Mixed duopoly in a Hotelling framework with cubic transportation costs

Abstract

We examine a two-stage location-price model of a mixed duopoly where a private profit-maximizing firm competes with a public welfare-maximizing firm in a Hotelling-type framework. A noteworthy result in this model is that, with quadratic transportation costs, which has become the usual assumption in the literature, the socially optimal locations are obtained in equilibrium. We show here that under the alternative assumption of cubic transportation costs this result no longer holds: equilibrium locations are socially suboptimal. We find that just as in the case of linear transportation costs, previously studied in the literature, for some locations there is price equilibrium in the second stage of the game and for other locations there is not. But, in contrast with such a case, there is a location pair for which there is price equilibrium in the second stage of the game and neither firm has an incentive to marginally change its location. We also find that, in contrast with the case of quadratic transportation costs, this location pair is not socially optimal.

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Notes

  1. 1.

    Information on these banks can be found at www.cgd.pt, www.bncr.fi.cr, www.corporativo.bancoestado.cl and https://www.the-brow.com/ for the cases of Portugal, Costa Rica, Chile and Uruguay, respectively. Barros and Modesto (1999) provide a detailed analysis of the Portugal banking system in the 90 s.

  2. 2.

    This literature has expanded to study a variety of issues, such as partial privatization (Matsumura 1998; Jain and Pal 2012), mergers (Bárcena-Ruiz and Garzón 2003; Artz et al. 2009; Méndez-Naya 2008) and capacity choice (Nishimori and Ogawa 2004; Lu and Poddar 2005), to name but a few.

  3. 3.

    See Cremer et al. (1991) and Matsumura and Matsushima (2004).

  4. 4.

    Modelling the public firm as maximizing social welfare is typical in the mixed oligopoly literature. For an alternative approach see Bennett and La Manna (2012) and Zhang and Zhong (2015).

  5. 5.

    D’Aspremont et al. (1979) show that in a private duopoly with quadratic transportation costs firms locate too far away from each other from a social point of view, at the edges of the line.

  6. 6.

    Hotelling (1929) original paper refers to this interpretation with the example of the sweetness of cider. For an exposition of the Hotelling model and alternative interpretations of it see also Tirole (1988) and Waterson (1994).

  7. 7.

    The literature of transportation costs takes into account not only the average travel time, but also its variability. See for example Lam and Small (2001), Small et al. (2005) and Asensio and Matas (2008).

  8. 8.

    An extreme example would be the choice of an obstetric unit or birth center.

  9. 9.

    This trade-off also appears in the private duopoly, as explained in Tirole (1988), and has been previously mentioned in the location choice of the private firm in a mixed duopoly in, for instance, Benassi et al. (2017).

  10. 10.

    Which in turns leads to the non-existence of equilibrium.

  11. 11.

    We assume for simplicity that firms choose different locations. Just as in Lu (2006) there would be infinite price equilibria in the second stage if they chose the same location. As we explain below, we restrict our attention to a subset of first-stage locations for which there is a unique price equilibrium in the second stage.

  12. 12.

    To see that Eq. (1) has a solution and it is unique, notice that the continuous function \(d\left( y \right) = p_{0} + \left| {y - x_{0} } \right|^{3} - p_{1} - \left| {y - x_{1} } \right|^{3}\): (1) approaches \(- \infty\) as y approaches \(- \infty\), (2) approaches \(+ \infty\) as y approaches \(+ \infty\), and (3) is strictly increasing.

  13. 13.

    We present in the appendix \(q_{0}\) written as an explicit function of \(p_{0}\) and \(p_{1}\).

  14. 14.

    As evidenced by the closely related fact that some cities have severely limited or completely ban most vehicles from the city center, putting special attention in the most polluting vehicles.

  15. 15.

    Also, for historic reasons, the city center in many places has an outstanding cultural value, which makes certain businesses incompatible with the preservation of its character.

  16. 16.

    See also Hinloopen and Van Marrewijk (1999) and Posada and Straume (2004).

  17. 17.

    The literature on private oligopolies uses a slightly different definition for equilibrium in the location-price game. It considers the whole set E of first-stage locations for which there is unique price equilibrium in the second stage, instead of considering only a subset of E as we do here. It then considers the zero-relocation locus (in our case is \(\frac{{\partial \Pi_{1} }}{{\partial x_{1} }} = \frac{\partial TC}{{\partial x_{0} }} = 0\)). The intersection of this locus with E defines an equilibrium (Economides 1986). It is easy to see that our results do not change if we use this alternative definition. Notice that with linear transportation costs this alternative approach underscores that whenever firms choose locations in E, they have a tendency to move away from this zone and into the zone where there is no equilibrium.

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Acknowledgements

I thank two anonymous referees for helpful comments and suggestions and J. Fernández for useful discussions and suggestions (in particular for footnote 12).

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Appendix

Appendix

Proof of Proposition 1

Let \(p^{*} = \frac{{3\left( {2 - x_{0} - x_{1} } \right)\left( {x_{1} - x_{0} } \right)^{2} }}{4}\) be the candidate equilibrium common price in Eq. (9) and \(\Pi_{1}^{*} = \frac{{3\left( {2 - x_{0} - x_{1} } \right)^{2} \left( {x_{1} - x_{0} } \right)^{2} }}{8}\) be the candidate equilibrium firm 1’s profits as given in Eq. (11). We now examine if given \(p_{0} = p^{*}\), firm 1 can increase its profits above \(\Pi_{1}^{*}\) by choosing a price \(p_{1} \ne p^{*}\).

(1) if firm 1 chooses a price in the interval \(p_{0} - \left( {x_{1}^{3} - x_{0}^{3} } \right) \le p_{1} \le p_{0} - \left( {x_{1} - x_{0} } \right)^{3} ,\) then \(q_{0} \in \left[ {0 ,x_{0} } \right]\) and we have

$$p_{0} + \left( {x_{0} - q_{0} } \right)^{3} = p_{1} + \left( {x_{1} - q_{0} } \right)^{3}$$
(16)

from where

$$\frac{{\partial q_{0} }}{{\partial p_{1} }} = \frac{1}{{3\left( {x_{1} - q_{0} } \right)^{2} - 3\left( {x_{0} - q_{0} } \right)^{2} }}$$

and therefore:

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = \frac{{ - p_{1} }}{{3\left( {x_{1} - q_{0} } \right)^{2} - 3\left( {x_{0} - q_{0} } \right)^{2} }} + 1 - q_{0}$$
(17)

Replacing \(p_{1}\) from (16) into (17) and using \(p_{0} = p^{*}\) we obtain

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = \frac{{36q_{0}^{2} - 24\left( {x_{0} + x_{1} + 1} \right)q_{0} + 7x_{1}^{2} + 4x_{0} x_{1} + 6x_{1} + x_{0}^{2} + 18x_{0} }}{{12\left( { - 2q_{0} + x_{0} + x_{1} } \right)}}$$
(18)

The denominator in the RHS of (18) is positive because \(q_{0} \le x_{0} < x_{1}\). The numerator is also positive because (a) it is decreasing in \(q_{0 }\) (its derivative with respect to \(q_{0}\) is equal to \(72q_{0} - 24\left( {x_{0} + x_{1} + 1} \right) \le 72x_{0} - 24\left( {x_{0} + x_{1} + 1} \right) = - 24\left( {x_{1} - x_{0} } \right) - 24\left( {1 - x_{0} } \right) < 0\)) and (b) it is positive when \(q_{0} = x_{0}\) (it is then equal to \(\left( {x_{1} - x_{0} } \right)\left( { - 13x_{0} + 7x_{1} + 6} \right) > 0).\)

It follows that \(\Pi_{1}\) is increasing over the whole range \(p_{0} - \left( {x_{1}^{3} - x_{0}^{3} } \right) \le p_{1} \le p_{0} - \left( {x_{1} - x_{0} } \right)^{3}\). Let \(\Pi_{1}^{a}\) be firm 1’s profits when \(p_{1} = p_{0} - \left( {x_{1} - x_{0} } \right)^{3}\)-which implies \(q_{0} = x_{0}\)- and \(p_{0} = p^{*}\). Then

$$\Pi_{1}^{a} = \left( {\frac{{3\left( {2 - x_{1} - x_{0} } \right)\left( {x_{1} - x_{0} } \right)^{2} }}{4} - \left( {x_{1} - x_{0} } \right)^{3} } \right)\left( {1 - x_{0} } \right)$$

If we substract from these profits the candidate equilibrium profits \(\Pi_{1}^{*}\) we obtain, after some simplifications,

$$\Pi_{1}^{a} - \Pi_{1}^{*} = \frac{{\left( {x_{1} - x_{0} } \right)^{3} \left( { - 3x_{1} + 5x_{0} - 2} \right)}}{8} < 0$$

which implies that firm 1 will not deviate to such a price, and will thus neither deviate to any price \(p_{1}\) with \(p_{0} - \left( {x_{1}^{3} - x_{0}^{3} } \right) \le p_{1} \le p_{0} - \left( {x_{1} - x_{0} } \right)^{3}\).

(2) If firm 1 chooses a price in the interval \(p_{0} - \left( {x_{1} - x_{0} } \right)^{3} \le p_{1} \le p_{0} + \left( {x_{1} - x_{0} } \right)^{3}\), then \(q_{0} \in \left[ {x_{0} ,x_{1} } \right]\) and we will have:

$$p_{0} + \left( {q_{0} - x_{0} } \right)^{3} = p_{1} + \left( {x_{1} - q_{0} } \right)^{3}$$
(19)

and thus

$$\frac{{\partial q_{0} }}{{\partial p_{1} }} = \frac{1}{{3\left( {q_{0} - x_{0} } \right)^{2} + 3\left( {x_{1} - q_{0} } \right)^{2} }}$$

Therefore:

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = \frac{{ - p_{1} }}{{3\left( {q_{0} - x_{0} } \right)^{2} + 3\left( {x_{1} - q_{0} } \right)^{2} }} + 1 - q_{0}$$
(20)

Replacing \(p_{1}\) from (19) into (20) and using \(p_{0} = p^{*}\) we obtain, after some simplifications:

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = \frac{{\left[ {2q_{0} - x_{0} - x_{1} } \right]\left[ {16q_{0}^{2} - \left( {10\left( {x_{0} + x_{1} } \right) + 12} \right)q_{0} + \left( {7x_{1} - 10x_{0} + 6} \right)x_{1} + 7x_{0}^{2} + 6x_{0} } \right]}}{{ - 12\left( {q_{0} - x_{0} } \right)^{2} - 12\left( {x_{1} - q_{0} } \right)^{2} }}$$

Therefore, \(\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = 0\) if \(q_{0} = \frac{{x_{0} + x_{1} }}{2}\), which implies \(p_{1} = p_{0} = p^{*} .\) Notice also that, from (20):

$$\frac{{\partial^{2} \Pi_{1} }}{{\partial p_{1}^{2} }} = \frac{{ - \left( {3\left( {q_{0} - x_{0} } \right)^{2} + 3\left( {x_{1} - q_{0} } \right)^{2} } \right) + 6p_{1} \left( {\left( {q_{0} - x_{0} } \right) - \left( {x_{1} - q_{0} } \right)} \right)\frac{{\partial q_{0} }}{{\partial p_{1} }}}}{{\left( {3\left( {q_{0} - x_{0} } \right)^{2} + 3\left( {x_{1} - q_{0} } \right)^{2} } \right)^{2} }} - \frac{{\partial q_{0} }}{{\partial p_{1} }}$$

and that this second derivative evaluated at \(p_{1} = p_{0} = p^{*}\) (and thus at \(q_{0} = \frac{{x_{0} + x_{1} }}{2}\)) is negative:

$$\frac{{\partial^{2} \Pi_{1} }}{{\partial p_{1}^{2} }} = \frac{ - 4}{{3\left( {x_{1} - x_{0} } \right)^{2} }} < 0$$

Therefore, \(p_{1} = p^{*}\) is a local maximum, and it is also the only value where \(\frac{{\partial \Pi_{1} }}{{\partial p_{1} }}\) vanishes (in the segment \(q_{0} \in \left[ {x_{0} ,x_{1} } \right]\)) unless the following equation has roots in \(q_{0} \in \left[ {x_{0} ,x_{1} } \right]:\)

$$16q_{0}^{2} - \left( {10\left( {x_{0} + x_{1} } \right) + 12} \right)q_{0} + \left( {7x_{1} - 10x_{0} + 6} \right)x_{1} + 7x_{0}^{2} + 6x_{0} = 0$$
(21)

Equation (21) is a second degree equation in \(q_{0}\) with discriminant equal to

$$12\left( { - 29x_{1}^{2} + \left( {70x_{0} - 12} \right)x_{1} - 29x_{0}^{2} - 12x_{0} + 12} \right)$$
(22)

If the discriminant in (22) is negative \(\frac{{\partial \Pi_{1} }}{{\partial p_{1} }}\) will only vanish at \(q_{0} = \frac{{x_{0} + x_{1} }}{2}\) (which corresponds to \(p_{1} = p^{*}\)) in the segment \(q_{0} \in \left[ {x_{0} ,x_{1} } \right].\) If this discriminat is positive, then Eq. (21) will have two roots:

$$q_{0}^{H} = \frac{{5x_{0} + 5x_{1} + 6 + \sqrt 3 \sqrt { - 29x_{1}^{2} + \left( {70x_{0} - 12} \right)x_{1} - 29x_{0}^{2} - 12x_{0} + 12} }}{16}$$
(23)
$$q_{0}^{L} = \frac{{5x_{0} + 5x_{1} + 6 - \sqrt 3 \sqrt { - 29x_{1}^{2} + \left( {70x_{0} - 12} \right)x_{1} - 29x_{0}^{2} - 12x_{0} + 12} }}{16}$$
(24)

But, \(q_{0}^{H} > x_{1}\), since the positiveness of the discriminant in (22) implies that (a) \(q_{0}^{H} > \frac{{5x_{0} + 5x_{1} + 6}}{11}\), and (b) \(x_{1} < \frac{{5x_{0} + 6}}{11}\), because this discriminat is strictly decreasing in \(x_{1}\) and it is negative when \(x_{1} = \frac{{5x_{0} + 6}}{11}\). These two facts imply \(q_{0}^{H} - x_{1} \ge \frac{{5x_{0} + 5x_{1} + 6}}{11} - x_{1} = \frac{{5x_{0} - 11x_{1} + 6}}{11} > 0\). Thus, \(q_{0}^{L}\) is the only possible root additional to \(q_{0} = \frac{{\left( {x_{0} + x_{1} } \right)}}{2}\) in the range \(q_{0} \in \left[ {x_{0} ,x_{1} } \right]\), which cannot therefore be a local maximum (since \(p_{1} = p^{*}\) is a local maximum and \(\frac{{\partial \Pi_{1} }}{{\partial p_{1} }}\) only vanishes at \(p^{*}\) and \(p^{L}\) (associated to \(q_{0}^{L}\)) then \(\Pi_{1}\) is strictly decreasing in \(p_{1} \in \left( {p^{*} , p^{L} } \right)\) if \(p^{*} < p^{L}\) and, similarly, it is strictly increasing in \(p_{1} \in \left( {p^{L} , p^{*} } \right)\) if \(p^{L} < p^{*}\)).

Since we proved above that firm 1 will not deviate to a price \(p_{1}\) such that \(q_{0} = x_{0}\), the only price that remains to be considered in this interval is \(p_{1} = p_{0} + \left( {x_{1} - x_{0} } \right)^{3}\) which corresponds to \(q_{0} = x_{1}\). If \(p_{0} = p^{*}\) and firm 1 chooses such a price, firm 1’s profits will be equal to

$$\Pi_{1}^{b} = \left( {\frac{{3\left( {2 - x_{0} - x_{1} } \right)\left( {x_{1} - x_{0} } \right)^{2} }}{4} + \left( {x_{1} - x_{0} } \right)^{3} } \right) \left( {1 - x_{1} } \right)$$

Subtracting the candidate equilibrium profits \(\Pi_{1}^{*}\) from \(\Pi_{1}^{b}\) we obtain:

$$gb\left( {x_{0} ,x_{1} } \right) = \Pi_{1}^{b} - \Pi_{1}^{*} = \left( {\frac{{3\left( {2 - x_{0} - x_{1} } \right)\left( {x_{1} - x_{0} } \right)^{2} }}{4} + \left( {x_{1} - x_{0} } \right)^{3} } \right)\left( {1 - x_{1} } \right) - \frac{{3\left( {2 - x_{0} - x_{1} } \right)^{2} \left( {x_{1} - x_{0} } \right)^{2} }}{8}$$

with

$$\frac{{\partial gb\left( {x_{0} ,x_{1} } \right)}}{{\partial x_{1} }} = \frac{{ - \left( {x_{1} - x_{0} } \right)^{2} \left( {10x_{1} - 7x_{0} - 3} \right)}}{4}$$

For any given \(x_{0}\),\(gb\left( {x_{0} ,x_{0} } \right) = 0\), \(gb\left( {x_{0} ,x_{1} } \right)\) reaches a maximum at \(x_{1} = \frac{{\left( {7x_{0} + 3} \right)}}{10}\) with \(gb\left( {x_{0} ,\frac{{\left( {7x_{0} + 3} \right)}}{10}} \right) = \frac{{27\left( {1 - x_{0} } \right)^{4} }}{16,000} > 0\), and \(gb\left( {x_{0} ,1} \right) = \frac{{ - 3\left( {1 - x_{0} } \right)^{4} }}{8} < 0\). Since \(gb\left( {x_{0} ,x_{1} } \right)\) is strictly increasing in \(x_{1}\) for \(x_{0} < x_{1} < \frac{{\left( {7x_{0} + 3} \right)}}{10}\), strictly decreasing in \(x_{1}\) for \(\frac{{\left( {7x_{0} + 3} \right)}}{10} < x_{1} < 1\), and \(gb\left( {x_{0} ,\frac{{\left( {3x_{0} + 2} \right)}}{5}} \right) = 0\), it follows that, when \(\frac{{3x_{0} + 2}}{5} \le x_{1} \le 1\), \(gb\left( {x_{0} ,x_{1} } \right) \le 0,\) and firm 1 does not find this deviation profitable, while for \(x_{1} < \frac{{3x_{0} + 2}}{5}\), \(gb\left( {x_{0} ,x_{1} } \right) > 0\), firm 1 does deviate from the candidate equilibrium price and there does not exist an equilibrium in the second stage of the game.

(3) if firm 1 chooses a price in the interval \(p_{0} + \left( {x_{1} - x_{0} } \right)^{3} \le p_{1} \le p_{0} + \left( {1 - x_{0} } \right)^{3} - \left( {1 - x_{1} } \right)^{3}\), then \(q_{0} \in \left[ {x_{1} ,1 } \right]\) and we have

$$p_{0} + \left( {q_{0} - x_{0} } \right)^{3} = p_{1} + \left( {q_{0} - x_{1} } \right)^{3}$$
(25)

from where

$$\frac{{\partial q_{0} }}{{\partial p_{1} }} = \frac{1}{{3\left( {q_{0} - x_{0} } \right)^{2} - 3\left( {q_{0} - x_{1} } \right)^{2} }}$$

and therefore:

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = \frac{{ - p_{1} }}{{3\left( {q_{0} - x_{0} } \right)^{2} - 3\left( {q_{0} - x_{1} } \right)^{2} }} + 1 - q_{0}$$
(26)

Replacing \(p_{1}\) from (25) into (26) and using \(p_{0} = p^{*}\) we obtain

$$\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = \frac{{ - 36q_{0}^{2} + 24\left( {x_{0} + x_{1} + 1} \right)q_{0} - x_{1}^{2} - 4x_{0} x_{1} - 18x_{1} - 7x_{0}^{2} - 6x_{0} }}{{12\left( {2q_{0} - x_{0} - x_{1} } \right)}}$$

The denominator of \(\frac{{\partial \Pi_{1} }}{{\partial p_{1} }}\) is positive because \(x_{0} < x_{1} \le q_{0}\). \(\frac{{\partial \Pi_{1} }}{{\partial p_{1} }}\) will vanish when the numerator does so, which yields a quadratic equation in \(q_{0}\) with discriminant equal to

$$144 \left( {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} \right)$$
(27)

The discriminant in (27) is strictly decreasing in \(x_{1}\) and it is equal to zero when \(x_{1} = x_{1}^{R}\), with

$$x_{1}^{R} = \frac{{\left( {\sqrt {13} - 2} \right)x_{0} + 5 - \sqrt {13} }}{3} \in \left( {\frac{{3x_{0} + 2}}{5},1} \right)$$

Therefore,

(a) when \(x_{1} > x_{1}^{R}\), the discriminant in (27) is negative and \(\frac{{\partial \Pi_{1} }}{{\partial p_{1} }}\) never vanishes (it is negative). It then suffices to consider the price deviation associated to \(q_{0} = x_{1}\) (which yields profits \(\Pi_{1}^{b}\)). Since \(x_{1}^{R} > \frac{{3x_{0} + 2}}{5}\), we have that \(x_{1} > \frac{{3x_{0} + 2}}{5}\) and thus \(gb\left( {x_{0} ,x_{1} } \right) < 0\), \(\Pi_{1}^{b} < \Pi_{1}^{*}\) and there is equilibrium.

(b) when \(x_{1} \le x_{1}^{R}\) the discriminant in (27) is positive and \(\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = 0\) has the following roots:

$$\begin{aligned} q_{0}^{c} & = \frac{{2x_{1} + 2x_{0} + 2 + \sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} }}{6} \\ q_{0}^{d} & = \frac{{2x_{1} + 2x_{0} + 2 - \sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} }}{6} \\ \end{aligned}$$

Now, from (26)

$$\frac{{\partial^{2} \Pi_{1} }}{{\partial p_{1}^{2} }} = \frac{{ - 3\left( {q_{0} - x_{0} } \right)^{2} + 3\left( {q_{0} - x_{1} } \right)^{2} + p_{1} \left( {6\left( {q_{0} - x_{0} } \right) - 6\left( {q_{0} - x_{1} } \right)} \right)\frac{{\partial q_{0} }}{{\partial p_{1} }}}}{{\left( {3\left( {q_{0} - x_{0} } \right)^{2} - 3\left( {q_{0} - x_{1} } \right)^{2} } \right)^{2} }} - \frac{{\partial q_{0} }}{{\partial p_{1} }}$$
(28)

Also, from (26), \(\frac{{\partial \Pi_{1} }}{{\partial p_{1} }} = 0\) implies that

$$p_{1} = \left( {1 - q_{0} } \right)\left( {3\left( {q_{0} - x_{0} } \right)^{2} - 3\left( {q_{0} - x_{1} } \right)^{2} } \right)$$
(29)

Replacing \(p_{1}\) from (29) and \(q_{0}\) with \(q_{0}^{c}\) in (28) we get:

$$\frac{{\partial^{2} \Pi_{1} }}{{\partial p_{1}^{2} }} = \frac{{ - 3\sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} }}{{\left( {x_{1} - x_{0} } \right)\left( {\sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} - x_{0} - x_{1} + 2} \right)^{2} }}$$

Similarly, when \(q_{0} = q_{0}^{d}\) we get:

$$\frac{{\partial^{2} \Pi_{1} }}{{\partial p_{1}^{2} }} = \frac{{3\sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} }}{{\left( {x_{1} - x_{0} } \right)\left( {\sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} + x_{0} + x_{1} - 2} \right)^{2} }}$$

Therefore \(\Pi_{1}\) reaches a local maximum at \(q_{0}^{c}\) and a local minimum at \(q_{0}^{d}\).

Let \(\Pi_{1}^{c}\) be firm 1’s profits when \(p_{1}\) is such that \(q_{0} = q_{0}^{c}\),

$$\Pi_{1}^{c} = \left( {\frac{{3\left( {2 - x_{0} - x_{1} } \right)\left( {x_{1} - x_{0} } \right)^{2} }}{4} + \left( {q_{0}^{c} - x_{0} } \right)^{3} - \left( {q_{0}^{c} - x_{1} } \right)^{3} } \right)\left( {1 - q_{0}^{c} } \right)$$

and let \(gc\left( {x_{0} ,x_{1} } \right) = \Pi_{1}^{c} - \Pi_{1}^{*}\). \(gc\left( {x_{0} ,x_{1} } \right)\) is the increase in profits when firm 1 chooses \(p_{1}\) such that \(q_{0} = q_{0}^{c}\) instead of the candidate equilibrium price \(p^{*}\).

Since we know that for \(x_{1} < \frac{{3x_{0} + 2}}{5}\) there is no equilibrium (because \(\Pi_{1}^{b} > \Pi_{1}^{*}\)), while for \(x_{1} > x_{1}^{R}\) there is equilibrium, we will focus our attention on the behaviour of \(gc\left( {x_{0} ,x_{1} } \right)\) in the interval \(x_{1} \in \left[ {\frac{{3x_{0} + 2}}{5},x_{1}^{R} } \right]\).

We have that \(gc\left( {x_{0} ,\frac{{3x_{0} + 2}}{5}} \right) > 0\), \(gc\left( {x_{0} ,x_{1}^{R} } \right) < 0\),and we will now prove that \(\frac{{\partial gc\left( {x_{0} ,x_{1} } \right)}}{{\partial x_{1} }} < 0\) for \(x_{1} \in \left[ {\frac{{3x_{0} + 2}}{5},x_{1}^{R} } \right]\). This implies that there exists \(x_{1}^{*} \in \left( {\frac{{3x_{0} + 2}}{5},x_{1}^{R} } \right)\) such that \(\Pi_{1}^{c} = \Pi_{1}^{*}\) if \(x_{1} = x_{1}^{*}\) and \(\Pi_{1}^{c} > \Pi_{1}^{*}\)\(\left( {\Pi_{1}^{c} < \Pi_{1}^{*} } \right)\) if \(x_{1} < x_{1}^{*}\)\(\left( {x_{1} > x_{1}^{*} } \right)\). This in turn implies that there does not exist (there exists) equilibrium if \(x_{1} < x_{1}^{*}\)\(\left( {x_{1} > x_{1}^{*} } \right).\)

We have:

$$\frac{{\partial gc\left( {x_{0} ,x_{1} } \right)}}{{\partial x_{1} }} = \frac{{f\left( {x_{0} ,x_{1} } \right)\sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} + h\left( {x_{0} ,x_{1} } \right)}}{{36\sqrt {3x_{1}^{2} + \left( {4x_{0} - 10} \right)x_{1} - 3x_{0}^{2} + 2x_{0} + 4} }}$$

with

$$f\left( {x_{0} ,x_{1} } \right) = - \left( {34x_{1}^{3} + \left( { - 12x_{0} - 90} \right)x_{1}^{2} + \left( { - 18x_{0}^{2} + 60x_{0} + 60} \right)x_{1} + 4x_{0}^{3} + 6x_{0}^{2} - 36x_{0} - 8} \right)$$

and

$$\begin{aligned} h\left( {x_{0} ,x_{1} } \right) & = 36x_{1}^{4} + \left( {51x_{0} - 195} \right)x_{1}^{3} + \left( { - 59x_{0}^{2} - 35x_{0} + 310} \right)x_{1}^{2} \\ & \quad + \,\left( { - 39x_{0}^{3} + 235x_{0}^{2} - 200x_{0} - 140} \right)x_{1} + 27x_{0}^{4} - 69x_{0}^{3} \\ & \quad - \,14x_{0}^{2} + 76x_{0} + 16 \\ \end{aligned}$$

Now, \(f\left( {x_{0} ,x_{1} } \right) < 0\) because, as can be easily checked, \(\frac{{\partial^{2} f}}{{\partial x_{1}^{2} }} > 0\), \(\frac{\partial f}{{\partial x_{1} }} < 0\) when \(x_{1} = \frac{{3x_{0} + 2}}{5}\), \(\frac{\partial f}{{\partial x_{1} }} > 0\) when \(x_{1} = x_{1}^{R}\) and \(f < 0\) when both \(x_{1} = \frac{{3x_{0} + 2}}{5}\) and \(x_{1} = x_{1}^{R}\).

Similarly, \(h\left( {x_{0} ,x_{1} } \right) < 0\) because \(\frac{{\partial^{2} h}}{{\partial x_{1}^{2} }} > 0\), \(\frac{\partial h}{{\partial x_{1} }} > 0\) when \(x_{1} = \frac{{3x_{0} + 2}}{5}\) and \(h = 0\) when \(x_{1} = x_{1}^{R}\).

Proof of socially optimal locations in Proposition 2

Since demand is inelastic, the socially optimal locations are those that minimize transportation costs as given in (5). To find them, remember first that, as shown above, \(q_{0} = \frac{{x_{0} + x_{1} }}{2}\) minimizes transportation costs for any locations \(x_{0} ,x_{1}\). Replacing this value back in (5) yields TC as a function of \(x_{0}\) and \(x_{1}\) as given in (10). The first-order conditions to minimize TC in (10) with respect to both \(x_{0}\) and \(x_{1}\) are:

$$\frac{\partial TC}{{\partial x_{0} }} = x_{0}^{3} - \left( {\frac{{x_{1} - x_{0} }}{2}} \right)^{3} = 0$$
$$\frac{\partial TC}{{\partial x_{1} }} = \left( {\frac{{x_{1} - x_{0} }}{2}} \right)^{3} - \left( {1 - x_{1} } \right)^{3} = 0$$

from where we can obtain, respectively, \(x_{0} = \frac{{x_{1} }}{3}\) and \(x_{0} = 3x_{1} - 2\) and, therefore, \(x_{0} = \frac{1}{4}\), \(x_{1} = \frac{3}{4}\). The second order conditions are also satisfied.

Firm 0’s demand \(q_{0}\) written as an explicit function of \(p_{0}\) and \(p_{1}\)

If \(- \left( {x_{1}^{3} - x_{0}^{3} } \right) \le p_{1} - p_{0} \le - \left( {x_{1} - x_{0} } \right)^{3}\) then:

$$q_{0} = \frac{{x_{0} + x_{1} }}{2} - \sqrt {\frac{{4\left( {p_{0} - p_{1} } \right) - \left( {x_{1} - x_{0} } \right)^{3} }}{{12\left( {x_{1} - x_{0} } \right)}} }$$

If \(- \left( {x_{1} - x_{0} } \right)^{3} \le p_{1} - p_{0} \le \left( {x_{1} - x_{0} } \right)^{3}\) then:

$$\begin{aligned} q_{0} & = \frac{{\left( {x_{0} + x_{1} } \right)}}{2} + \left( {\frac{{\sqrt {4\left( {p_{1} - p_{0} } \right)^{2} + \left( {x_{1} - x_{0} } \right)^{6} } }}{8} - \frac{{p_{0} - p_{1} }}{4}} \right)^{1/3} \\ & \quad - \,\frac{{\left( {x_{1} - x_{0} } \right)^{2} }}{{4\left( {\frac{{\sqrt {4\left( {p_{1} - p_{0} } \right)^{2} + \left( {x_{1} - x_{0} } \right)^{6} } }}{8} - \frac{{p_{0} - p_{1} }}{4}} \right)^{1/3} }} \\ \end{aligned}$$

If \(\left( {x_{1} - x_{0} } \right)^{3} \le p_{1} - p_{0} \le \left( {1 - x_{0} } \right)^{3} - \left( {1 - x_{1} } \right)^{3}\) then:

$$q_{0} = \frac{{x_{0} + x_{1} }}{2} + \sqrt {\frac{{4\left( {p_{1} - p_{0} } \right) - \left( {x_{1} - x_{0} } \right)^{3} }}{{12\left( {x_{1} - x_{0} } \right)}} }$$

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Fernández-Ruiz, J. Mixed duopoly in a Hotelling framework with cubic transportation costs. Lett Spat Resour Sci 13, 133–149 (2020). https://doi.org/10.1007/s12076-020-00249-y

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Keywords

  • Mixed duopoly
  • Hotelling line
  • Socially optimal locations
  • Product differentiation

JEl Classification

  • L13
  • L32
  • L33
  • H44