Abstract
This paper studies the optimal zoning of a mixed duopoly when the objective function of the public firm is a weighted sum of its profits and social surplus. We find that a regulator may attain the optimal locations of both firms by restricting the location of the private firm only. There is no need to limit the location of the public firm. In contrast, in a private duopoly, the regulator needs to restrict the locations of both firms.
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Notes
Economides (1986) shows that this result holds when the transportation costs function is highly convex. When it is moderately convex, equilibrium locations are interior points of the product space.
In some cases, this can be considered as a restriction that firms must locate within the linear city.
The “line” can be used to study firms’ locations or to analyze product differentiation.
Tsai et al. (2006) analyze how zoning affects firms’ locations and land rents.
Colombo (2012) extends this analysis by considering a non-discriminating Cournot duopoly and an asymmetric central zoning area.
If the two firms are private and the regulator maximizes social surplus, the regulator wants firms to locate in the first and third quartiles. However, when firms are free to locate within the city, they locate at the endpoints of the town. In this case, the regulator may get firms to locate optimally by not allowing either firm to locate at less than 1/4 from the nearest border of the city.
An alternative justification of this objective function is the following. The regulator seeks to maximize the weighted average of the social surplus and the profit of the public firm (see, for example, Matsumura (1998), Bárcena-Ruiz and Garzón (2003): \(\beta (\mathrm {SS})+(1-\beta )\pi _{0}\). Maximizing the above expression is equal to maximizing \(\frac{\beta }{\beta } \mathrm {SS}+\frac{1-\beta }{\beta }\pi _{0}=\mathrm {SS}+\alpha \pi _{0},\) where \(\alpha = \frac{1-\beta }{\beta }\ \)(when \(\beta =1\), then \(\alpha =0\); when \(\beta \rightarrow 0\), then \(\alpha \rightarrow +\infty \)). As a result: \(\alpha \in \left[ 0,+\infty \right) \).
It should be noted that the incomes of firms are a transfer from consumers to producers.
The second-order conditions of the problems that we analyze are always satisfied.
It can be easily checked that when \(\alpha \) goes to \(\infty \) (that is, when both firms are private), the equilibrium locations for firms allowed to locate outside the city boundaries are \(x_{0}=-\frac{1}{4}, x_{1}=\frac{5}{ 4}.\) This result is the same that obtained by Lambertini (1994) and Tabuchi and Thisse (1995). This is also the result obtained by Braid (2011) when consumers do not buy monopoly goods from the firms.
Matsumura and Matsushima (2003) extend this analysis by assuming that locations can be chosen sequentially and that the government may regulate prices. Ogawa and Sanjo (2007) consider that the private firm is a multinational. They show that as the share of foreign capital in the private (multinational) firm increases, the public firm moves toward the central place.
Note that, weighted welfare can be expressed as \(W=(1+\alpha )\pi _{0}+\pi _{1}+\mathrm {CS}\).
It can be shown that if firms are not allowed to locate in the interval [0, z), as in Lai and Tsai (2004), the regulator cannot make firms locate optimally. Moreover, if firms are private, the zoning designed in Proposition 3 does not assure optimal location by the two firms: it is necessary to restrict the locations of both.
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Acknowledgments
The authors gratefully acknowledge financial support from Ministerio de Ciencia y Tecnología and FEDER (ECO2009-07939, ECO2012-32299) and Gobierno Vasco-Eusko Jaurlaritza (GIC07/22-IT-223-07). We want to thank two referees for helpful comments and suggestions.