Sequential Location with Asymmetric Preferences

Sequential Location with Asymmetric Preferences Tore Nilssen¤ University of Oslo [email protected] Christian Riis Norwegian School of Management [email protected] February 12, 2003

Abstract We suggest a new approach to the theory of …rm location, where we work directly with the indi¤erent consumer in stead of with consumers’ transportation costs. Within the context of sequential location with non-price competition, we discuss how limiting earlier approaches are and present results in cases not considered earlier. In particular, we …nd that whether …rms locate around a circle in the clockwise or the counter-clockwise direction, or even in a more complicated pattern, depends on the size of the circle.

Preliminary and incomplete. Please do not quote without authors’ permission.

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Correspondence: Tore Nilssen, Department of Economics, University of Oslo, P.O.Box 1095 Blindern, N-0317 Oslo, Norway. Fax: +47 22 85 50 35.

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1

Introduction

The so-called address approach to the economic study of …rms’ locations in product space or in geographical space has for a long time centered around the concept of consumers’ transportation costs: A consumer’s preference for a …rm’s product is assumed to depend on the distance between the consumer’s ideal product and the product supplied by the …rm, and the larger this distance is, in product space or in geographical space, the higher is the consumer’s utility loss from consuming the …rm’s product; see, e.g., the surveys by Eaton and Lipsey (1989) and Gabszewicz and Thisse (1992). Underlying the use of transporation costs as a key concept in location studies is the presumption that these costs are symmetric: Independent of the direction in product space, a consumer’s utility loss is the same as long as the distance is the same. This presumption is hard to justify: In cases where product characteristics are easily measurable and comparable, symmetry around an ideal variety seems to be a very restrictive description of consumers’ preferences. One way around is to allow transportation costs to be asymmetric, as proposed by Nilssen (1997) and Nilssen and Sørgard (2000).1 In the present paper, however, we argue that the whole notion of consumers’ transportation costs must be discarded. In our view, the central concept is that of the indi¤erent consumer : The market shares of two rivaling …rms are determined by the location of the consumer who is indi¤erent between their products. How the identity of this consumer, or rather the location of the indi¤erent consumer’s ideal point, is a¤ected by a relocation of one of the …rms, is the key issue. In order to illustrate our approach, we study strategic sequential location in the case of non-price competition. The seminal work is by Vickrey (1964, 1999) and Prescott and Visscher (1977). Later, Dewatripont (1987) has stressed the importance of late-mover indi¤erence in this model. In our view, this indi¤erence serves as an illustration of how limiting the focus on consumer transportation costs is. Working directly with the indi¤erent consumer, we are able to discuss sequential location in cases not earlier covered by the literature. And we show that the case of symmetric transportation costs is non-robust in a sense that will become clear in the following. The paper is organized as follows. In Section 2, we present our model, introducing the notion of the indi¤erent consumer. In Section 3, we discuss a …rm’s optimum location on an interval between two other …rms and illustrate the limitations of earlier approaches to this problem. The rest of the paper is 1

These papers generalize the notion of location under a directional constraint, due to Cancian, et al. (1995).

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an application of our approach to the case of four …rms locating in sequence around a circle.2 The …rst of these …rms has no strategic decision to make, since all points around the circle are the same for the …rst mover. Thus, essentially, this is a three-…rm location model. The problem of the last mover is essentially the one discussed already in Section 3. In Section 4, we turn to the location decision of the next-to-last …rm. In particular, does it want to have the last mover on its right or on its left? Finally, in Section 5, we provide results for the equilibrium of the whole game. We are able to show how the way the …rms distribute themselves around the circle depends on the size of the circle. Section 6 concludes the paper.

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The Model

Consider a set of four …rms, F :={1, 2, 3, 4}, that locate sequentially on A := [0; c], which is a circle with a circumference equal to c > 0. The number of …rms is …xed at 4 since adding …rms would give little insight and only make derivations more complex. Firms are labelled according to the move sequence: …rm 1 moves …rst, and so on, and …rm 4 moves last. Let x := (x1 ; : : : ; x4 ) 2 A4 denote the list of locations chosen by the …rms, ranked by the order of moves. We assume that consumers are located uniformly on A; this has as a consequence that, for a given circumference c, an interval on A is completely characterized by its length, irrespective of where on A it is. Without loss of generality, we let the clockwise direction around the circle be equivalent to the rightward direction. With location on the circle, there are in reality only three strategic players, since the …rst …rm’s location decision is arbitrary and therefore will be normalized to be at position 0 ´ position c (or, equivalently, we can assume position 0 to be wherever …rm 1 locates). Apart from di¤erences in location, …rms produce a homogeneous good for which each consumer has a unit demand. Prices are determined exogenously; without loss of generality, we put prices equal to one. Thus, pro…t maximization entails maximization of sales volume, and a consumer’s net bene…t equals his gross bene…t minus any transportation costs. 2

The seminal work on the circle model is by Vickrey (1964, 1999) and Salop (1979); see also more recent contributions by Kats (1995), Frutos, et al. (1999), and Peitz (1999). Sequential location on the circle has been studied by Economides and Howell (1991), Pal (1998), Kats (1999), and Anderson and Engers (2001). The work by Kats (1999) is particularly relevant here, since he, like us, assumes …xed prices. But his concerns are very di¤erent from ours.

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For any two locations a0 , a00 2 A, where a0 · a00 , let m(a0 ; a00 ) 2 [a0 ; a00 ] denote the marginal consumer, i.e., the consumer who is indi¤erent between buying from a …rm at a0 and buying from one at a00 ; we shall assume preferences and circumference to be so related that, irrespective of …rms’ locations, the marginal consumer always prefers consumption of the good to no consumption, thus ensuring that …rms are always in direct competition with each other. It follows from the uniform distribution of consumers that, for any pair of locations a0 ; a00 2 A, m(a0 ; a00 ) = a0 + m(0; a00 ¡ a0 ). For an interval of length I > 0, with one …rm located at each of its endpoints and no …rm located in its interior, de…ne L(I) as that part of the interval served by the left …rm and, correspondingly, R(I) as the part served by the right …rm. Formally, L(I) := m(0; I) and R(I) := I ¡ L(I). The marginal consumer on the interval [a0 ; a00 ] is now given by: a0 + L(a00 ¡ a0 ). The function L(¢) is what we need in our analysis in order to describe consumer preferences. In the textbook case, which we here will denote the case of symmetric consumer preferences and where consumers’ transportation costs are symmetric, whether linear or non-linear, L(I) = R(I) = I=2. But we are presently more interested in cases where consumer preferences are not symmetric, i.e., where, in general, L(I) 6= R(I). We impose the following assumption on L(¢): Assumption A: The function L(¢) is continuous; it is twice di¤erentiable; and L0 (I) 2 (0; 1) for almost every I. The assumption that L0 2 (0; 1) implies that the marginal consumer moves monotonically as the interval between two …rms increases, and in a regular way in the sense that it moves in the same direction as the interval is increased but a distance that is shorter than the increase of the interval. The assumption is not completely innocuous, since it excludes the case of a ”directional constraint”, discussed by Cancian et al. (1995) and Nilssen and Sørgard (1998), where L(I) = 0 for any I. We will also explore alternative assumptions on the curvature of the L function: Assumption B1: L00 (I) ¸ 0, for all I. Assumption B2: L00 (I) = 0, for all I. Assumption B3: L00 (I) > 0, for all I. Assumption B1 is a generalization both of Assumption B2 and of Assumption B3, while the two latter assumptions are mutually exclusive. By imposing that L00 ¸ 0, as we do even in the most general formulation B1, we want to exclude cases where L(¢) is strictly convex for some intervals while strictly concave for others; without loss of generality, we exclude the discussion of a strictly concave L(¢) altogether. Note that Assumption B2 covers both the case of symmetric consumer preferences, where L0 = 1=2, and the 3

case of consumers having asymmetric but linear transportation costs, discussed in Nilssen (1997): Let tL (tR ) be consumers’ transportation cost per unit in the left (right) direction. Then, L0 (I) = tR =(tL +tR ), and L00 = 0 when transportation costs are linear. In fact, any exponential transportation-cost function will give the same result: Let leftwise (rightwise) transportation costs over a distance d be given by tL d® (tR d® ), where ® > 0. Then L0 (I) = 1+

1 ³ ´ ®1 ; tL tR

and L00 = 0. Thus, Assumption B2 implies a generalization of the analysis in Nilssen (1997). Most of our results make use of Assumption B3. A useful addition to it is the following: Assumption C: 14 · L0 (0) < 12 , and limI!1 L0 (I) > 12 . This assumption has two elements with distinctly di¤erent purposes. First, the assumption that L0 (0) ¸ 14 , together with Assumption B3, imply that there does not exist any interval length I for which L0 (I) < 14 . This property turns out to be useful in order to limit the number of cases to consider in our analysis. The assumption that L0 (0) < 12 and limI!1 L0 (I) > 12 , on the other hand, implying that there exists an interval length I for which L0 (I) = 12 , is made in order to ensure richness to our analysis. Without it, some of the cases in the Theorem below would collapse and the result become simpler.

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The Optimum In-Between Location

We start the analysis of the equilibrium of this game by studying the last stage, as is standard. To understand the behavior of …rm 4, suppose there are (or, more generally, will be) …rms located at a0 and a00 = a0 + I. The sales volume that a …rm can obtain by locating in-between a0 and a00 at a0 + ¯, where ¯ 2 [0; I], is given by: u(¯; I) := R(¯) + L(I ¡ ¯) = ¯ ¡ L(¯) + L(I ¡ ¯);

0 · ¯ · I;

since, by locating at ¯, the …rm obtains R(¯) = ¯ ¡ L(¯) to its left and L(I ¡ ¯) to its right. It is also useful to consider e¤ects on a …rm’s sales volume of a neighboring …rm’s relocation. Thus, de…ne: U (a; a0 ; a00 ) = u(a ¡ a0 ; a00 ¡ a0 ) = R(a ¡ a0 ) + L(a00 ¡ a); 4

0 · a0 · a · a00 ;

as the sales volume a …rm obtains when locating at a, given other …rms’ locations at a0 and a00 , and let D(a1 ; a2 ; a0 ; a00 ) := U (a1 ; a0 ; a00 ) ¡ U (a2 ; a0 ; a00 ) be the di¤erence in a …rms’ sales volume from locating at a1 rather than a2 , given other …rms’ locations at a0 and a00 . A function f(z) is symmetric around (z1 ; y1 ) if, for every z, f (z1 ¡z)¡y1 = y1 ¡ f(z1 + z), or f (z1 ¡ z) + f (z1 + z) = 2y1 . Using this de…nition, we note the following properties of the sales-volume function u(¢; ¢) and its relatives: Lemma 1 (i) The function u(¢; I) is symmetric around (I=2; I=2), for all I. (ii) If L000 (I) > [ I=2. (iii) @ 2 u=@¯@I = ¡ L"(I ¡ ¯) · 0, with a strict inequality under Assumption B3. (iv) @U=@a0 < 0, and @U=@a00 > 0. (v) If a1 < a2 , then @D=@a0 · 0 and @D=@a00 ¸ 0, with strict inequalities under Assumption B3. Proof. (i) Follows from the de…nition of symmetry and the observations that u(I=2; I) = I=2; 8 I, and: u(¯; I) + u(I ¡ ¯; I) = I; 8 ¯ 2 [0; I] and 8 I > 0.

(1)

(ii) We have: @u=@¯ = 1 ¡ L0 (¯) ¡ L0 (I ¡ ¯); @ 2 u=@¯ 2 = L00 (I ¡ ¯) ¡ L00 (¯)

(2) (3)

Suppose L000 > 0; the other case is treated equivalently. If ¯ < [>] I=2, then L00 (¯) < [>] L00 (I ¡ ¯). Now, the result follows from (3). (iii) Follows from di¤erentiation in (2). (iv) Follows from di¤erentiations in the expression for U (a; a0 ; a00 ). (v) Follows from di¤erentiations in the expression for D(a1 ; a2 ; a0 ; a00 ) . An illustration of the symmetry property in Lemma 1(i) is given in Figure 1. Note that symmetry is related Lemma 1(i) implies ¡ to Ioddity. ¢ I In particular, 3 that the function h (¯) := u ¯ ¡ 2 ; I ¡ 2 is odd. An illustration of Lemma 1(iii) is provided in Figure 2. Here, the u(¢; I) function is depicted for a low (solid line) and a high (dotted line) value of I. We see that an increase in I entails a tilted curve, as Lemma 1(iii) implies. 3

A function f is odd if f (¡x) = ¡f (x) ; 8x; see, e.g., Edwards and Penney (1990, p. 260).

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< FIGURES 1 AND 2 ABOUT HERE > The maximum sales volume obtainable by a location in-between a0 and a00 is given by: v(I) := max u(¯; I): ¯

The solution to this problem is an element of B(I) := arg max u(¯; I) µ [0; I]: ¯

An interior solution b 2 B(I) \ (0; I) satis…es the …rst-order condition, L0 (b) + L0 (I ¡ b) = 1, from (2), and the second-order condition, L00 (b) · L00 (I ¡ b); from (3). Note that v0 (I) > 0. For an interior solution b 2 B(I) \ (0; I), we have, by the envelope theorem, that v0 (I) = L0 (I ¡ b) > 0, where the inequality follows from Assumption A; and it is easily checked that v0 (I) > 0 also for corner solutions. In the case of symmetric consumer preferences, L(¯) = R(¯) = ¯2 and , so that u(¯; I) = I2 , independent of ¯, and the …rm considL(I ¡ ¯) = I¡¯ 2 ering locating in-between two other …rms is indi¤erent among all in-between locations: B(I) = [0; I]; this observation is known at least since Prescott and Visscher (1977). But the symmetric model is special in one respect: It is only when consumer preferences are symmetric that the mid-location I2 can be an optimum in-between location. This is essentially a consequence of the symmetry property of the u(¢; I) function noted in Lemma 1(i): The only case when the symmetry point ( I2 ; I2 ) can be a maximum point is when the function is constant in ¯. Formally, we thus have: Proposition 2

I 2

2 B(I) if and only if consumer preferences are symmetric.

This Proposition motivates an investigation of a location game where consumer preferences are not symmetric. Presently, we do this by imposing Assumption B3, which excludes symmetric consumer preferences, in the following analysis. For cases where L00 is monotonically decreasing or increasing, we have: 6

Proposition 3 If L000 (I) > [ [ 0 such that B(I) = I, i.e., B(I) is a singleton and equal to the right corner of the interval, if and only if I 2 [0; I B ]; or: (ii) B(I) = I; 8 I. Proof. If B(I) is interior, then, by Proposition 4, it increases as I decreases. However, it cannot increase beyond I. Thus, the two cases of the Proposition are the only two possible. The typical relation between I and B(I) is depicted in Figure 3. Note that the B(I) curve makes a jump over the I=2 line - this is a consequence of Proposition 2. 4

Note, however, that a corner solution on the left edge, i.e., one where B(I) = 0, may not exist for any I. A necessary condition for B(I) = 0 is clearly: @u(0; I)=@¯ = 1 ¡ L0 (I) ¡ L0 (0) < 0. Thus, if limI!1 L0 (I) < 1 ¡ L0 (0), then B(I) > 0; 8 I.

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Note that Assumption B3 is not necessary for the result in Proposition 5 either. However, if, as in Assumption B2, L00 (I) = 0; 8 I, then we are in case (ii) of Proposition 5, and the optimum in-between location is always the right corner: B(I) = I; 8 I.

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Optimum Positioning by the Penultimate Firm

In this section, we analyze the decision of …rm 3 and identify two interval lengths that, in addition to I B discussed in the previous section, are important for the description of the equilibrium of our game. Throughout the section, we consider the situation where the two …rst-moving …rms already are located on each end of an interval that is assumed to be of length 2I. This implies that, if …rm 3 locates in the middle of this interval, …rm 4 has two equal-sized intervals of length I to choose from. One crucial question is whether …rm 3 prefers having …rm 4 on its right or on its left. We shall show below that there exists, under our maintained assumptions, an interval length, which we call I C , such that …rm 3 prefers having …rm 4 on its right whenever I · I C . Furthermore, we have that I C > I B , where I B is de…ned in Proposition 5. Another crucial question is whether or not …rm 3, in cases where it wants to induce …rm 4 to go to its right, is constrained by this and would otherwise have preferred to be further to the right. We start out this section with characterizing an interval length I A < I B such that …rm 3 is so constrained except if I 2 (I A ; I B ). De…ne: I A := maxfI j u(I; 2I) ¸ maxfu(¯; 2I) j ¯ · Igg: Thus, 2I A is the largest interval length 2I such that the mid-location I is better for …rm 3 than all locations to the left of the middle. We have: Lemma 6 Suppose Assumptions A, B3, and C hold. Then: (i) I A < I B ; and (ii) L0 (I B ) > 12 . Proof. (i) Clearly, I A · IeA , where IeA solves: @u(I; 2I) = 0; @¯ 8

the possibility I A < IeA arises because the optimum in-between location B(I) in general is a discontinuous function of I. It follows from the de…nition of @u , in the proof of Lemma 1 above, that IeA is IeA and the expression (2) for @¯ the interval length such that: L0 (IeA ) = 1=2;

which exists by Assumption C. On the other hand, u(¯; IeA ) = 1 ¡ L0 (IeA ¡ ¯) ¡ L0 (¯) > 0; 8 ¯ · IeA ;

since, by Assumption B3,

1 L0 (IeA ¡ ¯) < L0 (IeA ) = ; 8 ¯ 2 (0; IeA ]; and 2 1 L0 (¯) < L0 (IeA ) = ; 8 ¯ 2 [0; IeA ): 2

But this implies: B(IeA ) = IeA . Thus, from the de…nition of I B , we have: IeA < I B , which again implies the statement in part (i) of the Lemma. (ii) This follows straightforwardly from Assumption B3, since IeA < I B by the above proof of part (i). Recall from Section 2 that xf denotes the location of …rm f 2 F .

Lemma 7 Suppose Assumptions A and B3 hold, that …rms 1 and 2 are located at each end of an interval of length 2I, and that, inside this interval, …rm 4 locates to the right of …rm 3. Then: (i) x3 < I only if x4 = 2I, and (ii) x4 < 2I only if x3 = I. Proof. If …rm 4 is to the right of …rm 3, then it follows from Proposition 5 that it locates at 2I if [x3 ; 2I] is su¢ciently short. Furthermore, it follows 4 from Proposition 4 that dx > 1 if B(2I ¡ x3 ) is singleton and interior, and dx3 that x4 makes a discrete jump to the right at a marginal increase in x3 if B(2I ¡ x3 ) contains multiple interior solutions. Thus, if …rm 3 moves slightly to the right, then …rm 4 either is already at 2I or moves to the right even more than …rm 3. From this, the Lemma follows. We are now ready to characterize …rms 3 and 4’s equilibrium locations x3 and x4 , given that …rm 4 locates to the right of …rm 3. Since v 0 (I) > 0, …rm 3 must obey the restriction x3 · I in order to keep …rm 4 to the right. This restriction is binding for …rm 3, except for interval lengths just below I B . We have: 9

Proposition 8 Suppose Assumptions A, B3, and C hold, that …rms 1 and 2 are located at each end of an interval of length 2I, and that, inside this interval, …rm 4 locates to the right of …rm 3. Then: (i) x3 < I if and only if I 2 (I A ; I Bh); in iparticular, there exists an £ ¢ b IB . Ib 2 I A ; I B such that x3 = 2I ¡ I B ; 8 I 2 I; (ii) If I · I B , then x4 = 2I, while if I > I B , then x4 < 2I.

Proof. By Proposition 5, x4 < 2I and x3 = I together imply that 2I ¡ I = I > I B . On the one hand, this and Lemma 7 prove part (ii) of the Proposition. On the other hand, it means that we can reformulate Lemma 7(i) as: x3 < I only if I · I B . Now, the …rst statement in part (i) follows from Lemma 6(i) and the de…nition of I A . The second statement in part (i) is true because, by Proposition 4, …rm 3 would not like …rm 4 to choose an interior solution. And this constraint is binding for I su¢ciently close to I B . It follows from Proposition 8 that, for the case when …rm 3 wants …rm 4 to its right, I A and I B delineate three subcases: For small interval lengths, i.e., when I · I A , …rms 3 and 4 locate at I and 2I, respectively. For large interval lengths, i.e., when I ¸ I B , …rm 3 is again constrained while …rm 4 is not: x3 = I and x4 < 2I. Finally, for intermediate interval lengths, i.e., when I 2 (I A ; I B ), …rm 3 locates, without being constrained by its preference for having …rm 4 to its right, strictly to the left of the middle, x3 < I, while x4 = 2I. However, …rm 3 will in this case have to obey another restriction: It cannot locate so far to the left that …rm 4 prefers a location left of 2I, i.e., if I · I B , then …rm 3 actually has two restrictions on its optimum choice: (2I ¡ I B ) · x3 · I. Moreover, it follows from Proposition 2 that the optimum interior solution on the interval [0; 2I] cannot be at I. Thus, at I = I A , …rm 3’s optimum location makes a downward jump. Thereafter, by Proposition 5, it possibly decreases further until the lower bound (2I ¡ I B ) is reached at some Ib 2 [I A ; I B ). b I B ), x3 increases at rate 2 as I increases until, again, x3 = I at For I 2 (I; I = IB. We have so far worked under the restriction that …rm 4 ends up to the right of …rm 3. We turn now to the other case, working under the restriction that …rm 4 ends up to the left of …rm 3. Let …rm 3’s sales volume when …rm 4 is to its left (implying x3 ¸ x4 ) be denoted V3L (x3 ; x4 ; I); it is given as: V3L (x3 ; x4 ; I)

: = U (x3 ; x4 ; 2I) = u(x3 ¡ x4 ; 2I ¡ x4 ) = L(2I ¡ x3 ) + R(x3 ¡ x4 ) = L(2I ¡ x3 ) + x3 ¡ x4 ¡ L(x3 ¡ x4 ): 10

In order to really get …rm 4 to its left, …rm 3 has to obey the restriction x3 ¸ I. Thus, we de…ne the maximized sales volume of …rm 3 as: v3L (I) := maxfV3L (x3 ; x4 ; I) j x3 ¸ I; x4 2 B(x3 )g:

(4)

If, in this case, …rm 3’s location, x3 , is interior, then it never increases as I increases; formally: Proposition 9 Suppose Assumptions A and B3 hold, that …rms 1 and 2 are located at each end of an interval of length 2I, and that, inside this interval, …rm 4 locates to the left of …rm 3. If x3 2 (I; 2I), i.e., …rm 3’s location is 3 interior, then dx · 0, or it jumps downward ( i.e., leftward). dI Proof. Note that, in order for x3 to be interior, we must have x4 < x3 ; dV L otherwise, i.e. if x4 = x3, then we would have dx33 < 0 and thus no interior x3 . Recall from Propositions 2 and 4 that B(x3 ) is decreasing with jumps. Consider values of x3 for which B(x3 ) does not jump. Thus, from Proposition 4 4, dx < 0. From the expression for V3L (x3 ; x4 ; I) above, we get the …rst-order dx3 condition: µ ¶ dV3L dx4 0 = [1 ¡ L (x3 ¡ x4 )] 1 ¡ ¡ L0 (2I ¡ x3 ) = 0: dx3 dx3 Also x3 may jump downwards at a slight increase in I, again following Propositions 2 and 4. But if it does not jump, then we can di¤erentiate the above …rst-order condition to …nd: 2L00 (2I ¡ x3 ) dx3 = < 0; d2 V3L dI 2 dx3

since L00 > 0 by Assumption B3, and since the second-order condition on …rm 3’s choice is satis…ed in an interior solution, so that the denominator is negative. Note, …nally, that, at a value for x3 at which x4 = B(x3 ) jumps, the optimum choice for …rm 3 is to hold back on x3 , at some x b3 , say, so that …rm 4’s jump is avoided; thus, the inequality in the Lemma is weak. As I increases, it may well be optimum for …rm 3 to have a low x3 , i.e., an x3 < x b3 , to use the language of the proof of this Proposition, even if this means a jump in x4 towards x3 . Being non-increasing, the interior x3 eventually reaches I, at some critical interval length I3L , and starts to increase, since I is the lower bound on …rm 3’s location given that it is going to have …rm 4 on its left. 11

On the other hand, a decrease in I means that an interior x3 increases or is constant. However, it cannot increase beyond 2I; a further decrease in I when x3 = 2I will have to entail a decrease also in x3 . Denote the critical interval length at which, if it exists, x3 = 2I for intervals of lengths just below it while x3 < 2I for such lengths just above, by I2L . Since, by Proposition 5, …rm 3 would not choose the extreme location 2I if (2I ¡ x4 ) > I B , and since, by Proposition 4, x4 is non-increasing if indeed x3 = 2I, it is not possible to have x3 = 2I when I ¸ I B . Thus, I2L < I B . Let I3L denote the critical interval length such that x3 = I if and only if I ¸ I3L . Clearly, since an IL interior x3 is non-increasing, we have that I2L ¸ 23 . Thus, since I2L < I B , we have: I3L < 2I B . B If the interval length is small, in particular if I · I2 , then, by Proposition 5, …rm 4’s optimum location to the left of …rm 3 is a co-location with the other, i.e., B(x3 ) = x3 , irrespective of where in the interval [I; 2I] …rm 3 is located. Thus, given the restriction that …rm 4 will be at its left, the optimum for …rm 3 is, in order to have as large sales volume at its right as possible, to move as far as it can to the left; thus, in this case, x3 = I. By continuity, this may hold also for interval lengths larger than, but close to, B IB ; in general, there exists a critical interval length I1L 2 [ I2 ; I B ] such that 2 x3 = I for I · I1L , while a slight increase in I beyond I1L implies an upward jump in x3 , either to x3 = 2I, if I1L < I2L , or to an interior location. In order to discuss whether …rm 3 prefers having …rm 4 to its right or to its left, some more notation is needed. With …rms 1 and 2 at each end of the interval of length 2I, de…ne therefore …rm 3’s sales volume - when …rms 3 and 4 are located on this interval at x3 and x4 , respectively, with …rm 4 to the right of …rm 3 (implying x3 · x4 ) - as V3R (x3 ; x4 ; I). Thus: V3R (x3 ; x4 ; I) := u(x3 ; x4 ) = L(x4 ¡ x3 ) + R(x3 ) = L(x4 ¡ x3 ) + x3 ¡ L(x3 ): Recall the restriction x3 · I that …rm 3 must obey in order to keep …rm 4 to the right. The maximized sales volumes of …rm 3, when …rm 4 is to its right, is: v3R (I) := maxfV3R (x3 ; x4 ; I) j x3 · I; x4 ¡ x3 2 B(2I ¡ x3 )g:

(5)

The corresponding de…nitions relevant for the case when …rm 3 has …rm 4 to its left, V3L (x3 ; x4 ; I) and v3L (I), have already been introduced above. For a su¢ciently small interval length, …rm 3 prefers intuitively having …rm 4 on its right: If …rm 4’s optimum in-between location is to the far right of an interval, i.e., if the interval has length I · I B , so that B(I) = I, then …rm 3 prefers having …rm 4 on its right in order to obtain distance between 12

the two …rms. On the other hand, if the interval length is large, then …rm 4, when it locates to the right of …rm 3, gets closer and closer to …rm 3 as the interval length increases, which deteriorates the pro…tability for …rm 3 of having …rm 4 to its right. At the same time, …rm 4 moving to the left in its interval has, relatively speaking, the opposite e¤ect on the pro…tability for …rm 3 of having …rm 4 to its left. Thus, for su¢ciently large interval lengths, …rm 3 prefers the latter. For intermediate levels of interval lengths, in particular for I > I B but not very much larger, …rm 3’s preferences cannot be determined in general. In particular, there does not seem to exist generally any critical interval length I 0 such that v3R (I) > v3L (I) if and only if I < I 0 . The problem arises because of inherent non-convexities in both the v3R (I) and the v3L (I) functions. We solve this by invoking that part of Assumption C that implies L0 ¸ 14 . Below, we show …rst that, for any I · I B , …rm 3 prefers having …rm 4 to its right. Thereafter, we show that, for any I ¸ 2I B , the opposite holds, with …rm 3 preferring …rm 4 to be to its left. For the statements and proofs of these results, let xL3 and xL4 denote the equilibrium locations of …rms 3 and R 4, respectively, when …rm 4 is to the left of …rm 3, with xR 3 and x4 as the corresponding locations when …rm 4 is to the right. Proposition 10 Suppose Assumptions A, B3, and C hold, and that …rms 1 and 2 are located at each end of an interval of length 2I. If I · I B , then v3R (I) > v3L (I), i.e., …rm 3 prefers having …rm 4 to its right. Proof. We consider two cases separately. Case (i): Suppose xL3 + xL4 > 2I. The claim in the Proposition follows from: v3L (I) = u(xL3 ¡ xL4 ; 2I ¡ xL4 ) < u(xL3 ¡ xL4 ; xL3 ) = min u(¯; xL3 ) · u(I; xL3 ) · u(I; 2I) · v3R (I): ¯

Here, the …rst inequality follows from Lemma 1(iv), since xL3 > 2I ¡ xL4 in this case. The second equality follows from the symmetry noted in Lemma 1(i): Since …rm 4 locates at x4 , it follows that u(xL4 ; xL3 ) = max¯ u(¯; xL3 ). By symmetry, then, u(xL3 ¡ xL4 ; xL3 ) = min¯ u(¯; xL3 ). The second inequality follows from the previous equality and the restriction xL3 ¸ I: If (xL3 ¡ xL4 ) is in the arg min of an interval of length xL3 , then I cannot be worse. The third inequality follows again from Lemma 1(iv), since xL3 · 2I. Finally, the fourth inequality follows by Proposition 8(ii): Since I · I B , xR 4 = 2I. At the same time, the location I is always available for …rm 3. Its maximum sales volume must therefore be at least as large as what it can get from locating at I, which is u(I; 2I). 13

Case (ii): Suppose xL3 + xL4 · 2I. The claim in the Proposition now follows from: v3L (I) = u(xL3 ¡ xL4 ; 2I ¡ xL4 ) · u(0; 2I ¡ xL4 ) < u(0; 2I) · v3R (I): In order to see that the …rst inequality holds, recall, from the previous paragraph, the observation that u(xL3 ¡ xL4 ; xL3 ) = min¯ u(¯; xL3 ), which implies u(0; xL3 ) ¸ u(xL3 ¡xL4 ; xL3 ), again as before. Thus, u(0; xL3 )¡u(xL3 ¡xL4 ; xL3 ) ¸ 0. It follows now from Lemma 1(v) and the supposition xL3 + xL4 · 2I, implying xL3 · 2I ¡ xL4 , that u(0; 2I ¡ xL4 ) ¡ u(xL3 ¡ xL4 ; 2I ¡ xL4 ) ¸ 0, which is equivalent to the …rst inequality above. The second inequality follows from Lemma 1(iv). The third inequality follows from the de…nition of v3R (I). The following Lemma provides a su¢cient condition for the balance to go the other way around, with …rm 3 preferring …rm 4 to be on its left rather than on its right. Lemma 11 Suppose Assumptions A, B3, and C hold, and that …rms 1 and 2 are located at each end of an interval of length 2I, with I > I B . If, in addition, B(I) ½ [0; I2 ), then v3L (I) > v3R (I), i.e., …rm 3 prefers having …rm 4 to its left. Proof. Recall from Proposition 8(i) that, since I > I B , …rm 3 locates at I. The condition in the Lemma thus implies that the segment between …rms R 3 and 4 is smaller than the one to the right of …rm 4: 2I ¡ xR 4 > x4 ¡ I, 3I R or: x4 < 2 : Suppose now that …rm 3, if …rm 4 were to locate at its left, would locate at x bL3 = 3I ¡ xR bL3 = xR 4 ;i.e, such that: 2I ¡ x 4 ¡ I. Note that, 3I 3I R L since x4 2 [I; 2 ), this implies that x b3 2 ( 2 ; 2I]. Denote …rm 3’s payo¤ from this location by: vb3L ; since this location is not necessarily the optimum given that …rm 4 is to the left, we have: b bL3 , …rm 4’s v3L · v3L . With …rm 3 at x L location would be at x b4 , which, by a slight abuse of notation (since B(I) is not necessarily a singleton), is characterized by: I x bL4 = arg max u(¯; x bL3 ) · arg max u(¯; I) < ; ¯ ¯ 2

where the …rst inequality follows from Proposition 5 and the second inequality follows from the condition B(I) ½ [0; I2 ). This implies that x bL3 ¡ x bL4 > I. Thus, locating at x bL3 , with …rm 4 to its left, …rm 3 has the same market to its right as when …rm 4 is at its right: 2I ¡ x bL3 = xR 4 ¡ I; at the same time, it L L has a larger market available to its left: x b3 ¡ x b4 > I. Thus, v3L ¸ b v3L > v3R , where the second inequality follows from Assumption A. As a straightforward consequence of Lemma 11, we obtain the following result: 14

Proposition 12 Suppose Assumptions A, B3, and C hold, and that …rms 1 and 2 are located at each end of an interval of length 2I. If I ¸ 2I B , then v3L (I) > v3R (I), i.e., …rm 3 prefers having …rm 4 to its left. Proof. By Proposition 4, B(I) is decreasing in I. Thus, since B(I) < I for I > I B , we must have that B(I) < I2 for I ¸ 2I B . Now, the result follows from Lemma 11. De…ne now I C as the largest interval length such that …rm 3 prefers to have …rm 4 to its right for all interval lengths smaller than I C , and de…ne IbC as the smallest interval length such that …rm 3 prefers to have …rm 4 to its left for all interval lengths larger than IbC . Formally: I C = supfI 0 : v3R (I) > v3L (I); 8 I · I 0 g, and IbC = inffI 0 : v3L (I) > v3R (I); 8 I ¸ I 0 g:

(6)

(7)

It follows from Propositions 10 and 12 and these de…nitions that: I B < I C · IbC < 2I B :

What we do not know in general is whether I C = IbC . Because of inherent non-convexities in the v3L (I) and v3R (I) functions, we cannot exclude, in general, the possibility of a strict inequality here, implying that, as I increases, …rm 3’s evaluation of having ³ …rm ´4 to its left versus to its right changes several times in the interval I C ; IbC . It turns out, however, that Assumption C is su¢cient for I C = IbC . We have: Proposition 13 Suppose Assumptions A, B3, and C hold, and that …rms 1 and 2 are located at each £end of an ¤ interval of length 2I. There exists a critical interval length I C 2 I B ; 2I B such that …rm 3 prefers having …rm 4 to its right if I < I C while it prefers having …rm 4 to its left if I > I C .

Proof. The proof is by contradiction. Suppose, contrary to what we claim in the Proposition, that I C < IbC , as they are de…ned in (6) and (7). Thus, there exists an I, call it I S , such that v3L (I S ¡ ") > v3R (I S ¡ ") and v3L (I S + ") < v3R (I S + "), for a su¢ciently small, but positive, ". Note that, if there exists an I for which v3R (¢) is discontinuous, then v3R (¢) makes a downward jump; a discontinuity may occur only when x3 = I and …rm 4’s optimum location jumps to the left because of a downward jump in B (I). Moreover, if there exists an I for which v3L (¢) is discontinuous, then v3L (¢) makes an upward jump; a discontinuity may again occur only when …rm 4’s 15

optimum location jumps to the left. The implication is that both v3R (¢) and v3L (¢) are continuous at I S , because this is the only way we can have v3R (¢) crossing v3L (¢) from below. If I > I B and …rm 4 locates to the right of …rm 3, then x3 = I by Proposition 8(i). Thus, from the de…nition in (5), v3R (I) = L (x4 ¡ I) + I ¡ L (I) Thus, when v3R (I) is continuous and I > I B , the slope is: ¶ µ dv3R dx4 1 0 0 = 1 ¡ L (I) + L (x4 ¡ I) ¡ 1 < 1 ¡ L0 (I) < , dI dI 2

(8)

4 where the …rst inequality follows from Assumption A and dx < 1, which is a dI consequence of Proposition 4, and the second inequality follows from I > I B and Lemma 6(ii). ¡ ¢ Thus, v3R I S < 12 . But this implies that, if I = I S and …rm 4 locates to the left of …rm 3, then …rm 3 is at an interior solution. To see this, suppose to the contrary that …rm 3 is at the corner location x3 = I. From the de…nition in (4),

v3L (I) = L (I) + I ¡ x4 ¡ L (I ¡ x4 ) , and µ ¶ dv3L dx4 1 0 0 = L (I) + [1 ¡ L (I ¡ x4 )] 1 ¡ > L0 (I) > , dI dI 2 4 where, as above, the …rst inequality follows from Assumption A and dx < 1, dI B and the¡second one from and Lemma 6(ii). But this would imply I > I ¢ ¡ ¢ that v3R I S < v3L I S , which is contrary to the de…nition of I S . Thus, …rm 3 is at an interior solution at I = I S when …rm 4 locates to its left; call …rm 3’s location xL3 . By the envelope theorem and the de…nition of v3L (I) in (4), we have

¡ ¢ dv3L = 2L0 2I ¡ xL3 dI

By the argument above, that

dv3R (I S ) dI

>

dv3L (I S ) . dI

This, together with (8), imply

¡ ¢ 1 2L0 2I ¡ xL3 < , 2 16

or ¡ ¢ 1 L0 2I ¡ xL3 < . 4

However, by Assumption C, no interval length exists for which L0 < 14 . Therefore, there cannot exist an interval length like I S , and we can conclude that I C = IbC . Naturally, …rm 3 would not want …rm 4 on its right if it would get too close. This leads to a simple condition for I B and I C to coincide. We have:

Lemma 14 Suppose Assumptions A, B3, and C hold, and that £ B…rms ¤ 1 and C 2 are located at each ¡ end¢ of an interval of length 2I. If I 2 I ; I , then x3 = I, and x4 2 32 I; 2I . Proof. We know already from Proposition 8 that x3 = I and x4 < 2I if I ¸ I B . So the only thing left to prove is that x4 > 32 I if I · I C . xxx Proposition 15 If L000 (I) < 0; 8I, then I B = I C . Proof. Follows from combining Lemma 14 and Proposition 3. With this, we are ready to proceed to the analysis of the whole game.

5

Equilibrium of the whole game

Since player 1 is not a strategic player, the …rst player to make a strategic move, taking into consideration the e¤ect of his choice on other players’ behavior, is player 2, moving in stage two. When …rm 2 makes its decision, …rm 1 is already located at position 0 on the circle, by construction. The circle has a circumference equal to c. Thus, with …rm 1 already in place, …rm 2’s decision is as if it is locating on a straight line of length c with one …rm already being located at each end of the line. We have so far worked with the interval length, I, as the critical variable delineating the various cases to consider. We now turn to the circle circumference, c, as the key variable. But there is a straightforward transformation between the two: c = 3I;

17

i.e., I is the length a segment along the circumference corresponding to a third, or 120± , of the whole circle. We will also have critical values of the circumference de…ned as: cA = 3I A cB = 3I B cC = 3I C Just as the three interval lengths I A , I B , and I C were critical in delineating equilibrium behavior in the stage-three and stage-four subgames, so are the corresponding concepts cA , cB , and cC critical in delineating the important cases for equilibrium behavior in the full game. Note that 0 < cA < cB < cC < 2cB . The details of equilibrium behavior in our model are contained in the following theorem: Theorem 16 Suppose Assumptions A, B3, and C hold, and that …rm 1 locates at position 0. The sequential location behavior around the circle by …rms 2, 3, and 4 depends on the size of the circle, i.e., on its circumference c: (i) If the circle is small, in particular if c 2 (0; cA ], then, in equilibrium, …rms 1 and 4 co-locate at 0 ´ c, one of …rms 2 and 3 locates at 3c , and the other at 2c . The equilibrium payo¤s for the …rms are: V1 = L( 3c ); V2 = V3 = 3 c ; V4 = R( 3c ); so that: V2 = V3 > V4 > V1 . 3 (ii) If the circle is a bit larger, so that c 2 (cA ; cB ), then there exists a unique equilibrium in which …rm 3 locates to the left of …rm 2: Firms 1 and 4 co-locate at 0 ´ c, while …rms 2 and 3 locate such that 0 < x3 < x2 < c. The …rms’ equilibrium payo¤s are ordered as follows: £ B min ¤(V2 ; V3 ) > V4 > V1 . C (iii) If the circle is even larger, so that c 2 c ; c , then there exists a unique equilibrium in which the …rms locate clockwise around the circle: , and V2 > V3 > 0 = x1 < x2 < x3 < x4 < c, where x2 · 3c , x3 = x22+c · 2c 3 V4 > V1 . (iv) If the circle is large, so that c > cC , then there exists a unique equilibrium in which the …rms locate counter-clockwise around the circle: 0 · . x4 < x3 < x2 < x1 = c, where x3 ¸ 3c and x2 ¸ 2c 3 Proof. The complicated part to prove is part (ii), while the proof of part (iv) is useful in the proof of part (iii). We thus prove the theorem by proving its parts in the sequence (i), (iv), (iii), and (ii). Part (i). In order to show that this is the unique equilibrium outcome, we need to show that everything else cannot be an equilibrium. Note …rst that the payo¤s in the candidate equilibrium outcome are straightforward to 18

deduce. In particular, …rm 4 can choose where to locate within an interval of length 3c and chooses the far right, by Proposition 5, since 3c < I B in this case. Therefore, …rm 4 must get more than …rm 1, who is at the far left of such an interval, and V4 > V1 . We want …rst to show that …rm 2 choosing a location x2 2 [0; 3c ) cannot occur in an equilibrium. To see this, note that, in this case, we would clearly have at least one of the two …rms 3 and 4 locating to the left of 2c . Thus: 3 V2 = u(x2 ; min[x3 ; x4 ]) < u(x2 ;

2c ) · u(I; 2I) = I; 3

where the …rst inequality follows from Lemma 1(iv) and the second from the de…nition of I A . Thus, x2 2 [0; 3c ) is not part of any equilibrium. Consider next the possibility of …rm 2 choosing x2 2 ( 3c ; 2c ). We need 3 to consider three subcases in turn. Suppose …rst that both …rms 3 and 4 subsequently locate to the right of …rm 2: min[x3 ; x4 ] > x2 . This cannot be an equilibrium, since, with x2 > 3c , the interval to the right of …rm 2 has a length c ¡ x2 < 2I A , implying that …rm 3 would locate at its mid-location, so that …rm 4 would be better o¤ to the left of …rm 2. The same argument can be used against the possibility that both …rms 3 and 4 subsequently locate to the left of …rm 2 following …rm 2 choosing an x2 2 ( 3c ; 2c ). Therefore, 3 the only possibility left is for a …rm-2 location decision x2 2 ( 3c ; 2c ) to be 3 followed by one of the …rms 3 and 4 locating to its left and the other to its right. In particular, the …rm to the left makes a location decision, call it xL , characterized by: xL = arg max u(¯; x2 ) > arg max u(¯; ¯

¯

2c c )¸ ; 3 3

where the …rst inequality follows from Proposition 5 and the second from the de…nition of I A . Thus, letting xR denote the location of the …rm to the right, we have V2 = u(x2 ¡ xL ; xR ¡ xL ) < u(x2 ¡ xL ;

2c c 2c c ) < u( ; ) = ; 3 3 3 3

where the …rst inequality follows from Lemma 1(iv) and the above observation that xL > 3c , and the second inequality follows from the de…nition of I A together with xL > 3c . It follows that …rm 2 locating somewhere in the interval ( 3c ; 2c ) cannot be an equilibrium. 3 Consider …nally the possibility that x2 2 ( 2c ; c]. Clearly, both …rms 3 and 3 B 4 will be locating to the left of it. If …rm 4 locates so that x2 2 ( 2c ; 2c3 ] = 3 (2I; 2I B ], then, by Proposition 8, …rm 4 locates in-between …rms 2 and 3 in 19

such a way that x4 = x2 , leaving …rm 2 with L(c ¡ x2 ) < 3c . If, on the other B hand, …rm 2’s location satis…es x2 2 ( 2c3 ; c], then we need to distinguish between the case where …rm 4 locates to the right of …rm 3, i.e., in-between …rms 3 and 4, and the one where it locates to the left of …rm 3, i.e. in-between …rms 1 and 3. In the former case, we have that x3 = x22 , by Proposition 8. Firm 4 locates on the interval [ x22 ; x2 ] and its optimum choice is characterized by: x4 ¡

x2 x2 x2 x2 c = arg max u(¯ ¡ ; x2 ¡ ) > arg max u(¯ ¡ ; 2I) ¸ I = ; ¯ ¯ 2 2 2 2 3

where both inequalities follow from the de…nition of I A . Thus, x4 > x22 + 3c > 2c , implying that V2 < 3c . 3 In the latter case, where …rm 4 locates to the left of …rm 3, …rm 3 will never be to the left of x22 ; suppose, therefore, that x3 = x22 , since any x3 > x22 would be even less pro…table for …rm 2. Firm 2’s payo¤ is then given by: ³x ´ ³x ´ x2 2 2 = L(c ¡ x2 ) + V2 = L(c ¡ x2 ) + R ¡L 2 2 2 This payo¤ is decreasing in x2 : ³ ´i 1h dV2 0 x2 = 1¡L ¡ L0 (c ¡ x2 ) < 0: dx2 2 2

Here, the second term is negative from Assumption A. The square-bracketed B term is negative by Lemma 6(ii) and Assumption B3, since, here, x2 > 2c3 , or: x22 > I B . Thus, what …rm 2 gets must be less than what it would get at B x2 = 2c < 2c3 : 3 ³c´ ³c´ c ³x ´ 2 V2 = L(c ¡ x2 ) + R 2c , then …rm 2 strictly prefers case (b’) to case (a), 3 since …rm 3 further to the right brings …rm 4 closer to …rm 2 in case (a). Suppose next that, in case (a), x2 < 3c . Now, the interval to the left of …rm 2 is smaller for …rm 2 in case (a) than in case (b’). Also, the interval to its right is smaller. This is because, now, the interval [x2 ; x3 ] in which …rm 4 locates in case (a) is larger than the interval [x2 ; c] in which …rm 4 locates in case (b’). Therefore, …rm 4 is closer to …rm 2 in the former case than in the latter, by Proposition 4. Thus, also now, case (b’) is preferred by …rm 2 to case (a). Finally, we show that the candidate equilibrium, case (c), is better for …rm 2 than case (b’). In case (c), with both …rms 3 and 4 to the left of …rm 2, …rm 2 must obey the restriction x2 ¸ 2c in order to make sure that the 3 C other …rms do locate to its left. Since c ¸ c , …rm 3 prefers …rm 4 to its left. In order to ensure this, …rm 3 is restricted by: x3 ¸ x2 ¡ x3 , or x3 ¸ x22 . It follows that, in case (c), x3 ¸ 3c . Suppose …rst that x3 = 3c if x2 = 2c . Now, 3 case (c) is clearly preferable for …rm 2 to case (b’), since the interval on its left is the same in the two cases and the interval on its right is larger in case (c) than in case (b’), since in the latter case, …rm 4 is located to the right of …rm 2. Suppose next that x3 < 3c if x2 = 2c ; i.e., …rm’s location is an interior 3 solution. 0 Let …rm 4’s location in case (b’) be denoted xb4 and …rm 3’s location in case (c) be denoted xc3 . We want to show that the total interval available to …rm 2 in case (c), c ¡ xc3 , is larger than the total interval available to it in 0

0

xb +xc

case (b’), xb4 ¡ 3c , i.e., that 4 2 3 · 2c . Let …rm 3’s optimum location in case 3 2c c (c), if …rm 2 is at 3 , be denoted x , we have that x b3 . Since x2 ¸ 2c bc3 ¸ xc3 , by 3 21

Proposition 9. With …rm 2 at 2c , …rm 3 maximizes 3 µ ¶ 2c c b V3 = L ¡ x3 + x3 ¡ x4 (x3 ) ¡ L (x3 ¡ x4 (x3 )) 3

with respect to x3 . Thus, …rm 3’s optimum location is determined by: µ ¶ dVb3c 2c dx4 0 c =1¡L b3 ¡ L0 (b xc3 ¡ x4 (b xc3 )) ¡ [1 ¡ L0 (b xc3 ¡ x4 (b xc3 ))] =0 ¡x dx3 3 dx3 4 It follows from Proposition 4 that, here, dx < 0, or …rm 4’s location makes dx3 a jump to the left. Thus, the last term in the above expression is positive, so that µ ¶ 2c 0 c (9) 1¡L xc3 ¡ x4 (b xc3 )) · 0. b3 ¡ L0 (b ¡x 3

From …rm 4’s …rst-order condition, we know that

1 ¡ L0 (b xc3 ¡ x4 (b xc3 )) = L0 (x4 (b xc3 )) Combining this with the condition in (9), we …nd that µ ¶ 2c 0 0 c c L (x4 (b b3 , x3 )) · L ¡x 3

which implies, under Assumption B3, that

2c bc3 ¸ x4 (b xc3 ) ¡x 3

Thus, x bc3 ¡

(10)

c c 0 xc3 ) < c ¡ xb4 , · ¡ x4 (b 3 3

where the …rst inequality follows from (10) and the second inequality from Proposition 4, since x bc3 > 3c . Combining this with the observation that x bc3 ¸ 0

xb +xc

, i.e., that total interval available to …rm xc3 , we can conclude that 4 2 3 · 2c 3 2 in case (c) is larger than the total interval available to it in case (b’). Consider now another hypothetical ¡ c0 ¢ case, case (c’), in which …rm 2 is c0 c0 located at x2 given by x2 ¡ x3 x2 = c; since we work here under the supposition that x3 < 3c if x2 = 2c , such a location exists by Proposition 9. 3 At this position, …rm 2 has the interval to its left as in case (b’). And by 22

the above argument, it must have a larger interval on its right. It follows that …rm 2 is better o¤ in case (c’) than in case (b’). And since case (c’) is a restriction relative to case (c), the latter cannot be worse for …rm 2 than the former. It follows that case (c) is preferred by …rm 2 to case (b’), and we have proved that the unique equilibrium when c > cC is the one described in part (iv) of the Theorem. and …rm 4 to the Part (iii). With …rm 2 locating at 3c , …rm 3 locates at 2c 3 c right of …rm 3. This way, …rm 2’s pay-o¤ is 3 . We need to show that …rm 2, by locating to the right of 3c , always obtains than 3c . First, consider a ¡ c less ¤ location by …rm 2 somewhere in the interval 3 ; 2c . The analysis of this case 3 corresponds to that of case (b) of the proof of part (iv) above. We showed there that …rm 2 prefers case (b’) (weakly) to case (b), and case (b’) clearly gives …rm 2 a lower payo¤ than 3c . ¡ ¤ Next, consider a location by …rm 2 in the interval 2c . In this case, ; c 3 2cC 2cC we cannot know whether or not x2 · 3 . If x2 · 3 , then x4 > x3 , while if C x2 > 2c3 , then x4 < x3 , by Proposition 13. In any case, max fx3 ; x4 g ¸ x22 , which gives us an upper bound on …rm 2’s payo¤ in this case: ³x ´ x2 2 (11) V2 · L (c ¡ x2 ) + ¡L · I. 2 2 The …rst inequality follows from the argument above. To see that the second inequality holds, di¤erentiate the expression in (11) with respect to x2 to get:

³ ´ i 1h 0 x2 0 1¡L ¡ 2L (c ¡ x2 ) < 0, 2 2 ¡ ¢ where the inequality follows from L0 x22 > 12 , by Lemma 6(ii) and x22 > 3c , and L0 (c ¡ x2 ) > 14 , by Assumption C. Part (ii). In order to prove this, we characterize …rm 2’s optimum location provided it will have both …rms 3 and 4 on its right and thereafter show that …rm 2 can do even better with …rm 3 on its left, while keeping …rm 4 on its right. We do this separately for two cases, depending on whether …rm 3’s location decision to the right of …rm 2 is unconstrained or not. Finally, we show that having both …rms 3 and 4 on its left cannot be optimum for …rm 2. Suppose that both …rms 3 and 4 will locate to the right of …rm 2. Firm 2’s decision is, in this case, subject to three restrictions: (1) a restriction ensuring …rm 3 that …rm 4 is at its right: x3 ¡ x2 · x4 ¡ x3 = c ¡ x3 , or: x3 · x22+c ; (2) a restriction ensuring …rm 3 that …rm 4’s location to the right is at c: x3 ¸ c ¡ I B ; and (3) a restriction ensuring …rm 2 that …rm 4 prefers its location at c to a location to the left of …rm 2: x2 · x4 ¡ x3 = c ¡ x3 , 23

or: x3 · c ¡ x2 . Note that restriction (3) implies restriction (1) whenever: c¡x2 · x22+c , or: x2 ¸ 3c , and vice versa when the opposite holds. Thus, …rm 2 chooses its optimum location, provided both …rms 3 and 4 are on its right, in such a way that …rm 3’s optimum £ location ¤on [x2 ; c] satis…es the combined restrictions c ¡ I B · x3 · min c ¡ x2 ; x22+c . Since, in our case, 3c < I B , there exist values for x3 satisfying both the lower and the upper restriction when x2 = 3c . Thus, we can disregard restriction (1) and write the relevant two conditions as: c ¡ I B · x3 · c ¡ x2 . Note that, taken together, the two restrictions imply: x2 · I B . With both …rms 3 and 4 at its right, the location by …rm 2 may be such that …rm 3’s location decision is constrained or not by the restriction x3 ¸ c ¡ I B . Consider …rst the case where it is unconstrained. If now …rm 2 moves rightward, it entails an even greater movement by …rm 3 rightward, since, with …rm 3’s location problem having an interior solution, dx3 ¡L00 (x3 ¡ x2 ) = 00 > 1; dx2 L (c ¡ x3 ) ¡ L00 (x3 ¡ x2 ) where the inequality follows from the numerator being negative, by Assumption B3, the denominator being negative by …rm 3’s second-order condition, and the absolute value of the numerator being greater than that of the denominator, since L00 (c ¡ x3 ) > 0 by Assumption B3; moreover, if …rm 3’s optimum location makes a jump following a small increase in x2 , then it makes a rightward jump. It follows that …rm 2 pro…ts from moving rightward, as long as x3 · c ¡ x2 , since we here assume that the other constraint, x3 ¸ c ¡ I B , does not bind. We cannot be sure, however, that a value of x2 exists for which the restriction x3 · c ¡ x2 holds with equality, since x3 may jump rightward at a slight increase in x2 . To start with, though, suppose that such a value does exist. Thus, given the provision that both …rms 3 and 4 are to its right, …rm 2 locates such that …rm 3’s subsequent decision is x3 = c ¡ x2 ; denote the two …rms’ locations in this case as x b2 and x b3 , respectively. Clearly, x b2 > 3c , 2c and it follows that x b3 = c ¡ x2 < 3 . Now, …rms 2 and 3 have the same distance between their neighboring …rms: …rm 3’s neighbors are …rm 2 at x b2 and …rm 4 at c, with a distance c ¡ x b2 between them; …rm 2’s neighbors are …rm 1 at 0 and …rm 3 at x b3 , with a distance x b3 = c ¡ x b2 . Firm 3 locates without restrictions on the interval [b x2 ; c], since its decision is an interior solution here. Thus, x b3 ¡ x b2 = arg max¯ u(¯; c ¡ x b2 ). However, it follows from the symmetry noted in Lemma 1(i) and the equality x b3 = c ¡ x b2 , implying x b2 = c ¡ arg max¯ u(¯; c ¡ x b2 ), that, actually, x b2 = arg min¯ u(¯; c ¡ x b2 )! Thus, …rm 2’s payo¤ is less than that of …rm 3 in this situation, and it would prefer changing places with that …rm. It follows that …rm 2 in this 24

case prefers having …rm 3 on its left to having it on its right. Firm 2’s 0 location now would be exactly x b2 = x b3 , as de…ned above, with …rm 3 at 0 x b3 = arg max¯ u(¯; c ¡ x b2 ) < x b2 , so that …rm 2 now has a larger payo¤ than with …rm 3 on its right. To see that this is, in fact, the optimum location of 0 0 …rm 2, note that both x b2 and x b3 are derived from …rm 3’s location decision on an interval of length c ¡ x b2 = x b3 ; thus, the intervals to the right of …rm 0 0 3 in the two situations must be of equal length: x b2 ¡ x b3 = c ¡ x b3 . Since, dx3 with …rm 3 to the left of …rm 2, dx2 < 0 by Proposition 4, a …rm-2 location 0 x2 > x b2 would therefore entail …rm 4 locating just to the left of …rm 2, at x2 . Nor is it optimum to locate further to the left: With …rm 4 at c, …rm 2’s payo¤ is: V2 = L(c ¡ x2 ) + R(x2 ¡ x3 ) = L(c ¡ x2 ) + x2 ¡ x3 ¡ L(x2 ¡ x3 ); and thus: µ ¶ dx3 dV2 0 0 = [1 ¡ L (c ¡ x2 )] + [1 ¡ L (x2 ¡ x3 )] 1 ¡ > 0; dx2 dx2 since the square-bracketed terms are positive by Assumption A and the curvebracketed term is positive (and even > 1) by Proposition 4. To see that V2 > V3 in this case, note that, while …rms 2 and 3 have the same interval 0 0 0 length at their respective right sides, as noted above: c ¡ x b2 = x b2 ¡ x b3 , …rm 0 0 0 2 has a larger interval on its left side: x b2 ¡ x b3 > x b3 , where the inequality follows from the above argument that …rm 3’s location decision is an interior solution. We also have that V3 > V4 , since: 0

0

0

V3 = u(b x3 ; x b2 ) = v(b x2 ) = v(b x3 ) > v(c ¡ x b3 ) ¸ V4 ;

where the …rst inequality follows from the envelope theorem and x b3 > 2c , implying c ¡ x b3 < x b3 ; and c ¡ x b3 is the length of the interval between …rms 2 and 1 on which …rm 4 locates, so that the second inequality must hold. Finally, V4 > V1 , since: 0

b3 ) > u(0; x V4 ¸ u(0; c ¡ x b3 ) = V1 ;

where the …rst inequality is a consequence of …rm 4’s being able to choose the left end of the interval but not doing it, while the second inequality follows 0 from @u > 0 and c ¡ x b3 > x b3 . @I Let us now return to the provision that both …rms 3 and 4 will be locating to the right of …rm 2, and suppose, contrary to the previous discussion, that a value of x2 does not exist for which the restriction x3 · c ¡ x2 holds with equality. Now, …rm 2’s optimum location, provided …rms 3 and 4 are to its 25

right, is some x e2 < x e3 < x b2 , with a corresponding location of …rm 3 at x b3 , such that a further increase in x2 would lead to a jump in x3 from x e3 to some location in the interval (c ¡ x e2 ; xe22+c ]. The lower limit of this interval follows from the argument above that an increase in x2 is pro…table for …rm 2 as long as x3 < c ¡ x2 and the present supposition that an equality is not feasible. The upper limitation follows from the restriction on …rm 3’s location necessary to keep …rm 4 at its right. The two restrictions imply that this jump will only happen if x e2 > 3c , since otherwise the interval into which …rm 3 may jump is an empty set. Notice that, since x e2 > 3c , 2c e2 ; xe22+c ]. 2 (c ¡ x 3 The alternative for …rm 2 to a location at x e2 with both …rms 3 and 4 at its right is to locate in such a way that …rm 3 locates at its left and …rm 4 at its 0 right. Consider the alternative location x e2 = c ¡ x e2 . We need to show that this location is better for …rm 2 than x e2 , which is the best it can have with 0 both …rms 3 and 4 at its right. Note …rst that x e3 . Therefore, …rm 4, e2 > x locating to the right of …rm 2, has a shorter interval to locate on, of length x e2 rather than c ¡ x e3 > x e2 . Thus, …rm 4 is strictly worse o¤ with …rm 2’s new location. Firm 3 is indi¤erent, since it locates on an interval of the same length, only to the left of …rm 2 rather than on its right. Firm 2’s location 0 at x e2 is such that …rm 3, locating to the left of …rm 2, is indi¤erent between x2 locating at x e3 ¡ x e2 and locating somewhere in the interval (c ¡ 2e x2 ; c¡e ]; in 2 any case, this brings …rm 3 at least as close to …rm 1 as …rm 2 would be in the alternative case of …rms 3 and 4 to its right. Thus, …rm 1 is made (weakly) 0 worse o¤ by …rm 2’s move from x e2 to x e2 . All in all, therefore, the e2 = c ¡ x …rms other than …rm 2 are in total made worse o¤ by this move. Since total payo¤ over all …rms is constant, this implies that …rm 2’s payo¤, also in the case of a jump in …rm 3’s behaviour, is larger with …rm 3 on its left and …rm 4 on its right than with both …rms on its right. Suppose next that the lower constraint on x3 binds when …rms 3 and 4 are to the right of …rm 2, so that x3 = c ¡ I B . Firm 2’s optimum location is: ¡ ¢ xL2 = arg max u ¯; c ¡ I B . ¯ ¤ £ subject to ¯ 2 c ¡ 2I B ; I B , where the restrictions are there to keep …rm 4 to the right of …rm 3. The payo¤ of …rm 2hat xL2 is lower than that of …rm 3 at c¡I B . To see this, suppose i B …rst that xL2 2 c ¡ 2I B ; c¡I2 , which is equivalent to: xL2 · c¡I B ¡xL2 · I B , with at least one strict inequality. The …rst inequality here implies that x3 ¡ xL2 ¸ xL2 , the second one that x4 ¡ x3 ¸ x3 ¡ xL2 , with at least one strict inequality. since L0 (I) 2 (0; 1), that V3 > V2 . Suppose next that ³ B It follows, i xL2 2 c¡I2 ; I B . In this case, 2xL2 > c ¡ I B . Since …rm 3 is at c ¡ I B , its 26

payo¤ must be at least as great as if it had been located at 2xL2 . But even at 2xL2 , …rm 3 would have a payo¤ greater than what …rm 2 can have with …rm 3 at c ¡ I B , since x3 ¡ xL2 = xL2 , c ¡ 2xL2 ¸ c ¡ I B ¡ xL2 , and L0 (I) 2 (0; 1). Thus, also in this case, V3 > V2 . It follows that …rm 2 can improve its payo¤ by locating at c ¡ I B and have …rm 3 at its left. The best …rm 3 now can do is locating at xL2 as just described. It follows that V2 > V3 in this case. Finally, we need to show that having both …rms 3 and 4 on its left cannot be optimum for …rm 2 in this case. To see this, suppose …rst that this would imply …rm 4 locating inbetween …rms 3 and 2, which would happen if x2 · 2I C , so that v3R (x2 ) ¸ v3L (x2 ), by Proposition 13. This would clearly not be optimum for …rm ¡ 2 unless ¤ x4 < x2 , which can only be the case if x2 > 2I B . With x2 2 2I B ; 2I C , we know from Lemma 14 that x4 > 3x42 . This provides an upper bound on …rm 2’s payo¤: ³x ´ x2 2 V2 = L (c ¡ x2 ) + x2 ¡ x4 ¡ L (x2 ¡ x4 ) < L (c ¡ x2 ) + =: Ve (x2 ) . ¡L 4 4

This upper bound is decreasing in x2 , since L0 ¸

1 4

by Assumption C:

³x ´ i dVe 1h 2 1 ¡ L0 = ¡ 4L0 (c ¡ x2 ) < 0. dx2 4 4

Thus, since x2 > 2I B in the present case, we have that: ¡ ¢ V2 < Ve 2I B :

If …rm 2 were to locate at 2I B and …rm 4 at 34 2I B = 32 I B , then …rm 2 would B have a market to its right equal to c¡2I B and a market to its left equal to I2 . From our previous discussion, on the other hand, we know that …rm 2 can ensure itself, by locating at I B so that both …rms 3 and 4 locate to its right with …rm 3 at or beyond c ¡ I B , a market to its right greater than or equal to B c ¡ 2I B and a market to its left equal to I > I2 , the latter inequality follows from the observation that, necessarily, x2 > 2I B , which can only occur if B 2I B < c = 3I, or I > 2I3 . Thus, this would be preferred by …rm 2, and both …rm 3 and 4 on its left cannot be optimum for …rm 2 if x2 · 2I C . It follows that, if it is to be optimum for …rm 2 to have both …rms 3 and 4 to its left, it must locate at x2 > 2I C , so that v3R (x2 ) < v3L (x2 ), implying that …rm 4 on its part locates to the left of …rm 3. In order for …rm 3 to keep …rm 4 on its left, …rm 3 must obey the restriction x3 ¸ x22 . This creates an upper bound on …rm 2’s payo¤ in this case: It cannot get more than it would do if …rm 3 locates at x22 ; thus: ³x ´ x2 2 V2 · L (c ¡ x2 ) + =: Vb (x2 ) ¡L 2 2 27

Like in the previous case, this upper bound is decreas ing in x2 : ³ ´ i dVb 1h 0 x2 0 = 1¡L ¡ 2L (c ¡ x2 ) < 0. dx2 2 2

x2 C B B where ¡ ¢the inequality follows from x2 > 2I ¸ 2I0 , so that 12 > I and 0 x2 L 2 by Lemma 6(ii) and Assumption B3, and L (c ¡ x2 ) ¸ 4 by Assumptions C and B3. this property of Vb (x2 ) holds for all x2 ¸ 2I B . Thus, ¡ Clearly, ¢ we can use Vb 2I B as an upper bound on V2 in this case: ¡ ¡ ¢ ¢ ¡ ¢ V2 · Vb 2I B = L c ¡ 2I B + I B ¡ L I B .

We need to show that this upper bound is less than the payo¤ that …rm 2 can get otherwise. ¡ B ¢ overlapping) £ ¤ subcases, ac¡ B ¢ We consider four (possibly B cording to B I . Suppose …rst¡ that 2 0; c ¡ 2I ; without loss ¢ B I B of generality, we proceed as if B I is a singleton. By locating at c ¡ I B , it is possible for …rm 2 to have …rm at I B and …rm 4£ at its ¢ ¡ B ¢¤ ¤ £ 3 Bat its ¡left B B right at c. Consider the interval c ¡ , I B I ; c B I I ; c ¡ ¡ ½ ¡ ¢ £ ¤ where ¡the¢inclusion follows from B I B 2 0; c ¡ 2I B . The restriction ¡ ¢ 0 · B I B · c ¡2I B also implies that c¡ I B ¡ B (I) · 2I B · c¡B I B , so that the interval contains 2I B . However, the optimum location in the interval, which has length I B , is c ¡ I B . Since c < cB , we have that c ¡ I B < 2I B . Thus, we can use Lemma 1(v) to establish that c ¡ I B is preferred £to 2I¤B when the above interval is extended in both directions to become I B ; c . Thus, with neighboring …rms at I B and c, locating at c ¡ I B is better for …rm 2 than locating at 2I B , and having …rm 3 at its left and …rm 4 at its right is better for …rm 2 than having both of them to the left with …rm 4 to the left of …rm 3. £ ¡ ¢ ¤ ¡ ¢ Suppose, secondly, that B I B 2 2 I B ¡ I ; I ; the interval is nonB empty because I > 2I3 , as shown above. By locating at I, it is possible for …rm 2 to have …rm 1 at its left at 0 and …rm 3 at its right at c ¡ I B . This is equivalent, by a rightward shift of I B , £to locating at¡ I B¢+I with neighboring ¡ B ¢¤ B B B B and Consider the interval …rms at c. I + I B I + I B I ¢ ½ ; 2I I ¡ ¡ £ B ¤ ¢ ¤ ¡ ¡ ¢ £ ¡ I ¡ ; c ¢; the inclusion follows ¡from B I¢B 2 ¡2 ¢I B ¡ I ; I . Since 2 I B ¡¡ I¢ · B B B B B that B IB · ¢ ¡ B2¢ 2I ¡ ¢ I · 2I · 2I + I ¡ BB I . ¡ I,B we have ¡ IB ¡ B Since 2 2I ¡ I ¡ B I > 2 I ¡ I , the interval contains 2I . The optimum location in the interval is I B + I. Since I < I B , we have that I B + I < 2I B , and we can again use Lemma 1(v) to establish £that I¤B + I is preferred to 2I B also when the interval is extended to become I B ; c . Thus, having both …rms 3 and 4 at its right, with …rm 4 to the right of …rm 3, is better for …rm 2 than having both …rms 3 and 4 at its left, with …rm 4 to the left of …rm 3. 28

¢¤ ¡ B¢ ¡ B £ B ; this Consider, thirdly, the case where I B I ; 2 I c 2I ¡ 2 ¡ ¡ ¤ case can only occur if IIB 2 23 ; 45 . xxx¡ ¢ ¡ ¤ Consider, …nally, the case where B I B 2 I; I B . xxx A number of comments to this theorem are in order. Note, …rst, that the result can be easily extended to more than four …rms. The …rms that are di¢cult to discuss are the last one, the next to last one, and the …rst strategic player (…rm 2). With more than four …rms, we get a string of …rms locating with equal distance between them the same way as …rm 3 does in the above Theorem. Secondly, we do not have an ordering of the …rms’ pay-o¤ in case (ii) of the Theorem. However, as is clear from the proof at this point, V2 > V3 is the most likely outcome here. Thirdly, we would like to re-emphasize the importance of our Assumption C. By assuming that an I exists such that L0 (I) = 12 , we open up for the relevance of all parts of the Theorem. In particular, if L0 (I) > 12 ; 8I, then there cannot exist any cA , and so case (i) of the Theorem disappears. It may even be the case that cB vanishes, so that also case (ii) disappears. If L0 (I) < 12 ; 8I, then cB cannot exist, and neither can cC , then. Thus, in this case, we are left with cases (i) and (ii) of the theorem. On the other hand, if we were to assume that L0 (I) < 14 for some I, we would open up for many complications. In particular, we would have to consider the possibility that I C < IbC . ³As the ´proof of Proposition 13 indicates, the analysis of cases where I 2 I C ; IbC is quite involved. A condition weaker than the one we introduce in Assumption C would ensure that the equilibrium in part (iii) ¡ ¢ of the Theorem extends also to c 2 cC ; b cC , where b cC = 3IbC . However, without restrictions on L beyond those stated in Assumptions A and B3, pinning down the equilibrium outcome in this region of the parameter space would be di¢cult.

6

Concluding remarks

We have in this paper suggested a new approach to strategic location problems: one not based upon consumers’ transportation costs, but rather upon the indi¤erent consumer. It is how the location of the indi¤erent consumer varies with the length of the interval between two …rms that drives our analysis. We apply this idea to the case of sequential loation around the circle. We are able to derive a unique equliibrium outcome for a very general formulation of the indi¤erent consumer on an interval. Firms are spread more or less evenly across the circle, except for the …rst and last …rms. But this

29

spread is the result of strategic consideration on behalf of each of the entering …rms, and not because of any rule of thumb imposed upon the …rms, such as locating in the middle of the available interval. As our analysis points out, locating in the middle is very often the best solution for strategic reasons, in order to keep the …rms entering after you at bay, although it is never optimum for the …rm without anyone entering after it.

References [1] Anderson, S.P. and M. Engers (2001), ”Preemptive Entry in Di¤erentiated Product Markets”, Economic Theory 17, 419-445. [2] Cancian, M., A. Bills, and T. Bergstrom (1995), “Hotelling Location Problems with Directional Constraints: An Application to Television News Scheduling”, Journal of Industrial Economics 43, 121-124. [3] Dewatripont, M. (1987), “The Role of Indi¤erence in Sequential Models of Spatial Competition: An Example”, Economics Letters 23, 323-328. [4] Eaton, B.C. and R.G. Lipsey (1989), “Product Di¤erentiation”, Handbook of Industrial Organization, Vol. 1 (R. Schmalensee and R.D. Willig, eds.), Elsevier, Amsterdam, 723-768. [5] Economides, N. and J. Howell (1991), ”Does It Pay To Be First? Sequential Choice of Locations of Di¤erentiated Products”, Working Paper EC-91-24, Stern School of Business, New York University. [6] Edwards, C.H. and D.E. Penney (1990), Calculus and Analytic Geometry, 3rd Ed., Prentice-Hall, Englewood Cli¤s, NJ. [7] Frutos, M.A.d., H. Hamoudi, and X. Jarque (1999), ”Equilibrium Existence in the Circle Model with Linear Quadratic Transport Cost”, Regional Science and Urban Economics 29, 605-615. [8] Gabszewicz, J.J. and J.-F. Thisse (1992), “Location”, Handbook of Game Theory with Economic Applications, Vol. 1 (R.J. Aumann and S. Hart, eds.), Elsevier, Amsterdam, 281-304. [9] Kats, A. (1995), ”More on Hotelling’s Stability in Competition”, International Journal of Industrial Organization 13, 89-93. [10] Kats, A. (1999), ”Spatial Oligopolies with Sequential Entry in Circular Spaces”, unpublished manuscript, Virginia Tech. 30

[11] Nilssen, T. (1997), ”Sequential Location When Transportation Costs Are Asymmetric”, Economics Letters 54, 191-201. [12] Nilssen, T. and L. Sørgard (1998), “Time Schedule and Program Pro…le: TV News in Norway and Denmark”, Journal of Economics and Management Strategy 7, 209-235. Erratum: 8, 161-162. [13] Nilssen, T. and L. Sørgard (2000), “A Public Firm Challenged by Entry: Duplication or Diversity?”, unpublished manuscript, University of Oslo and Norwegian School of Economics and Business Administration; available at http://www.uio.no/~toreni/research. [14] Pal, D. (1998), “Does Cournot Competition Yield Spatial Agglomeration?”, Economics Letters 60, 49-53. [15] Peitz, M. (1999), ”The Circular Road Revisited: Uniqueness and Supermodularity”, Working Paper 99-06, IVIE, Valencia. [16] Prescott, E.C. and M. Visscher (1977), “Sequential Location among Firms with Foresight”, Bell Journal of Economics 8, 378-393. [17] Salop, S.C. (1979), “Monopolistic Competition with Outside Goods”, Bell Journal of Economics 10, 141-156. [18] Vickrey, W.S. (1964), Microstatics, Harcourt, Brace and World, New York. [19] Vickrey, W.S. (1999), “Spatial Competition, Monopolistic Competition, and Optimum Product Variety”, International Journal of Industrial Organization 17, 953-963.

31

Figure 1

u(β,I)

I/2

β I

Figure 2 u(β,I)

I/2

β I

Figure 3

I/2

B(I) I