The locations of firms on intersecting roadways

For Bertrand-Nash mill-pricing models with linear transportation costs, see also the duopoly-line-segment models in part of d’Aspremont et al. (1979) and (Anderson et al. (1992), pp. 293–297), and the multi-firm and many-firm models (using circles, line segments, and infinite lines) of (Vickrey (1964), pp. 334–336, 1999, pp. 960–962), Salop (1979), Mulligan (1996), and Capozza and Van Order (1980). In Hotelling (1929), each firm moves toward the center if firms are not closer than (0.25, 0.75). However, d’Aspremont et al. (1979) show that a Bertrand-Nash equilibrium in mill prices does not exist once firms are closer than this (see also Vickrey 1964, p. 327), unless both firms are at the center and charge zero prices. Thus, no subgame-perfect Nash equilibrium in locations exists either.

Many authors have Bertrand-Nash mill pricing and transportation costs proportional to distance squared, including the duopoly-line-segment models in (d’Aspremont et al. (1979), pp. 1148–1149), (Anderson et al. (1992), pp. 298–299); Lambertini (1994); Lambertini (1997), Tabuchi and Thisse (1995), Meza and Tombak (2009), Matsumura and Matsushima (2009), Egger and Egger (2010) and Braid (2011), and the multi-firm and many-firm models in Neven (1987), (Anderson et al. (1992), pp. 299–301), Brenner (2005), Eaton and Wooders (1985), and Economides (1989).

With linear transportation costs, nonexistence presumably also applies if firms are on intersecting finite line segments. Bertrand-Nash mill-pricing models with linear transportation costs and networks of intersecting roadways include Fik (1991), Fik and Mulligan (1991), and Braid (1993), which assume exogenous firm locations, and Sect. 6 of Mulligan (1996), which presents an analysis with endogenous locations.

This is in fact similar to an exercise performed in Fik (1991). There is a complicated network of roadways, and the total length of all roadway segments (not the length of each segment) is held constant as the number of segments becomes large.

For example, with five firms, the socially optimal locations on a unit interval are (0.1, 0.3, 0.5, 0.7, 0.9), but the equilibrium locations are (0.104, 0.329, 0.500, 0.671, 0.896).

DeGraba (1987) also considers a national firm that charges different prices in different cities. This pricing assumption is not very relevant to my paper, since it would require that the central store can charge different mill prices to consumers living on different roadway segments.

My Sects. 3–4 can be used to extend DeGraba (1987) in a way not considered at all in his paper. My Proposition 2 (which has \(n = 2\)) can be interpreted as a national firm at location 0 in the middle of a line-segment product space of length 2, competing with one firm on each side in a single city. Suppose there are \(m = n/2\) cities (for any even \(n\)). My Proposition 3 can be interpreted as a national firm at location 0 competing with local firms on each side in \(m = 2\) cities (a total of four local firms), if the national firm is constrained to set a single mill price for both cities. Proposition 4 can be interpreted in an analogous way. Propositions 1 and 5, which present the socially optimal locations and compare them to the equilibrium locations, are applicable here.

See also Hurter and Lederer (1985), Lederer and Hurter (1986), MacLeod et al. (1988), Sect. 4 of Hamilton et al. (1989), (Anderson et al. (1992), pp. 325–330), Gupta (1992), Sect. 3 of Yu (2007), Braid (2008), and Heywood and Ye (2009). I use “firm” instead of “store,” since spatial price discrimination is more likely to be used by a firm that pours concrete or delivers a bulky item than a restaurant or a store where a consumer eats a meal or carries the goods home.

A peripheral firm will often choose to locate 2/3 of the way from the adjacent firm to the end of the market. See (Anderson et al. (1992), pp. 325–330) and (Lerner and Singer (1937), pp. 182–184). It also happens in Gupta (1992) for the last firm to locate.

It is useful to recall from Sect. 4 that with three roadway segments and a single intersection, and with a single store at the center, the equilibrium has \(a = b = 8/13 = 0.6154\).

If \(R = 1\), then \(a = b = 0.6207\), so an increase in \(R\) does not always decrease the equilibrium value of a, something that is also shown in the second model below.

From (15) of Sect. 4, with six roadway segments and a single intersection, and with a single store at the center, the equilibrium has a = b = 1/2.

Models of spatial competition with mill pricing usually assume that \(\alpha = 1\) or \(\alpha = 2\) because these are the most convenient to work with. I am not aware of empirical evidence, but for shopping trips, a value of \(\alpha \) significantly greater than 1 is likely to be most realistic when consumers must walk (particularly if they have to carry something home).

For example, see Hsu (1983), who considers spatial monopoly with a general population density function that might be negative exponential.

See also (Fujita and Thisse (2002), Chapter 7).