Strategic Sourcing for Entry Deterrence and Tacit Collusion

Yutian Chen∗ August 23, 2010

Abstract By sourcing key intermediate goods to a potential entrant, an incumbent firm can credibly and observably commit to an intense post-entry competition, thereby deterring the entry. At the same time, a collusive effect exists, whereby the entrant’s loss from staying out of the final-good market is compensated through their sourcing transaction. We find that entry-deterring sourcing in general has ambiguous effect on social welfare. However, there exist scenarios where it enhances not only social welfare, but also consumers’ surplus.

Keywords: Sourcing; Entry deterrence; Stackelberg Competition; Cournot Competition JEL Classiﬁcation: D41, L11, L13

∗Department of Economics, California State University, Long Beach, CA 90840, USA. Email: [email protected]; Tel: 562-985-5083; Fax: 562-985-5804

1 1 Introduction

While firms invariably endeavor to minimize their input cost when they configure the supply of critical intermediate goods, literature reveals many strategic elements which may play a pivotal role in firms’ sourcing decisions.1 In this work, we show that sourcing strategy can be utilized for an anti-competitive purpose. An incumbent firm may outsource key intermediate goods to a potential entrant even when the entrant is costlier with respect to the intermediate goods production. By doing so, the incumbent deters the entrant from producing the final good. Some sourcing practices in the business world could be anticompetitive. One example is Boeing and a Japanese consortium composed of the three biggest industrial giants of Japan: Mitsubishi Heavy Industries, Kawasaki Heavy Industries LTD, and Fuji Heavy Industries. These Japanese firms expressed an interest in entering the market for commercial aircrafts (Chicago Tribune, April 14, 1990). Consequently, agreements were signed between Boeing and the Japanese firms. According to the agreements, Boeing would purchase from them the 767-X fuselage during the 1990s (Chicago Tribune, April 14, 1990), and then wings, together with related research and development during the 2000s (Chicago Tribune, December 21, 2003). Boeing’s outsourcing cannot be easily justified based on cost-saving, since in the aircraft industry, costs in Japan “are just as high as or higher than at home” (Newsweek In- ternational Edition, May 15-22, 2006). Another example is Boeing and Lockheed. Although Lockheed exited the commercial aircraft market after 1981, it still possessed the production capability to reenter and compete with Boeing. Boeing signed a contract with Lockheed to purchase from Lockheed certain parts of commercial aircraft (The Wall Street Journal, May 10, 1989, p. 87). Subsequently, Lockheed never reentered the commercial aircraft market. In this work, we analyze a simple model with an incumbent and an entrant for a final good. An intermediate good is required in order to produce the final good. The incumbent can produce both goods, whereas the entrant can only produce the intermediate good before entrance. By paying some irreversible entry cost, the entrant can be equally efficient as the incumbent in producing the final good. To abstract from other determinants of sourcing, we assume that the incumbent is more efficient than the entrant in producing the intermediate good. These two firms interact in a three-stage game. In stage one, the incumbent and the entrant negotiate a sourcing contract. Once they sign a contract, the entrant is obligated to supply the incumbent the quantity of the intermediate good specified in their contract. In stage two, the entrant makes its entry decision. In stage three, active firms compete in quantities for the final good. We find that the model results in a unique equilibrium in terms of the quantity sourced by the incumbent to the entrant. In equilibrium, whenever there is a positive quantity outsourced, the entrant stays out of the final good market. The effect of sourcing from the incumbent to the entrant is two-fold. On one hand, the outsourced quantity constitutes an entry-deterring “capacity” for the incumbent. When producing within the established capacity, the incumbent is granted a Stackelberg leader’s advantage and is bent on taking a more aggressive response towards entry. The entrant, understanding that the incumbent will fully convert its acquired intermediate good into the

1See, e.g., Spiegel (1993), Chen (2001), Shy and Stenbacka (2003), Chen et al. (2004), Van Long (2005), Buehler and Haucap (2006), Arya et al. (2008a,b).

2 final good, is forced to act as a Stackelberg follower after its entry. Consequently, it has less incentive to enter the market for the final good. As such, there exists a threshold quantity for the incumbent to order from the entrant, which reduces the entrant’s post-entry profit to the point of deterrence. On the other hand, outsourcing facilitates tacit collusion.2 With entry deterred, the final good market is kept monopolized and the entry cost is avoided. Whenever the benefit of outsourcing is more than offsetting its cost, a joint surplus is generated and both firms are better off through their sourcing transaction. Although entry-deterring outsourcing leads to efficiency loss together with a monopolized final-good market, it does not necessarily reduce social welfare. In fact, outsourcing can improve social welfare either due to eliminating a large entry cost, or due to a large quantity produced for the final good. The latter case occurs when the entrant faces a relatively small entry cost, which gives the entrant a strong incentive to enter. Thus for entry deterrence, the incumbent needs to commit to a relatively large quantity. As a result, consumers’ loss is moderate and can be dominated by the gain of firms, resulting in enhanced social welfare. In fact, the aggressive stance taken by the incumbent can even benefit consumers. If the entry cost is very small, such that deterring entry requires a quantity exceeding the duopoly quantity, entry-deterring outsourcing enhances consumers’ surplus. Our finding indicates that, everything else being equal, providers who pose a real entry threat can be less likely to practice entry compared to entrants independent of the incumbent. It thus offers insight for a puzzling empirical finding. As literature points out, there are good reasons for firms to beware of the entry potential of their key suppliers. For example, Caves and Porter (1977) argue that, “important suppliers to an industry ... are often likely entry candidates”.3 However, empirical evidences tell quite a different story. Smiley (1988) summarizes an extensive survey across a broad range of industries regarding which source of entry concerns them the most. One finding is, “surprisingly few firms were concerned about new entrants ...from (among) their suppliers”. Such finding is consistent with an arising trend in outsourcing. Nowadays, many firms outsource their “core” activities, including R&D, in disregard of the potential that their providers might turn into fierce competitors. For instance, pharmaceutical giants such as GlaxoSmithKline and Eli Lilly outsource to Asian bio-tech research companies for bringing new drug to market. Dell, Motorola and Philips are buying complete designs of some digital devices from Asian developers, although “there is a lot of great capability that has grown in Asia to develop complete products” (BusinessWeek, March 21, 2005). While other factors could be at play, our work highlights that, the buyer-seller relationship offers a channel of implicit collusion. As a result, suppliers could be less inclined to enter and compete compared to independent entrants. Our work is in line with the literature where an incumbent builds entry-deterring capacity to secure monopoly profit (Spence (1977), Dixit (1979, 1980)). An explicit difference between

2The collusion is “tacit” because outsourcing and entry appear to be independent decisions in our model. Some abuse of terminology exists here. Usually collusion implies that firms maximize joint profit, but individual profit is not maximized. However, in this work, joint profit is maximized in collusion due to the maximization of individual profit, and no individual firm has incentive to deviate from their sourcing agreement. 3As pointed out in Caves and Porter (1977), suppliers are likely to possess key elements for a successful entry, including well-established distribution or service networks, and the ability to produce components transformable into other commodities. One can also name other reasons for key suppliers to be likely entrant candidates. For example, it could be relatively easier for providers to infer information regarding the downstream market demand or consumers’ tastes.

3 outsourcing and capacity-building in entry deterrence is that there is no tacit collusion when capacity is built for entry deterrence, and the entrant is a passive player there. More importantly, an implicit difference exists. Previous literature points out that if capacity is only observed with some noise, or if there is positive observation cost to the entrant, capacity built before entry totally loses its value in entry deterrence (Bagwell (1995) and Várdy(2004)).4 This might explain to some extent the lack of empirical evidence on capacity building as constituting an entry barrier (see, e.g., Hilke (1984), Lieberman(1987), Goolsbee and Syverson(2008)). Instead, with outsourcing to deter entry, through contracting, the entrant naturally gets informed about the exact quantity it needs to supply the incumbent. Observability problems are thus unlikely to arise, especially when we take into consideration that entry deterrence through outsourcing is in both firms’ interests. A rather sparse literature examines the viability of sourcing in entry deterrence in different contexts. Spiegel (1993) shows that when production exhibits diseconomies of scale, outsourcing to a potential entrant can enhance its future production cost therefore induce it to stay out. In our work, the occurrence of outsourcing does not rely on the strict convexity of cost. Outsourcing deters the entrant’s entry even when production exhibits economies of scale, as shown in Section 5. Arya et al. (2008b) find that outsourcing to a monopoly supplier pre-entry induces the supplier to favor the entrant less, which can make entry unprofitable to the entrant. Complementarily, in this work, we show that the incumbent can order directly from the entrant to deter its entry. Our work shares a common spirit with Gelman and Salop (1983) and Judd (1985), where they show that an incumbent can be induced to accommodate entry once the entrant commits to a less aggressive action. While entry accommodation in their works can be fragile when there exist multiple entrants, in our work outsourcing to one entrant can establish a capacity which deters the entry of multiple entrants, as discussed in Section 5. The anticompetitive effect of vertical integration has received lots of attention in literature on vertical foreclosure (see, e.g., Salinger (1988), Ordover et al. (1990), Rey and Tirole (2007)). As a complement, our work illustrates the anticompetitive effect of vertical disintegration. Literature which models the endogenous leader-follower relationship between buyer and seller includes Baake et al. (1999), in the context of cross-supplies among competitors; and Chen et al. (2010), in the context of outsourcing from a non-integrated final good producer to a vertically integrated rival. The rest of the paper is organized as follows. Section 2 establishes the baseline model. Section 3 analyzes the model and derives the major result. Section 4 gives welfare analysis. Section 5 discusses the result by considering several model variations. Section 6 then concludes. All proofs are relegated to an appendix.

2 The Model

Firm 0 is a monopolist and ﬁrm 1 is a potential entrant for a ﬁnal good F . The inverse demand of good F is given by P (Q), where Q is the total quantity of good F . For Q not

4Bagwell (1995) shows that if the incumbent’s capacity is only observed by the entrant with some non-zero noise, the commitment value of capacity may totally vanish regardless of how small that noise is. V´ardy (2004) ﬁnds that if observing the incumbent’s capacity incurs some positive cost to the entrant, then the incumbent’s commitment loses entirely its value, irrespective of the size of the observation cost.

4 too large such that P (Q) > 0, it holds that P 0(Q) < 0,P 00(Q) ≤ 0. To produce good F requires a key intermediate good, denoted as good I. Both firm 0 5 and firm 1 can produce good I, and the constant marginal costs for good I are c0 for firm 6 0 and c1 for firm 1. Assume c1 > c0 > 0. By investing a fixed fee K ≥ 0, firm 1 can acquire the same technology as firm 0 in converting good I into good F . Assume one unit of good I can be converted into one unit of good F . W.l.o.g, the constant average cost for producing good F is normalized to zero. The strategic interaction between firms 0 and 1 is modelled into a three-stage game: 1 1 Stage one. Firm 0 and firm 1 negotiate a sourcing contract {x0,S}, where x0 ≥ 0 is the quantity of good I firm 0 orders from firm 1,7 and S is the total payment firm 0 pays to firm 1.8 A binding contract is signed once they reach an agreement. Stage two. Firm 1 decides whether to enter the market of final good F by investing K.

Stage three. Firm 0 and firm 1 decide {x0, x1}, the quantity of good I each produces inside, which is unobservable to each other. If firm 1 enters, firms 0 and 1 simultaneously decide {q0, q1}, quantities of good F . If firm 1 does not enter, firm 0 chooses q0 as a monopolist. 9 1 1 The quantities are subject to q0 ≤ x0 + x0; q1 ≤ x1 − x0. The solution concept for the game is subgame perfect Nash equilibrium (SPNE). Since 1 c1 > 0, once firm 1 enters to produce good F , in any equilibrium x1 = x0 + q1. Let δ = 1 if firm 1 enters the market for good F and δ = 0 if it stays out. Profit for each firm at the terminal nodes of the game tree is given by ½ e 1 Π0(x0, S, x0, q0, q1) = P (q0 + q1)q0 − S − c0x0 if δ = 1 n 1 Π0 (x0, S, x0, q0, q1) = P (q0)q0 − S − c0x0 if δ = 0 ½ e 1 1 Π1(x0, S, x0, q0, q1) = P (q0 + q1)q1 + S − c1(x0 + q1) − K if δ = 1 n 1 1 Π1 (x0, S, x0, q0, q1) = S − c1x0 if δ = 0 3 Model Analysis and Major Result

In this section, we solve for SPNE of the game through backward induction. We start from finding the Nash equilibrium (NE) quantities and profits in stage three for given sourcing decision in stage one and entry decision in stage two. 5The linearity enables us to have a clear view of the central point. It is not critical to our analysis, though, as shown in Section 5. 6 Assuming c1 > c0 allows us to focus on the strategic aspect of sourcing. Our model is readily extended to the case when c1 ≤ c0. If c1 < c0, firm 0 is more eagerly sourcing to firm 1 for efficiency gains. Our key results will not be affected as long as c1 is not much smaller than c0. If c1 is much smaller, firm 1 will always enter the market of good F to drive firm 0 out and enjoy monopoly profit. 7Our key results are well preserved if we allow firm 1 to order good I from firm 0. Suppose firms 0 and 1 also negotiate in stage one regarding who is the supplier of good I. If firm 1 enters the market for good F , it would fully outsource good I to firm 0, due to firm 0’s lower cost of production. That increases the opportunity cost of entry deterrence, but has no impact on the qualitative part of our major results. 8To focus on our central point, we consider a lump sum payment for the shipment of good I, and do not explicitly assume how firms 0 and 1 split the surplus generated by outsourcing. Our major finding persists under alternative pricing schemes, including linear pricing and two-part tariff. Section 5 gives a discussion. 9The inequality here is equivalent to assuming free disposal of good I for both firms. I.e. no cost will occur if either firm leaves some of its acquired good I unused, without converting them into good F .

5 3.1 Stage Three There are two cases in stage three according to firm 1’s entry decision in stage two: either firm 1 has entered or it has stayed out. Case I. Firm 1 enters to produce good F . In this case, firms 0 and 1 simultaneously e 1 choose quantities of good F . Firm 1 maximizes Π1(x0, S, x0, q0, q1). Since producing x0 incurs 1 1 marginal cost c0 > 0 to firm 0, in equilibrium x0 = q0 − x0 if q0 > x0. Firm 0’s problem is ½ 1 1 e 1 P (q0 + q1)q0 − S − c0(q0 − x0) if q0 > x0 max Π0(x0, S, x0, q0, q1) = q0 P (q0 + q1)q0 − S otherwise.

1 1 If q0 > x0, firm 0’s marginal cost is c0; if q0 < x0, firm 0’s marginal cost is zero. Figure 1 displays the best response curves in stage three. Firm 1’s best response curve 0 1 is RR . Firm 0’s best response curve is kinked at q0 = x0, as shown by the heavy curve. If 0 0 firm 0’s marginal cost is c0, its best response curve is MM , intersecting RR at point W . If firm 0’s marginal cost is zero, its best response curve is OO0, intersecting RR0 at point V . W W V V Let quantities at point W and V be (q0 , q1 ) and (q0 , q1 ), respectively. Notice that since V W 10 the best response curves are downward-sloping, q0 > q0 > 0.

Figure 1: Post-entry Best Response Curves

We deﬁne the autarky duopoly proﬁts as

W W W W W W W W W W (π0 , π1 ) = (P (q0 + q1 )q0 − c0q0 ,P (q0 + q1 )q1 − c1q1 ). Also define the Cournot profits when firm 0 has zero marginal cost as

V V V V V V V V V (π0 , π1 ) = (P (q0 + q1 )q0 ,P (q0 + q1 )q1 − c1q1 ).

b For any given qi of firm i, denote firm j’s best response as qj (qi), i 6= j, i, j = 0, 1, solved 0 from the first order condition P (qi + qj)qj + P (qi + qj) − cj = 0. We have

b 00 b 0 d(qi + q (qi)) P q (qi) + P j = 1 − j > 0. (1) 00 b 0 dqi P qj (qi) + 2P

10 W We assume q1 > 0 throughout our analysis. Otherwise, ﬁrm 1 shall never enter the market for good F .

6 If firm 0 is a Stackelberg leader at zero marginal cost and firm 1 is a follower, their profits are l f b b b b (π0(q0), π1 (q0)) = (P (q0 + q1(q0))q0,P (q0 + q1(q0))q1(q0) − c1q1(q0)). b R Notice that the smallest q0 at which q1(q0) = 0 is given by point R and denoted as q0 . For R f q0 ∈ [0, q0 ), π1 (q0) is strictly decreasing in q0 since by the envelope theorem,

f dπ1 (q0) 0 b b = P (q0 + q1(q0))q1(q0) < 0. (2) dq0

1 If x0 > 0, the competition in stage three is a Cournot game where firm 0 has built a 1 1 capacity of x0. As in Figure 1, there are three subcases according to the value of x0: 1 W Subcase 1 (small capacity). If x0 < q0 , the best response curves intersect at point W W W . In equilibrium firms 0 and 1 produce (q0 , q1 ) respectively. 1 W V Subcase 2 (intermediate capacity). If x0 ∈ [q0 , q0 ], the best response curves inter- 1 1 sect at q0 = x0. Firm 0 produces x0 of good F . Understanding this, firm 1 will produce its b 1 best response q1(x0). Sourcing to firm 1 thus establishes firm 0 as a Stackelberg leader and forces firm 1 to act as a Stackelberg follower in the market for good F . 1 V Subcase 3 (large capacity). If x0 > q0 , the best response curves intersect at point V . V V 1 V Firms 0 and 1 produces (q0 , q1 ), and x0 − q0 amount of good I is left unused by firm 0. Case II. Firm 1 stays out for good F . As a monopolist of good F , firm 0’s problem is ½ 1 1 n 1 P (q0)q0 − S − c0(q0 − x0) if q0 > x0 max Π0 (x0, S, x0, q0, q1) = q0 P (q0)q0 − S otherwise. 1 1 Same as in Case I, if q0 > x0, firm 0’s marginal cost is c0; if q0 < x0, firm 0’s marginal O M cost is zero. Denote q0 ≡ arg maxq0 P (q0)q0. Denote qi ≡ arg maxqi [P (qi) − ci]qi and M M M πi ≡ [P (qi ) − ci]qi , i = 0, 1. The following lemma derives the NE and the corresponding profits in stage three.

1 1 Lemma 1 A unique NE exists in stage three where x0 = q0 − x0 if q0 > x0 and x0 = 0 otherwise, which is given as follows.

I Suppose ﬁrm 1 enters the market for good F .

1 W W W e 1 e 1 (a) If x0 < q0 , then (q0, q1) = (q0 , q1 ) and each firm obtains (π0(x0), π1(x0)) = W 1 W (π0 + c0x0, π1 ); 1 W V 1 b 1 e 1 e 1 (b) If x0 ∈ [q0 , q0 ], then (q0, q1) = (x0, q1(x0)) and each firm obtains (π0(x0), π1(x0)) = l 1 f 1 (π0(x0), π1 (x0)). Firm 0 is a Stackelberg leader and firm 1 is a Stackelberg follower; 1 V V V e 1 e 1 (c) If x0 > q0 , then (q0, q1) = (q0 , q1 ) and each firm obtains (π0(x0), π1(x0)) = V V (π0 , π1 ). II Suppose firm 1 stays out of the market for good F .

1 M M n 1 M 1 (a) If x0 < q0 , then q0 = q0 and firm 0 obtains π0 (x0) = π0 + c0x0; 1 M O 1 n 1 1 1 (b) If x0 ∈ [q0 , q0 ], then q0 = x0 and firm 0 obtains π0 (x0) = P (x0)x0; 1 O O n 1 O O (c) If x0 > q0 , then q0 = q0 and firm 0 obtains π0 (x0) = P (q0 )q0 .

7 1 Based on Lemma 1, at given {x0,S}, total equilibrium profit of each firm when firm 1 enters the market for good F is

e 1 e 1 e 1 e 1 1 (Π0(x0,S), Π1(x0,S)) = (π0(x0) − S, π1(x0) − K + S − c1x0). (3)

Instead, when firm 1 stays out, total equilibrium profit of each firm is

n 1 n 1 n 1 1 (Π0 (x0,S), Π1 (x0,S)) = (π0 (x0) − S, S − c1x0). (4)

We next move back to stage two to ﬁnd the entry rule of ﬁrm 1.

3.2 Stage Two W.l.o.g., if firm 1 is indifferent between entering or not, we assume that it stays out. By f 1 e 1 V W W Lemma 1 and the continuity of π1 (x0), we have π1(x0) ∈ [π1 , π1 ]. If K ≥ π1 , firm 1 never V enters. By Bain’s terminology (Bain (1956)), entry is blockaded. If K < π1 , firm 1 always 1 V W enters irrespective of the value of x0, and entry is unavoidable. For K ∈ [π1 , π1 ), there 1 exists a threshold value of x0, below which it is profitable for firm 1 to enter for good F . M R f Define function τ(K) : [0, π1 ] → [0, q0 ] satisfying π1 (τ(K)) = K. Notice that τ is R strictly decreasing in K and c1 since for τ < q0 , dτ 1 dτ 1 = < 0, = < 0. (5) 0 b 0 b b dc1 P (τ + q1(τ)) dK P (τ + q1(τ))q1(τ)

V W Throughout our following analysis, we focus on K ∈ [π1 , π1 ), the range of interests. The lemma below summarizes ﬁrm 1’s entry rule in stage two.

1 W V R Lemma 2 In SPNE, ﬁrm 1 enters if and only if x0 < τ, with τ ∈ (q0 , min{q0 , q0 }].

When firm 0 sources a quantity no less than τ to firm 1, firm 1’s post-entry profit is reduced to be no larger than its entry cost. Therefore, firm 1 will not enter the market for good F . The reason τ decreases in K and c1 is intuitive: a smaller value of K or c1 gives firm 1 a stronger incentive to enter and compete firm 0. As a result, firm 0 needs to take a more aggressive stance towards entry in order to keep firm 1 out.

3.3 Stage One We finally move back to stage one, where firms 0 and 1 discuss their sourcing transaction for good I. W.l.o.g, if both firms are indifferent between reaching an outsourcing agreement 1 1 (x0 > 0) and not (x0 = 0), we assume that no outsourcing occurs. 1 W When firm 0 anticipates firm 1’s entry, it has incentives to outsource quantity x0 > q0 to firm 1 in order to exploit its Stackelberg leader’s advantage in their future competition. However, the following lemma shows that no outsourcing can occur if firm 1 enters.

1 Lemma 3 In any SPNE, if firm 1 enters for good F , it must be x0 = 0, with profits for firm e e W W 0 and firm 1 being (Π0, Π1) = (π0 , π1 − K).

8 The intuition is that if outsourcing is followed by firm 1’s entry, although firm 0 can be 1 W better off ordering x0 > q0 to force firm 1 to be a Stackelberg follower for good F , industry profit is reduced because the Stackelberg quantity is larger than the Cournot quantity. Therefore, at least one firm will refuse to contract in stage one in order to secure its autarky duopoly profit. Sourcing from firm 0 to firm 1 only occurs for entry-deterring purpose. P1 n 1 P1 n 1 Denote the industry profit without entry as i=0 Πi (x0) ≡ i=0 Πi (x0,S). Since outsourcing entails efficiency loss and firms are maximizing industry profit in outsourcing, the following lemma is intuitive.

1 Lemma 4 In any SPNE, if ﬁrm 1 stays out of the market for good F , it must be x0 = τ. Moreover, outsourcing arises in equilibrium if and only if

X1 n W W Πi (τ) > π0 + π1 − K. (6) i=0

Lemma 4 says that outsourcing will arise in equilibrium whenever entry-deterrence generates a joint surplus compared to the autarky case, and the outsourced quantity is τ ≤ V R O min{q0 , q0 } ≤ q0 . Outsourcing facilitates tacit collusion between firms 0 and 1. By Lemma 1, with firm 1’s entry deterred, firm 0 produces

d M q0 ≡ max{q0 , τ}

¯ M as a monopolist for good F . Deﬁne K ∈ (0, π1 ) to be such that X1 n W W ¯ Πi (τ) = π0 + π1 − K; i=0

also deﬁnec ¯1 > c0 to be such that

W M W (¯c1 − c0)q0 = π0 − π0 .

¯ ¯ W Lemma 5 Condition (6) holds if and only if K > K. Moreover, K < π1 when c1 < c¯1. The intuition of Lemma 5 is spelled out below. In order to deter firm 1’s entry, the smaller the value of K, the larger the quantity firm 0 needs to commit to through sourcing to firm 1. However, a large quantity can reduce industry profit to be no larger than the joint autarky profit net of the entry cost, leading to no outsourcing. The lower bound of K for outsourcing to occur is given by K¯ , which solves Condition (6) at equality. At K = K¯ , industry profit is the same with and without entry deterrence. If we fix the value of K, then outsourcing occurs if c1 is not too big. A higher value of K allows for a larger scope of c1 where outsourcing arises in equilibrium. The upper bound W of c1 for outsourcing to be in equilibrium solves Condition (6) at equality with K = π1 . In this case, firm 1 gets zero profit whether it enters or not. To deter entry, firm 0 needs W W to outsource τ = q0 , leading to the efficiency loss (c1 − c0)q0 . On the other hand, entry deterrence generates a gross gain given by the difference between firm 0’s monopoly profit M W π0 and its autarky duopoly profit π0 . The upper bound of c1 thus is given byc ¯1, which balances the efficiency loss and the gross gain of entry deterrence.

9 3.4 SPNE Before we fully characterize the SPNE, it is useful to deﬁne ½ V W ¯ V [π1 , π1 ) if K < π1 Ψ ≡ ¯ W ¯ V (K, π1 ) if K ≥ π1 .

M f M W 11 By Lemma 5, Ψ 6= ∅ when c1 < c¯1. Also deﬁne K ≡ π1 (q0 ) ∈ (0, π1 ), such that M M τ(K ) = q0 . The following theorem then characterizes the SPNE.

V W Theorem 1 For K ∈ [π1 , π1 ), the SPNE is given as below.

1 1 I If K ∈ Ψ, firm 1’s entry is deterred by firm 0. In SPNE, x0 = τ and S ∈ [c1x0 + W P1 n W W M π1 − K, i=0 Πi (τ) − (π0 + π1 − K)]. Moreover, if K ≥ K , firm 0 produces M M M x0 = q0 − τ, q0 = q0 ; if K < K , firm 0 produces x0 = 0, q0 = τ.

1 II If K/∈ Ψ, ﬁrm 1’s entry is accommodated by ﬁrm 0. In SPNE, x0 = 0, q0 = x0 = W W q0 , q1 = x1 = q1 .

Our major result for c1 ∈ [c0, c¯1) is also illustrated in Figure 2, where K varies on the V W axis. To have the whole picture, we allow for K < π1 and K ≥ π1 .

Figure 2: SPNE

When c1 ∈ (c0, c¯1), Theorem 1 asserts that there exists a non-empty range of K where firm 0 outsources to firm 1 in equilibrium, and its sole purpose of outsourcing is to deter firm 1’s entry. Two effects work together for outsourcing to serve entry deterrence. The first is the “capacity-building” effect. The outsourced quantity τ builds up a sizable capacity for firm 0 prior to firm 1’s entry, which enables firm 0 to commit to a more aggressive response towards entry, hence raising the entry barrier for firm 1. The second is the collusive effect. Without firm 1’s entry, the profit in the market of good F exceeds the joint autarky profit net of the entry cost, giving both firms incentives to engage in outsourcing.

4 Welfare Analysis

We define social welfare as the summation of firms’ surplus and consumers’ surplus. It is clear that firms are better off under outsourcing, yet consumers may be worse off when

11See proof of Lemma 4 in the Appendix.

10 outsourcing reduces the quantity of good F . Let us take no outsourcing as the status quo, W W W then Q ≡ q0 + q1 is the status quo quantity. Status quo social welfare is

Z QW S W W SW = P dq − c0q0 − c1q1 − K. 0 When firm 1’s entry is accommodated in SPNE, by Theorem 1, each firm produces the autarky duopoly quantity, so that social welfare is given by the status quo social welfare. Consider K ∈ Ψ, where outsourcing occurs in SPNE and firm 1’s entry is deterred. Social welfare is Z d q0 O d SW = P dq − c1τ − c0(q0 − τ). 0 Outsourcing generates a distortion in social welfare, given by

D(K) = SW O − SW S ( R qM D (K) ≡ K + 0 P dq + [c (qW − qM + τ) + c (qW − τ)] if K ≥ KM (7) 1 RQW 0 0 0 1 1 = τ W W M D2(K) ≡ K + QW P dq + [c0q0 + c1(q1 − τ)] if K < K

D(K) includes three parts: the saving of the entry cost K; the distortion in welfare due W d d M to the quantity change from Q to q0 (recall that q0 = max{q0 , τ}); and the distortion in welfare due to the change in cost structure. While the ﬁrst part is positive, the second and last parts can be either positive or negative. Diﬀerentiating D(K) with respect to K, we have12 ( dD(K) dD1(K) = 1 + (c − c ) dτ > 0 if K ≥ KM = dK 0 1 dK (8) dD2(K) dτ M dK dK = 1 + [P (τ) − c1] dK < 0 if K < K

V W Proposition 1 For K ∈ [π1 , π1 ), the following holds. I When ﬁrm 1’s entry is accommodated (K/∈ Ψ), social welfare is the same as in the status quo.

II When ﬁrm 1’s entry is deterred (K ∈ Ψ), let K1,K2 solve D1(K1) = 0,D2(K2) = 0. Then compared to the status quo,

(a) Social welfare is smaller if K ∈ (K2,K1), and it is larger if K > K1 or K < K2. f W (b) Consumers’ surplus is larger if K < π1 (Q ). As long as c0 is not small, the range f W of c1 where π1 (Q ) ∈ Ψ is non-empty .

Outsourcing may increase social welfare in two cases. The first case is when the entry cost is big (K > K1). Outsourcing enhances social welfare mainly due to the saving of K. M However, consumers are worse off since the quantity produced of good F is q0 , firm 0’s monopoly quantity, which is less than QW . The second case is when the entry cost is small (K < K2). To deter entry, firm 0 outsources a large quantity of good I and fully converts its acquired good I into good F . Thus consumers faces a moderate loss. When the total amount of the consumers’ loss and the efficiency loss is less than the saving of the entry cost,

12See proof of Proposition 1 in the Appendix.

11 f W social welfare increases. Furthermore, when the entry cost is sufficiently low (K < π1 (Q )), outsourcing leads to a quantity of good F exceeding QW , implying that entry deterrence even benefits consumers. Notice that in either of these two cases, the less efficient producer, firm 1, is producing W more than its autarky duopoly quantity (τ > q1 ). Therefore, the reason for excessive entry in Lahiri and Ono (1988)—where social welfare is reduced when less efficient firms enter so that a larger quantity is produced at high cost—does not apply here. Instead, reasons for excessive entry in Mankiw and Whinston (1986) function here in both of the two cases. Namely, firm 1’s production is at the cost of a contraction in firm 0’s quantity (business- stealing effect) as well as a positive entry fee, resulting in a reduced social welfare. A new element exists in this work which strengthens the business-stealing effect in Mankiw and Whinston (1986). Entry diminishes the incumbent’s incentives for entry deterrence, which gives rise to a even larger contraction in the incumbent’s quantity. This occurs in the second case, when entry deterrence generates a quantity exceeding firm 0’s monopoly M W quantity (τ > q0 ). Entry implies that firm 0 contracts its quantity by τ − q0 , which is M W larger than the business-stealing given by q0 − q0 in Mankiw and Whinston (1986). The result is that entry into a monopoly market is excessive for the society. Furthermore, this f W work demonstrates that under certain circumstances (K < π1 (Q )), incentives for entry deterrence can lead firm 0 to take a very aggressive stance, i.e., τ > QW , such that entry is excessive even for consumers. Our finding suggests that a prohibition of entry-deterring sourcing may not be justified from an antitrust perspective. Although a careful analysis is required for antitrust decisions, Proposition 1 gives general policy implications. It indicates that when the entry cost falls in the intermediate range, antitrust concerns on entry deterrence are well posed and government should discourage entry-deterring sourcing. Instead, when the entry cost is relatively large or relatively small, entry-deterring sourcing should be judged favorably by antitrust authorities. ¥ We use a numerical example to graphically illustrate the welfare effect of outsourcing. 7 Let P = max{0, a − Q}, Q = q0 + q1, with a = 10, c0 = 2 . In the market of good F , firm M 13 M 169 0’s monopoly quantity is q0 = 4 and profit is π0 = 16 . The autarky Cournot profits are W c1 2 W 9 2 2 π0 = (1 + 3 ) , π1 = ( 2 − 3 c1) . If firm 0’s average cost is zero, firm 1’s Cournot profit is V 4 2 f 1 2 π1 = 9 (5 − c1) . If firm 1 is a Stackelberg follower, its profit is π1 (q0) = 4 (10 − q0 − c1) . V W Consider K ∈ [π1 , π1 ). To deter firm 1’s entry, the minimum quantity firm 0 needs to √ f outsource is τ = 10 − c1 − 2 K, which satisfies π1 (τ) = K. If entry deterrence occurs in P1 n W W equilibrium, it must generates a larger industry profit, i.e., i=0 Πi (τ) > π0 + π1 − K. ¯ ¯ Solving this condition at equality gives K, such that K > K holds when there is outsourcing.√ ¯ W W M W 39 5−9 By Lemma 5, the largest c1 for K ≤ π1 is solved from (c1 −c0)q0 = π0 −π0 asc ¯1 = 16 . Figure 3 illustrates this example. The shaded area is Ψ, where firm 1’s entry is deterred 1 by x0 = τ in SPNE. By (7), for K ∈ Ψ, outsourcing distorts social welfare by

( 2 √ √ 463 7c1 19c1 M D1(K) = 32 + 18 − 3 − 7 K + 2c1 K + K if K ≥ K D(K) = 2 109 c1 17c1 M D2(K) = 8 − 9 − 6 − K if K < K √ dD (K) dD (K) Notice that 1 = 2 K√+2c1−7 > 0 since c > c = 7 ; and 2 = −1 < 0. Figure 3 plots dK 2 K 1 0 2 dK K1,K2, such that within Ψ, K > K1 and K < K2 give the areas where outsourcing improves social welfare. Relative to the ﬁxed value of c0, c1 can not be too big for outsourcing to

12 f W beneﬁt the society. Moreover, again for c1 not too big, π1 (Q ) ∈ Ψ. The hatched area in f W Figure 3 gives K < π1 (Q ), where outsourcing beneﬁts also consumers.

Figure 3: A Linear Example

5 Model Variations

To gain insights on the robustness of our basic results, we consider several model variations including alternative pricing schemes in outsourcing, multiple potential entrants, price competition of good F , and economies of scale in the production of good I.

¤ Alternative Pricing Schemes. Our main result is well preserved with different pricing schemes in outsourcing such as linear pricing and two-part tariff. Here we only analyze the case of linear pricing. At the beginning of the game, firm 1 announces price p at which it 1 is willing to supply good I. Firm 0 then orders x0. After that, stage two and stage three follow the same as in the baseline model. 13 Consider the case when firm 1 enters. Since p ≥ c1 > c0, the only incentive for firm 0 to outsource is to gain the Stackelberg leader’s advantage in their ensuing competition in 1 W the market for good F . This implies x0 > q0 if there is outsourcing. Firm 1, foreseeing its loss as a Stackelberg follower when it supplies firm 0, will set p high enough to recoup its future loss in the market for good F . However, such a high price will drive firm 0 to turn to in-house production, and no outsourcing shall occur in equilibrium.14 Therefore, the

13 Firm 1 has no incentive to set p < c1. If it stays out, it loses money for each unit of good I sold at a price lower than its average cost. If it enters, it may suffer as a Stackelberg follower in the market of good F when supplying firm 0, thus will never offer p < c1 for good I. 14The case here when firm 1 enters is similar to the setting in Chen et al. (2010). In their work, firm 0 can not produce good I in-house. It chooses its supplier of good I among pure outside suppliers and a vertically integrated rival (firm 1). They find that, since firm 1 will charge a high price of good I in order to recoup its future loss as a Stackelberg follower, outsourcing from firm 0 to firm 1 occurs only if firm 1 is

13 W opportunity cost of staying out for firm 1 is still π1 − K, and Lemma 3 remains unaltered. At the same time, at the given value of p, firm 0’s opportunity cost of entry deterrence is W 1 1 W either π0 (its profit at x0 = 0), or its optimal profit when x0 ∈ (q0 , τ), the case when it lets firm 1 enter and then exploits its Stackelberg leader’s advantage. Although the second case complicates our analysis to some extent, it has no impact on the qualitative part of our finding. Entry-deterring sourcing can again arise in SPNE, where firm 1 sets its price p of 1 good I in a way that firm 0 finds it optimal to outsource x0 = τ to deter firm 1’s entry.

¤ Multiple Potential Entrants. Suppose firm 0 faces n > 1 symmetric potential entrants, denoted as firms 1, ..., n. 1 First, suppose these entrants appear in sequential periods. By ordering x0 = τ from firm 1 (the first entrant to appear), firm 0 is able to deter all the entrants from producing good F . The reason is that all other entrants, based on their observation that outsourcing occurs (although they may not directly observe the quantity outsourced) and firm 1 stays out of the market for good F , understand that firm 0 has committed to a quantity large enough for entry to be unprofitable. Second, suppose these entrants appear at the same time and make simultaneous decisions whenever firm 1 moves in the baseline model. The solution concept to this game is sequential equilibrium. We find that there exist multiple equilibria. It can occur in equilibrium that the incumbent accommodates all entrants, or it outsources to some of the entrants to deter their entry and lets the rest enter. Nevertheless, the strategy stated in Theorem 1, combined with an appropriate belief system, constitutes a sequential equilibrium where all entrants stay out of the market for good F . In this equilibrium, firm 0 outsources only to firm 1 the quantity τ. As a result, firm 1 will not enter to produce good F . Each of the other entrants, believing that firm 0 outsources to firm 1 the quantity τ, will not enter the market for good 1 F . Moreover, none of firms 0 and 1 wants to deviate to x0 ∈ [0, τ), as that will invite firm 1’s entry. In such an equilibrium, by outsourcing to firm 1 alone, firm 0 deters all the entrants from producing good F .

¤ Bertrand Competition. Our major finding in the baseline model goes through the case when firms 0 and 1 produce differentiated good F and compete in prices upon firm 1 1’s entry. First, outsourcing x0 > 0 before firm 1’s entry still establishes an entry-deterring capacity for firm 0. By producing within its established capacity, firm 0 faces a smaller marginal cost in the market for good F , thus is able to commit to a price of good F which is low enough for entry deterrence. Second, the collusive effect of outsourcing persists. With firm 0 alone producing good F and the entry cost saved for firm 1, industry profit can exceed the joint autarky duopoly profit net of the entry cost, giving both firms incentives to sign the outsourcing contract. In fact, since duopoly profit is lower under price competition than under quantity competition, price competition gives both firms even stronger incentives to engage in entry-deterring outsourcing.

¤ Economies of Scale. If the production of good I exhibits economies of scale, outsourcing from firm 0 to firm 1 strengthens firm 1’s competency, thus may induce firm 1 to enter

sufficiently more efficient than outside suppliers. However, in our work firm 1 is less efficient. As a result, in equilibrium firm 0 never orders from firm 1.

14 for good F . In light of this insight, we analyze a modified game where firms’ marginal cost of good I decreases in quantity.15 We find that the theme of our major finding is well-preserved when economies of scale exist. It is still the case that in equilibrium a non-empty range of parameters exists, where firm 0 outsources good I to deter firm 1 from producing good F .A new feature in SPNE under economies of scale is that whenever firm 0 outsources, it always exclusively orders good I from firm 1 and produces zero amount of good I in-house.

6 Conclusion

We examine the role of sourcing in entry deterrence in a setting where the potential entrant is able to provide key intermediate goods. We find that the incumbent in the market for a final good may order the intermediate good from the entrant for the sole purpose of preventing the latter from producing the final good. Two strategic effects exist with outsourcing. First, the outsourced amount commits to the incumbent’s quantity of the final good, hence functioning as an entry-deterring capacity of the incumbent. Second, tacit collusion exists, where firms utilize outsourcing to keep the final-good market monopolized. These strategic elements can lead outsourcing to occur even when the entrant is costlier in the intermediate good production, as long as its cost disadvantage is not too severe. We have exogenously assumed in our model a critical timing, namely, the outsourced quantity is determined before the entrant’s entry decision. In fact, this timing can arise endogenously if we allow the entrant to choose either to make its entry decision prior to or following the incumbent’s sourcing decision. It can be shown that once outsourcing is negotiated after the entry, no outsourcing can occur and each firm ends up with its autarky duopoly profit. Therefore, whenever outsourcing arises in our base setting, it is the case that the entrant will choose to put off its entry decision until they finish the sourcing negotiation, so that it can benefit from entry deterrence. Our model is readily extended to the case where a final good producer lacks comparable ability in producing the intermediate good and is choosing among multiple suppliers. Our finding stresses that an intermediate good supplier, by developing its entry potential for the final good, may acquire a competitive edge, hence win the upstream competition. How- ever, many factors are missing in our simple setting which could affect our major finding. To highlight the anticompetitive effect of sourcing, we have ruled out the possibility that outsourcing is accompanied by technology leakage, which could directly build the entrant’s entry potential. For example, the buyer may need to teach the seller techniques in order to guarantee high-quality supply.16 In addition, our finding is derived in a static setting. In a

15For simplicity, we assume that ﬁrms 0 and 1 have the same production cost of good I, given by ½ cq − vq2 for q ≤ c/2v C(q) = c2/4v for q > c/2v

Market demand of good F is P (Q) = max{0, a − Q}, with 0 < c < a ≤ c/2v. The last inequality guarantees that in equilibrium, production entails positive marginal cost. Moreover, cost function is not “too concave” so that the existence of post-entry Nash equilibrium is guaranteed. Everything else is kept the same as in the baseline model. 16Van Long (2005) analyzes a setting where outsourcing is associated with labor training in source country which can benefit rival firms. As a result, firms tend to retain part of their demand to inside production, irrespective of the high labor cost of internal sourcing.

15 dynamic framework, an originally innocent supplier may gradually accumulate its know-how and innovate a superior ﬁnal good through supplying the incumbent. However, it can also be the case that repeated interaction between incumbent and entrant in the long run facilitates a higher degree of collusion, leading to even less entry. All of these factors, including technology leakage and the sustainability of long-run collusion through outsourcing, are interesting avenues for future research.

Appendix

1 W Proof of Lemma 2. By Lemma 1, if x0 ≤ q0 , firm 1 should enter for good F since W 1 V R V π1 − K > 0; if x0 > min{q0 , q0 }, firm 1 should stay out, otherwise it gets π1 − K ≤ 0. 1 W V R f W W f V V For x0 ∈ (q0 , min{q0 , q0 }], by (2) and π1 (q0 ) = π1 > K, π1 (q0 ) = π1 ≤ K, there exists f 1 1 a unique intersection of π1 (x0) and K, given by x0 = τ(K), below which entry is profitable W W V V R W V R for firm 1. We have τ(π1 ) = q0 , τ(π1 ) = min{q0 , q0 }, and τ(K) ∈ (q0 , min{q0 , q0 }]. By dqb(q ) 1 1 < 0 and qW > 0, qW < qR holds, hence (qW , min{qV , qR}] 6= ∅. dq0 1 0 0 0 0 0 1 W W Proof of Lemma 3. Given firm 1’s entry, if x0 = 0, industry profit is π0 + π1 − K 1 by Lemma 1 and (3). Since either firm can opt out of outsourcing, x0 > 0 can arise in equilibrium only if it leads to a joint surplus. 1 1 W Suppose x0 > 0. There can be three cases: (i) x0 ∈ (0, q0 ]. By Lemma 1 and (3), W W 1 W W industry profit is π0 + π1 − K − (c1 − c0)x0 < π0 + π1 − K, a contradiction. (ii) 1 V V V 1 1 V x0 > q0 . Industry profit is π0 + π1 − c1x0 − K. By deviating to x0 = q0 , it can be V V V improved to π0 + π1 − c1q0 − K. At least one firm can be strictly better off with the other 1 W V firm no worse off, again a contradiction. (iii) x0 ∈ (q0 , q0 ]. For it to be in equilibrium, l 1 f 1 1 W W 1 b 1 1 b 1 π0(x0) + π1 (x0) − c1x0 > π0 + π1 must hold, implying [P (x0 + q1(x0)) − c1](x0 + q1(x0)) > W W 1 b 1 W b W W W b M π0 + π1 . By (1), x0 + q1(x0) > q0 + q1(q0 ) = q0 + q1 > 0 + q1(0) = q1 . By the strict 1 b 1 1 b 1 W W W W W W concavity of profit, [P (x0+q1(x0))−c1](x0+q1(x0)) < [P (q0 +q1 )−c1](q0 +q1 ) < π0 +π1 , 1 W a contradiction. Thus it must be x0 = 0 whenever firm 1 enters. By Lemma 1, profit is π0 W for firm 0 and π1 − K for firm 1. Proof of Lemma 4. By Lemma 1, Lemma 2 and (4), to have firm 1 stays out, firms’ problem in stage one is  M M 1 M 1 1 M X1  P (q0 )q0 − c1x0 − c0(q0 − x0) if x0 < q0 n 1 1 1 1 1 M O max Π (x ) = P (x )x − c1x if x ∈ [q , q ] 1 i 0 0 0 0 0 0 0 x0  O O 1 1 O i=0 P (q0 )q0 − c1x0 if x0 > q0 1 s.t. x0 ≥ τ

1 M f M M M R R b R It is solved at x0 = τ. Deﬁne K ≡ π1 (q0 ), then τ(K ) = q0 . By (1), q0 = q0 + q1(q0 ) > W W b M W f W f M M f R q0 + q1 > 0 + q0(0) = q0 . By (2), π1 = π1 (q0 ) > π1 (q0 ) = K > π1 (q0 ) = 0. I.e., M W M M 1 K ∈ (0, π1 ). By (5), τ(K) < q0 when K > K . At x0 = τ(K), industry proﬁt is ½ X1 M M M M n ξ1(τ) ≡ P (q0 )q0 − c1τ − c0(q0 − τ) if K ≥ K Πi (τ) = M (9) ξ2(τ) ≡ P (τ)τ − c1τ if K < K i=0

P1 n M i=0 Πi (τ) is continuous in K and kinked at K = K . By Lemma 3, outsourcing arises in P1 n e e equilibrium if and only if i=0 Πi (τ) > Π0 + Π1, rewritten as (6).

16 Proof of Lemma 5. First, we show that there exists a unique K¯ such that (6) holds if and only if K > K¯ . We have P ½ d( 1 Πn(τ) + K) dτ M i=0 i (c0 − c1) dK + 1 ≥ 1 if K ≥ K = 0 dτ M (10) dK (P (τ)τ + P (τ) − c1) dK + 1 > 1 if K < K The first inequality follows (5); the second inequality follows (5), the strict concavity of M M M M M profit, and theP fact that τ > q0 > q1 when K < K . When K = π1 > K , we have τ = 0 and 1 Πn(τ) + K = P (qM )qM − c qM + πM = πM + πM > πW + πW . When i=0 iP 0 0 0 0 1 0 1 0 1 K = 0, τ = qR and 1 Πn(τ) = [P (qR)− c ]qR < [P (qW + qW )− c ](qW +qW ) < πW +πW . 0 i=0 i P0 1 0 0 1 1 0 1 0 1 1 n W W ¯ M There exists a unique intersection of i=0 Πi (τ) and π0 + π1 − K, given by K ∈ (0, π1 ). Condition (6) holds if and only if K > K¯ . Second, we show the existence and uniqueness ofc ¯1 > c0. We start from showing that W W M W W (c1 − c0)q0 + π0 − π0 is monotonic in c1. Note that (q0 , q1 ) solve first order conditions ( 0 P q0 + P − c0 = 0 F 1 0 P q1 + P − c1 = 0 F 2

By implicit function theorem, dqW P 00qW + P 0 dqW P 00qW + 2P 0 0 = − 0 > 0, 1 = 0 < 0. 0 2 0 00 W W 0 2 0 00 W W dc1 (P ) + P P (q0 + q1 ) dc1 (P ) + P P (q0 + q1 )

W W W W W dπ0 ∂π0 dq1 ∂π0 0 W dq1 M We have = + = P q > 0. Since π is invariant in c1 and dc1 ∂q1 dc1 ∂c1 0 dc1 0 W W W W d[π0 +(c1−c0)q0 ] dπ0 W dq0 W W M = + q + (c1 − c0) > 0, (c1 − c0)q + π − π strictly increases dc1 dc1 0 dc1 0 0 0 W M in c1. When c1 goes to its lower bound c0, it goes to π0 − π0 < 0. When c1 goes to its W W M M upper-bound such that q1 = 0, q0 goes to q0 and it goes to (c1 − c0)q0 > 0. A unique W W M c¯1 > c0 exists such that (¯c1 − c0)q0 + π0 − π0 = 0. ¯ W M M M Third, at c1 =c ¯1, it holds that K = π1 > K . By the definition ofc ¯1, P (q0 )q0 − W M W W ¯ ¯ W c¯1q0 − c0(q0 − q0 ) = π0 . By the uniqueness of K at any given c1, K = π1 at c1 =c ¯1. ¯ W M W ¯ Finally, we prove that K < π1 if c1 < c¯1. Since K < π1 , when K is solved from W W ¯ W ¯ ξ2(τ) = π0 + π1 − K, K < π1 is trivially satisfied. Consider the case when K is solved W W W W ¯ M from ξ1(τ) = π0 + π1 − K. Define f ≡ ξ1(τ) − π0 − π1 + K. For K ≥ K , we have dK¯ df/dc1 df dτ W W W = − . By (10), = (c0 − c1) + 1 > 0. Recall that Q ≡ q + q . It is true dc1 df/dK dK dK 0 1 W W W W W W dQ 1 dπ1 ∂π1 dq0 ∂π1 0 W dq0 W that = 0 00 W < 0. Moreover, = + = P q − q < 0. Since dc1 P +P Q dc1 ∂q0 dc1 ∂c1 1 dc1 1 dqW d(πW +πW ) dqW dqW W qW > qW and 1 < 0, we have 0 1 = P 0qW 1 + P 0qW 0 − qW > P 0qW dQ − qW . 0 1 dc1 dc1 0 dc1 1 dc1 1 1 dc1 1 Thus W W W df dτ d(π0 + π1 ) W dτ 0 W W dQ = −τ + (c0 − c1) − < q1 − τ + (c0 − c1) − P (Q )q1 dc1 dc1 dc1 dc1 dc1 P 0(QW )(qW − qW ) dQW = qW − τ + 0 1 − P 0(QW )qW (by (5) and F 1,F 2) 1 0 b 1 P (τ + q1(τ)) dc1 W W W W 0 W W dQ b W P 0(QW ) ≤ q1 − τ + (q0 − q1 ) − P (Q )q1 (as τ + q1(τ) > Q , P 0(τ+qb(τ)) ∈ (0, 1]) dc1 1 W W 0 W W dQ W = q0 − τ − P (Q )q1 < 0. (by Lemma 2, τ > q0 ) dc1

17 dK¯ df/dc1 ¯ W Therefore = − > 0, proving K < π for c1 < c¯1. dc1 df/dK 1 Proof of Theorem 1. Proof of I. For K ∈ Ψ, Condition (6) holds and entry deterrence generates a joint surplus compared to the autarky case. In the market of good I, if firm 1 1 W is just recouped its loss from staying out, S = c1x0 + π1 − K; if firm 1 appropriates the P1 n W W 1 W whole surplus of entry deterrence, S = i=0 Πi (τ) − (π0 + π1 − K). For S ∈ [c1x0 + π1 − P1 n W W 1 K, i=0 Πi (τ) − (π0 + π1 − K)], no firm has incentive to deviate to x0 = 0. The rest part in I then follows Lemma 1-4. Proof of II. Condition (6) is violated for K/∈ Ψ. The rest part then follows Lemma 1 and Lemma 3. Proof of Proposition 1. Proof of I. It is straightforward. Proof of II. First, we prove (8) is true. The first inequality follows (5). By (5) and P 0 < 0, 0 b b b dD2(K) dτ P (τ+q1(τ))q1(τ)+P (τ)−c1 P (τ)−P (τ+q1(τ)) = 1 + [P (τ) − c1] = 0 b b = 0 b b < 0. Notice that 0 ∈/ Ψ, dK dK P (τ+q1(τ))q1(τ) P (τ+q1(τ))q1(τ) R b hence τ < q0 and q1(τ) > 0. Second, results in II(a) follow (8). f W W Third, we prove II(b). Since K < π1 (Q ) ⇒ τ > Q , consumers’ surplus is larger f W for K < π1 (Q ). We need to show that as long as c0 is not too small, the range of c1 f W f W W f W f W where π1 (Q ) ∈ Ψ is non-empty. Notice that π1 (Q ) < π1 = π1 (q0 ). Thus π1 (Q ) ∈ Ψ f W V f V f W ¯ requires (a) π1 (Q ) ≥ π1 = π1 (q0 ) and (b) π1 (Q ) > K. Consider (a) first. By (2), W V W V it requires Q ≤ q0 . It can be shown that Q is decreasing in c0. Since q0 is invariant in c0, at given c1, (a) is satisfied for c0 sufficiently large. Next, we consider (b). Since W M f W M f W W Q > q0 , we have π1 (Q ) < K by (2). Moreover, τ(π1 (Q )) = Q . Thus by (10), (b) W W W f W W W is satisfied if P (Q )Q − c1Q + π1 (Q ) > π0 + π1 . Reorganizing the inequality gives f W W f W W π1 (Q ) > (c1 − c0)q0 . Since π1 (Q ) increases in c0 and (c1 − c0)q0 decreases in c0, at f W W given c1, π1 (Q ) > (c1 − c0)q0 holds for c0 sufficiently large. We conclude that for c0 not f W too small, the range of c1 where π1 (Q ) ∈ Ψ is non-empty.

Acknowledgment

I am grateful to two anonymous reviewers for their comments and suggestions. I want to thank Sandro Brusco, Pradeep Dubey and Yossi Spiegel for their suggestions during the development of this work. I also want to thank Eva Carceles-Poveda, Yongmin Chen, Jianpei Li, Qihong Liu, Kristen Monaco, Wallace Mullin, Debapriya Sen, Konstantinos Serfes, Yair Tauman, Na Yang and the participants at the 17th International Conference on Game Theory 2006, the 5th Annual International Industrial Organization Conference 2007, the 5th Biennial Conference of HKEA 2008, for helpful discussions and comments.

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