Predation among supermarkets: An algorithmic locational analysis

JOURNAL

OF URBAN

ECONOMICS

15,246251

(1984)

Predation among Supermarkets: An Algorithmic Locational Analysis BALDER VON HOHENBALKEN AND DOUGLAS S. WEST Department of Economics,Universi@of Alberta, Edmonton,Alberta T6G 2H4, Canada Received May $1982; revised July 27,1982 First an algorithm that finds certain polygonal tesselations of R* is described; the polygons involved here are market areas of supermarkets in a city, defined by prices and transport cost proportional to Euclidean distance. Then statistics based on the adjacencies, boundaries, and sizes of all market areas (in Edmonton, Canada, during 1959-1973) are used to test hypotheses on locational predation.

1. INTRODUCTION The first part of this paper develops the theoretical framework and the tools for a locational analysis of predatory behavior among retailing firms; its centerpiece is an algorithm that computes market areas and adjacencies between them when distances are Euclidean and the locations of stores in a city are known. The method is, in spirit if not in its workings, a relative of procedures that calculate so-called Voronoi diagrams [3]; it matches these in efficiency but exceeds them in economic generality, because boundaries between stores can be shifted when relative prices change. Section 3 applies tests based on comparative size of market areas to a case of economic predation, as promulgated by Kreps and Wilson [2] and interpreted by West (in the spirit of [6-81). The basic hypothesis is that an established supermarket chain deterred entry into its growing Edmonton market by placing new stores so as to strangle the outlets of competing grocery chains. One typical test finds that the market areas of targeted “ victim” stores are on average smaller than those of not targeted ones in the same neighborhoods. Another test discovers that predator stores tend to have broad fronts against adjacent victim stores. 2. A BOUNDARY-FOLLOWING METHOD FOR MARKET POLYGONS 2.1 Notation,

Norms, and Boundaries

The paper makes sparing use of mathematical notation, so only a few items need to be mentioned. We denote both scalars and vectors by lower 244 0094-1190/84 $3.00 Copyright All rights

Q 1984 by Academic Press. Inc. of reproduction in any form reserved

PREDATION AMONG SUPERMARKETS

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case latin letters; scalars (and coordinates) are distinguished by subscripts, while vectors are superscripted, if at all. The simplest weak inequality sign “ < ” is used for both scalars and vectors. Sets are represented by (subscripted) capitals; R, R2 are the real line and the Euclidean two-space, respectively. The principal ingredient for our analysis is the (index) set of stores S = {1,2,. . . ,n}; their locations on the plane are given by {x’ E R*[i E S}; S \ {j, k,. . . ,r} denotes set difference. The convex hull of a set of points is the collection of all their convex combinations.

describes a family of distance functions; we mention the block metric (k = l), but use exclusively Euclidean distances (k = 2), that can also be written ]Jx - y]] = [(x - y)‘(x - r)]‘12. Let pi E R be the price of a basket of goods from store i, at its location xi, and let t > 0 be unit transport cost; we then define the cost of that basket of goods from store i at location y E R2 as p; + td(x’, y). If consumers are cost minimizers, the boundary between the market areas of two stores i and j is ( y E R2)p; + td(x’,

y) =pj + td(xi,

y)).

(1)

The shape of (1) depends on the distance function selected and on the parameters pi, pj, xi, xi. If, for instance, d is the block metric, (1) can be linear, piecewise linear, or even partially “thick,” depending on the relative position of xi and xi in the coordinate plane (see [l]). With d being Euclidean (1) is a hyperbola if pi # pi and a straight line if pi = pj. Rewriting the defining equation in (1) as

IW - Yll - llxj - Yll = (Pj - Pi)/t

(2)

one seesthat (1) is the locus of all points y that have constant difference of distances from the two foci xi and xj. If pi > pi, (pi - pi)/t is positive and (1) represents an economically meaningful boundary, namely that branch of the hyperbola that is closer to and bent toward the higher-priced store; see Fig. 1. If pi = pi (1) degeneratesand becomes the well-known perpendicular bisector of [x’, xJ] with (xi - xi)T(y

- rj)

= 0

(3)

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VON HOHENBALKEN

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\ Exact Boundaries for PI < P) = Pk

Exact Boundaries

/

Market Area of Store i

.

/

I

/

/ /

/

/

Market Area of Store k

I

/ /

I I I I \ I

/ ------

,iSL-----------.-

------

LXk

I FIG,

1. Market area boundaries between three stores at equal and selectively reduced prices.

where yj = *xi + ix’ E [xi, xj]; see Fig. 1. In this equal price case the powerful machinery of linear inequalities is directly applicable; it can also be used profitably in the general case, if the hyperbolic boundary is approximated by its tangent perpendicular to [xi, xj]; see Fig. 1. This tangent is also described by Eq, (3), but yj =

xxi

+(I - x)xj E

[xi,

~‘1.

By substituting yj into (2), one obtains the appropriate x = 4 -( pj - Pi)/2fllX’

- xq.

2.2 Shape and Size of Market Areas

We concentrate now on one store i E S that is embedded in an urban market, a “city,” which we describe analytically as a convex region in the

PREDATION AMONG SUPERMARKETS

247

plane bounded by piecewise linear “city limits.” Store i shares the city with n - 1 other stores, each of which has a (potential) market border with i. Given all borders are straight lines, store i’s actual market area A4 will be that piece of the plane which is common to all half-planes generated by the market and city boundaries that contain i. It follows that M is a nonempty convex polygon described by

iv= {yER21AyIb}.

(4

The linear inequalities in (4) are easily derived from the data on location of stores and city limits, on prices and transport cost, as shown in the last section. The above (external) representation of M does not, however, reveal the information we want, namely, (a) the set of neighbors of i, (b) the vertices of i’s market polygon, (c) the size of this polygon. There is a variety of general algorithms that obtain such information for polytopes in R”, and they are applicable here (see, e.g., [4]). Elegance and efficiency, however, demand special treatment for polygons in R2, because they have a closed simple curve as boundary whose linear pieces can be followed around in a complete circuit. We describe below such a boundary-following procedure which, in its general form, has served us well in our empirical investigations. The present exposition will be heuristic and technicalities are kept at bay by stipulating that the system Ay = b

(5)

contains no parallel lines, and that all pairs of lines intersect in distinct points. Without loss of generality we take the equations in (5) to be ordered such that the corresponding lines recede from xi as their index in the index set I = {1,2,... , T} increases, in other words, line 1 is closest to and line T farthest from store i’s position. The algorithm starts with the line nearest to xi (line l), because line 1 is certain to be part of the boundary of M (if not, another line would have to lie between line 1 and xi, contradicting nearestness).It then continues with row 1 in Table 1. Each row of the table corresponds to one iteration that yields a vertex of M; within each iteration the line whose index appears in the first column is consecutively intersected with the lines whose indices

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TABLE 1 Algorithmic Sequenceand Choice Sets Index of given line

Choice set of intersecting lines

1

I\{11

k m

I\{l,kl I\tkml

s

I\{k,m,r

,.... s}

Index of first feasible intersetting line

Vertex

k m r

Urn

1

US

appear in the second column; after each intersection the resulting point is checked for feasibility by

the index of the first line associated with an intersection point that satisfies (6) is placed into column 3; the intersection point itself, a vertex of M, appears in the last column. After an iteration is complete the (index of the) “successful” intersecting line goes to the first column of the next row, and simultaneously it is removed from the choice set in column 2; with this done a new iteration in search of the next edge of M can be started. Observe that beginning with the third row, line 1 is a member of the choice set in the second column; this ensures that in this or later iterations the intersection sequencealong M’s boundary will be able to “bite its own tail”; if this happens the starting line 1 appears in the third column; the circuit is then complete and the algorithm terminates. Termination cannot happen earlier than in the third iteration (the simplest polygon is a triangle), and it must occur at the latest in the Tth, because by then line 1 is the only choice left in column 2. It is clear from the above that the number of rows in Table 1 is equal to the number of vertices of M, which in turn is equal to the number of edgesof the market polygon. The indices of these edgesare given in column 1 (or 3) of Table 1 (in sequence around M), and refer to neighboring stores or to bounding pieces of the city limits. The ordering of the boundary lines in (5) by proximity to x’ was introduced above because if the lines in each iteration of the algorithm are brought up for intersection in this order, the average running time of the algorithm becomes independent of the total number of stores; this is so because in the present spatio-economic context there is a high correlation between being close and being adjacent, and therefore the wreath of neighbors will nearly always be completed from thefirst lo-12 stores in the ordering while all others never even enter the calculations. It follows that

PREDATION AMONG SUPERMARKETS

249

the whole algorithm depends on n (the number of stores) only through the ordering operation and thus assumesits computational complexity-a very competitive position indeed. The above calculations prepare the ground for an easy calculation of the area of M. Column 4 of Table 1 gives the positions of the vertices of M, and importantly, their sequencearound M, with this information some triangulations of A4 are immediately possible; e.g., if the set of ordered vertices is [u1,u2,..., us], a partitioning into triangles which are all incident on d is

[d, u2, d], [ul, I?, u4],. . .,[d, us-l, u”]. It is well known that the area of a triangle with vertices u’, u2, u3 E R2 is .51det[u2- ul, u3 - ~‘11,i.e., one-half of the parallelogram generated by the vectors u2 - ul, u3 - u1(for analogues in R" and related matters, see [5]). It follows that the area of our polygon is just the sum of such terms, i.e., areaM = + 6 Idet[uj-’

- ul, uj - u’]l.

j=3

2.3 Market Areas in a City

How do market areas fit together to cover or pack a given city? In general and with arbitrary (distance) functions d, market areas can have a variety of shapes and they may overlap (at thick boundaries) but they always cover the city completely, as long as no approximations are used (proof: at any point in the city at least one store will be the cheapest). In the case where d is Euclidean and pi = pj for all i, j E S the market areas iWi are convex and polygonal, and they have the dual representation Mi = convex hull of I$, all i E S (where K are the vertex sets computed by our algorithm). Such A4;‘s tile the city smoothly without overlaps or gaps; normally exactly three Mi’s will have one vertex in common, which is also the center of the circle defined by the associated store points xi (see Figs. 1 and 3). If pi # pi for some i, j E S, we replace hyperbolic borders with straightline approximations becausethen the resulting Mi’s are again polygonal and convex, and thus representable as convex hulls of vertices; the approximate Mi’s have the property that they never overstate the market size of stores with prices below those of its neighbors (see Fig. 1); they do not, however, tesselate the city perfectly: instead of joining at one common vertex, triples of Mi’s now abut on small triangular lacunae (see Fig. 1; the reason is that any three perpendiculars to sides of a triangle [xi, xi, x“] do not necessarily meet at the same point). For realistic price differences the relative size of these lesions is quite insignificant (around 0.1% in our example below).

250

VON HOHENBALKEN

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Finally a word on constructing city limits: if the city in question is basically convex its boundaries can be captured algorithmically by a set of linear inequalities that appear as additional constraints in the calculation of the vertex sets of all Mi’s. These inequalities can be entered directly or they are easily constructed (by a least squares step, for instance) from a series of “fence posts” (vertices) if these are given in cyclical order around the city. Nonconvexities in the city’s topography (like lakes, mountains, sinuous rivers, etc.) can be nicely modeled by “dummy” stores, because they occupy space and create boundaries only locully without the global effects of ordinary constraints. An example appears on the map of Edmonton in Fig. 3. 3. AN APPLICATION:

PREDATION IN A SPATIAL MARKET

We mention first some earlier results and then use the boundary-following algorithm to test some market area implications of a spatial theory of predation. 3.1 Rival Elimination and Entry Deterrence Implications In two recent papers West [7, 81 conducted tests of the rival elimination and entry deterrence implications of the Kreps and Wilson [2] theory of predation. The hypotheses tested were the following: if an established firm has been a successful locational predator, then there exists evidence that (a) new plants of an entering firm are challenged by plants of the established firm built in their neighborhood, (b) the predator’s new plants are located so as to force losses on the entrant’s plants, (c) the entrant ceasesto construct new plants, (d) the entrant eventually withdraws from the market. To test these hypotheses for the supermarket industry in Calgary and Edmonton (in Alberta, Canada) it was necessary to reconstruct the locational development of this industry over time from city directories and phone books. West’s tests confirmed that for the years 1959 to 1973 the data are consistent with Canada Safeway having used predatory location strategies against Dominion Stores, Tom Boy, and Loblaw in both cities. 3.2 Market Area Implications The tests for predation mentioned in the previous section were based on the ownership of and neighbor relations between stores. Tests involving the size of market areas had to await a suitable algorithm; one is now at hand and thus we develop several test statistics for locational predation involving

PREDATION AMONG SUPERMARKETS

251

contiguity and size of market polygons (based on the Euclidean metric; see Sect. 2). The empirical implementation and the (highly significant) results are presented in Section 3.3. Before embarking on this discussion we define some notation and abbreviated language. Let P be the set of “predator” stores (Canada Safeway’s in our example), I/ the set of “victim” stores (Loblaw’s, Tom Boy’s, and Dominion’s) and C the set containing stores of the “competitive fringe” (IGA’s, Woodward’s, etc.). To ease the discussion of spatial interplay between stores we shall sometimes say “a P store bounds a V store” or “ they are adjacent” or “ they are neighbors” when we mean that the market area of a store that is a member of P borders on the market area of a store belonging to the set I/. Taking these liberties we can now define p c P as the set of P stores bounding Y stores; v as the subset of V which contains neighbors of P stores; and C as the subset of C which contains neighbors of P stores. The membership of all the above sets changes over the sample period (1959-1973) and so we shall sometimes adjoin a time argument t to signal this dependence. The main tool in our tests will be the function m( K, q, t), which represents the average market area of all stores in set K at time t at prices q. The first set of hypotheses that we wish to test involves market areas of victim stores bounded by predators, as compared to those of other stores, i.e., we compare m(v,q, t) with m(K,q, t), K # v, for all t; q is here constant across all sectors and has been omitted: m(v, t) < ,(F, ,(t, ,(v,

t)

all t

(1)

t) < m(P, t) t) < m(C, t)

allt all t

(10 (III)

t)

all t

t) I m(v/, t)

all t.

(IV) (V)

m(v, t) < ,(C, ,(v,

Hypothesis (I) follows from the fact that the predator would be expected to locate its new stores in the neighborhoods of the victim stores in such a way as to inflict losses on the victim stores but maximize profits for its own stores. Hypothesis (II) is merely an extension of hypothesis (I). Implicit in hypothesis (II) is the subsidiary hypothesis that one would not expect those predator stores which are not members of P to have market areas so small as to reduce m(P, t) below m(p, t). Hypothesis (III) follows from the fact that the predator’s targeted victim stores would be expected to have an average market area which is less than that of nonvictim stores. Hypothesis (IV) extends hypothesis (III) in the sensethat one would expect the average market area of the predator’s intended victim stores to be less than the

252

VON HOHENBALKEN

AND WEST

average market area of those nonvictim stores which happen to be close enough to v stores to be bounded by members of P. Hypothesis (v) is the only hypothesis which is not stated in terms of a strict inequality. While one would expect V(t) = p(
the coordinates of all stores in operation in year t, the distribution of store ownership in year t (see Fig. 2), the price structure in year t, coordinate information on city boundaries and the dummy store.

To make our results comparable with earlier work on the same data, we

253

PREDATION AMONG SUPERMARKETS

11 1959

I I 19611963

I 1965

I 1967

I I 196919711973

I

+

Year FIG.

2. Store holdings by type of firm: 1959-1973, Edmonton

decided to let the market areas of Edmonton be constrained only by the city limits and its main topographic feature, the valley of the North Saskatchewan River. We considered the river valley a barrier which divides Edmonton into two separate subcities. Both South and North Edmonton are basically convex regions, whose true boundaries (the city limits and the river) are approximated by piecewise linear curves. One prominent incursion of the river into North Edmonton was simulated by a dummy store, placed into the bend, whose boundaries created the desired cavity in the otherwise convex region (see Fig. 3). Given the results of the previous tests for spatial predation conducted by West, we conjecture that the tests of the market area implications of spatial predation will be consistent with Canada Safeway being designated the predator, Loblaw, Dominion, and Tom Boy being designated the victims, and all other supermarket firms operating in Edmonton between 1959 and 1973 being designated competitive fringe firms. Table 2 presents the calculations of average market areasby type of firm for years 1959 to 1973 for both equal and selectively reduced prices. From Table 2, it is clear that hypotheses(I)-(IV) do in fact hold for all years. Indeed, with the exception of three years, we find that m( P, q) and m( P, q) are well over twice m( v, q). Also, with the exception of four years, m(c’, q) is at least 25% greater than m(v, q). Hypothesis (V) holds for 10 out of 15 years. For the 5 years in which hypothesis (V) does not hold, the differences between m( v, q) and m(V, q) are small. One result in Table 2 which might appear anomalous is the fact that m(P) > m(P) for 8 out of 15 years. However, the anomaly disappears

254

VON HOHENBALKEN

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FIG. 3. (a) Edmonton, 1959; (b) Edmonton, 1973.

when one realizes that the magnitude of m(P) depends upon where P stores are located relative to I/ stores (i.e., some p stores might be located relatively close to I/ stores, but may nonetheless have a portion of their market areas extending to the city limits). In addition, the stores which form the complement of P in P for various years might have been established prior to the date when a predatory location strategy was implemented, and could have an average market area below that of p stores. (Indeed, for the years in which m(P) > m(P), all but three stores which form the complement of p in P were already established in 1959.) This is why m(P) -c m(P) was not one of the tested hypotheses. To assist the reader in visualizing the spatial configurations of stores that have led to our results, we had our computer construct maps (Figs. 3a and b) that illustrate market areas and store locations in Edmonton for the years 1959 and 1973. In both years, it is apparent that the average market area of

PREDATION AMONG SUPERMARKETS

255

b

lkm

FIG. 3. (continued)

V stores was less than the average market area of P stores, and also that V stores tended to be bounded by P stores. One can also see that whereas there were quite a few C stores on the periphery of the market in 1959, by 1973 the periphery was dominated by P stores,just as theory suggests. To test the second set of market area predictions discussedin the previous section, we need to calculate Am(K)/m(K) for K = V, v, C, C:, where Am( K)/m( K) represents the relative reduction in average market area for firm K stores resulting from a five percent reduction in p-store prices. (The magnitude of the price reduction will not ah&t the qualitative results provided that it is not so large as to reduce a store’s market area to zero.) In Table 2, we report our recalculations of average market areas by type of firm given the five percent reduction in p-store prices. We find that Am(V)/m(V) > Am(C)/m(C) for all years, and Am(t)/m(v) > Am( C)/m( C) in all but two years. Both of these results are consistent with theoretical predictions.

256

VON HOHENBALKEN

AND WEST

TABLE 2 Average Market Areas by Type of Firm for Equal and Selectively Reduced Prices: 1959-1973, Edmonton”. ’ ____

m(P)

Year

m(P)

m(C)

m(V)

1959

147 139'

560 590

465 485

783 773

561 547

206 199

1960

144 133

368 388

498 511

654 647

452 443

234 227

1961

256 245

423 450

528 545

611 600

569 555

214 265

454

440

535 562

489 512

566 554

490 474

438 427

1963

418 405

516 543

473 496

585 571

488 412

418 407

1964

309 295

804 834

135 760

422 407

441 424

300 287

1965

256 241

886 915

810 834

347 333

359 344

250 237

1966

252 237

917 947

840 865

274 260

273 258

247 233

1961

266 251

918 947

841 866

282 26-l

281 265

262 247

1968

228 214

748 175

728 748

290 275

289 274

228 214

1969

204 191

578 606

669 689

347 333

347 332

204 191

1970

201 189

556 582

666 685

305 293

305 292

201 189

1971

228 215

496 523

656 673

295 283

300 288

228 215

1972

223 209

497 525

657 674

292 281

297 285

223 209

1973

229 214

508 538

658 675

298 285

304 290

229 214

1962

‘For each year the first-row entries represent average market areas for equal prices while second-row entries give them for selectively reduced prices; all entries are rounded to integers. ‘The table is expressed in units that correspond to the grid in the original city map we used; one such unit equals 1.8 acres; Edmonton comprises about 35,000 units = 100 square miles = 260 km*. ‘The 5% reduction of some prices prompted lacunae (as shown in Fig. 1) which cover only 0.1% of the total area.

PREDATION

AMONG SUPERMARKETS

251

From all of our test results, we conclude that the Edmonton market data are generally consistent with Canada Safeway having engaged in locational predation against Loblaw, Dominion, and Tom Boy. 4. CONCLUDING

REMARKS

In this paper we have described an algorithm for finding neighbor relations and market areas of stores located on a Euclidean plane. The method is flexible (it permits modeling price differences) and very efficient (over 2000 nearest point sets were computed in about 100 set CPU, in an APL environment on an AMDAHL V8). We have used our algorithm to test some market area implications of a theory of locational predation, and we have found that the data generally support the hypothesis that Canada Safeway engaged in locational predation against new entrants in the Edmonton city market. ACKNOWLEDGMENT We acknowledge the assistance of Alan Sharpe in the preparation of the computer plots contained in the text.

REFERENCES 1. B. C. Eaton and R. G. Lipsey, The block metric and the law of markets, J. Urban Econ. 7, 337-347 (1980). 2. D. M. Kreps and R. Wilson, Reputation and imperfect information, J. &on. Theory. 27, 253-279 (1982). 3. M. I. Shames, “Problems in Computational Geometry,” Springer-Verlag, Berlin (1976). 4. B. Von Hohenbalken, Least distance methods for the scheme of polytopes, Math. Programming 15, l-11 (1978). 5. B. Von Hohenbalken, Finding simplicial subdivisions of polytopes, Math. Programming 21, 233-234 (1981). 6. D. S. West, Testing for market preemption using sequential location data, Bell J. Econ. 12, 129-143 (1981). 7. D. S. West, Market predation in a spatially extended market, University of Alberta, Dept. of Economics Research Paper 82-3, February 1982. 8. D. S. West, Market predation and entry deterrence, mimeo, University of Alberta, Dept. of Economics, February 1982.