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Available online at www.sciencedirect.com International Journal of Industrial Organization 26 (2008) 1 – 16 www.elsevier.com/locate/econbase Outsourcing, vertical integration, and price vs. quantity competition ☆ Anil Arya a , Brian Mittendorf b , David E.M. Sappington c,⁎ a b The Ohio State University, United States Yale School of Management, United States c University of Florida, United States Received 12 December 2005; received in revised form 2 October 2006; accepted 14 October 2006 Available online 28 November 2006 Abstract We show that standard conclusions about duopoly competition can be reversed when the production of key inputs is outsourced to a vertically integrated retail competitor with upstream market power. Under such outsourcing, Bertrand competition can produce higher prices, higher industry profit, lower consumer surplus, and lower total surplus than Cournot competition. In addition to limiting the intensity of retail competition, Bertrand competition can limit the extent of wholesale competition by reducing the incentive of retail providers to produce key inputs themselves. © 2006 Published by Elsevier B.V. JEL classification: D43; L13; L40 Keywords: Competition; Outsourcing; Vertical integration 1. Introduction Outsourcing the production of key inputs to external suppliers is ubiquitous in today's economy, and outsourcing to retail competitors is common in many important industries. For example, in the telecommunications industry, vertically integrated incumbent operators routinely supply key inputs (e.g., telephone loops)1 to retail competitors. In addition, soft-drink producers, ☆ We thank the co-editor, Roman Inderst, and two anonymous referees for very helpful comments. ⁎ Corresponding author. E-mail address: [email protected] (D.E.M. Sappington). 1 Telephone loops are the wires that connect a telephone customer's residence to the central office of a telecommunications supplier. 0167-7187/$ - see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.ijindorg.2006.10.006 2 A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 cereal manufacturers, and gasoline refiners have long supplied key inputs both to their downstream affiliates and to retail competitors. More recently, the explosion in online commerce has brought manufacturers into direct competition with their own retailers. We demonstrate that standard conclusions about price and quantity competition (e.g., Singh and Vives, 1984) can be altered when the production of inputs is outsourced to retail rivals. We show that when the supplier of an input is also a retail rival, the vertically integrated producer (VIP) may set a higher input price under Bertrand competition than under Cournot competition. The higher input price is not designed to limit the competitive strength of the VIP’s retail rival or to drive the rival from the market. Rather, the higher input price increases the VIP's opportunity cost of aggressive retail price competition by increasing the wholesale profit the VIP foregoes when the retail output of its competitor declines. The higher opportunity cost serves as a credible commitment on the part of the VIP to refrain from aggressive retail competition. The higher retail prices that ensue can generate a lower level of consumer surplus under Bertrand competition than under Cournot competition, which reverses the standard conclusion regarding the welfare implications of price and quantity competition. Our demonstration of this conclusion and related observations proceeds as follows. Section 2 describes the key elements of our baseline model, in which a VIP is the monopoly supplier of an essential input to a non-integrated retail rival. Section 3 demonstrates that retail prices and industry profit are higher while consumer surplus and total surplus are lower under Bertrand competition than under Cournot competition in this setting. Section 4 demonstrates that the industry structure presumed in the baseline model can emerge naturally as the equilibrium of a simple game. The analysis in Section 4 reveals that a more efficient retail competitor will outbid its less efficient counterpart for the right to merge with a monopoly input supplier. The resulting vertical integration increases both consumer surplus and total surplus. The increase is more pronounced under Cournot retail competition than under Bertrand competition because of the more intense competition that prevails in the former regime. Section 5 extends the model to admit upstream competition. Entrants are presumed able to achieve the same upstream cost structure as the VIP by incurring an investment I N 0. Intense upstream pricing competition deters non-integrated firms from entering the input market. However, if I is sufficiently small, the retail rival will undertake the investment in order to ensure low-cost access to the input. The retail rival is not willing to pay as much under Bertrand as under Cournot competition to secure such access because of the less intense retail competition that prevails under Bertrand competition when production is outsourced to the VIP. As a result, the decision to seek alternate input sources may actually be less prevalent when external input prices are high (as under Bertrand competition) than when they are low (as under Cournot competition). Thus, in addition to reducing the intensity of retail competition, Bertrand competition can limit the extent of wholesale competition. As noted above, our analysis extends the extensive literature that compares outcomes under Bertrand and Cournot competition in standard duopoly settings (e.g., Singh and Vives, 1984; Okuguchi, 1987; Vives, 2005) by examining the role of outsourcing to a rival. As such, our analysis complements the standard literature on outsourcing (e.g., Shy and Stenbacka, 2003; Van Long, 2005) by analyzing outsourcing to a vertically integrated retail rival. Kamien et al. (1989), Spiegel (1993), and Baake et al. (1999), among others, consider outsourcing (or subcontracting) to a rival. In contrast to our model, these analyses focus on the effects of nonlinear cost structures and do not compare outcomes under alternative forms of retail competition. We also contribute to the literature on vertical integration (e.g., Salinger, 1988; Hart and Tirole, 1990; Ordover et al., 1990) by analyzing the incentives for and the effects of vertical integration in the presence of monopoly supply of an essential input. Biglaiser and DeGraba (2001) also consider monopoly supply A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 3 of an input, but assume the input price is regulated. Consequently, the equilibrium changes in input prices that are a central feature of vertical integration in our model do not arise in their model. Chen's (2001) analysis of vertical integration may be closest to our own in that it identifies the role of high input prices in relaxing downstream price competition. Other authors have noted that wholesale profit margins can affect the intensity of retail price competition. For example, Armstrong (1998) and Laffont et al. (1998a,b) do so in a setting where network operators set a reciprocal interconnection fee; Armstrong (1999) and Harbord and Ottaviani (2002) do so in a setting where cable television operators can sell programming to competitors; and Sappington (2005) does so in a setting where entrants decide whether to make an essential input themselves or purchase the input from a vertically-integrated incumbent supplier. In contrast to the present analysis, these studies do not contrast outcomes and welfare implications of Cournot and Bertrand competition. 2. The baseline model We modify the classic model of duopoly competition by allowing one of the retail competitors (firm 1) to be a VIP. In the baseline model of primary interest, firm 1 is the sole producer of an input that is essential for retail production. Each unit of retail output requires exactly one unit of the input. Firm 1 charges its retail rival, firm 2, unit price w for the input. We normalize firm 1's upstream cost to zero. After securing the required input supply, firm i's incremental cost of producing and selling its retail product is ci (i = 1,2). Consumer demand for the retail product of firm i is given by the (inverse) demand function pi = α − qi − γqj, where pi is the price of firm i's retail product, α is a strictly positive constant, and qi and qj are the outputs of firms i and j, respectively (i, j ∈ {1,2}, i ≠ j). The parameter γ ∈ (0,1) represents the degree of product homogeneity. As γ approaches 0, the products of the two retail providers become independent. As γ approaches 1, the products of the firms become completely homogeneous. The timing in the model is as follows. First, firm 1 sets the input price it will charge to firm 2. Then, under Cournot competition, firms 1 and 2 choose their retail output levels simultaneously and independently. Under Bertrand competition, the two firms set retail prices simultaneously and independently. Finally, under both forms of retail competition, consumers make their purchase decisions and all realized consumer demand is satisfied. The profits of firms 1 and 2 when firm i produces retail output qi, firm i's retail price is pi, and the input price is w are, respectively: P1 ¼ wq2 þ ½ p1 −c1 q1 ; and ð1Þ P2 ¼ ½ p2 −w−c2 q2 : ð2Þ The first term to the right of the equality in Eq. (1) captures firm 1's profit from selling the input to firm 2. The second term reflects firm 1's profit from its retail sales, just as Eq. (2) reflects firm 2's profit from its retail sales. Retail profit is the product of sales volume and the relevant retail profit margin. Firm 1's retail profit margin is the difference between the price ( p1) it charges for its retail product and its downstream production cost (c1). Firm 2's profit margin is the difference between the price of its product ( p2) and the sum of its input cost (w) and incremental downstream production cost (c2). Consumer surplus given retail outputs q1 and q2 is: CS ¼ ð½q1 2 þ 2cq1 q2 þ ½q2 2 Þ=2: ð3Þ 4 A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 The quadratic expression in Eq. (3) arises from the linear retail demand. Total surplus is the sum of CS, Π1, and Π2. Ensuing calculations are simplified by introducing the parameters α1 ≡ α − c1 and α2 ≡ α − c2. In words, αi is the difference between the intercept of firm i's inverse demand curve and its downstream marginal cost of production. The larger is αi, the more efficient is firm i in its retail operations. We assume α1 ≥ α2, so the VIP (firm 1) is at least as efficient a retail provider as its rival (firm 2). The analysis in Section 4 demonstrates that this industry structure arises endogenously as the equilibrium of a simple game. 3. Outcomes in the baseline model Because firm 1 is the only supplier of the essential input in the baseline model, firm 1 has the ability to foreclose firm 2 (i.e., to set the input price so high that firm 2 finds it unprofitable to participate in the retail market). Firm 1 can enhance its success in the retail market by foreclosing its only retail competitor. However, foreclosure also terminates firm 1's sale of the input to firm 2 and thereby eliminates firm 1's profit from wholesale operations. When the potential for wholesale profit is sufficiently large, firm 1 will not foreclose firm 2. As Lemma 1 indicates, firm 1 will prefer to maintain firm 2 as an active buyer of the input rather than foreclose firm 2 under both Bertrand and Cournot retail competition when firm 2 is a sufficiently efficient competitor (i.e., when α2 / α1 N γ). Lemma 1. Under both Bertrand and Cournot retail competition, firm 1 forecloses firm 2 (i.e., q2 = 0) if and only if α2 / α1 ≤ γ. Lemma 1 reveals that foreclosure of firm 2 is less likely when the firms' retail products are more heterogeneous (i.e., when γ is smaller). Greater product heterogeneity implies that firm 1's retail profit declines less rapidly as firm 2 enjoys greater retail success. Consequently, because firm 2's retail success secures greater wholesale profit for firm 1, firm 1 is more willing to support firm 2's operation. If firm 2 is foreclosed, the setting reduces to one where a vertically integrated monopolist supplies a retail market. Because our primary purpose is to study the competitive interactions between a vertically integrated producer and its retail rival, we will abstract from the foreclosure setting in the ensuing discussion by assuming that the following non-foreclosure (NF) condition holds: a2 =a1 Nc: ðNFÞ In the absence of foreclosure, firm 1 finds a high input price particularly appealing under Bertrand competition. A high input price increases firm 1's wholesale profit margin. The greater wholesale profit margin increases the (opportunity) cost that firm 1 incurs if it reduces firm 2's demand for the input by limiting firm 2's retail success. The opportunity cost of competing vigorously in the retail market endows firm 1 with a credible commitment to set a higher retail price. Because prices are strategic complements under Bertrand competition, firm 1's credible commitment to set a higher retail price induces firm 2 to set a higher retail price. Firm 1 benefits from the increased demand for its retail product that results from firm 2's higher price. Firm 1 also gains from the higher retail price that it sets because the higher price increases firm 2's retail output and thereby increases firm 1 wholesale profit by increasing firm 2's demand for the input. Firm 1 does not experience a corresponding commitment benefit under Cournot competition. When firms set retail quantities rather than prices, firm 1 takes the demand for its input (q2) as A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 5 given when it chooses its retail output. Therefore, firm 1 does not perceive that variations in its retail output will affect its wholesale profit under Cournot retail competition. Consequently, firm 1 will set a higher input price under Bertrand competition than under Cournot competition, as Lemma 2 states. Lemma 2. Given (NF), firm 1 sets a higher price input under Bertrand competition 2 ½a2 −ca1 a2 c2 ½a2 −ca1 than under Cournot competition a22 − c2½8−3c . 2 2 − 2½8þc2 The higher input price and reduced intensity of retail competition under Bertrand competition reverses the standard welfare comparisons of Bertrand and Cournot competition. As long as firm 2 is sufficiently efficient that it is not foreclosed by firm 1, both firms secure greater profit under Bertrand competition than under Cournot competition. In contrast, consumer surplus is higher under Cournot competition because it does not provide firm 1 with an incentive to raise the input price and thereby reduce the intensity of retail competition. The loss consumers suffer under Bertrand competition (relative to Cournot competition) outweighs the corresponding increase in industry profit. Consequently, total surplus is lower under Bertrand than under Cournot competition, as Proposition 1 reports. Proposition 1. Given (NF): (i) both firms secure greater profit under Bertrand competition than under Cournot competition; and (ii) consumer surplus and total surplus are both lower under Bertrand competition than under Cournot competition. In summary, standard conclusions regarding the effects of price and quantity competition can be reversed when a retail provider purchases an essential input from a vertically integrated retail rival. In particular, price competition can be less intense than quantity competition in the presence of such outsourcing. Before proceeding to consider extensions of this baseline model, we briefly discuss three natural modeling variations. First, the key qualitative conclusions drawn above persist when firm 1 does not have unrestricted ability to set its preferred input price. In particular, the equilibrium input price continues to be higher under Bertrand competition than under Cournot competition when the input price that firm 1 charges to firm 2 is determined by Nash bargaining, where the disagreement outcome is that in which firm 1 is the monopoly supplier of the retail product. Also, it can be shown that firm 1 and firm 2 both secure higher profit while consumer surplus is lower under Bertrand competition than under Cournot competition in this setting with Nash bargaining.2 Second, the main qualitative conclusions drawn above also can arise in the presence of nonlinear pricing. When nonlinear pricing of the input is feasible and the upstream VIP has all of the relevant bargaining power, it can foreclose the least efficient retail provider and extract the full monopoly retail surplus. Such a policy can be optimal when the retail products are sufficiently homogeneous (Chemla, 2003), much as foreclosure arises in the baseline model when γ is sufficiently large (recall Lemma 1). However, as long as the firms' retail products are sufficiently heterogeneous, the integrated supplier will sell the input to its rival and set the marginal price in a two-part input tariff above marginal cost. Furthermore, this marginal price will be higher under Bertrand competition than under Cournot competition. In this case, the higher marginal price under Bertrand competition translates into higher profit for the firms, lower consumer surplus, and lower total surplus under Bertrand competition than under Cournot competition. 2 See Chemla (2003), Inderst and Wey (2003), and de Fontenay and Gans (2004, 2005), for example, for related models of bargaining between upstream and downstream firms. 6 A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 Finally, we note that Proposition 1 is derived in a setting in which the duopolists' products are substitutes. When the products are independent, the two retailers effectively do not compete. Consequently, the input price plays no meaningful role in reducing the intensity of retail competition. When the duopolists' products are complements and the firms engage in Bertrand competition, the VIP realizes that by reducing the price of its retail product it can enhance the demand for its input by increasing the output of the non-integrated retail rival. The lower equilibrium prices that emerge under Bertrand competition secure a higher level of consumer surplus than arises under Cournot competition. 4. Endogenous market structure The analysis to this point has taken the industry structure as given. We now demonstrate that the industry structure in the baseline model can arise naturally as the equilibrium of a simple game. The game we consider proceeds as follows. Initially, an independent upstream supplier (U) announces that it will merge with the retail operator that bids the most for this opportunity, provided the winning bid exceeds the maximum profit (Π̄U) U can secure under vertical separation. The two independent retail operators, D1 and D2, each announce simultaneously the amount they are willing to pay to merge with U. If at least one of the announced bids exceeds Π̄U, U accepts the highest bid and the merger of U and the highest retail bidder is consummated (costlessly).3 If neither bid exceeds Π̄U, no merger takes place, and vertical separation prevails. Under vertical separation, U sets the profit-maximizing prices at which it will sell the input to each of the nonintegrated firms, D1 and D2. The two retail providers then compete to serve customers. Lemma 3 reports that under vertical separation, U charges the same input prices regardless of the form of the prevailing downstream competition. Lemma 3. Given (NF) and vertical separation, the independent upstream supplier sets its input price for Di equal to αi / 2 for i = 1,2 under both Bertrand and Cournot retail competition. It is well known that, holding constant the cost structures of the retail competitors, Bertrand competition produces lower retail prices than Cournot competition and thereby secures greater consumer surplus and total surplus while reducing industry profit (e.g., Singh and Vives, 1984). Lemma 3 states that the form of retail competition (Bertrand vs. Cournot) does not affect the cost structures of the retail competitors under vertical separation in the present setting. Hence, the standard welfare comparisons emerge if the retail rivals procure inputs from an independent wholesale producer. However, in part because it eliminates the double marginalization problem,4 vertical integration increases the joint profit of the merging parties. Therefore, vertical integration, not vertical separation, will arise in equilibrium. Because D1 has a lower downstream cost than D2 (i.e., α1 ≥ α2), D1 will produce more retail output than D2 under both Bertrand and Cournot competition if the two firms incur the same input costs. Because of its relatively large retail 3 Following the merger of U and Di, the combined entity acts to maximize the profit from its wholesale and retail operations, just as in the baseline model. The entire profit earned by the vertically integrated firm accrues to Di. Residual rights of control are not an issue in the simple game analyzed here because all relevant industry information is common knowledge and there are no contracting frictions. Bolton and Whinston (1993) and Heavner (2004), among others, provide interesting analyses of vertical integration in which non-trivial residual control rights arise. 4 A double marginalization problem arises when a retailer pays more for an input than the supplier's marginal cost of production and then raises its retail price above its own (already inflated) marginal cost of production to a level that is higher than that which maximizes the joint profits of the retailer and the supplier. A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 7 output, D1 derives a greater increase in profit than D2 from the ability to secure the input at marginal cost. Consequently, as Proposition 2 reports, D1 will bid more than D2 for the right to merge with U. Proposition 2. Given (NF), D1 successfully outbids D2 for the right to merge with U under both Bertrand and Cournot competition. Other authors have reported similar conclusions. For example, Armstrong (1999) provides a corresponding finding in a setting where an independent supplier markets a premium product to retailers of a basic product. Armstrong demonstrates that the conclusion persists under a variety of allocation procedures that admit different distributions of bargaining power.5 Chen (2001) considers mergers between a retail provider and multiple upstream producers. He finds that if a merger occurs, it will involve the most efficient upstream producer. Such a merger will occur if and only if the production cost of the most efficient upstream producer is strictly less than the production costs of the other upstream producers. The equilibrium merger between the upstream producer and the most efficient retail operator in the present setting helps to justify our focus on the baseline model. After U and D1 merge, the combined entity (the VIP) will sell the essential input to D2 as long as D2 is sufficiently efficient (i.e., as long as (NF) holds). Therefore, Proposition 2 implies that the outsourcing to a more efficient vertically integrated rival that was assumed to occur in the baseline model arises naturally as the equilibrium of a simple game that admits vertical mergers. Proposition 3 reports welfare implications of the merger between U and D1. Proposition 3. Given (NF), the merger of U and D1 increases consumer surplus and total surplus under both Bertrand competition and Cournot competition. The increases are most pronounced under Cournot competition. The merger of U and D1 increases surplus under both Bertrand and Cournot competition for two primary reasons. First, the merged entity avoids the double marginalization problem because the retail division of the VIP receives the essential input at cost. Second and perhaps more surprisingly, the VIP sets the input price for D2 below the corresponding price that D2 faces under vertical separation. One might suspect the VIP would charge its retail rival a particularly high price for the input in order to limit the rival’s competitive strength. However, as Lemmas 2 and 3 indicate, this is not the case in the absence of foreclosure. The VIP charges D2 a relatively low input price in order to partially offset the advantage the retail division of the VIP secures from obtaining the input at cost. The increase in D2’s competitive strength increases its retail sales and thereby increases D2's retail sales and so increases D2's purchase of the input from the VIP. The resulting increase in the VIP's wholesale profit outweighs the decline in its retail profit caused by the input price reduction.6 The lower input prices that both retailers face under vertical integration lead to an increase in both consumer surplus and total surplus, as Proposition 3 reports. It can also be shown that the merger of U and D1 produces greater total surplus than a merger of U and D2 would generate. 5 In Armstrong's analysis, the retailer that acquires the premium content does not sub-license the content to its retail rival. Harbord and Ottaviani (2002) show that sub-licensing will occur if per-subscriber license fees are admitted because such fees reduce the intensity of retail price competition and thereby enhance industry profit. 6 Although D2 enjoys a lower input price under vertical integration, it also faces a more formidable competitor that secures the input at cost. The latter drawback dominates the former benefit, leaving D2 with lower profit under vertical integration than under vertical separation. 8 A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 Therefore, in pursuing their own private interests, the industry participants secure a welfaremaximizing outcome. The larger increases in consumer surplus and total surplus that arise under Cournot competition reflect two factors. First, the integrated supplier sets a higher input price under Bertrand competition than under Cournot competition (recall Lemma 2). As explained above, the higher input price helps the VIP establish a credible commitment to compete less vigorously against D2. This high input price introduces a relatively severe double marginalization problem, which reduces welfare. Second, because the VIP is able to establish a credible commitment to compete less vigorously and because D2 faces a relatively high input price, retail prices are relatively high under Bertrand competition. The resulting decline in consumer surplus below the level achieved under Cournot competition outweighs the corresponding increase in industry profit. Consequently, the increases in surplus from vertical integration are less pronounced under Bertrand competition than under Cournot competition. Together, Lemma 3 and Proposition 3 reveal that the identified reversal of the standard welfare conclusions about Bertrand and Cournot competition is not due simply to the presence of outsourcing. Outsourcing is present under both vertical separation and vertical integration, and the standard conclusions prevail under vertical separation. Instead, the reversal results from outsourcing to a vertically integrated retail rival. 5. Upstream competition The analysis to this point has abstracted from competition among suppliers of the essential input. To assess the effects of upstream competition most simply, consider the following modification of the baseline model. Suppose that, as in the baseline model, firm 1 (the VIP) and firm 2 are the retail providers. Now, though, suppose that new upstream competitors can replicate the upstream cost structure of firm 1 by incurring a fixed investment cost, I N 0. After upstream participation is determined, firm 1 and any active upstream competitors simultaneously and independently choose the unit price at which they will sell the essential input to firm 2. (Firm 1 continues to supply all of its own input requirements.) Firm 2 then makes its procurement decision and competes with firm 1 for retail customers. A non-integrated producer will not enter the upstream industry by incurring I in this setting. The ensuing price competition with an equally efficient upstream rival (firm 1) would dissipate all variable profit, making it impossible for the non-integrated entrant to recover its investment cost, I. In contrast, firm 2 may find it profitable to incur I and then “in-source” production of the essential input at the same marginal cost as firm 1. If the net benefit associated with the reduction in input costs exceeds I, firm 2 will find it profitable to incur the up-front investment cost. Under Bertrand competition, firm 2's net benefit from reduced input costs is diminished by an opportunity cost of in-sourcing. Recall that under Bertrand competition, firm 1 sets a relatively high input price in the baseline model. The high input price reduces the intensity of retail competition and thereby increases firm 2's profit. Firm 2 foregoes this profit enhancement from softened competition if it chooses to in-source production of the input. Consequently, firm 2 views this foregone profit as an additional (opportunity) cost of in-sourcing. Proposition 4 characterizes firm 2's sourcing decision. The proposition refers to I B, which denotes the minimum investment cost such that firm 2 optimally relies on outsourcing under Bertrand competition. I C denotes the corresponding investment cost under Cournot competition. (The Appendix provides closed-form expressions for IB and I C.) A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 9 Proposition 4. Given (NF) in the setting with potential upstream entry, 0 b IB b IC. Consequently, firm 2 will always in-source production of the input if I is sufficiently small (I b IB) and always outsource production of the input to firm 1 if I is sufficiently large (I ≥ I C). For intermediate values of I (I ∈ [IB,I C)), firm 2 will in-source production of the input under Cournot retail competition and outsource production of the input to firm 1 under Bertrand competition. Proposition 4 reports that firm 2 is less likely to in-source production of the input under Bertrand competition than under Cournot competition. This is the case even though firm 2 faces a higher input price under Bertrand competition than under Cournot competition when it outsources production to firm 1. The reluctance to in-source production despite the presence of higher input prices reflects the aforementioned opportunity cost of in-sourcing (i.e., the foregone benefit of less intense retail competition) under Bertrand competition. Proposition 4 implies that in addition to promoting less intense retail competition, Bertrand competition can limit the extent of upstream competition. It can do so by reducing the incentive of retail providers to in-source production of essential inputs. These findings imply that standard welfare conclusions continue to be reversed (as they are in the baseline model) in the presence of potential upstream competition as long as upstream entry costs (I ) are sufficiently pronounced. Clearly, if the requisite entry cost (I ) is at least I C, firm 2 will outsource production of the input to firm 1, and so the baseline model prevails and all of the conclusions derived in the baseline model hold. Furthermore, as Proposition 5 reports, there exists w an up-front investment level I aðI B ; I C Þ such that the key conclusions of the baseline model also w C hold for all IaðI ; I Þ. Proposition 5. Given (NF) in the setting with potential upstream entry, there exists an w w I aðI B ; I C Þ such that for IN I : (i) firms 1 and 2 both secure greater profit under Bertrand competition than under Cournot competition; and (ii) consumer surplus is lower under Bertrand competition than under Cournot competition. To understand the conclusions drawn in Proposition 5, recall from Proposition 4 that if I ∈ [IB, I ), firm 2 will in-source production of the input under Cournot competition and outsource production of the input under Bertrand competition. The lower input costs that arise under Cournot competition and the less intense retail competition and lower fixed costs that arise under Bertrand competition (with outsourcing of production to a vertically-integrated retail rival) are the driving forces underlying Proposition 5.7 C 6. Conclusions It is well known that Bertrand competition typically produces lower retail prices, lower industry profit, and higher levels of consumer surplus and total surplus than Cournot competition. We have shown that these standard conclusions can be reversed when a retail competitor secures an essential input from a vertically integrated provider of substitute goods. In the presence of such 7 Of course, if entry costs are sufficiently low, firm 2 will always in-source production of the input. This is the case even when (NF) does not hold, as firm 2 can avoid foreclosure by producing the input itself. In the absence of w outsourcing to a vertically-integrated rival, the standard welfare conclusions prevail. Even when IaðI ; I C Þ , the standard conclusion that total surplus is higher under Bertrand competition than under Cournot competition persists. This is the case because the loss from the duplication of fixed costs outweighs the gains realized by consumers under Cournot competition. 10 A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 outsourcing, Bertrand competition can produce higher retail prices, higher industry profit, and lower levels of consumer surplus and total surplus than Cournot competition. These outcomes arise because the vertically integrated producer (VIP) sets a relatively high input price under Bertrand competition in order to establish a high opportunity cost of aggressive retail competition. The high opportunity cost provides the VIP with a credible commitment to engage in less aggressive retail competition, which serves to increase the retail rival's output and thus its demand for the VIP's (lucrative) input. The resulting diminished intensity of retail competition and the relatively high input price under Bertrand competition produce the higher retail prices that cause the reduction in consumer surplus and total surplus (and the increase in industry profit) relative to Cournot competition. These effects are particularly pronounced when the VIP is a monopoly supplier of the input. However, the effects continue to operate in the presence of potential upstream competition. Furthermore, when upstream competition is possible, Bertrand retail competition can reduce both the intensity of retail competition and the extent of wholesale competition. Future research might consider more general demand and cost structures, economies/ diseconomies of integration, relevant information asymmetries, and alternative forms of wholesale and retail competition. Although these extensions may provide new insights of interest, they seem unlikely to reverse the finding that outsourcing to vertically integrated rivals can alter standard conclusions about outcomes under price and quantity competition. Appendix A Proof of Lemma 1. We first characterize the equilibrium outcomes under both Cournot and Bertrand competition. The equilibria are derived using backward induction. The firms' strategic decision variables in the retail market are derived first, taking input prices as given. Then profitmaximizing input prices are determined. A.1. Cournot Competition When α2 / α1 N γ, equilibrium outcomes are as follows: pC1 ¼ ½8−c2 a1 −2ca2 þ c1 ; 2½8−3c2 qC1 ¼ ½8−c2 a1 −2ca2 ; 2½8−3c2 PC1 ¼ pC2 ¼ qC2 ¼ 2½a2 −ca1 ; 8−3c2 ½8 þ c2 a21 þ 4a22 −8ca1 a2 ; 4½8−3c2 CS C ¼ ½4−c2 ca1 −4½3−c2 a2 þ c2 ; 2½8−3c2 PC2 ¼ wC ¼ a2 c2 ½a2 −ca1 − ; 2½8−3c2 2 4½a2 −ca1 2 ½8−3c2 2 ; and ½64−64c2 þ 9c4 a21 þ 4½4−3c2 a22 þ 12c3 a1 a2 : 8½8−3c2 2 ðA1Þ When α2 / α1 = γ, q2C is 0. When α2 / α1 b γ, firm 2 again produces no retail output in equilibrium, and it is readily verified that firm 1 produces α1 / 2 units of output. A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 11 A.2. Bertrand Competition Again, when α2 / α1 N γ, there is no foreclosure (so q2 N 0) and equilibrium outcomes are readily shown to be the following: P1B ¼ ½8−c2 a1 þ 2ca2 þ c1 ; 2½8 þ c2 qB1 ¼ ½8−c2 −c4 a1 −6ca2 ½2 þ c2 ½a2 −ca1 a2 c2 ½a2 −ca1 ; qB2 ¼ ; ; wB ¼ − 2 4 2 4 8−7c −c 2½8−7c −c 2 2½8 þ c2 PB1 ¼ 2½6 þ c2 a2 −c½4 þ c2 a1 þ c2 ; 2½8 þ c2 pB2 ¼ ½8−3c2 −c4 a21 þ 4a22 −8ca1 a2 ½2 þ c2 2 ½a2 −ca1 2 B ; P ¼ ; and 2 4½8−7c2 −c4 ½1−c2 ½8 þ c2 2 CS B ¼ ½64−23c4 −5c6 a21 þ 4½4 þ 5c2 a22 −4½16 þ 3c2 −c4 ca1 a2 : 8½1−c2 ½8 þ c2 2 ðA2Þ When α2 / α1 = γ, q2B is 0. When α2 / α1 b γ, firm 2 again produces no retail output in equilibrium, and it is readily verified that firm 1 produces α1 / 2 units in this case. Proof of Lemma 2. The lemma follows directly from the expressions for input prices in Eq. (A1) and Eq. (A2). Proof of Proposition 1. From Eq. (A1) and Eq. (A2): PB1 −PC1 ¼ PB2 −PC2 ¼ c2 ½4 þ c2 ½a2 −ca1 2 N0; 64−80c2 þ 13c4 þ 3c6 c2 ½256−32c2 −8c4 þ 9c6 ½a2 −ca1 2 CS B −CS C ¼ ½1−c2 ½64−16c2 −3c4 2 c½a2 −ca1 ½ca2 A−a1 B 2½1−c2 ½64−16c2 −3c4 2 N0; and b0; ðA3Þ where A = 512 − 288γ2 + 4γ4 − 3γ6 and B = 1024 − 896γ2 + 80γ4 + 26γ6 − 9γ8. The inequalities in Eq. (A3) hold because γ b 1, γα1 b α2 ≤ α1, and γA b B. From the expressions for profit and consumer surplus in Eq. (A1) and Eq. (A2): TS B −TS C ¼ c½a2 −ca1 ½ca2 C−a1 D 2½1−c2 ½64−16c2 −3c4 2 b0; where ðA4Þ C = 1536 − 352γ2 − 68γ4 + 9γ6 and D = 1024 + 128γ2 + 16γ4 − 46γ6 + 3γ8 . The inequality in Eq. (A4) holds because γ b 1, γα1 b α2 ≤ α1, and γC b D. 12 A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 Proof of Lemma 3. The proof follows from examining equilibrium outcomes under vertical separation (VS) for both Cournot and Bertrand retail competition. A.3. VS and Cournot Competition Denote the equilibrium retail prices under VS and Cournot competition by p̄iC, quantities PC by q̄iC, input prices by w̄iC, firm profits by Π̄1C, Π̄2C, and Π̄UC, and consumer surplus by CS . Equilibrium outcomes are readily shown to be (for i, j ∈ {1,2}, i ≠ j): C p̄i ¼ C P̄ i ¼ ½6−c2 ai −caj þ ci ; 2½4−c2 ½2ai −caj 2 4½4−c2 2 ; C C q̄i ¼ P̄ U ¼ 2ai −caj ; 2½4−c2 C w̄i ¼ ai ; 2 a21 þ a22 −ca1 a2 ; and 2½4−c2 P C ½4−3c2 ½a21 þ a22 þ 2c3 a1 a2 : CS ¼ 8½4−c2 2 ðA5Þ A.4. VS and Bertrand Competition Now employ the same notation as above, but replace the superscript C with the superscript B to denote Bertrand competition. Equilibrium outcomes are: B p̄i ¼ B P̄ i ¼ 2½3−c2 ai −caj þ ci ; 2½4−c2 ½ð2−c2 Þai −caj 2 4½1−c2 ½4−c2 2 ; B q̄i ¼ B P̄ U ¼ ½2−c2 ai −caj ; 2½4−5c2 þ c4 B w̄i ¼ ai ; 2 ½2−c2 ½a21 þ a22 −2ca1 a2 ; and 4½4−5c2 þ c4 P B ½4−3c2 ½a21 þ a22 −2c3 a1 a2 : CS ¼ 8½1−c2 ½4−c2 2 ðA6Þ Proof of Proposition 2. Initially, consider the setting with Cournot retail competition. We show first that D1 will merge with U if, in the absence of their merger, U would not merge with D2. The profit of the merged firm (U and D1) is as specified in Eq. (A1). If U merges with neither D1 nor D2, the profits of U and D1 are as specified in Eq. (A5). Therefore, the incremental joint profit U and D1 secure from a merger in this case is: C C PC1 −½P̄ U þ P̄ 1 ¼ ½32 þ 4c2 −6c4 þ c6 a21 þ c4 a22 −2½16−6c2 þ c4 ca1 a2 : 4½8−3c2 ½4−c2 2 This expression is a convex function of α2. Furthermore, both its roots exceed α1 Therefore, the expression is positive for all relevant values of α2, since α2 ≤ α1. A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 13 We now show that D1 also will merge with U if, in the absence of their merger, U would merge with D2. If D1 merges with U, their combined profit is as specified in Eq. (A1). If D2 merges with U, D1's profit is as given by Π2C in Eq. (A1) with α1 and α2 interchanged. Denote ∼ this profit by Π 1C. These expressions reveal that D1's maximum bid in this case is: ∼C ½48−16c2 −3c4 a21 þ 4½8−7c2 a22 −8½4−3c2 ca1 a2 : PC1 −P1 ¼ 4½8−3c2 2 Analogous logic reveals that D2's maximum bid in this case is this same expression, but with α1 and α2 interchanged. This implies that the difference between D1's maximum bid and D2's ½16þ12c2 −3c4 ½a21 −a22 z0: maximum bid is 4½8−3c2 2 This inequality implies D1 is willing to bid at least as much as D2 to merge with U. The analysis for the setting with Bertrand competition is analogous. In this case, firm 1's maximum willingness to bid when firm 2 would not merge with U is: ½32−12c2 −2c4 þ c6 a21 þ c4 a22 −4½8−3c2 ca1 a2 4½8 þ c2 ½4−c2 2 N0: When firm 2 would merge with U, firm 1's maximum bid is readily shown to exceed the ½16−4c2 −3c4 ½a21 −a22 maximum bid of firm 2 by z0: 4½8þc2 2 Proof of Proposition 3. Finding 1. The merger of U and D1 increases total surplus under Cournot competition. Under vertical integration (VI), the surpluses of the relevant parties are as specified in Eq. (A1). Under VS, the surpluses are as specified in Eq. (A5). Using these expressions: P C Ea2 þ Fa22 −2Gca1 a2 TS C − TS ¼ 1 N0; 8½32−20c2 þ 3c4 2 ðA7Þ where E = 1280 − 896γ2 + 260γ4 − 43γ6 + 3γ8, F = [192 − 92γ2 + 9γ4]γ2, and G = 512 − 288γ2+ 48γ4 − 3γ6. The expression in Eq. (A7) is convex in α2 and both roots exceed α1. Therefore, the inequality in Eq. (A7) holds since α2 ≤ α1. Finding 2. The merger of U and D1 increases total surplus under Bertrand competition. Eq. (A2) and Eq. (A6) imply that: P B Ha21 þ J a22 −2Kca1 a2 TS B ¼ TS ¼ N0; 8½1−c2 ½32−4c2 −c4 2 ðA8Þ where H = 1280 − 640γ 2 + 36γ 4 + 77γ 6 − 25γ 8 + γ 10 , J = [960 − 188γ 2 − 47γ 4 + 4γ 6 ]γ 2 , and K = 1024 − 224γ2 − 80γ4 + 3γ6 + 6γ8. The inequality in Eq. (A8) holds because the expression in Eq. (A8) is convex in α2 and has no real roots. Finding 3. The merger of U and D1 increases total surplus more under Cournot competition than under Bertrand competition. From Eq. (A7) and Eq. (A8): PC PB TS C − TS −½TS B − TS ¼ c½−Lca21 −M ca22 þ 2N a1 a2 8½1−c2 ½256−128c2 þ 4c4 þ 3c6 2 N0; ðA9Þ 14 A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 where L = 16,384 + 4096γ2 + 512γ4 − 4224γ6 + 1716γ8 − 271γ10 + 12γ12, M = 49,152− 43,008γ2 + 12,544γ4 + 64γ6 − 572γ8 + 45γ10, and N = 32,768− 20,480γ2 + 5632γ4+ 704γ6 − 192γ8 −258γ10 + 51γ12. The expression in Eq. (A9) is a concave in α2 and is positive at the two boundary values for α2. Therefore, the expression is positive for all relevant values of α2. Finding 4. The merger of U and D1 increases consumer surplus under Cournot. Under VI, consumer surplus is as specified in Eq. (A1); under VS, it is as specified in Eq. (A5). These expressions imply: P C Pa2 þ Qa22 þ 2Rca1 a2 CS C − CS ¼ 1 N0; 8½32−20c2 þ 3c4 2 ðA10Þ where P = 768 − 1152γ2 + 540γ4 − 109γ6 + 9γ8, Q = [64−68γ2 + 15γ4]γ2, and R = [32 − 3γ4]γ2. The inequality in Eq. (A10) holds because the expression in Eq. (A10) is convex in α2 and has no real roots. Finding 5. The merger of U and D1 increases consumer surplus under Bertrand. From Eqs. (A2) and (A6): P B Sa2 þ T a22 −2U ca1 a2 CS B − CS ¼ 1 N0; 8½1−c2 ½32−4c2 −c4 2 ðA11Þ where S = 768 − 384γ2 − 260γ4 + 107γ6 + 17γ8 − 5γ10, T = [320 − 100γ2 + 23γ4]γ2, and U = [512− 224γ2 − 64γ4 + 21γ6 − 2γ8 ]. The expression in Eq. (A11) is convex in α2, and both roots of the expression exceed α1. Therefore, the inequality holds for all relevant values of α2, since α2 ≤ α 1 . Finding 6. The merger of U and D1 increases consumer surplus more under Cournot competition than under Bertrand competition. From Eqs. (A10) and (A11): h i PC PB CS C − CS − CS B − CS ¼ c½−V ca21 −W ca22 þ 2X a1 a2 8½1−c2 ½256−128c2 þ 4c4 þ 3c6 2 N0; ðA12Þ where V = 49,152 − 69,632γ 2 + 32,256γ 4 − 5248γ 6 − 836γ 8 + 419γ 10 − 36γ 12 , W = 16,384− 14,336γ2 + 5888γ4 − 2240γ6 + 364γ8 + 15γ10 , and X = 32,768 − 36,864γ2 + 9728γ4 + 1728γ6 − 1600γ8 + 330γ10 − 15γ12. The expression in Eq. (A12) is a concave function in α2 and is positive at the two boundary values. Therefore, the expression is positive for all relevant values of α2. Proof of Proposition 4. First, we show that under Cournot competition, firm 2 produces the input itself if and only if I b IC, where ICu −c4 ½16−5c2 a21 þ 32½6−5c2 þ c4 a22 −4c½32−32c2 þ 7c4 a1 a2 : ½32−20c2 þ 3c4 2 A. Arya et al. / Int. J. Ind. Organ. 26 (2008) 1–16 15 If firm 2 produces the input itself (at zero marginal cost), the equilibrium outcome is the outcome under VS with w = 0 . Corresponding profits and consumer surplus are: 2 C ½2a1 −ca2 ; P1 ¼ ½4−c2 2 2 C ½2a2 −ca1 P2 ¼ −I; and ½4−c2 2 j C ½4−3c2 a21 þ ½4−3c2 a22 þ 2c3 a1 a2 : CS ¼ 2½4−c2 2 C I C is derived by equating P 2 and Π2C. Similarly, under Bertrand competition, firm 2 will produce the input itself if and only if IbI B u −c4 ½16 þ 3c2 −c4 a21 þ 16½12−2c2 −c4 a22 −2c½64−2c4 þ c6 a1 a2 ½32−4c2 −c4 2 : As above, if firm 2 produces the input itself, the equilibrium outcome is the outcome under VS with w = 0, yielding profits for firms 1 and 2 and consumer surplus: 2 B ½ð2−c2 Þa1 −ca2 P1 ¼ ; ½4−c2 2 ½1−c2 2 B ½ð2−c2 Þa2 −ca1 P2 ¼ −I; and ½4−c2 2 ½1−c2 j B ½4−3c2 a21 þ ½4−3c2 a22 −2c3 a1 a2 : CS ¼ 2½4−c2 2 ½1−c2 B Equating P 2 and Π2B yields IB . Finally, observe that: I C −I B ¼ −c6 Y a21 þ 16c2 Za22 þ 2c5 Y a1 a2 ½256−128c2 þ 4c4 þ 3c6 2 N0; ðA13Þ where Y = 512 − 80γ4 + 9γ6 and Z = 256 − 160γ2 − 8γ4 + 11γ6. It is readily verified that the inequality in Eq. (A13) holds, given (NF). C C B Proof of Proposition 5. Let Î be the value of I that equates Π2B and P 2 . Î N IB because P 2 NP 2 . C B C Also, Î b I because Π2 N Π2 . Recall that firm 2 does not produce the input itself under either Cournot or Bertrand competition if I ≥ IC. In this case, conclusions (i) and (ii) follow from Proposition 1. If Î b I b I C, firm 2 produces the input itself only under Cournot competition, in which case conclusions (i) and (ii) are confirmed as C by construction of Î. Second, a comparison of firm 1's profits ̂ PB NP follows. First, when IbI; 2 2 yields: C c4 A Va21 þ 4B Va22 −24c3 ½2 þ c2 a1 a2 PB1 −P 1 ¼ N0; 4½4−c2 2 ½8−7c2 −c4 ðA14Þ where A′ = 32 + 5γ2 − γ4 and B′ = 16 − 16γ2 + 8γ4 + γ6. The inequality in Eq. (A14) holds because the expression in Eq. (A14) is convex in α2 and has no real roots. Finally: −c2 C Va21 þ 12D Va22 þ 4cE Va1 a2 jC N0; CS −CS B ¼ 8½1−c2 ½32−4c2 −c4 2 ðA15Þ 16 A. Arya et al. / Int. J. Ind. 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