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Endogenous Price Leadership and Product Positioning

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Abstract

This paper models the timing of price competition in a differentiated duopoly as endogenously determined and studies the timing’s impact on the first-mover advantage in establishing product position. It is shown that a firm's location advantage—being closer to a majority of consumers—leads to the strategic disadvantage of leading in the price game. When product positioning occurs sequentially, endogenous timing in pricing results in the second entrant’s locating farther away from its rival; and, although the first entrant becomes the price leader, the magnitude of the advantage for the first mover in the location (entry) game is larger, as compared with the case of simultaneous pricing.

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Notes

  1. We find that if firms simultaneously choose their product positions then no pure-strategy equilibrium of the game exists. Our primary interest is whether the advantage to a firm of being the first to enter a market and establish position is undermined by a resulting second-mover (in prices) advantage for the late-to-enter competitor when the timing of the price game is endogenously determined.

  2. If the firms are ex ante identical, first-mover advantage also implies the relative advantage of the leading firm vis-à-vis the following firm with regard to profitability.

  3. Among other factors, a larger number of loyal consumers (Deneckere et al. 1992) and higher product quality (Li 2014) confer price leadership. In the former, loyal consumers buy only from one of the firms; and in the latter, all consumers prefer the high-quality product. In our model, more (but not all) consumers prefer the product of the better-positioned firm but do not necessarily buy from it.

  4. Markham (1951), for example, discusses the patterns and implications of price leadership in a number of industries in the United States. Gasmi et al. (1992) find that Coca-Cola had been a price leader in the soft drink market. In the market for liquid laundry detergents, Kadiyali et al. (1996) reject the hypothesis of Bertrand-Nash price-setting behavior in favor of price leadership. See also Roy et al. (1994) and Seaton and Waterson (2013).

  5. If the firms are constrained to locate within the linear city, they choose to locate at the two ends—regardless of whether they choose a location sequentially or simultaneously.

  6. Tyagi (1999) notes that it is easily feasible to position a product outside the ideal range of consumer preferences. Also, shopping malls are often located outside the city, although one of the reasons is that it is land-intensive outside the city and land rents are generally cheaper. Matsumura and Matsushima (2012) and Li and Shuai (2018) both show that locating outside a linear city can actually benefit consumers despite the increase in transportation cost.

  7. See also Fleckinger and Lafay (2010).

  8. Price setting is a short-term commitment in response to the current market conditions, whereas location choice (the positioning of a product) is usually made in a long time horizon. It is unlikely that the firms agree on their roles in the price game before the locations are set.

  9. We have followed d'Aspremont et al. (1979) to denote firm B’s location as \(1-{s}_{B}\). When \({s}_{B}\) is negative, firm B’s location is greater than 1 which implies that it is located beyond the right end of the market.

  10. For instance, if a firm chooses T1 in the pre-stage, it will set its price in T1 and will not change the price in T2, regardless of the competitor’s strategies. Many firms send brochures that contain price information, on the basis of which the customers make purchases. Also, the prices that are announced at launch events are usually kept for a certain period of time.

  11. We focus on pure strategy equilibria, which is common in the literature on endogenous timing. A mixed strategy equilibrium, when it exists, is unstable (Amir and Stepanova 2006).

  12. Note that when \({s}_{A}-{s}_{B}\in (2\sqrt{2}-1, 3]\), based on Observation 3, both firms prefer the sequence with firm A leading and firm B following. However, the other sequence of play is also an equilibrium.

  13. Harsanyi and Selten (1988) also proposed payoff dominance as a criterion in equilibrium selection. It is easily verified that payoff dominance may not apply, and when it applies (as in the case \(2\sqrt{2}-1<{s}_{A}-{s}_{B}\le 3\)), the two criteria lead to consistent results.

  14. The other case with firm B being advantaged in location—\((-1, 0)\)—is similar.

  15. There does exist a mixed strategy equilibrium: Although the standard existence theorems do not apply due to discontinuous payoff functions, Theorem 5 of Dasgupta and Maskin (1986) can be used to show that a mixed-strategy equilibrium exists. If the location space of the firms is constrained to the unit interval where consumers reside—with \({s}_{i}\ge 0\)—then it can be readily shown that maximum differentiation—with each firm choosing to locate at one end of the market, (0, 0)—is the equilibrium in the location stage. No firm wants to deviate—to move closer to the center of the market even for the price leader—as \(\partial {\pi }_{i}^{L}/\partial {s}_{i}<0\).

  16. Using a different model, Tyagi (1999) obtained the same numerical result under sequential location choices. However, no formal proof was given as to whether this is a subgame perfect equilibrium and unique. In this paper, we show that—given the first mover’s choice of the center of the market—the first mover takes the leading role in the subsequent price game and no other location would give the first mover a higher payoff when the second mover’s response is taken into consideration.

  17. Total social welfare, which is determined by the cost of transportation, is smaller as transportation cost is increased.

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Acknowledgements

We wish to thank the Editor, Lawrence J. White, and two anonymous referees for their helpful comments and suggestions. The usual disclaimer applies.

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Appendix

Appendix

Proof of Proposition 1

From the payoff functions (2) and (4) , the reduced game in the pre-stage of timing choices takes three forms: This depends on whether \(|{s}_{i}-{s}_{j}|\in\) [0, 3]; \(|{s}_{i}-{s}_{j}|\in\) (3, 5); or \(|{s}_{i}-{s}_{j}|\ge\) 5. Solving these games, we obtain the equilibrium outcomes shown in the proposition. □

Proof of Proposition 2

The reduced game in the pre-stage is a 2 × 2 game with two strong equilibrium points. Following Harsanyi and Selten (1988), one equilibrium risk dominates the other if the product of deviation losses of the former is larger. Since \(\left({\Pi }_{A}^{L}-{\Pi }_{A}^{N}\right)\left({\Pi }_{B}^{F}-{\Pi }_{B}^{N}\right)>\left({\Pi }_{A}^{F}-{\Pi }_{A}^{N}\right)\left({\Pi }_{B}^{L}-{\Pi }_{B}^{N}\right)\) if and only if \({\mathrm{s}}_{\mathrm{A}}>{\mathrm{s}}_{\mathrm{B}}\), the proposition is proved. □

Proof of Proposition 3

First, any symmetric locations are not equilibrium under simultaneous location choices. One of the firms (the firm that leads in the price game) does strictly better by moving a bit away from the market center and becoming the price follower.

Next, suppose \({s}_{A}>{s}_{B}\) at the equilibrium. The relevant range of \({s}_{A}-{s}_{B}\) is (0, 5), otherwise firm B earns zero profit and has an incentive to deviate. Then by Proposition 1 and with the risk-dominance criterion, firm A plays the leader role in the subsequent price game, and firm B plays the follower role. Firm A’s best response strategy is defined by \(\left(3+{s}_{A}-{s}_{B}\right)(1+3{s}_{A}+{s}_{B})=0\); and firm B’s best response strategy is defined by \(\left(5-{s}_{A}+{s}_{B}\right)(3+{s}_{A}+3{s}_{B})=0\). Solving these equations yields the strategy profile \((0, -1)\), with associated payoffs of \({\Pi }_{A}^{L}=2c\) and \({\Pi }_{B}^{F}=c\). Given \({s}_{A}=0\), firm B does not want to change its role as the price follower by moving to \({s}_{B}>0\). However, given that \({s}_{B}=-1\), firm A wants to move to \({s}_{A}=-(1+\varepsilon )\) and follow in the price game, which yields a payoff greater than \(2c\). The other case with \({s}_{A}<{s}_{B}\) is similar. □

Proof of Proposition 4

Suppose firm A is the first positioner. Without loss of generality, assume that \({\mathrm{s}}_{\mathrm{A}}\le \frac{1}{2}\). If \({\mathrm{s}}_{\mathrm{A}}=\frac{1}{2}\), the profit maximizing location for firm B is \({\mathrm{s}}_{\mathrm{B}}=-\frac{7}{6}\), which yields a profit of \({\Pi }_{A}^{L}\left(\frac{1}{2}, -\frac{7}{6}\right)=\frac{245}{108}c\) for firm A. This is indeed the highest equilibrium payoff for firm A as we show next.

Suppose (\({s}_{A}^{^{\prime}}, {s}_{B}({s}_{A}^{^{\prime}})\)) is another sequential play equilibrium. Then it must satisfy one of the following conditions:

  1. 1.

    \(\frac{\partial {\Pi }_{B}^{F}}{\partial {s}_{B}}=0\quad {\text{if}}\,\,{s}_{B}\left({s}_{A}^{^{\prime}}\right)<{s}_{A}^{^{\prime}}\);

  2. 2.

    \(\frac{\partial {\Pi }_{B}^{F}}{\partial {s}_{B}}\ge 0 \quad {\text{if}} \,\,{s}_{B}\left({s}_{A}^{^{\prime}}\right)={s}_{A}^{^{\prime}}\);

  3. 3.

    \(\frac{\partial {\Pi }_{B}^{L}}{\partial {s}_{B}}=0 \quad {\text{if}} \,\, {s}_{B}\left({s}_{A}^{^{\prime}}\right)>{s}_{A}^{^{\prime}}\).

In the first case, we have \({s}_{B}\left({s}_{A}^{^{\prime}}\right)=-\frac{3+{s}_{A}^{^{\prime}}}{3}\) and \({\Pi }_{A}^{L}\left({s}_{A}^{^{\prime}}, -\frac{3+{s}_{A}^{^{\prime}}}{3}\right)=\frac{2c}{27}(3-{s}_{A}^{^{\prime}}){(3+{s}_{A}^{^{\prime}})}^{2}\le \frac{245}{108}c\) if \({s}_{A}^{^{\prime}}\le \frac{1}{2}\), with the equal sign holding if and only if \({s}_{A}^{^{\prime}}=\frac{1}{2}\). In the third case, we have \({s}_{B}({s}_{A}^{^{\prime}})=-\frac{1+{s}_{A}^{^{\prime}}}{3}\) and \({\Pi }_{A}^{F}({s}_{A}^{^{\prime}}, -\frac{1+{s}_{A}^{^{\prime}}}{3})=\frac{c}{27}\left(2-{s}_{A}^{^{\prime}}\right){\left(4+{s}_{A}^{^{\prime}}\right)}^{2}<\frac{245}{108}c\) if \({s}_{A}^{^{\prime}}\le \frac{1}{2}\). In the second case, \({\Pi }_{A}^{L}\left({s}_{A}^{^{\prime}}, {s}_{A}^{^{\prime}}\right)=\frac{9c}{16}\left(1-2{s}_{A}^{^{\prime}}\right)\ge \frac{245}{108}c\) only if \({s}_{A}^{^{\prime}}<-\frac{737}{486}\). But when \({s}_{A}^{^{\prime}}<-\frac{737}{486}\), \({\Pi }_{B}^{L}({s}_{A}^{^{\prime}}, -\frac{1+{s}_{A}^{^{\prime}}}{3})=\frac{2c}{27}{(2-{s}_{A}^{^{\prime}})}^{3}>{\Pi }_{B}^{F}({s}_{A}^{^{\prime}}, {s}_{A}^{^{\prime}})=\frac{25c}{32}(1-2{s}_{A}^{^{\prime}})\). That means, instead of choosing \({s}_{A}^{^{\prime}}\) and taking the following role in the price game, firm 2 will choose \({s}_{B}=-\frac{1+{s}_{A}^{^{\prime}}}{3}\) and lead the price game, a contradiction. □

Proof of Proposition 5

Based on the calculations shown in the text, with endogenous price leadership the difference in profit between the first mover and the second mover is \(\frac{365}{216}c\approx 1.69c\), which is much larger than \(\frac{2}{3}c\approx 0.67c\) (which is the difference in profit when prices are set simultaneously). □

Proof of Proposition 6

Based on (1) and (3) , the indifferent consumer is located at \(\stackrel{\sim }{\mathrm{x}}=7/12\) under endogenous price leadership and at \(\stackrel{\sim }{\mathrm{x}}=2/3\) under simultaneous price setting. It is easy to verify that total transportation cost is higher in the former case, which implies decreased total social welfare. As firm profits are higher, consumer surplus must be lower under endogenous price leadership. □

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Gilpatric, S.M., Li, Y. Endogenous Price Leadership and Product Positioning. Rev Ind Organ 58, 287–302 (2021). https://doi.org/10.1007/s11151-020-09752-4

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