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Incumbent positioning as a determinant of strategic response to entry

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Abstract

Existing research has identified different firm characteristics that determine an incumbent’s reaction toward market entrants. This study widens the perspective from a firm’s absolute attributes to its relative positioning within its competitive environment. The hypotheses for the empirical analysis are derived from game theoretic models with either vertical or horizontal product differentiation. In each of the two theoretical models two incumbents initially choose very different quality and variety levels, respectively, and move closer to the center when faced by entry of a competitor. From the vertical differentiation model we predict a stronger price reduction by high-quality firms. In our empirical analysis we are using a novel dataset of the high-quality restaurant market in Germany to test the theoretical findings. While the hypotheses with respect to adjustments of prices and varieties are supported, we do not find empirical evidence for the predicted quality adjustment.

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Notes

  1. In the high-quality restaurant market, reputation commits restaurants to their attribute choices. If a restaurant cheats, it will suffer a loss in reputation. This argument applies to the incumbents as well as the entrant (Snyder and Cotter 1998). A similar result could be obtained by expanding the models to include positioning-related adjustment costs.

  2. Because quality \(q_B\) is restricted to \(q_B\le 1\), the three-firm case requires \( \frac{13}{36}<\alpha \). See Appendix B for details.

  3. See Lancaster (1990) in regard to modeling product variety.

  4. Our assumption of quadratic disutilty is also crucial for the three-firm case because with linear disutility, an equilibrium in pure strategies does not exist (Chamberlin 1933; Lerner and Singer 1937; Economides 1993), but a Nash equilibrium in mixed strategies (Shaked 1982).

  5. As above, offering a medium variety level is indeed optimal for the entrant.

  6. Considering this yields \(p_{A}^{*}-p_{A}^{**}=\frac{917}{2160} <p_{B}^{*}-p_{B}^{**}= \frac{1605}{2160}\) for \(\alpha =\frac{5}{9}\).

  7. For \(\alpha =\frac{5}{9}\) we obtain \(q_{A}^{*}-q_{A}^{**}=-\frac{5}{20} <q_{B}^{*}-q_{B}^{**}=\frac{7}{20}\).

  8. Even though ratings might differ between Michelin and Schlemmer critics, it should not be a problem to combine both data sources since we consider changes in ratings rather than absolute values.

  9. This is a realistic case because the variable relative variety ranges between 0.24 and 1.61.

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Acknowledgments

We would like to express our thanks to the editor, Erik E. Lehmann, and two anonymous referees who helped to improve the manuscript tremendously. Further, we want to thank Markus Reisinger for invaluable comments and suggestions as well as Christoph Starke and discussants at the 14th Annual Meeting of the German Economic Association of Business Administration in Magdeburg, Germany.

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Correspondence to Peter-J. Jost.

Appendices

Appendix 1: Quality and price: two-firm case

Solving the game using backward induction implies maximization of profits given by Eq. (3) with respect to prices. The first-order conditions lead to the best price responses

$$\begin{aligned} p_{A}^{*}(p_{B})&= \frac{1}{2}(p_{B}+\alpha q_{A}^{2}) \\ p_{B}^{*}(p_{A})&= \frac{1}{2}\left( p_{A}+\alpha q_{B}^{2}+q_{B}-q_{A}\right) \end{aligned}$$

and equilibrium prices, given the quality levels \(\left( q_{A},q_{B}\right) \) are

$$\begin{aligned} p_{A}^{*}(q_{A},q_{B})&= \frac{1}{3}\left( \alpha \left( 2q_{A}^{2}+q_{B}^{2}\right) +q_{B}-q_{A}\right) \\ p_{B}^{*}(q_{A},q_{B})&= \frac{1}{3}\left( \alpha \left( q_{A}^{2}+2q_{B}^{2}\right) +2\left( q_{B}-q_{A}\right) \right) . \end{aligned}$$

Substituting these expressions into the firms’ profits gives

$$\begin{aligned} \pi _{A}&= \frac{1}{9}\left( q_{B}-q_{A}\right) (1+\alpha (q_A +q_B ))^{2} \\ \pi _{B}&= \frac{1}{9}\left( q_{B}-q_{A}\right) (2-\alpha (q_A +q_B ))^{2}. \end{aligned}$$

At the first stage of the game, firms maximize these profits with respect to quality. This yields first order conditions for an interior solution

$$\begin{aligned} \frac{\partial \pi _A}{\partial q_A}&= \frac{1}{9} \left( \alpha (q_{B}-3 q_{A})-1) (1+\alpha ( q_{A}+q_{B})\right) =0\\ \frac{\partial \pi _B}{\partial q_B}&= \frac{1}{9} \left( 4+\alpha ^2\left( 3 q_{B}^2-q_{A}^2\right) +2 \alpha q_{B} (\alpha q_{A} -4)\right) =0 \end{aligned}$$

and best responses

$$\begin{aligned} q_A(q_B)&= \left\{ \begin{array}{lll} \frac{-1 - q_B \alpha }{\alpha }\\ \frac{-1 + q_B \alpha }{3\alpha } \end{array}\right. \\ q_B(q_A)&= \left\{ \begin{array}{lll} \frac{2-q_A \alpha }{\alpha }\\ \frac{2+q_A \alpha }{3 \alpha }. \end{array}\right. \end{aligned}$$

Because qualities are restricted to a non-negative number on the quality scale \(0\le q_A<1\), we can rule out \(q_A(q_B) = \frac{-1 - q_B \alpha }{\alpha }\). Solving the best response equation system for qualities \(q_A\) and \(q_B\) yields two solutions

$$\begin{aligned} \left( q_{A},q_{B}\right)&= \left\{ \begin{array}{lll} \left( -\frac{1}{8 \alpha },\frac{5}{8 \alpha }\right) \\ \left( \frac{1}{4 \alpha }, \frac{7}{4 \alpha }\right) . \end{array}\right. \end{aligned}$$

Again, because \(q_A\ge 0\), we can rule out solution 1. The second solution constitutes a minimum for firm \(B\) as \(\frac{\partial ^2 \pi _B}{\partial q_B^2} >0\). Hence, there is no interior equilibrium.

Because our quality scale is given by \([0,1]\), there might be solutions where only one or both qualities are chosen at an end of the scale:

  • Case 1: \(q_A>0\) and \(q_B=1\)

  • Case 2: \(q_A=0\) and \(q_B<1\)

  • Case 3: \(q_A=0\) and \(q_B=1\)

Next, we analyse whether these corner solutions constitute equilibria.

1.1 Case 1: \( q_A>0\) and \( q_B=1\)

If firm B locates at the higher end of the quality scale, \(A\)’s best response simplifies to

$$\begin{aligned} q_A(q_B=1)&= \left\{ \begin{array}{lll} \frac{-1-\alpha }{\alpha } &{}\hbox {with} &{}\frac{\partial ^2 \pi _A}{\partial q_A^2} <0\\ \frac{-1+\alpha }{3 \alpha } \ \ \ &{}\hbox {with} &{} \frac{\partial ^2 \pi _A}{\partial q_A^2} <0. \end{array}\right. \end{aligned}$$

As above, we can rule out the first solution because qualities have to be non-negative. For \(\alpha \le 1\), the same applies to the second solution. If, however, \(\alpha >1\), \(A\)’s best response is \( q_A(q_B= 1)=\frac{-1+\alpha }{ 3\alpha }\). Given this, \(B\)’s best response is

$$\begin{aligned} q_B\left( q_A=\frac{-1+\alpha }{3 \alpha }\right)&= \left\{ \begin{array}{lllll} \frac{7-\alpha }{3\alpha }\ &{}\hbox {with} &{} \frac{\partial ^2 \pi _B}{\partial q_B^2} <0 \ \text {for}\ \alpha >4\\ \frac{5+\alpha }{9\alpha } &{}\hbox {with} &{}\frac{\partial ^2 \pi _B}{\partial q_B^2} <0 \ \text {for}\ 1<\alpha <4, \end{array}\right. \end{aligned}$$

that is, for \(\alpha >4\), \(q_B(q_A)=\frac{7-\alpha }{3\alpha }\) denotes \(B\)’s best response, and for \(0<\alpha <4\) it is \(q_B(q_A)=\frac{5+\alpha }{9\alpha }\). In neither of the two preceding cases, \(q_B=1\) is a solution because \(\frac{7-\alpha }{3\alpha }\ne 1\) and \(\frac{5+\alpha }{9\alpha } \ne 1\). As a consequence, \(B\) has an incentive to deviate from \(q_B=1\).

Thus, there is no equilibrium with \(q_A>0\) and \(q_B=1\).

1.2 Case 2: \( q_A=0\) and \( q_B<1\)

If firm A locates at the lower end of the quality scale, \(B\)’s best response is

$$\begin{aligned} q_B(q_A=0)&= \left\{ \begin{array}{lll} \frac{2}{3 \alpha } \ \ \ &{}\hbox {with} &{} \frac{\partial ^2 \pi _B}{\partial q_B^2} <0\\ \frac{2}{\alpha } &{}\hbox {with} &{}\frac{\partial ^2 \pi _B}{\partial q_B^2}>0. \end{array}\right. \end{aligned}$$
(7)

The second solution can be ruled out because it constitutes a minimum. As \(B\)’s quality is restricted to \(q_B\le 1\), the first solution is \(B\)’s best response for \(\alpha >\frac{2}{3}\). Is there an incentive for \(B\) to deviate from the underlying quality choice to the corner solution \(q_B=1\)? For \(\frac{2}{3}<\alpha <\frac{8}{3}\), there is no incentive to deviate from the proposed equilibrium because \(\pi _B(q_B=\frac{2}{3\alpha })>\pi _B(q_B=1)\).

Furthermore, firm \(A\)’s best response to \(B\)’s choice is \( q_A(q_B= \frac{2}{3 \alpha })=0\) as \(\frac{\partial \pi _A}{\partial q_A}<0\) for \(\frac{2}{3}<\alpha <\frac{8}{3}\).

Hence, for \(\frac{2}{3}<\alpha <\frac{8}{3}\), \(\left( q_{A}^{*},q_{B}^{*}\right) =(0,\frac{2}{3 \alpha })\) constitutes an equilibrium.

1.3 Case 3: \(\ q_A=0\) and \(\ q_B=1\)

We know from case 1, that \(A\) chooses \( q_A(q_B= 1)=\frac{-1+\alpha }{ 3\alpha }>0\) if \(\alpha >1\). Therefore, we restrict case 3, where we look for a corner solution, implying \(q_A=0\), to \(\alpha \le 1\).

In case 3, firm \(A\)’s optimal quality choice is to position itself at the lower end of the scale \(q_A(q_B=1)=0\) because \(\frac{\partial \pi _A}{\partial q_A}<0\).

From case 2, we know that \(B\) chooses best response \( q_B(q_A= 0)=\frac{2}{3 \alpha }\) if \(\alpha >\frac{2}{3}\) because this generates higher profits than \(q_B=1\). Based on this finding, we further restrict our attention to \(\alpha <\frac{2}{3}\). Here, \(B\) chooses \(q_B(q_A=0)=1\) because \(\frac{\partial \pi _B}{\partial q_B}>0\).

Hence, for \(\alpha <\frac{2}{3}\), \(\left( q_{A}^{*},q_{B}^{*}\right) =(0,1)\) constitutes an equilibrium. \(\square \)

Appendix 2: Quality and price: three-firm case

Using expressions \(\hat{t}_{AE}\) and \(\hat{t}_{EB}\) for the indifferent consumers, firms’ profits read as

$$\begin{aligned} \pi _{A}&= \left( \frac{p_{E}-p_{A}}{q_{E}-q_{A}}\right) \left( p_{A}-\alpha q_{A}^{2}\right) \\ \pi _{E}&= \left( \frac{p_{B}-p_{E}}{q_{B}-q_{E}}-\frac{p_{E}-p_{A}}{ q_{E}-q_{A}}\right) \left( p_{E}-\alpha q_{E}^{2}\right) \\ \pi _{B}&= \left( 1-\frac{p_{B}-p_{E}}{q_{B}-q_{E}}\right) \left( p_{B}-\alpha q_{B}^{2}\right) . \end{aligned}$$

Solving the game using backward induction implies maximizing profits with respect to prices. The best responses are

$$\begin{aligned} p_{A}^{**}(p_{E},p_{B})&= \frac{1}{2}(p_{E}+\alpha q_{A}^{2}) \\ p_{E}^{**}(p_{A},p_{B})&= \frac{1}{2\left( q_{B}-q_{A}\right) } (p_{B}(q_{E}-q_{A})+p_{A}(q_{B}-q_{E})+\alpha q_{E}^{2}(q_{B}-q_{A})) \\ p_{B}^{**}(p_{A},p_{E})&= \frac{1}{2}\left( p_{E}+\alpha q_{B}^{2}+q_{B}-q_{E}\right) . \end{aligned}$$

This system leads to the following equilibrium prices for given qualities \( \left( q_{A},q_{E},q_{B}\right) \)

$$\begin{aligned} p_{A}^{**}\left( q_{A},q_{E},q_{B}\right)&= \frac{1}{6\left( q_{B}-q_{A}\right) }\left( \left( q_{B}-q_{E}\right) \left( q_{E}-q_{A}\right) \right. \\&\quad \left. +\,\alpha \left( q_{B}-q_{A}\right) \left( 3q_{E}^{2}+3q_{A}^{2}+\left( q_{B}-q_{E}\right) \left( q_{E}-q_{A}\right) \right) \right) \\ p_{E}^{**}\left( q_{A},q_{E},q_{B}\right)&= \frac{1}{3\left( q_{B}-q_{A}\right) }\left( \left( \left( q_{B}-q_{E}\right) \left( q_{E}-q_{A}\right) \right) \right. \\&\quad \left. +\, \alpha \left( q_{B}-q_{A}\right) \left( 3q_{E}^{2}+\left( q_{B}-q_{E}\right) \left( q_{E}-q_{A}\right) \right) \right) \\ p_{B}^{**}\left( q_{A},q_{E},q_{B}\right)&= \frac{1}{6\left( q_{B}-q_{A}\right) }\left( \left( q_{B}-q_{E}\right) \left( 3q_{B}+q_{E}-4q_{A}\right) \right. \\&\quad \left. + \, \alpha \left( q_{B}-q_{A}\right) \left( 3q_{B}^{2}+3q_{E}^{2}+\left( q_{B}-q_{E}\right) \left( q_{E}-q_{A}\right) \right) \right) , \end{aligned}$$

with the second-order conditions being satisfied. Substituting these prices in firms’ profits gives

$$\begin{aligned} \pi _{A}\left( q_{A},q_{E},q_{B}\right)&= \frac{(q_{E}-q_{A})}{ 36(q_{A}-q_{B})^{2}}\left( q_{B}+\alpha \left( q_{B}-q_{A}\right) \left( 3q_{A}+q_{B}\right) -q_{E}\left( 1-2\alpha \left( q_{B}-q_{A}\right) \right) ^{2}\right) \\ \pi _{E}\left( q_{A},q_{E},q_{B}\right)&= \frac{(q_{E}-q_{A})(q_{E}-q_{B})}{ 9(q_{A}-q_{B})}\left( 1-\alpha (q_{B}-q_{A})\right) ^{2} \\ \pi _{B}\left( q_{A},q_{E},q_{B}\right)&= \frac{(q_{B}-q_{E})}{ 36(q_{A}-q_{B})^{2}}\left( 3q_{B}+q_{E}-\alpha \left( 3q_{B}^{2}+2q_{B}q_{E}-q_{A}^{2}\right) +2q_{A}\left( 2-\alpha \left( q_{B}-q_{E}\right) \right) ^{2}\right) . \end{aligned}$$

Maximizing the entrant’s profits with respect to \(q_{E}\) for given qualities \(\left( q_{A},q_{B}\right) \) of the incumbent firms then gives the first-order condition

$$\begin{aligned} \frac{\partial }{\partial q_{E}}\pi _{E}=\frac{\left( 1-\alpha (q_{B}-q_{A})\right) ^{2}}{9\left( q_{B}-q_{A}\right) }\left( q_{A}+q_{B}-2q_{E}\right) =0, \end{aligned}$$

with \(\frac{\partial ^2 \pi _E}{\partial q_E^2}<0\). Hence

$$\begin{aligned} q_{E}^{**}\left( q_{A},q_{B}\right) =\frac{1}{2}\left( q_{A}+q_{B}\right) . \end{aligned}$$

Substituting this best response of the entrant firm \(E\) into the incumbents’ profits gives

$$\begin{aligned} \pi _{A}\left( q_{A},q_{B}\right)&= \frac{1}{72(q_{B}-q_{A})}\left( \frac{1 }{2}(q_{B}-q_{A})-2(q_{A}-q_{B})(2q_{A}+q_{B})\alpha \right) ^{2} \\ \pi _{B}\left( q_{A},q_{B}\right)&= \frac{1}{72(q_{A}-q_{B})^{2}} (q_{B}-q_{A})\left( \frac{7}{2}(q_{B}-q_{A})+2(q_{A}-q_{B})(q_{A}+2q_{B}) \alpha \right) ^{2}. \end{aligned}$$

The incumbent’s optimal qualities are then given by the first-order conditions

$$\begin{aligned} \frac{\partial }{\partial q_{A}}\pi _{A}&= \frac{1}{288}\left( -(1+24q_{A}\alpha -12q_{B}\alpha )(1+8q_{A}\alpha +4q_{B}\alpha )\right) =0\\ \frac{\partial }{\partial q_{B}}\pi _{B}&= \frac{1}{288}\left( (7+12q_{A}\alpha -24q_{B}\alpha )(7-4q_{A}\alpha -8q_{B}\alpha )\right) =0, \end{aligned}$$

with \(\frac{\partial ^2 \pi _A}{\partial q_A^2}<0\) and \(\frac{\partial ^2 \pi _B}{\partial q_B^2}<0\). The only positive solution to these equations that implies profit maxima is \(\left( q_{A}^{**},q_{B}^{**}\right) =\left( \frac{5}{36\alpha },\frac{13}{ 36\alpha }\right) ,\) leading to \(q_{E}^{**}=\frac{9}{36\alpha }\) and equilibrium prices \(\left( p_{A}^{**},p_{E}^{**},p_{B}^{**}\right) =\left( \frac{203}{3888\alpha },\frac{331}{3888\alpha },\frac{ 635}{3888\alpha }\right) .\) Because quality \(q_B\) is restricted to \(q_B\le 1\), \(\alpha >\frac{13}{36}\) must hold. \(\square \)

Appendix 3: Positioning choice of the entrant

We next analyze whether the entrant has an incentive to deviate from the proposed three-firm case equilibrium by choosing a position on the quality scale different from \(q_A<q_E<q_B\) given that \(q_A=q_A^{**}\) and \(q_B=q_B^{**}\). We restrict our attention to the cases \(q_E=0\) and \(q_E=1\). Any other deviation (i.e., \(0<q_E<q_A\) or \(q_B<q_E<1\)) would lead to strictly lower profits than the extreme positions \(q_E=0\) and \(q_E=1\).

1.1 Case 1: \( 0= q_E<q_A<q_B\)

Inserting the expressions for the indifferent consumers \(\hat{t}_{AE}\) and \(\hat{t}_{AB}\), causes firms’ profits to read as

$$\begin{aligned} \pi _{E}&= \frac{(p_A-p_E) \left( p_E-q_E^2 \alpha \right) }{q_A-q_E}\\ \pi _{A}&= \frac{(p_B (q_A-q_E)+p_A (q_E-q_B)+p_E (-q_A+q_B)) \left( -p_A+q_A^2 \alpha \right) }{(q_A-q_E) (q_A-q_B)}\\ \pi _{B}&= \frac{(p_B-p_A+q_A-q_B) \left( p_B-q_B^2 \alpha \right) }{q_A-q_B}. \end{aligned}$$

Solving the game using backward induction leads to the following equilibrium prices for given qualities \(\left( q_{A},q_{E},q_{B}\right) \)

$$\begin{aligned} p_{E}^{**}\left( q_{A},q_{E},q_{B}\right)&= \frac{1}{6}\frac{(q_A-q_E) (q_A-q_B)}{q_E-q_B}+\frac{1}{6}\left( 2 q_A^2+q_E (3 q_E-q_B)+q_A (q_E+q_B)\right) \alpha \\ p_{A}^{**}\left( q_{A},q_{E},q_{B}\right)&= \frac{(q_A-q_E) (q_A-q_B)+(q_E-q_B) \left( 2 q_A^2-q_E q_B+q_A (q_E+q_B)\right) \alpha }{3 (q_E-q_B)}\\ p_{B}^{**}\left( q_{A},q_{E},q_{B}\right)&= \frac{(q_A-q_B) (q_A-4 q_E+3 q_B)}{6 (q_E-q_B)} \\&\quad +\frac{(q_E-q_B) \left( 2 q_A^2+q_A (q_E+q_B)+q_B (3 q_B-q_E)\right) \alpha }{6 (q_E-q_B)}, \end{aligned}$$

with the second-order conditions being satisfied. Substituting these prices together with \(q_E=0\) into \(E\)’s profits gives

$$\begin{aligned} \pi _{E}\left( q_{A},q_{B}\right)&= \frac{q_A \left( q_B-q_A+2 q_A q_B\alpha +q_B^2 \alpha \right) ^2}{36 q_B^2}. \end{aligned}$$

For \(q_A=\frac{5}{36 \alpha }\) and \(q_B=\frac{13}{36 \alpha }\), the entrant’s profit simplifies to

$$\begin{aligned} \pi _{E}=\frac{1722845}{283855104 \alpha }. \end{aligned}$$

For any \(\alpha \), \(\pi _{E}(q_E=\frac{9}{36 \alpha })=\frac{121}{13122 \alpha }>\pi _{E}(q_E=0)=\frac{1722845}{283855104 \alpha }\). Hence, \(E\) has no incentive to deviate from \(q_E=\frac{9}{36 \alpha }\).

1.2 Case 2: \( q_A<q_B<q_E=1\)

Inserting the expressions for the indifferent consumers \(\hat{t}_{AE}\) and \(\hat{t}_{AB}\), causes firms’ profits to read as

$$\begin{aligned} \pi _{A}&= \frac{(p_A-p_B) \left( p_A-q_A^2 \alpha \right) }{q_A-q_B}\\ \pi _{B}&= \frac{(p_B (-q_A+q_E)+p_E (q_A-q_B)+p_A (-q_E+q_B)) \left( p_B-q_B^2 \alpha \right) }{(q_A-q_B) (q_E-q_B)}\\ \pi _{E}&= \frac{(p_B-p_E+q_E-q_B) \left( p_E-q_E^2 \alpha \right) }{q_E-q_B}. \end{aligned}$$

Solving the game using backward induction leads to the following equilibrium prices for given qualities \(\left( q_{A},q_{E},q_{B}\right) \)

$$\begin{aligned} p_{A}^{**}\left( q_{A},q_{E},q_{B}\right)&= \frac{1}{6} \frac{(q_A-q_B) (q_E-q_B)}{q_A-q_E}+\frac{1}{6}\left( 3 q_A^2+q_A (-q_E+q_B)+q_B (q_E+2 q_B)\right) \alpha \\ p_{B}^{**}\left( q_{A},q_{E},q_{B}\right)&= \frac{1}{3}\frac{(q_A-q_B) (q_E-q_B)}{q_A-q_E}+\frac{1}{3}(q_A (-q_E+q_B)+q_B (q_E+2 q_B)) \alpha \\ p_{E}^{**}\left( q_{A},q_{E},q_{B}\right)&= \frac{(q_B-q_E) (3 q_E+q_B-4 q_A)}{6 (q_A-q_E)}\\&\quad -\frac{(q_E-q_A) \left( q_E (3 q_E-q_A)+(q_A+q_E) q_B+2 q_B^2\right) \alpha }{6 (q_A-q_E)}, \end{aligned}$$

with the second-order conditions being satisfied. Substituting these prices together with \(q_E=1\) into \(E\)’s profits gives

$$\begin{aligned} \pi _{E}\left( q_{A},q_{B}\right)&= \frac{(1-q_B) \left( 3+q_B-3 \alpha +q_A^2 \alpha -2q_B \alpha +2 q_A (-2+\alpha +q_B \alpha )\right) ^2}{36 (1-q_A)^2}. \end{aligned}$$

For \(q_A=\frac{5}{36 \alpha }\), \(q_B=\frac{13}{36 \alpha }\), the entrant’s profit simplifies to

$$\begin{aligned} \pi _{E}=\frac{(36 \alpha -13) \left( 97-3312 \alpha +3888 \alpha ^2\right) ^2}{1679616 (5-36 \alpha )^2 \alpha }. \end{aligned}$$

Respective demand is given by

$$\begin{aligned} x_E=\frac{97-3312 \alpha +3888 \alpha ^2}{1080-7776 \alpha } \end{aligned}$$

and requires \(\left. 0<\alpha <\frac{1}{108} \left( 46-5 \sqrt{73}\right) \vee \frac{5}{36}<\alpha <\frac{1}{108} \left( 46+5 \sqrt{73}\right) \right) \) so that \(x_E\ge 0\). For these \(\alpha \), \(\pi _{E}(q_E=\frac{9}{36 \alpha })=\frac{121}{13122 \alpha }>\pi _{E}(q_E=1)=\frac{(36 \alpha -13) \left( 97-3312 \alpha +3888 \alpha ^2\right) ^2}{1679616 (5-36 \alpha )^2 \alpha }\). Hence, \(E\) has no incentive to deviate from \(q_E=\frac{9}{36 \alpha }\). \(\square \)

Appendix 4: Variety: two-firm case

We solve the game using backward induction. Using the firms’ profits (6), the first-order conditions lead to the following second stage best price responses:

$$\begin{aligned} p_{A}^{*}(p_{B})&= \frac{1}{2}\left( \left( v_{B}-v_{A}\right) +p_{B}+c\right) \\ p_{B}^{*}(p_{A})&= \frac{1}{2}\left( \left( v_{B}-v_{A}\right) \left( 2-v_{A}-v_{B}\right) +p_{A}+c\right) . \end{aligned}$$

This system has the following solution in equilibrium prices given varieties \(\left( v_{A},v_{B}\right) ,\)

$$\begin{aligned} p_{A}^{*}\left( v_{A},v_{B}\right)&= c+\frac{1}{3}\left( v_{B}-v_{A}\right) \left( 2+v_{A}+v_{B}\right) \\ p_{B}^{*}\left( v_{A},v_{B}\right)&= c+\frac{1}{3}\left( v_{B}-v_{A}\right) \left( 4-v_{A}-v_{B}\right) . \end{aligned}$$

Substituting these expressions into the firms’ profits and maximizing with respect to a firm’s variety shows that marginal profits are

$$\begin{aligned} \frac{\partial }{\partial v_{A}}\pi _{A}&= -\frac{1}{18}\left( 2-v_{B}+3v_{A}\right) \left( 2+v_{A}+v_{B}\right) <0 \\ \frac{\partial }{\partial v_{B}}\pi _{B}&= \frac{1}{18}\left( 4-3v_{B}+v_{A}\right) \left( 4-v_{A}-v_{B}\right) >0, \end{aligned}$$

because \(0\le v_{A}<v_{B}\le 1.\) Hence, the optimal positioning in variety \(\left( v_{A}^{*},v_{B}^{*}\right) =\left( 0,1\right) \,\)and equilibrium prices are \(\left( p_{A}^{*},p_{B}^{*}\right) =\left( c+1,c+1\right) .\) \(\square \)

Appendix 5: Variety: three-firm case

Using expressions \(\hat{s}_{AE}\) and \(\hat{s}_{EB}\) for the indifferent consumers, profits read as

$$\begin{aligned} \pi _{A}&= \frac{1}{2}\left( v_{E}+v_{A}+\frac{p_{E}-p_{A}}{v_{E}-v_{A}} \right) \left( p_{A}-c\right) \\ \pi _{E}&= \frac{1}{2}\left( v_{B}-v_{A}+\frac{p_{B}-p_{E}}{v_{B}-v_{E}}- \frac{p_{E}-p_{A}}{v_{E}-v_{A}}\right) \left( p_{E}-c\right) \\ \pi _{B}&= \frac{1}{2}\left( 2-\left( v_{E}+v_{B}\right) -\frac{p_{B}-p_{E} }{v_{B}-v_{E}}\right) \left( p_{B}-c\right) . \end{aligned}$$

Solving the game using backward induction first implies maximizing profits with respect to prices. Second stage best responses are

$$\begin{aligned} p_{A}^{**}\left( p_{E},p_{B}\right)&= \frac{1}{2}\left( c+p_{E}+v_{E}^{2}-v_{A}^{2}\right) \\ p_{E}^{**}\left( p_{A},p_{B}\right)&= \frac{1}{2}\left( c+p_{A} \frac{v_{B}-v_{E}}{v_{B}-v_{A}}+p_{B}\frac{v_{E}-v_{A}}{v_{B}-v_{A}}+\left( v_{E}-v_{A}\right) \left( v_{B}-v_{E}\right) \right) \\ p_{B}^{**}\left( p_{A},p_{E}\right)&= \frac{1}{2}\left( c+p_{E}+\left( v_{B}-v_{E}\right) \left( 2-v_{B}-v_{E}\right) \right) . \end{aligned}$$

Given varieties \(\left( v_{A},v_{E},v_{B}\right) \), this system of equations results in equilibrium prices

$$\begin{aligned} p_{A}^{**}\left( v_{A},v_{E},v_{B}\right)&= c+\frac{\left( v_{E}-v_{A}\right) }{6\left( v_{B}-v_{A}\right) }\left( \left( v_{B}-v_{E}\right) \left( 2+v_{B}-v_{A}\right) +3\left( v_{B}-v_{A}\right) \left( v_{E}+v_{A}\right) \right) \\ p_{E}^{**}\left( v_{A},v_{E},v_{B}\right)&= c+\frac{\left( v_{B}-v_{E}\right) \left( v_{E}-v_{A}\right) }{3\left( v_{B}-v_{A}\right) } \left( 2+v_{B}-v_{A}\right) \\ p_{B}^{**}\left( v_{A},v_{E},v_{B}\right)&= c+\frac{\left( v_{B}-v_{E}\right) }{6\left( v_{B}-v_{A}\right) }\left( \left( v_{E}-v_{A}\right) \left( 2+v_{B}-v_{A}\right) +3\left( v_{B}-v_{A}\right) \left( 2-v_{B}-v_{E}\right) \right) . \end{aligned}$$

Substituting these equilibrium prices into the entrant’s profit function gives us

$$\begin{aligned} \pi _{E}=\frac{1}{54}\left( 2+v_{B}-v_{A}\right) \left( \frac{\left( v_{B}-v_{E}\right) \left( v_{E}-v_{A}\right) \left( v_{A}-v_{B}-2\right) }{ \left( v_{B}-v_{A}\right) }\right) ^{2}. \end{aligned}$$

Given the varieties \(\left( v_{A},v_{B}\right) \) of the incumbent firms, the entrant’s optimal positioning with respect to \(v_{E}\) results from the first-order condition

$$\begin{aligned} \frac{\partial }{\partial v_{E}}\pi _{E}=\frac{\left( v_{E}-v_{A}\right) \left( v_{B}-v_{E}\right) \left( v_{B}-v_{A}+2\right) ^{3}}{27\left( v_{B}-v_{A}\right) ^{2}}\left( v_{A}+v_{B}-2v_{E}\right) =0. \end{aligned}$$

Because for extreme positions \(v_{E}=v_{A}\) or \(v_{E}=v_{B}\), the second order condition is positive

$$\begin{aligned} \frac{\partial ^{2}}{\partial v_{E}^{2}}\pi _{E}=\frac{\left( v_{B}-v_{A}+2\right) ^{3}}{27}>0, \end{aligned}$$

the only solution is

$$\begin{aligned} v_{E}^{**}\left( v_{A},v_{B}\right) =\frac{1}{2}\left( v_{A}+v_{B}\right) . \end{aligned}$$

Substituting this best response of the entrant firm \(E\) into the incumbents’ profits gives

$$\begin{aligned} \pi _{A}&= \frac{1}{144}\left( v_{B}-v_{A}\right) \left( 4v_{A}+2v_{B}+1\right) ^{2} \\ \pi _{B}&= \frac{1}{144}\left( v_{B}-v_{A}\right) \left( 2v_{A}+4v_{B}-7\right) ^{2}. \end{aligned}$$

Maximizing with respect to firms’ variety gives the following first-order conditions

$$\begin{aligned} \frac{\partial }{\partial v_{A}}\pi _{A}&= \frac{\left( 4v_{A}+2v_{B}+1\right) }{144}\left( 6v_{B}-12v_{A}-1\right) =0 \\ \frac{\partial }{\partial v_{B}}\pi _{B}&= \frac{1}{144}\left( 12v_{B}-6v_{A}-7\right) \left( 4v_{B}+2v_{A}-7\right) =0. \end{aligned}$$

The first equation is satisfied for \(v_{A}=\frac{1}{2}\left( v_{B}-\frac{1}{6 }\right) \), implying that the second equation has two solutions, \(v_{B}=\frac{ 13}{18}\) and \(v_{B}=\frac{43}{30}.\) The second solution, however, turns out to be a minimum so that profits are maximized for \(\left( v_{A}^{**},v_{B}^{**}\right) =\left( \frac{5}{18},\frac{13}{18}\right) .\) This implies a best response \(v_{E}^{**}=\frac{1}{2}\) and equilibrium prices \(\left( p_{A}^{**},p_{E}^{**},p_{B}^{**}\right) =\left( c+\frac{32}{243},c+\frac{22}{243},c+\frac{32}{243} \right) \). \(\square \)

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Jost, PJ., Schubert, S. & Zschoche, M. Incumbent positioning as a determinant of strategic response to entry. Small Bus Econ 44, 577–596 (2015). https://doi.org/10.1007/s11187-014-9609-x

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