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Mixed duopoly under hotelling with convex production costs

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Abstract

Dragone and Lambertini (Reg Sci Urban Econ 84:103568, 2020) show that in a Hotelling duopoly with linear transport cost, sufficiently convex production cost generates an elevated price equilibrium with collocation at the middle. We examine the consequences of a mixed duopoly under these cost assumptions. No equilibrium exists in a mixed duopoly with simultaneous location. Equilibria exist under location leadership. Private firm leadership generates greater welfare than public firm leadership, a result different from previous examinations. A mixed duopoly with a foreign firm generates an equilibrium with firms usually located apart but with the public firm serving all interior customers. This typically increases welfare relative to the fully private market.

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Notes

  1. Xefteris (2013) has shown the existence of a subgame perfect price equilibrium in the original formulation of the model, with linear consumer preferences and linear production costs.

  2. While diseconomies of scale are a long-run concept, in the short-run assuming convexity clearly fits with the basic intuition of increasing opportunity costs.

  3. Price discrimination changes the agglomeration tendency of ordinary Hotelling model, making these papers yields different results from those we will present.

  4. Sequential location has been common in the spatial literature with exclusively private firms. Neven (1987) considers endogenized sequential entry and deterrence using a Hotelling model with quadratic transport cost. Götz (2005) reexamines this and shows asymmetric locations. Loertscher and Muehlheusser (2011) examine sequential entry given a nonuniform distribution of consumers.

  5. While our presentation is limited to quadratic costs, we have also reproduced virtually identical propositions using cubic costs and this presentation is available upon request.

  6. We also retain from the authors the assumption of "supply obligation" that firms supply the demand they face. Vives (1999) and Matsumura (2012) discuss the implications of this assumption.

  7. It differs from Gupta (1992) who demonstrates first mover advantages under delivered pricing that allows spatial price discrimination.

  8. We could not easily identify a mixed strategy equilibrium for our model. Unlike Matsumura and Matsushima (2009), our Firm B, which wants to locate near its rival, does not have a concave payoff in its rival’s location. Thus, when expecting its rival to locate at two symmetric points with equal probability, it prefers to locate next to its rival instead of in the middle of these two points. In addition, under certain locations, (11) yields one firm undercutting its rival fully, resulting in no inner solution to even the mixed strategy equilibrium.

  9. The symmetric location equilibrium would be \(y_{A}^{BR\prime}(0.5) = 1 - \frac{2k\, +\, t}{{8k \,+\, 6t}}\), and it is in all ways identical to that presented in (17).

  10. Again, when the private firm was domestic, Firm A set a price for sharing the market resulting in the difference between the two firms’ prices being not that large.

  11. In addition, there remains the condition that Firm B has no incentive to undercut Firm A, which means \(\pi_{B} > p_{B}^{E} - k/2\). This yields \(p_{A}^{E} < (2 - y_{B} )((k + 2t)y_{B} - 2ty_{A} )/2y_{B}\). This condition is more binding at some location than (28) when \(k < 4t\). However, as we want to compare with fully private case, we limit ourselves to \(k \ge 4t\).

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Acknowledgements

Financial support is gratefully acknowledged from the National Social Science Foundation of China (No. 19ZDA110).

Funding

National Planning Office of Philosophy and Social Science, 19ZDA110, Guangliang Ye.

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Authors and Affiliations

  1. University of Wisconsin-Milwaukee, Milwaukee, WI, USA

    John S. Heywood

  2. Renmin University of China, Bei**g, China

    Dongyang Li

  3. Hainan University, 58 Renmin Avenue, Haikou, 570228, China

    Guangliang Ye

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  1. John S. Heywood
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  2. Dongyang Li
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Correspondence to Guangliang Ye.

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Heywood, J.S., Li, D. & Ye, G. Mixed duopoly under hotelling with convex production costs. Ann Reg Sci 69, 487–510 (2022). https://doi.org/10.1007/s00168-022-01144-8

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