On Bundling and Entry Deterrence

A multiproduct incumbent firm faces the threat of entry from another multiproduct (generalist) firm or from single-product (specialist) firms. Assuming that the incumbent offers higher quality products than its rivals, we inquire whether the possibility of bundling by the incumbent is more effective in deterring entry in one setting or the other, and explore how the quality difference affects the comparison. For instance, for relatively high-quality differences the generalist is more vulnerable to bundling than are the specialists; but bundling is a credible action for the incumbent more often against specialists than against the generalist.


Introduction
At least since Whinston (1990) it is known that an incumbent multiproduct firm may use bundling in order to reduce an entrant's profit-possibly below the entry costand thus build an entry barrier. This paper investigates how bundling is effective to deter entry, depending on whether the incumbent faces a single multiproduct rival or multiple single product rivals.
In the proposed merger between GE and Honeywell, one argument of the European Commission against the merger was that the merged firm would have had more than half of market share in the markets for airplane engines and for avionics, and 1 3 could have used bundling to induce its rivals' exit (see Nalebuff 2002). 1 The merged entity would have competed with single-product firms, and we inquire whether a multiproduct opponent could better endure the effect of bundling; this also reveals whether single-product firms gain from merging. Choi (2008) addresses the same question, relative to mixed bundling, in a framework with symmetric firms that he uses to analyze the effect of the proposed merger between GE and Honeywell. Conversely, we assume vertical differentiation in the sense that the incumbent offers products with higher quality than its rivals' products. 2 We consider two games that differ because in one the incumbent faces an entry threat from a "generalist" multiproduct firm; in the other the entry threat comes from two "specialist" single product firms. After its rival(s) entry decision(s), the incumbent chooses whether to bundle or not its products; then price competition occurs.
We compare the two games and determine when, as a function of the extent of vertical differentiation, the incumbent's ability to bundle is more effective in deterring entry. About this, we notice that competition under bundling is fiercer, and less profitable for each firm, when the incumbent faces a generalist than when it faces specialists. Precisely, each specialist has a smaller incentive to cut its own price as it does not capture the full benefit from the price cut: the Cournot complement effect; hence specialists are less aggressive than is the generalist. This benefits the incumbent, and also induces it to be less aggressive (since prices are strategic complements); this makes the specialists' total profit higher than the generalist's profit.
Therefore, upon entry, the incumbent wants to bundle more often when it faces specialists; but when the incumbent bundles against a generalist, the reduction in the profit of the generalist is greater than the reduction in the profits of specialists. We show that as a consequence, for small-to-intermediate vertical differentiation, the incumbent's ability to bundle is a better entry deterrent against specialists because against a generalist, bundling reduces also the incumbent's profit, hence it is not credible. 3 Conversely, for intermediate-to-large vertical differentiation, bundling is a better foreclosure instrument against a generalist because it is profitable for the incumbent and reduces the generalist's profit more than the specialists' total profits. The paper is organized as follows: The next section introduces the model, and Sect. 3 (Sect. 4) examines the game in which the incumbent faces a generalist (faces specialists). Section 5 compares the two games, and Sect. 6 concludes. The study of some price competition subgames (Lemmas 1, 4) relies on Hurkens et al. (2019) (HJM henceforth). The other results in this paper are original.

The Setting
We consider, and later compare, two competition settings. In one of them a multiproduct firm A, which offers products A1 and A2, is already active and faces the threat of entry of two single-product firms B1 and B2 such that firm Bj offers only product Bj, for j = 1, 2 . Each firm Bj incurs a cost k > 0 if it decides to enter. In the other competition setting, firms B1 and B2 are merged into a single entity-B-which offers products B1 and B2. Firm A faces the threat of entry of firm B, for which the entry cost is 2k. Firms are Bertrand competitors in the sense that each firm chooses the price(s) at which it offers its product(s); but products of different firms are differentiated, hence the Bertrand paradox does not apply here.
After its rival(s) entry decisions(s), firm A decides whether to offer its products A1, A2 separately or as a bundle-in the latter case, each consumer either buys A1 and A2 or buys no product from A-that is, A chooses separate sales or joint sales (i.e., bundling). 4 Then firm A competes with its active rival(s), or acts as a monopolist if no rival entered. We use Γ s to denote the game in which A faces specialist firms B1, B2; Γ g denotes the game in which A faces the generalist firm B. One way to interpret the comparison between Γ g and Γ s is to assume that A's rival wants to maximize profit and can costlessly split into two firms if it is initially a generalist, can costlessly merge if it is initially split in two firms.
In either setting, each consumer has unit demand for good j, for j = 1, 2 . Moreover, the firms are vertically differentiated in the sense that product Aj has a higher quality than product Bj; the former yields to consumers a higher utility than does Bj, for j = 1, 2 . Precisely, each consumer has the same monetary gross utility v ij from consuming product ij, Thus, is the utility difference between products Aj and Bj and represents a measure of vertical differentiation between firms.
The firms are also horizontally differentiated in a standard Hotelling fashion: Relative to each good j, consumers are uniformly distributed over the interval [0, 1] (and independently across goods); firm A is located at 0; and firm B (or firm Bj) is located at 1. A consumer with location x j ∈ [0, 1] for good j incurs a transportation cost tx j ( t(1 − x j ) ) if he buys product Aj (product Bj), with t > 0 . This consumer's net utility from buying (for instance) product Aj is v + − tx j minus his payment to firm A. Utility is additive over the two goods.
For each firm i, let c denote the marginal production cost for product ij (for j = 1, 2 ). We assume that c is larger than t: If a consumer buys the bundle of A, then he will not also buy a single product Bj even though Bj yields a higher utility than Aj (including the transportation costs), because the utility difference is smaller than t, the price of Bj is no less than c, and c > t . Then, since marginal costs have an additive effect on prices, without loss of generality we simplify the notation by setting c = 0 and interpret prices as profit margins.
With regard to v, , t we suppose that v ≥ 3t > . The inequality v ≥ 3t implies that when both products Aj and Bj are offered, for j = 1, 2-when each rival of A enters-both markets are fully covered in equilibrium: Competition among firms leads to equilibrium prices such that each consumer buys one good 1 (product A1 or B1) and one good 2 (product A2 or B2). Hence, in this case we can view goods 1 and 2 as complements such that in order to obtain a positive utility, a consumer needs one unit of good 1 and one unit of good 2.
The property of fully covered markets does not hold if product B1 and/or B2 is not offered. For instance, if firm A is a monopolist for both goods, then A may want to charge prices such that not all consumers buy A1 and A2, as it is known from standard monopoly theory: see Sect. 3.1.2. This is socially inefficient because each consumer values one unit of good j more than its production cost, for j = 1, 2 ; thus a social welfare loss (a deadweight loss) results. 5 The inequality < 3t implies that each active firm has a positive profit (gross of the entry cost) both for the case of competition under separate sales and for competition under bundling. 6 In the rest of the paper, we derive our results in terms of ∈ (0, 3t) , the key parameter of our analysis, that reflects how strong vertical differentiation (measured by ) is with respect to horizontal differentiation (measured by t). In particular, affects not only price competition between firms, 7 but also firms' preferences between competition under separate sales or under joint sales, which eventually affects the entry decision(s) of firm B (of firms B1, B2).

Competition Between Firm A and Firm B: Game 0 g
In game Γ g , the players are firm A and firm B and the timing is as follows: • Stage one: Firm B chooses between entering and not entering, and A observes B's decision. • Stage two: After B's entry decision, A decides whether to offer its products separately (separate sales: "SS"), or in a bundle (joint sales: "JS"). 6 Both the inequalities v ≥ 3t and < 3t mainly simplify the exposition, and relaxing them does not yield additional insights. In Sect. 3.1.1 we illustrate the consequences of allowing ≥ 3t. With regard to v ≥ 3t , a weaker inequality suffices to generate the result of fully covered markets when all products are offered. However, v ≥ 3t simplifies our presentation in Sect. 4.3 and in Sect. 5. 7 As is intuitive, a higher favors firm A and harms its rival(s). Notice that if B has entered and A has chosen SS, then at stage three firm A (B) sets the prices p A1 , p A2 ( p B1 , p B2 ) for its single products and each consumer buys one good 1 and one good 2 (precisely, this is the case at the equilibrium prices). In particular, each consumer may "mix and match", that is buy one product from firm A and one from firm B, selecting for instance products A1, B2 and paying a total of p A1 + p B2 . Conversely, if A has chosen JS then competition occurs between bundles because each consumer either buys A's bundle, or buys the products B1, B2, which therefore are bundled in practice. In this case firm A (B) sets the price P A ( P B ) for the bundle of its products. 8 In this section we use backward induction to identify the subgame perfect Nash equilibria (SPNE) of Γ g : For each terminal subgame of Γ g (a subgame that starts at stage three) we identify the unique equilibrium and the equilibrium profits. These are used to deduce A's stage two choice about bundling or not, and ultimately B's entry decision at stage one.

Subgames Such That Firm B Has Entered
Given that firm B has entered, competition at stage three is affected by A's choice at stage two.
If competition occurs under SS, each market can be analyzed in isolation. In market j, a consumer located at x j buys product Aj if and only if his utility with Aj is higher than with Bj: if and only if v + − tx j − p Aj ≥ v − t(1 − x j ) − p Bj , which reduces to in which x m j is the location of the marginal consumer in market j. Moreover, F (f) denotes the c.d.f. (the density) for the uniform distribution with support [0, 1]; hence Therefore the demand for product Aj is F(x m j ) ; p Aj F(x m j ) is A's profit in market j; and p Bj (1 − F(x m j )) is B's profit in market j gross of the entry cost. (1) is the average price of the products in A's bundle (in B's bundle); and x gm is the average location of the marginal consumer.
Comparing (1) and (3) shows that competition under JS is analogous to competition under SS in the market for a single product, except that the distribution of the average location is not uniform: The c.d.f. F and the density f of the average location are Hence, the demand for the bundle of A is F (x gm ) ; P AF (x gm ) is the profit of A; and P B 1 −F(x gm ) is the profit of B gross of the entry cost. In the following, we denote with  (2) Under JS, the equilibrium prices and profits for firms A and B are: With respect to the analysis of game Γ g , Lemma 1(3) is important as it determines the choice of firm A at stage two between SS and JS, upon B's entry (see

On Bundling and Entry Deterrence
Lemma 3(1) below). Therefore, in the rest of this subsection we first provide some remarks about Lemma 1(1) and competition under SS, then we explain why JS leads to different prices and profits with respect to SS, and how determines whether SS or JS is more profitable for firm A. Lemma 1(1) identifies the equilibrium prices p * A , p * B for the single products under SS. Notice that p * A > p * B , which is intuitive given the higher quality of A's products; but p * A − p * B is smaller than the quality difference . We explain why by considering market 1: the demand for A1 is elastic, and A gains by reducing p A1 below p * B + , which is consistent with (5). Using (1) and (5) we see that in each market the marginal consumer is located at x * = 1∕2 + ∕(6t) , and x * ∈ (1∕2, 1) since ∈ (0, 3t) . A consumer who is located at Hence, consumers with location smaller than x * for one good but greater than x * for the other good mix and match: They buy one good from firm A, and one good from firm B.
As we mentioned in Sect. 2, when < 3t , each firm has a positive profit under SS. If instead ≥ 3t , then in each market j the equilibrium prices are p * Aj = − t , p * Bj = 0 ; each consumer buys product Aj, and B's profit is zero. The inequality ≥ 3t is necessary for this equilibrium as it guarantees that A has no incentive to increase p Aj above − t : When ≥ 3t , p * Aj is large enough that an increase in p Aj generates a profit decrease for A from consumers lost that is greater than the profit increase from the consumers that keep buying product Aj.
With respect to competition under JS: Lemma 1(2) shows that P g A ≠ 2p * A and P g B ≠ 2p * B : The equilibrium price for each bundle under JS is different from the sum of the equilibrium prices under SS for the products in the bundle. Moreover, also the profits under JS are different from the profits under SS. This occurs because under SS the demand functions are determined by F in (2), the c.d.f. of consumers' locations for a single good; but under JS F is replaced by F in (4), the c.d.f. of the consumers' average locations. This modifies the demand elasticities and demand sizes, which affect the firms' pricing incentives and profits.
In order to make clearer the difference between F and F , Fig. 1 represents f ,f and F,F from (2) and (4) (please see next page).
Both f and f are symmetric around 1/2 (the mean), but f is more peaked than f around 1/2, that is f puts more weight around 1/2 than does f, but puts less weight near 0 and near 1. This implies (see Fig. 1b) In order to examine the effect of JS, we first suppose that (P A , P B ) = (2p * A , 2p * B ) , that is the price for each bundle is the sum of the prices under SS of the products in the bundle. Under JS, the demand function for the bundle of for each x ∈ ( 1 2 , 1).
1 −F(x gm ) ); and given (P A , P B ) = (2p * A , 2p * B ) we have that x gm is equal to x * : the location of the equilibrium marginal consumer (in both markets) under SS. Therefore, firm A's demand changes from F(x * ) for each product of A to F (x * ) for A's bundle, and x * ∈ (1∕2, 1) plus (7) imply F (x * ) > F(x * ) . Hence, JS with unchanged unit prices increases the demand for A, and reduces the demand for B. This is the demand size effect, according to the terminology in HJM. It arises because (1) under SS, product Aj is purchased by consumers who are located in , because f puts more weight around 1/2 than f. Thus JS increases A's demand because it makes the distribution of consumers' locations more concentrated near 1/2, and consumers close to 1/2 buy from A because of A's quality advantage. But notice that this effect is weak if x * is close to 1/2 or close to 1 as then F (x * ) − F(x * ) is small.
Moving to JS also generates a demand elasticity effect that induces firm i to set the price P i of its bundle different from 2p * i , for i = A, B : The demand for A's bundle is F (x gm ) and its elasticity with respect to P A at (P A , In the following we identify the demand elasticity effect for firm A by comparing − When is about zero, we have that x * is about 1/2; thus (see (2), (4) or Fig. 1) A's demand is more elastic under JS than under SS, as under JS more consumers are located near the marginal consumer; this is a consequence of the fact that f puts more weight around 1/2 than does f. Conversely, when is large, close to 3t, x * is about 1, and (see (2), (4) or Fig. 1 , that is under JS firm A's demand is less elastic because fewer consumers are located near the marginal consumer, as f puts less weight near 1 than does f. . 1 a Density of the uniform distribution (thin) and of the average (thick). b c.d.f. of the uniform distribution (thin) and of the average (thick) 1 3

On Bundling and Entry Deterrence
For the demand elasticity effect for firm B we notice that 1 −F(x gm ) is the demand for B's bundle and its elasticity with respect to P B at (P A , (2), (4) we see that for each x * ∈ (1∕2, 1) , this is smaller than −1 , the demand elasticity −(p * B ∕2t)(f (x * )∕(1 − F(x * ))) for each single product of B under SS. This occurs because the demand size effect implies . Thus the demand elasticity effect makes B willing to reduce P B below 2p * B . Summarizing: For close to 0 the demand size effect is weak, and the demand elasticity effect induces both firms to reduce prices under JS. Then a more intense competition results, which makes both firms worse off under JS than under SS. In essence, if vertical differentiation is small, then JS intensifies competition and reduces profits because firms compete in a region with a high density of consumers (the marginal consumer x * is near 1/2 as firms are almost symmetric when is about zero, and f (x * ) > f (x * ) ), which provides incentives for aggressive pricing.
Conversely, for a large the demand size effect is still weak, but now the demand for A's bundle is inelastic. This induces firm A to increase P A , which benefits firm B because it softens the competition that B faces. Moreover, the increase in P A above 2p * A induces B to increase P B since prices are strategic complements, even though B's demand is elastic at (P A , P B ) = (2p * A , 2p * B ) . Then less intense competition results, which makes A and B better off under JS. In essence, if vertical differentiation is large, then JS softens competition and increases profits because firms compete in a region with a low density of consumers (the marginal consumer x * is near 1 since A's quality advantage is large, and f (x * ) ≃ 0 < f (x * ) = 1 ), which leads to less aggressive pricing.
Finally, for intermediate the demand size effect is non-negligible (unlike in the extreme cases above), and it dominates the demand elasticity effect. As a result, A is better off, and B is worse off under JS than under SS:

Subgames Such That Firm B Has Not Entered
If firm B has not entered, then A is a monopolist. Under SS, in market j a consumer who is located at x j buys product Aj if and only if his utility from the purchase is non-negative: if v + − tx j − p Aj ≥ 0 . Hence, the demand for product Aj is F((v + − p Aj )∕t) . Since v ≥ 3t , we find that the profit maximizing p Aj is v + − t , the lowest possible consumer's valuation for Aj; at this price, each consumer buys Aj. 9 Since a high implies that moving from SS to JS softens competition, we may investigate the incentive of firm B to reduce the quality of its own products, which increases , in order to induce A to choose JS instead of SS at stage two: After B has entered, its profit is . Hence, the demand for A's bundle is F ((2v + 2 − P A )∕(2t)).

Lemma 2
(1) If firm A is a monopolist under SS, then the optimal price in market j is p Aj = v + − t , for j = 1, 2 , and A's total profit is 2(v + − t).
(2) If firm A is a monopolist under JS, then the optimal price for the bundle is slightly greater than 2(v + − t) , and A's profit is greater than 2(v + − t).
About Lemma 2(2), notice that P A = 2(v + − t) under JS is equivalent to p A1 = p A2 = v + − t under SS, as it induces each consumer to buy A's bundle and makes A's profit equal to 2(v + − t) , which is also A's maximal profit under SS. However, A can do better by slightly increasing P A above 2(v + − t) , as we explain below. 10 Under monopoly by A, we can view the consumers' valuations for product A1 as uniformly distributed over the interval [v,v] , with v = v + − t and v = v + , 11 and the consumers' valuations for product A2 as uniformly distributed over the same interval [v,v] . Under SS, consider p A1 = v + with > 0 and small. This p A1 is less profitable than p A1 = v because it induces a consumer not to buy product A1 if his value v 1 for A1 is between v and v + : if v 1 is close to v , independently of the consumer's value v 2 for product A2. By Lemma 2(1), this profit decrease for firm A exceeds the profit increase from consumers that buy A1 after the price increase. A similar argument explains why p A2 = v + is less profitable than p A2 = v. Now consider JS and P A = 2v + 2 : a bundle price that is slightly above 2v . Then a consumer with values v 1 , v 2 does not buy the bundle if and only if v 1 + v 2 < 2v + 2 : if and only if v 1 , v 2 are both close to v . Thus the demand reduction that follows the price increase is much smaller under JS than under SS because under JS it comes only from consumers with both values close to v , whereas under SS it comes from each consumer with at least one value close to v . Then the profit increase from the consumers who buy the bundle after the increase in P A (this is about equal to the profit increase under SS from consumers buying the product(s) after the increases in p A1 , p A2 ) makes the price increase profitable under JSalthough it is not so under SS. 12

On Bundling and Entry Deterrence
The difference between SS and JS arises because under JS consumers are forced to buy the bundle or no product at all. This implies that a consumer does not buy the bundle after the increase in P A only if his values are both close to v . Conversely, SS allows each consumer to buy a single product. Hence, after the increases in p A1 , p A2 , a consumer with v 1 that is close to v and v 2 that is not too small-say high for brevity-chooses to buy only product A2, and a similar remark holds if v 2 is close to v and v 1 is high. Therefore, the price increase(s) lead to different allocations such that the demand reduction is greater under SS. For instance, a consumer with low v 1 and high v 2 buys only A2 under SS, but buys the bundle under JS because of the high v 2 .

Stage Two in 0 g
At stage two, firm A chooses between SS and JS. Such choice is determined by the profits A expects to earn under SS and under JS, and by whether firm B has entered or not. Lemma 1(3) shows that if firm B has entered, then the comparison between g A and Π g A depends on (as we illustrated after Lemma 1), whereas Lemma 2 shows that this is not the case if B has not entered. Using these results, we deduce A's choice at stage two.

Lemma 3 In game Γ g , at stage two (1) if firm B has entered at stage one, then A's best action is SS if
(2) if firm B has not entered, then A's best action is JS.

Stage One in 0 g
Here we study the entry decision of firm B, which relies on anticipating the equilibrium play at the successive stages, described by Lemmas 1, 3. When < g A , firm B knows that A will play SS in case of entry; since B 's entry cost is 2k, B compares

3 4 Competition Among Firm A and Firms B1, B2: Game 0 s
In game Γ s , the players are A and the specialist firms B1, B2. The timing is as follows: • Stage one: Firms B1 and B2 simultaneously and non-cooperatively decide whether to enter or not. • Stage two: After observing the entry decisions of B1 and B2, firm A chooses SS or JS. • Stage three: If no specialist has entered, then A acts as a monopolist. If B1 and/ or B2 has entered, then the active firms compete under SS or under JS, as determined by A.
In the event that B1, B2 have entered and A has chosen SS, competition occurs essentially as in game Γ g , except that firm B1 (firm B2 ) sets p B1 ( p B2 ). If instead A has chosen JS, then competition takes place between the bundle of A and the bundle of products B1&B2 ; firm A sets P A , firms B1, B2 choose P B1 , P B2 and P B1 + P B2 is the price of the bundle B1&B2.
In particular, under JS the so-called Cournot complement effect applies: If firm B1 reduces P B1 , then the demand for bundle B1&B2 increases since the goods can be seen as complements, as we remarked in Sect. 2. 13 However, firm B1 does not take into account that a reduction in P B1 increases the profit of B2 and chooses P B1 to maximize just its own profit. Conversely, an integrated firm B takes this externality into account and charges a bundle price that is lower than the total price charged by B1, B2. As a consequence, under JS firm A prefers to compete against B1, B2 rather than against B because B1, B2 are less aggressive. Under SS, there is no such effect because there is no link between the sale of B1 and the sale of B2.
As for Γ g , we apply backward induction to Γ s and begin with the study of terminal subgames.

Subgames Such That Both B1 and B2 Have Entered
When competition occurs under SS after both B1 and B2 have entered, the analysis of Sect. 3.1.1 still applies: There are no links between market 1 and market 2, hence when A faces the separate entities B1, B2, the equilibrium prices are still p * A , p * B in (5) and the equilibrium profit of B1 (B2) is half the equilibrium profit of B in Γ g . In order to simplify notation for the rest of the paper, we define ̂= g B ∕2 = (3t − ) 2 ∕(18t) ; therefore ̂ is the profit of both B1 and B2 under SS. Using ̂ , Proposition 1(1) can be stated as follows: When < g A , the unique SPNE of Γ g is such that firm B enters if and only if k <̂.

3
On Bundling and Entry Deterrence

When competition occurs under JS, (3) shows that a consumer who is located at (x 1 , x 2 ) buys A's bundle if and only if his average location x satisfies
Hence, x sm is the average location of the marginal consumer. The demand for the bundle of A is F (x sm ) , A's profit is P AF (x sm ) , and the profits for firms B1, B2 (gross of the entry costs) are P B1 [1 −F(x sm )] , P B2 [1 −F(x sm )] , respectively. We use s i ( Π s i ) to denote the equilibrium profit of firm i upon entry of B1, B2 under SS (under JS).

Lemma 4 (HJM) In game Γ s , suppose that firms B1 and B2 have entered. Then
(1) Under SS, the equilibrium prices and profits for firms A and B1, B2 are: (2) Under JS, the equilibrium prices and profits for firms A and B1, B2 are: For the analysis of Γ s , Lemma 4(3) is a key result, analogous to Lemma 1(3) for Γ g , as it determines the choice of firm A at stage two between SS and JS, upon entry of B1, B2. The main difference between Lemma 1(3) and Lemma 4(3) is that in Γ s it is more frequent that firm A's profit under JS is higher than under SS: The inequality Π Under JS, in Γ s firms B1, B2 charge a higher total price than firm B in Γ g ; this increases A's demand and profit: Moreover, B1, B2 are collectively better off in Γ s than is firm B in Γ g : The reason is that since P s B1 + P s B2 is higher than P g B , it is profitable for firm A to increase P A , as prices are strategic complements, and indeed P s A > P g A . This benefits firms B1, B2 as it increases the demand for bundle B1&B2 . Hence B1, B2 are hurt by their lack of coordination: for each given P A , their total profit is lower than the profit that an integrated firm B would earn. But B1, B2 benefit from the fact that A charges a higher P A than against an integrated firm B. We find that the latter effect dominates the former and implies (9). This explains why the inequality ̂< Π g B ∕2 holds for > g B ; but ̂< Π s B1 holds more frequently, for > s B12 . Therefore, for each firm competition under JS is more profitable in Γ s than in Γ g .

Subgames Such That Only One Specialist Firm Has Entered
Suppose that only one specialist firm has entered: say, B1. If A has chosen SS at stage two, then A faces competition in market 1 but is a monopolist in market 2. Then the equilibrium prices are described by Lemma 1(1) for market 1, and by Lemma 2(1) for market 2.
Matters are more complicated if firm A bundles because A's bundle competes with the single product B1 and each consumer chooses one of these alternatives. If the consumer buys product B1, then he does not consume any good 2 (and a social welfare loss occurs). Therefore, this game differs substantially from the pricing games considered above, as A's bundle includes product A2 but no competitor of A offers product B2.
Nevertheless, the purchase choice of each consumer is readily determined: If the consumer is located in (x 1 , x 2 ) , then he buys the bundle of A if and only if 2v The demand for product B1 is 1 − D A (P A , P B1 ) , and the profit functions are ) to denote the equilibrium profits under SS (under JS) when only B1 enters. 14 Lemma 5 In game Γ s , suppose that only firm B1 has entered. Then (1) Under SS, the equilibrium prices and profits for firms A and B1 are On Bundling and Entry Deterrence (2) Under JS, the equilibrium prices and profits for firms A and B1 are .
The results in Lemma 5(3) can be explained using the demand size effect and the demand elasticity effect as in Sect. 3.1.1: Moving from SS to JS with (P A , P B1 ) = (p s1 A1 + p s1 A2 , p s1 B1 ) makes firm A increase its market share in market 1, but A loses market share in market 2 because (1) under SS all consumers buy A2; (2) under JS a few consumers buy product B1 and no good 2 at all. Therefore the net effect on A's profit is ambiguous: The profit loss in market 2 may not be compensated by the profit increase in market 1. In fact, A's total profit increases if is large (or v is not too large), as then A's market share (and profit) loss in market 2 is small. The demand size effect reduces firm B1's profit since the sales of its single product decrease.
In addition, the reduced market share for B1 makes the demand for product B1 elastic, even though a small decrease in P B1 at (P A , P B1 ) = (p s1 A1 + p s1 A2 , p s1 B1 ) allows B1 to win over relatively few consumers (this principle applies also in Sect. 3.1.1 to firm B). Thus the demand elasticity effect induces firm B1 to reduce P B1 below p s1 B1 . This harms firm A and makes Π s1 A smaller than s1 A unless is larger than s1 A (v) in Lemma 5(3). Finally, we notice that for large-close to 3t-the demand for A's bundle is rigid (as in the games of competition under JS examined above), which induces A to increase P A above p s1 A1 + p s1 A2 . This benefits firm B1 and makes Π s1 B1 greater than s1 B1 .

Subgames Such That Neither B1 Nor B2 Has Entered
In this case A is a monopolist, as in Γ g when B does not enter. Hence Lemma 2 applies.

Stage Two in 0 s
At stage two, firm A chooses between SS and JS by comparing its profit under two alternatives, as in Γ g (see Sect. 3.2 and Lemma 3). However, now matters are more complicated than in Γ g because there are three possible cases for the specialists' entry decisions: both specialists enter; only one enters; no specialist enters. Lemmas 2, 4(3), 5(3) lead to the following results: (2) if only one specialist has entered, then A's best action depends on and v: for (3) if no specialist has entered, then A's best action is JS.

Stage One in 0 s
Here we examine the entry decisions of B1, B2 at stage one. In Γ g , the entry decision of firm B is straightforward given Lemmas 1, 3 because B is the sole entrant. But in Γ s there is strategic interdependence between B1 and B2: The profitability of entry for B1 may depend on whether B2 enters or not because that affects A's reaction at stage two (and also price competition at stage three). Therefore we examine an entry game between B1 and B2 in which the payoffs depend on the actions of B1, B2 and take into account the anticipated choice of A at stage two, which is described by Lemma 6. By Lemma 6(1), we distinguish below the case of < s A from the case of > s A . 15 For each specialist the gross profit from entering is ̂ , or Π s ), and since the entry cost is k we can conclude that no specialist enters if k > max{̂, Π s B1 }. Case of < s A In this case A chooses SS if at least one specialist has entered. Hence, the game at stage one between B1 and B2 has the following normal form, in which the action In means entry, the action Out means no entry: In this game the profit of B1 and of B2 is independent of the action of the other firm, hence there is no strategic interdependence between B1 and B2 in (10). This occurs because each specialist that enters competes with A under SS; and in the event that both enter there is no link between market 1 and market 2; hence for a specialist it does not matter whether the other specialist enters or not. Then for each firm In is strictly dominant if k <̂ ; in such case (In, In) is the unique Nash Equilibrium (NE (10)

B1�B2
In Out In̂− k,̂− k̂− k, 0 Out 0,̂− k 0, 0 15 By Lemma 6(2), we may also distinguish between < s1 A (v) and > s1 A (v) for the case in which only one specialist enters. But we show below that this is basically irrelevant from the point of view of specialists' entry.  (v) . Therefore we distinguish between these cases, starting with < s1 A (v) and recall that s1 1t . Then the game at stage one between B1 and B2 is described by (11) below.
For (11), we distinguish two subcases: The first one is such that Π s B1 <̂ : ) . This distinction is important because when Π s B1 <̂ , from (11) we see that a specialist firm that enters-say, B1-is better off if B2 does not enter: entry by B2 induces A to choose JS, which hurts B1 with respect to SS. Conversely, if Π s B1 >̂ , then an entrant B1 prefers that also B2 enters because B1 is better off if A chooses JS. These differences affect the structure of the NE of (11).
Subcase 1 When Π s B1 <̂ (i.e., < s B12 ) if k < Π s B1 <̂ , then In is strictly dominant in (11) for each firm, and (In, In) is the unique NE. If instead Π s B1 < k <̂ , then (11) is a hawk-dove game with two asymmetric NE, (Out, In) and (In, Out), and the following symmetric mixed strategy NE (in which each specialist's profit is zero): Since (11) is a symmetric game and (12) is its unique symmetric NE, we assume that B1, B2 play (12)

B1�B2
In Out ). If instead > s B12 , then Π s B1 >̂ , and JS actually favors entry as it increases each specialist's profit.

Comparison Between 0 g and 0 s
In this section we compare the outcomes of Γ g and Γ s in terms of the firms' profits. 17 For firms B and B1, B2 we compare the profit of B in Γ g with the sum of the profits of B1, B2 in Γ s . We first notice that Γ g and Γ s lead to the same outcome if < s A or k > max{̂, Π s B1 } : If < s A then A chooses SS whenever some entry occurs (Lemmas 3(1), 6 (1-2)), both in Γ g and in Γ s . Thus, in both games firm A is a monopolist if ̂< k , faces competition in both markets (under SS) if k <̂ , and the same outcome arises. When k > max{̂, Π s B1 } , in both Γ g and Γ s the high entry cost deters any entry, and A is a monopolist.
Therefore, in the following we consider > s A and k < max{̂, Π s B1 } , and recall that Lemmas 1,4 show that 17 In Greppi and Menicucci (2020) we compare social welfare and consumers' surplus in Γ g and in Γ s .

3
On Bundling and Entry Deterrence To this purpose we use Fig. 2 above, in the space , k , in which the thick curve is the graph of k = max{̂, Π s B1 } . Region X consists of the points such that ∈ ( s A , g A ) and k < max{̂, Π s B1 } . The dashed curve in Region X is the graph of k = Π s B1 ; by (13), it lies below the thick curve. The points ( , k) such that > g A and k < max{̂, Π s B1 } belong to Region Y or Z. The solid thin curve separating the two regions is the graph of k = Π g B ∕2 ; it lies below the thick curve since Π g B ∕2 < Π s B1 by (9).

Case of ˛∈ (˛s A ,˛g A ) (Region X in Fig. 2)
In Region X we first consider k smaller than Π s B1 : ( , k) below the dashed curve. Then in Γ s firms B1, B2 enter, A plays JS, and A (B1, B2) earns Π s A (earns Π s B1 − k ). In Γ g , firm B enters as k < Π s B1 <̂ , A plays SS, and A (B) earns g A (earns 2̂− 2k ). Hence, A prefers Γ s since competition under JS is softer in Γ s than in Γ g (see (8)), which makes JS more profitable for A than SS. Conversely, B prefers Γ g as JS in Γ s reduces the profit of B1, B2: Π s B1 <̂. The fact that JS in Γ s reduces the profit of B1, B2 is especially relevant if k is greater than Π s B1 but smaller than ̂- ( , k) in Region X is between the dashed and the thick curve-because then entry in Γ s is affected: For B1, B2 it is unprofitable to enter jointly (see (11)), and by Proposition 2b(1) each specialist enters with probability less than one, according to (12).
It follows that firm A is better off in Γ s than in Γ g in the whole Region X: A would play SS in Γ g , but in Γ s , (1) if k < Π s B1 , then B1, B2 enter and A prefers JS to SS; (2) if k > Π s B1 , then entry of B1, B2 is deterred with positive probability, hence A becomes a monopolist in one market, or both, with positive probability, which is more profitable than facing competition.
The root of the result is that in Γ s competition under JS is more profitable for A than under SS if entry occurs, and it reduces the profits of B1, B2. Hence, A's ability to bundle creates in Γ s an entry barrier against B1, B2 for some values of k. This is not the case for Γ g because bundling reduces B's profit, but reduces also A's profit given B's entry when < g A ; hence B profitably enters as long as k <̂ (Proposition 1(1)). Here the fact that competition under JS is tougher in Γ g than in Γ s is useful for B as it induces A to choose SS.
Case of > g A (Regions Y , Z in Fig. 2) For ( , k) in Regions Y or Z, in Γ g firm A chooses JS after B's entry; hence B enters if and only if k < Π g B ∕2 , that is if ( , k) is in Region Y. In Γ s , for ( , k) in Region Y, firms B1, B2 enter since k < Π s B1 holds. Therefore, the comparison between Γ g and Γ s in Region Y reduces to the comparison under JS of competition between A and B or among A, B1, B2. Then (8), (9) reveal that A and B both prefer Γ s .
Finally, for ( , k) in Region Z, in Γ g firm B stays out because its gross profit from entry Π g B is smaller than its entry cost 2k; thus Γ g leads to A's monopoly. Conversely, in Γ s both B1 and B2 enter with positive probability (actually, they enter with certainty if k < Π s B1 ), thus A is less often a monopolist in Γ s than in Γ g ; this makes A prefer Γ g and B prefer Γ s .
The difference with respect to Region X is that when > g A , JS is profitable for A in Γ g ; thus B enters if ( , k) is in Region Y, does not enter if ( , k) is in Region Z. But in Γ s both B1, B2 enter with positive probability if ( , k) is in Region Z. Therefore in this region A's ability to bundle is a more effective foreclosure instrument in Γ g than in Γ s . 18 Unlike in Region X, here the fact that competition under JS is tougher in Γ g than in Γ s is a disadvantage for B as it makes its entry more difficult.
The next proposition summarizes the three cases we have considered: Proposition 3 Suppose that > s A and k < max{̂, Π s B1 }, otherwise Γ g and Γ s lead to the same outcome. Then: (1) A prefers Γ s , B prefers Γ g if ∈ ( s A ,

Conclusions
This paper examines the effects of the ability to bundle of a firm that offers two higher-quality products and faces an entry threat from an integrated rival or from separate rivals. One significant extension, left to future research, would allow for more than two products, product-specific quality differences, and the possibility of mergers among a few, but not all, specialist firms. 18 This claim applies if < s B12 , because for > s B12 , JS actually favors entry in Γ s (and in Γ g if > g B ) with respect to SS, but entry remains more difficult in Γ g than in Γ s .