Coalition Building in a Spatial Economy

Journal of Urban Economics 47, 136᎐163 Ž2000. Article ID Juec.1999.2139, available online at http:rrwww.idealibrary.com on

Coalition Building in a Spatial Economy1 Joachim Henkel Institut fuer Inno¨ ationsforschung und Technologiemanagement, Uni¨ ersity of Muenchen, Ludwigstr. 28 RG, D-80539 Muenchen, Germany

Konrad Stahl Department of Economics, Uni¨ ersity of Mannheim, Seminargebaeude A5, D-68131 Mannheim, Germany

and Uwe Walz 2 Department of Economics, Uni¨ ersity of Tubingen, Mohlstrasse 36, ¨ D-72074 Tubingen, Germany ¨ Received December 9, 1997; revised September 21, 1998

We analyze the possibility and consequences of coalition formation among suppliers of retail services. We first provide a framework in which producers of substitutes have an incentive to cluster in marketplaces to attract consumers dispersed in space. Owing to spatial externalities, the resulting spatial equilibrium can be welfare suboptimal. We characterize regimes in which we find too little and those in which there is too much agglomeration of firms. We analyze the role of coalitions of firms Že.g., initiated by a land developer. in this framework and show that such coalitions can overcome the suboptimality of the decentralized spatial allocation. 䊚 2000 Academic Press Key Words: endogenous agglomerations; monopolistic competition; coalition building. 1 We gratefully acknowledge comments from participants at the 1996 Gerzensee Summer Workshop in Economic Theory, the Bellaterra Seminar in Barcelona, the 1996 EEA Meeting in Istanbul, the Annual Conference of the Verein fur ¨ Socialpolitik and its regional economics seminar in Vienna, in particular Salvador Barbera, ` Patrick Bolton, Raquel Fernandez, Jossi Greenberg, Kai-Uwe Kuehn, Patrick Rey, and Harald Uhlig. We also thank two anonymous referees and the editor for their very helpful comments on an earlier version of the paper. The usual disclaimer applies. 2 Author to whom correspondence should be addressed.

136 0094-1190r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.

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1. INTRODUCTION The spatial allocation of economic activity is governed by the interaction of agglomerative and disagglomerative forces. These forces are generated from what one might call spatial externalities. An example of an agglomerative force is the case in which the profitability of a location to a firm may be enhanced if additional households demanding its nontradable supply decide to locate there. However, upon locating, individual households typically do not take such an enhancement into account. Owing to such spatial externalities central to regional and urban economics analysis, suboptimality of decentralized spatial allocation decisions is the rule rather than the exception. To improve allocation decisions in space, the intervention of regional or urban government authorities is typically called for. These authorities have to internalize the spatial externalities, to restore the optimality of the spatial allocation. However, there are many obstacles to determining an optimal allocation and to implementing it. Information problems are probably most important. It is also not clear whether regional and urban government authorities have an incentive to target the suboptimal allocation. It might rather be in their parochial interest to even worsen decentralized allocation decisions from a welfare point, e.g., by attracting economic activity to the location under their jurisdictions Žsee, for example, Walz and Wellisch w24x.. In this paper, we focus on an alternative way to cope with the suboptimality of spatial allocation decisions. Rather than delegating this task to government authorities, we consider the possibility of coalition formation among private economic agents and their effects on spatial allocation decisions. We start from our earlier observation that some of the central market failures are due to spatial externalities not internalized by decentralized individual decision making. In addition we make use of the idea that coalitions of agents do have an incentive to internalize these externalities at least partially. Toward this end, we analyze the behavior of coalitions in a specific spatial context and investigate whether coalition formation can improve on the spatial allocation. If this is the case, then it might be preferable from a public policy point of view to allow for, or even to strengthen, the incentives for coalition formation. Indeed, we will show that some market failures can be resolved at least partially within the private sector. Below we focus on the retailing sector as an empirically relevant example of resolution of the allocation problem. There, shopping center developers take such a coordination function. Our analysis extends that of Henderson w12x, who looks at the role of land developers in a setting with perfect competition, to a situation in

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which imperfect competition prevails and spatial externalities exist. Whereas in our model the land developer has the role of overcoming a coordination failure of the market, the land developer in the Henderson w12x paper can avoid the consequences of ‘‘bad politics.’’ By stressing coordination failures of the market our paper also relates, at least in spirit, to the work of Rauch w16x, who analyzes the role of land developers to overcome initial spatial inertia. 3 We model the emergence of retailing markets involving specialized sellers of differentiated products, by combining two branches of the literature in which agglomeration incentives of firms are analyzed. On one hand we apply aspects of industrial organization approaches, in which the location decisions of small numbers of firms in space are analyzed via game-theoretic methods. They usually adopt a continuous space Hotelling-type framework.4 In these models, consumers are continuously distributed in space. On the other hand we make use of regional economics approaches in which monopolistically competitive firms select from a discrete set of locations. Typically, the by now classic framework of Dixit and Stiglitz w5x is adopted.5 We present a monopolistic competition model where consumers are distributed uniformly along a line and thus are differentiated by income net of their transaction costs of patronizing a market. For simplicity of exposition, firms are restricted to locating at the line’s end points. We concentrate on the formation of agglomerations by specialized Žone product. retailers. Upon patronizing one of these points called market places, at a cost purely dependent on distance, consumers buy the utility-maximizing commodity bundle. Just as in Dixit and Stiglitz w5x or Stahl w20x, consumers are differentially attracted to the marketplace offering a larger commodity bundle. We allow for an endogenous determination of the number of firms in each agglomeration. Our basic approach resembles the work of Church and Gandal w4x, who analyze the emergence of technical standards. We modify their approach in a number of ways Že.g., with respect to the formulation of transport costs. and adopt it to a spatial economics setting.

Similarly, Helsley and Strange w11x look at the implications of the actions of limited developers Ži.e., developers do not control the entire city size. who provide a local public good Žinfrastructure .. Since some of the benefits of infrastructure spill over to areas outside the developers’ control, this prevents an efficient allocation in their analysis. However, they also leave issues of imperfect competition in goods markets aside. 4 See, e.g., Eaton and Lipsey w6x, Stahl w19, 20x, de Palma et al. w15x, Economides w7x, Economides and Siow w8x, Gehrig w10x, and Schulz and Stahl w18x. 5 See, e.g., Fujita w9x, Krugman w14x, and Walz w22, 23x. 3

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We derive conditions under which in equilibrium a single agglomeration andror two Žsymmetric. agglomerations arise. We show that as consumers’ income increases and thus access costs become relatively less important, a single agglomeration is the likely equilibrium outcome. The same tendency arises with decreasing substitutability between the goods. We also characterize situations in which multiple equilibria arise. Equilibria with two agglomerations of different sizes are not found. We also determine the welfare preferred equilibria under a simple additive welfare criterion. For parameter regimes under which multiple equilibria occur, the one actually arising may be inferior as it regards to welfare. In this case, there is room for the welfare-increasing coalition formation considered then. Welfare improvement via coalitions is most promising if a welfare inferior singleagglomeration equilibrium arises. In this case a coalition of firms has an incentive to voluntarily defect from the single agglomeration equilibrium and establish the second marketplace. Welfare-improving coalition formation is more demanding when it comes to changing a symmetric two-market equilibrium into a welfare-superior single-market one. There, only in a few cases does coalition formation represent a way to overcome the suboptimal market outcome. The paper is structured as follows. Section 2 contains the model description. In Section 3 we deduce the equilibrium allocations. The welfare comparison of equilibria is conducted in Section 4. Section 5 contains the analysis of coalition formation and its welfare effects. We summarize and conclude with Section 6. 2. THE MODEL The geographical space considered here is a line of unit length. Possible marketplaces are restricted to locations at the ends of the line. This assumption is less restrictive than it seems to appear. Our approach is identical to a circle model in which firms can only locate on the circle line as long as at most two marketplaces arise. In the circle model the case of two symmetric locations Ži.e., locations lying on the ends of a radial line. is identical to our approach with both locations at the end of the line. Accordingly, the case in which a single marketplace is located on the circle line corresponds exactly to our case with one marketplace at one of the line’s ends. Rather than using the more realistic circular model Žan example is shopping malls in a circular suburbia surrounding a city center that is prohibitively costly to cross., we choose to use the line model with marketplaces restricted to the ends, which proves to be simpler from an analytical point of view. We refer to these markets as 0 and 1. A continuum of consumers of mass M is uniformly distributed along the line. Each consumer is endowed

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with income Y spent on transportation to one of the markets and on purchasing commodities there.6 Transportation costs tz are linear in distance z. Income is large enough to cover the expenses for the longest possible trip, i.e., Y ) t. Each consumer maximizes utility, 1r ␣

n

U Ž x1 , . . . , x n . s

x i␣

žÝ /

0 - ␣ - 1,

,

Ž 1.

is1

where n denotes the number of goods consumed and x i , i s 1, . . . , n, respresents the quantity bought of good i. The parameter ␣ describes the degree of substitutability between the goods: the larger ␣ is, the closer substitutes they are. The demand of a consumer living at z for good i Ž i s 1, . . . , n k . at location k Ž k g  0, 14. is xik s

␤ y1 piy1r k ␤ k Ý njs1 py1r jk

Ž Y y t < z y k <. ,

Ž 2.

where ␤ s 1 y␣ ␣ . pi k stands for the price of good i at location k. The term Ž Y y t < z y k <. equals total consumption expenditures Žincome minus transportation costs. of the consumer. Using Ž1. and Ž2., the indirect utility function of a consumer living at z and shopping at location k is given by V Ž p, z, k . s qk Ž Y y t < z y k < . ,

Ž 3.

r␤ .␤ k where qk [ ŽÝ nis1 py1 . ik The consumer indifferent between shopping at locations 0 and 1 is located at z*. With Ž3. we can express z* by 7

z* s 6

y Ž m ␤ y 1 . q1 m␤ q 1

,

Ž 4.

The assumption that consumers shop in one marketplace only is perfectly justified in the case of minimal diversity of products between the two locations Žproducts supplied in the smaller market are also supplied in the larger one.. In this case, no consumer would ever travel to both locations. This not unrealistic case displays the concept of hierarchical marketplaces Žsee Christaller w3x.. It can be shown that with maximum diversity of products between the locations Žthe products in the two locations are completely different., all equilibria identified in the following analysis still exist Žfor a detailed analysis, see Henkel w13x.. 7 Of course, this expression and the one above only hold if the r.h.s. of Ž4. lies in w0, 1x; otherwise, z* is either 0 or 1.

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with y [ Yty1 ) 1 and m [ Ž q0rq1 .1r ␤ . The market demand for good i at location 0 and location 1, respectively, is then X i0 s X i1 s

y1r ␤ y1 pi0

␤ 0 Ý njs1 py1r j0 y1r ␤ y1 pi1 1 Ý njs1

␤ py1r j1

ž ž

M

H0

z*

Ž Y y zt . dz ,

/

Ž 5.

1

Hz* Ž Y y Ž 1 y z . t . dz

M

/

.

Ž 6.

Our firms trade one good each and incur a fixed cost F as well as constant variable costs c. A firm trading good i at location k maximizes profits Gi k s Ž pi k y c . X i k y F ,

Ž 7.

by choosing the optimal price ps

c

␣

' c Ž ␤ q 1. ,

Ž 8.

which is derived under the assumption of a large number of firms, implying that strategic interactions are neglected. Backward substitution of Ž8. implies m s n 0rn1 , such that Ž4. can easily be reinterpreted: the larger the relative number of goods provided in location 0, the larger its market area, due to consumers’ preference for variety Ž ⭸ z*r⭸ m ) 0.. Furthermore, Ž4. reveals that the larger is Y, or the smaller is t, the more attractive the location with the larger number of goods, as income lost for transportation purposes becomes relatively less important: ⭸ z*r⭸ Y ) Ž-.0 and ⭸ z*r⭸ t - Ž).0 as m ) Ž-.1. Inserting Ž5., Ž6., and Ž8. in Ž7. gives us the typical firm’s profits at the respective location8 : G 0 Ž n 0 , n1 . s

␤ Mz*

Ž ␤ q 1. n0

ž

Yy

tz* 2

/

yF

Ž 9.

and G 1 Ž n 0 , n1 . s

␤ M Ž 1 y z* .

Ž ␤ q 1 . n1

ž

Yy

t Ž 1 y z* . 2

/

y F.

Ž 10 .

In principle we model the interaction between firms and consumers in three consecutive stages. In the first stage, firms decide simultaneously about entry and location. In the second, firms choose their prices, and in the last stage, consumers choose the locations to shop. Since we have 8

We drop the firm index, as firms operating at one location perform symmetrically.

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already analyzed the last two stages, we concentrate now on the first stage and the resulting spatial equilibria. 3. THE NUMBER OF MARKETPLACES 3.1. Equilibrium Candidates In our model, there are two basic types of configurations characterized by Ž n 0 , n1 ., the number of firms in either marketplace: those in which firms locate at both ends of the line Žwe call this a fragmented configuration. and those with concentration of all firms at one of the ends Žwe call this an agglomerated configuration.. A necessary condition for the former to constitute an equilibrium is that firms’ profits are equalized across both markets. We can express the profit differential with the help of Ž4., Ž9., and Ž10. as 9 G 0 y G1 s

␤ Mt 2 Ž ␤ q 1. n0 Ž m ␤ q 1.

2

Ž y Ž m ␤ y 1. q 1 .Ž y Ž m ␤ q 3. y 1 .

qm Ž Ž y y 1 . m ␤ y y .Ž Ž 3 y y 1 . m ␤ q y . . Ž 11 . Let g Ž m. denote the expression in square brackets.10 g Ž m. s 0 constitutes the necessary equilibrium condition. We indicate in Appendix 1 that this condition is satisfied for at most three values of m. Figure 1 displays the two potential shapes of the g Ž m. function. If goods are poor substitutes, g Ž m. intersects the m axis only once at m s 1 Žline A.. Conversely, if goods are close substitutes, we find three points of

FIG. 1. Examples for g Ž m.. 9 Like Eq. Ž4., the following equation is only correct if the r.h.s. of Ž4. lies in w0, 1x. However, it can be shown that G 0 y G1 s 0 can arise only for z* g Ž0, 1.. 10 We henceforth neglect the integer problem and treat n 0 and n1 as continuous variables.

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intersection of the g Ž m. function with the m axis Žline B.: one at m s 1 and the other two at m1 - 1 and the corresponding m 2 s 1rm1.11 Hence, we have three potential types of equilibria: symmetric fragmentation configurations Žwith an equal number of firms in the two locations., asymmetric fragmentation configurations Žwith an unequal but positive number of firms in the two marketplaces., and agglomeration configurations Žwith all firms concentrated in one marketplace.. We show in the following that the second type of configuration never constitutes an equilibrium, whereas the other two configurations may arise as spatial equilibria. 3.2. The Fragmentation Equilibria For a configuration involving n k firms at location k to be an equilibrium, no firm must have an incentive to deviate. Suppose n 0 and n1 firms decide to enter at locations 0 and 1, respectively, and profits are the same in both locations, i.e., g Ž m. s 0 Žwith m s n 0rn1 ., and are driven to zero: G 0 Ž n 0 , n1 . s G1Ž n 0 , n1 . s 0. Then, Ž n 0 , n1 . constitutes an equilibrium if no firm has an incentive to switch location and no potential entrant finds entry profitable. In Appendix 2 we prove PROPOSITION 1. Ži. There exists a function ␤ b Ž y ., characterizing the fragmentation threshold, with ⭸␤ br⭸ y - 0. If for gi¨ en y, the degree of substitutability exceeds that associated with the fragmentation threshold Ž ␤ ␤ b Ž y .., there exists a unique equilibrium in¨ ol¨ ing an identical number of firms at the two locations. Hence, a symmetric fragmentation equilibrium is more likely the lower the typical consumer’s income, the higher her transportation costs, and the higher the substitutability between the goods. Žii. In the symmetric fragmentation equilibrium, n 0 s n1 s ␤ M 2Ž ␤ q 1 . F Ž Y y tr4. \ nŽ2. firms will locate in either marketplace. Furthermore, we prove in Appendix 3 that an asymmetric fragmented configuration does not constitute an equilibrium: PROPOSITION 2. A configuration Ž n 0 , n1 . with g Ž m. s 0, g ⬘Ž m. ) 0 implies incenti¨ es for some firms to de¨ iate. Hence, there is no asymmetric fragmentation equilibrium, since g ⬘Ž m. ) 0 ᭙ m with g Ž m. s 0 and m / 1. 3.3. Agglomeration Equilibria We now address the possibility that the concentration of all firms in a single location Ž m s 0 or m s ⬁. constitutes an equilibrium. Let ␮ [ MtrF. We draw the lines only for the range m g w0, 1x, because for m g w1, ⬁x they are symmetric in 1rm. 11

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Appendix 4 contains the proof for PROPOSITION 3. Ži. For sufficiently large ␮ , there exists a function ␤ g Ž y, ␮ ., characterizing the agglomeration threshold, with ⭸␤ gr⭸ y - 0. A degree of substitutability smaller than the one associated with the agglomeration threshold Ž i.e., ␤ G ␤ g Ž y, ␮ .. implies that there exist agglomeration equilibria with all firms concentrated at one of the line’s ends. Hence, an agglomeration equilibrium is more likely the higher the typical consumer’s income, the lower her transportation costs, and the poorer the substitutability 1 Ž . firms between the goods. In the agglomeration equilibria, nŽ1. s ␤␤␮ q1 yy 2 locate in the single agglomeration. Žii. If the degree if substitutability is just between the agglomeration and fragmentation threshold Ž ␤ g w ␤ g Ž y, ␮ ., ␤ b Ž y .w ., an agglomeration and a symmetric fragmentation equilibrium coexist. Žiii. In that regime, consumers enjoy a larger ¨ ariety of goods with an agglomeration equilibrium. Howe¨ er, those li¨ ing farther away ha¨ e to bear the burden of higher transport costs. Figure 2 shows the function agglomeration threshold as the lower curve, where, as an example, ␮ s 500 has been chosen.12 The second curve, the fragmentation threshold, is the upper boundary of the area of stability of a symmetric configuration with zero profits. Hence, the parameter space is divided into three regimes.13 In region A, a symmetric fragmentation configuration constitutes an equilibrium, while an agglomeration configuration does not. In B, both configurations are equilibria, while in C an agglomeration configuration is an equilibrium and a symmetric fragmentation configuration is not.

FIG. 2. Areas of fragmentation Ž A, B . and agglomeration equilibria Ž ␮ s 500.. 12

Our numerical analysis provides ample evidence that for sufficiently large ␮ a unique ␤ g Ž y, ␮ .-curve exists that fulfills ␤ g Ž y, ␮ . - ␤ b Ž y .. 13 Any other value of ␮ yields qualitatively the same result, provided it is not too small. A higher value of ␮ shifts the boundary between areas A and B downward.

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Since only symmetric fragmentation and agglomeration equilibria exist, Figure 2 completely describes the equilibrium regimes in our game. In concluding the discussion of equilibrium regimes, we observe finally that a replication of the consumer sector leaves the fragmentation threshold unchanged but moves the agglomeration threshold toward the axes. With further replication, the parameter region in which the symmetric fragmentation equilibrium is the only one disappears and the agglomeration equilibrium always arises as at least one equilibrium type. This is due to the fact that a larger number of consumers increases the absolute difference between the equilibrium number of goods supplied in one market and in two markets, making the agglomeration disadvantage of transport costs increasingly unimportant. The same happens when the firms’ entry cost F is reduced: the firms need only a smaller market share to cover their cost, which implies an increase in the regime in which the agglomerated configuration is an equilibrium. The effect of independent variations of t cannot be studied as easily, as y varies in t as well. However, it can be shown that the effect of a change in t via ␮ is more than compensated by the effect via y. Hence the regime in which both the agglomerated and the fragmented equilibria exist increases, and that in which only the fragmented one obtains decreases with a decrease in consumers’ transaction costs. 4. WELFARE ASPECTS We now turn to a welfare comparison of the symmetric fragmentation and the agglomeration configuration. We employ a simple additive welfare function in which all consumers’ surpluses are weighed equally. Producer surplus does not arise, as profits are zero in both types of equilibria due to free entry. We derive in Appendix 5 PROPOSITION 4. Ži. The superior of the two types of configurations as regards welfare is always an equilibrium. Žii. There exists a degree of substitutability just between the ones associated with the fragmentation and agglomeration thresholds Ž ␤ w Ž y . gx ␤ g Ž y, ␮ ., ␤ b Ž y .w. such that for any gi¨ en y, ␤ - ␤w Ž y . implies a higher welfare le¨ el for the symmetric fragmentation equilibrium, and ␤ ) ␤w Ž y . implies a higher le¨ el of welfare for the agglomeration equilibrium. Hence, the decentralized decisions of firms lead to an inferior solution as regards welfare if, with ␤ g w ␤ g , ␤w w Ž ␤ gx ␤w , ␤ b w., an agglomeration Ž symmetric fragmentation. equilibrium emerges. Why do welfare inferior equilibria arise at all? The reasoning is as follows. In their respective location decisions, firms do not properly take into account consumers’ interests. They only consider the marginal con-

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sumer Žvia the market size effect., while the interests of the remaining consumers are left aside. Specifically, suppose that the symmetric fragmentation equilibrium is welfare preferred, but consider a firm’s move that increases the asymmetry between the agglomerations. This increases utility for those customers who live close to the larger location and decreases it for those far away who still prefer the smaller location. For a relatively low heterogeneity of the goods and low income, the second effect dominates such that the concentration of firms tends to be less preferable. Conversely, let the agglomeration equilibrium be welfare preferred, and let a deviating firm’s move increase the symmetry between the locations. Then, at relatively low degrees of substitutability, or high consumer income, the utility increase for those enjoying increased variety is outweighed by the disadvantages to the customers consuming less variety, such that the symmetric fragmentation equilibrium tends to be welfare inferior. Before turning to the analysis of coalitions we should note that the size of the regime in which an agglomeration equilibrium is welfare inferior increases with population density. This is due to the fact that ␤w is invariant in ␮ , whereas ␤ g decreases with ␮ Žsee our discussion in 3.3.. Hence, our analysis coincides very well with the intuition that it is not feasible to achieve welfare-improving decentralization of economic activity when population size increases. 5. COALITIONS In many real cases an urban developer or a big investor in a Žplanned. shopping mall coordinates the actions of single firms. In the preceding sections, such a coordinating agent was assumed away. Now, we consider the possibility of coalitions of firms. To do this, we proceed in three steps. In the first one, we introduce a game-theoretic concept which allows us to investigate the stability of spatial equilibria with respect to profitable deviations of coalitions. In the second step, we analyze the incentives of coalitions to deviate from an equilibrium as described in the preceding sections. In the third step, we ask for the welfare effects of such deviations. It seems straightforward to assume that coalition building becomes more difficult and costly when the size of the coalition increases. One might think about transaction costs which have to be incurred to get agents to agree on joint behavior and costs to set up enforceable contracts. Therefore, we use the size of the smallest coalition profitably altering its location as a measure of the stability of a configuration. The smaller the critical coalition, the easier it is for the intermediary coordinating the location decisions of firms to organize a sufficient number of firms to switch locations jointly. We assume the existence of binding and enforceable contracts.

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More specifically we use a modified version of Aumann’s w1x strong Nash equilibrium ŽSNE. to analyze the stability of configurations against deviations of coalitions of a given maximum size. A strategy profile s* characterizes a SNE, if no partition J of the set of players and a strategy profile ˜s J exist, such that each member of J will strictly gain by deviating from sUJ to ˜ s J . Thereby, gains are not transferable among players. In the present context, however, this does not restrict the generality of the analysis, because of the symmetry between the players.14 By definition a SNE in Aumann’s sense is stable against deviations of all feasible coalitions. However, considering only coalitions of maximum size k, 1 F k F N, we pursue matters based on the following. DEFINITION w k-SNEx. Suppose the existence of a game ⌫ s Ž I, Ž S i . i g I , Ž G i . i g I . with the set of players I s  1, . . . , N 4 , where G i denotes the profit function of player i and S i characterizes her set of strategies. Then the strategy profile s* g G i g I S i is a k-SNE if and only if for all J : I with < J < F k Ži.e., for all coalitions with at most k members. and for all s J g G j g J S j, no i g J exists such that G i Ž s J , sUyJ . ) G i Ž s*.. According to this definition the set of all 1-SNEs is equivalent to the set of Nash equilibria. In the case of N players the definitions of N-SNE and Aumann’s SNE coincide. Sinceᎏby definitionᎏthe set of k-SNEs is a subset of all Nash equilibria, only the equilibria derived in Section 3 are candidates for a k-SNE. The analysis of k-SNEs is facilitated by the observation that it suffices to look at coalitions which only include firms moving jointly from one location to the other Žsee the proof in Appendix 6.. Including entrants into the coalition andror firms switching location in the other direction weakens the profitability of the coalition. Given our definition of a k-SNE, we ask now for the minimum size of the coalition which can profitably deviate from a single marketplace ŽSection 5.1. and from a fragmentation equilibrium ŽSection 5.2.. 5.1. Coalition Building in the Presence of a Single Marketplace By definition, a single firm cannot profitably deviate from an agglomeration equilibrium with nŽ1. firms located in one marketplace and zero firms at the other location. This equilibrium with Ž nŽ1., 0. firms implies zero profits. However, the joint change of location of 50% of firms from Ž nŽ1., 0. is always profitable Žsee Stahl w19x.: the coalition receives, as before its move, half of the total revenues, which have, however, increased. The 14 We show later on that the only deviations of interest are the ones in which the strategy of all coalition members is the same before and after deviation. Due to the symmetry between firms, deviating is strictly profitable for all or for none.

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increase in revenues is due to an increase in total consumption expenditures from M Ž Y y tr4. to M Ž Y y tr2. as a consequence of lower total transport costs. Hence, the configuration Ž nŽ1., 0. can never be a nŽ1.-SNE and consequently does not constitute a SNE either. The crucial question is: for which k and which parameter combination is Ž nŽ1., 0. a k-SNE? We can restrict the analysis to the case of a location change of a coalition from 0 to 1. Suppose a fraction r of firms choose a coordinated location change from 0 to 1, the marketplace with initially zero firms. The condition that this move does not leave the firms worse off is G1 Ž Ž 1 y r . nŽ1. , r nŽ1. . G 0.

Ž 12 .

Recalling that equality in Ž12. is equivalent to Mt ␤ Ž m q 1 . 2 nŽ1. Ž ␤ q 1 . Ž m ␤ q 1 .

2

Ž m ␤ Ž 1 y y . q y .Ž m ␤ Ž 3 y y 1. q y . y F s 0, Ž 13 .

with m being defined here as Ž1 y r .rr, Eq. Ž13. leads us to Žsee Appendix 7.

␤ 1 Ž y, r . s ln

ž

r 1yr

y1

/

ln

ž'

2yy1

y y 2 ry q r 2

/

y1 .

Ž 14 .

Location changes are profitable if ␤ - ␤ 1Ž y, r . Žcf. Appendix 7.. The index delineates coalition building with initially one marketplace, in contrast to coalition building in the presence of two marketplaces Žcf. Section 5.2.. What are the properties and the shape of the ␤ 1-function? For the limit values r ª 0 and r ª 0.5 we can derive from Ž14. lim ␤ 1 Ž y, r . s 0,

rª0

lim ␤ 1 Ž y, r . s ⬁,

rª0.5

᭙ y.

Ž 15 .

Hence, for a given y and ␤ , there always exists an r g w0, 0.5x such that ␤ - ␤ 1. Figure 3 illustrates the function ␤ 1Ž y, r . for different values of r. For instance, with parameter combinations below the lowest curve, a coordinated change of location from the agglomerated Nash equilibrium is profitable if more than 10% of the firms cooperate. With more than 40% cooperating, a change in location is even profitable in a part of parameter space where an equilibrium with two marketplaces does not exist Ž ␤ g w ␤ b Ž y ., ␤ 1Ž y, 0.4.w ..

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FIG. 3. Below ␤ 1Ž y, r .: profitable change of location from 0 to 1 of a coalition of size r s 0.1, r s 0.4, respectively, starting from Ž nŽ1., 0..

In Appendix 7 we prove PROPOSITION 5. Consider the configuration Ž nŽ1., 0.. For all r gx0, 0.5w and all y ) 1 there exists a ␤ 1Ž y, r . such that G1ŽŽ1 y r . nŽ1., r nŽ1. . G 0 m ␤ F ␤ 1Ž y, r .. That is, a coalition of size rnŽ1. can change to location 1 without incurring losses if and only if ␤ F ␤ 1Ž y, r .. Furthermore, ⭸␤ 1Ž y, r .r⭸ y - 0, and lim y ª⬁ ␤ 1Ž y, r . s 0 holds. Therefore, the configuration Ž nŽ1., 0. represents for the parameter values Ž y, ␤ . a Ž rnŽ1. .-SNE if and only if no ˜ r F r exists such that ␤ - ␤ 1Ž y, ˜ r .. Numerical evidence indicates that ⭸␤ 1Ž y, r .r⭸ r ) 0. If this is the case, then Ž nŽ1., 0. is a Ž rnŽ1. .-SNE if and only ␤ G ␤ 1Ž y, r .. The economic intuition behind the effects of r and y on the profitability of defection is the following. With decreasing income, transport costs become relatively more important for consumers living farther away from the larger marketplace. Hence, attracting these consumers to the second Žsmaller. marketplace is facilitated, making the defection of a coalition of given size more likely to be profitable. On one hand a coalition of larger size Žlarger r . is able to attract more consumers Ža larger market area.. But on the other hand, for a given number of consumers, competition for market shares increases with r in the second location. However, our numerical analysis indicates that the first, profitability-enhancing effect dominates the second, which operates in the opposite direction. Taken together, we have found that the larger the degree of substitutability Žthe smaller ␤ . and the larger the coalition size Ž r ., the more likely a profitable deviation of a coalition of firms becomes. Furthermore, it has turned out that large enough coalitions can deviate from an agglomeration equilibrium even above the fragmentation threshold.

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5.2. Coalition Building with Two Marketplaces For which k, y, and ␤ is the symmetric fragmentation configuration with zero profits and nŽ2. firms in either location a k-SNE? A coalition of size rnŽ2. can profitably switch location from 0 to 1 if G1 Ž Ž 1 y r . nŽ2. , Ž 1 q r . nŽ2. . ) 0,

Ž 16 .

with m s Ž1 y r .rŽ1 q r .. Equality in Ž16. leads to the function Žsee Appendix 8 for details.

␤ 2 Ž y, r . [ ln

ž

1qr 1yr

y1

/

ln

ž'

2 Ž 2 y y 1.

2 Ž 2 y y 1. y r Ž 4 y y 1.

/

y 1 . Ž 17 .

Location changes are profitable if ␤ ) ␤ 2 Ž y, r . Žcf. Appendix 8.. This is a necessary condition for a profitable formation of a coalition of size rnŽ2.. By construction, it is sufficient if z* gx0, 1w . However, as soon as the market of decreasing size at 0 can no longer attract consumers Ž z* s 0., Ž17. is no longer valid. With a border solution with no consumer on location 0, profits of a representative member of the deviating coalition can be written as G1 Ž Ž 1 y r . nŽ2. , Ž 1 q r . nŽ2. . s ␤ Ž ␤ q 1.

y1

Mt Ž y y 1r2 . r Ž nŽ2. Ž 1 q r . . y F.

Ž 18 .

␤ Mt Hence, by using nŽ2. s 2 Ž ␤ q 1 . F Ž y y 1r4. Žsee part Žii. of Proposition 1. in Ž18., condition Ž16. can be expressed as

y ) ymin Ž r . [

3yr 4Ž 1 y r .

m r - rmax Ž y . [

4yy3 4yy1

.

Ž 19 .

The condition r - rmax Ž y . can readily be understood: the number of firms nŽ2. Ž1 q r . at the larger marketplace after the move is limited by consumers’ total net income, which decreases because of increased transportation costs. Together, the conditions Ž17. and Ž19. are necessary and sufficient. The limit values of ␤ 2 Ž y, r . for minimum and maximum Žsensible . coalition size

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COALITION BUILDING

FIG. 4. Above and to the right of yminŽ0.8. and ␤ 2 Ž y, 0.8.: profitable deviation of a coalition of 80% of nŽ2., starting from Ž nŽ2., nŽ2. ..

are lim ␤ 2 Ž y, r . s

rª0

lim

rª Ž4 yy3 .r Ž4 yy1 .

4yy1 2 Ž 2 y y 1.

␤ 2 Ž y, r . s ln

ž

y yy1

2

/

' ␤b Ž y . , ln Ž 2 Ž 2 y y 1 . .

Ž 20 . y1

.

Ž 21 .

Figure 4 illustrates the restrictions Ž17. and Ž19. for r s 80%, together with ␤ b Ž y . Ždotted.. The formation of a coalition and deviation from the initial equilibrium is profitable to the right and above the limiting curves Žfor instance, for parameter values given at point B in Fig. 4.. Given that ␮ is not too small there exists a parameter region below ␤ b Ž y . in which multiple equilibria arise. Our above results can be summarized in PROPOSITION 6. Consider a configuration Ž nŽ2., nŽ2. .. For any r gx0, 1w there exists a ␤ 2 Ž y, r . and a yminŽ r ., such that G1ŽŽ1 y r . nŽ2., Ž1 q r . nŽ2. . G 0 m ␤ G ␤ 2 Ž y, r . n y ) yminŽ r .. That is, a coalition of size rnŽ2. can profitably switch to location 1 if and only if ␤ ) ␤ 2 Ž y, r . and y ) yminŽ r .. Furthermore ⭸␤ 2 Ž y, r .r⭸ y - 0 and lim y ª⬁ ␤ 2 Ž y, r . s 0. The last sentence of this proposition is shown in Appendix 8. That ␤ 2 decreases with y Žor that for given ␤ a profitable defection of a coalition of size r becomes more likely with increasing y . is just the consequence of the fact that with increasing y, transport costs become relatively less important. This makes it easier for firms in the larger market to attract more consumers living farther away. In turn, this implies higher profits in the larger market. Defection of a coalition of a given size becomes more profitable.

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In a nutshell, we have shown that profitable defections of coalitions are possible for large enough income levels and sufficiently low degrees of substitution Žlarge enough ␤ ’s.. Our analysis also shows that coalitions may find it profitable to deviate from a fragmentation equilibrium. 5.3. Welfare Aspects of Coalition Formation By analyzing equilibrium regimes in Section 3, we found that in a certain portion of the Ž y, ␤ . parameter space both types of equilibria coexist. In the regime ␤ g w ␤ g , ␤w w ᎏ call it B1 ᎏthe agglomeration equilibrium is suboptimal. The same is true for the fragmentation equilibrium in the regime ␤ g w ␤w , ␤ b x called B2 . What implications does the possible formation of coalitions have for the implementation of preferred equilibria, i.e., the ones leading to a higher welfare level? Does the formation of coalitions favor welfare optimal equilibria? It does indeed, as the following proposition shows for the case of agglomeration equilibria. PROPOSITION 7. Ži. For any y ) 1 and any r gx0, 0.5w there exists a function ␤˜1Ž y, r . such that a configuration Ž nŽ1., 0. is a Ž r nŽ1. . ᎐SNE iff ␤ G ␤˜1Ž y, r . and ␤ G ␤ g Ž y, ␮ .. The function ␤˜1Ž y, r . is monotonically increasing in r, lim r ª 0 ␤˜1Ž y, r . s 0, and lim r ª 0.5 ␤˜1Ž y, r . s ⬁. Žii. If, for gi¨ en y, the formation of coalitions of the maximum relati¨ e size r makes an agglomeration equilibrium disappear that is superior to the fragmentation configuration Ž i.e., one in area B2 or C , ␤ ) ␤w Ž y .., then it makes all inferior ones Ž those in area B1 , ␤ g w ␤ g Ž y, ␮ . , ␤w Ž y .w . disappear. The proof of this proposition is contained in Appendix 9. Hence, the formation of coalitions is welfare improving in the sense that if the maximum relative size r of coalitions is increased from zero, then, for given y, the ‘‘first’’ concentrated equilibria to disappear are those that are suboptimal compared to the symmetric equilibrium. To grasp the economic intuition behind this result, it is helpful to recall the reasons for the suboptimality of the decentralized uncoordinated spatial allocation. The suboptimality is due to the fact that by deciding upon their location, individual firms consider only the marginal consumer and the marginal market area effect but disregard the impact on all other consumers and firms. The coalition, in turn, partially internalizes this external effect, by taking the market area effect on all coalition members into account. In doing so, the coalition gains most if the coordination problem is rather pronounced, i.e., if the spatial allocation in the uncoordinated equilibrium is suboptimal. The question arises if we can obtain a similar result for symmetric equilibria. The situation here is more complicated, since we have two conditions for a profitable deviation of a coalition ŽŽ17. and Ž19.. instead of

COALITION BUILDING

153

just one. Furthermore, the limit behavior of ␤ 2 Ž y, r . toward the minimum and maximum value of r ŽŽ20. and Ž21.. is not as simple as that of ␤ 1Ž y, r . Ž15.. Nevertheless, we can prove the following. PROPOSITION 8. Ži. For any y ) 1 and any r gx0, rmax Ž y .w there exists a function ␤˜2 Ž y, r . such that a configuration Ž nŽ2., nŽ2. . is a Ž r nŽ2. . ᎐SNE iff ␤ F ␤˜2 Ž y, r . and ␤ - ␤ b Ž y, ␮ .. The function ␤˜2 Ž y, r . is continuous and monotonically decreasing in r, lim r ª 0 ␤˜2 Ž y, r . s ␤ b Ž y ., and ␤˜2 Ž y, rmax Ž y .. F lnŽ yrŽ y y 1..rlnŽ2Ž2 y y 1... Žii. If, for gi¨ en y, the formation of coalitions of the maximum relati¨ e size r gx0, rmax Ž y .w makes a symmetric equilibrium disappear that is superior to the concentrated configuration Ž i.e., one in area A or B1 , ␤ - ␤w Ž y .., then it makes all inferior ones Ž those in area B2 , ␤ gx ␤w Ž y ., ␤ b Ž y .w . disappear. The proof of this proposition is presented in Appendix 10. Proposition 8 closely resembles Proposition 7; again, if the maximum relative size r of coalitions is increased from zero, then, for given y, the ‘‘first’’ symmetric equilibria to disappear are those that are suboptimal compared to the concentrated equilibrium. In this sense here, too, the possibility of coalition formation has a welfare-improving effect. For given y and coalitions of any size possible, a symmetric configuration with zero profits is stable against deviations of coalitions iff ␤ F ␤˜2 Ž y, rmax Ž y .. and ␤ - ␤ b Ž y .. Numerical evidence suggests that for y F y 1 f 1.984 we have ␤˜2 Ž y, rmax Ž y .. s ␤ b Ž y ., and hence no symmetric equilibrium is affected by coalition formation. In the range y 1 - y F y 2 f 3.852 we found ␤˜2 Ž y, rmax Ž y .. g w ␤w Ž y . , ␤ b Ž y .w , which means that only suboptimal symmetric equilibria disappear because of coalition formation. Finally, for y ) y 2 , we have ␤˜2 Ž y, rmax Ž y .. - ␤w Ž y .: all suboptimal equilibria and part of those superior to the agglomeration configuration disappear when coalitions of any size are possible. Taking the above findings together we can state that the possibility of coalition formation, by destroying suboptimal equilibria ‘‘more easily’’ Ži.e., with smaller maximum size of the coalition. than optimal ones, favors an efficient outcome of the game and, hence, has a welfare-improving effect. We focused in our analysis on the possibility of market-oriented solution of coordination failures in a spatial context by looking at coalition formation by private agents. It should be clear, however, that if one abstracts from information and incentive problems of regional authorities, it is theoretically feasible to propose a tax subsidy scheme which induces the establishment of the optimal spatial solution. The basic idea is to induce Ždeter. entry in Žfrom. the preferred Žnonpreferred. location via the use of subsidies Žtaxes.. Given our highly stylized model, this could be done in a number of ways. One could think about sales taxes and subsidies or land

154

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taxes and subsidies and the like. Since this is not the focus of our analysis, however, we will not elaborate on this in more detail. 6. SUMMARY We analyzed in this paper the endogenous formation of market places by Ži. uncoordinated decisions and Žii. by coalitions of firms. The central tradeoff discussed in the present model is between agglomeration advantages to consumers arising from increased product variety and the agglomeration disadvantages to the firms that arise from increased competition for market shares. More specifically, the larger the number of products available in an agglomeration, the larger the access costs consumers are willing to incur to participate in this marketplace, thereby increasing the market area captured by the agglomeration. However, the more firms that are located at the same location, the smaller the market share per firm. Both agglomerative as well as disagglomerative force are accompanied by spatial external effects. Hence, we find that in cases in which multiple equilibria coexist, equilibria resulting from the uncoordinated decisions of agents can be suboptimal. It turns out that allowing for Žor even promoting. coalition formation seems to have the most clear-cut welfare-enhancing effects in the case of a suboptimal agglomeration equilibrium Že.g., coordinated by an investor or a land developer.. Coordinated action will destroy the decentralized equilibria with the lower welfare level first. In a setting with suboptimal symmetric fragmentation equilibria, welfare-improving activities are also possible. Hence our model suggests that coordinated action via a coalition formation is a potential alternative to local authorities trying to improve on spatial allocation decisions. The story told here can be naturally varied and applied to a specific context. For instance, one could again start with an inefficient entry and location equilibrium, and then ask for the minimum size Žin terms of number of commodities. a multiproduct seller should take to distort that equilibrium toward the Žwelfare. preferred one.15 The essence of the story remains unchanged: almost by definition, coalitions have a greater incentive to internalize spatial externalities than individual agents making uncoordinated decisions. This leaves room for a private sector solution, even in the presence of spatial externalities. Furthermore, we could have allowed firms to locate at the center. However, beyond the fact that this would favor the tendency to agglomerate, this would not alter our basic principle: namely that in view of the ‘‘demand externalities’’ that generically arise in spatial economies, cooperation between firms can have a very strong potential to improve the state of affairs. 15

Observe that the resulting move to the new equilibrium would involve the exit of some specialized firms.

COALITION BUILDING

155

However, a final caveat is in order. So far, we have considered only the private coordination of location decisions, and not one involving, for instance, product selection or price decisions. All these decisions undertaken by specialized sellers generate externalities. In particular, the unilateral introduction of a product or a unilatural price decrease leads to an expansion of the market size captured by a marketplace.16 In contrast to the coalition of specialized sellers considered in our paper, the aforementioned multiproduct seller will have the potential to internalize these externalities, at the cost of Žpartially. monopolizing the Žlocal. market. In fact, while we have discussed here the interaction between sellers of final commodities and consumers, all of the principal arguments can be recast in a context of interacting firms. Thus there is much room for the analysis of an intriguing subject, namely that of finding an optimal structure for the organization of industry in space. APPENDICES Appendix 1: Deri¨ ation of  m4 such that g Ž m. s 0 Note that g Ž0. s Ž1 y y .Ž3 y y 1. - 0 Žsee Ž11.. and lim m ª⬁ g Ž m. s q⬁ ) 0. Furthermore, we find g Ž1. s 0, which follows from symmetry. If goods are poor substitutes Žlarge ␤ ., g Ž m. intersects the m axis exactly once, namely, at m s 1.17 Conversely, if goods are close substitutes Žsmall ␤ ., we find a local maximum in Ž0, 1. Žand a corresponding minimum in Ž1, ⬁... In this case, g Ž m. intersects the m axis three times; namely at mU1 - 1, at mU2 ) 1, with mU1 s 1rmU2 Žin either case from below., and at mU3 s 1 Žfrom above.. For y G 1 q 2y1 r2 we can show that g Ž m. has either one or three roots.18 While we have not been able to extend the proof to the full range of y, we derived ample numerical evidence that the result holds for any y gx1, ⬁w . Appendix 2: Proof of Proposition 1 Ži. We first note that profits per firm at, say, location 0 are higher the fewer firms there are at the other location. Hence, if further entry at that location is profitable, then all the more profitable is a switch from the other location. When looking for profitable deviations from a possible equilibrium configuration, we can thus restrict our attention to players switching locations. These externalities an described in detail in Stahl w21x and Schulz and Stahl w18x. By symmetry it suffices to look at the range m g w0, 1x in analyzing the zeros of g Ž m.. 18 A proof is available upon request. 16 17

156

HENKEL, STAHL, AND WALZ

For strictly positive n 0 , n1 to constitute an equilibrium, it is necessary and sufficient that Gi Ž n 0 , n1 . s 0

for i s 0, 1,

G 0 Ž n 0 q 1, n1 y 1 . F 0,

and

Ž A.1.

G1 Ž n 0 y 1, n1 q 1 . F 0.

Ž A.2.

Inequalities ŽA.1. and ŽA.2. stipulate that a firm, upon changing from a zero profit location, earns negative profits at its new location. Let n [ Ž n 0 q n1 .r2, ⌬ [ Ž n 0 y n1 .r2. We rewrite conditions ŽA.1. and ŽA.2. in differential notation as19

⭸ G 0 Ž n, ⌬ . ⭸⌬ ⭸ G1 Ž n, ⌬ . ⭸⌬

- 0,

Ž A.3.

) 0.

Ž A.4.

Let us now turn first to the situation where the number of firms is the same in the two locations. Clearly, this is a candidate for an equilibrium, since g Ž1. s 0 Žsee Ž11... In this case, conditions ŽA.3. and ŽA.4. are equivalent. Using z*Ž n, ⌬ s 0. s 0.5 and

⭸ z* Ž n, ⌬ . ⭸⌬

⌬s 0

s

␤ n

ž

yy

1 2

/

,

Ž A.5.

the necessary and sufficient condition for an equilibrium is

⭸ G 0 Ž n, ⌬ . ⭸⌬

⌬s 0

s

␤t 8 Ž ␤ q 1 . n2

2

2 ␤ Ž 2 y y 1 . y Ž 4 y y 1 . - 0.20

Ž A.6. 19 While in ŽA.1. and ŽA.2. equality is sufficient to guarantee that there are no incentives to switch locations, this is not the case in differential notation: even if a single firm, as compared to the whole marketplace, is small enough to justify the differential approach, the gains from a switch of location may be positive, if small in second order, when the derivatives in ŽA.3. or ŽA.4. vanish.

157

COALITION BUILDING

We can now deduce from Eq. ŽA.6. the region in the Ž y, ␤ .-parameter plane where a symmetric fragmentation configuration constitutes an equilibrium as the area below the function

␤b Ž y . s

4yy1 2 Ž 2 y y 1.

2

.

Ž A.7.

Žii. Plugging Ž4. in Ž9. and Ž10. yields for a zero-profit situation with an equal number of firms in the two locations the expression in Proposition 1. Q.E.D. Appendix 3: Proof of Proposition 2 Suppose there exists an asymmetric solution involving m ˜ [ ˜n 0rn˜1 and G0 Ž ˜ n0 , ˜ n1 . s G1Ž ˜ n0 , ˜ n1 . s 0. W.l.o.g. look at ˜ n0 - ˜ n1. Hence, z* - 0.5. With a deviation of a single firm from location 1 to location 0, m increases and the marginal consumer moves closer to the center Žsee Ž4... Hence, a smaller fraction of total income is spent on transportation, leading to larger overall profits for the given total number of firms. With g ⬘Ž m. ) 0 the profit differential between the two agglomerations becomes positive: G0 Ž ˜ n 0 q 1, ˜ n1 y 1. y G1Ž ˜ n 0 q 1, ˜ n1 y 1. ) 0. With increasing overall profits, G 0 Ž ˜ n 0 q 1, ˜ n1 y 1. q G1Ž ˜ n 0 q 1, ˜ n1 y 1. ) 0. It follows that G0 Ž ˜ n 0 q 1, ˜ n1 y 1. ) 0. Hence, g ⬘Ž m. ) 0 is a sufficient condition for profitable deviation and, thus, for the instability of the corresponding configuration. Since g ⬘Ž m. ) 0, ᭙ m / 1, asymmetric equilibria do not exist. Q.E.D. Appendix 4: Deri¨ ation of Proposition 3 Ži. W.l.o.g. consider the situation Ž0, nŽ1. ., where nŽ1. is determined from the zero profit condition nŽ1. s

␤ ␤q1

M Ž Y y tr2 . rF s

␤␮ ␤q1

ž

yy

1 2

/

,

Ž A.8.

with ␮ [ MtrF. No single firm changes location if this yields negative profits. As z* s 0 if nŽ1. y 1 G Ž yrŽ y y 1..1r ␤ , this is always the case if the number of firms is sufficiently large. The defecting firm is not able to attract consumers and thus will not be able to cover its fixed costs.21 For 20 It turns out that in the case of two symmetric markets, the condition g ⬘ - 0 is equivalent to ŽA.6.. This is due to the fact that total profits remain the same with a marginal deviation from the symmetric solution. Hence, the deterioration of profits in the agglomeration the defecting firm switches to, relative to the profit situation in the agglomeration it defects from, guarantees that defection is unprofitable. In general, however, ŽA.6. implies g ⬘ - 0, but not vice versa. Hence, g ⬘ ) 0 is a sufficient condition for a configuration not to be an equilibrium; see Proposition 2.

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HENKEL, STAHL, AND WALZ

simplicity of exposition we restrict the defecting firm to set its price to p s cr␣ .22 From Ž9. we obtain as the condition for an unprofitable deviation of a single firm

␤␮ z* y y

ž

z*

/

2

F ␤ q 1,

Ž A.9.

where z* is implicitly defined by Ž4. and ŽA.8.. Equality in ŽA.9. implicitly defines a function ␤ g Ž y, ␮ ., such that an agglomerated configuration with zero profits is an equilibrium iff ␤ G ␤ g Ž y, ␮ .. Žii. Combing part Žii. of Proposition Ž1. and part Ži. of Proposition Ž3. gives us Žii.. Žiii. nŽ1. ) nŽ2. Žsee Proposition Ž3. and ŽA.8... Q.E.D. Appendix 5: Deri¨ ation of Proposition 4 Denote by W Ž1. and W Ž2. the total welfare in the case of one and two locations, respectively. Total welfare under the alternative equilibrium regimes is easily calculated from Ž3.,23 W

Ž1.

s

M Ž nŽ1. .

␤

c Ž ␤ q 1.

ž

t

Yy

2

/

s

M 1q ␤␤ ␤ c Ž ␤ q 1.

Ž ␤ q1 .

F␤

ž

Yy

ž

Yy

t 2

1q ␤

/

, Ž A.10.

and W

Ž2.

s

M Ž nŽ2. .

␤

c Ž ␤ q 1.

ž

Yy

t 4

/

s

M 1q ␤␤ ␤ 2 ␤c Ž ␤ q 1.

Ž ␤ q1 .

F␤

t 4

1q ␤

/

. Ž A.11.

Hence, the condition for the agglomeration equilibrium to be superior Žinferior. concerning welfare to the fragmentation equilibrium is 2 ␤Ž y y

1 1q ␤ 2

.

) Ž -. Ž y y

1 1q ␤ 4

.

,

Ž A.12.

21 The condition for nŽ1. to be ‘‘large’’ is different, however, from the condition justifying the approximation p s cr␣ . From a numerical analysis we obtained that in a considerable portion of the parameter space a single firm, defecting from an agglomeration of 100 firms, will obtain positive profits when defecting to location 0, while the approximation p s cr␣ is less than 1% off the correct value. Hence, it is not inconsistent to maintain p s cr␣ , while taking the finiteness of n1 into account when considering a defecting firm. 22 It would do better when correctly optimizing its price. While this would slightly diminish the area of stability of a single agglomeration in parameter space, the results would remain qualitatively unchanged. 23 Strictly speaking, ŽA.10. and ŽA.11. show the consumers’ aggregate utility. However, as there is no outside good, the utility derived from not shopping at all can be considered as zero.

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COALITION BUILDING

which can be solved for ␤ :

␤ ) Ž - . ␤w Ž y . \

ln Ž 1 q 1r Ž 4 y y 2 . . ln Ž 2 y 2r Ž 4 y y 1 . .

.

Ž A.13.

The curve ␤w Ž y . separating the two welfare regimes divides region B into two parts. It is decreasing in y, and for y ª ⬁ it runs asymptotically to Ž y4 ln 2.y1 . In all, there are two instances in which the free market outcome can be suboptimal: in regime B1Ž ␤⑀ w ␤ g , ␤w w . the agglomeration equilibrium is inferior with relation to welfare, and in B2 Ž ␤⑀ x ␤w , ␤ b w. the same is true for the symmetric fragmentation equilibrium. Q.E.D. Appendix 6 Because of symmetry among firms a strategy profile may be described by a triple s [ Ž n 0 , n1 , n ¨ ., with n 0 q n1 q n ¨ s N and n␶ Ž␶ s 0, 1, ¨ . describing the number of firms choosing location 0 or 1, respectively, or stay out of the market altogether. Once again for symmetry reasons a coalition of size k is described by a triple Ž ˜ n0 , ˜ n1 , ˜ n ¨ ., with 0 F ˜ n0 F n0 , 0 F ˜ n1 F n1 , 0 F ˜ n ¨ F n ¨ , und ˜ n0 q ˜ n1 q ˜ n ¨ s k. A deviation of a coalition from the strategy profile s can be described with the help of the matrix

˜n 00 n10 M[ ˜ ˜n ¨ 0



˜n 01 ˜n11 ˜n ¨ 1

˜n 0 ¨ ˜n1¨ , ˜n ¨ ¨

0

whereby, for example, ˜ n 01 denotes the number of coalition members using initially the strategy 0 and choosing, after deviation, strategy 1. Hence, we find for the sum of the first Žsecond, third. row of M ˜ n0 , Ž ˜ n1 , ˜ n ¨ .. If there are coalition members who play the same strategy before and after deviation, their participation in the coalition is without consequence for the payoffs of the other players. This implies that we can restrict our search for profitable deviations of coalitions to the case ˜ n 00 s ˜ n11 s ˜ n¨ ¨ s 0. A deviation is profitable for a coalition if each member of the coalition gains from the deviation. Starting from a situation with zero profits, no coalition member can gain by abandoning market entry, since this would yield zero profits, too. Therefore, it suffices to consider the case ˜ n0 ¨ s ˜ n1¨ s 0. Suppose ˜ n 01 G ˜ n10 ) 0 Žor, equivalently, ˜ n10 G ˜ n 01 ) 0. and a profitable deviation of the coalition. Then, the deviation of the smaller coalition with Ž˜ n 01 y ˜ n10 . firms switching from 0 to 1 and none from 1 to 0 is profitable, too. This enables us to focus on situations in which either ˜ n 01 s 0 or ˜n10 s 0.

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HENKEL, STAHL, AND WALZ

If the deviation of a coalition with ˜ n 01 ) 0 and ˜ n¨ 0 ) 0 Ž ˜ n10 ) 0 and ˜n ¨ 1 ) 0. is profitable, then this is even more the case with a deviation of a smaller coalition with no further market entry ˜ n¨ 0 Ž ˜ n ¨ 1 .. We can limit ourselves on matrices M which contain either solely positive ˜ n 01 and ˜ n¨ 1 or solely positive ˜ n10 and ˜ n¨ 0 . Finally, suppose it is profitable for a coalition of ˜ n 01 firms to deviate from location 0, with ˜ n ¨ 1 entrants choosing location 1. Then, deviation is even more profitable if min n 0 , ˜ n 01 q ˜ n ¨ 14 firms switch from location 0 to 1 and no market entry occurs. It thus suffices to analyze matrices M in which only ˜ n 01 or ˜ n10 is positive. Q.E.D. Appendix 7: Proof of Proposition 5 With the help of ŽA.8. we can rewrite Ž13. as 2

Ž m q 1 . Ž m ␤ Ž 1 y y . q y .Ž m ␤ Ž 3 y y 1 . q y . G Ž 2 y y 1 . Ž m ␤ q 1 . . Ž A.14. From this, we obtain the limiting curve ␤ 1Ž y, r . Žwith m s Ž1 y r .rr .

␤ 1 Ž y, r . s ln

ž

r 1yr

y1

/

ln

ž'

2yy1

y y 2 ry q r 2

/

y1 .

Ž A.15.

The coalition does not loose after its switch of location if ␤ F ␤ 1Ž y, r .. The limitation to r - 0.5 is straightforward since a change of location is always profitable with r s 0.5. Accordingly, ␤ 1Ž y, r . diverges with r ª 0.5 Žfrom below. m m ª 1 Žfrom above., since ln m ª 0. The slope of the limiting curve therefore is

⭸␤ 1 Ž y, r . ⭸y

s

yŽ1 y 2 r . q r ln Ž rr Ž 1 y r . . Ž rr Ž 1 y r . .

␤ 1Ž y , r .

Ž y 2 y 2 ry q r .

3r2

- 0,

Ž A.16. since 1 y 2 r ) 0. Finally, we have to show that lim y ª⬁ ␤ 1Ž y, r . s 0. This follows straightforwardly from ŽA.15., since the argument of the second logarithmic term converges to 1 with y ª ⬁. Q.E.D.

161

COALITION BUILDING

Appendix 8: Proof of Proposition 6 With the help of Ž10. and Proposition Ž1Žii.. we can rewrite G1ŽŽ1 y r . nŽ2., Ž1 q r . nŽ2. . G 0 as 2

2 Ž m q 1 . Ž m ␤ Ž 1 y y . q y .Ž m ␤ Ž 3 y y 1 . q y . G Ž 4 y y 1 . Ž m ␤ q 1 . .

Ž A.17. With an equality sign in ŽA.17. the following limiting curve emerges ␤ 2 Ž y, r . Žwith m s Ž1 y r .rŽ1 q r ..:

␤ 2 Ž y, r . s ln

ž

1qr 1yr

y1

/

ln

2 Ž 2 y y 1.

ž'

2 Ž 2 y y 1. y r Ž 4 y y 1.

/

y 1 . Ž A.18.

After the change of location a coalition does not lose if ␤ G ␤ 2 Ž y, r .. A further condition guarantees y ) yminŽ r ., that total maximum revenues which can be achieved at the enlarged marketplace Ž Mt Ž y y 1r2.. suffice to cover fixed costs of Ž1 q r . nŽ2. firms. We find for the slope of ␤ 2 Ž y, r .

⭸␤ 2 Ž y, r . ⭸y

2 Ž 2 y y 1. q r

s y4 ln

ž

1qr 1yr

/ž

1qr 1yr

␤ 2Ž y , r .

/

Ž Ž 2 y y 1. 2 y r Ž 4 y y 1. .

- 0.

3r2

Ž A.19.

Because the argument of the second logarithmic term in ŽA.18. converges with y ª ⬁ to 1, lim y ª⬁ ␤ 1Ž y, r . s 0 holds. Q.E.D. Appendix 9: Proof of Proposition 7 Define

␤˜1 Ž y, r . [

max ␤ 1 Ž y . .

␳ g x0, r x

Ž A.20.

By definition this function has all the features required by Proposition 7Ži.. Proposition 7Žii. follows directly from monotonicity of ␤˜1Ž y, r . in r and from the fact that, for given y, the points Ž y, ␤ . in parameter space, where a suboptimal agglomeration equilibrium exists Ž ␤ - ␤w Ž y .., lie below those where the agglomeration equilibrium is optimal Ž ␤ ) ␤w Ž y ... Q.E.D.

162

HENKEL, STAHL, AND WALZ

Appendix 10: Proof of Proposition 8 Define

␤˜2 Ž y, r . [

min ␤ 2 Ž y . .

␳ g x0, r x

Ž A.21.

␤˜2 Ž y, r . fulfills by definition all the requirements of Proposition 8Ži. Žsee Ž20. and Ž21... The second part of the proposition follows from the monotonicity of ␤˜2 Ž y, r . in r and from the fact that, for given y, the points Ž y, ␤ . in parameter space, where a suboptimal fragmentation equilibrium exists Ž ␤ ) ␤w Ž y .., lie above those where the symmetric equilibrium is optimal Ž ␤ ) ␤w Ž y ... Q.E.D.

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