Competitive Location and Pricing on Networks with Random Utilities

Abstract

In this paper we analyze the effect of including price competition into a classical (market entrant’s) competitive location problem. The multinomial logit approach is applied to model the decision process of utility maximizing customers. We provide complexity results and show that, given the locations of all facilities, a fixed-point iteration approach that has previously been introduced in the literature can be adapted to reliably and quickly determine local price equilibria. We present examples of problem instances that demonstrate the potential non-existence of price equilibria and the case of multiple local equilibria in prices. Furthermore, we show that different price sensitivity levels of customers may actually affect optimal locations of facilities, and we provide first insights into the performance of heuristic algorithms for the location problem.

Appendices

Appendix A: Determining $\tilde p_{q}$

Appendix B: Vanishing Profit Derivatives

Lemma 6

$\underset {p_{q} \to \infty }{\lim } {\frac {\partial {\Pi }_{q}}{\partial p_{q}}} = 0$ for q∈{I,E}.

Proof

Making use of the results of Section 3.1 and defining Z _q , q∈{I,E}, as in Eq. 4, we get

$$\begin{array}{@{}rcl@{}} \lim_{p_{q} \to \infty } {\frac{\partial {\Pi}_{q}}{\partial p_{q}}} &=& \lim_{p_{q} \to \infty } \left( \sum\limits_{i \in V }{\sum\limits_{j \in Z_{q} }{\pi(i)P_{ij}^{q}}} + (p_{q} - c_{q}) \sum\limits_{i \in V }{\sum\limits_{j \in Z_{q} }{\pi(i)} \cdot \frac{\partial P_{ij}^{q}}{\partial p_{q}}} \right) \\ &=& -s \beta \lim_{p_{q} \to \infty } {(p_{q} - c_{q}) \cdot \sum\limits_{i \in V }{\sum\limits_{j \in Z_{q} }{\pi(i)} P_{ij}^{q} \left( 1 - \sum\limits_{k \in Z_{q} } {P_{ik}^{q}} \right)}} = 0, \end{array} $$

because, by applying L’Hospital’s rule, we derive

$$\begin{array}{@{}rcl@{}} \underset{p_{q} \to \infty }{\lim } {p_{q} P_{ij}^{q}} &=& \underset{p_{q} \to \infty }{\lim } \frac{p_{q}}{\sum\limits_{k \in Z_{q} } { e^{s({a_{k}^{q}}-{a_{j}^{q}}-\alpha(d_{ik}-d_{ij}))} } + \frac{c}{e^{s({a_{j}^{q}} - \alpha d_{ij} - \beta p_{q})}} }\\&=& \underset{p_{q} \to \infty }{\lim } \frac{ e^{s({a_{j}^{q}} - \alpha d_{ij} - \beta p_{q})} }{c s \beta} = 0 \end{array} $$

for any i∈V, j∈Z _q and c as defined in Eq. 17. □