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Sequential product positioning and entry timing under differential costs in a continuous-time model

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Abstract

We investigate the product positioning decisions of two firms that enter a market sequentially under a duopoly competition. Two important assumptions are made: differential marginal production costs between the two firms, and endogenous entry timing of a follower in a continuous-time setting. We analyze a standard location-pricing Hotelling game with quadratic transportation costs with the points of departure being that the two firms (i) are allowed to have different (constant) marginal costs; (ii) enter sequentially in a pre-determined order in a market in which the consumers are growing over time; and (iii) are not restricted to choosing positions inside the interval in which consumer preferences are located. For the first mover, the dynamic market growth can give rise to a trade-off between exploiting short-run monopoly and long-run duopoly profits. This trade-off affects the equilibrium positions when the first mover has a larger marginal cost as well as a larger discount rate, in which case the first mover chooses its position at the center of the interval along which consumers are located. We also introduce uncertainty regarding entry and marginal costs to examine their effects on positioning and entry timing. If the entry cost of the follower firm is uncertain for the leader firm, then the leader firm is likely to choose its position farther away from the most attractive point. Moreover, we show that the follower firm enters the market earlier if the leader firm faces such uncertainty.

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Notes

  1. Location point is interpreted as a firm’s differentiation selection because the distance between a firm’s location point and a consumer’s address corresponds to that between a firm’s attribute and a consumer’s ideal point. This interpretation is standard in operations research and management science literature.

  2. Lieberman and Montgomery (2013) state that the market entry timing has been a major focus for management, economics, and business strategy researchers over the past three decades.

  3. Following Tyagi (2000), we use the term “differential costs” to represent the difference between the marginal costs of firms in production, although we introduce two types of costs into our model.

  4. Among these, Meza and Tombak (2009) show a similar motivation to the current study, constructing a theoretical model to consider entry timing under differential costs for two firms. Unlike our model, theirs is stable, as economic conditions, such as demand or market growth, do not change over time. Such a stable model is useful for a short-run situation or one in which economic conditions do not change. However, if economic conditions change over time, a dynamic model like ours is beneficial. Hence, their model and focus are different from ours.

  5. According to Baah and Bohaker (2015), in addition to differentiation, Coca-Cola maintains low-cost production and distribution by owning most of its bottling companies. They quote the statement by Euromonitor, “Coca-Cola’s production strength lay in its widespread bottling capability which is at the heart of the company’s success.”

  6. Following Tyagi (2000), we use the terminology “the most attractive point" as the center of the Hotelling’s linear interval.

  7. Our formulation that consumers repeatedly purchase the product is suitable for perishable or non-durable goods, as in Footnote 7 of Ebina et al. (2015). Refer to the related description in the Introduction.

  8. We will show in Lemma 1 that \(T_2\) is in fact strictly positive if F is sufficiently high. Note that firm 2 does not necessarily enter the market even when its total profit function is positive, because \(T_2\) is chosen to maximize the total profit function.

  9. Parameter \(\alpha \) represents the instantaneous growth rate of consumer demand due to population increase. Several previous studies assume a time-varying market growth. However, we consider that the growth rate \(\alpha \) is constant for the following reasons. Let N denote the population of consumers at time 0. Then, the population at time t equals \(N\textrm{e}^{\alpha t}\), and we normalize \(N=1\) in the current setting. The validity of exponential market growth is stated in many industries, such as telecommunication services, DRAM, and pharmaceuticals. Refer to Footnote 14 in Ebina et al. (2015) for more detail.

  10. Kress and Pesch (2012) conduct a survey on sequential competitive location, and Turkoglu and Genevois (2020) conduct a survey on facility location problems. Many studies have investigated the issue of spatial competition with sequential entry (e.g. Eiselt, 1992; Loertscher & Muehlheusser, 2011; Serfes, K., & Zacharias, 2012; Marianov, & Eiselt, 2016; Wang & Lyu, 2020). These studies emphasize the importance of considering a sequential product positioning (location) problem. However, they do not study entry timings, which is one of the novelties in the current study.

  11. Lambertini (2002) introduces a dynamic model considering that time is continuous. However, he assumes that the follower does not choose its entry timing strategically, in that the entry timing of the follower is exogenously given under his Assumption A or is probabilistically determined under Assumption B. Hence, Lambertini (2002) considered a continuous time model; however, an endogenous entry-timing model with continuous time has not been considered in locational models (Ebina et al., 2015). Related discussions are presented in Remark 4.

  12. Following the previous studies employing the continuous-time setting (Lambertini, 2002; Ebina et al., 2015), we restrict our attention to competitive pricing rather than tacit collusion.

  13. We do not consider \(x_1 = x_2\), as the profit functions are as in a Bertrand game, and we can easily verify that it does not happen in equilibrium.

  14. The optimal price is such that it maximizes the instantaneous profit. See the proof of Lemma 1 in the Appendix of Ebina et al. (2015).

  15. From Tyagi (2000)’s and Ebina et al. (2015)’s analyses, we can verify that the second-order conditions (SOCs) are satisfied with respect to \(x_2\) and \(T_2\), respectively.

  16. Lambertini (2002) assumes that the firms can position their products in a unit interval, whereas we assume that their products can be positioned in a real line (Lambertini, 1997; Tyagi, 2000).

  17. We only present the case of \(x_1^T <x_2^T\).

  18. We omit the analysis on the effect of \(\bar{u}\) of F, as it produces non-surprising results.

  19. Because the firms in our study can choose their positions on a real line, not in a unit interval and consumers purchase at least one product by Assumption 1, our model corresponds to the largest market (8 miles \(\times \) 8 miles) in Thomadsen (2007).

  20. Couturier and Sola (2010) conducted interviews in several industry sectors and found that the choice of market entry strategy is dependent on the stage of market evolution. We argue that the current study supports their finding with interviewees CEOs and other high-level managers, who are in charge of important managerial decisions.

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Acknowledgements

The authors are very thankful to the anonimous referee for his/her comments on the manuscript which helped us to provide additional explanations and examples to improve the presentation of the paper.

Funding

We gratefully acknowledge the financial support from JSPS KAKENHI Grant Numbers JP20K01742, JP20H00088, JP21K01468, JP23H01632, JP23K01465.

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Authors and Affiliations

  1. School of Commerce, Meiji University, 1-1, Kanda-Surugadai, Chiyoda-ku, Tokyo, 101-8301, Japan

    Takeshi Ebina

  2. Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, 186-8601, Japan

    Katsumasa Nishide

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  1. Takeshi Ebina
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  2. Katsumasa Nishide
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Both authors contributed equally to the manuscript, read and approve the final manuscript.

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Correspondence to Takeshi Ebina.

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Ebina, T., Nishide, K. Sequential product positioning and entry timing under differential costs in a continuous-time model. Ann Oper Res 332, 277–301 (2024). https://doi.org/10.1007/s10479-023-05665-z

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