The locations of firms on intersecting roadways

Abstract

With Bertrand-Nash mill-price competition, travel costs proportional to distance squared, and three firms on an interval, the equilibrium locations of the peripheral firms are further from the center than is socially optimal. If there is a central intersection, with four (or more) finite roadway segments radiating outward from the center (a small city spread along two intersecting roadways), and with one firm at the center and one on each radial segment, then the equilibrium locations of the peripheral firms are closer to the center than is socially optimal. Extensions include competition with spatial price discrimination, a more complicated system of intersecting roadways, and more than one firm on each roadway segment.

Appendix

In this “Appendix”, condensed from one in the previous version of the paper, I consider the validity of the Bertrand-Nash equilibrium in mill prices in Sect. 3, and then the validity of the subgame-perfect Nash equilibrium in locations in Sect. 4. I focus on the case $n = 4$, but in the next paragraph I consider $n = 2$, and at the end I consider $n>4$. Differences from the model in DeGraba (1987), in addition to those mentioned in Sect. 5, are discussed below.

It is well known that everything is fine in the case $n = 2$ (this is a special case of Caplin and Nalebuff 1991). It is very helpful that the demand curve for store 1 (and each other store) as a function of its own price is a straight line all the way from its vertical intercept to its horizontal intercept, even in cases where (for example) store 0 (in the middle) sells to some consumers to the left of store 1, or where store 1 sells to all consumers to the left of and some to the right of store 0. As will be seen below, the demand curve facing a store can be more complicated if $n>2$.

Suppose now that $n = 4$, which is the case of primary interest in this paper. The revenues (and profits) of store 0, as a function of the prices of all stores and the locations of the four peripheral stores, are given by substituting $n = 4$ into (3).

$$\begin{aligned} \pi _{0} = [(1/2)a + (3/2)b + (2ta)^{-1 }(P_{1} - P_{0}) + 3 (2tb)^{-1 }(P_{2} - P_{0})] P_{0}. \end{aligned}$$

(20)

Substituting $n = 4$ into (5), we have $2(3a + b)P_{0} - bP_{1}-3aP_{2} = tab(3b + a)$. From this equation and Eqs. (7) and (9), the following Bertrand-Nash equilibrium mill price functions are derived.

$$\begin{aligned} P_1 \left( a, b, 4 \right)&= \left( ta/6 \right)\left[ 9a\left( 2 - a \right)+ b\left( 14 - 2a + 3b \right) \right]\left( 3a + b \right)^{-1},\end{aligned}$$

(21)

$$\begin{aligned} P_0 \left( a, b, 4 \right)&= 2P_1 \left( a, b, 4 \right)- ta\left( 2 - a \right)=\left( tab/3 \right)\left( 8 + a + 3b \right)\left( 3a + b \right)^{-1},\nonumber \\ \end{aligned}$$

(22)

$$\begin{aligned} P_2 ( a, b, 4 )&= ( 1/2 )[ P_0 ( a, b, 4 )+ tb( 2 - b )]=( tb/6)[ 3b( 2 - b )\nonumber \\&+\, a( 26 + a - 6b ) ]( 3a + b )^{-1}. \end{aligned}$$

(23)

Since (20) is a strictly concave function of $P_{0}$, (6) is a strictly concave function of $P_{1}$, and (8) is a strictly concave function of $P_{2}$, the (local) second-order conditions are satisfied for all stores if they are choosing the prices (21)–(23).

According to (15) with $n = 4$, the Nash equilibrium values of a and b are $a = b = 5/9$. I now fix b at 5/9 and check whether (or not) a pure-strategy Bertrand-Nash equilibrium in mill prices exists for various values of a in the unit interval [0, 1] that are not necessarily equal to 5/9.

The number of consumers served by store 1 is given by $1 -z_{1}$, which equals the term in square brackets in (6) (since $z_{1}$ is given by (1)), which yields the equation.

$$\begin{aligned} {1}-{z}_1 ={1}-\left( {{1}/{2}} \right){a}-\left( {{2ta}} \right)^{-{1}}\left( {{P}_1 -{P}_0 } \right)\!. \end{aligned}$$

(24)

This is a linear function of $P_{1}$, and thus (ignoring the complication described in the next paragraph for the time being), the profits of store 1 as a function of its own price would seem to be a strictly concave function, from the vertical intercept (the value of $P_{1}$ for which the number of consumers is 0) down to the horizontal intercept ($P_{1}= 0$), for any value of a, given that $b = 5/9$.

The complicating feature of the analysis is that, with some values of a, and with b fixed at 5/9, there may be a positive value of $P_{1}$ for which (24) gives $1 - z_{1} = 1$ (or $z_{1} = 0$), so the market area of store 1 is the line segment it is located on, but no more. I denote this value of $P_{1}$ by $P_{1u}(a, b, n)$ for general values of b and n, and by $P_{1u}$(a, 5/9, 4) for the case here. In DeGraba (1987), for any value of n (including his main case $n = 2$, and the case $n>2$ in his extensions section), the demand curve is continuous, but it has a kink and becomes vertical for prices below this (from $P_{1} =P_{1u}(a, b, n)$ down to $P_{1}= 0)$, since the quantity demanded from a local firm can never be greater than 1 in his model. In my model, something very different happens for some values of a. For $n = 4$ and $b = 5/9$, and if $0\le P_{1}<P_{1u}(a, 5/9, 4)$, then $z_{1} < 0$, so the profits of store 1 are no longer given by $\pi _{1} = (1 - z_{1})P_{1}$, which is the formula in the text just before (6), and thus are no longer given by (6). Whenever $z_{1} < 0$, store 1 sells to all consumers on its own segment of length 1, plus consumers in [$0, z_{1}$] on each of the other three segments. There is a kink in the demand curve for store 1 at this price, and, for $P_{1} <P_{1u}(a, 5/9, 4)$, demand is only 1/3 as steep as it is for higher prices. It is possible (see Appendix of previous version of paper) to derive the equation $2P_{1}- P_{0} = ta[(2/3) - a]$. Assuming $P_{0}$ is fixed at its equilibrium value, given by (22), this is easily solved for $P_{1}= P_{1v}(a, 5/9, 4)$, the profit-maximizing price for store 1 if store 1 is on this low-price segment of the demand curve. A similar complication exists for any $n>2$. Note the difference between DeGraba 1987, where (as pointed out above) there is a “downward kink” for any value of n, and my model, where there is an “upward kink” if $n>2$.

If $a = b = 5/9$, then, using (21), (22), $P_{1} = P_{1}(5/9, 5/9, 4) = (155/243)(t)$ and $P_{0} = P_{0}(5/9, 5/9, 4) = (115/243)(t)$, in agreement with Proposition 3. As $P_{1}$ decreases, starting from $P_{1}(5/9, 5/9, 4)$, the profits of store 1 decrease monotonically to 0 at $P_{1}= 0$, so store 1 maximizes its profits by choosing $P_{1}= P_{1}(5/9, 5/9, 4) = (155/243)(t)$. Everything works out nicely, and the Bertrand-Nash equilibrium in prices described by (21)–(23) is exactly correct.

Suppose now that $a = 1/3$ (and $b = 5/9$). It can be shown (see Appendix of previous version of paper) that everything works out nicely (just as it did when $a = 5/9$), and once again the Bertrand-Nash equilibrium in prices described by (21)–(23) is exactly correct.

However, suppose that $a = 2/9$. It can be shown (see Appendix of previous version of paper) that store 1 reaches its maximum profits, for the given location pattern (and for the given $P_{0})$, by choosing $P_{1}= P_{1v}(2/9, 5/9, 4)$. In this case, therefore, things do not work out neatly, and the pure-strategy Bertrand-Nash equilibrium in prices described by (21)–(23) is not valid.

More generally, for $n = 4$, and with three of the peripheral stores at their equilibrium locations ($b = 5/9$), the Bertrand-Nash equilibrium in mill prices described in Sect. 3, and by (21)–(23), is valid if the distance of store 1 from the central intersection is $a>a^{*}$, but not if $a<a^{*}$, for some $a^{*}$ such that $2/9<a^{*}<1/3$. In fact, $a^{*}$ is just very slightly greater than 1/4.

Consider existence of a subgame-perfect Nash equilibrium. Anderson (1988) states what must be shown for the simpler case of spatial duopoly. First, I confine attention to pure strategies (see pp. 480, 482, and fn. 2 of Anderson 1988). In my model with $n = 4$, in order to show that the set of locations stated in Proposition 3 ($a = b = 5/9$) is a part of a pure-strategy subgame-perfect Nash equilibrium, it is necessary to show that if three of the peripheral stores are at their equilibrium locations, $b = 5/9$, then a pure-strategy Bertrand-Nash equilibrium in mill prices exists for any location of store 1 (i.e., for any value of a in the interval [0, 1]) and that the resultant profits of store 1 are higher if $a = 5/9$ than for any other value of a in [0, 1]. However, in my model with $n = 4,$ this works only if the location space for each peripheral store is restricted to [$a^{*}, 1$], where $a^{*}\approx 1/4,$ because only for these values of a does a pure-strategy Bertrand-Nash equilibrium in mill prices exist (given that $b = 5/9$), as discussed above.

Some spatial duopoly models (Osborne and Pitchik (1987); Anderson (1988)) consider “the case when the strategy space is enlarged to include mixed strategies at the price stage” (Anderson 1988, p. 480). (Anderson (1988), p. 490) states “as long as the mixed-strategy price equilibrium is unique in terms of payoffs, the location game is now well defined. If there is an equilibrium to this game, it may well be such that mixed strategies are never played at equilibrium.” The second sentence applies to my model for $n = 4$, since at $a = b = 5/9$ there is a pure-strategy Nash equilibrium in prices. I have not derived mixed-strategy Nash equilibria in prices when $b = 5/9$ and $0<a <a^{*}$. As observed by (Anderson (1988), p. 489), Osborne and Pitchik (1987) have solved their case only by using very complicated computer simulations. Their equilibrium locations are in the region where there is no pure-strategy Nash equilibrium in prices. In my case with $n = 4$, and with $b = 5/9$, it is only if a is much less than 5/9 that it is necessary to resort to mixed-strategy price equilibria, and the mixed-strategy profits of store 1 for these values of a are almost certainly much less than the pure-strategy equilibrium profits when $a = 5/9$. If this is indeed true, then if mixed-price strategies are allowed, the subgame-perfect Nash equilibrium in Proposition 3 (with $a = b = 5/9$) is valid if the location space for each peripheral store is the whole interval [0, 1].

Consider $n>4$. For a sufficiently large $n$, and for the equilibrium value of $b$, given by (15), a pure-strategy Bertrand-Nash equilibrium in prices might fail to exist even if a assumes its equilibrium value, given by (15). In fact, this is almost certainly the case in the limit that $n$ goes to infinity, which is the case considered in Proposition 4. However (see Appendix of previous version of paper) this potential problem does not exist at all with the model in footnote 7.