Chapter 5. Competitive Location and Pricing with Random Utilities
consequence, this chapter also contributes to numerical approaches for com- puting equilibrium prices under multinomial logit demand and is therefore very closely related to Morrow (2008); Morrow and Skerlos (2011). Related models can be found in the ﬁeld of product positioning, cf. Choi et al. (1990) and Rhim and Cooper (2005). This chapter proceeds as follows. First, a detailed problem formulation is given in Section 5.1 with results concerning the existence of price equilibria and the computational complexity in Sections 5.1.1 and 5.1.2, respectively. In Section 5.2 we show that, given the locations of all facilities, standard numerical approaches can be combined to reliably and quickly determine local price equilibria. Finally, in Section 5.3, we show that price competition may actually aﬀect optimal locations of facilities, and we provide ﬁrst insights into the performance of heuristic algorithms for ﬁnding solutions to the location problem. The chapter closes with a conclusion in Section 5.4. 5.1 Problem Formulation Consider a network N = (V,E, λ) with vertex demand only. A ﬁrm L (leader) acts as a monopolist with multiple facilities in this spatial market. L’s facilities are located at p > 0 distinct vertices Xp ⊆ V of the network. A competitor F (follower) wants to enter the market with an a priori ﬁxed number of facilities r > 0.1 F’s potential facility sites are restricted to the vertex set of the network. Hence, F solves a discrete (r|Xp)-medianoid problem. At most two facilities, one of the leader’s and one of the follower’s...
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