UNIVERSITAT ROVIRA I VIRGILI DEPARTAMENT D’ECONOMIA  WORKING PAPERS Col·lecció “DOCUMENTS DE TREBALL DEL DEPARTAMENT D’ECONOMIA - CREIP” The strategic value of partial vertical integration Raffaele Fiocco Document de treball n.28- 2016 DEPARTAMENT D’ECONOMIA – CREIP Facultat d’Economia i Empresa UNIVERSITAT ROVIRA I VIRGILI DEPARTAMENT D’ECONOMIA  Edita: Departament d’Economia www.fcee.urv.es/departaments/economia/publi c_html/index.html Universitat Rovira i Virgili Facultat d’Economia i Empresa Av. de la Universitat, 1 43204 Reus Tel.: +34 977 759 811 Fax: +34 977 758 907 Email: sde@urv.cat CREIP www.urv.cat/creip Universitat Rovira i Virgili Departament d’Economia Av. de la Universitat, 1 43204 Reus Tel.: +34 977 758 936 Email: creip@urv.cat Adreçar comentaris al Departament d’Economia / CREIP ISSN edició en paper: 1576 - 3382 ISSN edició electrònica: 1988 - 0820 DEPARTAMENT D’ECONOMIA – CREIP Facultat d’Economia i Empresa The strategic value of partial vertical integration Raffaele Fiocco∗ Abstract We investigate the strategic incentives for partial vertical integration, namely, partial ownership agreements between manufacturers and retailers, when retailers privately know their costs and engage in differentiated good price competition. The partial misalignment between the profit objectives within a partially integrated manufacturer-retailer hierarchy entails a higher retail price than under full integration. This ‘information vertical effect’ translates into an opposite ‘competition horizontal effect’: the partially integrated hierarchy’s commitment to a higher price induces the competitor to increase its price, which strategically relaxes competition. Our analysis provides implications for vertical merger policy and theoretical support for the recently documented empirical evidence on partial vertical acquisitions. Keywords: asymmetric information, partial vertical integration, vertical mergers, vertical restraints. JEL Classification: D82, L13, L42. ∗ Department of Economics and CREIP, Universitat Rovira i Virgili, Avinguda de la Universitat 1, 43204, Reus, Spain. Email address: raffaele.fiocco@urv.cat. 1 1. Introduction Most of the practical and theoretical debate about the firms’ organizational structure in vertically related markets has focused on two extreme alternatives: full vertical integration and separation. However, it is quite common to observe partial vertical integration, namely, partial ownership agreements in which a firm acquires less than 100% of shares in a vertically related firm (e.g., Allen and Phillips 2000; Fee et al. 2006; Reiffen 1998). Emphasizing the relevance of partial vertical integration, Riordan (2008) reports that in 2003 News Corp., a major owner of cable programming networks in the US, acquired 34% of shares in Hughes Electronics, which operates via its wholly-owned subsidiary Direct TV in the downstream market of direct broadcast satellite services. Gilo and Spiegel (2011) provide empirical evidence that partial vertical integration is much more common than full integration in telecommunications and media markets in Israel. For instance, Bezeq operates in the broadband Internet infrastructure market and holds a share of 49.77% in DBS Satellite Services that competes in the downstream multi-channel broadcast market. Partial acquisitions have recently received great attention in antitrust control.1 Despite the practical relevance of this phenomenon, relatively little theoretical research has been devoted so far to partial vertical acquisitions. The aim of this paper is to investigate the strategic incentives of vertically related firms to partially integrate and their competitive effects. We address this question in a setting where two manufacturer-retailer hierarchies engage in differentiated good price competition and retailers are privately informed about their production costs. The economic literature has emphasized since Crocker’s (1983) seminal contribution that a major problem within a supply hierarchy is that a firm can access privileged information about some relevant aspects of the market. In our framework, a manufacturer exclusively deals with its retailer, which is reasonable in the presence of product-specific investments that have to be sunk before production decisions take place.2 Moreover, in line with the main literature on competing hierarchies under asymmetric information (e.g., Coughlan and Wernerfelt 1989; Katz 1991; Martimort 1996; Martimort and Piccolo 2010), bilateral contracting within a supply hierarchy is secret. This reflects the natural idea that the trading rules specified in a contractual relationship are not observed by competitors and therefore cannot be used for strategic purposes. Alternatively, these rules can be easily (secretly) renegotiated if both parties agree to do so. In the benchmark case of full information within a supply hierarchy, a manufacturer that uses non-linear (secret) contracts is indifferent about the ownership stake in its retailer. This is because the manufacturer makes the retailer residual claimant for the hierarchy’s profits and appropriates these profits through a fixed fee. The outcome of vertical integration is achieved irrespective of the ownership stake, and therefore vertical ownership arrangements are inconsequential. This well-known ‘neutrality result’ (Coughlan and Wernerfelt 1989; Katz 1991) does not hold in the presence of asymmetric information. To begin with, consider a successive monopoly framework where a manufacturer-retailer pair operates in isolation and the retailer is privately 1 In the sequel, we discuss the antitrust approach to partial acquisitions. For instance, bilateral exclusive relationships are common in the video rental market. Blockbuster provides each downstream retailer with the exclusive right to sell its brand in a geographical area where competing retailers distribute alternative brands. 2 2 informed about its costs. It is well established in the economic literature (e.g., Gal-Or 1991c) that asymmetric information within a supply hierarchy entails a higher retail price in order to curb the (costly) informational rents to the retailer. Full vertical integration guarantees the owner of the hierarchy complete control and removes the problem of asymmetric information, which improves the hierarchy’s joint profits. We show that the strict preference for full vertical integration does not carry over in a competitive environment. In a setting where two manufacturer-retailer pairs engage in differentiated good price competition, partial vertical integration can emerge in equilibrium. In line with the successive monopoly framework, a partial vertical ownership agreement entails an information vertical effect: the partial misalignment between the profit objectives of the manufacturer and the retailer leads to a higher retail price than under full integration in order to reduce the informational rents to the retailer. For a given price of the competitor, this form of double marginalization from asymmetric information reduces the hierarchy’s profitability relative to full integration. In a competitive environment, however, the information vertical effect translates into an opposite competition horizontal effect: the partially integrated hierarchy’s commitment to a higher price induces an accommodating behavior of the rival that increases its price as well. Therefore, partial vertical integration is profitable since it constitutes a strategic device to relax competition. The trade-off between the benefits of softer competition and the informational costs drives the equilibrium degree of vertical integration. To better appreciate the rationale for our results, it is important to realize that, when a manufacturer is partially integrated with its retailer, it is common knowledge that there exists no contract which can ‘solve’ the problem of asymmetric information within the supply hierarchy. It follows from the seminal paper of Katz (1991) that, even when contracting is secret, this can affect the play of the continuation game. In our model, the rival firm — the ‘outside party’ in Katz’s (1991) terminology — anticipates that the partially integrated hierarchy’s retail price will be higher than under full integration, which induces the rival to increase its price in a game of strategic complementarity. Therefore, partial vertical integration exhibits a commitment value ` la Katz (1991) that relaxes competition. a Partial ownership agreements have been examined by the US antitrust law since a long time. Section 7 of the Clayton Act of 1914 (currently, Section 18 of Title 15 of the US Code) provides that “no person engaged in commerce or in any activity affecting commerce shall acquire, directly or indirectly, the whole or any part of the stock or other share capital and no person subject to the jurisdiction of the Federal Trade Commission shall acquire the whole or any part of the assets of another person engaged also in commerce or in any activity affecting commerce, where [...] the effect of such acquisition may be substantially to lessen competition, or to tend to create a monopoly”.3 In the Horizontal Merger Guidelines revised in 2010 a section has been introduced which is explicitly devoted to partial acquisitions. The seminal articles of Bresnahan and Salop (1986), Reynolds and Snapp (1986) and O’Brien and Salop (2000) provide a formal foundation for the 3 The quotation (with emphasis added) is available at http://www.law.cornell.edu/uscode/text/15/18. 3 antitrust control of partial acquisitions between rival firms since they can entail a dampening of competition. More recently, Gilo et al. (2006) show the collusive effects of partial cross ownership. Foros et al. (2011) find that rival firms can prefer partial acquisitions to full mergers, which leads to softer competition. The main contribution of our paper is to unveil the strategic incentives for partial ownership agreements between firms that do not compete with each other but are vertically related. Our results indicate that partial vertical acquisitions can emerge in equilibrium and mitigate competition relative to full vertical mergers. As discussed in Section 8, our analysis extends to vertically related markets Foros et al. (2011)’s recommendation for antitrust investigations of partial divestitures. In the same vein, we also provide theoretical support for antitrust policies that favor full mergers over partial acquisitions. In other countries, such as Austria, Germany, the UK, Australia, Canada, Japan and New Zealand, antitrust authorities are also entitled to scrutinize partial ownership agreements. For instance, the German Bundeskartellamt can investigate partial acquisitions that exhibit a competitively significant influence. However, the European Commission does not have any explicit competence in this area under the current merger control rules. Recent proposals aim at expanding the remit of the merger control function to enable the European Commission to examine partial acquisitions that entail non-controlling minority shareholdings. In the 2013 consultation document ‘Towards more effective EU merger control’ (p. 3), the European Commission recommends a reform of the current European merger control system in order to “extend the scope of the Merger Regulation to give the Commission the option to intervene in a limited number of problematic cases of structural links [i.e., partial acquisitions], in particular those creating structural links between competitors or in a vertical relationship”.4 The results of our paper suggest that there is scope for antitrust intervention of the European Commission in this area. Our analysis is presented in a fairly general setting without making any particular assumption on functional forms. Remarkably, it also provides theoretical corroboration for the empirical evidence recently documented in Ouimet (2013) that partial equity stakes are more likely to be preferred to full integration in industries requiring relationshipspecific investments, such as vertically related markets. The predictions of our model may serve as guidance for the empirical work on the competitive effects of partial vertical integration. 2. Related literature As discussed in the introduction, the economic literature has extensively explored the private and social effects of partial acquisitions in horizontally related markets. Conversely, the literature on partial vertical acquisitions is still in its infancy. Dasgupta and Tao (2000) show that partial vertical ownership may perform better than take-or-pay contracts if upstream firms make investments that benefit downstream firms. More recent contributions are Greenlee and Raskovich (2006), Hunold et al. (2012) and Levy et al. (2016). In Section 8 we compare these contributions with our work when discussing the antitrust policy implications of our results. 4 The document (including the quotation with emphasis added) can http://ec.europa.eu/competition/consultations/2013 merger control/merger control en.pdf. 4 be found at Our analysis is also related to the literature on the strategic delegation in a competitive environment. The idea that, in a full information framework, delegation can act as a strategic commitment device to relax competition traces back to Fershtman and Judd (1987), Sklivas (1987), Bonanno and Vickers (1988) and more recently Jansen (2003). In particular, Bonanno and Vickers (1988) find that in a differentiated good price competition model a manufacturer prefers to sell its product through an independent retailer rather than directly to consumers if it can publicly commit to a wholesale price above marginal costs, which induces a more lenient behavior of rivals. However, as shown by Coughlan and Wernerfelt (1989) and Katz (1991), this result no longer holds when contracts cannot be observed by rivals. Katz (1991) specifies that the strategic value of delegation is restored if it is common knowledge that there exists no contract that can solve the agency problem. Caillaud and Rey (1995) provide an overview of the strategic use of vertical delegation. Gal-Or (1992, 1999) shows in alternative settings of asymmetric information that firms may follow different strategies of integration and separation. Along these lines, Barros (1997) demonstrates that in an oligopolistic industry some firms may profit from a commitment to face asymmetric information about their agents’ operations, since they are prevented from extracting full surplus and can provide the agents with a credible incentive to invest. Contrary to the aforementioned contributions, we allow for a partial degree of vertical integration and show that partial vertical ownership agreements trade off the benefits of softer competition against the informational costs. Our paper also belongs to the literature on vertical restraints. For our purposes, early relevant contributions are Rey and Stiglitz (1988, 1995) and Gal-Or (1991a, 1991b) that explore the impact of vertical restraints on competition. Martimort (1996) investigates the choice of competing manufacturers between a common and an exclusive retailer in a setting of adverse selection. Martimort and Piccolo (2007) qualify the results of Gal-Or (1991c) about the choice between resale price maintenance and quantity fixing contracts, according to the retailers’ technology for providing services. In a model with competing manufacturer-retailer pairs, Martimort and Piccolo (2010) and Kastl et al. (2011) show that manufacturers may strategically prefer quantity fixing to resale price maintenance and explore the welfare consequences of these contractual relationships. Piccolo et al. (2014) investigate the allocation of residual claimancy in a setting with competing principal-agent hierarchies and demonstrate that a principal may find it optimal to retain a share of surplus from production with an inefficient agent because this reduces the mimicking incentives of the efficient agent. Our paper provides novel insights into the interaction between competition and the organizational structure of vertically related firms, and shows that a manufacturer can prefer to partially integrate with its retailer since the partial internalization of the retailer’s rents leads to a dampening of competition. The rest of the paper is structured as follows. Section 3 sets out the formal model. Section 4 considers the benchmark case of a manufacturer fully informed about the costs of its retailer. Section 5 shows that, in the presence of asymmetric information, partial vertical integration can emerge in equilibrium. Using explicit functions, Section 6 derives the equilibrium degree of vertical integration and performs a comparative statics analysis. Section 7 investigates alternative assumptions and the robustness of the results. Section 8 discusses some antitrust policy and empirical implications. Section 9 concludes. All formal proofs are provided in the Appendix. 5 3. The model Setting We consider a vertically related market where two manufacturers, M1 and M2 , provide symmetrically differentiated goods through two retailers, R1 and R2 , which engage in price competition. As discussed in the introduction, we assume that each manufacturer is in an exclusive relationship with one retailer. In the spirit of Martimort and Piccolo (2010), we examine a setting where manufacturer M1 and retailer R1 exclusively deal with each other, while manufacturer M2 is fully integrated with retailer R2 . As explained in Section 7.5, our results carry over in a more symmetric setting where both supply hierarchies decide on the degree of vertical integration. We denote by qi (pi , p−i ) the (direct) demand function for good i = 1, 2, which satisfies the following assumption. Assumption 1 − ∂qi (pi ,p−i ) > ∂pi ∂qi (pi ,p−i ) ∂p−i ≥ 0 (product substitutability). Goods are imperfect substitutes (the second condition holds with equality for independent goods), and own-price effects are larger than cross-price effects. Manufacturer M1 offers retailer R1 a contract that specifies a retail price p1 for the good and a fixed franchise fee t1 paid by the retailer to the manufacturer for the right to sell the good. The practice of dictating the final price to a retailer is commonly known as resale price maintenance. In Section 7.3 we show that our main results remain valid under a two-part tariff specifying a unit wholesale price and a fixed fee. Notably, resale price maintenance yields the manufacturer higher profits than a two-part tariff, and therefore our analysis does not depend on any restriction on the contract set that limits the manufacturer’s profits. Let θ1 ∈ {θl , θh } be R1 ’s (constant) marginal costs, whose realization is R1 ’s private information at the time the contract is signed with M1 . Costs are θl with probability ν ∈ (0, 1) and θh with probability 1 − ν. We define by ∆θ ≡ θh − θl > 0 the spread of the cost distribution. Retailer R1 ’s interim expected profits are πR1 = (p1 − θ1 ) Eθ2 [q1 (p1 , p2 ) |θ1 ] − t1 , (1) where Eθ2 [q1 (p1 , p2 ) |θ1 ] represents the expected quantity of R1 . The profits in (1) are interim expected, since they are evaluated at the contractual stage where R1 is informed about its own costs θ1 but does not know the costs θ2 of the competitor M2 − R2 . We allow for positive correlation between retail costs (e.g., Gal-Or 1991b, 1999; Martimort 1996), and therefore R1 ’s uncertainty on θ2 depends on the realization of θ1 . This reflects the idea that in competitive markets the costs of rival firms are usually subject to common trends. Notably, our results fully apply with independent costs. In the example provided in Section 6, we consider the case of perfect cost correlation, which implies θ1 = θ2 . In Section 7.4 we investigate the impact of cost correlation on the equilibrium ownership stake that the manufacturer holds in its retailer. Manufacturer M1 ’s interim expected profits are πM1 = t1 + ρ {(p1 − θ1 ) Eθ2 [q1 (p1 , p2 ) |θ1 ] − t1 } , (2) which is a weighted sum of the upstream profits from the franchise fee t1 (manufacturing costs 6 are normalized to zero) and the downstream profits πR1 in (1) from retail operations. When offering a contract to R1 , M1 is concerned about the profits in (2). The parameter ρ ∈ [0, 1] denotes the ownership stake acquired by M1 in R1 . Following O’Brien and Salop (2000), ρ captures the financial interest of the acquiring firm, which is entitled to receive a share of the profits of the acquired firm. If ρ = 0, the two firms are fully separated. If ρ ∈ (0, 1), M1 has a partial ownership stake in R1 , and therefore the two firms are partially integrated. If ρ = 1, M1 wholly owns R1 and the two firms are fully integrated. It is worth noting that the ownership stake ρ does not appear in R1 ’s profit function in (1). Therefore, R1 maximizes the full profits arising from the retail activities irrespective of the ownership stake acquired by M1 . In other terms, all the shareholders of R1 are treated equally. In line with some relevant contributions (e.g., Farrell and Shapiro 1990; Greenlee and Raskovich 2006; Hunold et al. 2012), this assumption can be justified on several grounds. The acquisition of a passive (non-controlling) ownership stake ensures the acquiring firm a participation in the acquired firm’s profits but it does not entail any corporate control. This ‘silent financial interest’ does not lead to any change in the incentives of the acquired firm (e.g., Bresnahan and Salop 1986). Moreover, corporate law or antitrust law can impose a legal requirement — known as ‘fiduciary obligation’ —, which provides that the managers of the acquired firm must act in the interest of the firm as an independent, stand-alone entity. The main purpose of this requirement is the protection of the minority shareholders, in particular those with no other holdings. This implies that, even when the acquiring firm holds a large financial interest, the acquired firm continues to maximize its stand-alone profits. O’Brien and Salop (2000) provide an accurate discussion of how the requirement of fiduciary obligation can be implemented in practice.5 We wish to derive the equilibrium degree of vertical integration between manufacturer M1 and retailer R1 , namely, the ownership stake ρ that M1 decides to acquire in R1 . In line with the main literature on partial acquisitions (e.g., Foros et al. 2011; Greenlee and Raskovich 2006; Hunold et al. 2012), we assume that M1 chooses the ownership stake ρ in R1 that maximizes the (expected) joint profits of the two firms. This ensures that M1 can design an offer to R1 which makes the shareholders in both firms better off, so that they will find it mutually beneficial to sign such an agreement.6 A joint profit maximizing ownership agreement does not leave any scope for mutually beneficial renegotiations and exhibits a commitment value. In Section 7.2 we consider the case in which the manufacturer maximizes its own profits when deciding on the ownership stake. In order to focus on the strategic effects of acquisition, we abstract from any cost saving that may arise from the ownership arrangement. 5 We recognize that a sufficiently high ownership stake may influence the decisions of the acquired firm in favor of the acquiring firm. However, as long as the acquired firm does not fully internalize the objectives of the acquiring firm, the conflict of interests within the supply hierarchy and the problem of asymmetric information persist, so that our qualitative results apply. If the acquiring firm obtains full corporate control above a certain threshold that induces the acquired firm to maximize joint profits (or which allows the acquiring firm to access the acquired firm’s relevant information), the problem of asymmetric information disappears above this threshold. The benefits of partial vertical integration still arise, but the equilibrium ownership stake depends on this threshold. 6 In a similar vein, Farrell and Shapiro (1990) suggest the criterion of joint profits to derive the equilibrium ownership stake. Notably, this approach seems to reflect the practice of takeovers and acquisitions. For instance, in the US a bidder that makes an offer to purchase less than 100% of the shares of a firm must accept all shares tendered on a pro-rated basis. For further discussion on this point, we refer to Foros et al. (2011). 7 The interim expected profits of the fully integrated supply hierarchy M2 − R2 are π2 = (p2 − θ2 ) Eθ1 [q2 (p1 , p2 ) |θ2 ] , (3) where Eθ1 [q2 (p1 , p2 ) |θ2 ] denotes the expected quantity of M2 − R2 .7 The two competing hierarchies do not know the costs of each other but, as discussed previously, their costs can be positively correlated. As it will become clear in the sequel, since M2 − R2 is fully integrated, our results are unaffected if M2 does not know the costs of its downstream division R2 . In order to characterize the equilibrium of the game, we impose the following conditions on the functional forms of profits πi , i = 1, 2, where π1 ∈ {πR1 , πM1 } and π2 are given by (1), (2), (3), and on the functional form of M1 − R1 ’s joint profits πM1 −R1 (e.g., Vives 2001, Ch. 2). Assumption 2 ∂ 2 πi ∂pi ∂p−i Assumption 3 ∂ 2 πi ∂p2 i < 0, Assumption 4 ∂ 2 πi ∂p2 i + > 0 (strategic complementarity). ∂ 2 πM1 −R1 ∂ρ2 ∂ 2 πi ∂pi ∂p−i < 0 (concavity). < 0 (contraction). Assumption 2 indicates that the firms’ best-response functions are positively sloped (Bulow et al. 1985).8 Assumption 3 guarantees that the second-order optimality conditions are satisfied and ensures together with Assumption 4 that the equilibrium of the game is globally stable. Contracting In line with relevant contributions on competing hierarchies under asymmet- ric information (e.g., Coughlan and Wernerfelt 1989; Gal-Or 1999; Kastl et al. 2011; Katz 1991; Martimort 1996; Martimort and Piccolo 2010), bilateral contracting within a hierarchy is secret. We invoke the revelation principle (e.g., Myerson 1982) in order to characterize the set of incentive feasible allocations. In our setting, this means that, for any strategy choice of M2 − R2 , there is no loss of generality in deriving the best response of M1 within the class of direct incentive compatible mechanisms. Specifically, M1 offers R1 a direct contract menu t1 θ1 , p1 θ1 θ1 ∈{θl ,θh } that determines a fixed franchise fee t1 (.) and a retail price p1 (.) contingent on R1 ’s report θ1 ∈ {θl , θh } about its costs θ1 . This contract menu must be incentive compatible, namely, it must induce R1 to report truthfully its costs, which implies θ1 = θ1 in equilibrium.9 In our setting contracts are incomplete, since M1 cannot contract upon either the retail price of the competitor M2 − R2 or any report of M2 − R2 about its costs. This assumption has 7 In the baseline model we do not impose any particular restriction on θ2 that can take values either within a discrete set or an interval. Moreover, we only require standard regularity conditions on the probability distribution function for θ2 . ∂ 2 qi (pi ,p−i ) 8 Sufficient (albeit not necessary) condition on the demand function for Assumption 2 is ∂pi ∂p−i ≥ 0. In the sequel, we sometimes make use of this condition. 9 Since the manufacturer can obtain (a part of) the retailer’s profits, it might infer the value of the retail costs and implement a penalty that extracts the full profits of the retailer arising from cost misreporting. However, this penalty is unfeasible in a range of reasonable circumstances. The profit realization may be affected by (independent) random shocks which, for instance, occur after the firms’ decisions. In this case, retail costs cannot be directly inferred from the retailer’s profits and, especially in the presence of limited liability, it would be unfeasible to design any penalty that deters cost misreporting. Furthermore, the fine implemented by the manufacturer would have the only effect of expropriating the profits of the other shareholders of the retailer. This would be interpreted as a violation of their rights and condemned by antitrust authorities. 8 a solid foundation in the literature (e.g., Gal-Or 1991a, 1991b, 1992, 1999; Kastl et al. 2011; Martimort 1996; Martimort and Piccolo 2010) and can be justified on several grounds. For instance, a contract contingent on the retail price of the competitor may be condemned as a collusive practice by antitrust authorities.10 Timing The sequence of events unfolds as follows. (I) M1 decides on which ownership stake ρ ∈ [0, 1] to acquire in R1 . (II) R1 and M2 − R2 privately learn their respective retail costs θ1 ∈ {θl , θh } and θ2 . (III) M1 secretly makes an offer t1 θ1 , p1 θ1 θ1 ∈{θl ,θh } to R1 . The offer can be either rejected or accepted by R1 .11 If the offer is rejected, each firm obtains its outside option (normalized to zero), while M2 − R2 acts as a monopolist. If the offer is accepted, R1 picks one element within the contract menu by sending a report θ1 ∈ {θl , θh } about its costs. (IV) Competition takes place in the downstream market and payments are made. The manufacturer’s decision on the ownership stake in the retailer is observable and takes place before the retailer learns its costs. This reflects the idea that the decisions on the firms’ ownership rights — mainly when scrutinized by antitrust authorities — become public and are harder to alter than the (generally flexible) production activities that can be adjusted to the realization of costs. Observability and commitment value are standard features of the ownership stake in the literature on partial acquisitions. As Foros et al. (2011) emphasize, the use of the financial and corporate structure of a firm to affect competition is a widespread phenomenon. In Section 7.1 we consider the case in which the ownership stake can be made contingent on retail costs and is incorporated into the vertical contract between the manufacturer and the retailer. The solution concept we adopt is Perfect Bayesian Equilibrium.12 Proceeding backwards, we first compute the retail prices at the competition stage for a given ownership stake. Afterwards, we derive the equilibrium ownership stake. 4. Benchmark: full information within the supply hierarchy To better appreciate how the strategic value of partial vertical integration follows from the presence of asymmetric information, we first consider the benchmark case in which M1 is fully informed about R1 ’s costs. We formalize the main results in the following remark. Remark 1 If M1 is fully informed about R1 ’s costs θ1 ∈ {θl , θh }, the equilibrium retail price pi charged by Mi − Ri , i = 1, 2, satisfies Eθ−i [qi (pi , p−i ) |θi ] + (pi − θi ) ∂Eθ−i [qi (pi , p−i ) |θi ] = 0. ∂pi 10 (4) Alternatively, the retail price charged by the rival can be hard to observe or verify because of the lack of proper auditing rights. We refer to Martimort (1996) for a discussion of this assumption. 11 A take-it-or-leave-it offer is a standard assumption in the literature on competing hierarchies. 12 As a standard equilibrium refinement, we require a ‘no signaling what you do not know’ condition (e.g., Martimort 1996). Whenever R1 receives an unexpected offer from M1 , it does not change its beliefs about the equilibrium strategy of M2 − R2 . This condition reflects the idea that a manufacturer cannot signal to its retailer information that it does not know about the competitor, since the supply hierarchies are independent and act simultaneously. 9 The equilibrium ownership stake that M1 holds in R1 is any ρ ∈ [0, 1]. The retail price of each supply hierarchy is set above marginal costs in order to equate (expected) marginal revenues with (expected) marginal costs from retail activities.13 The problem of manufacturer M1 coincides with the problem of the fully integrated hierarchy M2 − R2 . Since contracting is secret and cannot be used for strategic purposes, a fully informed manufacturer using non-linear contracts finds it optimal to remove the double marginalization problem by making its retailer residual claimant for the hierarchy’s profits, which are extracted via a fixed fee. Hence, the outcome of full integration is achieved irrespective of the ownership stake ρ, and the choice of the degree of vertical integration is inconsequential. This well-known ‘neutrality result’ (Coughlan and Wernerfelt 1989; Katz 1991) no longer holds in the presence of asymmetric information. 5. The case of asymmetric information As discussed in Section 3, when R1 privately knows its costs, M1 can restrict attention to a direct incentive compatible contract menu {(t1l , p1l ) , (t1h , p1h )}, where (t1l , p1l ) and (t1h , p1h ) are the contracts designed for the efficient and inefficient retailer, with costs θl and θh respectively. 5.1. Competition stage We first derive the retail prices for a given ownership stake. In addition to the participation constraints πR1l ≥ 0 and πR1h ≥ 0 for the efficient and inefficient retailer, the contract offered by M1 to R1 must satisfy the following incentive compatibility constraints πR1l = (p1l − θl ) Eθ2 [q1 (p1l , p2 ) |θl ] − t1l ≥ (p1h − θl ) Eθ2 [q1 (p1h , p2 ) |θl ] − t1h (5) πR1h = (p1h − θh ) Eθ2 [q1 (p1h , p2 ) |θh ] − t1h ≥ (p1l − θh ) Eθ2 [q1 (p1l , p2 ) |θh ] − t1l . (6) Conditions (5) and (6) ensure that R1 does not benefit from misreporting its costs. As implied by the Spence-Mirrlees (single-crossing) property, the relevant incentive constraint is the one for the efficient retailer in (5), which is binding in equilibrium together with the participation constraint πR1h ≥ 0 for the inefficient retailer.14 Given the expression for πR1h in (6) and πR1h = 0 in equilibrium, we can rewrite the binding constraint (5) after some manipulation as follows πR1l = p1h {Eθ2 [q1 (p1h , p2 ) |θl ] − Eθ2 [q1 (p1h , p2 ) |θh ]} + θh Eθ2 [q1 (p1h , p2 ) |θh ] − θl Eθ2 [q1 (p1h , p2 ) |θl ] = ∆θEθ2 [q1 (p1h , p2 ) |θh ] − (p1h − θl ) × {Eθ2 [q1 (p1h , p2 ) |θh ] − Eθ2 [q1 (p1h , p2 ) |θl ]} , (7) which captures the informational rents that the efficient retailer commands to reveal truthfully its costs. Using πR1h = 0 and the fact that the constraint (5) is binding, M1 ’s problem of 13 Throughout the analysis we assume interior solutions at the competition stage. Otherwise, M1 could increase the franchise fee and be better off. For further technical details we refer to the proof of Lemma 1 in the Appendix. 14 10 maximizing its (expected) profits in (2) becomes max ν {(p1l − θl ) Eθ2 [q1 (p1l , p2 ) |θl ] − (1 − ρ) πR1l (p1h )} p1l ,p1h + (1 − ν) (p1h − θh ) Eθ2 [q1 (p1h , p2 ) |θh ] . (8) The supply hierarchy M2 − R2 maximizes its profits in (3) as follows max (p2 − θ2 ) Eθ1 [q2 (p1 , p2 ) |θ2 ] . (9) p2 After taking the derivative of πR1l in (7) with respect to p1h ∂πR1l ∂Eθ2 [q1 (p1h , p2 ) |θh ] ≡ Ω (p1h ) =∆θ − Eθ2 [q1 (p1h , p2 ) |θh ] + Eθ2 [q1 (p1h , p2 ) |θl ] ∂p1h ∂p1h ∂Eθ2 [q1 (p1h , p2 ) |θh ] ∂Eθ2 [q1 (p1h , p2 ) |θl ] < 0, (10) − − (p1h − θl ) ∂p1h ∂p1h we can formalize the equilibrium retail prices for a given ownership stake ρ.15 Lemma 1 If R1 is privately informed about its costs θ1 ∈ {θl , θh }, the retail price charged by M1 − R1 is p∗ ∈ {p∗ , p∗ }, where p∗ and p∗ respectively satisfy 1 1h 1l 1l 1h Eθ2 [q1 (p∗ , p∗ ) |θl ] + (p∗ − θl ) 1l 1l 2 ∂Eθ2 [q1 (p∗ , p∗ ) |θl ] 1l 2 =0 ∂p1l Eθ2 [q1 (p∗ , p∗ ) |θh ] + (p∗ − θh ) 1h 1h 2 with φ (ν) ≡ ν 1−ν . (11) ∂Eθ2 [q1 (p∗ , p∗ ) |θh ] 1h 2 − φ (ν) (1 − ρ) Ω (p∗ ) = 0, 1h ∂p1h (12) Furthermore, the retail price p∗ charged by M2 − R2 satisfies 2 Eθ1 [q2 (p∗ , p∗ ) |θ2 ] + (p∗ − θ2 ) 1 2 2 ∂Eθ1 [q2 (p∗ , p∗ ) |θ2 ] 1 2 = 0. ∂p2 (13) Equipped with Lemma 1, we can show how the ownership stake affects the equilibrium retail prices. To simplify notation, we define p1k as the full information equilibrium price of R1 with costs θk , k = l, h. Proposition 1 If ρ = 1, then p∗ = p1l , p∗ = p1h , p∗ = p2 . Furthermore, 2 1l 1h ∂p∗ 2 ∂ρ ∂p∗ 1l ∂ρ ≤ 0, ∂p∗ 1h ∂ρ < 0, ≤ 0, where equalities follow if consumer demands are independent. We illustrate the results of Lemma 1 and Proposition 1 with the help of Figure 1.16 Note ∗ that M1 ’s asymmetric information best-response function r1l for low costs (θ1 = θl ) coincides with the corresponding full information best-response function r1l , regardless of the ownership stake. This is because R1 ’s informational rents in (7) are independent of p1l and therefore M1 does not find it profitable to implement any price distortion for the efficient retailer. The ownership stake ρ affects the position of M1 ’s asymmetric information best-response ∗ function r1h for high costs (θ1 = θh ), relative to the corresponding full information best-response function r1h . Specifically, with a full acquisition of R1 (ρ = 1), the two best-response functions 15 16 The sign of (10) follows from Assumptions 1-2 and positively correlated (or independent) costs. The figure considers the case of linear demand. 11 p2 ∗ r1l = r1l ˆ r1 ˆe ∗ r1h e∗ r1 r1h ˆ ρ<1 ρ<1 ∗ r2 = r2 ˆ p∗ 2 p2 ˆ p1l p∗ ˆ 1l p∗ 1h p1h ˆ p1 Figure 1: Best-response functions under full and asymmetric information coincide. In this case, M1 maximizes the hierarchy’s joint profits in (8) and fully internalizes R1 ’s informational rents in (7). Since these rents are costless, M1 does not want to implement any price distortion and, as Proposition 1 indicates, the asymmetric information prices in (11), (12) and (13) reflect the full information levels in (4). An ownership stake ρ lower than 1 shifts M1 ’s asymmetric information best-response func∗ tion r1h outwards, and the price charged by the inefficient retailer is higher than under full information for any price of the competitor M2 − R2 . A lower ρ makes the informational rents in (7) more costly for M1 , because they are internalized to a lesser extent in M1 ’s objective function in (8). As (10) reveals, in order to curb the informational rents M1 can induce an upward price distortion for the inefficient retailer.17 Since the ownership stake ρ measures the degree of M1 ’s internalization of R1 ’s rents, the magnitude of this form of double marginalization from asymmetric information increases when ρ declines. The outward shift in M1 ’s best-response ∗ e e∗ function r1h for ρ < 1 moves in the same direction M1 ’s expected best response (from r1 to r1 ), ∗ while M2 − R2 ’s best response is clearly unaffected (r2 = r2 ).18 Therefore, the two competing hierarchies can coordinate on higher prices. As Proposition 1 shows, a lower ownership stake ρ induces a higher equilibrium price p∗ of the inefficient retailer, which translates into a higher 1h equilibrium price p∗ of the competitor M2 − R2 in the presence of strategic complementarity. 2 In response, the equilibrium price p∗ of the efficient retailer increases as well. 1l It is worth noting that this latter result crucially depends on the fact that M2 − R2 cannot distinguish between the efficient and inefficient retailer, and it determines the price on the basis of the (conditional) distribution of R1 ’s costs. As we will see in Section 6, with perfect cost correlation the prices of the two supply hierarchies reflect the full information values in case of low costs, since M2 − R2 knows R1 ’s costs and anticipates that M1 ’s best-response function is unaffected for low costs. 17 This result is reminiscent of the rent extraction-efficiency trade-off in optimal regulation (e.g., Baron and Myerson 1982). 18 This holds even under asymmetric information within M2 − R2 since M2 maximizes joint profits and does not find it profitable to distort the price of the privately informed division R2 . For the sake of clarity, the best-response function of M2 − R2 is depicted in Figure 1 for a given (and commonly known) θ2 . 12 5.2. Equilibrium ownership stake Having derived the equilibrium retail prices for a given ownership stake, we can go back to the first stage of the game and determine the equilibrium ownership stake. Since M1 chooses the ownership stake in R1 in order to maximize the hierarchy’s joint profits, M1 ’s problem is given by max ν [p∗ (ρ) − θl ] Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θl ]+(1 − ν) [p∗ (ρ) − θh ] Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θh ] . 2 2 1l 1l 1h 1h ρ∈[0,1] (14) We are now in a position to show our main results. Proposition 2 If R1 is privately informed about its costs, partial vertical integration is more profitable for M1 − R1 than full vertical integration when consumer demands are interdependent. The equilibrium ownership stake that M1 holds in R1 is ρ∗ < 1. If consumer demands are independent, full vertical integration arises in equilibrium, i.e., ρ∗ = 1. Under asymmetric information M1 is no longer indifferent about the ownership stake in R1 . If demands are independent and therefore each supply hierarchy acts as a monopolist, full integration is optimal because it removes any negative informational externality and maximizes the profits of the supply hierarchy taken in isolation. Notably, we obtain this result without the need to assume that the manufacturer exogenously acquires any relevant information about the retailer when they are fully integrated. As Proposition 1 indicates, under full integration the manufacturer completely internalizes the retailer’s informational rents and does not find it optimal to implement any price distortion. Proposition 2 shows that the strict preference for full vertical integration does not carry over in a competitive environment. In this case, partial vertical integration ensures the supply hierarchy M1 − R1 higher profits than full integration, and the equilibrium ownership stake that M1 acquires in R1 is ρ∗ < 1. In order to substantiate the rationale for this result as provided in the introduction, it is helpful to recall from Proposition 1 that a partial misalignment between the profit objectives within a partially integrated hierarchy (ρ < 1) leads to an upward price distortion for the inefficient retailer to reduce the (costly) informational rents to the efficient retailer. This form of double marginalization from asymmetric information generates an information vertical effect that reduces ceteris paribus the profitability of the supply hierarchy. With price competition, the information vertical effect translates into an opposite competition horizontal effect. Since there exists no vertical contract that can ‘solve’ the problem of asymmetric information, it follows from Katz (1991) that even with secret contracting a partially integrated hierarchy can commit vis-`-vis the rival to a higher retail price than under a full integration, which induces the rival to increase its price as well in the presence of strategic complementarity. Therefore, partial vertical integration constitutes a commitment device ` la a Katz (1991) to relax competition. The equilibrium degree of vertical integration trades off the benefits of softer competition against the informational costs. To better appreciate the rationale for our results, it is convenient to write a second-order 13 Taylor approximation for M1 − R1 ’s joint profits around ρ = 1 as follows πM1 −R1 (ρ)|ρ=ρ<1 ≈ πM1 −R1 (ρ)|ρ=1 −(1 − ρ) As the proof of Proposition 2 shows, we have ∂πM1 −R1 (ρ) ∂ρ + ρ=1 ∂πM1 −R1 (ρ) ∂ρ ρ=1 (1 − ρ)2 ∂ 2 πM1 −R1 (ρ) 2 ∂ρ2 . ρ=1 < 0, and therefore a departure from full vertical integration entails first-order benefits for the supply hierarchy M1 − R1 . A reduction in ρ from ρ = 1 has no first-order information vertical effect associated with the costs of the double marginalization from asymmetric information (minimized at ρ = 1), but it induces a first-order competition horizontal effect which is beneficial for M1 − R1 in terms of coordination with M2 − R2 on higher prices. However, a lower ρ also entails second-order losses for M1 − R1 , i.e., ∂ 2 πM1 −R1 (ρ) ∂ρ2 ρ=1 < 0 (by Assumption 3), which stem from the informational costs. The resulting trade-off implies that the equilibrium ownership stake diverges from the full integration outcome until the level that equates the marginal benefits of relaxing competition with the marginal informational costs. Since we know from Proposition 1 that a lower ownership stake leads to higher prices, in a competitive environment there exists a conflict of interests between consumers (and the society as a whole), whose welfare is maximized under full integration, and the supply hierarchy, which prefers to partially integrate. In Section 8 we discuss the antitrust policy implications of our results. 6. An illustrative example Using explicit functions, we now derive the equilibrium degree of vertical integration and conduct a comparative statics analysis. The consumer demand for good i = 1, 2 takes the following form qi = α − βpi + γp−i , (15) where α and β are positive parameters, and γ ∈ [0, β) denotes the degree of substitutability between goods.19 The profits of R1 , M1 and M2 − R2 are respectively given by (1), (2) and (3), with retail costs being now perfectly correlated, which implies θ1 = θ2 ∈ {θl , θh }. The assumption of perfect correlation between the retailers’ types is relatively common in the literature on competing hierarchies (e.g., Kastl et al. 2011; Martimort 1996; Martimort and Piccolo 2010). As discussed at the end of Section 5.1, with perfectly correlated costs M2 − R2 knows whether it faces an efficient retailer, whose price is not distorted for the purpose of reducing the informational rents. This implies that for θ1 = θ2 = θl the retail prices reflect the full information values, i.e., p∗ = pil = il α+βθl 2β−γ , i = 1, 2. For θ1 = θ2 = θh the retail prices charged by M1 − R1 and M2 − R2 for a given ownership stake ρ are respectively p∗ = 1h (α + βθh ) 4β 2 − γ 2 + φ (ν) (1 − ρ) 4β 3 ∆θ − γ 2 (α + βθh ) (2β − γ) {4β 2 − γ 2 [1 + φ (ν) (1 − ρ)]} 19 (16) The demand system in (15) follows from the optimization problem of a unit mass of identical consumers characterized by a quasi-linear utility function y + U (q1 , q2 ), where y is the composite good and U (q1 , q2 ) = g 2 2 1 b a (q1 + q2 ) − 2 bq1 + bq2 + 2gq1 q2 , with a > 0, b > g ≥ 0, and α ≡ a(b−g) , β ≡ b2 −g2 , γ ≡ b2 −g2 (e.g., Vives b2 −g 2 2001, Ch. 6). 14 p∗ = 2h (α + βθh ) 4β 2 − γ 2 + γφ (ν) (1 − ρ) 2β 2 ∆θ − γ (α + βθh ) . (2β − γ) {4β 2 − γ 2 [1 + φ (ν) (1 − ρ)]} (17) These results illustrate with explicit solutions the main insights gleaned from Lemma 1 and Proposition 1.20 The price in (16) of the inefficient retailer is inflated above the full information level when the amount of ownership stake ρ is lower than 1. Differentiating p∗ in (16) with 1h respect to ρ yields ∂p∗ 4β 3 (2β − γ) ∆θφ (ν) 1h < 0, =− ∂ρ {4β 2 − γ 2 [1 + φ (ν) (1 − ρ)]}2 (18) which indicates that a lower ρ exacerbates the upward price distortion. As the discussion after Proposition 1 reveals, this is because M1 internalizes to a lesser extent the informational rents in (7) and therefore it is more inclined to curb these rents with a price increase. For a given price charged by the competitor M2 − R2 , this form of double marginalization from asymmetric information reduces the profits of M1 − R1 relative to full integration. However, in the presence of strategic complementarity in prices, M2 − R2 reacts with an accommodating behavior. Differentiating p∗ in (17) with respect to ρ yields 2h ∂p∗ 4β 2 γ (2β − γ) ∆θφ (ν) 2h =− ≤ 0, ∂ρ {4β 2 − γ 2 [1 + φ (ν) (1 − ρ)]}2 (19) where the equality holds for γ = 0. A lower ρ allows the two competing hierarchies to coordinate on higher prices. Note from (18) and (19) that the price response of M2 − R2 to a change in ρ is smoother than the price response of M1 − R1 , and it vanishes when consumer demands are independent (γ = 0). Even though the two supply hierarchies share the same costs, M1 − R1 sets a higher price than M2 − R2 for ρ < 1. The following proposition illustrates the result of the trade-off between the benefits of softer competition and the informational costs. Proposition 3 If R1 is privately informed about its costs θ1 ∈ {θl , θh }, the equilibrium ownership stake that M1 holds in R1 is ρ∗ = max 1 − γ 2 4β 2 − γ 2 [α − (β − γ) θh ] ;0 . φ (ν) {8β 3 ∆θ (2β 2 − γ 2 ) + γ 4 [α − (β − γ) θh ]} (20) It holds ρ∗ < 1 when consumer demands are interdependent (γ = 0). In particular, we have γ 2 (4β 2 −γ 2 )[α−(β−γ)θh ] (i) partial vertical integration, i.e., ρ∗ ∈ (0, 1), if φ (ν) > 8β 3 ∆θ(2β 2 −γ 2 )+γ 4 [α−(β−γ)θ ] ; (ii) full vertical separation, i.e., ρ∗ = 0, otherwise. h Full vertical integration, i.e., ρ∗ = 1, is preferred if consumer demands are independent (γ = 0). Proposition 3 reveals that in a competitive environment partial vertical integration emerges in equilibrium if the probability ν of the efficient retailer is relatively high (recall that φ (ν) ≡ ν 21 1−ν ). In this case, a partial ownership stake optimally trades off the benefits of softer com- 20 We refer to the proof of Proposition 3 in the Appendix for the formal derivation of these results. The condition α − (β − γ) θh > 0, which ensures positive quantities under full information, implies that the ratio in (20) is positive for γ = 0. 21 15 ρ∗ ρ∗ 1 1 0.75 0.75 0.5 0.5 0.25 0.25 0.25 0.5 0.75 1 γ β 0.25 0.5 0.75 1 γ β (a) High probability of the efficient retailer (b) Intermediate probability of the efficient retailer Figure 2: Product differentiation and ownership stake petition against the costs of the double marginalization from asymmetric information. When the probability ν of the efficient retailer is low enough, the hierarchy M1 − R1 prefers full separation. To understand the rationale for this result, we write a first-order Taylor approximation for ∂πM1 −R1 ∂ρ where ∂πM1 −R1 ∂ρ ≈ ν=ν>0 ∂πM1 −R1 ∂ρ ν=0 around ν = 0 as follows ∂πM1 −R1 ∂ρ +ν ν=0 ∂ 2 πM1 −R1 ∂ρ∂ν = −ν ν=0 2β 2 γ 2 [α − (β − γ) θh ] ∆θ < 0, (2β − γ)3 (2β + γ) = 0 since any ρ ∈ [0, 1] maximizes M1 − R1 ’s profits under full information (see Remark 1). For small values of ν, a lower ρ increases M1 − R1 ’s joint profits, and therefore full separation arises in equilibrium. This is because, when ν is low enough, the problem of asymmetric information is relatively mild and the benefits of softer competition driven by a reduction in ρ dominate the informational costs. A comparison between panel (a) and panel (b) of Figure 2 indicates that a higher probability of the efficient retailer shifts the equilibrium ownership stake ρ∗ in (20) upwards.22 A more probable efficient retailer increases the expected informational rents, which exacerbates the upward price distortion for the inefficient retailer and inflates the informational costs. A higher level of integration is preferred since it mitigates these costs. As Figure 2 illustrates, there exists a non-monotone relation between the degree of product differentiation γ and ρ∗ , which implies that for intermediate values of the probability of the efficient retailer full separation can still occur in equilibrium for a range of product differentiation. When each supply chain acts as a monopolist (γ = 0), a pattern of full integration that removes any informational externality within the hierarchy is optimal. If γ increases, ρ∗ declines over an initial range of γ. As competition intensifies, the benefits of softer competition become more attractive and this induces a reduction in ρ∗ . However, above a certain threshold this pattern is reversed and a higher γ translates into a higher ρ∗ . This is because, when competition is relatively fierce, the two supply hierarchies are reluctant to coordinate on high 22 In Figure 2 the parameter values are β = 1, α = 32 (1 − γ), θh = ∆θ = 1.5. Moreover, φ (ν) = 1.2 in panel (a), and φ (ν) = 0.7 in panel (b). 16 prices. At the extreme, when goods are close substitutes and competition tends to be perfect, prices converge to marginal costs and the equilibrium ownership stake approaches the outcome of full integration (if γ → β, then α → 0 and ρ∗ → 1). Note from (20) that a higher spread of the cost distribution ∆θ ≡ θh − θl increases the equilibrium ownership stake ρ∗ . A higher ∆θ aggravates the problem of asymmetric information, which induces a higher ρ∗ in order to mitigate the informational costs. The result in Proposition 3 that the equilibrium ownership stake is lower than 1 holds whenever consumer demands are interdependent (γ = 0). Hence, partial vertical integration can emerge even with complementary goods (γ < 0). As (18) and (19) indicate, a higher price of the inefficient retailer due to a lower ownership stake than under full integration leads to a lower price for the complementary good of M2 − R2 , since prices are now strategic substitutes. This stimulates the output of M1 − R1 and improves its profits. 7. Robustness We now discuss some assumptions of the model to gain insights into the robustness of the results. 7.1. Type-contingent ownership stake In our model the manufacturer’s decision on the ownership stake is observable and takes place before the retailer learns its costs. In line with the literature on partial acquisitions, observability and commitment value of the ownership stake are relevant ingredients of our model. Now, suppose that the ownership stake is chosen after costs have materialized and it is incorporated into the vertical contract between the manufacturer and the retailer. In this case, M1 secretly offers R1 a contract of the form ρ θ1 , t1 θ1 , p1 θ1 . It follows from Piccolo et al. (2014) that partial vertical integration can still emerge in this alternative scenario. This occurs when R1 ’s profits directly depend on the ownership stake ρ, and a higher ρ induces R1 to internalize to a larger extent the hierarchy’s joint profits. With positive cost correlation, R1 anticipates that, after a report of high costs, M1 conjectures that the competitor is more likely to be inefficient, which entails hierarchy’s lower profits in the presence of strategic complementarity. Hence, M1 is willing to compensate R1 for this decline in the hierarchy’s profits. A lower ownership stake designed for the inefficient retailer reduces the interest of the efficient retailer (that reports high costs) in the hierarchy’s profits and mitigates its incentives to manipulate costs. Differently from our setting, a partial ownership stake does not arise from the benefits of softer competition but from a ‘competing-contracts effect’ ` la Martimort (1996) and Gal-Or (1999). a 7.2. Manufacturer’s profit maximizing ownership stake In line with the main literature, we derive the ownership stake of M1 in R1 from the joint profit maximization problem. We now consider the case in which M1 chooses the ownership stake that maximizes its own profits in (2). The equilibrium value for the ownership stake ρ is the 17 solution to the following program max ν {[p∗ (ρ) − θl ] Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θl ] − (1 − ρ) πR1l (p∗ (ρ))} 2 1h 1l 1l ρ∈[0,1] + (1 − ν) [p∗ (ρ) − θh ] Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θh ] , 2 1h 1h (21) where πR1l (p∗ (ρ)) is given by (7) evaluated in the competition stage equilibrium. As a compar1h ison between (14) and (21) indicates, the strategic incentives to partially integrate are weaker than under joint profit maximization. A manufacturer that maximizes its own profits when choosing the ownership stake in its retailer internalizes not only the allocative costs of the double marginalization from asymmetric information but also the distributional costs arising from the inability to fully appropriate the retailer’s rents. The informational costs of partial integration are higher than under joint profit maximization and full integration becomes more attractive. Partial vertical integration can still emerge in equilibrium as the manufacturer’s profit maximizing outcome if the retailer’s (expected) informational rents are not too large, which is typically the case when the spread of the cost distribution ∆θ ≡ θh − θl is relatively small. 7.3. Two-part tariff The contract that M1 offers to R1 directly specifies the retail price, which is known as resale price maintenance. Even though this type of vertical arrangements is sometimes viewed with skepticism by antitrust authorities, some countries (e.g., New Zealand) traditionally allow this practice if the beneficial effects can be shown to outweigh the detrimental effects. Remarkably, in the 2007 case ‘Leegin Creative Leather Products, Inc., v. PSKS, Inc.’ the US Supreme Court replaced the well-established doctrine of per se unlawfulness of resale price maintenance with a rule of reason which allows a firm to produce evidence that an individual resale price maintenance agreement is justified.23 Our qualitative results do not depend on the use of resale price maintenance. Suppose that this form of vertical contracting is not allowed, and M1 secretly offers R1 a two-part tariff {f1 , w1 } that specifies a fixed franchise fee f1 and a wholesale price w1 for each unit of input that M1 provides to R1 . Manufacturing costs are still normalized to zero. Using a standard assumption (e.g., Martimort and Piccolo 2007), R1 converts M1 ’s input with a one-to-one technology into a final product supplied on the retail market. The interim expected profits of R1 and M1 are respectively given by πR1 = (p1 − θ1 − w1 ) Eθ2 [q1 (p1 , p2 ) |θ1 ] − f1 (22) πM1 = f1 + w1 Eθ2 [q1 (p1 , p2 ) |θ1 ] + ρ {(p1 − θ1 − w1 ) Eθ2 [q1 (p1 , p2 ) |θ1 ] − f1 } . (23) The game exhibits the same features as the baseline model, with the difference that M1 cannot dictate the retail price to R1 . This implies that, after the offer {f1 , w1 } from M1 , R1 chooses the price that maximizes its own profits. The following proposition summarizes the main results when a two-part tariff is adopted. 23 For some empirical evidence on resale price maintenance in Europe, we refer to Bonnet and Dubois (2010). 18 Proposition 4 Suppose that M1 secretly offers R1 a two-part tariff {f1 , w1 }. Then, (i) if M1 is fully informed about R1 ’s costs θ1 ∈ {θl , θh }, the equilibrium wholesale price is w1 = 0. The equilibrium ownership stake that M1 holds in R1 is any ρtp ∈ [0, 1]; (ii) if R1 is privately informed about its costs θ1 ∈ {θl , θh }, the equilibrium wholesale price ∗ ∗ ∗ ∗ ∗ ∗ is w1 ∈ {w1l , w1h }, where w1l = w1 and w1h ≥ w1 , with w1h = w1 if ρ = 1 and The equilibrium ownership stake that M1 holds in R1 is ρ∗ tp ∗ ∂w1h ∂ρ < 0. < 1 when consumer demands are interdependent. If consumer demands are independent, full vertical integration arises in equilibrium, i.e., ρ∗ = 1. tp Proposition 4 shows that our main results are robust to the form of vertical contracting. Under full information within the supply hierarchy, a two-part tariff is equivalent to a contract specifying a retail price and a fixed fee, since either contractual form removes the double marginalization problem and therefore the ownership stake that the manufacturer acquires in the retailer is inconsequential. More relevantly, we find that, in the presence of asymmetric information, partial vertical integration can still emerge in equilibrium. A manufacturer using a two-part tariff inflates the wholesale price for the inefficient retailer above the full information level in order to curb the informational rents to the efficient retailer (e.g., Gal-Or 1991c). This occurs as long as the manufacturer does not fully own the retailer and therefore the informational rents are costly. Since a higher wholesale price translates into a higher retail price, partial vertical integration still constitutes a commitment device to relax competition. The following corollary shows that the form of vertical contracting affects the equilibrium degree of vertical integration in a systematic manner. Corollary 1 It holds ρ∗ ≥ ρ∗ , where the equality follows if costs θ1 and θ2 are independent. tp A supply hierarchy prefers a higher level of integration under a two-part tariff than under resale price maintenance when retail costs are (positively) correlated. As the expressions for the informational rents (7) and (30) reveal, under either contractual form the efficient retailer envisages lower profits from a report of high costs in the presence of cost correlation, since it anticipates that the rival is more likely to be efficient and to set a relatively low price. The retailer’s lower profits depend on the reduction in the expected quantity weighted by the pricecost markup. Cost correlation mitigates the retailer’s incentive to manipulate costs, but it does so to a lower extent under a two-part tariff. This is because the upward distortion of the wholesale price for the inefficient retailer driven by asymmetric information leads to a lower price-cost markup and therefore alleviates the profit reduction that the efficient retailer expects from a report of high costs. In other terms, the double marginalization associated with a twopart tariff aggravates the manufacturer’s incentive problem, which induces the acquisition of a higher ownership stake in order to mitigate the informational costs. Resale price maintenance can replicate the outcome of a two-part tariff at a lower cost in terms of informational rents, which makes the manufacturer better off. Hence, a two-part tariff will be adopted only when resale price maintenance arrangements are banned. 19 7.4. Cost correlation and ownership stake The investigation of the impact of correlation between retail costs on the equilibrium ownership stake delivers results of some interest. Suppose that the retail costs θi of the supply hierarchy Mi −Ri , i = 1, 2, can be either θl or θh with ex ante probability ν ∈ (0, 1) and 1−ν, respectively. Following Piccolo and Pagnozzi (2013) and Piccolo et al. (2014), the vector of the retail costs (θ1 , θ2 ) is drawn from a joint cumulative distribution function such that Pr (θl , θl ) = ν 2 + µ, Pr (θl , θh ) = Pr (θh , θl ) = ν (1 − ν) − µ, and Pr (θh , θh ) = (1 − ν)2 + µ. The parameter µ ∈ [0, ν (1 − ν)] measures the degree of (positive) correlation between θ1 and θ2 . Using Bayes’ rule, µ posterior probabilities are Pr (θl |θl ) = ν + µ , Pr (θl |θh ) = ν − 1−ν , Pr (θh |θl ) = 1 − ν − µ , and ν ν Pr (θh |θh ) = 1 − ν + µ 1−ν . The incentive constraints (5) and (6) for the efficient and inefficient retailer become πR1l = (p1l − θl ) ≥ (p1h − θl ) πR1h = (p1h − θh ) ≥ (p1l − θh ) µ µ q1 (p1l , p2l ) + 1 − ν − q1 (p1l , p2h ) − t1l ν ν µ µ q1 (p1h , p2l ) + 1 − ν − q1 (p1h , p2h ) − t1h ν+ ν ν ν+ µ µ q1 (p1h , p2l ) + 1 − ν + q1 (p1h , p2h ) − t1h 1−ν 1−ν µ µ ν− q1 (p1l , p2l ) + 1 − ν + q1 (p1l , p2h ) − t1l . 1−ν 1−ν ν− Using the expression for πR1h and the fact that πR1h = 0 in equilibrium, we can reformulate the binding incentive constraint for the efficient retailer as follows πR1l = ∆θ [νq1 (p1h , p2l ) + (1 − ν) q1 (p1h , p2h )] µ (p1h − θl − ν∆θ) [q1 (p1h , p2h ) − q1 (p1h , p2l )] , − ν (1 − ν) (24) which captures the retailer’s informational rents. For the sake of tractability, we formally derive the impact of cost correlation µ on the equilibrium ownership stake when µ is relatively small and consumer demand takes the linear form in (15). Proposition 5 Suppose that the degree of correlation µ between costs θ1 and θ2 is small. If R1 is privately informed about its costs θ1 ∈ {θl , θh }, the equilibrium ownership stake ρ∗ that M1 holds in R1 increases (at an increasing rate) with µ. The numerical simulations illustrated in Figure 3 show that the result of Proposition 5 carries over for large values of µ.24 As (24) indicates, a higher price for the inefficient retailer makes a report of high costs less attractive for the efficient retailer and it does so to a larger extent when costs are more closely correlated ( ∂πR1l ∂p1h < 0 and ∂ 2 πR1l ∂p1h ∂µ < 0). In line with the discussion in Section 7.3, this is because the efficient retailer realizes that the rival is more likely to be efficient and to set a relatively low price, which reduces the retailer’s rents from cost manipulation. As a consequence, a larger degree of cost correlation increases the manufacturer’s benefits of a higher price for the inefficient retailer in terms of rent extraction. In other words, 24 In Figure 3 the parameter values are ν = 0.5, β = 1, γ = 0.4, α = 19.2, θl = 0. Moreover, θh = 1.5 (the bottom line), θh = 2 (the middle line), and θh = 2.5 (the top line). 20 ρ∗ 1 θh = 2.5 θh = 2 θh = 1.5 µ 0 Figure 3: Cost correlation and ownership stake for a given ownership stake, the manufacturer is more eager for an upward price distortion when cost correlation is higher. Since this increases the informational costs within the supply hierarchy, a higher ownership stake is preferred in equilibrium in order to mitigate these costs. 7.5. Competitor’s vertical integration decision Throughout the analysis we assume that manufacturer M1 faces the fully integrated competitor M2 −R2 when deciding on the ownership stake ρ in its retailer R1 . This assumption is innocuous and the outcome of partial vertical integration carries over in alternative settings. It can be immediately seen from the proof of Proposition 2 that the result ∂πM1 −R1 (ρ) ∂ρ ρ=1 < 0 does not depend on the degree of vertical integration of M2 − R2 . A similar result occurs if the decision on the level of ownership stake simultaneously takes place in the two supply hierarchies. This is because the benefits of softer competition arise regardless of what the rival does, and therefore each supply hierarchy has a unilateral incentive to depart from full integration. 8. Antitrust policy and empirical implications In line with the theoretical literature, the empirical research on vertically related markets (exhaustively surveyed in Lafontaine and Slade 2007) has mainly focused on the binary choice between separation and integration. However, as documented in some relevant empirical works (e.g., Allen and Phillips 2000; Fee et al. 2006; Reiffen 1998), partial vertical acquisitions are a common phenomenon. The predictions of our model about the impact of partial vertical integration on retail prices lend themselves to empirically testable validations. As discussed in Section 5.2, a higher degree of vertical integration in our model is welfare enhancing. A natural policy implication of this result is that any proposal of vertical acquisition should be approved by a myopic antitrust authority, since it improves welfare relative to full separation. However, a more sophisticated antitrust policy can achieve better outcomes. A forward-looking antitrust authority should block partial ownership agreements when it anticipates that firms will prefer full merger to separation. Alternatively, the antitrust authority should commit to only approving vertical acquisitions above a certain threshold that equates the marginal benefits of inducing a higher degree of vertical integration with the marginal costs of discouraging a vertical acquisition altogether. 21 We are aware that this kind of strategic commitment of the antitrust authority could be difficult to implement in practice. A more relevant implication of our results for vertical merger policy concerns the antitrust scrutiny of partial vertical divestitures. Our analysis suggests that the decision of a manufacturer to sell a fraction of the shares in its retailer can dampen competition. Remarkably, this is the case even when the acquirer is a silent investor or a firm operating in another market. Our conclusions complement the results of Foros et al. (2011) that show the anticompetitive effects of partial mergers relative to full mergers in horizontally related markets and recommend antitrust investigations of partial divestitures. By the same token, takeover regulations could be implemented, which favor full acquisitions over partial acquisitions. The policy implications of our model are also in line with the results of Hunold et al. (2012), which show that, in a full information setting, passive ownership of downstream firms in their suppliers entails higher retail prices and is profitable with sufficiently intense competition. Greenlee and Raskovich (2006), however, find that a passive ownership interest of a downstream firm in an upstream monopoly is generally inconsequential and may harm consumers only in some circumstances, which limits the scope for antitrust intervention. A well-known caveat of an antitrust policy recommendation in favor full integration is that it might induce anticompetitive input foreclosure. However, the empirical evidence suggests that lower retail prices tend to be associated with full integration (Lafontaine and Slade 2007), which is consistent with our results. Remarkably, Levy et al. (2016) show that under certain conditions partial integration is more likely to lead to input foreclosure than full integration. The predictions of our model provide further corroboration for the anticompetitive effects of partial vertical integration. Our results indicate that partial vertical ownership emerges when it can induce an accommodating behavior of rivals, which is typically the case in markets where firms compete in prices with differentiated goods. This mode of competition can naturally arise in sectors with relationship-specific investments, such as vertically related markets. Hence, our analysis provides theoretical support for the empirical investigation of Ouimet (2013) that shows the preference for partial equity stakes over full integration in these sectors, where the number of patents is used as a proxy for relationship-specific investments. Conversely, we do not generally expect partial vertical ownership for strategic purposes when capacity constraints induce Cournot competition, since the partially integrated hierarchy’s output reduction to curb the retailer’s informational rents triggers a more aggressive behavior of rivals. 9. Concluding remarks In this paper we investigate the strategic incentives for partial acquisitions in vertically related markets where two manufacturer-retailer pairs engage in differentiated good price competition and retailers are privately informed about their production costs. A partial ownership stake of a manufacturer in its retailer introduces a misalignment between the profit objectives of the two firms and entails an upward price distortion for the inefficient retailer in order to reduce the (costly) informational rents to the efficient retailer. This form of double marginalization from asymmetric information generates an information vertical effect that reduces the profitability 22 of the supply hierarchy taken in isolation. In a competitive environment, the information vertical effect translates into an opposite competition horizontal effect. The partially integrated hierarchy’s commitment to a higher price than under full integration induces the rival to increase its price as well. Therefore, partial vertical integration constitutes a strategic device to relax competition. The equilibrium degree of vertical integration trades off the benefits of softer competition against the informational costs. Our analysis provides theoretical support for the empirical evidence on partial vertical integration and formulates antitrust policy recommendations for mergers and acquisitions in vertically related markets. Acknowledgments I thank the Associate Editor and two anonymous referees for valuable comments and suggestions. I also thank Fabio Antoniou, Helmut Bester, Giacomo Calzolari, Matthias Dahm, Vincenzo Denicol`, Liliane Giardino-Karlinger, Mario Gilli, Morten Hviid, o Chrysovalantou Milliou, Martin Peitz, Patrick Rey, Klaus Schmidt, Nicolas Schutz, Giancarlo Spagnolo, Konrad Stahl, Roland Strausz and Nikos Vettas, as well as the participants in the seminars at Athens University of Economics and Business, University of Ioannina, Aristotle University of Thessaloniki, and the participants in the SFB-TR15 Conference 2013 in Munich, MaCCI Competition and Regulation Day Workshop 2013 in Mannheim, EARIE Conference 2014 in Milan, UECE Conference 2014 in Lisbon, EIEF-UNIBO-IGIER Bocconi Workshop in Industrial Organization 2014 in Bologna, CLEEN Workshop 2015 in Tilburg, CRESSE Conference 2015 in Rethymno, EEA Conference 2015 in Mannheim, SAEe 2015 in Girona. Appendix This Appendix collects the proofs. Proof of Remark 1. Substituting t1 with πR1 from (1), M1 ’s problem of maximizing (2) becomes max (p1 − θ1 ) Eθ2 [q1 (p1 , p2 ) |θ1 ] − (1 − ρ) πR1 s.t. πR1 ≥ 0, p1 ,πR1 where the constraint ensures that R1 is willing to participate in the contractual relationship with M1 . Since the maximand decreases with πR1 for any ρ ∈ [0, 1], we find πR1 = 0 in equilibrium.25 Taking the first-order condition for p1 yields Eθ2 [q1 (p1 , p2 ) |θ1 ] + (p1 − θ1 ) ∂Eθ2 [q1 (p1 ,p2 )|θ1 ] ∂p1 = 0. Using (3), M2 − R2 solves max (p2 − θ2 ) Eθ1 [q2 (p1 , p2 ) |θ2 ] , p2 which yields Eθ1 [q2 (p1 , p2 ) |θ2 ] + (p2 − θ2 ) ∂Eθ1 [q2 (p1 ,p2 )|θ2 ] ∂p2 = 0. Solving the system of the first- order conditions for the maximization problems of M1 and M2 − R2 gives the expression (4). Since the equilibrium prices do not depend on the ownership stake ρ, we have ρ ∈ [0, 1] in equilibrium. 25 Indeed, πR1 vanishes for ρ = 1 and πR1 ≥ 0 can be supported in equilibrium. However, this does not affect the first-order condition for p1 . 23 Proof of Lemma 1 . The results in the lemma immediately follow from the first-order conditions for p1l and p1h in the maximization problem (8), and from the first-order condition for p2 in the maximization problem (9). We now show that the incentive constraint (6) is satisfied in equilibrium. Combining terms in (6) yields πR1h = (p1h − θh ) Eθ2 [q1 (p1h , p2 ) |θh ] − t1h ≥ (p1l − θh ) Eθ2 [q1 (p1l , p2 ) |θh ] − t1l = πR1l + p1l {Eθ2 [q1 (p1l , p2 ) |θh ] − Eθ2 [q1 (p1l , p2 ) |θl ]} + θl Eθ2 [q1 (p1l , p2 ) |θl ] − θh Eθ2 [q1 (p1l , p2 ) |θh ] . Using (7) evaluated in equilibrium and the fact that πR1h = 0, we obtain after some manipulation 0 ≥ ∆θ {Eθ2 [q1 (p∗ , p∗ ) |θh ] − Eθ2 [q1 (p∗ , p∗ ) |θl ]} − (p∗ − θl ) {Eθ2 [q1 (p∗ , p∗ ) |θh ] 1h 2 1l 2 1h 1h 2 −Eθ2 [q1 (p∗ , p∗ ) |θl ]} + (p∗ − θh ) {Eθ2 [q1 (p∗ , p∗ ) |θh ] − Eθ2 [q1 (p∗ , p∗ ) |θl ]} . 1h 2 1l 1l 2 1l 2 Given the following first-order Taylor approximation Eθ2 [q1 (p∗ , p∗ ) |θk ] ≈ Eθ2 [q1 (p∗ , p∗ ) |θk ] + 1l 2 1h 2 ∂Eθ2 [q1 (p∗ , p∗ ) |θk ] ∗ 1l 2 (p1h − p∗ ) , k = l, h, 1l ∂p1l we find ∂Eθ2 [q1 (p∗ , p∗ ) |θh ] 1l 2 ∂p1l ∂Eθ2 [q1 (p∗ , p∗ ) |θl ] 1l 2 + (p∗ − p∗ ) Eθ2 [q1 (p∗ , p∗ ) |θl ] + (p∗ − θl ) (p∗ − p∗ ) 1l 1h 1h 1h 1l 1l 2 ∂p1l 0 ≥ − (p∗ − p∗ ) Eθ2 [q1 (p∗ , p∗ ) |θh ] − (p∗ − θh ) (p∗ − p∗ ) 1l 1h 1l 1l 2 1h 1h = − (p∗ − p∗ ) {Eθ2 [q1 (p∗ , p∗ ) |θh ] − Eθ2 [q1 (p∗ , p∗ ) |θl ] 1h 1l 1l 2 1l 2 + (p∗ − θh ) 1h ∂Eθ2 [q1 (p∗ , p∗ ) |θh ] ∂Eθ2 [q1 (p∗ , p∗ ) |θl ] 1l 2 1l 2 − (p∗ − θl ) 1h ∂p1l ∂p1l . Since p∗ − p∗ > 0 (see (11) and (12)) and the expression in curly brackets is also positive (by 1h 1l Assumptions 1-2 and positively correlated, or independent, costs), the constraint (6) is satisfied in equilibrium. Finally, we check that the participation constraint πR1l ≥ 0 is also fulfilled in equilibrium. Using the incentive constraint (7), sufficient (albeit not necessary) condition is that either the degree of cost correlation or the degree of substitutability is not too large. Proof of Proposition 1 . The proof of the first sentence of the proposition immediately follows from a comparison between (4) and (11), (12), (13) for ρ = 1. To prove the second sentence, denoting by e ∂πM ∂π e 1 , ∂pM1 ∂p1l 1h and ∂π2 ∂p2 the left-hand side of (11), (12) and (13) respectively, the implicit function theorem yields ∂p∗ 1l ∂ρ  ∂p∗   1h   ∂ρ  ∂p∗ 2 ∂ρ    e ∂ 2 πM 1 ∂p2 1l e ∂ 2 πM 1   = −  ∂p1h ∂p1l ∂2π 2 ∂p2 ∂p1l e ∂ 2 πM 1 ∂p1l ∂p1h e ∂ 2 πM 1 ∂p2 1h ∂ 2 π2 ∂p2 ∂p1h It follows from Assumptions 2-4 and −1  2 e  e ∂ 2 πM ∂ πM 1 1 ∂p1l ∂p2  ∂p1l  ∂ 2 πe∂ρ  e ∂ 2 πM   M1  . 1   ∂p1h ∂ρ  ∂p1h ∂p2  ∂ 2 π2 ∂ 2 π2 ∂p2 ∂ρ ∂p2 2 e ∂ 2 πM 1 ∂p1h ∂p1l = e ∂ 2 πM 1 ∂p1l ∂p1h 24 = 0 that the determinant of the Jacobian matrix is negative. Moreover, it can be immediately seen from (11), (12) and (13) that ∂ 2 π2 ∂p2 ∂ρ = 0 and e ∂ 2 πM 1 ∂p1h ∂ρ sign ∂p∗ 1l ∂ρ sign ∂p∗ 1h ∂ρ sign ∂p∗ 2 ∂ρ e ∂ 2 πM 1 ∂p1l ∂ρ = < 0. Hence, standard computations show that = sign e e ∂ 2 πM1 ∂ 2 π2 ∂ 2 πM1 ∂p1l ∂p2 ∂p2 ∂p1h ∂p1h ∂ρ = sign = sign − ≤0 e e ∂ 2 πM1 ∂ 2 π2 ∂ 2 πM1 ∂ 2 π2 − ∂p1l ∂p2 ∂p2 ∂p1l ∂p2 ∂p2 2 1l e e ∂ 2 πM1 ∂ 2 π2 ∂ 2 πM1 ∂p2 ∂p2 ∂p1h ∂p1h ∂ρ 1l e ∂ 2 πM1 <0 ∂p1h ∂ρ ≤ 0, where the values of the signs follow from Assumptions 2-4, and the equalities hold when con∂ 2 πe sumer demands are independent ( ∂p1l M12 = ∂p ∂ 2 π2 ∂p2 ∂p1h = 0). Proof of Proposition 2. Differentiating the objective function in (14) with respect to the ownership stake ρ yields ∂Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θl ] ∂p∗ (ρ) 2 1l 1l Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θl ] + [p∗ (ρ) − θl ] 2 1l 1l ∂ρ ∂ρ ∂p∗ (ρ) ∂Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θh ] 2 1h 1h + (1 − ν) Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θh ] + [p∗ (ρ) − θh ] 2 1h 1h ∂ρ ∂ρ ν . Applying the chain rule ∂Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θk ] ∂p∗ (ρ) ∂Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θk ] 2 2 1k 1k = 1k ∂ρ ∂ρ ∂p1k ∂q1 (p∗ (ρ) , p∗ (ρ)) ∂p∗ (ρ) 2 2 1k |θk , k = l, h, + Eθ2 ∂p2 ∂ρ we obtain ∂Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θl ] ∂p∗ (ρ) 2 1l 1l Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θl ] + [p∗ (ρ) − θl ] 2 1l 1l ∂ρ ∂p1l ∂q1 (p∗ (ρ) , p∗ (ρ)) ∂p∗ (ρ) 2 2 1l + [p∗ (ρ) − θl ] Eθ2 |θl + (1 − ν) 1l ∂p2 ∂ρ ∂Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θh ] ∂p∗ (ρ) 2 1h 1h × Eθ2 [q1 (p∗ (ρ) , p∗ (ρ)) |θh ] + [p∗ (ρ) − θh ] 2 1h 1h ∂ρ ∂p1h ∂q1 (p∗ (ρ) , p∗ (ρ)) ∂p∗ (ρ) 2 2 1h |θh . + [p∗ (ρ) − θh ] Eθ2 1h ∂p2 ∂ρ ν Substituting (11) and (12) finally gives ∂q1 (p∗ (ρ) , p∗ (ρ)) ∂p∗ (ρ) 2 2 1l |θl + (1 − ν) ∂p2 ∂ρ ∂p∗ (ρ) ∂q1 (p∗ (ρ) , p∗ (ρ)) ∂p∗ (ρ) 2 2 1h 1h φ (ν) (1 − ρ) Ω (p∗ (ρ)) + [p∗ (ρ) − θh ] Eθ2 |θh 1h 1h ∂ρ ∂p2 ∂ρ ν [p∗ (ρ) − θl ] Eθ2 1l × If the second condition in Assumption 1 holds with inequality (interdependent demands), it follows from Proposition 1 (and p∗ − θl > 0, p∗ − θh > 0) that 1l 1h ∂πM1 −R1 (ρ) ∂ρ ρ=1 < 0, which implies that the equilibrium ownership stake is ρ∗ < 1. If the second condition in Assumption 25 . 1 holds with equality (independent demands), we have the equilibrium ownership stake is ρ∗ ∂πM1 −R1 (ρ) ∂ρ ρ=1 = 0, which implies that = 1. Proof of Proposition 3. Under full information, M1 ’s problem of maximizing (2) can be written as max p1k ,πR1k (p1k − θk ) (α − βp1k + γp2k ) − (1 − ρ) πR1k s.t. πR1k ≥ 0, k = l, h, where the constraint ensures that R1 (with costs θl or θh ) accepts the contractual offer of M1 . Since the maximand decreases with πR1k for any ρ ∈ [0, 1], we have πR1k = 0 in equilibrium. Taking the first-order condition for p1k yields α − 2βp1k + γp2k + βθk = 0. Using (3), M2 − R2 solves max (p2k − θk ) (α − βp2k + γp1k ) , k = l, h, p2k which yields α − 2βp2k + γp1k + βθk = 0. Solving the system of the first-order conditions for the maximization problems of M1 and M2 − R2 yields pik = α+βθk 2β−γ , i = 1, 2, k = l, h. Now, we turn to the case of asymmetric information. Replacing (15) into (5) and (6), the incentive constraints write as πR1l = (p1l − θl ) (α − βp1l + γp2l ) − t1l ≥ (p1h − θl ) (α − βp1h + γp2l ) − t1h = πR1h + γp1h (p2l − p2h ) + θh (α − βp1h + γp2h ) − θl (α − βp1h + γp2l ) = πR1h + ∆θ (α − βp1h + γp2h ) − γ (p2h − p2l ) (p1h − θl ) (25) πR1h = (p1h − θh ) (α − βp1h + γp2h ) − t1h ≥ (p1l − θh ) (α − βp1l + γp2h ) − t1l = πR1l + γp1l (p2h − p2l ) − θh (α − βp1l + γp2h ) + θl (α − βp1l + γp2l ) = πR1l − ∆θ (α − βp1l + γp2l ) + γ (p2h − p2l ) (p1l − θh ) . (26) Since the constraint (25) is binding and πR1h = 0 in equilibrium, substituting (15) into (2) M1 ’s problem becomes max ν {(p1l − θl ) (α − βp1l + γp2l ) − (1 − ρ) [∆θ (α − βp1h + γp2h ) − γ (p2h − p2l ) p1l ,p1h × (p1h − θl )]} + (1 − ν) (p1h − θh ) (α − βp1h + γp2h ) . The first-order conditions for p1l and p1h are respectively α − 2βp1l + γp2l + βθl = 0 and α − 2βp1h + γp2h + βθh + φ (ν) (1 − ρ) [β∆θ + γ (p2h − p2l )] = 0. Substituting (15) into (3), M2 − R2 solves max (p2k − θk ) (α − βp2k + γp1k ) , k = l, h, p2k which yields α−2βp2k +γp1k +βθk = 0. The first-order conditions for the maximization problems of M1 and M2 −R2 yield p∗ = il α+βθl 2β−γ , i = 1, 2, and the expressions in (16) and (17). To check that 26 the incentive constraint (26) is satisfied in equilibrium, we substitute the binding constraint (25) into (26), which yields after some manipulation 0 ≥ − (p∗ − p∗ ) [β∆θ + γ (p∗ − p∗ )]. This 1h 1l 2h 2l condition is fulfilled since p∗ − p∗ > 0 and p∗ − p∗ > 0. Moreover, the binding constraint (25) 1h 1l 2h 2l implies that sufficient (albeit not necessary) condition for the participation constraint πR1l ≥ 0 to be satisfied is that γ is not too high. The optimal ownership stake is the solution to following program max ν (p∗ − θl ) (α − βp∗ + γp∗ ) + (1 − ν) [p∗ (ρ) − θh ] [α − βp∗ (ρ) + γp∗ (ρ)] . 1l 1l 2l 1h 1h 2h ρ∈[0,1] Assuming for the time being an internal solution and using (16) and (17), the first-order condition for ρ can be written as −φ (ν) 4β 3 ∆θ − γ 2 (α + βθh ) 4β 2 − γ 2 [1 + φ (ν) (1 − ρ)] (2β − γ) {4β 2 − γ 2 [1 + φ (ν) (1 − ρ)]}2 −γ 2 φ (ν) α− × × (α + βθh ) 4β 2 − γ 2 + φ (ν) (1 − ρ) 4β 3 ∆θ − γ 2 (α + βθh ) (2β − γ) {4β 2 − γ 2 [1 + φ (ν) (1 − ρ)]}2 (β − γ) 4β 2 − γ 2 (α + βθh ) − φ (ν) (1 − ρ) (2β − γ) {4β 2 − γ 2 [1 + φ (ν) (1 − ρ)]} 2β 2 2β 2 − γ 2 ∆θ − γ 2 (α + βθh ) (β − γ) (2β − γ) {4β 2 − γ 2 [1 + φ (ν) (1 − ρ)]} + (α + βθh ) 4β 2 − γ 2 + φ (ν) (1 − ρ) 4β 3 ∆θ − γ 2 (α + βθh ) − θh (2β − γ) {4β 2 − γ 2 [1 + φ (ν) (1 − ρ)]} × φ (ν) + γ 2 φ (ν) × 2β 2 2β 2 − γ 2 ∆θ − γ 2 (α + βθh ) (β − γ) 4β 2 − γ 2 [1 + φ (ν) (1 − ρ)] (2β − γ) {4β 2 − γ 2 [1 + φ (ν) (1 − ρ)]}2 (β − γ) 4β 2 − γ 2 (α + βθh ) (2β − γ) {4β 2 − γ 2 [1 + φ (ν) (1 − ρ)]}2 2β 2 2β 2 − γ 2 ∆θ − γ 2 (α + βθh ) (β − γ) (2β − γ) {4β 2 − γ 2 [1 + φ (ν) (1 − ρ)]}2 + φ (ν) (1 − ρ) = 0. Combining terms implies φ (ν) (1 − ρ) 4β 2 − γ 2 4αβ 3 γ 2 (2β − γ) ∆θ + 2β 2 2β 2 − γ 2 ∆θ − γ 2 (α + βθh ) (β − γ) × 4β 3 ∆θ − γ 2 (α + βθh ) + γ 2 2β 2 2β 2 − γ 2 ∆θ − γ 2 (α + βθh ) (β − γ) (α + βθh ) +2β 2 2β 2 − γ 2 + 4β 2 − γ 2 2 4β 3 ∆θ − γ 2 (α + βθh ) ∆θ + 2β 2 γ 2 (2β − γ) 2β 2 − γ 2 ∆θθh −4αβ 3 (2β − γ) ∆θ + 4β 3 (β − γ) (α + βθh ) ∆θ +2β 2 2β 2 − γ 2 (α + βθh ) ∆θ − 2β 2 (2β − γ) 2β 2 − γ 2 ∆θθh = 0, which gives after further manipulation φ (ν) (1 − ρ) 8β 3 2β 2 − γ 2 ∆θ + γ 4 [α − (β − γ) θh ] − γ 2 4β 2 − γ 2 [α − (β − γ) θh ] = 0. This yields the optimal ownership stake in (20). The remaining part of the proposition follows from straightforward computations. Proof of Proposition 4. Proceeding backwards, R1 ’s problem of maximizing (22) with respect 27 to p1 gives the following first-order condition Eθ2 [q1 (p1 , p2 ) |θ1 ] + (p1 − θ1 − w1 ) ∂Eθ2 [q1 (p1 , p2 ) |θ1 ] = 0. ∂p1 (27) Under full information within M1 − R1 , after substituting f1 with πR1 from (22) into (23), M1 solves max [p1 (w1 ) − θ1 ] Eθ2 [q1 (p1 (w1 ) , p2 ) |θ1 ] − (1 − ρ) πR1 s.t. πR1 ≥ 0. w1 ,πR1 Since the maximand decreases with πR1 for any ρ ∈ [0, 1], we find πR1 = 0 in equilibrium. The first-order condition for w1 yields ∂p1 (w1 ) ∂w1 Eθ2 [q1 (p1 (w1 ) , p2 ) |θ1 ] + [p1 (w1 ) − θ1 ] 1 (w Using (27), this expression reduces to w1 ∂p∂w1 1 ) 0 (see (27)) and ∂Eθ2 [q1 (p1 (w1 ),p2 )|θ1 ] ∂p1 ∂Eθ2 [q1 (p1 (w1 ) , p2 ) |θ1 ] ∂p1 ∂Eθ2 [q1 (p1 (w1 ),p2 )|θ1 ] ∂p1 = 0. = 0. It follows from ∂p1 (w1 ) ∂w1 > < 0 (by Assumption 1) that we obtain w1 = 0 in equilib- rium. Since equilibrium prices do not depend on the ownership stake ρ, we have ρtp ∈ [0, 1] in equilibrium. When R1 is privately informed about its costs θ1 ∈ {θl , θh }, M1 offers R1 a direct incentive compatible contract menu {(f1l , w1l ) , (f1h , w1h )} that satisfies the following incentive constraints πR1l = (p1l − θl − w1l ) Eθ2 [q1 (p1l , p2 ) |θl ] − f1l ≥ (p1h − θl − w1h ) Eθ2 [q1 (p1h , p2 ) |θl ] − f1h (28) πR1h = (p1h − θh − w1h ) Eθ2 [q1 (p1h , p2 ) |θh ] − f1h ≥ (p1l − θh − w1l ) Eθ2 [q1 (p1l , p2 ) |θh ] − f1l . (29) Using the expression for πR1h and the fact that πR1h = 0 in equilibrium,26 the binding constraint (28) can be rewritten as πR1l = p1h {Eθ2 [q1 (p1h , p2 ) |θl ] − Eθ2 [q1 (p1h , p2 ) |θh ]} + (θh + w1h ) Eθ2 [q1 (p1h , p2 ) |θh ] − (θl + w1h ) Eθ2 [q1 (p1h , p2 ) |θl ] = ∆θEθ2 [q1 (p1h , p2 ) |θh ] − (p1h − θl − w1h ) × {Eθ2 [q1 (p1h , p2 ) |θh ] − Eθ2 [q1 (p1h , p2 ) |θl ]} . (30) Hence, M1 ’s problem of maximizing its (expected) profits in (23) for a given ownership stake ρ 26 Note from (28) and (29) that the efficient (inefficient) retailer that declares high (low) costs will set a price p1h (p1l ), as defined by (27) for high (low) costs. Otherwise, M1 would discover the retailer’s cost misrepresentation and implement an adequate penalty. Moreover, in line with the proof of Lemma 1, the incentive constraint (29) for the inefficient retailer and the participation constraint πR1l ≥ 0 for the efficient retailer can be checked ex post. 28 becomes max ν {[p1l (w1l ) − θl ] Eθ2 [q1 (p1l (w1l ) , p2 ) |θl ] − (1 − ρ) πR1l (p1h (w1h ) , w1h )} w1l ,w1h + (1 − ν) [p1h (w1h ) − θh ] Eθ2 [q1 (p1h (w1h ) , p2 ) |θh ] , (31) where p1l (w1l ) and p1h (w1h ) are given by (27). Taking the derivative of (30) with respect to w1h yields ∂πR1l ∂p1h (w1h ) ∂Eθ2 [q1 (p1h (w1h ) , p2 ) |θh ] ≡ Ψ (p1h (w1h ) , w1h ) = ∆θ ∂w1h ∂w1h ∂p1h ∂p1h (w1h ) − 1 {Eθ2 [q1 (p1h (w1h ) , p2 ) |θh ] − Eθ2 [q1 (p1h (w1h ) , p2 ) |θl ]} − ∂w1h ∂p1h (w1h ) − [p1h (w1h ) − θl − w1h ] ∂w1h ∂Eθ2 [q1 (p1h (w1h ) , p2 ) |θh ] ∂Eθ2 [q1 (p1h (w1h ) , p2 ) |θl ] . × − ∂p1h ∂p1h (32) Given Assumptions 1-2 and positively correlated (or independent) costs, sufficient (albeit not necessary) condition for Ψ (.) to be negative is that the price-cost pass-through ∂p1h ∂w1h (which generally ranges between 0 and 1 from the combination of the first-order condition (27) and the associated second-order condition) is relatively large or costs are not highly correlated, which ensures that the positive expression in the first curly brackets is small enough. Taking the first-order conditions for w1l and w1h in the maximization problem (31) yields after some manipulation ∂p1l (w1l ) ∂w1l ∂p1h (w1h ) ∂w1h Eθ2 [q1 (p1l (w1l ) , p2 ) |θl ] + [p1l (w1l ) − θl ] ∂Eθ2 [q1 (p1l (w1l ) , p2 ) |θl ] ∂p1l Eθ2 [q1 (p1h (w1h ) , p2 ) |θh ] + [p1h (w1h ) − θh ] = 0 (33) ∂Eθ2 [q1 (p1h (w1h ) , p2 ) |θh ] ∂p1h − φ (ν) (1 − ρ) Ψ (p1h (w1h ) , w1h ) = 0. (34) ∗ Substituting (27) into (33) and (34) we find w1l = w1 = 0 and w1h ∂p1h (w1h ) ∂Eθ2 [q1 (p1h (w1h ) , p2 ) |θh ] − φ (ν) (1 − ρ) Ψ (p1h (w1h ) , w1h ) = 0, ∂w1h ∂p1h ∗ ∗ which yields w1h = w1 = 0 for ρ = 1 and w1h > w1 for ρ < 1. Denoting by e ∂πM 1 ∂w1h and ∂π2 ∂p2 the left-hand side of (27) for p1k , k = l, h, (34) and (13) respectively, the implicit function theorem yields  ∂p∗  1l ∂ρ  ∂p∗  1h  ∂ρ   ∗ =  ∂w1h   ∂ρ  ∂p∗ 2 ∂ρ ∂πR1l ∂πR1h ∂p1l , ∂p1h , ∂ 2 πR1l ∂p2  2 1l  ∂ πR1h  ∂p1h ∂p −  ∂ 2 πe 1l  M1   ∂w1h ∂p1l ∂ 2 π2 ∂p2 ∂p1l  ∂ 2 πR1l ∂p1l ∂p1h ∂ 2 πR1h ∂p2 1h e ∂ 2 πM 1 ∂w1h ∂p1h 2π ∂ 2 ∂p2 ∂p1h ∂ 2 πR1l ∂p1l ∂w1h ∂ 2 πR1h ∂p1h ∂w1h e ∂ 2 πM 1 2 ∂w1h ∂ 2 π2 ∂p2 ∂w1h 29 −1   ∂ 2 πR1l ∂ 2 πR1l ∂p1l ∂p2   ∂p1l ∂ρ  ∂ 2 πR1h   ∂ 2 πR1h  ∂p1h ∂p2    ∂p1h ∂ρ  ,  ∂ 2 πe  e ∂ 2 πM   M1  1   ∂w1h ∂ρ  ∂w1h ∂p2  ∂ 2 π2 ∂ 2 π2 ∂p2 ∂ρ ∂p2 2 ∗ where p∗ , k = l, h, is the equilibrium price in (27) evaluated at w1k and p∗ is the equilibrium 2 1k price arising from M2 − R2 ’s maximization problem (9). It follows from Assumptions 2-4 (where M1 ’s choice variables are w1l and w1h ), the first-order condition in (27) and ∂2π R1l = ∂p1l ∂w1h ∂2π R1h = ∂p1h ∂p1l e ∂ 2 πM 1 ∂w1h ∂p1l = e ∂ 2 πM 1 ∂w1h ∂p1h = e ∂ 2 πM 1 ∂w1h ∂p2 = ∂ 2 π2 ∂p2 ∂w1h ∂ 2 πR1l ∂p1l ∂p1h = = 0 that the determinant of the Jacobian matrix is positive. Moreover, it can be immediately seen from (13), (27) and (34) that ∂ 2 πR1l ∂p1l ∂ρ = ∂ 2 πR1h ∂p1h ∂ρ ∂ 2 π2 ∂p2 ∂ρ = = 0 and e ∂ 2 πM 1 ∂w1h ∂ρ < 0. Hence, standard computations show that = sign e ∂ 2 πR1l ∂ 2 πR1h ∂ 2 π2 ∂ 2 πM1 ∂p1l ∂p2 ∂p1h ∂w1h ∂p2 ∂p1h ∂w1h ∂ρ sign ∂p∗ 1h ∂ρ = sign ∂ 2 πR1l ∂ 2 π2 ∂ 2 πR1l ∂ 2 π2 − ∂p1l ∂p2 ∂p2 ∂p1l ∂p2 ∂p2 2 1l sign ∗ ∂w1h ∂ρ = sign ∂ 2 πR1l ∂ 2 πR1h ∂ 2 π2 ∂ 2 πR1l ∂ 2 πR1h ∂ 2 π2 + ∂p1l ∂p2 ∂p2 ∂p2 ∂p1l ∂p2 ∂p1h ∂p2 ∂p2 ∂p1h 1h 1l sign ∂p∗ 1l ∂ρ − ∂p∗ 2 ∂ρ e ∂ 2 πR1h ∂ 2 πM1 <0 ∂p1h ∂w1h ∂w1h ∂ρ e ∂ 2 πM1 <0 ∂w1h ∂ρ ∂ 2 πR1l ∂ 2 πR1h ∂ 2 π2 ∂p2 ∂p2 ∂p2 2 1l 1h sign ≤0 = sign − e ∂ 2 πR1l ∂ 2 πR1h ∂ 2 π2 ∂ 2 πM1 ∂p2 ∂p1h ∂w1h ∂p2 ∂p1h ∂w1h ∂ρ 1l ≤ 0, (35) where the values of the signs follow from Assumptions 2-4 (where M1 ’s choice variables are w1l ∂ 2 πR 1l and w1h ), and the equalities hold when consumer demands are independent ( ∂p1l ∂p2 = ∂ 2 π2 ∂p2 ∂p1h = 0). Differentiating (14) with respect to ρ and using (27), (33) and (34) yields after some manipulation ν [p∗ (ρ) − θl ] Eθ2 1l × ∂q1 (p∗ (ρ) , p∗ (ρ)) ∂p∗ (ρ) 2 2 1l |θl + (1 − ν) ∂p2 ∂ρ ∗ Ψ (p∗ (ρ) , w1h (ρ)) 1h ∂p∗ (ρ) 1h ∂w1h + [p∗ (ρ) − θh ] Eθ2 1h ∂p∗ (ρ) 1h φ (ν) (1 − ρ) ∂ρ   ∗ (ρ) , p∗ (ρ)) ∗ (ρ) ∂q1 (p1h ∂p2 2 |θh .  ∂p2 ∂ρ Proceeding along the same lines as in the proof of Proposition 2, if the condition in Assumption 1 holds with inequality (interdependent demands), it follows from (35) (and p∗ − θl > 0, 1l p∗ − θh > 0) that 1h ∂πM1 −R1 (ρ) ∂ρ ρ=1 < 0, which implies that the equilibrium ownership stake is ρ∗ < 1. If the condition in Assumption 1 holds with equality (independent demands), we have tp ∂πM1 −R1 (ρ) ∂ρ ρ=1 = 0, which implies that the equilibrium ownership stake is ρ∗ = 1. tp Proof of Corollary 1. Suppose that the equilibrium under resale price maintenance is replicated with a two-part tariff. Substituting (12) into (34) yields φ (ν) (1 − ρ∗ ) ∂p∗ ∗ 1h Ω (p∗ ) − Ψ (p∗ , w1h ) . 1h 1h ∂w1h As the inspection of (10) and (32) reveals, this expression is negative and vanishes when costs θ1 and θ2 are independent. It follows from Assumption 3 (where M1 ’s choice variables are w1l and 30 w1h ) that a reduction in w1h from the equilibrium under resale price maintenance is profitable unless costs are independent. Since we know from the proof of Proposition 4 that the optimal w1h decreases with ρ, the equilibrium ownership stake under a two-part tariff is higher than the one under resale price maintenance, and they coincide with independent costs. Proof of Proposition 5. With the demand function in (15), substituting (24) and πR1h = 0 into (8), M1 ’s maximization problem becomes ν2 + µ ν (1 − ν) − µ p2l + p2h ν ν max ν (p1l − θl ) α − βp1l + γ p1l ,p1h + (1 − ν) (p1h − θh ) α − βp1h + γ − (1 − ρ) πR1l (p1h ) ν (1 − ν) − µ (1 − ν)2 + µ p2l + p2h 1−ν 1−ν , where πR1l = ∆θ {α − βp1h + γ [νp2l + (1 − ν) p2h ]} − µγ (p2h − p2l ) (p1h − θl − ν∆θ) . ν (1 − ν) The first-order conditions for p1l and p1h are α − 2βp1l + γ ν(1−ν)−µ p2l 1−ν and α−2βp1h +γ 2 + (1−ν) +µ p2h 1−ν ν 2 +µ ν p2l + ν(1−ν)−µ p2h ν +βθh +φ (ν) (1 − ρ) β∆θ + µγ ν(1−ν) + βθl = 0 (p2h − p2l ) = 0, respectively. Taking the first-order conditions for p2l and p2h of M2 − R2 ’s maximization program (9) yields α − 2βp2l + γ γ ν(1−ν)−µ p1l 1−ν ν 2 +µ ν p1l + ν(1−ν)−µ p1h ν + βθl = 0 and α − 2βp2h + 2 + (1−ν) +µ p1h 1−ν + βθh = 0, respectively. Plugging these conditions into M1 ’s maximization program (14), we can find analytically the equilibrium value ρ∗ (µ). 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