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A comment on mixed oligopoly spatial model: the non-uniform consumer distribution

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Abstract

We allow a general (non-uniform) consumer distribution in a mixed duopoly Hotelling model. We characterize respective conditions under which socially optimal pricing and location are achieved. We find that, while the condition for socially optimal pricing is fairly general, that for a socially optimal location is much more restrictive. Consequently, using a public firm to regulate a market may continue to yield a socially optimal price, but not the socially optimal location. This implies that privatization of the public firm may be desirable.

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Notes

  1. Some other studies on mixed oligopoly use a different approach, such as mixed quantity duopoly model with endogenous timing (Amir and De Feo 2014), vertical differentiation model (Grilo 1994), and circular city model (Matsushima and Matsumura 2003).

  2. Other assumptions are the same as Cremer et al. (1991): \(x_1 \le x_2\), zero marginal cost, quadratic transport cost, and covered market.

  3. A general definition of \(\rho \)-concavity is provided in Caplin and Nalebuff (1991) p. 29. Our one-dimensional Hotelling model is a special case with \(\rho =-\frac{1}{2}\). Note that \(\rho \)-concave is a weaker condition than log-concave assumed in Anderson et al. (1997).

  4. The proof that T is quasi-convex in \(p_1\) can be done directly from the definition of quasi-convex. Details are available upon request.

  5. In C&N, the two firms’ objective functions (maximizing profit) are quasi-concave in firms’ own prices. In our model, both firms’ objective functions are still quasi-concave in their own prices, despite that the public firm maximizes welfare instead of its profit. Theorem 2 in C&N continues to hold in our case. One may write a proof similar to the proof of Theorem 2 in C&N. We leave this as an exercise to interesting readers.

  6. This assumption is a quite common in location model, see Liu and Shuai (2013), and papers cited there. One exception is Anderson et al. (1997), in which consumer distribution is allowed be asymmetric. However, in Anderson et al. (1997), both firms are private firms, their objective functions are symmetric. In our mixed duopoly model, given firms are already asymmetric, introducing another asymmetry in consumer distribution will make the analysis too complex and is beyond the scope of this paper. Nevertheless, it is an interesting direction for future research.

  7. The proof are the same as Anderson et al. (1997), (section 5 on p. 123), and we skip it.

  8. The first inequality comes from the definition of \(\rho \)-concave and the first equality comes from the symmetry assumption.

  9. Liu and Shuai (2013) distinguish these two effects in a private duopoly model. Interesting readers may refer their paper for more details.

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Acknowledgments

We would like to thank the Editor (Nicholas Yannelis) and one anonymous referee whose comments improved this paper substantially.

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Authors and Affiliations

  1. Wenlan School of Business, Zhongnan University of Economics and Law, 182 Nanhu Ave., East Lake High-tech Development Zone, 430073, Wuhan, People’s Republic of China

    Jie Shuai

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  1. Jie Shuai
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Shuai, J. A comment on mixed oligopoly spatial model: the non-uniform consumer distribution. Econ Theory Bull 5, 57–63 (2017). https://doi.org/10.1007/s40505-016-0101-4

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  • DOI: https://doi.org/10.1007/s40505-016-0101-4

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