Abstract
Under spatial price discrimination a critical cost of an upstream monopoly is the resulting distortion in downstream locations. While this is well-established for linear production costs it is examined here for the first time for convex production costs. Convex production costs create their own location distortions yet it is shown that the presence of the upstream monopoly has a smaller influence on efficiency with such costs.
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Notes
While inelastic demand is a critical assumption, Hamilton et al. (1989) have examined cases of downward sloping demand at each location.
Transport cost is normalized to one.
Setting delivered marginal cost of both firms equal, \(kx^{*}+x^{*}-L_{1}=k(1-x^{*})+L_{2}-x^{*}\), and rearranging yields the result in (3).
There are four potential cases in which both firms are critical: (a) one critical customer is located at the center of the market, (b) two critical customers, each at one edge of the market, (c) two critical customers, one at an edge and one at the market division, and (d) three critical customers, two at the edges and one at the market division. In each of these cases, at least one firm can improve its profit by deviating. This demonstration is available upon request.
Within the case when firm 2 is the critical firm, there are six possible deviations in location. If firm 1 deviates left or firm 2 deviates left while remaining to the right of firm 1, the critical firm and customer do not change. Firm 2 clearly will not locate to the left of firm 1. If either firm deviates far enough right, the critical firm and/or critical customer could change. By comparing the firms’ highest profits from any of these deviations to their original locations reveals no firm can increase its profit by deviating so as to change the critical firm or critical customer.
These conditions are \(\frac{\partial ^{2}\pi _{1}}{\partial L_{1}^{2}}=\frac{-(8k^{2}+15k+6)}{4k^{2}+8k+4}<0;\frac{\partial ^{2}\pi _{2}}{\partial L_{2}^{2}}=\frac{-(24k^{2}+31k+10)}{4k^{2}+8k+4}<0\). Together with the demonstrations that no more than one firm is critical and, given that only one firm is critical, no firm will deviate so as to change the critical firm or customer, these imply that the locations in (5) are a Nash equilibrium.
References
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The author thanks Barnali Gupta, John S. Heywood and two referees for helpful comments.
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Courey, G. Spatial price discrimination, monopoly upstream and convex costs downstream. Lett Spat Resour Sci 9, 137–144 (2016). https://doi.org/10.1007/s12076-015-0147-1
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DOI: https://doi.org/10.1007/s12076-015-0147-1