Abstract
This paper is the first to model consistent location conjectures under spatial price discrimination. With linear production cost, the well-known association of spatial price discrimination with efficiency vanishes as duopolists with consistent conjectures collocate at the center. With convex production cost, the duopolists do not collocate but continue to locate closer to the center than under Nash conjectures. Yet, with sufficient cost convexity, this movement to the center can actually increase welfare relative to Nash. We extend the results with linear costs to multiple private firms.
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Notes
We note that we do not attempt to model issues of evolutionary stability and use this only as motivation for our inquiry.
Thus airline flights between city pairs differ by departure time from early morning to late evening, the editorial policies of newspapers differ from liberal left to conservative right, and breakfast cereals differ in their sugar content.
A somewhat different approach to establishing price conjectures is outlined by Norman (1989).
Greenhut and Norman (1992) examine firms’ incentives to agglomerate under spatial price discrimination for several specific conjectures about rivalry but neither derives nor considers consistent location conjectures. Thus, they vary whether the price and location happen in one stage or two and whether the competition includes the more traditional price or, instead, quantity competition. They provide a useful presentation of the wide variety of possible spatial models but, again, are not concerned with consistent location conjectures.
Indeed, even without invoking the symmetric location assumption we will show that when the linear case is thought of as the limit of decreasing convexity, the two firms can be shown to collocate at the middle.
Uniform delivered pricing provides another alternative in which firms bear the total transport costs and charge a constant price to all consumers. Yet, De Palma et al. (1987) show that the locate first, price second equilibrium does not exist in our case of duopoly competition with inelastic demand.
Equating the delivered prices from the two firms allows solving for \(x^*\)as follows: \(kx^*+t( {x^*-L_1 })=k( {1-x^*})+t( {L_2 -x^*})\Rightarrow x^*=\frac{k+t( {L_1 +L_2 })}{2( {k+t})}\).
The other two roots yield consistent conjectures that imply two of the three firms locate at the same place and even if we locate the three firms optimally (given the relationship between them), they have higher social cost than that implied by the locations in (29).
Our efforts to establish evolutionary stability among of the multiple equilibria has proven fruitless. Possajennikov (2009) emphasizes the importance of imposing symmetry among identical firms as part of showing which conjecture is likely to persist. In the absence of the ability to impose symmetry (because of the unit line segment), we have been unable to follow the Possajennikov methodology.
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Heywood, J.S., Wang, Z. Consistent location conjectures under spatial price discrimination. J Econ 117, 167–180 (2016). https://doi.org/10.1007/s00712-015-0447-3
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DOI: https://doi.org/10.1007/s00712-015-0447-3