Abstract
Hamilton et al. (Regional Science and Urban Economics 19:87–102, 1989) claim that under certain conditions Cournot spatial discrimination always yields agglomeration of firms. However, I provide a counter-example that shows that this claim is not correct.
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Notes
See Puu (2008) for the derivation of the hyperbolic demand function from standard utility functions. The hyperbolic demand has been adopted for example in “rent-seeking” games [Szidarovsky and Okuguchi (1997); Chiarella and Szidarovsky (2002)], equilibrium (in)stability models [Puu (1991, 2008)] and differential oligopoly models [Lambertini (2010); Lamantia (2011)]. Colombo (2014) considers equilibrium locations under the hyperbolic demand function in the case of linear transportation costs. He considers both the case of Bertrand and Cournot competition. The present note is related to Hamilton et al. (1989) generalization for the case of quadratic transportation costs. While under linear transportation costs agglomeration of firms never emerges in equilibrium [Colombo (2014)], under quadratic transportation costs both dispersion and agglomeration of firms may emerge in equilibrium (see later).
This assumption guarantees that in equilibrium each consumer has a non-negative utility even when the hyperbolic demand function is derived from a log-linear utility function as: \(U(Q)=\ln [Q]+y\), where \(y\) is the numeraire good. Indeed, under the above parameter assumption, the quantity served at each location is never less than one.
By running \(\frac{\partial \Pi _A }{\partial x_A }\), then substituting \(x_B =1-x_A \), and finally equating to zero, we get the first-order conditions in the case of symmetric locations of firms, that is:
\(\frac{\sqrt{t\Gamma } }{2}\left[ {\frac{\sqrt{t} [16c^2+tc(11-28x_A +28x_A ^2)+t^2\Gamma (2-3x_A +3x_A ^2)]}{(4c+t\Gamma )[2c+t-2tx_A (1-x_A )]^2}-\frac{[8c+t\Gamma \arctan \sqrt{t /{(4c+t\Gamma )}} ]}{\sqrt{(4c+t\Gamma )^3} }} \right] =0\), where \(\Gamma \equiv (1-2x_A )^2\). As the first-order-conditions involve transcendental equations, an analytical solution is impossible to derive. Therefore, we need to adopt numerical simulations.
The second-order condition and the global-stability condition [Sydsaeter and Hammond (1995); Liang et al. (2012)] are also verified numerically. Due to symmetry, I plot only Firm \(A\)’s location. In Fig. 1 it is also reported the maximum admissible level of parameter \(c\), given the value of \(t\) and the restriction \(c<{(1-t)} / 2\).
Along these lines, it can be argued that the incentive to locate closer to each other in the space is stronger under concave demand functions than under linear or convex demand functions.
The non-monotonic impact of the marginal costs is subtle. When \(c\) increases, the quantity served by Firm \(A\) to closer (farther) consumers diminishes (increases), as \(\frac{\partial q_{I,x} *}{\partial c}=\frac{-2c+T_I (x)-3T_{-I} (x)}{(2c+T_I (x)+T_{-I} (x))^3}\), which is positive (negative) when \(T_I (x)\ge (\le )2c+3T_{-I} (x)\). Therefore, Firm \(A\) has a lower incentive to move to the right to increase the quantity at locations \(x\ge x_A \), but at the same time the locations \(x\ge x_A \) are now more relevant in terms of overall transportation costs. For low levels of \(c\), the former effect prevails and greater dispersion emerges, whereas for high levels of \(c\), the latter effect prevails and lower dispersion emerges.
In traditional articles of Cournot spatial discrimination as Hamilton et al. (1989) and Anderson and Neven (1991), the equilibrium locations of the firms are mainly explained by using the transportation cost argument. For example, [Hamilton et al. (1989), p. 92] claim that “...each firm is located so as to minimize transportation costs associated with its sales patters. No benefits from differentiated locations are therefore obtained” [a similar explanation is provided by Anderson and Neven (1991), p. 800]. However, this argument may be not sufficient under convex demand functions, as under convex demand the strategic effect at the locations near to the left endpoint of the segment becomes stronger, and this may determine a dispersed equilibrium.
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Colombo, S. A comment on the locations of firms in Cournot spatial discrimination models. Lett Spat Resour Sci 8, 119–124 (2015). https://doi.org/10.1007/s12076-014-0117-z
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DOI: https://doi.org/10.1007/s12076-014-0117-z