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Annals of Operations Research ISSN 0254-5330 Ann Oper ResDOI 10.1007/s10479-014-1704-5

On agglomeration in competitive locationmodels

Vladimir Marianov & H. A. Eiselt

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Ann Oper ResDOI 10.1007/s10479-014-1704-5

On agglomeration in competitive location models

Vladimir Marianov · H. A. Eiselt

© Springer Science+Business Media New York 2014

Abstract Agglomeration of facilities that compete with each other is common in practice,which suggests the existence of forces driving facilities to locate in clusters. Shopping centersand food courts are everyday examples. Although these agglomeration forces have beenadequately analyzed and explained in the economic literature, operational research locationmodels have not taken them into consideration as of today. This is particularly troublesome, aslocations prescribed by these models are rather dispersed, which is in blatant disagreementwith the examples that can be observed in real life. We present a selective review of theeconomic literature dealing with agglomeration forces acting in a linear market, classifyingthese forces into weak and strong. This paper demonstrates the sensitivity of competitivelocation models with respect to some assumptions that cause agglomeration or dispersion.

Keywords Competitive location · Agglomeration · Dispersion · Von Stackelberg and Nashequilibria

1 Introduction

The literature on facility location deals with finding optimal sites for both private and publicfacilities. Since the seminal work Stability in Competition by Hotelling (1929), attention hasbeen paid to the location of competing facilities. Hotelling found that stable equilibriumlocations, at which none of the firms could increase its profit by changing its location or priceexists, with both facilities sited next to each other at the center of the market, i.e., a case ofcentral agglomeration. The fact that an equilibrium is reached with both facilities clusteredhas been called the Principle of Minimum Differentiation. This term derives its name from

V. Marianov (B)Department of Electrical Engineering, Pontificia Universidad Católica de Chile, Santiago, Chilee-mail: marianov@ing.puc.cl

H.A. EiseltFaculty of Business Administration, University of New Brunswick, Fredericton, NB, Canadae-mail: haeiselt@unb.ca

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brand positioning. Hotelling’s own example comprises two cider makers that offer one brandeach. On an axis representing the degree of sweetness, the two brands of cider are located atpoints that correspond to their sweetness. Similarly, customers are located at different pointsof the axis, representing their taste for sweetness, usually referred to as their “ideal point”.Assuming that customers choose the cider that is closer to their individual taste, it is easyto see that, in order to maximize their market share, both cider producers will try to changetheir content of sweetness so to move towards each other over the line, reaching equilibriumon the middle point of the line. Note that this model does not involve prices, although sucha feature could potentially be included as well. Furthermore, it is also worth mentioning thatthis brand positioning model is similar to the political positioning problem, in which twoparties attempt to position themselves or their candidates on a “left-right” dimension, so asto maximize their respective number of votes.

In practice, competitive facilities agglomerate. Assuming that past planners have (mostlyintuitively) chosen good solutions (see, e.g., Burkey et al. (2012) for the case of hospitallocations), a good measure of the quality of a prescriptive theory is whether or not it canreasonably well recreate the observed solutions. Some authors found in Hotelling’s principleof minimum differentiation the ultimate reason for all kinds of agglomeration. Hotellinghimself provided examples ranging from politics (where the political platforms of Democratsand Republicans are deemed close to each other), to product characteristic/brand positioningsuch as furniture, shoes, or the aforementioned cider), and religious organizations, such asMethodists and Presbyterians. Even more enthusiastic though, were some of Hotelling’sdisciples, e.g., Boulding (1966), who claims the principle to be of the “utmost generality,”explaining the spatial clustering of dime stores; agglomeration of industrial facilities; andthe similarity of automobile brands and religions.

However, shortly after Hotelling published his paper, the principle of minimum differen-tiation was shown to depend critically on some of the features of the model. Modifying someof the settings of the model resulted in very different solutions. For example, retaining alinear market with two competing facilities, but now assuming that the uniformly distributedcustomers are no longer willing to pay just anything for the product under consideration,dramatically changes the results. In the cider analogy, this would mean that people are notwilling to drink any cider, if both brands have a sweetness that is too far from the customer’staste and, in this case, facilities do not cluster, unless the linear market is very short. Thus,agglomeration no longer appeared to be the natural order of things.

Hundreds of papers have followed Hotelling’s original contribution, authored byresearchers in different fields and focusing on different aspects of competitive location.In general, researchers in the economic field have contributed with theoretical, descriptivepapers, aiming at finding whether there is locational equilibrium under different settings, andanalyzing the resulting location patterns. Marketing researchers and geographers have beeninterested in the best models for finding the trade areas of competitive facilities, by studyinghow customers react to attractiveness of the facilities/products and to the travel distance.Researchers in the transportation field have analyzed the influence of facility locations onthe choice of activities and travel patterns of consumers. Operations researchers’ (and thus,by extension, our) goal is to prescribe locations that optimize some objective.

Whatever the particular angle of the researchers is, most contributions in the field ofcompetitive location analysis use many of Hotelling’s original assumptions. While mostresearchers (except for economists, who need simple spaces for their very involved analyses)use more general spaces, such as two-dimensional Euclidean spaces or networks, manycontributions still assume that customers patronize the store closest to them (or the least

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expensive, in case prices differ among firms) and that they make special trips to the facilitythey patronize.

However, practical problems include factors that lead to a much stronger agglomerationthan that prescribed by operations researchers’ models. Optimal locations obtained fromprescriptive models, on the other hand, do not show any special tendency to agglomeration,even though in real life, clusters are very common.

The objective of this paper is to show that there are forces that lead to agglomeration,which have not been taken into consideration in prescriptive models. It is our hope that thiswork will generate a discussion about the subject of agglomeration and dispersion, and maybenew lines of research for facility locators in the field of operations research, leading to theinclusion of the neglected forces. With this goal in mind, we here present a selective reviewof the literature, highlighting what, in our view, are the main results related to agglomerationand dispersion in competitive location, as it has been described using models on a line, byresearchers mainly in the economic field.

The remainder of this paper is organized as follows. Section 2 discusses agglomeration(and, to a lesser extent, dispersion) in practice. Section 3 describes the main components ofcompetitive location models, and Sect. 4 surveys the basic competitive models of Hotellingand von Stackelberg, along with some extensions. Section 5 is the main part: it first introducesforces that govern the location of facilities and its two subsections explore the workings ofthese forces in detail. Further issues on the subject are explored in Sect. 6 and a summaryand some further thoughts are offered in Sect. 7.

2 Agglomeration in practice

The agglomeration of facilities is everything but new. Take, for instance, the case of cities.Their present shape is the result of a very long dynamic process, in which different forceshave been exerted on the facilities. At first, people found it convenient to cluster so as to con-duct their affairs in one place and avoid long and costly travel: proximity fosters interactionsbetween manufacturing firms offering jobs and exchanging intermediate goods, commercialfacilities offering goods and services to people, and people (consumers) looking for jobs,shopping and leisure (Anas and Kim 1996). This is the first sign of agglomeration based ontravel costs. In addition to transportation costs, the place where agglomeration occurs couldbe determined by immobile factors of production and economies of scale in production(Krugman 1991a, b). Another issue involves security: walled cities resulted from smaller,more compact, places being much easier to defend as opposed to drawn-out settlements.Different combinations of these factors cause cities to organize around one or more concen-trations of employment, production, shopping and leisure. The resulting mono- or polycen-tric structures are consequence of history and equilibria between centrifugal and centripetalforces. A strong interaction between consumers, firms and commercial facilities pushes activ-ities to agglomerate in one Central Business District, to take advantage of economies of scale.

As cities grew, people moved to the suburbs and started commuting. With that behavior,congestion, and with it travel time and costs, increased when customers patronized the cities’central business districts. This resulted in new population centers being established outside ofthe traditional city. The firms followed and located their facilities at these new centers, makingit unnecessary for the customers—except for very specialized goods—to travel to the citiesat all. (Historically, Lösch (1954), described this process for lower-order goods, i.e., thosethat were purchased frequently, and higher-order goods, i.e., those that are more expensiveand less-frequently purchased). Other, more recent, developments concern the establishment

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of new shopping centers in many places in Europe. Based on the lower costs of land at oreven beyond the outskirts of larger cities, individual (typically big box) stores were located.Outliers so far, and seemingly incidents of dispersion. However, soon other stores locatednext to the pioneering facility and a new center emerged.

Similar observations can be made on outliers in other fields. One example concerns fastfood outlets and their “location” on the customers’ taste scale. The first fast-food restaurantsopened in the early 1920s (for an interesting account, see, e.g., WiseGeek 2014), and theyall sold hamburgers. Much later, customer tastes diversified and other foods, such as pizza,chicken, and subs were introduced to fast food. Again, a pioneering firm first became anoutlier, but soon agglomeration started again as other firms gathered around (here in thesense of similar tastes) the pioneer. Pioneering behavior is, however, very costly and may noteven be possible: consider computer technology. As Windows and Mac operating systemsexisted, it proved to be very difficult to introduce a new operating system such as Linux, evenas freeware. An interesting paper in the context of brand positioning is by DePalma et al.(1987). They describe the design of vehicles and come to the conclusion that economy carswill always cluster (meaning that they become virtually indistinguishable from each other),while luxury cars will find their respective niche by distinguishing themselves from others bypositioning themselves at a distinct and different position, thus fostering dispersion. Whilethis result was verifiable in the late 1980s and early to mid-1990s, it appears that it does notlonger hold.

Back to the physical location of industries, there are economies of aggregation that pushfirms or commercial facilities together. In industry, there are well-known examples of clus-ters: automotive industry in Michigan, leather industry in the city of Offenbach, Germany,cutlery in Sheffield, England, semiconductor industry in California, and footwear in northernItaly (Guimarães et al. 2004), demonstrating that, in spite of a potential stronger competitionbetween industries located close to each other, there appear to exist significant advantagesof clustering. Marshall (1920) was among the first to attribute industrial clustering mainlyto the preexistence of physical conditions or natural resources in certain areas. However,this explanation does not account for the increased attractiveness of these areas as moreindustries co-locate in the dynamic process alluded to above. Gordon and McCann (2000)studied the perceptions, by sector, of businesses’ leaders about advantages and disadvantagesof co-location (or proximity) with related activities, and found that although co-location isdisadvantageous from the point of view of mutual competition (based on increased price com-petition, as customers are easily able to compare product prices and features) in most sectorsthe advantages of proximity far overweighs the disadvantages. In the case of manufacturingindustries, advantages of agglomeration are production related and they include shared intel-ligence and knowledge spillover (for example, in the biotechnology industry); interactionpotential (co-operation, sub-contracting, potential contacts and meetings), greater access toinputs (intermediate products), and labor pooling and poaching, especially if the industryneeds highly trained personnel. From a conceptual point of view, agglomeration economiesthat operate in the industrial case have been classified (see the survey by Kilkenny and Thisse1999) into internal economies of scale (due to mass production in one place); localizationeconomies (formation of specialized labor and availability of input products), and urbaniza-tion economies (development of the region as a consequence of an increased scale of activity.)See Porter (1998) for an analysis of these production related economies. A good survey (upto that point) of clustering and its reasons in the high tech industry can be found in Magionni(1999).

Clustering of competitive facilities, which is our focus, has occurred since markets andbazaars appeared in ancient times. Recent examples are food courts in shopping malls, spe-

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cialized shopping districts (antique stores, garments, jewelry, etc.), lodging, retail stores, andsimilar facilities. Note that, in these examples, a cluster could include stores that competeagainst each other, as well as stores that are not in competition. The benefits of agglomerationare mainly related to demand, rather than production savings or economies of scale, as it wasin the industrial case. Central place theory (see, e.g., Christaller 1933; Lösch 1954, and Fis-cher 2011) offers an early explanation to clustering, as it postulates that facilities agglomerateaccording to a hierarchical pattern, consisting of a number of lower order centers regularlyspaced over the region, selling the products that are required more frequently by customers,and fewer, centrally located higher order places, in which providers with higher economiesof scale would locate, offering the products that are bought less frequently. Such a patternsaves transportation costs.

However, there are also a number of drawbacks to clustering, some of which have alreadybeen addressed above. The first such drawback is the congestion that clustering inevitablygenerates, and with it longer travel times, for customers. This effect will undoubtedly decreasethe attractiveness of agglomerations, if not the added attraction of multiple stores in theagglomeration counteract this effect and dominate it. Price competition is another effect thattypically causes facilities to disperse. As a matter of fact, some contracts (e.g., those fortenants in malls) deliberate specify that no other facility that offers similar products canbe in the direct vicinity of a tenant. This is an attempt to increase customers’ travel costsand thus lower the level of price competition that exists when comparison shopping is used.A third reason for dispersion is the use of delivered pricing or, equivalently, spatial pricediscrimination (Díaz-Báñez et al. 2011). Actually, as already pointed out by Gabszewicz andThisse (1986a, b), spatial price discrimination results in min-cost locations, a fact reiteratedby Díaz-Báñez et al. (2011), Pelegrín-Pelegrín et al. (2011).

Section 5 will offer additional thoughts on agglomeration and its reasons to occur inpractice. More specifically, that section will review some of the explanations that authorshave offered to explain agglomeration by analyzing consumers’ behavior and firms’ reactionto it.

3 Components of competitive location models

Competitive location can be seen as a game in which competitors are firms. Competitivemodels are distinguished from their noncompetitive counterparts by the feature that an actionby one of the players (firms) affects the other players that are participating in the game. Atypical example is the price change at, say, a gas station. Lowering the price will typicallyincrease some of the traffic, which, in turn, will reduce traffic at some of the firm’s competitors.While competition in general competitive models involving firms can take place upstream ofthe firm (involving its suppliers), between firms, and downstream (involving customers), ourfocus is much narrower. In this contribution, we will deal only with downstream competitionthat involves firm’s (optimized) behavior and rational customers. Furthermore, most of thedescribed and prescribed behaviors of the firm involve location, even though price, quantity,and quality competition also play a role.

In order to be able to properly embed the existing models in the literature, we will firstexamine some of the main components of competitive location models, followed by a shortdescription of Hotelling’s contribution. We then examine a variety of extensions and theirpropensity to result in agglomerated facilities. Eiselt et al. (1993) have provided a taxonomythat includes the five main components of competitive location models. A more detaileddescription is found in Eiselt and Marianov (2014). The space, in which the competing firms

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locate, can either be a linear market (i.e., a line segment), a two-dimensional plane, or anetwork. The number of players, i.e., competing firms, is either fixed to any number between2 and any other finite number, or it is unknown, as it will be determined endogenously basedon the size of the market and the costs of entering it. The third component concerns thepricing policy chosen by the firm. Many models use mill pricing (also referred to as f.o.b.pricing), in which case customers are responsible for the transportation of the good from thefacility. Other policies include spatial price discrimination (the firm sets prices that dependon the site where the good is offered, a policy heavily curtailed by legislation), uniformdelivered pricing (a policy that delivers the good to the customer and charges the same priceeverywhere), and zone pricing, where different prices are charged in different zones. Mostof our arguments will deal with mill pricing.

The fourth component describes the rules of the game. Loosely speaking, it deals withwhat it is the firms are trying to achieve. If firms have located their facilities in a specific way,we may be interested in finding whether or not the present situation is stable. The conceptthat is applied in this case is a Nash equilibrium. A Nash equilibrium is a situation, in whichnone of the firms has an incentive to unilaterally change its present situation. We are a bitvague in our terminology, as we want to encompass a variety of a firm’s strategies, not justlocations, but also price decisions, quality choices, and others. In other words, if all firmshave set their locations, prices, qualities, and everything else, and none of them can improvetheir objective by changing any one of these decisions, we then have a Nash equilibrium.Most researchers who attempt to determine whether or not a Nash equilibrium exists in aspecific situation are using simpler spaces, such as line segments or trees. This is due to thecomplexity of the resulting model.

On the other hand, if a firm has to decide whether or not to make the first move whendeciding on its location or quality (price is not that relevant in this context, as it can easily bechanged), we are dealing with a prescriptive problem, called a leader-follower model. Thistype of model is due to the work of the German economist (von Stackelberg 1943), who firstdescribed it. The leader (which actually may be a group of firms) makes its decisions first andirrevocably. On the other hand, the follower or followers will wait until the leader has madeits move and then make the decisions regarding its own facilities. Inherent in this process isa basic asymmetry between leader and follower: while the leader has to take the presumedactions of the follower into consideration and guard against them, the follower must onlyobserve what the leader does and then act in his own best interest. In order to plan, the leadermust know the follower’s reaction to each of its actions, which is commonly called a reactionfunction. Based on the reaction function, the leader will choose the best action for himself.It is apparent that the leader’s problem—a minimax problem, as he tries to guard against theactions of the follower—is much more difficult to solve than the follower’s problem, which isa conditional optimization problem, i.e., the follower will optimize his own objective, giventhat the leader’s actions are known and given.

Finally, the last –but most important for this paper—component deals with customerbehavior. In the simplest case, customers patronize the closest facility. This is, of course,only reasonable, if the products are homogeneous (e.g., standardized), and the prices at thefacilities are comparable. If the prices are no longer equal, customers will reasonably choosethe one that offers the good at the lowest full price, i.e., the lowest purchase price plustransportation costs. These are the customer choice rules in Hotelling.

Customers, however, not necessarily choose the closest or lowest full price facility. Reilly(1931) proposed a gravity-like formula for drawing trade area boundaries of competing cities,which assumes that customers at an intermediate point between the cities are attracted to themin proportion to their respective population and in inverse proportion to the square of their

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respective distances. In consequence, customers at the intermediate point would choose fortheir purchases the city with the highest attractiveness.

Huff (1963, 1964) used Reilly’s formula for stores or shopping centers or agglomerationof firms. He argued that the business of competing stores is not restricted to their tradeareas as implied by Reilly’s formula. Instead, customers with several available competingstores within reach are patronizing all of them probabilistically, in proportions that could becomputed using Reilly’s formula as a building block. Furthermore, according to Huff, thepopulation in Reilly’s formula had to be replaced by the square footage of the store, an easier-to-measure proxy of the number of items available in it. Thus, the probability of consumers atan intermediate point patronizing one of several available stores is proportional to the utilityobtained from visiting that store (its square footage divided by the squared distance or traveltime to it), divided by the sum of the potential utilities received from all possible stores, i.e.,

Pi j = S j/Ti j∑

k Sk/Tik,

where Pi j is the probability of customers at point i traveling to facility j ; S j is the size of thefacility (square footage of selling area); and Ti j the travel time between i and j . Huff’s (1964)behavioral model, either as is or extended to accommodate other attributes of the stores inthe multiplicative competitive interaction model (Nakashani and Cooper 1974), was adoptedby a number of operational research locators such as Hodgson (1978) in a non-competitivecontext, Drezner and Drezner (1996); Fernández et al. (2007) and Joseph and Kuby (2011).For a complete review, see Drezner (2014) and Kress and Pesch (2012).

The Huff gravity model is an aggregate demand model, as it assumes that there are groupsof homogeneous customers that react in the same way to attributes of the stores available tothem, e.g., attractiveness and travel time. The random discrete choice models, on the otherhand, are econometric models of population choice behavior that are constructed from dis-tributions of individual decision rules. One such model is the Logit model (McFadden 1974),which is used to represent either individual choices or choices of homogeneous customers,and has been extensively used in the field of transportation. Logit models are not as popular inlocation literature as gravity-related models, but they have been used in some location modelsby both economists and operational researchers,; see, e.g., DePalma et al. (1985); Hodgson(1981); Drezner et al. (1998); Aros-Vera et al. (2013) and Lüer-Villagra and Marianov (2013).Logit models assume that a customer has a utility function he wants to maximize. Consider-ing different facilities j will generate different utilities for a customer k, given by the utilityfunction u j (k). Utilities can depend on the attributes of the stores, attributes of the products,or socio-economic parameters of the customers themselves. A popular utility function is

u j (k) =∑

i

βiv j i (k) + με j

where v j i is the i th attribute of the facility or product j , βi is a coefficient that measures therelative influence or weight of that attribute on the utility of customers, ε j is a random vari-able with zero mean and unit variance (usually assumed to be independently and identicallyGumbel distributed), which reflects customer’s taste variations in time (or lack of consis-tency); or particularities of different customers in a homogeneous group, or modeling errors.Finally, μ is a positive weight of the random component. Following McFadden (1974), theexpression of the probability of a customer or group of customers k visiting facility j is

pkj = eu j (k)/μ

∑

�

eu�(k)/μ.

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1- b a

pB + t x – b

0 A BY

pB

pA

pA + t x – a

Fig. 1 Hotelling’s original model

Note that μ is a sensitivity parameter. The smaller μ, the higher the customers’ sensitivity toutility differences, and more customers will choose the facility with the largest utility. Thisis the multinomial logit model; for more details on Logit models, see Ortúzar and Willumsen(2011).

All prescriptive competitive location models assume that customers engage in travelingwith the objective of buying a specific item, starting the trip at their homes, visiting thechosen facility, and returning home. The factors explaining competition are price, distanceand attractiveness, and, in some instances, waiting time, as in Marianov et al. (2008). Theproducts are assumed to be homogeneous, and the customers have perfect information onprices and qualities at the facilities. In practice, however, customers’ behaviors are morecomplex, including comparison and window shopping, and multi-stop shopping. We analyzethese behaviors in Sects. 5.2, 6 and 7 below.

4 Hotelling’s and von Stackelberg’s original models and some extensions

4.1 Hotelling’s model

Hotelling’s original model uses a linear market on which demand is uniformly distributed, twocompeting firms that intend to locate a single facility each, and they first locate simultaneously,then after being able to observe the location of their competitor, they simultaneously set millprices. Customers will patronize the facility that offers the lowest full price, and the researchquestion is whether or not an equilibrium exists. To formalize, the market extends from 0 to�, the facilities are denoted by A and B, and their locations are a and b units away from theleft end of the market, respectively. The two firms charge mill prices pA and pB , respectively,and the unit transportation costs are t . Transportation costs are assumed to be linear in thedistance and in the quantity. If we were to define the utilities vA(x) and vB(x) for a customerlocated at a point x units from the left boundary of the market, and assuming that he purchasesthe good at firm A or B, respectively, we can write vA(x) = −pA −|a − x | and similarly forthe customer’s purchase from firm B. Given that the firms’ costs have been normalized tozero, both firms’ objectives of profit maximization reduce to the maximization of revenues.The scenario can be visualized in Fig. 1, where the horizontal axis shows the market, whilethe ordinate indicates the full price paid by any customer on the market.

The “V”-shaped functions rooted at A and B indicate the full prices customers have topay if they purchase the good at firms A and B, respectively. Given that customers purchase

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the good from the lower-priced source (the good is assumed to homogeneous), their pricefunction is shown as the broken line, i.e., the lower envelope of the two V-shaped functions.At the point Y , the full prices of firms A and B are equal, this is the site of the marginalcustomer. Firm A’s market area then extends from the left end of the market at “0” to themarginal customer, while firm B’s market area extends from “Y ” to the right end of themarket at �. The revenues of the two firms are then represented by the area of the rectangleswith base lines of (0, Y ) and (Y, �) and height pA and pB , respectively. To further facilitatethe discussion, we will refer to the area to the left of firm A as A’s hinterland and similarly,the area to the right of B will be referred to as B’s hinterland, while the market between Aand B is the competitive region.

We will now demonstrate that the situation shown in Fig. 1 is not stable, given thatfirms are competitive and can costlessly move their facilities. For simplicity, first assumethat the two firms have fixed their prices at the same level and that they compete only inlocations. Moving away from its component means that a firm loses markets in the competitiveregion and retains customers in its own hinterland for a net loss of customers and, giventemporary fixed prices, thus a loss of revenue. The opposite happens, if a firm moves towardsits opponent. Once that has happened, both firms are located next to each other, i.e., minimumdifferentiation. However, the firm whose hinterland is smaller (a competitive region does notexist anymore) will then jump over its opponent and temporarily locate on its other side,thus effectively exchanging hinterlands. Kohlberg (1983) remarked that this “leapfrogging”creates a discontinuity in the revenue function, which he blames for the lack of robustnessof Hotelling’s model and which he proposed to cure by using transportation costs that arenot only a function of distance, but also waiting time. However, as Gabszewicz and Thisse(1986a, b) point out, “Contrary to widespread opinion, however, we see that the nonexistenceof a price equilibrium is not necessarily related to the existence of these discontinuities; ratherit is the non-quasiconcavity of the profit function which may pose problems.” Leapfroggingcontinues until both firms are located at the center of the market. At this point, each firmcaptures half of the market and neither firm has an incentive to change its location. Hence,a locational equilibrium has been reached. Note that this locational push force results in anarrangement that exhibits agglomeration.

If we were to allow price competition at this point, we notice that either firm could lower itsprice by an infinitesimal amount and thus cut out its opponent and capture the entire market.The firm’s competitor will retaliate in kind until it is no longer profitable for at least one firm,so that it moves away, looking for a local monopoly. The process continues, demonstratingthat a stable solution does not exist.

This flaw in Hotelling’s reasoning was formally uncovered by D’Aspremont et al. (1979),who proved that there is no equilibrium in Hotelling’s original model when competitors can setthe prices. However, if the linear transportation costs are replaced by quadratic transportationcosts (that only make sense if transportation costs are defined as disutilities in general, notcosts in the narrow sense), not only would an equilibrium exist, but it would have one inwhich maximal (rather than minimal) differentiation exists, i.e., competitors locate as faras possible from each other. Actually, some researchers have considered linear-quadraticcost functions (i.e., functions that comprise linear and quadratic parts). Anderson (1988) hasshown that only the pure quadratic case has an equilibrium.

Shortly after it was proposed, the principle of minimum differentiation was challenged, asother authors noted that there is no clustering when some of the assumptions of Hotelling’smodel are relaxed. Lerner and Singer (1937), for example, found that the existence of areservation price would radically change the result. A reservation price denotes the maximalprice customers are able or prepared to pay. It is, in a sense, a customer’s valuation of the

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(a) (b) 0 A B

pB

pA

’DC C’ D

R

0 A B Y

pB

pA

C D

R

Fig. 2 Introduction of a reservation price R

product in question. For simplicity and in line with common custom, we have assumed thatthe reservation price is independent of the location. In Fig. 2, customers have a reservationprice R.

Let facility K = A or B, be located at point k = a or b. In the situation of Fig. 2a, themarket boundaries of firm K are defined by the points at which the utility vi (x) = R − pK −t |k − i | of a customer located at that point is exactly zero, which are the points C and C ′ forfirm A, and the points D and D′ for firm B. Customers located in the segments 0−C , C ′− D,and D′ − � do not make any purchases, since their utility is negative, no matter which of thefacilities they choose to patronize. As long as point D is to the right of C ′ and their outerboundaries stay clear of the extremes of the line, there is an infinite number of equilibriumlocations: in all of these location combinations, firm A captures the market between C andC ′, while firm B captures the market between D and D′. In these situations the market islarge enough for both firms that do not have to compete. More specifically, as long as eachfirm has a market area that equals that of a monopolist, there is a dispersed equilibrium, evenfor problems with different (but fixed) prices pA �= pB . Since in these situations the firmsstay at least (2R − pA − pB)/t units away from each other, there are an infinite number ofequilibria, but there is no agglomeration.

If the reservation price increases, so do the market areas of the two facilities. Once R =1/4(t� + 2pA + 2pB), there exists only a single equilibrium that has the ends of the firms’market areas coinciding with the ends of the market at 0 and �, respectively. For any higherreservation price, firms cannot avoid obtaining market areas that interfere either with theends of the market or their competitor (or both). It is easy to show that losses in a firm’s ownhinterland are more severe than those in the competitive region, so a firm behaves optimally,if it keeps is outside market boundary at the end of the market 0 or �. This means that the firmsare moving together, until it is either beneficial for the lower-priced company to undercut itsopponent (in which case equilibria do not exist anymore), or, in case of fixed and equal prices,once the duopolists are located next to each other. This case of central agglomeration is thesame, if we were to use Hotelling’s model with fixed and equal prices and no reservationprice (or, equivalently, one whose value approaches infinity).

We would also like to note that shrinking values of R have the same effect as increases ofthe unit transportation rate t and a lengthening of the market �. Figure 2b shows the firms’need to disperse, as by doing so, they can gain more market area in their respective hinterlandsthan they lose in the competitive region.

In summary, the existence of a reservation price, together with a sufficiently long market,leads to dispersion. Note that as the transportation cost t increases, the market segment(or market radius) of both facilities becomes shorter. This does mean—other things beingequal—that equilibria with two separate monopolies are more likely as unit transportationcosts increase.

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4.2 von Stackelberg’s model

Consider now sequential competition as described by von Stackelberg. Without loss of gen-erality, assume that firm A is the leader and firm B is the follower. Furthermore, suppose thatthe firms charge fixed prices pA and pB , and, at least for the time being, let pA = pB . FirmA will now attempt to construct its opponent’s reaction curve. Let A tentatively consider itslocation some a units away from the left end of the market. Firm B’s market share at point0 is 1/2a and, as B approaches A, its market share will rise to a as B locates just next toA. Directly to the right of A, Firm B’s market share is 1 − a, and it decreases as B movesfarther to the right, until it reaches a value of 1/2(1 − a) as B locates at the right end ofthe market. In other words, firm B’s market share function for A’s given location is triangu-lar with a maximum right next to A on the longer side of A. This result clearly prescribesagglomeration. Suppose now that pA > pB . Firm B’s capture function is then again linearlyincreasing from the ends of the market. However, once B gets closer than (pB − pA) to firmA’s location, its market share jumps up to l, the length of the entire market, as now B will cutout the more expensive facility A. Eiselt (1992) coined the phrase sufficient spatial separation(SSS), which, if facilities stay at least SSS apart, no undercutting will occur. Incidentally, theconcept, even though not by this name, was already known by Fetter (1924), who stated that“…the prices in the two markets cannot … differ by more than the amount of the freight…between the two points.” As a result, agglomeration is no longer assured: firm B’s optimallocation is anywhere within the area, in which the transportation costs are smaller than theprice difference between the two firms. Firm A now takes this result into consideration, butthere is no way to avoid being cut. Hence firm A will not obtain any market share, given thatit enters the market at all.

Finally, suppose that pA < pB . Given any location of the leader firm A, its competitorwill locate as close as possible to A, but no closer than the price differential of the two firms,as it would be cut out otherwise. If the market is sufficiently long, then the leader firm canlocate anywhere, as the follower will locate on its longer side SSS/t distance units away, sothat the leader captures the shorter side of the market plus half the distance to its opponent.This market is maximized, if the leader locates at the center of the market. That way, thereis no agglomeration, and the leader has a slight advantage (how slight depends on the pricedifferential). If the market length is � ≤ 2SSS/t , the leader locates at the center of the marketand the follower will be cut out anywhere, so that again, he may not enter the market at all.

In summary, we observe that whenever prices are temporarily fixed, the sequential locationprocess does not necessarily lead to agglomeration except in the case of equal prices.

Drezner (1982) appears to have been among the first to solve a von Stackelberg problem.He located the leader and the follower in the Euclidean plane, both in polynomial time. Ayear later, Hakimi (1983) introduced the terms centroid and medianoid for the locations thatresult from the leader’s and follower’s decisions. The reason for the terminology is that theleader, considering the follower’s course of action, will have to guard against it and usea maximin objective (similar to the type of objective used for center problems), while thefollower will be able to use a conditional maxisum objective (similar to median problems inlocation planning).

Still using Hotelling’s assumptions, i.e. mill pricing and customers choosing the least fullprice, but framing the scenario on a network, ReVelle (1986) formulated the follower problemfor the first time as an integer programming formulation using equal prices. The locationsobtained using this model do not show agglomeration, although co-location of facilitiesappears in some instances. This contribution was quickly extended in several directions,e.g., Eiselt and Laporte (1989) introduced attraction functions in ReVelle’s model, while

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ReVelle and Serra (1991) allowed the firms to relocate their facilities sequentially and tochoose the role of leader or follower; the follower role dominating that of the leader. Serraet al. (1992) solved the problem considering a hierarchical facility structure, similar to thatpredicted by central place theory, while Serra and ReVelle (1994) solved the leader problemon a network. Furthermore, Drezner (1994) proposed a model for the location of competitivefacilities on a plane. Good reviews can be found in Santos-Peñate et al. (2007); Younies andEiselt (2011) and Dasci (2011).

In some models referred to above, under specific assumptions, agglomeration occurs,while in others, dispersion results. There is no general rule or dominance of any of the two,which is in contrast to real-world observations, where agglomeration appears to dominate,at least for stores selling some types of products.

A lot of work has been done concerning extensions of Hotelling’s model, specificallyusing probabilistic models, mostly with Huff-like attraction functions. Earlier attempts werethose by Aboolian et al. (2007) and Fernández et al. (2007), followed by Blanquero et al.(2011); Drezner et al. (2012) and Küçükaydin et al. (2011, 2012). The last two contributionsare probably the most ambitious attempted so far, as they deal with existing facilities, anew chain entering the market, planning locations and qualities for its facilities, while theexisting facilities have an opportunity to react by closing existing facilities, opening new ones,or adjusting quality levels. It is no surprise that this bilevel, integer, nonlinear program is verydifficult to solve. Von Stackelberg and Nash equilibrium problems were also investigated bySaíz et al. (2009); Sáiz et al. (2011). An interesting comparison of some of the results hasbeen provided by Fernández and Hendrix (2013).

Other noteworthy extensions are those that deal with Nash and von Stackelberg problemsin the presence of delivered prices (i.e., spatial price discrimination), such as Díaz-Báñez etal. (2011) and Pelegrín-Pelegrín et al. (2011), while the contribution by García-Pérez andPelegrín (2003) considers a tree as the space, in which customers and facilities locate.

The next section will first define two different types of forces and then discuss how theyaffect the location of facilities.

5 Agglomeration and dispersion in theoretical models

This section will discuss the main forces that lead to agglomeration and dispersion in com-petitive location models. More specifically, we will distinguish between weak and strongforces that lead to the agglomeration of facilities. A weak force leading to agglomeration isa force that has facilities located in close proximity to each other as a secondary effect: theprime reason for a facility’s location is its proximity to customers. On the other hand, a strongforce towards agglomeration has facilities locate close to each other because there exists adirect benefit of locating close to other facilities. The existence (or the lack of the existence)of these forces will be examined in a variety of theoretical models below.

5.1 Weak forces of agglomeration

Hotelling and, for a long time its followers, considered only weak forces of agglomerationin their duopoly models. In these models, rather than obtaining a benefit from being closeto or far from their competitor, firms increase their profit by being close to customers, aptlyreferred to as “capturing them.” However, while clustering occurs in some cases, it doesso only in special cases: in Hotelling’s model firms cluster only in case of fixed and equalprices (otherwise there is no equilibrium), and the von Stackelberg solution forces clustered

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solution only if, again, prices are fixed and equal. (Fixed and unequal prices may, but donot have to, result in clustering). The analysis below examines how weak forces act in someextensions of Hotelling’s model.

Lerner and Singer (1937) and Eaton and Lipsey (1975) found that there is no equilibriumif there are three facilities in case prices are fixed and equal. As the three facilities locateat the center of the market, the one in the middle does not get any customers. If prices arefixed and equal, it leapfrogs one of its neighboring facilities, leaving another facility as themiddle one. Alternatively, the middle facility can reduce its price below their competitors’,but then, it is either price war, or the competitors run away from the middle facility, with noequilibrium situation.

For more than three facilities, Eaton and Lipsey (1975) performed a detailed analysis underthe condition that prices are fixed and equal. Their models include n > 2 firms that operateindependently and without collusion. The authors found equilibrium locations for 4 and morefacilities. For 4 facilities, the equilibrium situation is two pairs of facilities located each pair at�/4 away from each end of the market. In case of five facilities one facility locates at the centerof the market, while two firms pair up at �/6 and the remaining two at 5�/6. For six facilities,there is an infinite number of equilibria. To outline two of them, one features three pairs, andthe other comprises two pairs and two single facilities. The authors determine that for morethan five facilities, there are multiple equilibria. Common to all equilibrium solutions is thatwhile there may be (partial) pairing, there is no clustering. In an extension of their analysis,Eaton and Lipsey found that on a circle, there exist infinite equilibria for any number offacilities, and the facilities can be paired, but they never form clusters of three or more.

Sequential location, in which a firm locates its facility or facilities considering the reactionof other firms, was also addressed by Eaton and Lipsey (1975), and in detail by Prescott andVisscher (1977). When there is foresight (a situation Eaton and Lipsey refer to as non-zeroconjectural variation), firms adopt a minimax strategy, i.e., they locate so to minimize themaximum damage to their market due to other firms’ subsequent reaction. Note that thislogic applies as long as the total market is fixed. The rule is similar to the leader’s strategy ina von Stackelberg game in which each new entrant uses the information of already locatedfacilities, and foresees possible new entrants. Once the facility is located, it stays there. Ifthe total number of facilities to be located is known, by taking into account the reaction usedby the nth facility, then the (n − 1)st facility, and so on, the first facility can determine itsbest location by using backward recursion. Consider the three-facility case and solve theproblem recursively for the firms A, B, and C . Once facility A is located, say at a ≤ �/2,the second facility B should locate at a point b so to avoid the third facility C capturing themajority of its customers. Then, the second facility must locate to the right of the first, at 2/3of the distance between A and the right end of the market, �. Using this strategy, the thirdfacility can capture at most one half of the second facility’s market, no matter at what sideof B it locates. Knowing that this is the strategy, and that facility C could possibly locate ateither side of it, the first facility must locate at the point c = �/4. Thus, the facilities end uplocated at �/4, �/2, and 3�/4. Actually, the third facility can locate anywhere between thefirst and the second. For n facilities, the same type of reasoning shows that the i th facility ofn is always at i�/(n − 1). The general rule is, interior facilities choose the longest interiormarket and locate at the middle point between two facilities (except for the last one, that canlocate anywhere in between), while peripheral facilities locate at a point that is one third ofthe distance from the market boundary to the closest facility. The rule is the same for circularmarkets. Again, in this case there is no agglomeration, but full dispersion.

The consequences of a non-uniform distribution of the demand are also studied by Eatonand Lipsey (1975), who reach the conclusion that, for equilibrium to exist in a free entry

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market, the number of facilities cannot exceed twice the number of modes of the demanddistribution. In this case and when there is equilibrium, peripheral facilities are paired, butthere is no full agglomeration.

Until the late 1970s customers and products were assumed to be perfectly homogeneous,thus choosing the facility that offered the goods for the least full price was the only rationalbehavior. No information was required by consumers other than the location of the facilities.Customers were assumed to make specific trips to purchase one product at a time. However, inreal life neither products nor consumers are homogeneous. Even substitute products typicallyfeature some heterogeneity, with few exceptions, as in the case of franchises. Customers areheterogeneous, too: in a group of customers, there are different preferences (taste dispersion),and individual customers have a taste for variety, i.e., the same consumer can choose differentbrands or product varieties on different occasions.

The consequence of these heterogeneities is that customers not always choose the productat the least full price, but their purchases are distributed among all the available facilities.Based on this observation, Papageorgiou and Thisse (1985) in their n-firm model on a linearmarket explained agglomeration as a consequence of the dispersion of taste of consumers andthe existence of boundaries, as they do not observe agglomeration on a circular market. Theyassumed that consumers have complete information on prices and features of the productsat all facilities. If the market is divided into areas, the number of visits to an area decreaseswith the distance the customer has to travel, and increases with the number of facilities inthe area (more variety makes the area more attractive.) This last assumption, alone, impliesthat agglomeration will increase the demand, so their conclusion is not unexpected.

DePalma et al. (1985) formalized Papageorgiou and Thisse’s conclusions representingconsumers’ taste dispersion through a random utility choice rule in their duopoly model. Thesetting is similar to that of Hotelling: the market is a line, along which, customers are distrib-uted. Each of the two facilities can choose their location on the line. What is new in De Palmaet al.’s work is that they allow each individual customer to change taste randomly or simi-larly, to express different degrees of brand loyalty. In their model, an individual’s consumertaste heterogeneity is considered by adding a random term to Hotelling’s utility function, i.e.,using a utility function like those in multinomial logit models. The new expression for theutility is

u j (x) = v j (x) + με j = R − p j − t∣∣x − x j

∣∣ + με j .

Recall that ε j is a zero-mean, unit variance, independent, identically distributed randomvariable, and μ ≥ 0 a positive constant that reflects the degree of taste heterogeneity, with 0being perfectly homogeneous and ∞ absolutely heterogeneous. In other words, the choiceof μ = 0 means that each customer will purchase the good with the highest value of v j (x)

all the time, there is no tolerance for deviations from that. As the value of μ increases, acustomer will still prefer the product with the highest value of v j (x) most of the time, but forthe sake of variety will sometimes choose a different good. Finally, if μ → ∞, customerswill choose each of the two products half of the time, regardless of the value of v j (x).

Give the above utility function, the probability PA(x) of a customer purchasing fromfacility A is given by the logit model (see Sect. 3 for an explanation of the Logit model):

PA(x) = e−pA−t |x−xA|/μ

e−pA−t |x−xA|/μ + e−pB−t |x−xB |/μ .

and PB(x) = 1 − PA(x). Consumers visit both facilities performing single-purpose trips,in proportions given by the formula. Figure 3 shows this function when pA = pB , for threevalues of μ. Facilities are located at xA and xB , shown by the vertical lines. The left and

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xB

µ = 1

µ = 5

µ = 20

xA

0

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0.6

0.7

0.8

0.9

1

xB

µ = 1

µ = 5

µ = 20

xA

PA(x

)

PA(x

)

Fig. 3 Random utility functions for two distances between facilities

right figures differ only in the position of facility A. The solid line shows a case in whichcustomers are less heterogeneous (lower μ) and consequently, more sensitive to travel costdifferences, while the dashed line shows an intermediate case and the dash-dotted line a casein which customers are more heterogeneous and willing to travel far for variety (higher μ),so they visit both facilities in similar proportions. As μ → 0, the curve approaches the stepfunction of Hotelling. As μ → ∞, the curve approaches a horizontal line at PA(x) = 0.5∀x .

For a given value of μ (say, the solid line) and considering a uniform distribution ofcustomers over the market, the integral of the curve is the market share of facility A. SincePB(x) = 1 − PA(x), the area above the curve is the market share of facility B. For anynonzero, finite value of μ, De Palma et al. prove that the optimal location of facility A is apoint between �/2 and facility B. Since the same is true for facility B respect to facility A,there is a Nash equilibrium with both facilities clustered at �/2, i.e., there is agglomeration.If μ = 0, the situation is just Hotelling’s setting (agglomeration at �/2), and when μ =∞, PA(x) = 1/2∀x , and the facilities capture 1/2 of the market regardless where they arelocated.

For n > 2 facilities, the reasoning requires facility 1 locating at xA and the remaining(n − 1) facilities at xn . The probability of patronizing facility 1 becomes:

PA(x) = e−t |x−xA|/μ

e−t |x−xA|/μ + (n − 1)e−t |x−xn |/μ

Under these conditions, for a finite value of μ, an agglomerated equilibrium can exist only atthe center of the market, since if there is an off-center agglomeration of facilities, one of thefacilities in this agglomeration would benefit by moving towards the center (its profit wouldchange from �/n to larger than �/2. This is obvious for μ = 0). Furthermore, for more than2 facilities, agglomeration depends on the value of μ. If taste is rather homogeneous (lowμ), a facility that is off-center (say, at the left) while the remaining ones are at the center, willcapture its hinterland, because when μ is very small, customers in that hinterland will veryseldom need to travel past it to purchase at the cluster, and in the limit μ → 0, nobody inthe hinterland would go past the lonely facility. When this is the case, the off-center facilitywill capture �/2, while, if locating with the remaining ones, its market would only be �/n.To the contrary, if taste is heterogeneous (high μ), by returning to the center, the “prodigal”facility will gain more market from the right half of the market than the market it loses onits left. Finally, with μ large enough, the only possible equilibrium is central agglomeration.The same result holds if firms set both locations and prices.

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When firms are free to set prices, DePalma et al. (1985) show that the only possible pricefor a central agglomerated set of n facilities is p∗ = μn/(n − 1) for all facilities. Thus,price competition is softened by heterogeneity, which is intuitively appealing, since productsbecome imperfect substitutes, as opposed to perfect substitutes.

Note that recognizing the heterogeneity in customers’ tastes, and representing these het-erogeneities by a random choice function, eliminates the discontinuities in the profit functionthat cause the lack of equilibrium in Hotelling’s approach, and restores equilibrium andagglomeration at the center of the market. In this case though, the markets of the facilities arenot monopolistic as in Hotelling, i.e., market areas are shared by more than one facility. Theforces that draw facilities together, though, are weak and the same as in Hotelling’s setting.

Fujita and Thisse (1996) point out that when there is differentiation of products, pricecompetition between agglomerated facilities is decreased. To the contrary, if there is notenough differentiation, it is advantageous for firms to move their facilities away from thecluster and set higher prices. There is a substitution effect between product differentiationand geographical dispersion. Eiselt (2011) offers a good review of equilibria results in papersdealing with weak forces.

5.2 Strong forces of agglomeration

As explained in the previous section, when weak forces act, the benefit for firms to agglom-erate does not come from locating their facilities close to each other, but from locating closerthan the competitors to as many customers as possible. There are also strong forces that makeit convenient for firms to locate their facilities in a cluster with competitors’ stores. Considerthe case of consumers who prefer evaluating products in store rather than searching for infor-mation and purchasing online. One pertinent example is the purchase of shoes. Heterogeneityof products, imperfect information and selectivity force customers to visit more than one storeuntil they find an available product that suits their taste at an acceptable price. Furthermore,consumers can have uncertainty of taste, i.e., they need to see, touch, try on, and compareseveral varieties of the product before deciding on a purchase. This is true even in case someof the customers use the web for gathering information. In synthesis, consumers must travelto several stores, searching for suitable products at a good price and making up their minds ona purchase before buying. As economies of scale in transportation and consumers’ time makesingle purpose trips to each store inefficient, they make multistop trips, in which they visitmore than one store, i.e., comparison shopping. Geographers, market researchers and spaceeconomists agree that one of the main causes of agglomeration (strong forces) is the firms’response to customers’ need to perform comparison shopping, a feature already observedearly by Lösch (1954). By locating their facilities together in a cluster, firms facilitate thissearch in an attempt to maximize their profit.

Eaton and Lipsey (1979) formulate a simple model that shows how comparison shoppingcreates a force that pushes facilities together. They assume a linear, bounded market, witha uniform density of customers, without price competition (prices are fixed and equal), inwhich a fixed number of competing facilities A, B, C, . . . located at a, b, c, . . . sell the samecommodity. Co-location of facilities is not permitted, but facilities may locate an arbitrarilysmall distance δ from each other. The firms’ planners have no foresight. Consumers visitexactly two stores each time they make a purchase. Once the two stores are visited, thecustomer returns back home and orders the good from the preferred store, which delivers atno cost. The probability of purchasing from each one of the two visited stores is 1/2. Thestores to be visited are chosen so that the cost of the transportation (which is assumed to beproportional to the customer-facility distance) is minimized. Suppose a customer is at a point

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0 c d a e …b

Fig. 4 Eaton and Lipsey’s linear market

x of the line, and visits stores located at a and b. The cost will be |a − x | + |b − a| + |x − b|.Note that maximizing profit is equivalent to maximizing the number of visits which, in turn,comes from maximizing the market segment of each facility (now, facilities share marketsegments). Figure 4 shows the setting.

In Fig. 4, assume that there are more than three facilities, and that facilities C, D, . . . arefixed, while A and B can move. The market segment of facility B consists of all customersto its left (they must visit A and B) and the customers for whom is cheaper visiting B and Crather than C and D. If B moves to its right towards C , it keeps all its customers on its left,and captures a larger market segment on its right. The optimal location is at c−δ. For facilityA now, the optimal strategy will be also to move to its right, since customers to its left haveno choice, while more customers to its right are attracted to visiting A and B as opposed to Band C . Consequently, firm A will locate at b − δ. In conclusion, the three left hand peripheralfirms and the three right hand peripheral firms are grouped, at least, in triplets.

Following similar reasoning, Eaton and Lipsey (1979) conclude that groups cannot containmore than four facilities; interior firms can never be unpaired; and a facility obtains a largermarket if it is joined by a competing facility. The market segment of each group extendshalfway to the next group (customers go to the closest group); no facility’s market is lessthan half the market segment of other facility; and the markets on both sides of a group areof equal size. Also, as the first group is at a distance a from the boundary, the second mustbe at 3a, and so on and that, each group is at the middle point of its market.

The preceding rules apply to any number of facilities. If there are only two facilities,they are both visited, no matter where they are. If there are three facilities, they locate inthe middle of the market. Eaton and Lipsey did not comment on the fact that there is noequilibrium for three facilities, since customers will always visit the interior facility togetherwith a peripheral one. Peripheral facilities receive fewer visits than the interior facility, sothey will always be jumping over the interior facility, to become one. This is also true for anytriplet, but not for groups of four. The equilibrium is re-established if customers are allowedto visit any two facilities in a group. In this case, however, some of the results do not hold, asfor example, that clusters are in the middle point between other clusters. The same happensfor four facilities, but for five facilities there is no equilibrium. Six facilities will group intwo triplets and seven in a triplet and a group of four. For eight or more, the equilibrium isnot unique. In other words, the authors have found some local agglomeration.

As Eaton and Lipsey (1979) pointed out, their assumption that customers’ trips include atmost two individual facilities, is clearly an oversimplification. Intuitively, as customers visitmore stores, the clusters become larger. A more elaborated model was proposed by Stahl(1982), who analyzes the case in which n firms with one facility each, sell n differentiatedproducts at the same fixed price, substitutes of each other and hence, in competition. Thefirms locate their facilities in marketplaces with one or more facilities, i.e., we again obtainagglomeration, at least to some degree.

In Stahl’s model there are n types of customers. Among the n product varieties sold bythe different facilities, customers of type i have exactly mi varieties that suit their needs(selectivity), ranked in a strict order. A customer i obtains a utility ui ( j) from the j th varietyin his ranking. All of the n types of customers are distributed uniformly over the linear market.There are different marketplaces K located at k and housing sK facilities each. Customersknow what the location and number of facilities in each marketplace is, but they do not know

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what product is sold in each. Each customer visits exactly one marketplace and once there,he can visit all facilities in that marketplace at no cost. If the utility of not finding a suitableproduct is zero, the expected utility of a trip is:

mi∑

j=1

π j (sK )ui ( j) − t |k − x |

where π j (sK ) is the probability of finding the j th preferred variety or product (and notfinding the first ( j − 1)st preferred ones). If he finds products he likes, he buys one unit ofthe product that best suits his requirements. The market radius of each marketplace, for eachtype of customer, is defined by the distance at which the utility of visiting the marketplace isequal to zero, i.e.,

∑mij=1 π j (sK )ui ( j) = t |k − x |.

Using this setting, Stahl (1982) found that as the number of facilities in a marketplaceincreases, the market radius (and with it the demand that the firm captures) also increases, i.e.,larger clusters attract more demand. As customers become more selective (mi decreases),the probability of finding suitable products at a stand-alone facility decreases, and facilitiesare drawn to agglomerate. If each customer has only one variety he likes, the only possibleequilibrium is all facilities in a single marketplace. Stahl also shows that once a minimumnumber of facilities in the marketplace is exceeded, it will always be convenient for a newfacility to join the marketplace, as opposed to locate alone.

In synthesis, Stahl (1982) shows that, whenever consumers have a taste for variety but theyare selective, the optimal locational pattern for facilities is agglomeration. Note that, in thiscase, there is an implicit recognition of the fact that, if a customer must visit more than onestore, agglomeration reduces travel time and, as a consequence, customers are attracted toagglomerated facilities. Stahl finally warns that price competition could become a centrifugalforce, especially when there is a large concentration of facilities.

Along similar lines, Wolinsky (1983) explains agglomeration by taste diversity and imper-fect information on homogeneous but differentiated products. Imperfectly informed cus-tomers need to search for their individual best choice, and agglomeration is the best settingfor this activity, for both customers and stores. Agglomeration does not necessarily happen inthe center of the market, but at any point of the market, since there is benefit for the facilitiesof being close to each other.

In Wolinsky’s model, each consumer has an ideal brand, located at a point on a circularproduct space. A customer’s willingness to pay for other brands is described by a decreas-ing concave function of the distance to the ideal brand, over the product space. There isa reservation brand, at an arc-distance R from the ideal over the product space, which isthe distance at which the expected improvement obtained from searching one more store isequal to the marginal cost of searching. It is assumed that the consumer expects to find aprice p∗ everywhere, and he is willing to accept a brand priced p, if the utility of the brandminus p exceeds the utility of the reservation brand minus p∗. Each consumer buys oneunit of product. Each facility sells only one variety. Consumers do not know what productsare available, and they visit facilities at a marginal cost k each, plus travel cost, to find out.The travel cost has a fixed component t0 (e.g., parking) and a unit cost per distance, t . Themodel assumes two marketplaces: n clustered facilities in one, and a single facility, locatedcloser to the customer, in the other one. The consumer can choose to start the search at eithermarketplace, and the search ends when the customer finds an acceptable product, or whenall facilities have been visited, at which time the customer can choose to buy or not. In thiscase, he will not necessarily buy the best product, because it could mean going back to analready visited marketplace.

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Under these conditions, Wolinsky proves that the customer will always go to the clusterfirst, provided that the number n of stores in the cluster exceeds some N , and that the travelcost differences do not exceed some threshold �(n), an increasing function of n. This proofis based on the fact that consumers have imperfect information, which makes them searchthe market before purchasing whenever it is convenient (the cost of searching is low). Then,a consumer will try to perform the search without visiting both marketplaces and having topay twice the fixed cost. The strategy is to start at the place where the probability of having totravel to the other location is lowest—the cluster, whenever the travel cost to the cluster is nottoo high. The probability of having to visit the lonely store and, consequently, the distancethe consumer is willing to travel to the cluster, decreases with n. This proof appears also inFujita and Thisse (2002).

When consumers have at least two products that suit their requirements, if the consumerpopulation S exceeds a critical value S, and if there exists a location h within distance D(S)

for all customers, Wolinsky finds that there is an equilibrium at which all stores cluster ath. If only one brand meets the customer’s taste, the lonely store could be the one that sellsit, and the uninformed customer may have to travel to both the cluster and the lone store.This is an extension of the same reasoning. The critical number of customers guarantees anequilibrium in both price and brand (stores cannot improve their profit by changing price orthe brand they sell).

Note that, in Wolinsky’s model, an increasing fixed travel cost is an increasing incentivefor not visiting more than one of the marketplaces, and it contributes to increase the distancethat customers are willing to travel to visit the cluster: if parking—a fixed travel cost—costsmore, consumers will try to park only as often as they absolutely have to, which will drawthem to a cluster, where the likelihood of finding a suitable product is higher, even if it isfarther away than a single store. A strong taste for variety is also a positive incentive forstarting at the cluster. High values of search cost and transportation cost, though, reduce theattractive of searching and hence the attractive of the cluster. In this case, a lonely store canhave a local monopoly (a large market share of a smaller market).

Konishi’s (2005) two-dimensional spatial competition model based on the work by Stahl(1982) and Wolinsky (1983) emphasizes taste uncertainty. A customer does not know a prioriif he will find a suitable product, and clustering of homogeneous stores increases the proba-bility of finding such product. Together with taste uncertainty, Konishi (2005) also analyzesthe market size-effect due to lower price expectations: customers expect a stronger price com-petition and hence, lower prices, at a place where there are more homogeneous stores. Notethough, that Schulz and Stahl (1996) find that agglomeration can increase equilibrium prices.Intuitively, this is because, as clustering increases the demand (although the market shareof clustered stores is smaller, their common market is larger, and this effect dominates), theparticipants can internalize part of this externality by successfully increasing prices. Konishiproposes a three-stage game: in stage I, firms decide in which shopping center to locate; instage II, customers decide to which center to go, and in stage III, firms set their prices, afterwhich, consumers buy from some store at the chosen center. They visit several stores in thesame trip, because they do not know what commodity they will buy, and the probability offinding and purchasing an item increases with the number of visited stores. Clustered storesreduce the cost of travel and increase the probability of finding a suitable item. As agglom-eration pushes lower prices (at the third stage, there is price competition between clusteredstores), the market area of the cluster increases with the number of stores in the cluster.

However, we must point out that there are not only centripetal forces at play: there arealso forces that push facilities apart to locate in disperse patterns. Fujita and Thisse (2002)suggest that firms entering a market face the trade-off between joining an established cluster

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and obtaining a small market share of a larger market (as seen in Stahl and Wolinsky’smodels), or locating alone and monopolizing a small market.

A stand-alone location seems better if there is no product differentiation. Homogeneousfacilities located in disperse patterns obtain each a monopolistic market composed by localcustomers for which it is not worth traveling farther away than their nearby store for a productthat is basically the same. These monopolistic markets are sustainable as long as travel costsare high, and products are truly homogeneous. Due to the dispersion there is no strong pricecompetition. On the other hand, since the most common search pattern among customersconsists of choosing the marketplace first and subsequently a store within it (Konishi 2005),homogeneous stores in a cluster will most likely enter in price competition with each other.

When firms introduce differentiation in their products, price competition decreases, evenbetween nearby facilities, since taste heterogeneity guarantees some proportion of the marketto all of them. In the limit, when the products are totally different, they do not compete. Inpractice, attempts have been made to take advantage of this effect by selling the same productto chain stores and individual retailers, using different product numbers. This alleviates pricecompetition and attempts to protect individual/specialty stores.

Finally, if products are differentiated but mutual substitutes, the lack of selectivity onpart of the consumers also favors dispersion. If consumers are not too selective, they willpurchase whatever variety of product their local store offers and will not be willing to travelfar for variety, so that a stand-alone store will survive. On the contrary, if a consumer’s set ofacceptable product varieties decreases, he must travel farther away to find what he looks for.

6 Further issues on agglomeration

While our selective review focuses on linear market models, there is additional work in theliterature that deals with agglomeration in spaces other than linear markets, and from differentpoints of view. For reviews, see Mulligan (1984), covering research on agglomeration andcentral places until the early 1980s, and Eppli and Benjamin (1996) who look at the problemfrom the point of view of valuation of lease in shopping centers and review the literature untilthe 1990s. Furthermore, Krider and Putler (2013) survey the latest works on agglomeration ofdifferent types of stores and provide an analysis from an empirical point of view. From a dif-ferent angle, Thill and Thomas (1987) review the literature on trip-chaining behavior, whileMulligan (1987) proposes a model on a two-dimensional space for shopping trip behaviorincluding multistop, multipurpose trips, and trip chaining. O’Kelly (1981) models multipur-pose, multistop, trips to existing facilities using Markov chains. O’Kelly (1983) examinesthe relationship between multipurpose shopping trips and sizes of retail facilities at fixedlocations, in any space. Fotheringham (1988) discusses consumer choice models and hierar-chical decision making, i.e., the process of choosing first a shopping center or marketplace,and then a facility in it, in an undetermined space. Fotheringham (1985) replaces the gravitymodel by a “competing destinations” model that takes into account that facilities have anattractiveness that depends on their location in relation to other facilities, and uses this modelto find the location of a single facility, implicitly assuming a two-dimensional space. Thecompetitive destinations model is discussed by Roy and Thill (2004) in the context of spatialinterdependences. Fotheringham and Knudsen (1986), again assuming a two-dimensionalspace, analyze the relationship between facility locations and facility sizes using simulationand propose a gravity model that explicitly accounts for the fact that a retail store located invicinity with other stores attracts more customers.

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7 Conclusions

Agglomeration of competitive facilities is common in real life. Retail stores and food courtsare everyday examples. Shopping centers with two or more stores selling the same type ofproduct (such as shoes, clothing, jewelry, and others) are standard. The fact that competitivefacilities agglomerate in spite of the stronger price competition that comes with competitors’proximity, suggests the existence of strong underlying forces that drive competitive facilitiesto agglomerate almost as if it were the only possible locational pattern. Yet, operationalresearch models prescribing locations in competitive contexts result mostly in patterns thatexhibit dispersion, with occasional co-location of some of the facilities.

This discrepancy between practice and modeling derives from the fact that the onlyagglomeration forces that are present in operational research models are those in Hotelling’sprinciple of minimum differentiation. Although for some time this principle was enthusias-tically thought to be the explanation of all kinds of agglomeration, in Hotelling’s settingsthere is actually no benefit for the facilities of being close to each other. Whenever facilitiescluster, it is because there is a benefit for them of being closer to the customers than thecompetitor. In some cases, this is achieved by locating near to the competitor. As we showin this paper, the corresponding forces are weak, and critically dependent on the particulardetails and context.

On the contrary, when strong agglomeration forces appear, there is a clear benefit forfacilities of being close to each other, i.e., to agglomerate. These forces, although neglectedin prescriptive models, have been described in the literature by authors in other fields. Inparticular, economists, using linear markets, have developed models in which these strongforces are explained an analyzed. The strong forces are related to consumers’ heterogeneityand uncertainty of taste, heterogeneity of product features, lack of information, taste forvariety and the need to compare before purchasing.

This paper offers a selective review of results related to these strong forces, as analyzed intheoretical models on linear markets. Our goal is to direct the attention of operational researchlocators to the strong agglomeration forces, to start including them in our prescriptive models.

Together with including strong forces in prescriptive models, some related issues need tobe addressed:

• The prevalence of comparison shopping, as compared to single-purpose trips. O’Kelly(1981) in his study in Ontario, Canada, found that in non-grocery shopping trips over70 % were multistop or multipurpose, while grocery trips were multipurpose in more than60 % of the cases.

• Where and when clusters appear, and their development over time. Mulligan et al. (2012)point out that there is a probabilistic component in the location of clusters, and the temporalpath in firm location is very important. In fact, later entrants will co-locate with incumbents,even if the original location decision is not optimal, while leaders have more options. Inother words, it is the first entrant who determines where clustering occurs. Yang (2013) usesthe case of fast food industry in Canada to argue that the only way of clear the uncertaintythat firms have about the profitability of the markets, is by entry of the firm itself, or by theentry of competitors. Under high uncertainty, followers are more compelled to co-locatewith the leader if he has done well and less inclined to co-locate if the leader failed. Heremarks that on occasion, it is more convenient for the chains to wait and enter after acompetitor has done his move.

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This leaves a number of important questions:

• What is the effect of the (n + 1)st store in a cluster. Is it convenient for a new store tolocate in a cluster, independently of the number n of stores already present in it? Or thereis a decreasing gain of joining the cluster?

• What is the effect of on line shopping on agglomeration: the information about productscan be obtained on line for internet visitors, and the need for physically visiting storesdecreases.

• In what businesses are agglomeration forces particularly strong?

We like to note that there are other agglomeration forces that act when locating competitivefacilities, mainly related to economies of production (as opposed to demand). For example,those that appear in the case of lodging industry, in which hotels and motels agglomeratein places with natural features as beaches, airports, football stadiums, proximity to majorfreeways, or the availability of trained personnel (Canina et al. 2005) or proximity to otherfacilities, such as restaurants, e.g., Motel 6 and Denny’s restaurants.

Finally, although we focus here on competitive facilities, there is also agglomeration innon-competitive contexts. Multipurpose trips for example, make agglomeration of hetero-geneous facilities convenient for customers, as it allows them to make use of economiesof scale in transportation and use of time. As a consequence, agglomerated facilities get ahigher demand. A theoretical analysis is available in Eaton and Lipsey (1982). McLaffertyand Ghosh (1987) review the literature and propose a location-allocation procedure whenthere is multipurpose shopping, and Drezner and Eiselt (2002) discuss consumers’ behaviorin competitive location models, including multipurpose shopping.

Acknowledgments This research was in part supported by a grant from the Natural Sciences and EngineeringCouncil of Canada under Grant Number 0009160, by Grant FONDECYT 1130265, and by Institute ComplexEngineering Systems through Grants ICM P-05-004-F and CONICYT FBO16. This support is gratefullyacknowledged. The detailed and insightful comments by two referees are much appreciated.

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