Abstract
Guo and Lai (Ann Reg Sci 52(1):309–324, 2014) argue that, under quadratic transport costs, when two offline firms compete with one online firm, the two offline firms locate in such a way that they occupy unconnected regions in the market. However, we offer a counterexample to show that their provided condition is not sufficient to support existence of subgame perfect equilibria, because location deviations to different market structures are not taken into account.
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Notes
In Fig. 1 (market structures A and B), we consider \( p_{1} = p_{2} \) as the two traditional retailers are symmetric.
Note that this problem does not arise under market structure A, because in this case the locations of the traditional retailers are not relevant for the equilibrium prices.
The locations of the traditional firms are irrelevant for the equilibrium profits in market structure A. However, the location of Firm 2 is relevant in the case of a deviation of Firm 1 from the putative equilibrium.
Note that \( 1 > n_{2}^{R} > n_{12} > n_{1}^{L} > 0 \) implies that it cannot be \( 1 > n_{2}^{R} > n_{2}^{L} > n_{1}^{R} > n_{1}^{L} > 0 \).
All numerical examples we simulated through the software Mathematica confirm that a deviation from market structure A to induce market structure B is always profitable. Instead, Guo and Lai (2014) only consider one possible deviation, with Firm 1 locating at the left endpoint (\( x_{1} = 0 \)), and they show that this particular deviation is not profitable.
Indeed, when there is no online competition, the profits of each traditional retailer are larger.
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Colombo, S., Hou, Z. On spatial competition with quadratic transport costs and one online firm. Ann Reg Sci 63, 241–247 (2019). https://doi.org/10.1007/s00168-019-00934-x
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DOI: https://doi.org/10.1007/s00168-019-00934-x