Marie-Laure Allain
email: marie-laure.allain@polytechnique.edu

Claire Chambolle
email: claire.chambolle@polytechnique.edu

Patrick Rey

We Thank

John Asker

Susan Athey

Eric Avenel

Aaron Edlin

Joe Farrell

Joshua Gans

Rich Gilbert

Jerry Hausman

Ken Heyer

Louis Kaplow

Michael Katz

Kai-Uwe Kuhn

Volker Nocke

Eric Ramseyer

Mike Riordan

Bill Rogerson

Nicolas Schutz

Marius Schwartz

Glen Weyl

Mike Whinston

Lucy White

Dennis Yao

Vertical Integration, Innovation and Foreclosure 1 Preliminary draft

Keywords: L13, L41, L42 Vertical Integration, Foreclosure, Innovation, Imitation, Firewall

This paper studies the potential effects of vertical integration on downstream firms' incentives to innovate. Interacting efficiently with a supplier may require information exchanges, which raises the concern that sensitive information may be disclosed to rivals. This may be particularly harmful in case of innovative activities, as it increases the risk of imitation. We show that vertical integration exacerbates this threat of imitation, which de facto degrades the integrated supplier's ability to interact with unintegrated competitors. Vertical integration may thus lead to input foreclosure, thereby raising rivals' cost and limiting both upstream competition and downstream innovation. A similar concern of customer foreclosure arises in the case of downstream bottlenecks.

Introduction

In this paper, we investigate whether vertical integration may trigger input foreclosure through a risk of information leakage and imitation. Efficiency reasons may require firms to exchange sensitive information with their suppliers, which raises the concern that this information can then be disclosed to rivals. 1 Vertical integration exacerbates this concern, since an integrated supplier can be more tempted to pass on such information to its downstream subsidiary. This issue is particularly serious in the case of innovative activities, as it creates a risk of imitation and thus tends to make the integrated supplier less reliable when dealing with downstream rivals. In other words, vertical integration may result in input foreclosure, not because the integrated firm will refuse to supply unaffiliated rivals but simply because it becomes less reliable. 2 As a result, vertical integration strengthens the market power of alternative suppliers, thereby "raising rivals' costs" and impeding innovation. 3 This issue is a growing concern for the European Commission, who mentions for example in its recent Guidelines on the assessment of non horizontal mergers: "The merged entity may, by vertically integrating, gain access to commercially sensitive information regarding the upstream or downstream activities of rivals. For instance, by becoming the supplier of a downstream competitor, a company may obtain critical information, which allows it to price less aggressively in the downstream market to the detriment of consumers. It may also put competitors at a competitive disadvantage, thereby dissuading them to enter or expand in the market." 4 This issue has also been raised in a number of merger cases. 5

1 Asker and Ljungqvist (2010) show for instance that such concerns impact the choice of firms with respect to their investment banks, as firms appear reluctant to use the same investment bank as their direct competitors.

2 While we focus here on input foreclosure, brand manufacturers voice similar concerns in connection with the development of private labels. As the promotional activities associated with the launch of new products generally require advance planning with the main retailers, manufacturers have expressed the fear that this may give these retailers an opportunity to reduce or even eliminate the lead time before the apparition of "me-too" private labels.

3 For an early discussion of "raising rivals' costs" strategies, see Krattenmaker and Salop (1986). 4 Guidelines on the assessment of non-horizontal mergers under the Council Regulation on the control of concentrations between undertakings adopted by the European Commission on 18.10.2008 (O.J. 2008/C 265/07), at §78. 5 Milliou (2004) mentions for example a number of US cases in R&D intensive sectors such as defense, pharmaceuticals, telecommunications, satellite and energy. In Europe, the issue was for example discussed in such merger cases as Boeing/Hughes (Case COMP/M.1879), Cendant/ Galileo (Case 1 A recent European example is the merger between TomTom and Tele Atlas. 6 Tom-Tom manufactures portable navigation devices (or "PNDs"), whereas Tele Atlas was one of the two main providers of digital map databases for navigation in Europe and North America. In its decision, the European Commission states that "third parties have expressed concerns that certain categories of information considered confidential which they currently pass to Tele Atlas, for instance during technical consultations, could, after the merger, be shared with TomTom." This concern was based on the premise that "Tele Atlas's customers have to share information on their future competitive actions with their map supplier. [...] In a number of examples provided [...] by third parties, companies voluntarily passed information about their estimated future sales, product roadmaps and new features included in the latest version of their devices. They did this for four main reasons, firstly, to negotiate better prices, secondly, to incorporate existing features in new products, thirdly to encourage the map suppliers to develop new features, and finally, in order to ensure technical interoperability of new features with the core map and the software." 7 Third parties feared that " [a]ccess to information about the future behavior of its downstream customers, would allow the merged firm to preempt any of their actions aimed at winning more customers (through better prices, innovative features, new business concepts, increased coverage of map databases). This would in turn reduce the incentive of TomTom's competitors to cooperate with Tele Atlas on pricing policy, innovation and new business concepts, all of which would require exchange of information. This would strengthen the market power of NAVTEQ, the only alternative map supplier, with regards to these PND operators and could lead to increased prices or less innovation". 8 In the US, the FTC recently put conditions on two vertical mergers on the market for carbonated soft drinks. In February 2010, PepsiCo acquired its two largest bottlers and distributors in North America who were also acting as bottlers and distributors for its rival Dr Pepper Snapple (henceforth "DPSG"). Yet the FTC expressed his concern that "PepsiCo will have access to DPSG's commercially sensitive confidential marketing and brand plans. Without adequate safeguards, PepsiCo could misuse that information, leading to anticompetitive conduct that would make DPSG a less effective competitor[• • •]". 9 The FTC ordered PepsiCo to set up a firewall in order to regulate the use of this commercially sensitive information. 10 The FTC put similar conditions on Coca Cola's acquisition of its largest North American bottler in september 2010. 11 Our analysis supports these concerns. In a simple successive duopoly framework in which downstream firms must exchange sensitive information with their suppliers in order to implement innovation, we first show that vertical integration can indeed lead to foreclosure when it exacerbates a risk of imitation through information leakages. By making the supplier less "reliable", vertical integration forces the downstream competitor to share the value of its innovation with the other supplier; this discourages the rival' innovation efforts and expands the merging parties' market shares and profit at the expense of independent rivals. We then check that this insight is robust to various changes in the basic framework and that such strategic motive can make vertical integration attractive and hurt rivals even if these could in theory "fight back" and become vertically integrated themselves. Finally, we show that, through such foreclosure, vertical integration harms consumers and reduces total welfare.

We also discuss several reasons why an integrated firm may indeed be more likely to pass on sensitive information to its own subsidiary. Vertical integration may for example make it easier to transmit such information discreetly (or more difficult not to take advantage of this possibility). It may also enhance coordination between the upstream and downstream efforts required for successful imitation. But, maybe more to the point, vertical integration moreover drastically alters the merged entity's incentives to protect customers' information; as a result, strategic motives do exacerbate the risk of imitation. An integrated firm may for example choose to invest in reverse engineering technology where an independent supplier would not do so. An integrated firm has also less incentives to build effective firewalls or provide financial guarantees 9 See FTC 2010; The FTC was also concerned by the risk of facilitated coordination in the industry. 10 See FTC's decision and order "In the Matter of PepsiCo Inc", case 0910133 of February 26, 2010. 11 See FTC's decision and order "In the Matter of The Coca-Cola Company", case 1010107 of that the innovation will not be imitated. We first present these ideas in a static framework before showing, in a dynamic setting, how vertical integration affects the merged entity's incentives to build a reputation of reliability.

Our analysis is first related to the literature on market foreclosure and in particular to the seminal paper by Ordover, Saloner and Salop (1990), henceforth referred to as OSS. They argue that a vertical merger could be profitable as it allows the integrated firm to raise rivals' costs, by degrading their access to its own supplier and increasing in this way the market power of alternative suppliers. 12 Salinger (1988) has obtained the same result in a successive Cournot oligopoly framework where integrated firms are supposed to exit the intermediate market. As highlighted by Hart and Tirole (1990) or Reiffen (1992), both OSS and Salinger's analysis rely however on the assumption that the integrated firm could somehow commit itself to limiting its supplies to downstream rivals, since otherwise it would have an incentive to keep competing with the alternative suppliers. By contrast, in our article the integrated supplier needs not commit itself to refusing to deal with or limiting its supplies to rivals: by exacerbating the risk of information leakages, a vertical merger de facto degrades the perceived quality of the integrated supplier, which suffices to increase the market power of the alternative suppliers. Reiffen (1992) also mentions that the analysis of OSS relies on the assumption that suppliers can only charge linear prices on the intermediate market, otherwise the increased market power of the independent suppliers need not result into higher, inefficient marginal input prices. In our article, even if supply contracts are ex-post efficient (with cost-based marginal prices), increasing alternative suppliers' market power adversely affects unintegrated rivals' R&D incentives. 13 Several papers have explored ways to dispense with the commitment assumption. For example, Gaudet and Long (1996) have shown in a successive Cournot oligopoly framework that an integrated firm can find profitable to buy some inputs in order to raise the input price, and thus its downstream rivals' cost. Ma (1997) shows that foreclosure obtains without any commitment when the suppliers offer complementary components of downstream bundles. 14 In case of vertical separation, the competitive downstream industry makes no profit and offers at prices reflecting input costs. In contrast, when one of the suppliers integrates downstream, it has an incentive to stop supplying its component to downstream rivals, so as to monopolize the market for the bundle. Choi and Yi (2000) revisit the commitment issue by showing that an integrated supplier could find profitable to offer an input specifically tailored to the needs of its downstream unit, rather than a generic input that could be sold to other firms as well. In a close spirit, Church and Gandal (2000) have shown that a software supplier vertically integrated with a hardware firm could find profitable to stop providing a software compatible with the rival's hardware in order to depreciate the latter's product. Again foreclosure arises in equilibrium as long as the compatibility decision is not too easily reversible. Finally, imperfect competition in the upstream market (combined with input linear prices) could also restore some foreclosure effects even in the absence of commitment not to supply. Most of the literature on the foreclosure effect of vertical integration claims that foreclosure stems from the integrated firm's incentive to stop supplying (or to limit its supplies to) downstream competitors. Chen (2001) and Chen and Riordan (2007) stress however that the independent downstream firm may well favor the integrated supplier, in order to relax downstream competition (the downstream subsidiary becoming less aggressive as it then internalizes the impact of its marketing decision on its upstream division business). By contrast, we point out that foreclosure may arise from the independent downstream firm's reluctance to deal with the integrated supplier due to the risk of information leakage.

Our paper is also related to the literature on innovation and product imitation. Bhattacharya and Guriev (2006) investigate for example the link between the vertical market structure and the risk of imitation when information can be leaked. In a framework where a research unit bargains with two competitive development units, they compare the efficiency and R&D incentives generated by alternative modes of licensing: "open sale" (the usual form of patents) vs. "closed sale" and partial vertical integration (the licensor then holding a stake in the licensed firm's post-invention revenues). Although patenting is socially preferable, when the invention is highly profitable the parties may instead opt for a "closed sale", which limits the risk of leakage by reducing the incentives for secretly selling the information to downstream competitors.

Several papers have more specifically studied the impact of firewalls who prohibit internal transfers of the proprietary information that a subsidiary may receive from third parties. However, these papers do not analyze the foreclosure issue. For instance, Hughes and Kao (2001) consider a market structure where an integrated firm and less efficient upstream rivals compete to supply downstream firms among which one has private information about the demand. By supplying that firm, the integrated supplier obtains its private information and shares it with its downstream subsidiary, which thus becomes more efficient. In spite of this risk of information disclosure, the informed downstream firm buys from the efficient integrated supplier if the latter sets a sufficiently attractive input price. Here the risk of information disclosure leads to lower equilibrium input prices and a higher welfare. By eliminating the risk of information disclosure, a firewall would enable the integrated firm to raise its price towards the cost of its inefficient rival, and lower welfare.

Our paper is also close to Milliou (2004), who studies the impact of a firewall on downstream firms' R&D incentives; she considers the case of a pure bottleneck (the integrated supplier has full control of the intermediate market) and shows that a firewall enhances rivals' incentives to innovate but reduces the incentives of the integrated firm (in the case of complementary R&D paths) or enhances them (in the case of substitutes). In both cases, the integrated firm innovates more frequently in the absence of a firewall, however, due to the fact that it then benefits from the information flow (and the downstream rivals moreover face inefficient input prices). In contrast, we consider an R&D race in which competitors can turn to an alternative supplier, and indeed do so in the absence of a firewall; as a result, the integrated firm never actually benefits from any information flow and the adoption of a firewall would therefore not affect its behavior in the race for innovation (that is, its "best response" is not affected -the actual R&D effort however adapts to the change in rivals' R&D efforts). It follows that information flows always reduce the overall intensity of R&D.

The article is organized as follows. Section 2 develops a simple R&D model in which the risk of information leakages and imitation is treated as exogenous; we first use this model to show how vertical integration results in foreclosure, before providing robustness checks and discussing welfare implications. Section 3 discusses several reasons, most notably strategic ones, why vertical integration can indeed increase the threat of imitation. Section 4 explores more formally a reputation argument in the context of a dynamic model. Section 5 concludes.

Foreclosure through the risk of imitation

We develop in this section a very simple model capturing the main intuitions. Our working assumption here is that, contrary to independent suppliers, an integrated supplier will always make use of any confidential information it can obtain from its customers in order to try and imitate their innovation. We show that this creates an incentive for vertical mergers, motivated by input foreclosure, and analyze the welfare consequences. As mentioned, we show in the next sections how this working assumption can be validated in contexts where both integrated and independent suppliers choose whether to disclose customers' sensitive information.

Framework

Two upstream firms U A and U B supply a homogenous input to two downstream firms D 1 and D 2 , which transform it into a final good and compete for customers. Unit costs are supposed to be constant and symmetric at both upstream and downstream levels, and are normalized to 0; we moreover assume that technical constraints impose single sourcing. Upstream competition for exclusive deals then leads the suppliers to offer efficient contracts, which boils down to supply any desired quantity in exchange for some lump-sum tariff T . 15 Downstream firms may innovate, which increases the value of the final good they offer. When one firm innovates, its comparative advantage generates an additional profit ∆ > 0. However, when both firms innovate, competition dissipates part of this profit and each firm then obtains δ < ∆/2. 16 Normalizing to zero the profits achieved 15 Since suppliers compete here for exclusive deals, whether the contract terms are public or secret does not affect the analysis: in both instances, each supplier will have an incentive to propose an efficient contract, in which the marginal transfer price reflects the marginal cost (normalized here to 0). 16 Suppose for instance that the innovation allows a downstream firm to create a new good or to address a new market segment. If only one firm innovates, it obtains the corresponding monopoly profit, π M ; if instead both firms innovate, then they share a lower duopoly profit π D < π M . We then have ∆ = π M and δ = π D /2 < ∆/2. Consider for example a Cournot duopoly with linear demand P (Q) = d -Q, in which innovation reduces the unit cost c from d (so that the market is barely viable) to 0; a firm that does not innovate obtains zero profit, while the monopoly profit is π M = d 2 /4 and the duopoly profit is

π D = 2d 2 /9 < π M .
in the absence of innovation, the payoff matrix is thus as follows, where I and N respectively denote "Innovation"and "No innovation":

D 1 \D 2 I N I δ, δ ∆, 0 N 0, ∆ 0, 0 (1)
Each D i decides how much to invest in innovation. More precisely, we suppose that D i can innovate with probability ρ i by investing an amount C (ρ i ) -we will refer to ρ i as D i 's R&D effort. We will adopt the following regularity conditions:

Assumption A (unique, stable and interior innovation equilibrium). The cost function C (.) is twice differentiable, convex and satisfies:

• A(i) C 00 (.) > ∆ -δ; • A(ii) 0 ≤ C 0 (0) ≤ δ; • A(iii) C 0 (1) > ∆.
A(i) ensures that best responses are well behaved; A(ii) and A(iii) moreover imply that equilibrium probabilities of innovation strictly lie between 0 and 1.

In the absence of any vertical integration, the competition game is as follows:

• In stage 1, D 1 and D 2 simultaneously choose their R&D efforts and then innovate with probabilities ρ 1 and ρ 2 ; the success or failure of their innovation efforts is observed by all firms.

• In stage 2, U A and U B simultaneously offer lump-sum tariffs to each downstream firm; we will denote by T hi the tariff offered by U h to D i (for h = A, B and i = 1, 2); each D i then chooses its supplier.

We also consider a variant of this game in which U A is vertically integrated with D 1 . Throughout this section, we assume that this vertical integration creates a risk for D 2 to see its innovation imitated by D 1 if it chooses U A for supplier: in that case, with probability θ > 0 the integrated firm successfully mimics the innovation (at no cost).

Vertical separation

Since the two suppliers produce the same input with the same constant unit cost, in the second stage Bertrand-type competition yields T Ai = T Bi = 0. In the first stage, each D i chooses its R&D effort ρ i so as to maximize its expected profit, which is given by:

π i = Π ¡ ρ i , ρ j ¢ ≡ ρ i ¡ ρ j δ + ¡ 1 -ρ j ¢ ∆ ¢ -C (ρ i ) . (2)
It follows that R&D efforts are strategic substitutes:

∂ 2 Π i ∂ρ i ∂ρ j = -(∆ -δ) < 0. (3)
Let

ρ i = R ¡ ρ j ¢ denote D i 's best response to ρ j ∈ [0, 1]
(by construction, these best responses are symmetric); Assumption A ensures that it is uniquely characterized by the first-order condition:

C 0 (ρ i ) = ρ j δ + ¡ 1 -ρ j ¢ ∆, (4)
and that it yields a unique equilibrium,17 which is symmetric, interior and stable:18

Lemma 1 In case of vertical separation, under Assumption A the best response R (ρ) is differentiable and satisfies:

0 ≤ R (ρ) < 1, (5)
where the first inequality is strict whenever ρ < 1, and:

-1 < R 0 (ρ) < 0. ( 6
)
As a result there exists a unique equilibrium, which is symmetric and such that (where the superscript V S refers to Vertical Separation):

0 < ρ V S 1 = ρ V S 2 = ρ * < 1. (7)
Proof. The convexity assumption, together with the boundary conditions A (ii) and A (iii), ensures that the best response to

ρ j ∈ [0, 1], ρ i = R ¡ ρ j ¢
, is uniquely characterized by the first-order condition (4) and satisfies (5), with R (ρ) > 0 whenever ρ < 1. Differentiating the first-order condition yields:

R 0 (ρ) = -(∆ -δ) C 00 (R (ρ)) < 0.
We thus have: (i) R 0 (ρ) < 0, (ii) R (0) > 0, and (iii) R (1) < 1. These properties imply that there is a unique value ρ * , which moreover lies strictly between 0 and 1, such that ρ * = R (ρ * ). By construction, ρ 1 = ρ 2 = ρ * constitutes a symmetric equilibrium. Conversely, condition A (i) implies R 0 (ρ) > -1, which in turn implies that this equilibrium is stable and that there is no other equilibrium.

Vertical integration

Suppose now that U A and D 1 merge, and denote by U A -D 1 the resulting integrated firm. In the second stage of the game, the two suppliers are again equally effective when either D 2 does not innovate, or both D 1 and D 2 innovate; in both cases, Bertrand-like competition among the suppliers leads them to offer cost-based tariffs to D 2 . When instead D 2 is the sole innovator, dealing with the integrated supplier exposes D 2 to see its innovation imitated with probability θ > 0. Thus, while D 2 's expected gross profit is again

∆ if it buys from U B , it is only θδ + (1 -θ) ∆ if it buys from U A -D 1 ;
U A is however willing to offer a discount equal to the expected value from imitation, θδ. This asymmetric competition leads U A to offer T A2 = -θδ and U B to win19 with T B2 = θ (∆ -2δ), which gives D 2 a net profit:

θδ + (1 -θ) ∆ -T A2 = ∆ -T B2 = ∆ -θ (∆ -2δ) .
In the first stage, D 2 's expected profit is now given by:

π 2 = Π θ (ρ 2 , ρ 1 ) ≡ ρ 2 (ρ 1 δ + (1 -ρ 1 ) (∆ -θ (∆ -2δ))) -C (ρ 2 ) , (8)
whereas the integrated firm U A -D 1 's expected profit is as before equal to:

π A1 = π 1 = Π (ρ 1 , ρ 2 ) , (9)
where Π (., .), given by (2), coincides with Π θ (., .) only for θ = 0. Best responses are thus respectively given by ρ 1 = R (ρ 2 ) and ρ 2 = R θ (ρ 1 ), characterized by:

C 0 (ρ 2 ) = ρ 1 δ + (1 -ρ 1 ) (∆ -θ (∆ -2δ)) . (10)
R θ (.) coincides with R (.) for θ = 0 and is identically equal to zero when θ = 1 and δ = 0. Furthermore, for ρ < 1, R θ (ρ) strictly decreases as θ increases. As a result:

Lemma 2 In case of vertical integration, under Assumption A there exists a unique, stable equilibrium, in which R&D efforts are asymmetric for any θ > 0 and of the form (where the superscript V I refers to Vertical Integration):

ρ V I 1 = ρ + θ , ρ V I 2 = ρ - θ , (11)
where ρ + 0 = ρ - 0 = ρ * , and ρ + θ and ρ - θ respectively increase and decrease as θ increases from 0 to 1.

Proof. In the polar case (θ = 1, δ = 0), D 2 never invests in R&D since: (i) if both firms innovate, competition entirely dissipates their profits; and (ii) if only D 2 innovates, the threat of imitation by the integrated firm allows U B to extract the full value of the innovation. As a result, the integrated firm behaves as a monopolist and invests ρ 1 = ρ m ≡ R (0).

Suppose now that θ < 1 and/or δ > 0. The convexity assumption, together with the boundary conditions A (ii) and A (iii), ensures that D 2 's best response to ρ 1 ∈ [0, 1], R θ (ρ 1 ), is uniquely characterized by the first-order condition (10) and satisfies:

0 ≤ R θ (ρ) < 1,
with R θ (ρ) > 0 whenever ρ < 1. Differentiating (10) yields:

R 0 θ (ρ) = - ∆ -δ -θ (∆ -2δ) C 00 (R θ (ρ)) < 0. ( 12
)
It thus satisfies again R θ (1) < 1, R θ (0) > 0, R 0 θ (0) < 0, and (using condition A (i)) R 0 θ (ρ) > -1. The same reasoning as above thus implies the existence of a unique, stable equilibrium, in which the R&D efforts satisfy ρ ). Finally, differentiating the first-order conditions (4) and (10) with respect to ρ + θ , ρ - θ and θ yields:

+ θ = R ¡ ρ - θ ¢ and ρ - θ = R θ ¡ ρ + θ ¢ . Clearly, ρ + 0 = ρ - 0 = ρ * since R 0 (.) coincides with R (.
dρ + θ dθ = ¡ 1 -ρ + θ ¢ (∆ -δ) (∆ -2δ) C 00 ¡ ρ + θ ¢ C 00 ¡ ρ - θ ¢ -(∆ -δ) (∆ -δ -θ (∆ -2δ)) > 0, (13)
since assumption A (i) implies that the denominator is positive, whereas A (iii) implies that the numerator, too, is positive (i.e., ρ + θ < 1); similarly:

dρ - θ dθ = - ¡ 1 -ρ + θ ¢ C 00 ¡ ρ + θ ¢ (∆ -2δ) C 00 ¡ ρ + θ ¢ C 00 ¡ ρ - θ ¢ -(∆ -δ) (∆ -δ -θ (∆ -2δ)) < 0. ( 14
)

The foreclosure effect of vertical integration

Note first that vertical integration would have no impact here in the absence of R&D investments: with or without integration, both input providers would offer to supply at marginal cost. In contrast, when innovation matters, then whenever integration creates a risk of imitation (θ > 0) it de facto reduces the "quality" of the integrated supplier for the independent competitor, leaving it in the hands of the remaining, independent supplier. This "input foreclosure" enhances the independent supplier's market power, thereby raising the cost of supply for the downstream rival, who must share with the supplier the benefit of its R&D effort. This discourages the independent firm from investing in R&D, which in turn induces the integrated subsidiary to increase its own investment. The quality gap, and thus the foreclosure effect, increases with the risk of imitation θ. As long as this risk remains limited (θ < 1 and/or δ > 0), the integrated supplier still exerts a competitive pressure on the upstream market. As a result, the independent downstream competitor retains part of the value of its innovation and thus remains somewhat active on the innovation market ("partial foreclosure"). In contrast, when the imitation concern is maximal (θ = 1 and δ = 0), the integrated supplier provides no value for the independent firm; the independent supplier can then extract the full benefit of any innovation by the independent firm, which thus no longer invests in R&D. The integrated firm then de facto monopolizes the innovation market segment ("complete foreclosure"). Formally, a comparison of the investment levels with and without integration yields:

Proposition 3 Compared with the case of vertical separation, a vertical merger between U A and D 1 replicates the effect of input foreclosure:

(i) it leads the independent firm D 2 to invest less, and the integrated subsidiary to invest more in innovation -all the more so as the probability of imitation, θ, increases; in particular, when vertical integration triggers imitation with certainty (θ = 1) and competition fully dissipates profits (δ = 0), the integrated firm monopolizes the innovation market.

(ii) it increases the joint profit of the merging parties, U A and D 1 , at the expense of the downstream independent rival D 2 ; while the independent supplier U B benefits from its enhanced market power over D 2 , the joint profit of the independent firms also decreases.

Proof. Part (i) follows from the fact that ρ - θ and ρ + θ respectively decrease and increase as θ increases, and that they both coincide with ρ * for θ = 0, whereas ρ - θ = 0 for θ = 1 and δ = 0. As for part (ii), it suffices to note that ρ - θ < ρ * < ρ + θ implies:

π V I A1 = π + θ ≡ max ρ 1 Π ¡ ρ 1 , ρ - θ ¢ > π * ≡ max ρ 1 Π (ρ 1 , ρ * ) = π V S 1 = π V S A + π V S 1 ,
and:

π V I B + π V I 2 = Π ¡ ρ - θ , ρ + θ ¢ < max ρ 2 Π ¡ ρ 2 , ρ + θ ¢ < max ρ 2 Π (ρ 2 , ρ * ) = π V S 2 = π V S B + π V S 2 ,
where the first inequality stems from the fact that ρ - θ is chosen by D 2 so as to maximize its own profit,

Π θ ¡ ρ 2 , ρ + θ ¢ , rather than the joint profit Π ¡ ρ 2 , ρ + θ ¢ of the independent firms. Since π V I B ≥ π V S B = 0, the the last inequality also implies π V I 2 > π V S
2 . Note that imitation never occurs in equilibrium, since the independent downstream competitor always ends up dealing with the independent supplier. Yet, the threat of imitation suffices to increase the independent supplier's market power at the expense of the independent downstream firm, who reduces its innovation effort.

This input foreclosure effect benefits the integrated firm, U A -D 1 , who faces a less aggressive rival. Due to strategic substitution, the integrated firm moreover responds by increasing its investments which not only further degrades D 2 's profit but also degrades the joint profits of the independent firms.20

Robustness

This analysis is robust to various changes in the modeling assumptions.

Information leakages. The analysis still applies for example when information flows already exist in the absence of any merger, as long as vertical integration increases these flows and the resulting probability of imitation, e.g., from θ to θ. The distortion term θ (2∆δ) then simply becomes ¡ θθ ¢ (2∆δ). Bilateral bargaining power. The same logic applies when downstream firms have significant bargaining power in their bilateral negotiations with the suppliers, as long as suppliers obtain a positive share of the specific gains generated by the relationship.

Suppose for example that suppliers obtain a share λ < 1 of these specific gains from trade. This does not affect the outcome in case of vertical separation, since both suppliers are equally effective in that case: there is thus no specific gain to be shared and downstream firms still obtain the full benefit of their innovation; R&D efforts are therefore again given by ρ V S 1 = ρ V S 2 = ρ * . In contrast, in case of vertical integration the independent supplier obtains a share λ of its comparative advantage over the integrated rival whenever D 2 is the only innovator (that is, T B2 = λθ (∆ -2δ) in that case); D 2 's expected profit thus becomes:

π 2 = Π λθ (ρ 2 , ρ 1 ) ≡ ρ 2 (ρ 1 δ + (1 -ρ 1 ) (∆ -λθ (∆ -2δ))) -C (ρ 2 ) . ( 15
)
The same analysis then applies, replacing the probability θ with the "adjusted probability" λθ, which now depends on the relative bargaining power of the supplier as well as on the risk of imitation. As long as λ > 0, innovation efforts are again distorted compared with the case of vertical separation. Imperfect imitation. In practice, an imitator may not be as effective a competitor as a genuine innovator; the imitator may for example lag behind the innovator, who can moreover take steps to protect further its comparative advantage. Yet, the analysis applies as long as imitation reduces the value of the innovation by L, say. In case of vertical integration, whenever D 2 is the sole innovator the independent supplier can still charge a positive markup reflecting its comparative advantage, T B2 = θL > 0.

Imperfect competition in the downstream market. When both firms innovate, limiting factors such as product differentiation, capacity constraints, competition in quantities rather than prices, and so forth, may limit competition and thus increase the resulting profit δ. This increases the incentives to invest in R&D (since an innovator obtains more profit when the rival innovates as well) and attenuates the foreclosure effect, both because imitation is less costly and because the integrated supplier is willing to offer a larger discount, reflecting the increased value from duplication, and thus exerts a tougher pressure on the alternative supplier. Yet, our analysis shows that partial foreclosure still arises as long as imitation reduces total industry profit (that is, as long as ∆ > 2δ).

Imperfect competition in the upstream market. The above reasoning carries over to the case where suppliers produce imperfect substitutes, as long as vertical integration renders the integrated supplier less reliable for the independent downstream firms. Suppose for example that each downstream firm has a favored supplier: D 1 (resp. D 2 ) obtains an additional surplus γ when dealing with U A (resp. U B ), say. If U A and D 1 vertically integrate, and D 2 is the sole innovator, U A is then less attractive than before. It offers D 2 a subsidy T A2 = -θδ, reflecting the expected gain from imitation, but U B now wins the competition for D 2 with an even higher tariff, T B2 = θ (∆ + γ -2δ). Conversely, if U A were D 2 's favored supplier, U B would still be able to extract a positive rent from D 2 's innovation as long as the comparative advantage does not offset reliability concerns (i.e., as long as γ < ∆ -2δ). The foreclosure effect is however stronger when a downstream firm merges with its own favored supplier. 21 Number of competitors. It should be clear that the analysis does not rely critically on the restriction to duopolies. If for example there were additional stand-alone downstream firms, vertical integration would enhance the market power of the independent supplier over these other firms as well, thus discouraging their R&D efforts to the benefit of the integrated firm. Likewise, the argument still applies when there are more than two suppliers, as long as upstream competition remains imperfect, so that degrading the perceived quality of the integrated supplier enhances the market power of the others over the independent downstream firms.

Timing of negotiations. We assumed so far that negotiations take place only once an innovation materializes (ex post contracting). This makes sense, for example, when it is difficult to specify ex ante the exact nature of the innovation. The same analysis however applies when negotiations take place earlier on, as long as R&D efforts are observed beforehand. The suppliers then still offer cost-based tariffs in case of vertical separation; and in case of integration, the independent supplier again imposes a tariff reflecting its (expected) comparative advantage over the integrated supplier, T B2 = θ(1ρ 1 )ρ 2 ∆, and this has exactly the same impact on D 2 's incentives to invest in R&D.

Both timings thus result in creating a "hold-up" effect on a downstream firm's investment, and vertical integration then generates foreclosure by exacerbating this hold-up problem. If instead suppliers could commit themselves before downstream firms take their investment decisions, they could avoid hold-up problems, and foreclosure would no longer arise. Suppose for example that firms can agree on lump-sum payments, not contingent on the success of innovation efforts. While vertical integration might still increase the market power of independent suppliers, and thus their tariffs, this would no longer translate into lower investments, and thus the foreclosure effect would disappear. Such arrangements however raise several concerns. Liquidity constraints may for example call for deferred payments, which in turn triggers credi-bility issues, particularly when downstream firms have limited access to credit. To see this, suppose that downstream firms are initially cash constrained, and have moreover no access to credit. Downstream firms must therefore pay their suppliers out of realized profits. The best contracts then boil down to milestone payments, conditional upon the success or failure of (both) innovation efforts. Consider for example the case where δ = 0 and θ = 1. With ex post contracting there is then complete foreclosure: since U B would fully appropriate the benefit from innovation, D 2 does not investand U B thus obtains zero profit. With ex ante contracting, U B can instead commit itself to not appropriating the full value of innovation. Yet, since D 2 's payment can only come out of its innovation profit, U B 's market power still reduces investment incentives. To see this, let T denote D 2 's payment in case it is the sole innovator; D 2 's expected profit becomes ρ 2 (1ρ 1 ) (∆ -T ) and the resulting investment levels are of the form (ρ 1 (T ), ρ 2 (T )), where ρ 1 (.) and ρ 2 (.) respectively increase and decrease with T , and ρ 2 (∆) = 0. Ex ante, U B then sets T so as to maximize its expected profit, π B (T ) = ρ 2 (T ) (1ρ 1 (T )) T . The optimal tariff then satisfies T * < ∆, as it takes into consideration the negative impact of T on the probability of D 2 being the sole innovator, ρ 2 (T ) (1ρ 1 (T )); U B and D 2 thus both obtain a positive profit even when δ = 0 and θ = 1. More generally, ex ante contracting is more efficient than ex post contracting whenever T * < θ (∆ -2δ). Yet, the hold-up problem remains, even if to a more limited extent, and foreclosure still arises.

Customer foreclosure. The analysis can also be readily transposed to the case where upstream manufacturers need to exchange information with their distributors in order to launch new products. Vertical integration, as in the case oh the acquisition of downstream bottlers and wholesalers by PepsiCo or CocaCola's, or the development of private labels by large retail chain, may there again exacerbates the risk of information leaks and discourage manufacturer's innovation. 22 Suppose for example that: (i) upstream, two manufacturers U A and U B create a new product with probabilities ρ A and ρ B by investing C (ρ A ) and C (ρ B ); (ii) when a manufacturer innovates, it can choose either D 1 or D 2 to launch and distribute the new product; and (iii) a successful launch requires early communication of confidential information about the characteristics and new features of the product, which facili-tates the development of "me-too" substitutes. Concerns about information leaks then militate for relying on a single distributor, in which case the situation is essentially the same as the one studied above. Consider the following competition game, which mirrors the previous one:

• In stage 1, U A and U B simultaneously choose their R&D efforts and then innovate with probabilities ρ A and ρ B ; the success or failure of their innovation efforts is observed by all firms.

• In stage 2, D 1 and D 2 simultaneously offer lump-sum tariffs to each manufacturer, who then chooses its distributor (on an exclusive dealing basis).

Adopting similar cost and profit conditions as above, this competition game yields again a symmetric outcome of the form ρ A = ρ B = ρ * in case of vertical separation, and an asymmetric outcome reflecting a foreclosure effect, of the form ρ A = ρ + θ > ρ B = ρ - θ , when for example U A merges with D 1 . As a result, vertical integration increases the profit of the merging parties, at the expense here of the independent manufacturer. Manufacturers have often voiced such type of concern in reaction to the growing development of private labels by large retailers.

Productivity investments, expansion projects and business strategies. Finally, while we have focused on risky innovation projects, our analysis applies as well to less uncertain productivity gains, development plans, capacity investments, and so forth, that enhance firms' competitiveness but require prior communication and information exchanges with upstream or downstream partners. Suppose for example that:

• Downstream competition depends on firms' "effective capacities", δ 1 and δ 2 : each D i obtains Cournot-like profits of the form π (δ i , δ j ) ≡ P (δ 1 + δ 2 )δ i , where the "inverse demand function" satisfies P 0 (.) < 0 and P 0 (δ) + P 00 (δ) δ < 0, which in particular ensures the concavity of the joint profit function. 23 • Each δ i depends on D i 's investment decision, ρ i , but also requires the cooperation from D i 's supplier: thus, δ i = ρ i in the case of vertical separation, whereas if U A and D 1 are vertically integrated, then

δ 2 = ρ 2 but δ 1 = ρ 1 + θ i ρ 2 , where i = A, B denotes D 2 's supplier choice, and θ A = θ > θ B = 0; that is, D 1 benefits from D 2 's investment if D 2 deals with U A .
• The timing is as follows: first, the downstream firms make their capacity investment decisions, ρ 1 and ρ 2 (for simplicity, the costs of these decisions are born ex post and are embodied in the function P (.)); second, U A and U B compete for the development of D i 's effective capacity; third, downstream competition yields the above-described profits.

When both suppliers are independent, upstream competition leads them to supply at cost; thus, each D i chooses ρ i so as to maximize π(ρ i , ρ j ) = P ¡ ρ i + ρ j ¢ ρ i , which yields:

P (ρ 1 + ρ 2 ) + P 0 (ρ 1 + ρ 2 ) ρ i = 0. ( 16
)
The above regularity conditions then imply that capacity decisions are strategic substitutes (i.e., D i 's best response decreases when ρ j increases) and that there is a unique, stable symmetric equilibrium ρ 1 = ρ 2 = ρ * . 24 When instead U A and D 1 are vertically integrated, then U A is willing to offer a subsidy of up to π (ρ 1 + θρ 2 , ρ 2 )π (ρ 1 , ρ 2 ), which would give D 2 a profit equal to:

π 2 (ρ 2 , ρ 1 ; θ) ≡ π (ρ 2 , ρ 1 + θρ 2 ) + π (ρ 1 + θρ 2 , ρ 2 ) -π (ρ 1 , ρ 2 ) = P (ρ 1 + (1 + θ) ρ 2 ) (ρ 1 + (1 + θ) ρ 2 ) -P (ρ 1 + ρ 2 ) ρ 1 .
As long as total capacity ρ 1 + ρ 2 exceeds the monopoly level (implying that total profit, P (ρ 1 + ρ 2 ) (ρ 1 + ρ 2 ), decreases with any further increase in either investment), π 2 (ρ 2 , ρ 1 ; θ) remains lower than π (ρ 2 , ρ 1 ) = P (ρ 1 + ρ 2 ) ρ 2 , and thus U B wins the competition at a price that leaves D 2 with exactly π 2 (ρ 2 , ρ 1 ; θ). Conversely, D 2 's buying from U B leads U A -D 1 to maximize as before π(ρ 1 , ρ 2 ) = P (ρ 1 , ρ 2 ) ρ 1 ; thus, its behavior remains characterized by the first order condition (16), which in turn implies that ρ 1 + ρ 2 indeed exceeds the monopoly level. 25 By contrast, maximizing π 2 (ρ 2 , ρ 1 ; θ) rather than π 2 (ρ 2 , ρ 1 ) = π 2 (ρ 2 , ρ 1 ; 0) leads D 2 to limit its investment, 24 The slope of D i 's best response is equal to

∂ρ r i ∂ρ j = - P 0 (ρ 1 + ρ 2 ) + P 00 (ρ 1 + ρ 2 ) ρ i 2P 0 (ρ 1 + ρ 2 ) + P 00 (ρ 1 + ρ 2 ) ρ i ,
and thus lies between -1 and 0 when P 0 (ρ) + P 00 (ρ) ρ < 0 and P 0 (ρ) < 0.

25 The monopoly level, ρ M , is defined by

P ¡ ρ M ¢ + P 0 ¡ ρ M ¢ ρ M = 0, whereas P 0 < 0 and (16) imply: P (ρ 1 + ρ 2 ) + P 0 (ρ 1 + ρ 2 ) (ρ 1 + ρ 2 ) < P (ρ 1 + ρ 2 ) + P 0 (ρ 1 + ρ 2 ) ρ 1 = 0; from the concavity of the joint profit function, we thus have ρ 1 + ρ 2 > ρ M . since: ∂ 2 θρ 2 π 2 (ρ 2 , ρ 1 ; θ) = [P (ρ 1 + (1 + θ) ρ 2 ) + P 0 (ρ 1 + (1 + θ) ρ 2 ) (ρ 1 + (1 + θ) ρ 2 )] + (1 + θ) 2 [2P 0 (ρ 1 + (1 + θ) ρ 2 ) + P 00 (ρ 1 + (1 + θ) ρ 2 ) (ρ 1 + (1 + θ) ρ 2 )] ,
where the first term is negative because the total quantity

ρ 1 + (1 + θ)ρ 2 exceeds the monopoly level (ρ 1 + (1 + θ)ρ 2 > ρ 1 + ρ 2 > ρ M
) and the second term is negative from the concavity of the joint profit function. Therefore, in equilibrium D 2 invests less than in case of vertical separation, which benefits D 1 (as it faces a less aggressive rival) and makes vertical integration profitable -in addition, since investments are strategic substitutes, D 1 invests more than in the separation case, which reduces independent rivals' joint profit.26

Rivals' counter-fighting strategies

Since input foreclosure increases the profit of the merging firms at the expense of their rivals, it may encourage these rivals to merge as well. Indeed, the situation with two vertical mergers is similar to the initial, no-merger situation, since there is again no risk of imitation: the two integrated suppliers supply at cost their subsidiaries, which will thus invest ρ 1 = ρ 2 = ρ * . Since each integrated firm then obtains Π * , in the absence of any specific cost of integration the rivals would have an incentive to merge in response to a first vertical merger. Note however that the two situations (with zero or two mergers) would be different if there were any remaining independent downstream competitor. In case of vertical separation, the two suppliers would then sell at cost to all downstream firms, resulting in a level-playing field competition in the downstream market. To be sure, a first vertical merger between, say, U A and D 1 , may encourage a second merger between U B and, say, D 2 . But while the two suppliers would again sell at cost to all downstream

ρ 2 = 1 3 + 2t < ρ * = 1 3 < ρ 1 = 1 + t 3 + 2t ,
where t = 3θ + 2θ 2 > 0, and total capacity indeed satisfies:

ρ 1 + ρ 2 = 2 + t 3 + 2t > ρ M = 1 2 .
firms, they would become less reliable for the independent ones; downstream competition would therefore be biased in favor of the integrated firms, who would still enjoy a reliable access to the upstream market. Such integration wave would thus confer a strategic advantage to the merging parties to the detriment of the independent rivals, who would again decrease their R&D efforts. 27 But even in our duopoly model, a first merger can be profitable when integration is costly, in such a way that the initial merger does not lead the rivals to integrate; letting K denote the cost of integration, this will be the case when:

K ≡ π * - ¡ π V I B + π V I 2 ¢ < K < K ≡ π V I A1 -π * . ( 17
)
The interval

£ K, K ¤ is empty when π V I A1 + π V I B + π V I 2 < 2π * , i.e.
, when a merger decreases total industry profit. In that case, a vertical merger either is unprofitable or triggers a counter-merger that eliminates any strategic advantage for the first merging firms. Otherwise, we have: Proposition 4 When partial integration raises total industry profit, there exists a non-empty range £ K, K ¤ such that, whenever the integration cost K lies in this range, the remaining independent firms have no incentive to merge in response to a first vertical merger; as a result, the first merger creates a foreclosure effect that confers a strategic advantage to the merging firms, at the expense of the independent downstream rival.

The scope for counter-fighting strategies thus depends on the impact of partial integration on industry profits, which itself is ambiguous. To see this, consider the following benchmark case, in which duplication dissipates profit and R&D costs follow a standard quadratic specification:

Assumption B:

δ = 0, C (ρ) = k 2 ρ 2 .
Assumption A then boils down to:

η ≡ k ∆ > 1.
We have:

Proposition 5 Under assumption B, partial vertical integration raises total industry profit when and only when innovation is not too costly (η < η ≡ 1 + √ 2) or the risk of imitation is not too large (θ < θ (η), where θ (η) < 1 for η > η).

Proof. Straightforward computations yield:

• In case of vertical separation:

ρ V S 1 = ρ V S 2 = ρ * = 1 1 + η , ( 18
)
π V S 1 = π V S 2 = π * = k 2 µ 1 1 + η ¶ 2 . ( 19
)
• In case of vertical integration between U A and D 1 :

ρ V S 1 = ρ + θ = η -(1 -θ) η 2 -(1 -θ) , ρ V S 2 = ρ - θ = (1 -θ) (η -1) η 2 -(1 -θ) , (20)
π V I A1 = k (ρ + ) 2 2 = k 2 µ η -(1 -θ) η 2 -(1 -θ) ¶ 2 , π V I B + π V I 2 = k 2 ¡ 1 -θ 2 ¢ µ η -1 η 2 -(1 -θ) ¶ 2 .
It can then be checked that partial vertical integration always increases total industry profit when η < η = 1 + √ 2; when instead η ≥ η, vertical integration increases total industry profit if and only if θ < θ (η) ≡ 2(η-1) 2 (η+1) (η 2 -3)η 2 -2(η-1) , where θ (η) ∈ [0, 1] and θ0 (η) < 0.

To understand the impact of vertical integration on total industry profit, it is useful to consider what would be the optimal R&D efforts for the downstream firms if they could coordinate their investment decisions (but still compete in prices). 28 When innovation efforts are inexpensive (namely, η < 2), the firms would actually find it optimal to have one firm (and only one) invest

1 η ¡ > 1 2 ¢
, so as to avoid the competition that arises when both firms innovate. If instead innovation efforts are expensive (η ≥ 2), the decreasing returns to scale make it optimal to have both firms invest 1 η+2 < ρ * . Compared with this benchmark, in the absence of integration, downstream competition leads the firms to overinvest in innovation, since each firm neglects the negative externality that its investment exerts on the rival's expected profit. Consider now the case of partial integration and for the sake of exposition, let us focus on the polar case of complete foreclosure θ = 1. Vertical integration then de facto implements the integrated industry optimum when η < 2. When instead innovation efforts are expensive, i.e. η is large, the resulting asymmetric investment levels and the underlying decreasing returns to scale reduce industry joint profits. From proposition 4, a vertical merger then generates a profitable foreclosure effect without triggering a counter-merger.

Welfare analysis

We first study here the impact of vertical integration on investment levels and on the probability of innovation,

≡ 1 -(1 -ρ 1 ) (1 -ρ 2 ) = ρ 1 + ρ 2 -ρ 1 ρ 2 ,
before considering its impact on consumer surplus and total welfare. Proposition 6 Partial vertical integration reduces total investment; it also reduces the probability of innovation when θ is not too large, but can increase it for larger values of θ. For example, under Assumption B it decreases the probability of innovation if and only if innovation is very costly (η ≥ η, where η > 1) or when the risk of imitation is not too large (θ < θ (η), where θ (η) < 1 for η < η).

Proof. By construction, the probability of innovation is θ ≡ ρ + θ +ρ - θ -ρ + θ ρ - θ in the case of partial integration and * ≡ 0 in the case of separation. Under Assumption A, total investment decreases when θ increases:

d(ρ - θ + ρ + θ ) dθ = ¡ 1 -ρ + θ ¢ ¡ ∆ -δ -C 00 ¡ ρ + θ ¢¢ (∆ -2δ) C 00 ¡ ρ + θ ¢ C 00 ¡ ρ - θ ¢ -(∆ -δ) (∆ -δ -θ (∆ -2δ)) < 0,
since from A (i) the denominator is positive, and given A (iii) (which yields ρ + < 1), the numerator, too, is positive. However, the probability that both firms innovate also decreases with θ:

d(ρ - θ ρ + θ ) dθ = ¡ ρ - θ (∆ -δ) -ρ + θ C 00 ¡ ρ + θ ¢¢ ¡ 1 -ρ + θ ¢ (∆ -2δ) C 00 ¡ ρ + θ ¢ C 00 ¡ ρ - θ ¢ -(∆ -δ) (∆ -δ -θ (∆ -2δ)) < 0.
The overall effect on the probability of innovation is therefore:

d θ dθ = ¡¡ 1 -ρ - θ ¢ (∆ -δ) - ¡ 1 -ρ + θ ¢ C 00 ¡ ρ + θ ¢¢ ¡ 1 -ρ + θ ¢ (∆ -2δ) C 00 ¡ ρ + θ ¢ C 00 ¡ ρ - θ ¢ -(∆ -δ) (∆ -δ -θ (∆ -2δ)) .
This expression is negative for small values of θ since, for θ = 0, ρ + = ρ -= ρ * and thus:

d θ dθ ¯θ=0 = (∆ -δ -C 00 (ρ * )) (1 -ρ * ) 2 (∆ -2δ) C 00 (ρ * ) C 00 (ρ * ) -(∆ -δ) (∆ -δ -θ (∆ -2δ)) < 0.
It then follows that, for low values of θ, partial integration decreases the probability of innovation (that is, θ < * = 0 ). For larger values of θ, however, the impact may be positive. Indeed, under Assumption B straightforward computations yield d θ /dθ < 0 as long as θ < θ (η) ≡ (η -1) 2 , where θ (η) is positive and increases with η in the relevant range η > 1; in contrast, d θ /dθ > 0 when θ > θ (η). As a result, partial integration reduces the overall probability of innovation if and only if θ < θ (η) ≡ (η 2 -1) (η -1), where θ (η) is strictly higher than θ (η), θ0 (η) > 0, and θ (η) < 1 as long as η < η = 1+ √ 5

2 . An increase in the risk of imitation θ reduces the investment of the independent firm. Under A(i), this direct negative effect always dominates the indirect positive effect on the investments of its rival; therefore total investment decreases. As for the effect on the probability of innovation, the impact of an increase in θ can be written as

0 = (1 -ρ 1 ) ρ 0 2 + (1 -ρ 2 ) ρ 0 1
; that is, a change in innovation of one firm only affects the probability of innovation when the other firm fails to innovate. When the two firms invest to a similar extent (e.g., when θ is close to zero), the effect of an increase in θ on the probability of innovation is similar to the impact on the sum of investments. When instead, the vertically integrated firm invests much more in R&D than its independent rival, the effect of an increase in θ on the probability of innovation is mainly driven by its positive (indirect) effect on the integrated firm's effort.

In order to study the impact of vertical integration on consumers and welfare, we need to specify the impact of duplication on consumers. For the sake of exposition, let us interpret our model as follows:

• the downstream firms initially produce the same good at the same cost c, and face an inelastic demand of mass M as long as their prices does not exceed consumers' valuation v;

• innovation allows the firms to produce a better product, which increases the net value vc by ∆/M .

Absent innovation, Bertrand competition yields zero profit. If instead one firm innovates, it can appropriate the full added value generated by the new product and thus obtains ∆. In contrast, when both firms innovate, Bertrand competition leads the firms to pass on the added value ∆ to consumers, and thus δ = 0. The (expected) consumer surplus S and total welfare W are then:

S ≡ ρ 1 ρ 2 ∆, W ≡ (ρ 1 + ρ 2 -ρ 1 ρ 2 ) ∆ -C (ρ 1 ) -C (ρ 2 ) .
As shown in the proof of proposition 6, vertical integration always reduces the probability that both firms innovate simultaneously, and thus unambiguously reduces expected consumer surplus. For the quadratic cost specification, it can further be checked that vertical integration reduces total welfare: Proposition 7 Suppose that firms serve initially an inelastic demand with the same good, and that innovation uniformly increases consumers' willingness to pay by some fixed amount; then vertical integration:

(i) always lowers consumer surplus.

(ii) always lowers total welfare when R&D costs are quadratic.

Proof. Part (i) follows from the proof of proposition 6, which shows that the probability that both firms innovate under partial integration decreases with θ and coincides for θ = 0 with that obtained with vertical separation. 29 For part (ii), it suffices to note that vertical integration has no impact on innovation and welfare when θ = 0 and that, for δ = 0 and

C (ρ) = k 2 ρ 2 , W V I θ = (ρ + θ + ρ - θ - ρ - θ ρ + θ )∆ -k ρ + θ 2 2 -k ρ - θ 2 2 satisfies dW V I θ dθ = -(η-1) 3 η(η-1+θ) (η 2 +θ-1) 3
< 0. The framework developed in this section is, of course, restrictive and the scope of the welfare analysis is thus limited. Vertical integration may also create welfareenhancing effects that would appear in a more general framework. For instance, if the decision to disclose information is endogenous, a vertically integrated firm might prefer not to disclose information from its own subsidiary: this protection effect might increase the investment of the integrated firm, thus increasing the likelihood of innovation and thereby welfare.

3 Does vertical integration raise the threat of imitation?

To reflect concerns voiced in certain markets, in the previous section we postulated that vertical integration exogenously creates a risk of information leakage and imitation. We now relax this assumption and allow suppliers, integrated or not, to decide whether to exploit their customers' information. Indeed, since such information would be valuable to downstream competitors, even independent suppliers may choose to "sell"30 it to (some of) these competitors. As we will show, vertical integration drastically affects the ability of the firms, as well as their incentives,31 to do so.

First, vertical integration may facilitate information flows between the upstream and downstream units of the integrated firm -and may make it easier to keep such information flows secret. For example, the merged entity may wish to integrate their IT networks, which may not only facilitate information exchanges but also make it more difficult to maintain credible firewalls. As a result, an integrated supplier may be unable to commit itself not to disclose any business secret even when an independent supplier could achieve that.

Second, an integrated firm may be more successful in coordinating the upstream and downstream efforts required to exploit rivals' information. Suppose for example that the probability of successful imitation is equal to θ U θ D , where θ U and θ D are unobservable and respectively controlled by the upstream and downstream firms. Suppose further that each θ i can take two values, θ and θ > θ, and that opting for the low value θ yields a private, non-transferable benefit b, whereas successful imitation gives the downstream firm the monetary profit δ. It is then easier for an integrated firm to align upstream and downstream incentives in order to achieve the highest probability of successful imitation, θθ; as a result, vertical integration can indeed increase the likelihood of imitation. More precisely:

Reverse engineering

In order to benefit strategically from "unreliability", a supplier may make irreversible decisions facilitating imitation, for example by investing in reverse engineering capability. To capture this possibility, suppose that, in stage 0, each supplier must decide whether to invest publicly in a reverse engineering technology, which costs F but then allows to duplicate any innovation with probability θ.

By construction, suppliers who do not invest in reverse engineering capability cannot disclose their customers' information. Consider now the case of an unreliable supplier who did invest in such capability. If the supplier is integrated, it will never provide internal information to its independent rival, since the gain from doing so cannot exceed δ, and thus never compensates for the resulting loss in downstream profit, ∆δ. In contrast, any supplier (integrated or not) would have an incentive to sell the information from an unaffiliated customer since doing so yields a gain δ.

An independent supplier will however never invest in reverse engineering technology, as this would put its business at risk. Suppose for example that the rival does not invest in reverse engineering. Not investing then leads to symmetric competition and zero profit, whereas investing would cost F without bringing any benefit, since the rival would win the competition for customers. Suppose instead that the rival invests, and consider first the competition for independent customers. Investing as well leads to symmetric competition between equally unreliable suppliers, resulting in a net loss F , whereas not investing saves that cost and moreover confers a comparative advantage. As for an integrated customer, investing as well is costly and yields a comparative disadvantage whereas not investing yields symmetric competition.

Therefore, if both suppliers are vertically separated, the only equilibrium is such that no one invests in reverse engineering. By contrast, an integrated firm might find it profitable to invest in reverse engineering, in order to benefit from the resulting foreclosure effect:32 Proposition 9 Independent suppliers never invest in reverse engineering. In contrast, as long as the technology is not too costly, an integrated supplier invests in reverse engineering in order to benefit from input foreclosure.

Proof. See Appendix B.

Guarantees

Suppliers can also provide financial and non-financial guarantees against information leakages. They can for example offer a financial compensation in case of imitation. To be effective, such compensation must exceed δ (covering the innovator's loss in case of imitation, ∆δ, would e.g. be sufficient). For example, signing a confidentiality agreement makes the supplier legally liable to some compensation; additional protection can also be offered, by increasing the amount to be paid and/or expanding the set of circumstances under which such compensation would be awarded. This may however expose the firms to potential losses arising from the uncertainty of legal proceedings, the risk of default, and so forth, and thus raises the associated transaction costs.

Alternatively, suppliers can provide non-financial guarantees such as "firewalls " -internal information barriers designed to ensure that confidential information is not passed on from one unit to another. This can for example consist in assigning distinct teams to competing customers, setting-up specific routines and procedures, adopting compliance programs prohibiting employees' communication of sensitive information, and so on.

These guarantees come at a cost, such as legal fees and damages, transaction costs, or ad hoc organizational choices (e.g., duplication of tasks, internal auditing teams, ...). Firms may choose to provide such costly guarantees in order to enhance their reputation; our analysis however suggests that integrated suppliers may lack such incentive.

To explore this issue, consider the same situation as above except that, in stage 0, the suppliers no longer need to invest in reverse engineering but can instead provide guarantees at a cost ϕ. 33 To avoid equilibrium multiplicity issues, we introduce some upstream differentiation along the lines discussed in section 2.5: in case of innovation, D 1 (resp. D 2 ) obtains a small additional surplus γ when dealing with U A (resp. U B ). have:

Proposition 10 As long as the benefit from differentiation γ is not too large and the cost ϕ is not excessive, it is a dominant strategy for any independent supplier to offer guarantees, while an integrated supplier offers no guarantee in order to benefit from foreclosure.

Proof. See Appendix C. Consider first the case of an independent supplier facing a reliable rival. If it is unreliable, it obtains a profit (corresponding to its comparative advantage γ) only when both downstream firms innovate; in contrast, if it is reliable it obtains this profit whenever its "best customer" innovates. Offering guarantees thus brings a benefit.

When facing instead an unreliable rival, an independent supplier -reliable or not -obtains its comparative advantage γ whenever its "best customer" innovates. Becoming reliable however allows the supplier to earn additional profit when its best customer is the sole innovator. In the case of an independent rival, superior reliability may moreover allow the supplier to win the competition even when its rival's best customer is the innovator. However, this extra pressure on the rival supplier benefits its best customer and fosters that customer's R&D efforts; by strategic substitutability, this results into lower R&D efforts by the reliable supplier's own best customer, which tends to reduce the supplier's expected profit. The overall effect on the reliable supplier's profit remains positive, however, as long as reliability matters more than the comparative advantage γ; in that case, it is a dominant strategy to offer guarantees as long as their cost is not excessive.

Suppose now that the integrated firm U A -D 1 competes against a reliable U B . The integrated firm then supplies its own subsidiary (and protects its innovation from imitation) but never wins the competition for the independent downstream firm, who always favors the rival. Therefore, U A -D 1 's variable profit is the same, whether or not it offers guarantees. Offering no guarantee however saves the cost ϕ and moreover increases U B 's market power over D 2 , which as before reduces D 2 's innovation effort. Therefore, when facing a reliable rival, the integrated supplier prefers to offer no guarantee.

Finally, note that focusing on a merger between D 1 and its favorite supplier, U A , is not restrictive since Appendix C shows that D 1 is indeed better off merging with U A rather than with the other, less favored supplier.

Strategic foreclosure in a dynamic context

We presented so far our analysis in a simple and rather static framework, in which the suppliers could somehow commit themselves to be able (or unable) to imitate. In a dynamic setting, however, the same insights apply even in the absence of any commitment capacity. Whenever imitation creates a profitable foreclosure effect, a vertically integrated firm has an incentive to exacerbate the threat of imitation and, as a result, vertical integration drastically affects suppliers' incentives to appear reliable.

To see this, we now develop a dynamic framework in which suppliers must decide whether to invest in costly reverse engineering or duplication capability. We first consider the case where investment has long-term effects. By undertaking such investment, even if it is costly and not observed by customers, and then exploiting its customers' information, an integrated firm can demonstrate its capability and enjoy the resulting foreclosure benefits in subsequent periods. We then consider a variant in which, in each period, the suppliers can exploit their customers' information (without being observed, but at a cost); this setting, in which suppliers must bear a cost each time they want to exploit their customers' information, thus rules out any "pre-commitment" on behalf of the suppliers. To introduce reputation concerns, we also assume that suppliers can be of two types, "bad" suppliers having a lower cost of imitation than "good" ones; we then show that, while independent suppliers would imitate their customers' innovation only when being bad, vertical integration gives good suppliers an incentive to do so as well, in order to degrade customers' perceptions and benefit from the resulting foreclosure effects.

Reverse engineering with repeated interaction

We start with the framework described in section 3.1, in which suppliers can invest F > δ to acquire reverse engineering capability, except that investment is no longer observable; we assume instead that it has long-lasting effects: firms now interact over two periods and, while the investment can take place at any point of time, once it is made reverse engineering becomes available in all (current and future) periods. In addition, duplication, and/or its impact on the innovator's profit, is observable; thus, a supplier who exploits its customer's information in the first period reveals that it is in a position to do so again in the second period. Finally, all firms maximize the sum of their expected discounted profits, using the same discount factor β.

Formally, the timing of the game is as follows:

• First period: t = 1 -In a first stage, the two downstream firms simultaneously choose their investments, denoted ρ 1 1 and ρ 1 2 . Innovation then succeeds or fails accordingly. -In a second stage, the two upstream firms simultaneously offer fixed price tariffs to each downstream firm, who then selects a supplier. The selected supplier decides whether to invest in reverse engineering capability: * if it does not invest, the supplier cannot decipher the relevant information; * if it invests, or its customer provides the information, the supplier can sell it (through a take-it-or-leave-it offer) to the other downstream firm.

• Second period: t = 2. The same two stages apply, with the caveat that any supplier who has invested in reverse engineering at t = 1 can decipher at no cost any customer's relevant information.

Note first that a supplier who has not invested in reverse engineering in the first period will not invest in the second. This is true whether the supplier is integrated or not, and, if it is, whether its customer is affiliated or not. The reason is that investing in the second period costs F , and cannot generate more than the maximum price the downstream rival is ready to pay for the innovation, i.e. δ < F .

We now study the first period technology choices, starting with the case of vertical separation.

Vertical separation

In the first period, an independent supplier will not invest in reverse engineering, as it requires a cost F that cannot be compensated by the additional profits in the two periods: exploiting the customer's information in the second period only would bring less than βδ < F , whereas exploiting the information in the first period (which makes sense only when one of its customers is the sole innovator) would bring at most δ < F and degrade the supplier's reputation, thus wiping out any future profit. Therefore, if all firms are independent, no supplier ever invests in reverse engineering. Both suppliers are thus always equally reliable and obtain zero profit in both periods. The equilibrium outcomes are thus constant over time: in each period t, each D i 's expected profit from innovation is

π i i = ρ t i ¡ ρ t j δ + ¡ 1 -ρ t j ¢ ∆ ¢ -C ¡ ρ t i ¢ .
Investment behaviors are thus

ρ t i = R ¡ ρ t j ¢
, which yields for both firms and both periods the same investments and profits as in the static case: ρ t i = ρ * and π t i = π * .

Vertical integration

Assume now that U A and D 1 have merged, and first consider the second period competition stage. As noted above, the integrated firm protects its own subsidiary even if it has already invested in reverse engineering, and since the independent U B never invests in reverse engineering, it thus never exploits any customer's information. However, D 2 's procurement decision (when being the sole innovator) depends on its beliefs about the integrated supplier's ability to exploit its innovation. If D 2 believes that U A did not invest in reverse engineering in the first period (and thus will not invest either in the second period), then upstream competition remains symmetric, among reliable suppliers; suppliers thus obtain zero profit in the second period, whereas downstream firms invest ρ 2 i = ρ * and obtain π 2 i = π * . Suppose instead that D 2 , being the sole innovator, believes that U A previously invested in reverse engineering. Assuming passive beliefs,34 asymmetric upstream competition then leads U A to offer a discount -θδ and U B to win with a positive tariff reflecting its comparative advantage, thus giving D 2 the same expected profit as U A 's offer. The expected profits of the investing firms are therefore:

π 2 A1 = π 2 1 = Π (ρ 1 , ρ 2 ) and π 2 2 = Π θ (ρ 1 , ρ 2 ).
A foreclosure effect thus arises and, as a result, in the second period the investments are ρ 2 1 = ρ + θ > ρ * and ρ 2 2 = ρ - θ < ρ * , and the profits become:

π 2 A1 = π V I A1 > π * , π 2 2 = π V I 2 < π * , and π V I B = ρ - θ ¡ 1 -ρ + θ ¢ θ (∆ -2δ) .
Consider now the first period. When both firms innovate, or none of them innovates, upstream competition is symmetric and leads the suppliers to supply at cost. The two firms obtain δ in the former case and 0 in the latter case, and in both cases no supplier has an incentive to invest in reverse engineering (U B never invests anyway, and U A would not be able to demonstrate its capacity to imitate D 2 's innovation). In contrast, U A may be tempted to invest in reverse engineering when selected by a downstream firm that is the sole innovator; more precisely:

• If the innovator is D 1 , U A cannot benefit from investing in reverse engineering: even if it wants to sell its subsidiary's innovation, it is cheaper to simply obtain it from D 1 ; therefore, selling the information will not be interpreted as "having invested in reverse engineering ", which in turn implies that it is not worth selling it (it only brings δ and reduces downstream profit by ∆δ > δ).

• If the innovator is D 2 , investing in reverse engineering entails a net loss Fθδ at t = 1, but gives U B extra market power at t = 2 and thus increases the profit of the integrated firm in the second period by π V I A1π * ; therefore, if:

F -θδ < β ¡ π V I A1 -π * ¢ , ( 21
)
the integrated supplier will invest in reverse engineering if selected by the downstream rival.

Thus, under (21), when D 2 is the only innovator at t = 1, it will anticipate that selecting the integrated supplier will lead it to invest in reverse engineering. U B thus benefits from a comparative advantage over U A ; however, U A is willing to offer a discounted tariff, TA , reflecting not only the value from duplication in period 1, but also the additional profit it would obtain in period 2 if selected in period 1 and investing in reverse engineering:

TA = F -θδ -β ¡ π V I A1 -π * ¢ < 0.
In contrast, the best tariff that U B is willing to offer, TB , takes into account the additional profit it could achieve in period 2 if its rival, U A , is instead selected in period 1, and is thus such that:

TB = βπ V I B > 0.
Finally, U B wins the competition when its best offer dominates:

∆ -TB + βπ * > ∆ -θ (∆ -δ) + βπ V I 2 -TA , which amounts to: θ (∆ -2δ) > β ¡ Π V I -Π V S ¢ -F, where Π V I -Π V S = π V I A1 + π V I 2 + π V I B -2π * .
denotes the impact of foreclosure on total industry profit. This condition thus amounts to saying that the industry loss resulting from duplication in period 1 exceeds the increase in profit (if any) resulting from foreclosure in period 2 (in particular, it is satisfied whenever foreclosure reduces industry profit).

Two cases can thus be distinguished. When θ (∆ -2δ) > β ¡ Π V I -Π V S ¢ -F , U B wins the competition with a tariff, T B > TB , that leaves D 2 (almost) indifferent with accepting U A 's best tariff, TA . The integrated supplier then never invests in reverse engineering, since it is not selected when D 2 is the sole innovator. Therefore, both investing firms expect a symmetric profit π * in period 2, whatever the R&D outcome in period 1; as a result, R&D incentives are solely driven by the expected profits obtained in period 1. Foreclosure however arises in that period: when it is the only successful innovator, U A would invest in reverse engineering if selected, which allows U B to charge a higher tariff (T B > TB > 0).

When instead θ (∆ -2δ) < β ¡ Π V I -Π V S ¢ -F , U
A wins the competition in period 1 (and then invests in reverse engineering) with a tariff that leaves D 2 (almost) indifferent with accepting U B 's best tariff. Compared with the case of vertical separation, both downstream firms are less willing to invest. In addition, since U A wins the competition and invests in reverse engineering, foreclosure arises again in period 2.

The following proposition summarizes this discussion:

Proposition 11 Suppose that (21) holds. Then:35

• when θ (∆ -2δ) > β ¡ Π V I -Π V S ¢ -F , no firm ever invests in reverse engineering but the threat of doing so generates foreclosure in period 1;

• when θ (∆ -2δ) < β ¡ Π V I -Π V S ¢
-F , in period 1 both firms are less willing to invest in R&D than in the absence of integration, and the integrated firm moreover invests in reverse engineering when the independent rival is the sole innovator; foreclosure then arises in period 2.

Foreclosure thus arises (either in period 1 or 2) whenever (21) holds. Repeating the interaction over T > 2 periods further weakens this condition, which becomes:

F -θδ < 1 -β T 1 -β β ¡ π V I A1 -π * ¢ . ( 22
)
The right-hand side increases in T , which thus relaxes the condition. In particular, if β is close enough to 1, then condition ( 22) is always satisfied for T large enough.

Reputation

In the previous section, investment in reverse engineering was not observable but had long-lasting effects, which somehow allowed (integrated) suppliers to "commit" themselves to being unreliable in future periods. We now consider an alternative situation in which, in each period, suppliers must invest in order to exploit their customer's information in that period. In this context, we show that, even if its investment decisions are unobserved by customers, an integrated supplier has an incentive to build a reputation of exploiting such information. To this aim, we now assume that, while some suppliers must spend an amount F > δ in order to exploit a customer's information (e.g., by investing in specific reverse engineering), others can do so at no cost. We will refer to the former as "good " types and to the latter as "bad " types. 36 For the sake of exposition, we assume that only one supplier may be unreliable: U A , say, is good with probability p and bad with probability 1p, whereas U B is good with probability 1.

We extend the two-stage game of section 2.1 by adding a last stage where suppliers, reliable or not, may choose to sell the information:

• In stage 1, D 1 and D 2 simultaneously choose their R&D efforts and then innovate with probabilities ρ 1 and ρ 2 ; the success or failure of their innovation efforts is observed by all firms.

• In stage 2, U A and U B simultaneously offer lump-sum tariffs to each independent downstream firm; we will denote by T hi the tariff offered by U h to D i (for h = A, B and i = 1, 2); each D i then chooses its supplier.

• In stage 3, suppliers (at cost F if "good", at no cost otherwise) can sell a customer's information to its unsuccessful downstream rival, through a takeit-or-leave-it offer, in which case the downstream rival is able to duplicate the innovation.

We assume that this game is played over two periods, 1 and 2, and that U A privately learns its own type in the third stage of period 1, thus after price competition but before deciding whether to exploit its customers' information. 37 Besides the outcomes of the R&D projects, the other firms only observe whether innovation eventually takes place. Thus, if only one firm has innovated but both firms launch a new product, it becomes clear that the innovator's information has been exploited. For the sake of exposition, we make the following simplifying assumptions: (i) there is no discounting (β = 1); (ii) the imitation process is perfect (θ = 1); and (iii) the gain from duplication is "negligible": that is, we will set δ = 0, but suppose that a bad supplier chooses to exploit its customer's information whenever this yields the same expected payoff as not exploiting the information). 38 We consider below two scenarii, in which U A is either independent or integrated (U B is independent in both scenarii), and show that integration drastically affects U A 's incentive to build a reputation of being reliable:39 whereas a good independent supplier wants to maintain a good reputation, an integrated firm always prefers instead to appear as a bad supplier, so as to exacerbate the threat of imitation and benefit from the resulting strategic foreclosure effect. We only sketch the intuition here, starting with the second period before turning to the first one; the detailed analysis is presented in Appendix D.

Second period

Let p A denote the revised probability that U A is good at the beginning of period 2.

• Price competition. Since δ = 0, profits can only be earned when a single firm, D i , say, innovates. If D i is vertically integrated, then its upstream unit will supply it at cost and protect its innovation. Suppose now that D i is an independent firm and selects U A . Whether U A is integrated or not then does not affect its reliability: since exploiting D i 's information brings only a negligible revenue, U A prefers not to do so when it is "good" (to avoid the cost F ), but chooses to do so when it is "bad" (since it then faces no cost). δ = 0 also implies that U A obtains the same gain whatever its type; it is therefore natural to focus on pooling equilibria (that is, both types of U A offer the same T A ) with passive beliefs (that is, a deviating offer does not affect D i 's posterior beliefs). The price competition stage then boils down to a standard asymmetric Bertrand duopoly, in which U A offers T A = 0 while U B wins with a tariff reflecting its comparative advantage, T B = (1p A ) ∆. In the limit case p A = 1, T B = T A = 0 and we can assume that U B still wins the competition -selecting U A would actually be a weakly dominated strategy for D i .

• R&D decisions. Given the outcome of price competition, in the case of vertical separation each D i 's expected profit is equal to:

Π i = ρ i ¡ 1 -ρ j ¢ p A ∆ -C (ρ i ) . (23)
The resulting equilibrium R&D efforts are symmetric but lower than ρ * :

ρ 1 = ρ 2 = ρ * (p A ) < ρ * = ρ * (1) . (24)
Each downstream firm then obtains:

π * (p A ) ≡ ρ * (p A ) (1 -ρ * (p A )) p A ∆ -C (ρ * (p A )) . (25)
If U A is vertically integrated with D 1 , D 2 's expected profit remains given by (23), but D 1 benefits from the protection of its innovation and its expected profit is thus again given by (2). The resulting equilibrium is thus of the form

ρ 1 = ρ+ (p A ) > ρ * (p A ) > ρ 2 = ρ-(p A )
, characterized by the first-order conditions:

C 0 (ρ 1 ) = (1 -ρ 2 ) ∆, C 0 (ρ 2 ) = (1 -ρ 1 ) p A ∆. (26)
The resulting profits are then of the form π A1 = π+ (p A ) ≥ π * (p A ) (with a strict inequality whenever p A < 1), π 2 = π-(p A ) ≤ π * (p A ) (with a strict inequality whenever 0 < p A < 1), and πB (p

A ) ≡ ρ-(p A ) ¡ 1 -ρ+ (p A ) ¢ (1 -p A )
∆ (which is positive whenever 0 < p A < 1, and zero otherwise).

An increase in U A 's reputation fosters upstream competition and thus benefits downstream independent firms; in contrast, the integrated firm U A -D 1 benefits from a reduction in p A , since it raises its rival's cost. Indeed, we have:

Proposition 12 In the second period, an independent U A always obtains zero profit. All other equilibrium investments and profits are continuous in the revised belief p A ; they coincide with the benchmark levels ρ * and π * when p A = 1, and a reduction in p A :

(i) reduces independent downstream firms' investments and profits, down to 0 for p A = 0.

(ii) benefits instead U A -D 1 in case of integration, raising its investment and profit up to the monopoly level for p A = 0.

Proof. See Appendix D.1.

First period

Consider now the first period. From proposition 12, under vertical separation U A 's profit in the second period does not depend on its reputation; as a result, U A behaves as in the last period and, while U B benefits from a comparative advantage when a single firm innovates, it does not appropriate the entire value of the innovation, and thus both downstream firms invest in R&D. In contrast, a vertically integrated firm benefits from a bad reputation. Building on this insight, we now show that, when F is not too large, if selected by D 2 when it is the sole innovator, U A -D 1 would exploit D 2 's information even when it is of a good type. As a result, there is complete foreclosure in the first period: D 2 does not invest in R&D, and only the integrated firm is active in that period.

Vertical separation. Consider first the case of vertical separation. In the price competition stage, symmetric Bertrand competition yields zero profit for the suppliers when either both or none downstream firm innovates. Suppose now that D i is the sole innovator and selects U A . Since U A always obtains zero profit in the future, it then behaves as if this were the last period:

• With probability p, U A learns that it can costlessly exploit D i 's information and chooses to do so; this leads to p A = 0 in the second period, and thus to zero profit for all suppliers and downstream firms.

• With probability 1p, U A learns that exploiting D i 's information would cost F and thus refrains from doing so; this leads to p A = 1 in the second period, and thus again to zero profits for both suppliers but positive expected profits, π * , for the downstream firms.

Since U A also obtains zero profits if not selected, it is willing to supply at cost ( TA = 0), thereby giving D i an expected profit equal to p (∆ + π * ). This is better than what D i would obtain by rejecting all offers, namely π * (p) (< p∆). However, U B : (i) is more reliable (D 2 obtains ∆ with probability 1 rather than p); and (ii) if needed, would be willing to offer a discount, in order to avoid U A 's type being revealed: U B would then obtain zero profit, whatever the realized type ( πB (0) = πB (1) = 0) whereas it obtains πB (p) > 0 if U A 's type remains uncertain. Appendix D.2 shows that, as a result, U B wins the competition but, due to the competitive pressure exerted by U A , cannot extract all the value from the innovation. Each downstream firm then invests an amount ρV S (p), which is positive as long as p > 0, and obtains a total expected discounted profit of the form πV S (p) + π * (p), where πV S (p) > 0 for any p > 0.

Vertical integration. We now turn to the case where U A is vertically integrated with D 1 . U A then always protects the innovation of its own downstream division D 1 : selling the innovation to D 2 would reduce the first period profit (from ∆ to 0) and, since the integrated firm has direct access to D 1 's information, would not convey any relevant information on U A 's ability to exploit D 2 's innovation in period 2. We now study U A -D 1 's decision to imitate D 2 's innovation, before turning to the price competition stage; we then draw the implications for the overall equilibrium of the game.

Suppose that D 2 is the only successful innovator and has selected U A as supplier. Let denote by

F (p) ≡ π+ (p) -π * > 0 (27)
the expected gain that the integrated firm obtains in period 2 from exploiting D 2 's information in period 1. Intuitively, when this expected gain exceeds the actual cost F , the integrated supplier has an incentive to exploit its customer's information even when being good, in order to maintain the ambiguity and benefit from the resulting foreclosure effect. Selecting U A then leads to imitation with probability 1, and thus brings no information about U A 's type. It follows that, when F < F , U A -D 1 and D 2 are actually better off not dealing with each other: (i) the value of D 2 's innovation would be dissipated via imitation; (ii) future profits are unaffected since D 2 would not learn anything about U A 's type; but (iii) by not supplying D 2 , U A avoids the risk of having to incur the cost F to maintain its (bad) reputation, in case it turns out being a good type. As a result, U B can extract the whole value from D 2 's innovation, ∆; it follows that D 2 never invests in the first period, and thus U A -D 1 benefits from a monopoly position in that period. It thus invests ρ m = R (0) and obtains a total expected discounted profit equal to π m + π+ (p), where π m = max ρ (1ρ) ∆-C (ρ) > πV S (p) and π+ (p) > π * (p) whenever p < 1.

We thus have:

Proposition 13 In the case of vertical separation, U A obtains zero profit while both downstream firms invest a positive amount in the first period and obtain an expected profit equal to π * (p) in the second period. In contrast, in the case of vertical integration, if F < F (p) the integrated firm completely forecloses the market in period 1.

Proof. See Appendix D.2. Thus, U A (who obtains zero profit in the case of separation) and D 1 obtain larger joint profits when they are vertically integrated, since they benefit from strategic foreclosure in both periods. Note that, in the first period, complete foreclosure can arise even when U A is initially perceived as quite reliable (i.e., p close to 1 -the threshold F (p) however goes down to 0 as p goes to 1).

Lessons

Welfare implications When F is not too large, a vertical merger between U A and D 1 generates complete foreclosure in the first period, thereby discouraging any rival R&D investment in that period. Vertical integration however protects the integrated firm against the risk of imitation, which fosters its own incentives to invest in R&D. We now discuss the impact of these two effects on innovation and consumer surplus.

Consumer surplus in periods 1 and 2 is respectively equal to:

SC V I 1 = 0, SC V I 2 = ρ+ (p)ρ -(p)∆. (28)
In the case of vertical separation, D 2 buys from U B in the first period, which brings no information about U A 's type. As a result, consumer surplus is equal to:

SC V S 1 = (ρ V S ) 2 ∆, SC V S 2 = ρ * (p) 2 ∆. ( 29
)
It can be checked that, in the second period, consumer surplus is higher in the case of vertical integration; this comes from the "protection" effect just mentioned: while D 2 behaves in the same way in the two scenarii (in both cases, U B supplies D 2 with a positive tariff reflecting its comparative advantage over U A , who is perceived to be reliable only with probability p < 1), when vertically integrated D 1 obtains the full value ∆ when it is the sole innovator, which fosters its own R&D effort as well as the probability that both firms innovate: ρ+ (p)ρ -(p) > (ρ * (p)) 2 . However, the difference tends to disappear when p is large (since ρ+ (1) = ρ-(1) = ρ * (1) = ρ * ).

In contrast, when F < F (p), then in the first period consumers obtain zero surplus in case of vertical integration, since the independent rival is then entirely foreclosed, whereas they obtain a positive surplus in the case of separation, which moreover increases with p.

This yields:

Proposition 14 As long as F < F (p), vertical integration harms consumer surplus when p is large enough.

A similar insight applies to total welfare: when p is large, vertical integration has not much impact on innovation and thus on welfare in the second period, whereas (as long as F < F (p)), it has a drastic impact on the rival's innovation and thus on welfare in the first period.

Which merger? A related question concerns the choice of the merger partner. Suppose for example that D 1 merges instead with the more reliable supplier, U B . In the second period, U B supplies D 1 at cost whenever it innovates. In contrast, if D 2 is the sole innovator, than asymmetric Bertrand competition leads U A to offer T A = 0 and U B to win with T B = (1p A )∆, where p A denotes as before the revised probability, at the beginning of the second period, that U A is reliable. D 2 's expected profit thus remains given by (23), but D 1 's investment now maximizes:

π B1 = ρ 1 (1 -ρ 2 )∆ + (1 -ρ 1 )ρ 2 (1 -p A )∆ -C(ρ 1 ).
Since the vertically integrated firm now benefits from supplying its rival when it is the sole innovator, it invests less than before. The resulting investment levels are such that ρ

1 = ρ+ (p A ) < ρ+ (p A ) and ρ 2 = ρ-(p A ) > ρ-(p A ). Similarly, the profit of D 2 , π-(p A ), satisfies π-(p A ) > π-(p A )
; for the integrated firm, the resulting profit π+ (p A ) may also exceed π+ (p A ) since, while D 1 now faces a more aggressive rival, it also benefits from supplying it. Note that the equilibrium outcome coincides with the benchmark case (ρ i = ρ * and profit π i = π * ) for p A = 1 and with the monopoly case (ρ 1 = ρ m , ρ 2 = 0 and π B1 = π m , π 2 = 0) for p A = 0.

Let us now turn to the first period. If D 2 is the sole innovator, the independent U A would be willing to supply at cost, and if selected would imitate only when being bad, thus giving D 2 an expected profit equal to: πA 2 = p(∆ + π * ). This offer is attractive (rejecting all offers would give D 2 an expected profit only equal to π-(p) < p∆), which implies that the integrated supplier, U B , can no longer extract the full value of D 2 's innovation. As a result, complete foreclosure does not arise anymore in period 1. While D 1 might enjoy a greater profit in the second period when merging with U B rather than U A (if π+ (p) > π+ (p)), when p is close to 1 (in which case π+ (p) and π+ (p) are both close to π * ), this cannot offset the profit loss stemming from the reduced foreclosure effect in the first period. As a result, we have:

Proposition 15 When p is large enough, and F < F (p), the most profitable vertical merger involves the supplier whose reputation is uncertain, so as to benefit from a larger foreclosure effect.

Proof. See Appendix D.3.

Remark: the distinctive nature of imitation. In this section, we validate our previous working assumption, by showing that vertical integration indeed fosters imitation concerns. One could question whether a similar analysis might apply to the original raising rivals' cost arguments, in which the integrated firm supplies independent rivals at a higher price. This could for example be the case if a supplier could take irreversible decisions (as in sections 3 and 4.1) that affect the cost or the quality of their input. While an independent supplier would have the incentives to maintain good quality or low cost, an integrated firm might instead degrade its cost or quality conditions in order to benefit from the resulting foreclosure effect. In the absence of such irreversibility, however, the reputation argument developed here for imitation concerns is less easily transposed to cost or quality considerations. If for example the uncertain type concerns the cost of "being unreliable" (i.e., degrading quality or cost conditions), 40 then a "bad" supplier, namely, a supplier who could degrade performance at little cost, would have no incentive to do so anyway in the last periods, which defeats the reputation argument. If, by contrast, the type concerns the cost of "being reliable" (i.e., having the capacity of delivering good quality at low price), an integrated firm could be tempted to pretend being unreliable, but to be consistent this would require degrading the performance of its own subsidiary, which would reduce and possibly offset the benefit from foreclosure.

Conclusion

This article shows that vertical integration may generate foreclosure. The seminal paper by Ordover Saloner and Salop (1990), which provided a first consistent vertical foreclosure theory, relied on two critical assumptions. First, the vertically integrated firm had to be able to commit itself not to supply downstream rivals, in order to give greater market power to remaining suppliers. Second, in order to weaken downstream competition, this enhanced market power had to translate into higher input prices (as opposed to higher fixed fees or profit-based royalties, say). In our framework, foreclosure relies instead on innovation incentives and on the threat of information leakages between the integrated supplier and its downstream subsidiary. Thus, whenever vertical integration creates or exacerbates such security problems, foreclosure arises even absent any commitment capability -the integrated supplier's reliability concern suffices to weaken its ability to supply downstream rivals -or any ex post contractual inefficiency -downstream rivals must therefore share the value of their innovation with the remaining suppliers, which suffices to discourage their R&D efforts.

We further show that vertical integration indeed drastically affects a supplier's incentive to protect or exploit its customers' innovation. Where an independent supplier has an incentive to protect its customers' innovation, so as to maintain its reputation as a reliable supplier, an integrated supplier can instead prefer to degrade that reputation, in order to enjoy the resulting strategic foreclosure benefit. This analysis has direct policy implications for antitrust or merger policy. For example, even in an industry where (possibly costly) instruments exist for protecting customers' innovation (such as firewalls, compensating guarantees, and so forth), a merged entity may lack the incentives to invest in such instruments -and may rather choose to invest in (possibly costly) reverse engineering technology or other ways to exploit its customers' innovation. Therefore, such protective instruments should be required for merger approval. Besides, our results speak in favor of ex-ante rather than ex-post merger control. In our model, no imitation happens in equilibrium, as the very threat of information disclosure is sufficient to create foreclosure. This tends to highlight the inefficiency of a merger authorization followed by an ex-post control of anticompetitive behavior: if protective measures are not required at the time of the merger, the integrated firm has no incentives to provide such measures and foreclosure may arise without any ex post anticompetitive behavior.

While this paper emphasizes the adverse impact of vertical integration on information leaks and foreclosure, the same analysis could have different implications in different industry situations.

For instance, in markets where the risk of information leaks already exists even in the absence of vertical integration, a vertical merger would again exacerbate this risk for the independent rivals, but would also induce the integrated firm to better protect its own subsidiary, the overall impact of vertical integration on industry innovation, consumers, and welfare would then be more ambiguous. Also, if the upstream market is quasi-monopolized, then vertical integration and the associated foreclosure effect may well distort downstream competition in a way that reduces the merging parties' profit. This concern has for instance been mentioned in 1999 by General Motors (GM) as a motivation for spinning-off its auto parts subsidiary Delphi, so as to enable it to contract with other automakers, which were reluctant to rely on Delphi as long as it was a unit of GM. 41 A similar concern may underlie AT&T's 1995 voluntary divestiture of its manufacturing arm, AT&T Technology (now Lucent), as the 1996 Telecommunication Act was due to allow the RBOCs to compete with AT&T on the long distance market. 42 Finally, while we focus on situation where the information leaks intensify competition and dissipate profits, Milliou and Petrakis (2010) consider an alternative situation in which information flows increase industry profit: Namely, imitation expands demand more than it intensifies competition. In this context, the integrated firm may well choose to communicate information from its own subsidiary to the downstream rival and vertical integration may benefit consumers as well as firms.

investing firms are then respectively Π A1 = Π (ρ 1 , ρ 2 ), and Π 2 = Π θ (ρ 2 , ρ 1 ) .The equilibrium investments are thus ρ

1 = ρ + θ > ρ * > ρ 2 = ρ - θ , and U A -D 1 's expected profit is Π + θ > Π * .
U A -D 1 therefore invests in reverse engineering whenever F < Π + θ -Π * .

C Guarantees

In this Appendix, we prove Proposition 10, assuming that firm D 1 (resp. D 2 ) obtains a small surplus γ (in case of innovation) when buying from his favored supplier U A (resp. U B ). Suppliers' reliability is irrelevant when both downstream firms' innovation efforts are successful. In that case, for each D i , asymmetric Bertrand competition leads D i 's favored supplier to win the competition with a tariff appropriating the surplus γ: letting "f " designate the favored supplier and "n" refer to the other, non-favored supplier, U n offers D i a tariff T n = 0, but U f wins with a tariff (slightly below) T f = γ. As a result, each D i obtains a profit equal to δ.

Suppliers' reliability instead matters when only one downstream firm successfully innovates. While an integrated supplier will always protect the information from its own subsidiary, unreliable suppliers would be willing to trade the information obtained from their independent customers. We now study the implications under vertical separation and partial integration.

Vertical separation.

• If both suppliers are reliable, and only D i innovates, then asymmetric Bertrand competition leads D i 's favored supplier to win with a tariff reflecting its comparative advantage; D i thus obtains ∆ while its favored supplier obtains γ. Each D i 's expected profit is therefore given by Π i = Π ¡ ρ i , ρ j ¢ , and equilibrium investments are thus ρ 1 = ρ 2 = ρ * . Since suppliers obtain γ whenever the downstream firm that favors them innovates, their equilibrium expected profits are both equal to:

Π V S rr ≡ ρ * γ.
• Suppose now that both suppliers are unreliable, and that D i is the only successful innovator. Asymmetric Bertrand competition leads the non-favored supplier, U n , to offer T n = -θδ, while the favored supplier wins with T f = γθδ, and then sells (at "full" price θδ) the information to the downstream rival, who duplicates the innovation with probability θ. Thus,

D i obtains θδ + (1 -θ) ∆ + γ -T f = θδ + (1 -θ) ∆ -T n = ∆ -θ (∆ -2δ) ,
while its favored supplier obtains

T f + θδ = γ. Ex ante, each D i 's expected profit is thus Π i = Π θ ¡ ρ i , ρ j ¢ . Both best responses are thus of the form ρ i = R θ ¡ ρ j ¢ < R ¡ ρ j ¢
, and equilibrium investments are symmetric:

ρ 1 = ρ 2 = ρ *
θ < ρ * . Suppliers' equilibrium expected profits are thus lower than before and now equal to Π V S uu ≡ ρ * θ γ.

• Suppose now that U A , say, is unreliable whereas U B is reliable. As long as reliability matters more than suppliers' differentiation (namely, as long as γ < θ (∆ -2δ)), then when D i is the only successful innovator Bertrand competition results in U A offering T Ai = -θδ and U B winning with a tariff that leaves D i almost indifferent between the two offers. Thus, when D 1 is the sole innovator, U B charges T B1 = θ (∆ -2δ)γ and D 1 obtains ∆θ (∆ -2δ) + γ; when instead D 2 is the only successful innovator, then U B wins by offering T B2 = θ (∆ -2δ) + γ and D 2 obtains ∆θ (∆ -2δ). The expected profits of the two downstream firms are thus respectively:

Π 1 = Π γ θ (ρ 1 , ρ 2 ) ≡ ρ 1 (ρ 2 δ + (1 -ρ 2 ) (∆ -θ (∆ -2δ) + γ)) -C (ρ 1 ) = Π θ (ρ 2 , ρ 1 ) + ρ 1 (1 -ρ 2 ) γ, and Π 2 = Π θ (ρ 2 , ρ 1 ) .
Best responses are therefore of the form ρ 2 = R θ (ρ 1 ) and ρ 1 = R γ θ (ρ 2 ), which is characterized by the first-order condition:

C 0 (ρ 1 ) = ρ 2 δ + (1 -ρ 2 ) (∆ -θ (∆ -2δ) + γ) ,
and thus satisfies R θ (ρ) < R γ θ (ρ) < R (ρ). Note that D 1 benefits from U B 's superior reliability, as it forces its favorite supplier, U A , to concede better terms (that is, U A gives back γ). As a result, equilibrium investments are asymmetric and such that ρ 1 = ρ+ > ρ * θ > ρ 2 = ρ-: U B 's superior reliability actually reduces its best customer's R&D effort, since its rival, D 1 , who benefits from U B 's competitive pressure on U A , becomes more aggressive.

Note that U A now obtains a positive profit only when both downstream firms' innovation efforts are successful. Its expected profit is equal to:

Π A = Π V S ur ≡ ρ-ρ+ γ,
whereas U B 's expected profit is equal to:

Π B = Π V S ru ≡ ρ-ρ+ γ + ρ-¡ 1 -ρ+ ¢ (θ (∆ -2δ) + γ) + ¡ 1 -ρ-¢ ρ+ (θ (∆ -2δ) -γ) . U A 's expected profit is lower than Π V S rr , since ρ-ρ+ < ρ-< ρ * θ < ρ * . As for U B 's expected profit, it exceeds Π V S
uu whenever reliability matters sufficiently more than product differentiation. For example, when γ < γ V S ≡ θ (∆ -2δ) /2, then ex post U B obtains at least γ whenever at least one firm innovates, and thus

Π V S ru > ρ+ γ > ρ * θ γ = Π V S uu .
Therefore, as long as γ < γV S we have:

Π V S uu < Π V S ru and Π V S ur < Π V S rr .
This, in turn, implies that providing guarantees constitutes a dominant strategy whenever ϕ < ϕ

V S ≡ min © Π V S rr -Π V S ur , Π V S ru -Π V S uu ª .
Vertical integration. Suppose now that U A and D 1 are vertically integrated whereas U B and D 2 remain independent. Vertical integration protects D 1 against imitation and moreover allows it to internalize the full value of its innovation.

• Suppose first that the independent supplier is at least as reliable as the integrated supplier (that is, both suppliers are reliable, both are unreliable, or U A is unreliable whereas U B is reliable). U A -D 1 's expected profit is then equal to:44

Π A1 = Π γ (ρ 1 , ρ 2 ) ≡ ρ 1 (ρ 2 δ + (1 -ρ 2 )∆ + γ) -C (ρ 1 ) ,
The corresponding best response, ρ 1 = R γ (ρ 2 ), is characterized by the first-order condition:

C 0 (ρ 1 ) = ρ 2 δ + (1 -ρ 2 ) ∆ + γ.
and

Π 2 = Π γ θ (ρ 2 , ρ 1 ) = ρ 2 (ρ 1 δ + (1 -ρ 1 ) (∆ -θ (∆ -2δ) + γ)) -C (ρ 2 ) . D 2 's best response is thus ρ 2 = R γ θ (ρ 1 ), whereas U A -D 1 's best response is of the form ρ 1 = R (ρ 2 )
, characterized by the first-order condition:

C 0 (ρ 1 ) = ρ 2 δ + (1 -ρ 2 ) ∆ + γ -ρ 2 (θ (∆ -2δ) -γ) = ρ 2 (δ -(θ (∆ -2δ) -γ)) + (1 -ρ 2 ) ∆ + γ.
We will denote by (ρ 1 ,ρ 2 ) the corresponding equilibrium investments. U B 's expected profit is then equal to:

Π B = Π V I ur ≡ ρ1 ρ2 γ. • Let us now study the reliability decisions. If U A -D 1 chooses not to be reliable, then as long as ρ γ- θ > 0 (that is, as long as there is only partial foreclosure, or θ < 1), U B benefits from being reliable, since this increases its expected profit from Π V I uu to

Π V I ru = Π V I uu +ρ γ- θ ¡ 1 -ρ γ+ θ ¢ θ (∆ -2δ) > Π V I uu .
If instead U A -D 1 chooses to be reliable, U B 's benefit from reliability is equal to:

Π V I rr -Π V I ur = ¡ ρ γ--ρ 1 ρ2 ¢ γ.
When γ tends to zero, ρ γ-converges to ρ * θ , solution to ρ = R θ (ρ), whereas (ρ 1 ,ρ 2 ) tends to ¡ ρ0 1 ,ρ 0 2 ¢ , which in particular satisfies ρ 2 = R θ (ρ 1 ). In the limit, the difference ρ γ--ρ1 ρ2 thus converges towards ρ * θ -ρ0 1 ,ρ 0 2 , which is positive: since both (ρ * θ , ρ * θ ) and ¡ ρ0 1 ,ρ 0 2 ¢ lie on the best response ρ 2 = R θ (ρ 1 ), which has a negative slope, ρ * θ is greater than either ρ0 1 or ρ0 2 , and thus (since moreover ρ0 i ≤ 1) exceeds their product. Therefore, there exists γ V I such that Π V I rr -Π V I ur > 0 as long as γ < γ V I . In this range, it is a dominant strategy for the independent supplier to offer guarantees as long as ϕ < ϕ V I ≡ min © Π V I rr -Π V I ur , Π V I ru -Π V I uu ª . Consider now the reliability decision of the integrated firm, when facing a reliable rival. Being reliable yields an expected profit equal to Π γ (ρ γ+ , ρ γ-)ϕ, whereas being unreliable yields:

Π γ ¡ ρ γ+ θ , ρ γ- θ ¢ = max ρ 1 Π γ ¡ ρ 1 , ρ γ- θ ¢ > max ρ 1 Π γ ¡ ρ 1 , ρ γ-¢ = Π γ ¡ ρ γ+ , ρ γ-¢ ,
where the inequality stems from ρ γ- θ < ρ γ-. It follows that it is best for U A -D 1 to be unreliable (by denying guarantees), so as to benefit from the foreclosure effect.

To recap:

• when γ < min © γ V S , γ V I ª , it is always a dominant strategy for an independent supplier to provide guarantees as long as the cost of doing so does not exceed min © ϕ V S , ϕ V I ª ).

• by contrast, when facing a reliable independent supplier, an integrated firm finds it optimal to appear unreliable by denying guarantees. Therefore, as p A increases from 0 to 1, the equilibrium profits increase from π * (0) = 0 to π * (1) = π * .

D Reputation

D.1.2 Vertical integration

If U A is vertically integrated with D 1 , the equilibrium profits are then of the form π A1 = π+ (p A ), π 2 = π-(p A ), and π B = ρ-(p A ) ¡ 1 -ρ+ (p A ) ¢ (1p A ) ∆. In particular, the effort and the profit of the vertically integrated firm increase as its perceived quality, p A , decreases; indeed, as p A decreases from 1 to 0:

• ρ-(p A ) decreases from the symmetric competitive level ρ * to 0 ;

• ρ+ (p A ) therefore increases ρ * to ρ m , the monopoly level satisfying C 0 (ρ m ) = ∆;

• as a result, π+ (p A ) increases from the competitive level π * to the monopoly level, π m = max ρ ρ∆ -C (ρ).

D.2 Proof of Proposition 13

We consider in turn the separation and integration cases.

D.2.1 Vertical separation

Suppose that D i , being the sole innovator, selects U A as an independent supplier. U A then behaves as if this were the last period, since it obtains zero future profit anyway; it thus exploits D i 's innovation only when that it is of a bad type.

The expected gross profits of D i , U A and U B are therefore respectively equal to:

π A i ≡ (1 -p) × 0 + p (∆ + π * ) = p (∆ + π * ) , π A A ≡ 0, π A B ≡ 0 + p × πB (1) + (1 -p) × πB (0) = 0,
where the superscript A denotes the selected supplier. Since U A also obtains zero profits if not selected, it is willing to supply at cost ( TA = 0), which would give D i an expected profit equal to: Therefore, U B wins the competition, by offering a tariff that gives i the same expected profit as πA i = p (∆ + π * ). Ex ante, each D i 's expected profit is therefore equal to:

πA i = π A i -TA = p (∆ + π * ) .
π i = ρ i (1 -ρ j )π A i + (1 -ρ i (1 -ρ j ))(0 + π * (p)) -C(ρ i ) = π * (p) + ρ i (1 -ρ j )(p(∆ + π * ) -π * (p)) -C(ρ i ).
It follows that the R&D equilibrium is symmetric:

ρ 1 = ρ 2 = ρV S (p) ,

D.2.2 Vertical integration

As discussed in the text, when U A is vertically integrated with D 1 , U A always protects the innovation of its own downstream division D 1 . If instead D 2 is the only successful innovator and selects U A , we have:

Lemma 16 When F < F , if D 2 is the sole innovator and selects U A , then the integrated firm imitates D 2 's innovation, whatever U A 's type.

if it turns out to be of a good type. Therefore, D 2 and U A -D 1 are better off not dealing with each other. In contrast, D 2 and U B can together generate an extra profit ∆. Thus, U B wins the competition but, since D 2 second-best option is to reject all offers, U B extracts all the value from D 2 's innovation, by offering a tariff T B = ∆. It follows that D 2 never invests in the first period, and thus U A -D 1 benefits from a monopoly position in that period; it thus maximizes:

π A1 = ρ 1 ∆ -C(ρ 1 ) + π+ (p),
and chooses the investment level ρ m .

Compared with the case of vertical separation, whenever p < 1, U A and D 1 joint profit increases in the second period, from π * (p) to π+ (p), and it also increases in the first period, since:

D.3 Proof of Proposition 15

In the second period, the investment levels, ρ 1 = ρ+ (p A ) and ρ 2 = ρ-(p A ), are characterized by the following first-order conditions:

C 0 (ρ 1 ) = (1 -ρ 2 (2 -p A ))∆, C 0 (ρ 2 ) = (1 -ρ 1 )p A ∆, (33)
and the resulting expected profits are: The conclusion then follows, since ψ (1) = 0 and ψ 0 (1) < 0 imply πB 2 > πA 2 for p smaller than but close to 1.

Whenever πB 2 > πA 2 , U B wins the competition with a tariff T B that leaves D 2 indifferent between accepting that or U A 's best offer, namely, such that:

T B = ∆ + π-(p) -p(∆ + π * ) = (1 -p) ∆ + π-(p) -pπ * .
Therefore, investing firms' total expected discounted profits become: These investment levels converge respectively to ρ * when p tends to 1, and in the limit the integrated firm's simply obtains π * in each period. In contrast, when D 1 merges with U A , as long as F < F (p), their joint profit is equal to π m + π+ (p), which tends to π m + π * as p tends to 1. Since U A moreover obtains zero profit when remaining independent, integrating U A is more profitable than integrating U B when p is close to 1.

π B1 = ρ 1 (1 -ρ 2 )∆ + (1 -ρ 1

D. 1

 1 Proof of Proposition 12 D.1.1 Vertical separationGiven the outcome of price competition, in the case of vertical separation the equilibrium profits are thenπ 1 = π 2 = π * (p A ) ≡ ρ * (p A ) (1ρ * (p A ))p A ∆ -C (ρ * (p A )) , π A = 0, π B = 2ρ * (p A ) (1ρ * (p A )) (1p A ) ∆.Note that the equilibrium profits increase with p A . Indeed, the envelope theorem yields:π * 0 (p A ) = ρ * (p A ) (1ρ * (p A ))∆ρ * (p A ) ρ * 0 (p A ) p A ∆, while differentiating the first-order condition C 0 (ρ * (p A )) = (1ρ * (p A ))p A ∆ yields: ρ * 0 (p A ) = (1ρ * (p A )) ∆ C 00 (ρ * ) + p A ∆ 0) . Therefore: π * 0 (p A ) = ρ * (p A ) (1ρ * (p A )) ∆C 00 (ρ * (p A ))C 00 (ρ * (p A )) + p A ∆ > 0.

  This is better than what D i would obtain by rejecting all offers, namely π *(p) = ρ * (p) (1ρ * (p)) p∆ -C (ρ * ) < p∆.If instead D i selects U B , then these expected profits depend on the prior belief (which remains unchanged for the second period) and become respectively:2ρ * (p) (1ρ * (p)) (1p) ∆ = 2ρ * (p) (1ρ * (p)) (1p) ∆.In the price competition stage, U B is thus willing to offer up to:* (p) (1ρ * (p)) (1p) ∆ < 0,which would give D i an expected profit equal to:πB i ≡ π B i -TB = ∆ + π * (p) + 2ρ * (p) (1ρ * (p)) (1p) ∆.This best offer beats U A 's one, since:πB i -πA i = ∆ + π * (p) + 2ρ * (p) (1ρ * (p)) (1p) ∆p (∆ + π * ) ≥ φ (p) ≡ (1p) ∆ + π * (p)pπ * ,where φ (p) > 0 for p < 1, since φ (1) = 0 and φ 0 (p) = -∆ µ 1ρ * (1ρ * ) C 00 (ρ * ) C 00 (ρ * ) + p∆ ¶ π * < 0.

  characterized by the first-order condition:C 0 (ρ) = (1ρ) [p (∆ + π * )π * (p)] .ρV S (p) moreover strictly increases from 0 to ρ * as p increases from 0 to 1:dρ V S dp = ¡ 1 -ρV S ¢ (∆ + π *π * 0 (p)) C 00 ¡ ρV S ¢ + p (∆ + π * )π * (p) ,where the numerator is positive since:π * 0 (p) = C 00 (ρ * ) C 00 (ρ * ) + p∆ ρ * (1ρ * ) ∆ < ∆,whereas the denominator is also positive since π * (p) < p∆. Each downstream firm then obtains a total expected discounted profit equal to πV S (p) + π * (p), where:πV S (p) ≡ ρV S (1 -ρV S )(p(∆ + π * )π * (p)) -C(ρ V S ).

  πV S (p) = max ρ ρ(1 -ρV S )(p(∆ + π * )π * (p)) -C(ρ) < max ρ ρ (p(∆ + π * )π * (p)) -C(ρ) < max ρ ρ∆ -C(ρ) = π m ,where the last inequality stems fromd ((p(∆ + π * )π * (p))) dp = ∆ + π *π * 0 (p) > 0,and: (p(∆ + π * )π * (p))| p=1 = ∆.

π

  B1 = π+ (p A ) ≡ ρ+ (p A ) (1ρ-(p A ))∆ + (1 -ρ+ (p A ))ρ -(p A ) (1p A )∆ -C(ρ + (p A )), π 2 = π-(p A ) ≡ ρ-(p A ) (1 -ρ+ (p A ))p A ∆ -C(ρ + (p A )).As noted in the text, we haveρ+ (p A ) < ρ+ (p A ), ρ-(p A ) > ρ-(p A ), and π-(p A ) > π-(p A ).In addition, the outcome coincides with the benchmark case (ρ * and π * ) for and thus:d (π + + π-) dp ¯p=1 = -(1ρ * )ρ * ∆ρ * ∆ρ -0 (1) + (1ρ * )ρ * ∆ρ * ∆ρ +0 (1) = -ρ * ∆ C 00 (ρ * ) -∆ (C 00 (ρ * )) 2 -∆ 2 ∆ = -ρ * ∆ 2 C 00 (ρ * ) + ∆ < 0.

  )ρ 2 ((1p) ∆ + π-(p)pπ * ) + π+ -C(ρ 1 ), π 2 = ρ 2 (1ρ 1 )(p (∆ + π * )π-(p)) + π-(p) -C(ρ 2 ).The corresponding investment levels are thus characterized by the following first-order conditions:C 0 (ρ 1 ) = (1 -(2p) ρ 2 ) ∆ρ 2 ¡ π-(p)pπ * ¢ , C 0 (ρ 2 ) = (1ρ 1 )(p (∆ + π * )π-(p)).

September 27, 2010.

[START_REF] Hart | Vertical Integration and Market Foreclosure[END_REF],[START_REF] O'brien | Vertical Control with Bilateral Contracts[END_REF] and[START_REF] Mcafee | Opportunism in Multilateral Vertical Contracting: Non Discrimination, Exclusivity and Uniformity[END_REF] offer a different foreclosure rationale, in which vertical integration allows a bottleneck owner to exert more fully its market power over independent downstream firms. See[START_REF] Rey | The Role of Exclusive Territories in Producers' Competition[END_REF] for an overview of that literature.

Note however that, as long as the integrated firm stops supplying the downstream rival, efficient contracting (e.g., two-part tariffs) among the independent firms need not result into cost-based marginal input prices, as the rivals could "dampen competition" by maintaining above-cost transfer prices -see[START_REF] Bonanno | Vertical Separation[END_REF],Rey and Stiglitz (1995) and[START_REF] Shaffer | Slotting allowances and retail price maintenance: a comparison of facilitating practices[END_REF].

In Ma's paper, the inputs are differentiated substitutes, but complementarity arises from uncertainty about consumers' relative preferences, which leads the downstream firms to offer "bundles" in the form of option contracts.

We assume that fixed costs, if any, are small enough to ensure that expected profits are always positive (assuming C (0) = 0 would ensure that this is always the case) and thus that entry and exit considerations are not an issue.

That is, the slope of the best responses is lower than 1 in absolute value.

Note that, contrary to[START_REF]On Vertical Mergers and Their Competitive Effects[END_REF] and[START_REF] Chen | Vertical Integration, Exclusive dealing and Expost cartelization[END_REF], upstream tariffs do not influence here the intensity of downstream competition; the risk of opportunistic behavior then insures that in equilibrium D 2 always favors U B .

The joint profit of U B and D 2 is furthermore impaired by coordination failure in D 2 's investment decision (that is, ρ -< R (ρ + )). Also, while U B always benefits here from foreclosure (since it obtains no profit in the benchmark case of vertical separation), in more general contexts, foreclosure may have an ambiguous impact on U B , who obtains a larger share of a smaller pie. In contrast, in the OSS foreclosure scenario, the profit of the independent suppliers as well as the joint profit of the independent rivals can increase, since the integrated firm raises its price in the downstream market.

A formal derivation is presented at the end of Appendix C.

In a recent market study, DIW reports that new national brand products are imitated more quickly by private labels (average delay of 10, 9 month) than by other national brands (12, 3 months). Similar observations applies for packaging imitation (Zunehmende Nachfragemacht des Einzelhandels, Eine Studie fur den Markenverband (DIW Econ)).

The second order derivative of the joint profit function is 2P 0 (δ)+P 00 (δ), which is indeed negative under these assumptions.

For example, for a linear "demand" P (ρ) = 1ρ, the equilibrium capacities are:

This discussion applies for example to the TomTom/TeleAtlas and Nokia/Navteq mergers discussed in the introduction.

These R&D efforts thus maximize a joint profit equal to:(ρ 1 (1-ρ 2 )+ρ 2 (1-ρ 1 ))∆-kρ 2 1 /2-kρ 2 2 /2.

The argument also applies to the case δ > 0, implying that vertical integration reduces consumer surplus whenever an innovator fully appropriates the added value it generates if the other firm does not innovate. If for example consumers have heterogenous reservation prices -so that demand is elastic -this is the case when the innovation uniformly increases these reservation prices.

The "price"can take several forms: a higher input price, the extension of the customer's contract, the introduction of exclusive dealing or quota provisions, and so forth.

The recent battle between Google and Apple illustrates this concern. Google and Apple initially cooperated to bring Google's search and mapping services to Apple's iPhone. However, following Google's entry into the mobile market, with products similar to the iPhone, Apple started a legal fight in 2010, claiming that HTC, a Taiwanese maker of mobile phones which use Google's Android operating system, violates iPhone patents.

The risk of opportunistic behavior highlighted by[START_REF] Hart | Vertical Integration and Market Foreclosure[END_REF] may also impede independent suppliers' ability to exploit the information acquired through reverse engineering (as they would be tempted to sell the information to all downstream rivals). By contrast, the integrated supplier dos not face the same risk of opportunistic behavior and would only exploit the information internally.

ϕ corresponds here to the cost of setting-up and operating the guarantees system. In particular, in the case of financial guarantees, it does not include the stipulated compensations, since they will never be actually paid in equilibrium.

That is, assuming that D 2 does not revise its belief when receiving an out-of-equilibrium offer in period 2.

In the boundary case ∆ -2δ = βφ -F , foreclosure may arise in either the first or both periods.

An alternative interpretation is that exploiting confidential information exposes to prosecution and thus to some penalty later on; "good" types can then simply be interpreted as putting more weight on future profits. The following analysis corresponds formally to the case where bad types put no weight on the future, but would apply as well to situations where bad types have a significantly lower discount factor than good ones.

This simplifies the analysis, by ruling out signalling issues in the first period price competition stage.

Accounting for discounting or imperfect imitation is straightforward but notationally cumbersome. The extension to the case δ > 0 is more involved (in particular, it requires a careful analysis of signalling issues at the price competition stage) but is available upon request

.39 We show below that a downstream firm would indeed rather integrate with the unreliable supplier than with its competitor.

For example, in order to degrade the quality offered to rivals, the supplier might need to set-up distinct production lines and face diseconomies of scale as well as increased organizational costs.

In their discussion of this industry,Hausman and Kohlberg (1989) note that " The BOCs will not want to be in a position of technological dependence on a competitor, nor will they want to discuss further service plans with the manufacturing affiliate of a competitor.", p214

Note that D 1 does not make any additional profit when U B is unreliable and only D 2 's R&D project succeeds, since U B then sells the information at its full value θδ.

D 2 obtains δ if both downstream innovation efforts are successful. If it is the sole innovator, it obtains ∆ if both suppliers are reliable. If U A is not reliable, then U B will extract its comparative advantage (γ if it is unreliable, and γ+θ (∆ -2δ) if instead it is reliable) and leave only ∆-θ (∆ -2δ) to D 2 .

Note that, since ρ 2 affects Π B1 in an additive separable way, it does not affect D 1 's innovation behavior, which remains given by ρ 1 = R (ρ 2 ).

This is obvious if ρθ and ρ+ θ are both lower than ρ * ; if instead one -ρ i , say -exceeds ρ * , than the other one satisfiesρ j = R j (ρ i ) < R j (ρ * ) < R (ρ * ) = ρ * .

It can for example be shown that this is always the case when C 00 (.) > 2∆.

gratefully acknowledge support from the Ecole Polytechnique Business Economics Chair and from the French-German cooperation project "Market Power in Vertically related markets" funded by the Agence Nationale de la Recherche (ANR) and Deutsche Forschungsgemeinschaft (DFG).

Proposition 8 If θ < δ 2b(θ-θ) ≤ θ + θ, only vertical integration allows the firms to achieve the maximal probability of successful imitation.

Proof. See Appendix A. Third, while independent suppliers have incentives to maintain a good reputation, the incentives of integrated suppliers are drastically altered by strategic considerations, since entertaining the fear of information leakage and imitation yields foreclosure benefits. To see this, in what follows we compare the outcome of partial vertical integration to the outcome that prevails in a vertically separated industry, and consider several ways in which a supplier can affect the risk of information leakage and imitation: it may for example exacerbate this risk by investing in costly reverse-engineering technology, or attenuate it by offering guarantees, e.g. in the form of firewalls or compensations in case of information leakage.

We present here the main arguments in a simple way, by assuming that in a preliminary stage, suppliers publicly choose to be "reliable" or not. We thus consider the following type of game:

• In stage 0, both suppliers, vertically integrated or not, decide whether to be reliable (which option is more costly depends on the context, e.g., reverse engineering versus guarantees; more on this below).

• In stage 1, D 1 and D 2 simultaneously choose their R&D efforts and then innovate with probabilities ρ 1 and ρ 2 ; the success or failure of their innovation efforts is observed by all firms.

• In stage 2, U A and U B simultaneously offer lump-sum tariffs to each downstream firm; we will denote by T hi the tariff offered by U h to D i (for h = A, B and i = 1, 2); each D i then chooses its supplier. Finally, unreliable suppliers have the opportunity to sell their customers' information to unsuccessful downstream rivals, through a take-it-or-leave-it offer, in which case the downstream rival is able to duplicate the imitation with probability θ > 0.

In the next section, we dispense with the commitment assumption (i.e., stage 0) and show that the same insights apply in a dynamic framework.

Appendix

A Complementary investments

We prove here Proposition 8. Suppose that the probability of successful imitation is equal to θ U θ D , where θ U and θ D are unobservable and respectively controlled by the upstream and downstream firms. Suppose further that: (i) each θ i can take two values, high ¡ θ ¢ or low (θ), with 0 < θ < θ ≤ 1; and (ii) opting for the low value θ gives the controlling firm a private, non-transferable benefit b > 0, whereas successful imitation gives the downstream firm a monetary benefit δ > 0.

• If the firms are vertically separated, in order to provide adequate incentives the downstream firm can pay some amount φ to the supplier in case of successful imitation. The risk of imitation is then maximal (that is, θ U = θ D = θ) if and only if:

-the upstream firm prefers θ to θ, that is:

-the downstream firm does the same, that is:

Summing-up these two conditions, the risk of imitation can be maximal only if:

• If instead the two firms are vertically integrated, the risk of imitation is maximal whenever the integrated firm prefers both divisions providing a high effort rather than:

-only one doing so, which requires:

-none doing so, which requires:

Of these two constraints, the latter is the most demanding 43 and can be rewritten as:

which is less demanding than the condition (30) required in the absence of vertical integration. The conclusion follows.

B Reverse engineering

As already established in Section 3.1, no independent supplier will ever invest in reverse engineering. Therefore, when both suppliers are vertically separated, standard Bertrand competition among equally reliable suppliers yields T Ai = T Bi = 0 (even when only one downstream firm innovates), the downstream firms invest ρ 1 = ρ 2 = ρ * , characterized by the first-order condition (4), and obtain an expected profit equal to Π * ≡ Π (ρ * , ρ * ), whereas upstream firms make no profit. Suppose now that U A and D 1 , say, have merged, whereas U B remains independent -and thus chooses to be reliable. As already noted in Section 3.1, the integrated firm never provides internal information to its independent rival; that is, vertical integration de facto protects D 1 against imitation. Moreover, if both firms innovate, a customer's information has no market value; whether a supplier is reliable is therefore irrelevant: standard Bertrand competition among the suppliers always yields T Ai = T Bi = 0 and thus each downstream firm obtains a profit equal to δ. The only remaining relevant case is when D 2 is the sole successful innovator:

• If both U A -D 1 and U B are reliable suppliers, Bertrand competition drives again tariffs to zero. Expected downstream profits are thus again Π i ¡ ρ i , ρ j ¢ and both investments are equal to ρ * . U A -D 1 's expected profit is thus still equal to Π * .

• If instead U A -D 1 is an unreliable supplier, it offers D 2 a subsidy of up to T A2 = -δ but U B wins by charging T B2 = ∆ -2δ. The expected profits of the 43 To see this, note that they are respectively equivalent to

The conclusion then follows from θ > θ.

It thus satisfies R γ (ρ) > R (ρ), R γ (0) > 0, and:

D 2 's expected profit is equal to Π (ρ 2 , ρ 1 ) if both suppliers are reliable, and to Π θ (ρ 2 , ρ 1 ) if the integrated firm is not reliable; 45 therefore:

Proof. Consider a candidate equilibrium in which U A -D 1 imitates D 2 's innovation with probability µ b when it is bad, and with probability µ g when it is good. If µ g > µ b , imitating enhances the reputation of the firm: in the second period, D 2 's updated belief, p i A , satisfies

In contrast, by not imitating D 2 's innovation, the integrated firm would strategically benefit from a downgraded reputation in the second period: D 2 's updated belief, p n A , would then satisfy

Since the expected continuation profit π+ (p A ) increases as p A decreases, a good firm would rather not imitate, as this moreover saves the cost F , contradicting the initial assumption µ g > µ b . We can thus suppose µ g ≤ µ b , which in turn implies p n A ≥ p ≥ p i A . Imitating cost nothing to a bad firm and, by downgrading the reputation of the firm, can only increase its expected profit in the second period. Therefore, according to our tie-breaking assumption, a bad firm chooses to imitate D 2 's innovation. We thus have µ g ≤ µ b = 1, which implies

Imitating then costs F to a good firm but increases second-period profits from π+ (1) = π * to π+ (p i A ) ≥ π+ (p). Therefore, as long as F < F , even a good integrated firm chooses to imitate D 2 's innovation ¡ µ g = µ b = 1 ¢ : the integrated firm always imitates D 2 's innovation, whatever U A 's type, leading to unchanged beliefs in the second period: p i A = p. Thus, if F < F , then if D 2 selects U A the expected profits of U A -D 1 , D 2 and U B are respectively equal to:

If D 2 was to reject all offers, it would obtain the same profit π-(p), whereas U A -D 1 would obtain π+ (p) and thus save the expected cost pF that it may have to face it p A = 1 and with the monopoly case (ρ 1 = ρ m , ρ 2 = 0 and π B1 = π m , π 2 = 0) for p A = 0.

Let us now turn to the first period, and suppose that D 2 is the sole innovator. Selecting U A would lead it to exploit D 2 's innovation only when being bad. The expected profits of U A , D 2 and U B -D 1 are then:

If instead D 2 selects U B , these expected profits become: The latter is likely to be higher, 48 and is indeed so when p is close to 0, since then πB 2 = ∆ > πA 2 = 0. In addition, we have: Furthermore, differentiating the first-order conditions (33) yields: