Decision Support
Product differentiation and entry timing in a continuous time spatial competition model

https://doi.org/10.1016/j.ejor.2015.06.049Get rights and content

Highlights

  • We consider a continuous time spatial competition model with endogenous entry timing.

  • The leader locates closer to the center than when the follower’s timing is exogenous.

  • The leader locates closer to the center as the transport cost parameter decreases.

  • The leader’s profit has a non-monotonic relationship to the transport cost parameter.

Abstract

We extend the well-known spatial competition model (d’Aspremont, Gabszewicz & Thisse, 1979) to a continuous time model in which two firms compete in each instance. Our focus is on the entry timing decisions of firms and their optimal locations. We demonstrate that the leader has an incentive to locate closer to the center to delay the follower’s entry, leading to a non-maximum differentiation outcome. We also investigate how exogenous parameters affect the leader’s location and firms’ values and, in particular, numerically show that the profit of the leader changes non-monotonically with an increase in the transport cost parameter.

Introduction

Researchers in economics and marketing have emphasized the importance of (horizontal) product differentiation in the context of firm competition (e.g. Brown, 1989, d’Aspremont, Gabszewicz, Thisse, 1979, Lancaster, 1990). When firms launch their new products into markets, timing and product characteristics are some of the important factors for their profits (e.g. Krishnan & Ulrich, 2001). Taking into account firms’ decisions regarding product differentiation, researchers theoretically and/or empirically investigate how firms determine the timing of launching their products and those characteristics (e.g. Lambertini, 1997, Thomadsen, 2007).

From the theoretical point of view, Lambertini (2002) presented pioneering work that discusses the strategic interaction between the optimal locations of the inventor (the market leader), who anticipates subsequent entry and the location choice of the follower in a Hoteling-type spatial competition model, as in d’Aspremont et al. (1979).3 He was the first to introduce a dynamic model in the sense that time is continuous, the firm locations are fixed once entry is made and that firms earn their profits in each instance. Regarding the time structure, several papers deal with sequential locational entry in a discrete time model, which allows qualitative analyses such as how many steps the timing of investment would change given a change in other parameters (e.g. Prescott & Visscher, 1977). However, a more rigorous quantitative analysis, such as determining the percentage change in the investment time attributable to a percentage change in a parameter, requires a continuous time model.4

This novel point is from Lambertini (2002) and differs significantly from those in related theoretical papers discussing sequential location choices based on Hoteling-type spatial competition models (e.g. Götz, 2005, Neven, 1987).5 Those related papers are static Hoteling models in the sense that each firm has only one profit earning chance.6 Lambertini (2002) considered two scenarios: (i) the follower’s timing of entry is exogenous and (ii) the follower’s timing of entry is probabilistically determined. Therefore, the follower does not endogenously determine its optimal timing of entry in either scenario. To summarize, Lambertini (2002) considered a continuous time model, but an endogenous entry timing model with continuous time has not been considered in locational models. Because the entry timing of followers significantly influences market leaders as well as followers (Kalyanaram, Robinson, Urban, 1995, Vakratsas, Rao, Kalyanaram, 2003), we need to overcome the weakness in the model given by Lambertini (2002) and endogenize the follower’s entry-timing decision. Therefore, our paper substantially extends the model of Lambertini (2002).

We incorporate several aspects into the standard Hoteling duopoly model in d’Aspremont et al. (1979). The time horizon is infinite, as in Lambertini (2002). Each firm sets a price and earns a profit in each instance if it exists in the market, implying that a delay of entry causes a loss of profit opportunity. In anticipation of subsequent entry by the follower, the market leader initially sets its location. Because the leader’s location decision influences the profits of the follower, it also affects the timing of the entry (the length of the monopoly period), thus representing an additional value of our paper. After the location choice of the market leader, the follower determines the timing of entry and its location. When the follower enters the market, it incurs an investment cost that exponentially decreases with the standard discount rate. In contrast, consumer size increases with a growth rate lower than the discount rate. By balancing the benefit and cost of staying outside, the follower determines its entry timing and location. We also note that this formulation is suitable for perishable goods as consumers repeatedly purchase the good.7

Compared with Lambertini (2002), our contributions in this paper are threefold. The first contribution is that we endogenize the follower’s timing. The second contribution follows the first, as we introduce investment costs and a growth rate in consumer size to make the model more realistic. In addition, the growth rate ensures that the entry occurs within a finite time8 and, in turn, affects the leader’s location. The third contribution is a strategic interaction between the leader’s location and the follower’s entry timing. In addition to the effects considered by Lambertini (2002), the leader’s moving closer to the center increases the follower’s incentive to delay its entry, prolonging the monopoly regime. Thus, by endogenizing the follower’s timing, the leader has a stronger incentive to move closer to the central point. Although this strategic interaction among the leader’s location, the follower’s location and its entry timing is an important aspect of this problem, Lambertini (2002) does not take into account this strategic interaction because of his assumption of an exogenous entry timing by the follower.

We also show that the follower always chooses to maximize the distance between the firms whereas the leader has an incentive to locate closer to the center to delay the follower’s entry, possibly leading to a non-maximum differentiation outcome. Furthermore, the location interval between the leader and the follower is negatively correlated with the length of time for which the follower stays outside. These results are similar to those in Lambertini (2002), although the mechanism behind these results definitely differs between the two papers.

Finally, we show that the entry timing becomes earlier as the growth rate of consumer size or the parameter of consumer transport cost increases, and becomes later as the discount rate increases. We numerically investigate how those exogenous parameters influence their profits. A notable result is that the profit of the market leader non-monotonically changes with an increase in the consumer transport cost parameter.9

Section snippets

The model

Two firms, i ∈ {1, 2}, produce homogeneous goods. Consumers are uniformly distributed over the unit segment [0, 1] as proposed by Hoteling (1929).10 Each consumer at point x ∈ [0, 1] repeatedly purchases at each instance [t,t+dt) at most one unit of the good and decides from which firm to purchase if he does make a purchase.11

Equilibrium

In this section, we derive the price, location and timing outcomes in the subgame perfect equilibrium. First, given locations x1 and x2, we consider the problem of prices at each time t before and after the entry of firm 2. Then, we derive the local profits of the leader and the follower at each time t.

The following are the equilibrium prices. Notably, the maximization of the instantaneous profit flows is equivalent to the maximization of the total profits. In other words, firm 1 maximizes the

Numerical analysis

In this section, we investigate in detail the underlying properties of our model using numerical analysis. First, we investigate the effects of the key parameters, α, c and F2, on firm 1’s equilibrium location, which we denote as x1E{x1*,x1**,x1***}. In particular, focusing on the equilibrium-path behavior, we illustrate that all three types of equilibrium location patterns actually exist. Then, we show the importance of the endogeneity of T2, namely the first term of Eq. (21). Finally, we

Summary and discussion

In this paper, we develop a duopoly model that determines the follower’s entry timing, firms’ locations and their prices. Examining the timing of investments is important when considering firms’ entry strategies. Hence, we extend Lambertini (2002), which in turn extends the location-price competition model (d’Aspremont et al., 1979) by using a continuous time model in which firms earn profits in each instance and the follower’s entry timing is given exogenously. Our model endogenizes the

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    We thank Immanuel Bomze (editor) and two anonymous referees for invaluable comments and suggestions. We are also grateful to Reiko Aoki, Takahiro Watanabe, session participants at 2014 Autumn Meeting of Japan Association for Applied Economics, and seminar participants at Tokyo Metropolitan University and Yokohama National University for helpful comments. The authors gratefully acknowledge financial support from a Grant-in-Aid for Young Scientists (24730224, 15K17047) and Basic Research (24530248, 24530264, 15H03349) from the MEXT and the Japan Society for the Promotion of Science. The second author thanks the warm hospitality at MOVE, Universitat Autònoma de Barcelona where part of this paper was written and a financial support from the “Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation” by JSPS. Needless to say, we are responsible for any remaining errors.

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