On Bundling and Entry Deterrence

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 \(v+\alpha -tx_{j}-p_{Aj}\ge v-t(1-x_{j})-p_{Bj}\), which reduces toin which \(x_{j}^{m}\) 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]; henceTherefore the demand for product Aj is \(F(x_{j}^{m})\); \(p_{Aj}F(x_{j}^{m})\) is A’s profit in market j; and \(p_{Bj}(1-F(x_{j}^{m}))\) is B’s profit in market j gross of the entry cost.

If competition occurs under JS, a consumer located at \((x_{1},x_{2})\) buys A’s bundle if and only if \(2v+2\alpha -t(x_{1}+x_{2})-P_{A}\ge 2v-t(1-x_{1}+1-x_{2})-P_{B}\), which is equivalent toin which: \({\bar{x}}=(x_{1}+x_{2})/2\) is the consumer’s average location; \( p_{A}=P_{A}/2\) (\(p_{B}=P_{B}/2\)) is the average price of the products in A’s bundle (in B’s bundle); and \({\bar{x}}^{gm}\) 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. \({\bar{F}}\) and the density \({\bar{f}}\) of the average location areHence, the demand for the bundle of A is \({\bar{F}}({\bar{x}}^{gm})\); \(P_{A} {\bar{F}}({\bar{x}}^{gm})\) is the profit of A; and \(P_{B}\left( 1-{\bar{F}}({\bar{x}}^{gm})\right) \) is the profit of B gross of the entry cost. In the following, we denote with \(\pi _{i}^{g}\) (with \(\Pi _{i}^{g}\)) the equilibrium profit—upon entry of B—of firm i in \(\Gamma ^{g}\) under SS (under JS), for \(i=A,B\):

With respect to the analysis of game \(\Gamma ^{g}\), 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 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 \(\alpha \) 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 \(\alpha \). We explain why by considering market 1: The demand for product B1 is \(D_{B1}(p_{B1})=1-F((\alpha +t+p_{B1}-p_{A1})/2t)\) , with elasticity \(p_{B1}D_{B1}^{\prime }(p_{B1})/D_{B1}(p_{B1})\). This must be \(-1\) at \(p_{B1}=p_{B}^{*}\) in order for B’s profit to be maximized; and if \(p_{A}^{*}\) were equal to \(p_{B}^{*}+\alpha \), then \( D_{B1}(p_{B}^{*})=1/2\), \(D_{B1}^{\prime }(p_{B}^{*})=-1/(2t)\); thus \( p_{B}^{*}D_{B1}^{\prime }(p_{B}^{*})/D_{B1}(p_{B}^{*})=-1\) implies \(p_{B}^{*}=t\). For product A1, the elasticity is \( p_{A1}D_{A1}^{\prime }(p_{A1})/D_{A1}(p_{A1})\) and \(p_{A}^{*}=p_{B}^{*}+\alpha \), \(p_{B}^{*}=t\) imply \(p_{A}^{*}D_{A1}^{\prime }(p_{A}^{*})/D_{A1}(p_{A}^{*})=-1-\alpha /t<-1\): the demand for A1 is elastic, and A gains by reducing \(p_{A1}\) below \( p_{B}^{*}+\alpha \), which is consistent with (5).

Using (1) and (5) we see that in each market the marginal consumer is located at \(x^{*}=1/2+\alpha /(6t)\), and \(x^{*}\in (1/2,1)\) since \(\alpha \in (0,3t)\). A consumer who is located at \(x_{j}\) for good j buys Aj if \(x_{j}<x^{*} \), or buys Bj if \(x_{j}>x^{*}\). 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 \(\alpha <3t\), each firm has a positive profit under SS. If instead \(\alpha \ge 3t\), then in each market j the equilibrium prices are \(p_{Aj}^{*}=\alpha -t\), \(p_{Bj}^{*}=0\); each consumer buys product Aj, and B’s profit is zero. The inequality \( \alpha \ge 3t\) is necessary for this equilibrium as it guarantees that A has no incentive to increase \(p_{Aj}\) above \(\alpha -t\): When \(\alpha \ge 3t \), \(p_{Aj}^{*}\) 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 \(P_{A}^{g}\ne 2p_{A}^{*}\) and \(P_{B}^{g}\ne 2p_{B}^{*}\) : 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 \({\bar{F}}\) 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 \({\bar{F}}\), Fig. 1 represents \(f,{\bar{f}}\) and \(F,{\bar{F}}\) from (2) and (4) (please see next page).

Both f and \({\bar{f}}\) are symmetric around 1/2 (the mean), but \({\bar{f}}\) is more peaked than f around 1/2, that is \({\bar{f}}\) 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 A (of B) is \({\bar{F}}( {\bar{x}}^{gm})\) (is \(1-{\bar{F}}({\bar{x}}^{gm})\)); and given \( (P_{A},P_{B})=(2p_{A}^{*},2p_{B}^{*})\) we have that \({\bar{x}}^{gm}\) 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 \({\bar{F}}(x^{*})\) for A’s bundle, and \(x^{*}\in (1/2,1)\) plus (7) imply \({\bar{F}} (x^{*})>F(x^{*})\). 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 \([0,x^{*}]\) relative to good j; (2) under JS with \( (P_{A},P_{B})=(2p_{A}^{*},2p_{B}^{*})\), A’s bundle is purchased by consumers with average location in \([0,x^{*}]\); and (3) A’s quality advantage implies \(x^{*}>1/2\) and there are more consumers in \( [0,x^{*}]\) under the c.d.f. \({\bar{F}}\) than under F: \({\bar{F}}(x^{*})>F(x^{*})\), because \({\bar{f}}\) 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 \(\bar{F }(x^{*})-F(x^{*})\) 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 \({\bar{F}}({\bar{x}}^{gm})\) and its elasticity with respect to \(P_{A}\) at \((P_{A},P_{B})=(2p_{A}^{*},2p_{B}^{*})\) is \(-(p_{A}^{*}/2t)({\bar{f}}(x^{*})/{\bar{F}} (x^{*}))\), whereas under SS the elasticity of \(F(x_{j}^{m})\) (the demand for product Aj) with respect to \(p_{Aj}\) at \( (p_{Aj},p_{Bj})=(p_{A}^{*},p_{B}^{*})\) is \(-(p_{A}^{*}/2t)(f(x^{*})/F(x^{*}))\), equal to \(-1\) since \(p_{A}^{*}\) is a best reply for A to \(p_{B}=p_{B}^{*}\). In the following we identify the demand elasticity effect for firm A by comparing \(-(p_{A}^{*}/2t)( {\bar{f}}(x^{*})/{\bar{F}}(x^{*}))\) with \(-(p_{A}^{*}/2t)(f(x^{*})/F(x^{*}))\).

When \(\alpha \) is about zero, we have that \(x^{*}\) is about 1/2; thus (see (2), (4) or Fig. 1) \({\bar{F}}(x^{*})\) and \( F(x^{*})\) are both close to 1/2 but \({\bar{f}}(x^{*})\) is about twice as large as \(f(x^{*})\). Hence \(-(p_{A}^{*}/2t)({\bar{f}}(x^{*})/{\bar{F}}(x^{*}))<-(p_{A}^{*}/2t)(f(x^{*})/F(x^{*}))\): 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 \({\bar{f}}\) puts more weight around 1/2 than does f. Conversely, when \(\alpha \) is large, close to 3t, \(x^{*}\) is about 1, and (see (2), (4) or Fig. 1) \({\bar{F}}(x^{*})\), \(F(x^{*})\) are both about 1 but \({\bar{f}}(x^{*})\) is close to 0, smaller than \(f(x^{*})=1\). Thus \(-(p_{A}^{*}/2t)({\bar{f}}(x^{*})/{\bar{F}}(x^{*}))>-(p_{A}^{*}/2t)(f(x^{*})/F(x^{*}))\), that is under JS firm A’s demand is less elastic because fewer consumers are located near the marginal consumer, as \({\bar{f}}\) puts less weight near 1 than does f.

For the demand elasticity effect for firm B we notice that \(1-{\bar{F}}({\bar{x}}^{gm})\) is the demand for B’s bundle and its elasticity with respect to \( P_{B}\) at \((P_{A},P_{B})=(2p_{A}^{*},2p_{B}^{*})\) is \(-(p_{B}^{*}/2t)({\bar{f}}(x^{*})/(1-{\bar{F}}(x^{*})))\). From (2), (4) we see that for each \(x^{*}\in (1/2,1)\), 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 that \(1-{\bar{F}}(x^{*})<1-F(x^{*})\), which makes \(-(p_{B}^{*}/2t)({\bar{f}}(x^{*})/(1-{\bar{F}}(x^{*})))\) smaller than \(-(p_{B}^{*}/2t)(f(x^{*})/(1-F(x^{*})))\) even when \( {\bar{f}}(x^{*})<f(x^{*})\). Thus the demand elasticity effect makes B willing to reduce \(P_{B}\) below \(2p_{B}^{*}\).

Summarizing: For \(\alpha \) 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 \(\alpha \) is about zero, and \({\bar{f}}(x^{*})>f(x^{*})\)), which provides incentives for aggressive pricing.

Conversely, for a large \(\alpha \) 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 \({\bar{f}} (x^{*})\simeq 0<f(x^{*})=1\)), which leads to less aggressive pricing.

Finally, for intermediate \(\alpha \) 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: \(\Pi _{A}^{g}>\pi _{A}^{g}\) and \(\Pi _{B}^{g}<\pi _{B}^{g}\) for \( \alpha \in (\alpha _{A}^{g},\alpha _{B}^{g})\).⁹

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 \(v+\alpha -tx_{j}-p_{Aj}\ge 0\). Hence, the demand for product Aj is \(F((v+\alpha -p_{Aj})/t)\). Since \(v\ge 3t\), we find that the profit maximizing \(p_{Aj}\) is \(v+\alpha -t\), the lowest possible consumer’s valuation for Aj; at this price, each consumer buys Aj.

Now suppose that A has chosen JS at stage two. A consumer who is located at \((x_{1},x_{2})\) buys the bundle of A if and only if \(2v+2\alpha -t(x_{1}+x_{2})-P_{A}\ge 0\): if \((2v+2\alpha -P_{A})/(2t)\ge {\bar{x}}\), in which \({\bar{x}}=(x_{1}+x_{2})/2\). Hence, the demand for A’s bundle is \({\bar{F}}((2v+2\alpha -P_{A})/(2t))\).

About Lemma 2(2), notice that \( P_{A}=2(v+\alpha -t)\) under JS is equivalent to \(p_{A1}=p_{A2}=v+\alpha -t\) under SS, as it induces each consumer to buy A’s bundle and makes A’s profit equal to \(2(v+\alpha -t)\), which is also A’s maximal profit under SS. However, A can do better by slightly increasing \(P_{A}\) above \( 2(v+\alpha -t)\), as we explain below.¹⁰

Under monopoly by A, we can view the consumers’ valuations for product A1 as uniformly distributed over the interval \([{\underline{v}},{\bar{v}}]\), with \( {\underline{v}}=v+\alpha -t\) and \({\bar{v}}=v+\alpha \),¹¹ and the consumers’ valuations for product A2 as uniformly distributed over the same interval \([{\underline{v}},{\bar{v}}]\). Under SS, consider \(p_{A1}= {\underline{v}}+\varepsilon \) with \(\varepsilon >0\) and small. This \(p_{A1}\) is less profitable than \(p_{A1}={\underline{v}}\) because it induces a consumer not to buy product A1 if his value \(v_{1}\) for A1 is between \({\underline{v}}\) and \({\underline{v}}+\varepsilon \): if \(v_{1}\) is close to \({\underline{v}}\), 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 \(p_{A2}={\underline{v}}+\varepsilon \) is less profitable than \( p_{A2}={\underline{v}}\).

Now consider JS and \(P_{A}=2{\underline{v}}+2\varepsilon \): a bundle price that is slightly above \(2{\underline{v}}\). Then a consumer with values \( v_{1},v_{2}\) does not buy the bundle if and only if \(v_{1}+v_{2}<2{\underline{v}}+2\varepsilon \): if and only if \(v_{1},v_{2}\) are both close to \( {\underline{v}}\). 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 \({\underline{v}}\), whereas under SS it comes from each consumer with at least one value close to \({\underline{v}}\). 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 JS—although it is not so under SS.¹²

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 \({\underline{v}}\). 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 \({\underline{v}}\) 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 \({\underline{v}}\) 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 \(v_{2}\).

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 \(\Gamma ^{g}\) . In order to simplify notation for the rest of the paper, we define \({\hat{\pi }}=\pi _{B}^{g}/2=(3t-\alpha )^{2}/(18t)\); therefore \({\hat{\pi }}\) is the profit of both B1 and B2 under SS. Using \({\hat{\pi }}\), Proposition 1(1) can be stated as follows: When \(\alpha <\alpha _{A}^{g}\), the unique SPNE of \(\Gamma ^{g}\) is such that firm B enters if and only if \(k<{\hat{\pi }}\).

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 \({\bar{x}}\) satisfiesHence, \({\bar{x}}^{sm}\) is the average location of the marginal consumer. The demand for the bundle of A is \({\bar{F}}({\bar{x}}^{sm})\), A’s profit is \( P_{A}{\bar{F}}({\bar{x}}^{sm})\), and the profits for firms B1, B2 (gross of the entry costs) are \(P_{B1}[1-{\bar{F}}({\bar{x}}^{sm})]\), \(P_{B2}[1-{\bar{F}}( {\bar{x}}^{sm})]\), respectively. We use \(\pi _{i}^{s}\) (\(\Pi _{i}^{s}\)) to denote the equilibrium profit of firm i upon entry of B1, B2 under SS (under JS).

For the analysis of \(\Gamma ^{s}\), Lemma 4(3) is a key result, analogous to Lemma 1(3) for \( \Gamma ^{g}\), 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 \(\Gamma ^{s}\) it is more frequent that firm A’s profit under JS is higher than under SS: The inequality \(\Pi _{A}^{g}>\pi _{A}^{g}\) holds for \(\alpha >\alpha _{A}^{g}\), whereas \(\Pi _{A}^{s}>\pi _{A}^{s}\) holds for \(\alpha >\alpha _{A}^{s}\), and \(\alpha _{A}^{s}<\alpha _{A}^{g}\). Since \( \Gamma ^{g}\) and \(\Gamma ^{s}\) are equivalent under SS, we compare the two games under JS. In this case the Cournot complement effect applies and determines a difference between \(\Gamma ^{g}\) and \(\Gamma ^{s}\).

Under JS, in \(\Gamma ^{s}\) firms B1, B2 charge a higher total price than firm B in \(\Gamma ^{g}\); this increases A’s demand and profit:Moreover, B1, B2 are collectively better off in \(\Gamma ^{s}\) than is firm B in \(\Gamma ^{g}\): \(\Pi _{B1}^{s}+\Pi _{B2}^{s}>\Pi _{B}^{g}\), or equivalently, since \(\Pi _{B1}^{s}=\Pi _{B2}^{s}\),The reason is that since \(P_{B1}^{s}+P_{B2}^{s}\) is higher than \(P_{B}^{g}\), it is profitable for firm A to increase \(P_{A}\), as prices are strategic complements, and indeed \(P_{A}^{s}>P_{A}^{g}\). 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 \({\hat{\pi }} <\Pi _{B}^{g}/2\) holds for \(\alpha >\alpha _{B}^{g}\); but \({\hat{\pi }}<\Pi _{B1}^{s}\) holds more frequently, for \(\alpha >\alpha _{B12}^{s}\). Therefore, for each firm competition under JS is more profitable in \(\Gamma ^{s}\) than in \(\Gamma ^{g}\).

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 \(2v+2\alpha -tx_{1}-tx_{2}-P_{A}\ge v-t(1-x_{1})-P_{B1}\), which reduces to \(x_{2}\le -2x_{1}+(v+2\alpha +t+P_{B1}-P_{A})/t\). Therefore the demand for A’s bundle is equal to 1 if \( P_{A}-P_{B1}<v+2\alpha -2t\); is 0 if \(P_{A}-P_{B1}>v+2\alpha +t\); and if \( v+2\alpha -2t\le P_{A}-P_{B1}\le v+2\alpha +t\) it isThe demand for product B1 is \(1-D_{A}(P_{A},P_{B1})\), and the profit functions are \(P_{A}D_{A}(P_{A},P_{B1})\), \(P_{B1}[1-D_{A}(P_{A},P_{B1})]\) for A and for B1, respectively. We use \(\pi _{A}^{s1},\pi _{B1}^{s1}\) (\( \Pi _{A}^{s1},\Pi _{B1}^{s1}\)) to denote the equilibrium profits under SS (under JS) when only B1 enters.¹⁴

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 \( (P_{A},P_{B1})=(p_{A1}^{s1}+p_{A2}^{s1},p_{B1}^{s1})\) 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 \( \alpha \) 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 \( (P_{A},P_{B1})=(p_{A1}^{s1}+p_{A2}^{s1},p_{B1}^{s1})\) 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 \(p_{B1}^{s1}\). This harms firm A and makes \(\Pi _{A}^{s1}\) smaller than \(\pi _{A}^{s1}\) unless \(\alpha \) is larger than \(\alpha _{A}^{s1}(v)\) in Lemma 5(3). Finally, we notice that for \(\alpha \) 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 \( p_{A1}^{s1}+p_{A2}^{s1}\). This benefits firm B1 and makes \(\Pi _{B1}^{s1}\) greater than \(\pi _{B1}^{s1}\).

In this case A is a monopolist, as in \(\Gamma ^{g}\) when B does not enter. Hence Lemma 2 applies.