## 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.
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.
- 3.
- 4.
- 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.
- 7.
- 8.
An extreme example would be the choice of an obstetric unit or birth center.

- 9.
- 10.
Which in turns leads to the non-existence of equilibrium.

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

- 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.
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.
- 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.

## References

Artz, B., Heywood, J.S., McGinty, M.: The merger paradox in a mixed oligopoly. Res. Econ.

**63**(1), 1–10 (2009)Asensio, J., Matas, A.: Commuters’ valuation of travel time variability. Transp. Res. Part E Logist. Transp. Rev.

**44**(6), 1074–1085 (2008)Bárcena-Ruiz, J.C., Garzón, M.B.: Mixed duopoly, merger and multiproduct firms. J. Econ.

**80**(1), 27–42 (2003)Barros, F., Modesto, L.: Portuguese banking sector: a mixed oligopoly? Int. J. Ind. Organ.

**17**(6), 869–886 (1999)Benassi, C., Chirco, A., Colombo, C.: Mixed spatial duopoly, consumers’ distribution and efficiency. Econ. Lett.

**156**, 74–77 (2017)Bennett, J., La Manna, M.: Mixed oligopoly, public firm behavior, and free private entry. Econ. Lett.

**117**(3), 767–769 (2012)Chen, C.S., Lai, F.C.: Location choice and optimal zoning under Cournot competition. Reg. Sci. Urban Econ.

**38**(2), 119–126 (2008)Cremer, H., Marchand, M., Thisse, J.F.: Mixed oligopoly with differentiated products. Int. J. Ind. Organ.

**9**(1), 43–53 (1991)D’Aspremont, C., Gabszewicz, J.J., Thisse, J.F.: On Hotelling’s “stability in competition”. Econometrica

**47**(5), 1145–1150 (1979)Economides, N.: The principle of minimum differentiation revisited. Eur. Econ. Rev.

**24**(3), 345–368 (1984)Economides, N.: Minimal and maximal product differentiation in Hotelling’s Duopoly. Econ. Lett.

**21**(1), 67–71 (1986)Economides, N.: Quality variations and maximal variety differentiation. Reg. Sci. Urban Econ.

**19**(1), 21–29 (1989)Hinloopen, J., Van Marrewijk, C.: On the limits and possibilities of the principle of minimum differentiation. Int. J. Ind. Organ.

**17**(5), 735–750 (1999)Hotelling, H.: Stability in competition. Econ. J.

**39**(153), 41–57 (1929)Jain, R., Pal, R.: Mixed duopoly, cross-ownership and partial privatization. J. Econ.

**107**(1), 45–70 (2012)Kitahara, M., Matsumura, T.: Mixed duopoly, product differentiation and competition. Manch. Sch.

**81**(5), 730–744 (2013)Lai, F.C., Tsai, J.F.: Duopoly locations and optimal zoning in a small open city. J. Urban Econ.

**55**(3), 614–626 (2004)Lam, T.C., Small, K.A.: The value of time and reliability: measurement from a value pricing experiment. Transp. Res. Part E Logist. Transp. Rev.

**37**(2–3), 231–251 (2001)Lu, Y.: Hotelling’s location model in mixed duopoly. Econ. Bull.

**8**(1), 1–10 (2006)Lu, Y., Poddar, S.: Mixed oligopoly and the choice of capacity. Res. Econ.

**59**(4), 365–374 (2005)Matsumura, T.: Partial privatization in mixed duopoly. J. Public Econ.

**70**(3), 473–483 (1998)Matsumura, T., Matsushima, N.: Mixed duopoly with product differentiation: sequential choice of location. Aust. Econ. Pap.

**42**(1), 18–34 (2003)Matsumura, T., Matsushima, N.: Endogenous cost differentials between public and private enterprises: a mixed duopoly approach. Economica

**71**(284), 671–688 (2004)Matsumura, T., Matsushima, N.: Locating outside a linear city can benefit consumers. J. Reg. Sci.

**52**(3), 420–432 (2012)Méndez-Naya, J.: Merger profitability in mixed oligopoly. J. Econ.

**94**(2), 167–176 (2008)Nishimori, A., Ogawa, H.: Do firms always choose excess capacity? Econ. Bull.

**12**(2), 1–7 (2004)Posada, P., Straume, O.R.: Merger, partial collusion and relocation. J. Econ.

**83**(3), 243–265 (2004)Small, K.A., Winston, C., Yan, J.: Uncovering the distribution of motorists’ preferences for travel time and reliability. Econometrica

**73**(4), 1367–1382 (2005)Tirole, J.: The Theory of Industrial Organization. MIT Press, Cambridge, MA (1988)

Waterson, M.: Models of product differentiation. In: Cable, J. (ed.) Current Issues in Industrial Economics, pp. 105–133. Palgrave, London (1994)

Zhang, Y., Zhong, W.: Are public firms always less innovative than private firms? Jpn. Econ. Rev.

**66**(3), 393–407 (2015)

## 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

from where

and therefore:

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

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

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

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:

and thus

Therefore:

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

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):

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:

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 \(q_{0} \in \left[ {x_{0} ,x_{1} } \right]:\)

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

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:

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

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

with

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

from where

and therefore:

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

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

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

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:

Now, from (26)

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

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

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

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 \(q_{0} = q_{0}^{c}\),

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:

with

and

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:

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:

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

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:

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### Cite this article

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