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The Annals of Regional Science

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01-09-2014 | Special Issue Paper

A note on networks of collaboration in multi-market oligopolies

Authors: Pascal Billand, Christophe Bravard, Subhadip Chakrabarti, Sudipta Sarangi

Published in: The Annals of Regional Science | Issue 2/2014

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Abstract

In this note, we extend the Goyal and Joshi’s model of collaboration networks in oligopoly to multi-market situations. We examine the incentive of firms to form links and the architectures of the resulting equilibrium networks in this setting. We then present some results on efficient networks.

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Considering that firms can form links on one market only is admittedly restrictive. However, it allows us to deal with situations where products innovation is a strategic issue for only one market, and there is little to gain by forming collaborative links that will increase the quality of the product on the second market. Besides this, restrictive assumption simplifies the analysis and allows us to provide a first insight into the impact of multi-market competition on the collaborative networks.

We thank an anonymous referee for pointing out this general characteristic of efficient networks in our framework.

Formally, let \(\pi _{i} (q_{1},\ldots ,q_{i},\ldots ,q_{n}, Q_{1},\ldots , Q_{i},\ldots ,Q_{n})\) be the profit function of firm \(i\). Competitors’ products are strategic substitutes on market \(1\) whenever \(\frac{\partial ^{2} \pi _{i} }{\partial q_{i} \partial q_{j} } <0\). Let \(C_{i}(q_{i},Q_{i})\) be the cost function of a firm \(i\). There are diseconomies (economies) of scope across markets whenever \(\frac{\partial ^{2} C_{i}(q_{i},Q_{i}) }{\partial q_{i} \partial Q_{i} } > (<)\ 0\).

This definition comes from Belhaj et al. (2013).

Our definition can be seen as a generalization of stable networks given in GJ (2003), since in GJ no firm has an incentive to delete either one of its links, or all its links in a stable network.

Goyal and Joshi (2003) do not provide efficiency results for the one-market case stating that this is a hard problem.

Detailed calculations are available in the working paper version of this paper: http://mpra.ub.uni-muenchen.de/28188/1/MPRA_paper_28188.pdf.

Belhaj M, Bervoets S, Deroïan F (2013) Network design under local complementarities. Working Papers halshs-00796487

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Title: A note on networks of collaboration in multi-market oligopolies
Authors: Pascal Billand
Christophe Bravard
Subhadip Chakrabarti
Sudipta Sarangi
Publication date: 01-09-2014
Publisher: Springer Berlin Heidelberg
Published in: The Annals of Regional Science / Issue 2/2014
Print ISSN: 0570-1864
Electronic ISSN: 1432-0592
DOI: https://doi.org/10.1007/s00168-013-0577-x

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