities (Shapiro, 2001;Denicolò, 2002;Gilbert, 2004;Lerner and Tirole, 2008;Gallini, 2014;WIPO, 2014;Gallini, 2017). 2  We show in this paper that the welfare effects of a merger of complements are not so straightforward in a dynamic set up with endogenous product differentiation in the final goods market.The merger of complements reduces prices of the complementary products, and increases product differentiation in the final goods market. 3However, whether it increases or decreases consumer surplus and welfare depends on the market expansion following product differentiation, the number of merged input suppliers and the intensity of competition.
A merger among the complementary input suppliers 4 creates three effects in our analysis.First, like the existing literature, it reduces the prices of the complementary inputs, which helps to increase consumer surplus and welfare.However, it creates two more effects by increasing product differentiation in the final goods market.On the one hand, a higher product differentiation between the final goods increases local monopoly power of the final goods producers, which tends to reduce consumer surplus and welfare.On the other hand, a higher product differentiation tends to increase consumer surplus and welfare by increasing the market size.A merger decreases (increases) consumer surplus and welfare if the adverse effect of the higher monopoly power of the final goods producers dominates (is dominated by) the positive effects of the lower input prices and the higher market size.
Consider a situation where the final goods industry is matured so that higher product differentiation mainly increases monopoly power of the final goods producers and does not increase the market size since the value creation content of product differentiation is limited (as mentioned in Zanchettin and Mukherjee, 2017).In this situation, a merger of complements reduces consumer surplus and welfare if there are two complementary inputs and the final goods are sufficiently differentiated, which creates significant monopoly power of the final goods producers.
Although mergers of complements and vertical integration are different from the organisational point of view, the economic effects of these situations are very similar (Salop and Culley, 2015), as both situations help to reduce the input prices. 5Hence, our results can be compared with Matsushima (2009) and Zanchettin and Mukherjee (2017), which consider the effects of vertical integration on product differentiation and welfare.The result mentioned in the previous paragraph is similar to Matsushima (2009) and Zanchettin and Mukherjee (2017), where vertical integration reduces the input prices and welfare while increases product differentiation.
However, the welfare effects in our analysis differ significantly from those papers if all the input suppliers merge and there is either at least three complementary inputs or higher product differentiation between the final goods increases the market size for the final goods significantly.If there are at least three complementary inputs and all the input suppliers merge, consumer surplus is higher under merger compared to no merger (Proposition 1(iii)).If a higher product differentiation between the final goods increases the market size for the final goods significantly and all the input suppliers merge, consumer surplus is higher under merger compared to no merger (Proposition 2).Merger between the input suppliers also increase welfare in these situations.These results are in contrast to Matsushima (2009) and Zanchettin and Mukherjee (2017), and show the implications of the number of inputs, the number of merged input suppliers, and the market expansion effect of product differentiation. 6Hence, in a dynamic setup with endogenous product differentiation, the antitrust authorities may need to be more careful about mergers of complements.
We contribute to the literature in two ways.First, in contrast to the extant literature on the mergers of complements (which we review in the next section), we show that even if the merger reduces the input prices and solves the complements problems, it may reduce consumer surplus and welfare by increasing product differentiation.Second, although our analysis and the extant literature on vertical integration with endogenous product differentiation (Matsushima, 2009 andZanchettin andMukherjee, 2017) create similar effects on the input prices and product differentiation, the welfare effects in our analysis are significantly different from that literature if all the input suppliers merge and there is either at least three complementary inputs or higher product differentiation between the final goods increases the market size for the final goods significantly.Hence, the welfare effects and the policy implications change significantly depending on the number of inputs and the market expansion effect of product differentiation (which may vary from industry to industry as per their maturity).
The empirical motivation for our paper may come from the coexistence of mergers among the complementary input suppliers or complementary patent holders and product differentiation by the respective final goods producers.For example, smartphone companies purchase patents from complementary patent pools (Trappey et al., 2016;Galetovic et al., 2018) and adopt product differentiation strategies to reduce competition in the product market (Cecere et al., 2015).Trappey et al. (2016), pp.67) also provide a more direct piece of evidence of this relationship by mentioning, "Since so many patents are involved in the creation of a smartphone, most designers buy into patent pools to reduce the risk of infringement….Patent pools facilitate the reuse of SEPs [standard essential patents], which enables engineers to focus on the creation of new designs and new features (e.g., user interfaces and applications) that better differentiate the phone in the marketplace".
The co-existence of mergers among the complementary input suppliers or complementary patent holders and product differentiation occurs in other industries also.Firms in the pharmaceutical industry purchase patents from complementary patent pools (Osipchuk, 2018) and adopt strategies to differentiate their products (Kumar and Srivastava, 2013).
The remainder of the paper is organised as follows.Section 2 provides the literature review.Section 3 describes the basic model.Section 4 shows the results under Bertrand competition in the final goods market.Section 5 concludes.We show in the Online Appendix the implications of a merger among the complementary input suppliers under Cournot competition in the final goods market.

Literature review
As mentioned in the introduction, concerns over mergers among the complementary input suppliers or complementary patent holders are far from non-existence.Economides and Salop (1992) and Choi (2008) show that mergers among the producers of complementary products affect the consumers adversely by creating some forms of bundling.Vives and Staffiero (2009) discuss the problems of bundling and market foreclosure when analysing the merger between General Electric and Honeywell, producing complementary products.See also Motta (2004) and Spulber (2017) on this issue.As mentioned in Etro (2019), "a proposed merger between Qualcomm and NXP has been cleared by the European antitrust authorities under conditions aimed at avoiding risks of foreclosure for actual and potential rivals".Etro (2019) shows that a merger among the complementary input suppliers (called "mergers of complements" from now on) increases R&D investments of the merged firms but reduces R&D investments of the potential input suppliers, which, in turn, may make the consumers worse off. 7Galetovic and Gupta (2020) point out that royalty stacking is not a serious issue in the presence of within-functionality competition, which allows competition from alternative technologies even after setting up a standard.Llanes and Poblete (2014) show that complementary patent pools may reduce welfare if they reduce quality of the patented technology by reducing the number of firms involved in setting the technology standard.Choi and Gerlach (2015) show that complementary patent pools 5 Horizontal mergers are the mergers among firms competing at the same level of production or distribution and vertical mergers combine firms at different levels of production or distribution (Salop and Culley, 2015).Hence, from the organisational point of view, we consider horizontal mergers, and Matsushima (2009) and Zanchettin and Mukherjee (2017) consider vertical mergers.However, the economic effects of the merger in our analysis are very similar to the economic effects of mergers in those papers, as mergers in all these papers reduce the input prices and increase product differentiation.Since the economic effects of mergers of complements are very similar to the economic effects of vertical integration, sometime mergers of complements are described as vertical mergers (Quint, 2014).We thank a referee for bringing this point to us. 6In the next section, we compare with Matsushima (2009) and Zanchettin and Mukherjee (2017) in more detail. 7See Garcia and de Azevedo (2019) for concerns on conglomerate mergers.
may reduce social welfare if the patents are sufficiently weak.Reisinger and Tarantino (2019) show that a patent pool among the perfectly complementary patent holders can be anticompetitive in the presence of vertical integration among the patent owners and the final good producers if the share of the integrated firms is significant.
In contrast to the above-mentioned papers, we provide a new reason based on endogenous product differentiation by the final goods producers for the adverse welfare effects of the mergers of complements.
Complementary patent pools are often encouraged due to their beneficial effects on non-production activities, such as innovation, which helps to increase production efficiency.For example, as documented in Heller and Eisenberg (1998), USPTO (2001), WHO (2006), UNITAID (2008) and Cox (2012), patent pools of fundamental technologies make it easier in the biotechnology industry to develop enduser products or services.Langinier (2015) also supports the viewpoint by showing that a patent pool of basic innovations is helpful for follow-up innovations by the developers, especially in biotechnology.
These findings are similar to ours in the sense that a patent pool encourages a final goods producer to invest more in a non-production activity like product differentiation.However, our analysis shows that if the higher investment in the non-production activities increases monopoly power of the final goods producers, complementary patent pools may not be good for the consumers and the society.
To the best of our knowledge, there are two other papers considering the welfare effects of complementary patent pools in the presence of product differentiation.Gallini (2014) considers a situation where owners of essential patents develop a new product, which competes with a substitute product that is produced by one of the owners of the essential patents.These products are produced in perfectly competitive markets and the owners of the essential patents set the standard of their products to horizontally differentiate their products from the substitute product.A higher product differentiation creates complete market expansion in that paper.In such a framework, it shows that complementary patent pools increase product differentiation and welfare.
Our paper differs from that paper in some important ways.First, unlike that paper, we consider only products which are produced with the essential patents and product differentiation is among these products.Second, no patent owners are the producers of any products.Third, we show the implications of zero and complete market expansion following product differentiation.Finally, there is imperfect competition in the product market.Like that paper, we find that a complementary patent pool increases product differentiation.However, unlike that paper, we show that a complementary patent pool may reduce consumer surplus and welfare.Different economic environment and different market expansion following a higher product differentiation makes our results different from Gallini (2014).
Jeitschko and Zhang (2014) consider a situation where they assume that a patent pool reduces product differentiation since the patent holders share more information with the final goods producers, thus, making the products more similar.Like Gallini (2014), a higher product differentiation creates complete market expansion in their paper.They further consider that the final goods producers invest to increase the base demand and the final goods producers benefit from competitors' investments due to knowledge spillover.In this framework, they show that complementary patent pools may reduce welfare if knowledge spillover is large.
We differ from their paper in some important ways.First, unlike them, we do not assume that patent pool reduces product differentiation.Instead, we determine how patent pool affects product differentiation, and unlike them, we find that it increases product differentiation.Second, unlike them, we show that complementary patent pools may create adverse effects on the consumers and welfare in the absence of knowledge spillover.
Our paper was motivated by the extant literature on the mergers of complements.In contrast to that literature, we show that endogenous product differentiation in the final goods market significantly changes the welfare effects of the mergers of complements.Due to the similar input-price and product differentiation effects of the mergers of complements and vertical integration, our paper is also related to Matsushima (2009) and Zanchettin and Mukherjee (2017), which consider the effects of vertical mergers on product differentiation and welfare.However, our paper differs significantly from those papers.Matsushima (2009) considered a Hotelling model with inelastic demands and duopoly market structure in the input market and in the final goods market.In contrast, we consider elastic demands with oligopoly in both the input and final goods markets.Hence, unlike Matsushima (2009), we can also show the effects of the market structure, the number of the merged entities, and the market expansion effects of product differentiation.Further, Matsushima (2009) considers price competition among the final goods producers, while we consider the effects of both price and quantity competition in the final goods market.
Although Zanchettin and Mukherjee (2017) considered elastic demand functions like us, they considered a duopoly final goods market, and monopoly input supplier and duopoly input suppliers with substitutable inputs.Hence, unlike our paper, they neither considered complementary inputs nor showed the effects of the market structure, the number of the merged entities, and the market expansion effects of product differentiation.They were interested to examine how competition for vertical integration allows rent extraction by the input supplier, which affects the final goods producers' incentive for product differentiation and welfare.In contrast, like the literature on the mergers of complements, our purpose is to show how the mergers among the complementary input suppliers affect rent extraction by the input suppliers, which affects product differentiation by the final goods producers and welfare.In this respect, we show the implications of the market expansion following product differentiation, the number of merged input suppliers and the intensity of competition.Hence, our paper and Zanchettin and Mukherjee (2017) focus on two different aspects.While their results are not influenced by the mergers of complements, our results are not influenced by the competition for vertical integration.
Not only the structure but, as mentioned in the introduction, our results also differ from those papers.For example, in Matsushima (2009), if the costs of supplying inputs are low, which is similar to our normalisation of zero cost of input production, the mergers can increase consumer surplus but reduce welfare.In contrast, we show that a merger may increase or decrease the consumer surplus and welfare depending on the market expansion effect, the market structure, and the number of the merged entities.Zanchettin and Mukherjee (2017) show under strong market expansion effect that the threat of merger decreases the incentive for product differentiation under price competition but it may increase or decrease the incentive for product differentiation under quantity competition.In contrast, merger in our paper increases the incentive for product differentiation under both price and quantity competition.While the lower product differentiation decreases consumer surplus and welfare in their paper, higher product differentiation in our analysis may increase or decrease consumer surplus and welfare.
If there is no market expansion effect, Zanchettin and Mukherjee (2017) find that the threat of merger increases product differentiation.While higher product differentiation decreases welfare in their paper, higher product differentiation in our paper under no market expansion effect may increase or decrease welfare.
In general, our paper falls in the area of innovation in a vertical structure.There is a strand of literature showing the dynamic effects of patent pools or cooperation among the complementary input suppliers due to innovation by the patent holders or complementary input suppliers (Lampe and Moser, 2010;Gilbert and Katz, 2011;Joshi and Nerkar, 2011;Llanes and Trento, 2012;Dequiedt and Versaevel, 2013).Unlike these papers, we consider investments in product differentiation by the final good producers.
There is a literature showing how patent trolls or non-practising T.-D.Han and A. Mukherjee patent holders create the hold up problem (see, e.g., Geradin et al., 2011 and the references therein).In contrast, there are no patent trolls in our analysis.
There is another strand of literature showing the effects of cooperation among the substitutable input suppliers in the presence of process innovation by the final good producers (Calabuig and Gonzalez-Maestre, 2002;Haucap and Wey, 2004;Manasakis and Petrakis, 2009;Mukherjee and Pennings, 2011).8Unlike these papers, our focus is on complementary input suppliers.Han and Mukherjee (2017) show that cooperation among the complementary labour unions may create the adverse welfare effects by affecting investment in process innovation, which is different from the effects of product differentiation and the market size expansion considered in this paper.Further, they considered duopoly in the input market and in the final goods market.Hence, that paper neither showed the effects of a merger among the subset of input suppliers nor the effects of competition in the input market and in the final goods market.
Protecting consumers is one of the main goals of antitrust authorities (Motta, 2004).However, it is quite intriguing that for various reasons, antitrust authorities can be blinded by some incidences, which apparently look beneficial for consumers.This echoes related concerns found in other recent papers in different contexts.For example, Brito and Catalão-Lopes (2019) show that even when a merger exhibits significant cost-saving synergy, consumers can still be made worse off.Mukherjee and Wang (2012) and Stuck (2013) show that competition is not necessarily enhancing the well-being of consumers.Houba et al. (2018) propose a novel insight showing that current antitrust approach can be refined to enhance welfare and effectiveness of antitrust rules.We show another case where mergers of complementary input suppliers solve the complements problem but they may not make the consumers and the society better off.

The basic model
Consider an industry with n(≥ 2) final good producers, firms x 1 , …, x n , competing with homogeneous products.Assume that each firm invests I to differentiate the product horizontally.We assume that the firms invest cooperatively to create horizontal product differentiation, as in Gallini (2014) and Zanchettin and Mukherjee (2017).
The reason for considering cooperative investment to create horizontal product differentiation is to avoid the non-cooperative investment game, which may not allow us to use a symmetric demand structure that helps to convey our message in the easiest way.Under non-cooperative investments for product differentiation, only a subset of final goods producers may investment to create product differentiation.It also follows from Lambertini and Rossini (1998) that the noncooperative investment game may create the problem of prisoner's dilemma as horizontal product differentiation by one final goods producer creates positive externalities on other final goods producers.Since cooperative investment helps to consider a symmetric demand structure for our analysis, we avoid the non-cooperative investment game, which is not important for our main purpose.
We assume that the final goods production requires m(≥ 2) perfectly complementary inputs.For simplicity, assume that each final goods producer requires one unit of the input to produce one unit of the final goods.The analysis will be similar if we have considered m perfectly complementary patents rather than inputs.However, we will call them as inputs throughout the paper.
Each input is produced by a single input producer.Hence, the ith input is produced by firm y j , j = 1, …, m.For simplicity, we normalise the marginal costs of input production to 0. However, assume that each input supplier incurs a fixed cost T > 0 when there is no merger.The presence of economies of scale in the input sector is well-known (Besanko et al., 2013) and T > 0 ensures it.We assume that a merger of k input suppliers allows the merged firm to save on the fixed cost.Hence, we assume that the merged firm incurs the fixed cost T ′ (k), which is less than kT, while each of the non-merged firm incurs the fixed cost T.
For simplicity, we assume that T ′ (k) = T, i.e., the merged firm incurs only one fixed cost to produce k inputs.This type of fixed cost saving is similar to Davidson and Mukherjee (2007).This may happen if, say, the merged firm can produce all their inputs in one plant for which it incurred the fixed cost T.Alternatively, if the fixed cost represents the cost of contracting with the final goods producers, the merged firm can write a single contract with the final goods producers for all their inputs, thus incurring only T. 9The assumption of a positive T is not important for the effects of a merger on the input prices, product differentiation by the final goods producer and consumer surplus.However, a suitable value of T will help us to ensure that merger among the input suppliers always occurs.Our purpose is not to examine the profitability of a merger among the input suppliers but to see the effects of a merger among the input suppliers on the input prices, product differentiation by the final goods producers and consumer surplus.Hence, we assume T > 0, and consider an appropriate value of T to ensure a profitable merger among the input suppliers.
We will consider two scenarios: No merger: Here all input suppliers decide their prices simultaneously and non-cooperatively to maximise own profits.
Merger of complements: Assume that k input suppliers merge, where 2 ≤ k ≤ m.The merged firm determines their input prices to maximise the total profits of the merged input suppliers.Each of the nonmerged firms determines the price of its input to maximise its own profit.The merged and the non-merged firms determine the respective input prices simultaneously.If k = 1, it will be equivalent to the situation under no merger.
We consider a representative consumer's utility function as: where q i is the output of firm x i .The term ξ represents the numeraire good and its price is normalised to 1.The parameter g ∈ [0, 1] represents the degree of product differentiation.If g = 1, the final goods are perfect substitutes, and if g = 0, they are isolated.The market expansion brought on by a higher product differentiation is denoted by the parameter s ∈ [0, 1].As shown below, product differentiation does not affect the market size s = 1 but a higher product differentiation expands the market size completely for s = 0.As s falls, a higher product differentiation increases the market size more.The above-mentioned utility function generates the following inverse demand function for the ith final goods producer: (2) To understand how s captures market expansion following product differentiation, add the inverse demand functions, which gives , the total output is independent of g.Hence, the demand function is like Shubik and Levitan (1980) for s = 1, where product differentiation does not affect the market size.If s = 0, we get a demand function like Bowley (1924).In case of s = 0, we get Q = [1 + g(n − 1)] − 1 n(1 − P), i.e., a higher product differentiation increases the market size completely.As s reduces from 1 to 0, the term (1 + s(n − 1)(1 − g)) reduces from (1 + (n − 1)(1 − g)) to 1. Thus, s helps to capture a wide range of demand functions from Shubik and Levitan (1980) to Bowley (1924), depending on the extent of market expansion. 10 The corresponding demand functions are: ] . (3) We consider the following game: Stage 1: Conditional on no merger and merger among the input suppliers, the final goods producers decide cooperatively whether to invest to create product differentiation g ∈ [0, 1) among the final goods.If the final goods producers decide not to invest, the final goods will be homogeneous.If the final goods producers decide to invest, each firm invests I, thus generating total investment of nI, which creates the symmetric product differentiation g ∈ [0, 1) between all the final goods.This type of binary choice of investment and no investment is often used in the literature for analytical simplicity (see, e.g., Gallini and Winter, 1985;Bester and Petrakis, 1993;Lambertini and Rossini, 1998;Calabuig and Gonzalez-Maestre, 2002;Haucap and Wey, 2004;Mukherjee and Pennings, 2011;Zanchettin and Mukherjee, 2017;Panagopoulos and Park, 2018).As mentioned in Zanchettin and Mukherjee (2017), we can view that the cost of differentiation is a L-shaped convex function with the property that firms can differentiate up to g by investing I and the cost of differentiation is prohibitive after that.
Stage 2: Conditional on merger and investments in product differentiation, the input prices are determined.
Stage 3: The final goods producers determine the respective prices or outputs simultaneously and the profits are realised.
We solve the game through backward induction.
Like the papers examining the effects of mergers between the input suppliers on the investments by the final goods producers (see, e.g., Calabuig and Gonzalez-Maestre, 2002;Haucap and Wey, 2004;Mukherjee and Pennings, 2011;Han and Mukherjee, 2017), we also consider a game structure where the merger decision precedes investment that precedes the input price determination. 11The difference between those papers and our paper is that we consider investment in product differentiation, while they consider investment in process innovation.Further, we consider mergers among the complementary input suppliers whereas those papers, except (Han and Mukherjee, 2017), consider substitutable input suppliers.
Our analysis follows the tradition of ex-post (or short-term) bargaining (see, e.g., Ulph and Ulph, 1994), where the input suppliers cannot commit to the input prices before the investments made by the firms.In other words, investments in product differentiation are more irreversible than the choice of the input prices.
We will consider that an input supplier charges the same input price to all final goods producers.Given the symmetric final good producers we are considering, it is natural that we will get the same input prices in equilibrium.Moreover, the literature on patent pools (Shapiro, 2001;Lerner and Tirole, 2004;Aoki and Nagaoka, 2004;Schmidt, 2008;Choi, 2010, to name a few) provides the justification for non-discriminatory linear prices charged by the input suppliers, which are supported by empirical evidence.Layne-Farrar and Lerner (2011) show that linear prices are used by all the patent pools they investigated.

Bertrand competition
If k input suppliers merge, the gross profit functions of the ith final goods producer is π i = (P i − ∑ k u=1 w u − ∑ m v=k+1 w v )q i , where g = 1 under no investment by the final goods producers and 0 ≤ g < 1 under investments by the final goods producers.
Due to the well-known Bertrand paradox, we do the analysis for 0 ≤ g < 1 and g = 1 separately.
If the firms invest in product differentiation, i.e., 0 ≤ g < 1, we get the price charged by the ith final goods producer as: (4) The corresponding output of the ith final goods producer is ) . (5) The total profit of the merged input suppliers and the profit of the vth non-merged input supplier are respectively The merged input suppliers choose the values of w u to maximise (6) and the vth non-merged input supplier determines w v to maximise (7).If k = m, (7) is not relevant, since there will be no non-merged input suppliers.The equilibrium input prices can be found as These input prices are valid for 1 ≤ k ≤ (m − 1).If k = m, the input prices are w u = 1 2m .The equilibrium input prices are independent of g and s, and the reason follows from Dhillon and Petrakis (2002). 12 If the firms invested in product differentiation, given the equilibrium 10 One may look at Choné and Linnemer (2020), which review the development of the linear demand systems following the introduction of a quasilinear quadratic utility model as the foundation of a linear demand system by Levitan and Shubik in the 1960s. 11Etro (2019) also considers a similar sequence of moves but the input owners invest in that paper.
12 Dhillon and Petrakis (2002) show that the price charged by an industrywide input supplier does not depend on the market parameters, such as the number of firms, the degree of product substitutability, and the intensity of market competition, if the firm's equilibrium output and profit are log-linear in the price charged by the input supplier and the market parameters.
input prices, the equilibrium net profits of the ith final goods producer are shown by Eq. ( 9) given in Box I.The equilibrium profits of the merged input suppliers are shown by Eq. ( 10) given in Box II.
The equilibrium profit of each of the non-merged input supplier is shown by Eq. ( 11) given in Box III.
The equilibrium consumer surplus is shown by Eq. ( 12) given in Box IV.Now consider the situation under no investment by the final goods producers.Each final goods producer pays the total input price ∑ k u=1 w u + ∑ m v=k+1 w v .The corresponding output of the ith final goods producer is The total profit of the merged input suppliers and the profit of the vth non-merged input supplier are respectively The merged input suppliers choose the values of w u to maximise ( 13) and the vth non-merged input supplier determines w v to maximise ( 14).
If k = m, (14) is not relevant, since there will be no non-merged input suppliers.
The equilibrium input prices can be found as These input prices are valid for 1 ≤ k ≤ (m − 1).If k = m, the input prices are w u = 1 2m .Given the equilibrium input prices, the equilibrium profit of each final goods producer is 0. The equilibrium profits of the merged input suppliers are The equilibrium consumer surplus is If k = 1, the above mentioned equilibrium values correspond to the situation of no merger among the input suppliers.

The effects of a merger on the input prices
It follows from (8) that if there is a merger among the input suppliers, the prices of the merged inputs fall compared to the situation with no merger, as . Since the inputs are complements, a lower input price increases the demand and profit of that input supplier and the other input suppliers.The merger allows the merged input suppliers to internalise such an effect, which, in turn, encourages them to reduce their input prices compared to no merger.Thus, merger helps to solve the complements problems for the merged inputs.
However, a merger of some input suppliers allows the non-merged input suppliers to increase their input prices compared to no merger.Lower input prices charged by the merged input suppliers help to increase the demand and the profitability of the non-merged input suppliers, which encourage them to increase their input prices.
While looking at the total input prices, we find that the total input price is , which decreases under merger compared to no merger.Further, it decreases more as the number of merged firms increases.Since the total input prices are the marginal costs of the final goods producers, a merger among the complementary input suppliers helps the final goods producers by reducing the marginal costs for the final goods.Further, this benefit increases as more input suppliers merge.
The following lemma summarises the above discussion.
Lemma 1.Compared to no merger, a merger among the input suppliers decreases the prices of the inputs produced by the merged suppliers, increases the prices of the inputs produced by the non-merged suppliers and decreases the total input prices.The total input prices decrease more as more input suppliers merge.

The effects of a merger on the incentive for product differentiation
If the final goods producers do not invest in product differentiation, the profit of the ith final goods producer is 0. Hence, if k input suppliers merge, due to symmetry, the final goods producers invest in product differentiation if π i > 0 or the condition shown in ( 18) given in Box V. Condition (18) given in Box V shows the ith final goods producer's maximum willingness or the incentive to invest in product differentiation.If k reduces, it reduces Ī, implying that the ith final goods producer's incentive for investment is higher under merger compared to no merger (suggesting k = 1) among the input suppliers.A lower marginal cost for the ith final goods producer under merger compared to no merger among the input suppliers gives the final goods producers a higher incentive for product differentiation under the former than the latter.
The following result is immediate from the above discussion.

Lemma 2. A merger among the input suppliers increases the incentive for product differentiation compared to no merger among the input suppliers.
If k input suppliers merge, the above discussion suggests that the final goods producers invest in product differentiation irrespective of a merger among the input suppliers if I < Ī(k = 1), but does not do so irrespective of a merger among the input suppliers if I > Ī(k), and they invest in product differentiation only under a merger among the input suppliers if Ī(k = 1) < I < Ī(k).

Merger profitability
. We assume that the values of T are such that these conditions hold.

It is worth noting that
hold for any k, g and s.So, the fixed cost saving through a merger is important for creating a profitable merger in our analysis.As mentioned above, since our purpose is not to analyse merger profitability among the input suppliers but to show the implications of a merger among the input suppliers, we assume that the values of T are such that the merger is profitable.
T.-D.Han and A. Mukherjee

The effects of a merger on the consumer surplus
Before going to analyse the effects of a merger among the input suppliers, let us first see how the consumer surplus is affected by the total input price and product differentiation.Given the utility function in (1), the inverse demand functions in (2) and the equilibrium outputs in (5), the equilibrium consumer ) , which decreases with the total input price, ∑ k u=1 w u + ∑ m v=k+1 w v , but may increase or decrease with product differentiation, i.e., a lower g. 13 Since a merger among the input suppliers reduces the total input price (as shown in Lemma 1), Box IV 13 For example, ∂CS ∂g T.-D.Han and A. Mukherjee which increases the output q and reduces the prices of the final goods, such an effect tends to increase consumer surplus.On the other hand, since a merger among the input suppliers increases the incentive for product differentiation by the final goods producers (as shown in Lemma 2), it may decrease or increase consumer surplus by reducing g.Hence, a merger among the input suppliers may create opposing or similar effects on the consumer surplus by reducing the total input price and reducing g.These effects will determine the results shown in this subsection.
The following result is immediate from the above discussion.
Lemma 3. A reduction in the total input price tends to increase the consumer surplus, while an increase in product differentiation indicated by a reduction in g tends to decrease or increase the consumer surplus.Hence, (i) if a lower g increases the consumer surplus, a merger among the input suppliers increases the consumer surplus by reducing the total input price as well as g, but (ii) if a lower g decreases the consumer surplus, a merger among the input suppliers decreases the consumer surplus if the positive effect of a lower total input price on the consumer surplus is dominated by the adverse effect of a lower g on the consumer surplus.
It is easy to see that if the final goods producers either invest or do not invest in product differentiation under both merger and no merger among the input suppliers (i.e., I < Ī(k = 1) or I > Ī(k) respectively), the consumer surplus is higher under merger than under no merger among the input suppliers.This happens since the merger does not affect product differentiation but reduces the marginal costs of the final goods producers, which helps to reduce the prices of the final goods.Now we consider the situation where the final goods producers invest in product differentiation only under a merger among the input suppliers, i.e., Ī(k = 1) < I < Ī(k).
If the final goods producers invest in product differentiation under a merger of k input suppliers, we get the equilibrium consumer surplus from ( 12) is given in Box IV as However, if the final goods producers do not invest in product differentiation under no merger among the input suppliers, we get the equilibrium consumer surplus from (17) as Hence, a merger among the input suppliers decreases (increases) the consumer surplus if CS * (k ≥ 2, g < 1) < (>)CS * (k = 1, g = 1).For our analysis, we will focus on two special cases: s = 1 and s = 0.
First, consider the case of The shaded (white) region in Fig. 1 shows that the merger decreases (increases) the consumer surplus compared to no merger if g is suffi- for n ≥ 2 and g ∈ [0,1], more final goods producers increase the possibility of a higher consumer surplus under merger compared to no merger.A merger among the input suppliers increases cost efficiency in the final goods market by reducing the marginal costs of the final goods producers, and the presence of more final goods producers increases the beneficial effects of higher production efficiency.
It follows from Fig. 1 that a merger of complements may reduce the consumer surplus even if there are two final goods producers.If there are two final goods producers, differentiation between the products occurs if at least one of them invests to create differentiation.Hence, our assumption of cooperative investment to create differentiation is not crucial for our result, but that assumption helps us to work with a symmetric demand system.Now consider the case where m ≥ 3 and 2 ≤ k ≤ m.It is immediate from ( 19) that if k increases, i.e., (mk) decreases, it increases ΔCS * (s = 1), implying that as the number of merged firms increases, the possibility of a higher consumer surplus under merger compared to no merger increases.As k increases, it reduces the total input prices and therefore, the marginal costs of the final goods producers more, as shown in Lemma 1.This benefit from a merger of more input suppliers increases the possibility of a higher consumer surplus under merger compared to no merger as the number of merged input supplier increases. If ) . This expression increases with m and is positive at m = 3, implying that if there are three input suppliers and all of them merge, the consumer surplus is higher under merger compared to no merger.When there are more input suppliers, a merger among them helps to increase the cost efficiency of the final goods producers more by reducing the prices of more inputs.As a result, a merger of all input suppliers increases the consumer surplus compared to no merger.However, if 2 ≤ k < m when m ≥ 3, a merger among the input suppliers may decrease the consumer surplus compared to no merger.As an example, consider the case of m = 3 and k = 2.We get )〈 0 for g < 2n 3n− 1 .The following proposition summarises the above discussion.
(i) Compared to no merger, a merger among the complementary input suppliers reduces the consumer surplus for sufficiently lower values of g if there are either two input suppliers or a subset of input suppliers undertakes merger if the number of input suppliers is more than two.
(ii) An increase in the number of merged input suppliers increases the possibility of a higher consumer surplus under merger compared to no merger among the input suppliers.
(ii) The consumer surplus is always higher under merger compared to no merger if all the input suppliers merge and the number of input suppliers is at least 3.
(iv) Higher competition in the final goods market (i.e., an increase in the number of final goods producers) increases the possibility of a higher consumer surplus under merger compared to no merger.T.-D.Han and A. Mukherjee Now we consider the case of s = 0. Defining ΔCS * (s = 0) = CS * (s = 0, k, g < 1) − CS * (s = 0, k = 1, g = 1), we get ) n > 0 for g ∈ [0, 1) and n ≥ 2.
Now consider the situation with m ≥ 3.If k = m, we get from ( 20) n.This expression increases with m and is positive at m = 3, implying that if s = 0, merger among all the input suppliers always increases consumer surplus compared to no merger.The reason for this effect is similar to the reason mentioned for the case of s = 1.
Finally, consider the case of m ≥ 3 and 2 ≤ k < m.It is immediate from (20) that if k increases, i.e., (mk) decreases, it increases ΔCS * (s = 0), implying that as the number of merged input suppliers increases, the possibility of a higher consumer surplus under merger compared to no merger increases.Hence, a consumer surplus reducing merger among a subset of input suppliers will occur only if the merger reduces consumer surplus compared to no merger for k = 2; otherwise, the merger will always increase consumer surplus.If k = 2, we get from (20) ΔCS * (s = 0, ) n, which we plot in Fig. 2 for n = 3.The shaded (white) region in Fig. 2 shows that the merger decreases (increases) the consumer surplus compared to no merger if g is sufficiently low (high).Hence, a merger among a subset of input suppliers may reduce the consumer surplus compared to no merger if s = 0.
The reason for this result is as follows.If s = 0 and n = 2, we get the consumer surplus as CS = ( . Hence, as mentioned in Lemma 3, on the one hand, merger helps to increase consumer surplus by reducing the total input price.However, on the other hand, a lower g tends to reduce consumer surplus.If g is sufficiently small, the effect of product differentiation dominates the effect of total input price and the merger reduces the consumer surplus compared to no merger.
The following proposition summarises the above discussion.
Proposition 2. If s = 0, a merger may reduce the consumer surplus compared to no merger only if a subset of input suppliers undertakes merger.

Welfare implications
If a merger of k input suppliers is profitable, it is easy to understand that welfare will be higher under merger compared to no merger if the final goods producers either invest or do not invest irrespective of merger.This happens since merger reduces the total input prices without affecting product differentiation among the final goods, thus increasing the net profits of the input suppliers and final goods producers, and increasing consumer surplus.
However, a merger among the input suppliers may reduce welfare if product differentiation occurs only under merger, which happens for Ī(k = 1) < I < Ī(k).As shown in the previous subsection, in this situation, a merger may reduce consumer surplus, thus creating a trade-off between higher profits of the merged input suppliers and final goods producers and lower consumer surplus.Hence, it is intuitive that if the loss of consumer surplus is more than the gain the profits, the merger reduces welfare.We will use examples here to show the welfare reducing merger. Assume , n = m = k = 2, we get the equilibrium welfare difference under "a profitable merger with product differentiation" and "no merger with no product differentiation" as The shaded (white) area in Fig. 3 shows that the merger decreases (increases) welfare.If s is large (i.e., market expansion following product differentiation is very small), the merger decreases (increases) welfare compared to no merger among the input suppliers if g is low (high) (i.e., the final products are very (not very) differentiated).On the other hand, if s is small, the merger increases welfare compared to no merger among the input suppliers for g ∈ , n = 2, m = 5 and k = 2, we get the equilibrium welfare difference under "a profitable merger with product differentiation" and "no merger with no product differentiation" as We plot ΔW * = W * (k = 2, g < 1) − W * (k = 1, g = 1) in Fig. 4(a) for g ∈ [0, 1] and s ∈ [0, 1].
The shaded (white) area in Fig. 4(a) shows that the merger decreases (increases) welfare.If s is large (i.e., market expansion following product differentiation is very small), the merger decreases (increases) welfare compared to no merger among the input suppliers if g is low (high) (i.e., the final products are very (not very) differentiated).On the other hand, if s is small, the merger increases welfare compared to no merger among the input suppliers for g ∈ [0, 1].If s is moderate (around s = 0.33), merger decreases (increases) welfare compared to no merger among the input suppliers if g is moderate (very small or not very small) -this case is shown more clearly in Fig. 4(b), where the shaded (white) area shows that the merger decreases (increases) welfare.
The reasons for the above findings are similar to the reasons discussed under consumer surplus.Put it simply, there are three effects.One is due to the reduction in the input prices.The other two are owing to a higher product differentiation.On the one hand, a merger helps to increase welfare by reducing the total input prices.On the other hand, if the merger increases product differentiation, it creates two opposing effects on welfare: (1) increased differentiation tends to reduce welfare by increasing the local monopoly power of the final goods producers, and (2) increased differentiation tends to increase welfare by increasing the market size.The interactions of these effects create the abovementioned results.
If the market expansion effect is very weak (i.e., s is large), the adverse effects of a higher monopoly power of the final goods producers following the merger dominates (are dominated by) the other two positive effects of merger if g is low (high), which creates a significant (small) loss from a higher monopoly power.
If the market expansion effect is very strong (i.e., s is small), the adverse effect of a higher monopoly power of the final goods producers following the merger is always dominated by the other two positive effects of the merger.We find the above cases in Figs. 3 and 4(a).However, we get another case in Fig. 4(a).If the market expansion effect is moderate, a higher monopoly power of the final goods producers following the merger dominates the other two positive effects of the merger if the final goods are moderately differentiated so that the monopoly power is sufficiently strong but the market size does not increase much.14However, if the final goods are not very differentiated, it does not increase the monopoly power significantly, or if the final goods are very much differentiated, it expands the market size significantlyboth these effects reduce the relative adverse effects of a higher product differentiation significantly to make the merger welfare improving compared to no merger.
The above discussions show that if the final goods producers invest in product differentiation only under a merger among the complementary input suppliers, the merger may reduce welfare compared to no merger if the market expansion effect is not very high, i.e., s is not close to 0.
It can be argued that the merger will not reduce welfare even if there is no market expansion effect, i.e., s = 1, but there are at least three complementary inputs and all the input suppliers merger.This happens for the following reason.We have seen in Proposition 1(iii) that merger increases the consumer surplus in this situation.If product differentiation is profitable only under merger, it implies that the final goods producers earn more under merger and product differentiation compared to no merger and no product differentiation. 15Hence, whenever merger is profitable, it increases welfare compared to no merger.
It follows from Figs. 3 and 4(a) that a merger of complements may reduce the welfare in the presence of two final goods producers.In the case of two final goods producers, differentiation occurs if at least one of them invests to create differentiation, suggesting that our assumption of cooperative investment to create differentiation is not crucial for our result.
We have considered in the above analysis that the final goods producers compete like Bertrand oligopolists.We show in the Online  Appendix the implications of a merger among the complementary input suppliers under Cournot competition in the final goods market.The analysis in the Online Appendix shows that our result of a welfare reducing merger among the complementary input suppliers may occur even under Cournot competition among the final goods producers.

Conclusion
It is believed that mergers among the complementary producers or patent holders benefit the consumers and the society by reducing the input prices, suggesting that the antitrust authorities may not need to be concerned about mergers of complements.We show that such a belief should be treated with a caution since mergers of complements may make the consumers and the society worse off compared to no merger by increasing product differentiation in the final goods market, even if the mergers reduce the total input prices.This can happen under both Bertrand and Cournot competition in the final goods market.We show the implications of market expansion following product differentiation, competition in the final goods and input markets, and the number of merged input suppliers for our results.
Our results provide important insights for the policymakers and suggest that, in a dynamic setup, where mergers of complements increase monopoly power of the final goods producers by increasing product differentiation in the final goods market, the antitrust authorities may need to be more careful about the effects of mergers of complements on the consumers and the society.Thus, our paper contributes to the growing literature showing concerns about the welfare effects of mergers of complements.
As explained, our paper and the papers considering vertical integration in the presence of endogenous product differentiation in the final goods market have similar effects on the input prices and product differentiation.However, we have shown that our results differ from that literature depending on the number of inputs and the market expansion effect following product differentiation.
Future research can extend our analysis in the following ways.We have considered a situation like ex-post (or short-term) bargaining where the input suppliers cannot commit to the input prices before the investments made by the firms.Hence, an extension of our paper will be to consider the case where the input suppliers commit to the input prices before the investments made by the firms.In this situation, depending on the input suppliers' profitability under product differentiation and no product differentiation in the final goods market, the input suppliers will set the input prices to influence the investment decisions of the final goods producers so that those decisions are consistent with the input suppliers' preference for product differentiation in the final goods market.If the final goods producers bargain with the input suppliers for the input prices, one may also consider ex-ante (or long-term) bargaining (see, e.g., Ulph and Ulph, 1998) where the bargaining involves both the input prices and the investment decisions.
Like the papers on mergers of complements mentioned in Section 2, we considered the effects of mergers among the complementary input suppliers or complementary patent holders only.However, there are situations where mergers occur among the complementary input suppliers but some final goods producers can also use those inputs independently.Hence, another extension of our paper will be to consider mergers among the input suppliers as well as between the input suppliers and the final goods producers.In this situation, the effects shown in our paper and the effects due to competition for vertical mergers shown in Zanchettin and Mukherjee (2017) will be present to determine the final outcome.