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The Annals of Regional Science

Erschienen in:

01.12.2013 | Original Paper

Spatial Cournot competition with non-extreme directional constraints

verfasst von: Stefano Colombo

Erschienen in: The Annals of Regional Science | Ausgabe 3/2013

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Abstract

The circular city model and the linear city model are extended to allow for asymmetric directional transportation costs. A two-stage location-then-quantity model is proposed. We show that in the circular city model, maximal dispersion arises in equilibrium, while in the linear city model, the unique equilibrium is represented by both firms agglomerating in a non-central point of the segment.

Vorheriger Artikel The national and regional effects of fiscal decentralisation in China

Nächster Artikel Demand creation and location: a variable consumer-distribution approach in spatial competition

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Here and after, we shall refer to asymmetric transportation costs to describe transportation costs which differ across directions, and not across firms.

We are grateful to an anonymous reviewer for suggesting this example.

This example has been provided by another reviewer.

Other examples of directional constraints regard TV broadcasting times, bus/air schedules, one-way streets, highways, flows of natural resources as water and oil. For other examples, see Lai (2001).

Our model coincides with the traditional symmetric circular city model when \(r=t\) (Pal 1998). Instead, if the movement in one direction is prohibited (extreme directional constraints), it coincides with Sun (2010).

Sun (2010) also allows for the possibility that the two firms have opposite directional constraints, in the sense that one firm can only move in the clockwise direction, while the rival can only move in the anticlockwise direction. In this case, it is shown that the unique equilibrium is characterized by firms agglomerating in one point of the circle. In the present article, the case of opposite non-extreme directional constraints is not considered.

Our model coincides with the traditional symmetric linear city model when \(r=t\) (Hamilton et al. 1989). The case where the movement in one direction is prohibitively costly can be found in Colombo (2011).

When \({\partial \bar{{\Pi }}_A }/{\partial a}=0\) is solved with respect to \(a\), there are two roots: \(a*(b)\) and: \(\hat{{a}}(b)=\frac{2r^{2}-r-t-brt-bt^{2}-\sqrt{(r-2r^{2}+t+brt+bt^{2})^{2}+r(2r^{2}-rt-3t^{2})[2-r(1+2b-b^{2})+tb^{2}]}}{2r^{2}-rt-3t^{2}}\). However, \(\hat{{a}}(b)\) can be excluded as \(\hat{{a}}(b)\notin [0,b]\).

Similarly, when \({\partial \bar{{\Pi }}_B }/{\partial b}=0\) is solved with respect to \(b, \) there are two roots: \(b*(a)\) and \(\hat{{b}}(a)=\frac{2r^{2}-r-t+art+ar^{2}-\sqrt{[r^{2}(2+a)-t+r(at-1)]^{2}-(3r^{2}+rt-2t^{2})[r^{2}(1+2a)+a^{2}t^{2}+r(a^{2}t-2)]}}{3r^{2}+rt-2t^{2}}\). However, \(\hat{{b}}(a)\) can be excluded as \(\hat{{b}}(a)\notin [a,1]\).

It can be easily verified that the second-order conditions are satisfied in equilibrium.

Note that when \(b=1/2\), only Configuration 2 can arise.

Andree K (2011) Product differentiation and spatial agglomeration in a unidirectional Cournot model. Lett Sp Resour Sci 4:167–172CrossRef

Cancian M, Bills A, Bergstrom T (1995) Hotelling location problems with directional constraints: an application to television news scheduling. J Ind Econ 43:121–124CrossRef

Colombo S (2009) The unidirectional Hotelling model with spatial price discrimination. Econ Bull 29:3031–3040

Colombo S (2011) Spatial price discrimination in the unidirectional Hotelling model with elastic demand. J Econ 102:157–169CrossRef

Hamilton JH, Thisse JF, Weskamp A (1989) Spatial discrimination: Bertrand vs. Cournot in a model of location choice. Reg Sci Urban Econ 19:87–102CrossRef

Kharbach M (2009) A unidirectional Hotelling model. Econ Bull 29:1814–1819

Lai FC (2001) Sequential location in directional markets. Reg Sci Urban Econ 31:535–546CrossRef

Nilssen T (1997) Sequential location when transportation costs are asymmetric. Econ Lett 54:191–201CrossRef

Pal D (1998) Does Cournot competition yields spatial agglomeration? Econ Lett 60:49–53CrossRef

Sun CH (2010) Spatial Cournot competition in a circular city with directional delivery constraint. Ann Reg Sci 45:273–289CrossRef

Sun CH (2012) Sequential location in a discrete directional market with three or more players. Ann Reg Sci 82:569–580

Titel: Spatial Cournot competition with non-extreme directional constraints
verfasst von: Stefano Colombo
Publikationsdatum: 01.12.2013
Verlag: Springer Berlin Heidelberg
Erschienen in: The Annals of Regional Science / Ausgabe 3/2013
Print ISSN: 0570-1864
Elektronische ISSN: 1432-0592
DOI: https://doi.org/10.1007/s00168-013-0560-6

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