Abstract
I analyze the relation between market size and number of firms when an endogenous number of firms chooses the market strategy and (simultaneously or sequentially) an R&D investment. I generalize the linear Cournot model with an endogenous cost-reducing activity and show that, as long as exogenous fixed costs are positive, the market structure is naturally characterized by an inverted-U relation between market size and number of firms, in line with the celebrated hypothesis of Sutton. However, the increase of the market size reduces the prices and expands individual investment and production exactly as in endogenous market structure only with exogenous fixed costs.
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Notes
See Etro (2009) for a wide discussion of the macroeconomic implications.
See Vives (2008) for a recent generalization.
As well known, their analysis was about monopolistic competition. Strategic interactions add a standard competition effect, but the elasticity of the equilibrium number of firms \(n\) with respect to the market size \(S\) can still be above or below unity.
Dasgupta and Stiglitz (1980) assumed homogenous goods with a constant elasticity of demand \(\epsilon >0\), which insures size neutrality again. Our case of hyperbolic demand corresponds to their example with \(\epsilon =1\).
Indeed, in both the Dasgupta-Stiglitz model and the Sutton model, the introduction of an exogenous fixed cost would lead to a general result: an increase in the size of the market would increase the number of firms with an upper bound given respectively by (2) and (4). See Senyuta and Žigić (2012) for an interesting extension of the Sutton approach to the case of R&D spillovers showing that \(n^{*}\) increases with the spillovers.
One can verify that \(S<2\theta (n^{*}+1)\) for any \(S<4\theta \).
This imposes an upper bound on size:
$$\begin{aligned} S<\bar{S}=\frac{4\theta }{1+\frac{F}{\theta c^{2}}} \end{aligned}$$Since \(\bar{S}<4\theta \), what follows requires only the assumption \(S<\bar{S}\).
The idea of this bound on the number of firms, and the same formula, was present already in Dasgupta and Stiglitz (1980).
As a referee has noticed, the inverted-U relation emerges only for market sizes below the cut-off \(\bar{S}\). However, beyond this cut-off, all firms invest the maximum amount in R&D and the number of firms is increasing in the market size. Therefore, the unrestricted model strengthens non-monotonicity of the number of firms and delivers three different regimes when \(S\) increases: an initial increase in the number of firms, an intermediate phase of market concentration, and a final phase characterized again by an increase in the number of firms.
The investments of the average firm and of the least productive firms are decreasing in the number of firms as in the above model with homogenous firms, but an inverted-U curve between number of firms and investment can emerge for the most productive firms.
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Acknowledgments
I am grateful to the Editor Giacomo Corneo and to two anonymous referees, to Paolo Bertoletti, Alberto Bucci, Sergei Izmalkov, Sergey Kokovin, Dmitry Krutikov, Jacques Thisse, Evgeny Zhelobodko, Krešimir Žigić and other participants to seminars at the Higher School of Economics in St. Petersburg, and the New School of Economics in Moscow and the Cresse Conference in Corfù for useful comments.
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Etro, F. Some thoughts on the Sutton approach. J Econ 112, 99–113 (2014). https://doi.org/10.1007/s00712-013-0349-1
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DOI: https://doi.org/10.1007/s00712-013-0349-1