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Spatial competition and nonlinear responses in marketing Krider, Robert E. 1993

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We accept this thesis as conformingto the required stan daTHE UNIVERSITY OF BRITISH COLUMBIAJULY 1993SPATIAL COMPETITION AND NONLINEAR RESPONSES IN MARKETINGbyROBERT E. KRIDERB.Sc., The University of British Columbia, 1975M.Sc., The University of British Columbia, 1985A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Commerce and Business Administration)© Robert E. Krider, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) Department of ^6)tq &—EC^E1A5 i{ilt ^/MS 1771/11-7-1(.‘The University of British ColumbiaVancouver, CanadaDateDE-6 (2/88)AbstractSpatial competition, in the context of industry-wide changes in retailing formats andstrategies, is addressed in this dissertation from a theoretical modelling perspective. Chapter2 develops a normative individual choice model to explore how "power retailers" affectgrocery shopping behaviour, and, consequently, market share. Power retailers are very largeretail outlets that compete primarily on price, and are known variously as warehouse clubs,category killers, and superstores. The model shows that consideration of consumerstockpiling can lead to an "increasing returns" nonlinear response of market share to pricereductions, and that the effect is not noticeable when competitors have small pricedifferences. The model also differentiates between perishable and nonperishable goods, andshows that this may drive planned multistore shopping. Chapter 3 starts with the observationthat competent management in many sectors of retailing, including grocery retailing, requiresan ability to respond quickly and effectively to unexpected adversity. This dynamic isincluded in an oligopolistic spatial interaction model, and the system is shown to evolve toa novel and robust stochastic steady state known as self-organized criticality (SOC). Onecharacteristic of the SOC state is that it allows small exogenous shocks to produce largeresponses at a rate greater than would be expected if the law of large numbers applied. Thiswork represents the first known investigation of SOC in a marketing setting.iiTable of ContentsAbstract ^  iiList of Tables  viList of Figures ^  viiAcknowledgement  xiiChapter 1: Introduction ^  1Chapter 2: Grocery Shopping Behaviour in the Presence of a Power Retailer . . . 62.1 Introduction ^  62.2 Literature Review  92.3 Model Development ^  182.4 Results for Some Simple Cases ^  242.4.1 Non-perishable goods and one store only ^ 242.4.2 Perishable goods at one store only  28^2.4.3 Perishable and non-perishable goods   302.5 Two Goods at Two Stores: Feasible Normative Behaviour ^ 372.6 Optimal Shopping Behaviour: Solution to a Special Case ^ 572.6.1 Shopping at the small store only ^  572.6.2 Shopping at the large store only  582.6.3 Shopping at Both Stores ^  58iii2.7 A Numerical Application: Two-Dimensional Trading Areas ^ 722.8 Discussion ^  892.9 Contributions and Research Opportunities ^  95Chapter 3: Spatial Competition and Self-Organized Criticality ^ 1003.1 Introduction ^  100^3.2 Self-Organized Criticality   1023.3 Literature Review ^  1103.4 Model Development  1243.4.1 Customer Choice ^  1253.4.2 Firm Revenues  1293.4.3 Market Configuration ^  1293.4.4 Dynamics ^  1293.4.5 Software  1313.5 Results of Numerical Experiments ^  1333.5.1 Transient Behaviour  1343.5.2 Size Distribution of Avalanches of Innovation in the SteadyState ^  1383.5.3 Robustness to model parameters ^  1433.5.4 Robustness to Dynamic Structure  1573.5.5 Sensitivity to Initial Conditions ^  159iv3.6 Summary of Numerical Experiments ^  1633.7 Contributions, Limitations, and Research Opportunities ^ 165Chapter 4: Conclusions and Contributions, Limitations, and Future Research ^ 1704.1 Grocery Shopping Behaviour in the Presence of a Power Retailer ^ 1714.1.1 Conclusions and Contributions ^  1714.1.2 Limitations ^  1744.1.3 Future Research  1754.2 Spatial Competition and Self-Organized Criticality ^ 1784.2.1 Conclusions and Contributions ^  1784.2.2 Limitations ^  1794.2.3 Future Research  180Bibliography ^  183Appendix A: Calculation of Market Shares^  193Appendix B: C Code ^  195List of TablesTable 1: Notation ^  22Table 2: Parameter values used in numerical calculations. ^ 73Table 3: Small store shares under the three scenarios  88Table 4: Model Characteristics in Chatterjee and Eliashberg's (1985)Classification. ^  112Table 5: Parameters Investigated in Sensitivity Tests ^  133Table 6: Parameter values for transient test.  135Table 7: Summary of parameter sensitivity tests. ^  157viList of FiguresFigure 1: Stockpiling Parameters ^  20Figure 2: A complete and mutually exclusive classification of shopping behaviour.^37Figure 3: Timing of first purchase of non-perishable at small store. Diagram refersonly to lemma 3n. Subsequent results will further restrict this generalpicture. ^  43Figure 4: Visits to the small store to purchase both goods more than once per periodcannot be optimal. ^  45Figure 5: Shopping Behaviour Possibilities ^  46Figure 6: The amount of perishable purchased at the large store in the^case(diagram shows case (a) of Proposition 4n)^  48Figure 7: End-of-period effects in "n" case.  50Figure 8: Notation for the "p" case. ^  54Figure 9: The complete set of shopping possibilities^  56Figure 10: Mixed Store Shopping Pattern  59Figure 11: Long-run average trip costs to the small store as a function of length ofthe periodic interval. ^  61Figure 12: Perishability constraint binds as trip costs increase (fixed m). ^ 66Figure 13: Possible transitions from m trips to m +1 trips as period Atn ; increases:(a) perishability constraint binding for both m and m +1 (b) constraint bindingfor m and not binding for m +1.   68viiFigure 14: Perishability constraint binds outside circular region (small-store onlyshopping behaviour)^  74Figure 15: Perishability constraint is binding outside circle (large-store-only shoppingbehaviour)^  74Figure 16: Single-store-only, binding constraint, optimal cost surfaces. ^ 75Figure 17: Progressively larger regions around the small store where the perishabilityconstraint is not binding, for m = 5, 6, and 7 ^  76Figure 18: Optimal cost surfaces for mixed-store, binding constraint, shopping behaviou78Figure 19: Comparison of binding, mixed-trip costs for various values of m giveregions where each is minimal. From left to right the regions are for m =0, 1, 2, 3, and 4  79Figure 20: Contours of intersection of the small-store only cost surface with themixed trip (from left to right: m = 0,1,2, and 3) surfaces. ^ 80Figure 21: Superposition of Figure 20 on Figure 19. The Figure 20 intersections^occur to the right of the corresponding Figure 19 regions    80Figure 22: Intersection of mixed, m=4, binding optimal cost surface with small-only, nonbinding optimal cost surface ^  82Figure 23: Perishability constraint for small-only cost function is nonbinding insidethe circle. ^  82Figure 24: Intersection of nonbinding cost surfaces for small-store only, and mixed,m = 5 ^  83viiiFigure 25: Perishability constraint is nonbinding, for small-only shopping, inside thesmall circle. ^  83Figure 26: Perishabilty constraint is nonbinding on the mixed, m = 5, case inside thesmall circle. ^  83Figure 27: The intersection of the nonbinding, mixed, m = 6 contour with thenonbinding, small only, contour. ^  84Figure 28: Market areas of the large and small store, located at (-5,0), and (5,0), forthe parameters given in the text^  85Figure 29: Market areas with a 10% price difference between stores. From left toright, the regions are exclusively the large store's; mixed with m = 1 andm= 2; and exclusively the small store's.   87Figure 30: A greater consumption rate increases the tendency to stockpile. From leftto right market areas are exclusively the large store's; mixed with m = 1 andm = 2; and exclusively the small store's.   88Figure 31: Share response of the large store to its own price reductions, using thenumerical example of the last section. At small price differences, no mixedshopping occurs, and the two goods have the same share.   90Figure 32: Trading area defined by the gravity model, (42), with a 10% pricedifference; cf. Figure 32. ^  92Figure 33: Trade areas with 10% price difference using the stockpiling model (fromFigure 29)^  92ixFigure 34: Gravity model prediction of market areas with a 20% price difference.93Figure 35: Trade areas from stockpiling model with 20% price difference (fromFigure 19)^  93Figure 36: Weekly prices of a food basket of approximately $100 value, at twosupermarkets, relative to the price of the same basket purchased at RCS, thepower retailer, in Edmonton, Alberta, in 1992. (Week 1 = Jan 6)   94Figure 37: Transient behaviour: convergence of total profits in 64 store system.Parameters as in Table 5. ^  136Figure 38: Convergence of system profits with shock and innovation magnitude 50times those in Figure 37. ^  137Figure 39: Transient behaviour with increments 500 times those in Figure 37 is thesame as with increments 50 times those in Figure 37, indicating that largeshocks and innovations quickly remove the effect of the initial conditions. . 138Figure 40: Size Distribution of avalanches in the steady state, for the base case, withapproximately 60% of shocks producing avalanches ^ 140Figure 41: Avalanche size distribution when shocks are relatively large^ 142Figure 42: Size distribution when shocks are relatively small. ^ 142Figure 43: The distribution appears to fall off faster than a power law when the sizeexponent is 0 . 5^  144Figure 44: Avalanche size distribution when size exponent is 2.0 ^ 145Figure 45: Avalanche size distribution when size exponent is 3.0. ^ 145xFigure 46: The adjustment algorithm restores the power law (cf Fig. 43) . ^ 148Figure 47: Distribution with adjustment algorithm for alpha = 3.0 (cf Fig 45). . . ^ 148Figure 48: Log-linear plot of the data in Figure 47. Note the curvature at smallsizes. ^  149Figure 49: Size distribution, Q= 0.5. ^  150Figure 50: Size distribution when (3= 2 . 0 .  150Figure 51: Size distribution when j3 = 3.0. ^  151Figure 52: A small array: The distribution departs from a power law at around 8firms. ^  152Figure 53: With 64 stores, distribution breaks from a power law at around 12firms. ^  152Figure 54: The break in the power law for the large array is also around 12stores^  153Figure 55: For an 18 x 18 array, the break is around 16 stores^ 153Figure 56: Distribution of avalanche sizes with a larger reservation distance. ^ 156Figure 57: Distribution of sizes with a smaller reservation distance. ^ 156Figure 58: Changing the form of the innovation dynamic and the threshold fromprofits to revenues does not affect the avalanche size distribution. ^ 159Figure 59: Separation in store size space after one store is perturbed. ^ 160Figure 60: As in Figure 59, with initial perturbation 4 times greater  161Figure 61: Separation in store size space after the larger perturbation in Figure 59is applied to 3 stores. ^  162xiAcknowledgementsIt is with pleasure that I acknowledge the support and assistance of my committee members,Dr. Murray Frank and Dr. Ken Denike. It has been especially a privilege to work with mycommittee chairman and mentor, Dr. Charles Weinberg, whose encouragement, advice, andresponsiveness were invaluable for the completion of this dissertation. Dr. Derek Atkinsgave freely of his time to discuss inventory control theory, and an observation of his led toLemma 7n.I gratefully acknowledge the Social Sciences and Humanities Research Council, andthe Faculty of Commerce and Business Adminstration for financial support.Finally, to my wife Clair, who alone knows her contribution: thank you.I dedicate this dissertation to my parents, Evert and Inga.xii1CHAPTER ONEINTRODUCTIONMarketing academics and practitioners have an interest in spatial competition that dates backseveral decades. Of the two main branches of spatial competition research, namelypositioning in product attribute space, and location of facilities, particularly retail outlets, ingeographic space, this research is concerned with the latter. In this chapter, the structureof the dissertation is first briefly outlined, and then the two major sections discussed in moredetail.The context motivating the work is the grocery retailing business, a particularlydynamic, even volatile, sector of the retailing industry. It is characterized by intensecompetition and continually changing formats. This research explicitly recognizes thischaracter and explores its consequences. Specifically, Chapter 2 develops a theoreticalmodel to look at the impact of a retailing format--the "power retailer"--that has dramaticallyaltered the nature of retail competition in many categories across North America over thelast few years. In spite of power retailing's importance, only recently has it receivedresearch attention from marketing academics (e.g., Farris and Ailawadi, 1992).In Chapter 2, a consumer decision model is developed that is particularly applicablein situations where consumers have a choice between power retailers (for example,"superstores") and more conventional outlets (e.g., supermarkets). One consequence of themodel developed in Chapter 2 is the presence of increasing returns to scale to pricereductions in the market studied. This nonlinearity suggests a number of interesting areasChapter One: Introduction 2to pursue and Chapter 3 examines one of them. In particular, Chapter 3 tackles the natureof the long-run organization of an industry with the characteristics of the retail trade, andespecially of food retailing. The rapid and major changes in formats, and the need torespond quickly and effectively to these changes because of the intense competition, arefeatures that are captured in a simple model. The model, and the resulting stochastic steadystate, are in the spirit of traditional theories like the "Wheel of Retailing" (McNair, 1958).The common thread that runs through the traditional work and the research reported here isthe attempt to formalize the intuition that, in spite of the turbulent nature of retailing, thereis some underlying order. Chapters 2 and 3 will now be discussed in more detail.In Chapter 2, the impact of power retailers on the grocery trade is examined. "Powerretailer" is a term referring to a retail format that has appeared in many retail categories overthe last decade. It refers to very large retail outlets that compete primarily on price, andincludes formats known as "superstores", "warehouse clubs", and "category killers". Thesestores become destinations in their own right, with large trade areas and dramatic impactson suppliers, competitors, and customers. While channel relationships between suppliers andpower retailers have been the topic of some research (e.g., Farris and Ailawadi, 1992), andthe impact on competitors and customers is commonly discussed in the trade and popularpress (e.g., Business Week, December 21, 1992; Financial Times of Canada, April 3, 1993),theoretical work on customers and competitors is very limited. In particular, there is a needto examine the fundamental issue of individual consumer choice behaviour in the presenceof a power retailer, as it is consumer behaviour and the firm's ability to respond to it, whichultimately determines success or failure.Chapter One: Introduction^ 3To address this gap, a model is developed and analyzed to determine timing andpurchase quantities of optimizing consumers who trade off the costs of purchase price,travelling to and from the store, and holding inventory. Consumer stockpiling has long beenan issue as a response to short-term promotions (e.g., Meyer and Assuncao, 1992), and hasbeen modelled as a planned behaviour in the context of multipurpose shopping, and theresulting agglomeration of different types of retailers (that is, if consumers need to do somebanking, drop off the dry cleaning, and pick up something for supper, it is more efficientfor them to drive to one location where all three outlets are located close together, than tomake several stops; see, for example, Ingene and Ghosh, 1990). The inclusion of consumerstockpiling as a factor affecting long-term choices between directly competing retailers isnovel.Food products have a characteristic which complicates the stockpiling problem: somegoods are perishable and have much shorter storage lifetimes than others. This aspect ofgrocery retailing is very important to shopping behaviour and is explicitly taken into accountin the model. While much research on perishable goods has been done in the operationsresearch literature on inventory control, the inclusion in this context is unique.In summary, Chapter 2 addresses the following questions:• How does consumer grocery shopping behaviour change in the presence of apower retailer?• How is market share affected when a power retailer enters a grocery market?Chapter One: Introduction 4One of the consequences of the model is a possible explanation of why incumbentsin grocery retailing did not expect the new superstores to be as successful as they were: itis possible for the market share of a store to exhibit nonlinear increasing returns to pricereductions, but only when the price differences between stores become great enough. Thesenonlinear returns were possibly not anticipated by traditional retailers (or by traditionalmodels, as shown in Chapter 2). This leads to a more general observation, namely thatretailing is such an intensely competitive and dynamic industry, that unexpected adverseconsequences are the rule rather than the exception. To the extent that this is true, it followsthat a component of successful management must be the ability to respond creatively andeffectively to unexpected adversity. This leads to the main question addressed in Chapter3:• What is the nature of long-term industry structure when the dominant dynamicis unexpected adversity, followed by creative and effective response?This response to competitive and environmental attacks is very nonlinear, and anunusual dynamic from the point of view of theoretical modelling; however, it will be arguedin Chapter 3, that it is a commonly acknowledged characteristic of retailing (see, forexample, Corstjens and Doyle, 1989).This dynamic is formalized and embedded in a spatial interaction model. Spatialinteraction models are well established probabilistic market share models used in empiricaland decision-oriented work in retail outlet (and other facility) location problems (e.g., JainChapter One: Introduction^ 5and Mahajan, 1979).Chapter 3 contributes not only by modelling an unusual and important dynamic, butalso by showing that in the long run, this dynamic, operating in a spatial competitioncontext, leads to a novel steady state. This state, known as self-organized criticality (SOC),has only once before been described in an economic context (Bak, Chen, Scheinkman, andWoodford, (1992). It has, however, received a very large amount of attention as a robustand general organizing principle in the physical and life sciences since it was first describedin Bak, Tang, and Weisenfeld (1988).The economic model of Bak et. al. (1992) is a highly stylized inventory andproduction model. While Bak et. al. (1992) is important in showing the possibility of SOCemerging in economics, it is mathematically isomorphic with the prototype model instatistical physics, with an economic interpretation imposed on it. The model developed inChapter 3, in contrast, has its origins solidly in existing marketing models. As such, it isa first example of the possibility of self-organized criticality arising from a purely economicmodel.The dissertation closes with Chapter 4. The results and contributions of the previouschapters are summarized. Limitations are noted, and resulting directions for future researchsuggested.Research into spatial competition in marketing has a long and rich history. It isbelieved that the research reported here adds to our understanding of the field, particular inthe complex and dynamic area of retailing.6Chapter 2Grocery Shopping Behaviour in the Presence of a Power Retailer2.1 Introduction"Clout! More and More, Retail Giants Rule the MarketPlace", is the headline on a recentBusiness Week cover story. To quote, "In category after category, giant "power retailers"are using sophisticated inventory management, finely tuned selections, and, above all,competitive pricing to crowd out weaker players". Marketing academics, too, haverecognized the phenomenon. The academic emphasis, however, has been primarily on thechannel relationships. For example, Farris and Ailawadi (1992) discuss the apparentredistribution of channel power to the retailer. The nature of horizontal competition, andthe impact on consumer shopping behaviour has yet to be investigated. The research in thischapter addresses those issues in the context of food retailing.Specifically, a modelling approach is taken to answer two related questions: first,"How does consumer shopping behaviour change when a power retailer enters the grocerymarket?"; and, second, "What is the effect of the presence of a power retailer on share ofmarket?". Assuming cost-minimizing shoppers, for whom three costs are important (tripcosts, inventory costs, and price), and recognizing that some goods are perishable, the modelprovides several important insights. Two are that, first, stockpiling behaviour becomesdramatically more important as the price reduction of the power retailer becomes greater,leading to an "increasing returns" effect of price reductions; and, second, the differencebetween perishable and nonperishable goods can lead some consumers to have normativestrategies of shopping at more than one store, even when there is no uncertainty. CombiningChapter 2: Grocery Shopping Behaviour^ 7these two results, it is shown that perishable goods are much less susceptible to the influenceof a power retailer entering the market, than are nonperishables.Food retailing is an appealing industry to initially investigate the power retailingphenomenon for two reasons. First, total demand for groceries is relatively inelasticcompared to many other retail goods (Ghosh and Harche, 1992). This allows modelling tofocus on the allocation of the food dollar to the competing retailers. Second, the food anddrug category of retailing is relatively well defined. It will be helpful, for subsequentempirical work, to have well defined market boundaries.One case which helps illustrate and motivate this research is the entry of the RealCanadian Superstore (RCS) to the western Canadian grocery retailing industry, during thelast decade. In the major cities of western Canada (Winnipeg, Calgary, Edmonton, andVancouver), the retail grocery business was dominated by supermarkets, led by Safeway, inthe early eighties. The supermarkets were typically 20,000 to 40,000 square feet, and tradedin a relatively localized area of a few kilometres in diameter. Price competition appearedto be healthy, with much promotion and advertising. RCS entered various cities in the mid-eighties, with an expansion westward, opening in Edmonton in 1984, and in Vancouver in1989. RCS stores were over 100,000 square feet, of which about 60% to 70% was devotedto food and drugs. Labour and land costs of the RCS outlets were kept dramatically lowerthan Safeway's, by employing non-union staff, and locating in industrial areas. The resultingreduction in prices, as well as some increase in variety, appeared to make the outlets adestination, so that the unfavourable locations were not a disadvantage. RCS, like powerretailers in other categories, drew tremendous market share with very few outlets. OneChapter 2: Grocery Shopping Behaviour^ 8of the interesting aspects of the entry of RCS to the market, and one which a model shouldbe able to explain, is Safeway's early response. According to private communications withconsultants and academics in Winnipeg, Edmonton, and Vancouver, Safeway was surprisedby the impact of RCS on the market. The modelling approach in this chapter suggests areason: the market's highly nonlinear response to price differences that arises whenconsumers are allowed to stockpile. Consumers at unusually long distances from theSuperstore can make large purchases, in order to make the long trip less frequently, and thuskeep long-run average trip costs down. If the price advantage is great enough, thestockpiling costs will be offset, and the power grocery retailer can capture a huge tradingarea.Chapter 2: Grocery Shopping Behaviour^ 92.2 Literature ReviewThe model developed in this chapter draws on literature in three areas: consumer stockpilingin response to promotions, normative inventory control theory, and multipurpose shopping.First the relevance of these areas will be stated, and then the literature reviewed.The literature on promotions in retailing establishes, first, the value ofnormative inventory models in consumer shopping (eg., Meyer and Assuncao, 1992).Second, it establishes that consumers make tradeoffs between the costs of the goodspurchased ("price costs") and inventory holdings (eg., Blattberg, Eppen, and Lieberman,1981). Third, it recognizes the "fill-in trip", a minor trip that occurs between major trips,which has a parallel with the shopping behaviour studied in this research (eg., Kahn andSchmittlein, (1992). Finally, this literature is also useful for structuring consumer's holdingcosts. Although perishability is recognized as a relevant factor in the promotion literature,it has not previously been explicitly treated as an inventory cost. There is, however,extensive work on perishable goods in the operations research inventory control literature(Nahmias, 1982). The form of the perishability costs are taken from this literature.As an inventory control problem, the situation would fall into the class of jointreplenishment problems (JRP). Joint replenishment refers to the problem of maintaininginventory of more than one good, when there is the possibility of reducing replenishmentcosts by ordering the goods simultaneously. The problem analyzed here is different fromthe solved JRP problems in the types of goods (perishable and nonperishable) analyzed, andin differences in cost structure associated with consumers, as opposed to firms. An exampleof a JRP that, like this problem, has periodic solutions, is Andres and Emmons, 1975.Chapter 2: Grocery Shopping Behaviour^ 10The notion that consumers trade off price against travel costs has a long history. Inthe last decade inventory costs have also been included in this tradeoff, in the multipurposeshopping literature (Ghosh and McLafferty, 1986; Ingene and Ghosh, 1990). Multipurposeshopping refers to the observation that consumers will make a single trip to a central locationto purchase more than one good or service (Christaller, 1933). The model developed in thisresearch is similar in spirit to the multipurpose shopping literature in that consumers areassumed to make a choice on the basis of cost minimization over transportation, inventory,and price costs. It differs in its objectives (choice between stores differentiated by price,rather than between "order", or level of agglomeration, of centre), the detailed coststructures, and types of goods (perishable vs nonperishable) considered.Now, each of these three areas will be discussed more fully.Promotions and StockpilingA large literature exists on consumer grocery shopping behaviour in the presence ofpromotions, and a substantial proportion of this deals with consumer stockpiling as aresponse to promotions. The usual concern is that if purchases are simply accelerated intime or quantity in response to promotions, they may not be beneficial to the manufactureror retailer (Neslin, Henderson and Quelch, 1985; Gupta, 1988). This concern, and thecontext of short term responses to promotions, is quite different from the long termbehaviour addressed in this chapter. (For an excellent current review of the promotionsliterature see Blattberg and Neslin, forthcoming). However, some of the results that arerelevant to this study will be reviewed here.Chapter 2: Grocery Shopping Behaviour^ 11Blattberg, Eppen, and Lieberman (1981) present a theoretical model where consumerstockpiling may be beneficial to the retailer. Under the assumptions of a single brand anda single optimizing retailer, and optimizing consumers (trading off price against holdingcosts) who have holding costs less than half the retailer's holding costs, the retailer mayincrease profits by reducing holding costs through dealing. This reason for dealing iscontrasted with an alternate explanation based on manufacturer's trade deals offered toretailers, to force retail price reductions and encourage consumer trial. An empirical testfinds the inventory explanation consistent with the data, and, more importantly for the modeldeveloped in this chapter, suggests that consumers do trade off price against holding costs.Another important reason for dealing is to attract customers to the store, and thenhope that they will buy non-deal items while they are there. Kumar and Leone (1988)investigate this store substitution effect in the disposable diapers product category. They findthat price promotions in one store negatively affects the same-category sales in competingstores, indicating that promotions do attract customers to the store. However, the effect wasdependent on geographic proximity of the competing stores, implying that there was a limitthe customers would travel to take advantage of a promotion. Whether this is a plannedtradeoff between price and travel costs, as assumed in the long-term context of the researchin this chapter, or simply opportunistic shopping within a local set of stores, is notaddressed. Walters (1991) found similar, although weaker, insterstore effects in spaghettiand cake mix categories.Customers may neutralize the promotional objective described above in a number ofways, such as "cherry-picking", (shopping at different stores and purchasing only the dealChapter 2: Grocery Shopping Behaviour^ 12items, e.g., Bucklin and Lattin 1992) or stockpiling deal items that would have beenpurchased in any case. Much of the research in this area involves testing for stockpiling orcherry-picking, based on panel data, and initially did not find consistent patterns. WhileBlattberg et. al. (1981) found evidence of purchase quantity and timing acceleration,McAlister (1985) found minimal acceleration. This led to various attempts to sort out whysome studies found stockpiling and some did not (eg., Helsen and Schmittlein, 1992).Litvack, Calantone, and Warshaw (1985) suggested that some goods were more susceptibleto deals because they were "stock-up" goods. A "stock-up item" was defined as "anynonperishable good in a unit size that is consumed frequently by a purchaser's household".Similarly, a non-stock-up item was either perishable or infrequently consumed. Using expertjudges to select 36 stock-up and 36 non-stock-up items, a four-store price experiment wasrun to determine elasticities. The hypothesis, that elasticity for stock-up products would begreater than for non-stock-up products, was strongly supported. This suggests not only thatconsumers stockpile in response to price reductions, but that the response varies accordingto "stockpilability", one component of which is perishability.Raju (1992) examines the effect of promotional activity in a product category, as wellas category characteristics, on sales variability. In particular, he found that "bulkiness"reduces sales variability, consistent with the notion that "stockpilability" is important toconsumers when considering price reductions.Another predictor of the effectiveness of promotions is whether the trip is "major"or "fill-in". The notion of a filler trip has a long history in the marketing literature (Kollatand Willet 1967; MacKay 1973) and is relevant because the model in this chapter predictsChapter 2: Grocery Shopping Behaviour^ 13a similar (although not identical) type of behaviour. Filler trips are defined in various waysin the literature, but always on the basis of expenditure. Major trips are large expendituretrips, and filler trips involve smaller expenditures. In applying Ehrenberg's negativebinomial buying model to filler trips, Frisbie (1980) considers three definitions of a fillertrip. One is based on an absolute dollar threshold as a cutoff; one is a threshold percentageof monthly food expenditure and varies by household; and one is a threshold based on apercentage of annual income, and also varies by household. Frisbie finds that the differentdefinitions have little effect on the ability of the NBD to describe the shopping behaviour.Kahn and Schmittlein (1992) define the cutoff on a household basis by constructing ahistogram of amounts spent on each trip for the household. If the histogram is unimodal,the mode is used as the cutoff. If it is bi-modal, the midpoint between modes is used as thecutoff. They then use the definition to correlate the type of trip with various types ofpromotions. In all of this research, the distinction between trip types is a defined quantity,based on expenditures on goods. In the research of this chapter, the different trip types arethe result of optimizing behaviour; and not all customers will find it optimal to engage in fill-in shopping. Furthermore, the differences in trip types are not only size of expenditures.They are also related to expected long-term price differences, and the perishability of thegoods purchased.Fill-in trips also occur at shorter intervals than major trips. MacKay (1973) foundthat major trips tended to occur weekly, with fill-in trips more frequently. This pattern ofregularly weekly shopping trips and irregular fill-in trips is commonly observed (eg., Kahnand Schmittlein, 1992). The general pattern is consistent with the results of this chapter;Chapter 2: Grocery Shopping Behaviour^ 14however, the time scales are substantially increased. It will be argued that in the presenceof a power grocery retailer, there are some consumers who add another level of trip--a"super trip" for which the previous "major trip" becomes a fill-in trip.Normative Inventory Theory One of the unique features of the model in this chapter, compared to existingconsumer shopping models, is that perishable goods are treated explicitly and are a criticalfactor in governing shopping trip behaviour among competing stores. Perishable goods,however, have been treated extensively in the theoretical inventory management literature.Of the many kinds of perishability studied (eg., deterministic and stochastic demand,deterministic and stochastic product lifetimes, and continual deterioration; see Nahmias, 1982for a review) this chapter considers the case of fixed deterministic lifetimes and demand ratesfor the perishable good. This case is analyzed in the perishable inventory literature whenthe perishable good also has time-dependent holding costs, and under fairly generalconditions it is shown that ordering will occur such that no items expire, that is, becomenon-usable. In contrast, the model here has one good with only holding costs, and a secondgood with only perishability costs. Results for this case are derived in section 2.4, andshown to be parallel to the case of one good with both holding and perishability costs.Another inventory theory aspect of the model presented her concerns the costsassociated with buying more than one good. In the model developed here, once customershave incurred one trip cost, they may purchase either or both goods with no additional trip,or "ordering cost". The inventory management literature refers to this as the jointreplenishment problem (JRP). The consumer-based variation considered here could beChapter 2: Grocery Shopping Behaviour^ 15considered as an example of the infinite horizon JRP problem. There are two importantdifferences, however. First, the JRP literature assumes variable costs are constant and aretherefore ignored. In the consumer model here, we are concerned explicitly with thecompetitive situation where there are price differences between stores, and prices of thegoods translate to the variable costs in inventory management. Second, the JRP literatureassumes there is a "major set-up cost" for an order, plus an additional "minor set-up cost"for each type of good included. For the consumer problem, there is only a "trip cost",which corresponds to the major set-up cost.The consumer model may be considered as a member of the subclass of JRPproblems with continuous time, an infinite horizon, and deterministic constant parameters.Quoting from Iyogun (1987),The common denominator [of this subclass] is that they use equally spacedreplenishment epochs. These policies are not necessarily optimal... Theoptimum referred to in all these papers where 'optimum' is specificallymentioned should be qualified by the word 'periodic'. It is still an openproblem what problem parameters ensure the existence of periodic optimumpolicies, except two cases. The first case is obvious and this is the case whenall items have equal parameter values...The second case...has item-dependentset-up costs but with a fixed saving when all items are replenished together.For this problem, Andres and Emmons (1975) showed that an optimum policyhas equally spaced replenishment epochs for each item. Even the problem offinding optimal periodic policies is not a trivial task. Because of this,researchers have focused mainly on developing approximate methods forfinding good periodic policies.In this chapter, it is argued that the optimal policy for the consumer stockpiling model is"almost always" (in the sense of the probability approaching unity) periodic, and that if thereis a non-periodic optimal policy, there is also a periodic optimal policy. This allows thederivation of exact numerical solutions to the problem.Chapter 2: Grocery Shopping Behaviour^ 16There are few examples of normative inventory theory applied to consumerstockpiling. Exceptions include Blattberg et. al. (1981), Jeuland and Narasimhan (1985), andMeyer and Assuncao (1992). The latter uses a set of results derived by Golabi (1985) ininventory control policy to describe optimal buying strategy. The application, again, isconsumer buying strategy in the presence of promotions and the opportunity for stockpiling,which differs from the context of this chapter. However, the main thrust of the research isto empirically validate the normative policy in an experimental setting. As Meyer andAssuncao state, "It is a central tenet of this research that purchase quantity decisions aremade through the use of heuristics which, while perhaps not optimal in structure,nevertheless serve to mimic the central properties of optimal inventory control policies." Forthe purposes of this chapter their main conclusion is that, while some systematic biases doexist, purchasing patterns were positively correlated with those predicted by the normativemodel.Multipurpose ShoppingA final stream of literature that has much in common with this chapter is themultipurpose shopping literature. With its origins in Central Place theory (Christaller,1933), this literature considers "high order" and "low order" centres, or shopping locations.The high order centre has more goods, some of which may be purchased less frequently, andare located further apart from each other than low order centres. In the last decade, severalpapers have appeared which model consumer shopping behaviour in order to show how thesecentres develop (McLafferty and Ghosh, 1986; McLafferty and Ghosh 1987; Ingene andGhosh 1990). The consumers are trading off travel costs against storage costs (andChapter 2: Grocery Shopping Behaviour^ 17occasionally price of goods) in the presence of at least two goods, that have differentconsumption rates. As a result, the two different kinds of centres develop. While thecontext and objectives of these models differ from the model in this chapter, the form, asa cost minimization of travel, inventory, and price costs for more than one good at morethan one location, is quite similar. The models also differ in the explicit form of the costs.In the papers referred to above, inventory costs are related to price. In the model here, itis argued that it is preferable to consider inventory costs as proportional to quantity, for non-perishable goods. Furthermore, the second good is differentiated by being perishable, so thatthe inventory cost for it is the loss associated with any expired goods. A second differenceis that prices for both goods differ by store in this model. In the multipurpose shoppingliterature, price does not have a critical role, and may be left out. Finally, the solutionmethod in the multipurpose shopping literature involves relaxing the constraint that thenumber of trips is a nonnegative integer, so that calculus may be applied. In this chapter,the integer constraint will be retained. In summary then, the multipurpose shoppingliterature is relevant primarily because the general structure of the consumer's costminimization problem is the same as that used here.Chapter 2: Grocery Shopping Behaviour^ 182.3 Model DevelopmentAssumptions, notation, and rationale for the model components follow.Goods: Two goods, one perishable and one non-perishable, are available toconsumers. Subscripts *p and *„ denote quantities attached to the perishable and non-perishable goods respectively.Stores: Each of the two goods is available at two stores, which, in keeping with thepower retailer theme, are identified as the "large" and the "small" store, with subscripts *,and *$. The two stores are differentiated by price, with the small store the most expensiveon both goods, and location. Thus, there are four prices: Pn,1, Pn,59 Pp,19 and Pp,s•Cost Minimization: Consumers minimize long-run average costs over an infinitehorizon. Total costs consist of trip costs C„ plus storage costs C s , plus the price of thegoods Cp .Trip Costs: Trip costs consist of a fixed amount c s or c1 incurred for each trip to thesmall or large store. Later, trip costs will be assumed proportional to the Euclidean distancebetween customer and store, but in the early sections of this chapter, they will remaingeneral. It should be noted that trip costs correspond to order costs or set-up costs in theinventory management literature, but that, unlike the inventory management literature, thecosts are not tied to the goods. In particular, once the customer is in the store, there is nofurther cost whether she purchases one or both of the goods.Non-perishable good inventory costs:  Consumers have quantity-dependentinstantaneous storage costs s•Q,(t) for the non-perishable good, where s is the cost per unitquantity per unit time, and Q,(t) is the quantity of the non-perishable good on hand at timeChapter 2: Grocery Shopping Behaviour^ 19t. This serves to limit the quantity purchased, which would be unlimited if only trip costswere considered. Note that since price appears explicitly elsewhere in the model, thisformulation makes storage costs NOT dependent on price. This is in contrast to theinventory management literature, where it is implicitly assumed that the holding costs includethe cost of capital; and in contrast to the multipurpose shopping models (eg., Ghosh andMcLafferty, 1987), where the holding cost is proportional to the monetary value of thestock. Quantity-dependent holding costs are used here for three reasons. First, given thatthe frequency of grocery shopping is on the order of weekly, it seems unlikely that the costof capital would play a significant role in limiting the amount of consumers' purchasescompared to storage and transportation capacity limitations (how big is the car and thecupboard?), and time limitations for a single trip (how many hours can the customer standbeing in a grocery store?). Second, an objective of this research is to investigate the effectsof price differences between stores, and that objective is facilitated by keeping price as aseparate component. The third reason is empirical. Litvack, Calantone and Warshaw (1985)and Blattberg, Eppen and Lieberman (1981) both conclude that "bulkiness" lowers theincentive to stockpile in the presence of promotions. More recently, Raju (1992)investigated how the variability in category sales, where the variability is largely due to pricediscounting, depends on category characteristics. While the relationship between categoryexpensiveness and variability was not statistically significant, bulky categories hadsignificantly lower variability. To the extent that sales variability reflects a stockpilingresponse to discounts, this supports the modelling of storage costs as quantity-dependent andprice-independent. Note that if there were more than one nonperishable good, and the goodsChapter 2: Grocery Shopping Behaviour^ 20differed in bulkiness, this argument would require that a scale factor on the Q variable beapplied in the storage cost formulation to account for the difference. However, here thereis only one nonperishable, so the relation between bulkiness and quantity may be ignored.Storage costs over time are given byJ oTsQ(Odt^(1)An amount Q. purchased and immediately consumed at the constant rate D. will havea storage cost (Figure 1): to + QnIDos(Q n - D nt)dt =toQn^QnAt Dn(At)2 (2)2D n^2^2Figure 1: Stockpiling ParametersPerishable Good Inventory Cost:  The perishable good has a lifetime Ate afterpurchase, and incurs an inventory cost equal to the price paid for any quantities that expirebefore they are consumed. If the expired quantities, which must be disposed of, are givenChapter 2: Grocery Shopping Behaviour^ 21by Qp,expired, the cost is PpQp ,expn,i . Inventory management models consider both suddenexpiry and gradual deterioration. This model only considers the former, which, for grocerieshas face validity, in that many items are marked with an expiry date. It also seemsreasonable that even for goods like vegetables that do not have marked expiry dates, theconsumer has to decide whether to consume or dispose of the product. It is assumed thatthe quantity-dependent storage costs associated with the perishable good are negligiblecompared to the other costs in the model, but nonzero. The rationale for this negligibilityis that the consumer always has enough room to transport and store (eg., in the refrigerator)more perishables than she can consume before they expire. In other words, the expiry time,rather than the quantity-dependent storage costs, impose the limitation on the amount ofperishables purchased on a single trip, which simplifies the analysis. The nonzeroassumption is also reasonable, since the perishable goods do take up some room, and allowsthe derivation of analytic results.Stockouts: No stockouts of either good are allowed.Discounting: The time value of money is assumed neglible compared to other costs,and not considered. This is consistent with the promotions literature, the multipurposeshopping literature, and much of the JRP literature. Although the model is infinite horizon,grocery shopping trips occur frequently enough and expenditures are large enough that it isdifficult to imagine the time value of money entering into the consumer's calculations.Demand: Demands are determined by consumption rates, denoted D n and Dp , and areconstant. Ingene and Ghosh (1990) discuss this assumption in the spatial shopping context:A number of authors have worked with inelastic demand models (see, forexample, Bacon 1984; Ghosh and McLafferty 1984; Hotelling 1929;Chapter 2: Grocery Shopping Behaviour^ 22McLafferty and Ghosh 1986; and Thill 1985) One interpretation of the fixedquantities model is that it is representative of a rectangular demand curve,provided that delivered price does not exceed the maximal demand price(Ingene, 1975). The alternative approach of utility maximization has beenexplored in the multipurpose shopping context by Mulligan (1987), amongothers. However, its complexities have caused Thill and Thomas, citing thework of Bacon (1984), to write that "it seems that the neoclassical approach--has few prospects, unless the utility-maximizing behaviour is relaxed andreplaced by a cost-minimizing one" (1987, p. 8). We recognize, of course,the importance of extending the analysis in the future to consider utility-maximizing behaviour.Table 1: NotationSubscripts:n: non-perishablep: perishables: small (expensive) store1: large (cheap) storei: visit index to large storej: visit index to small storeQ, 1 , Q,,s , Qp , 1 , Qp , s : Quantities of each good purchased at each store(time subscript omitted)P„,„ Pio , Po , Pp,, : Prices of each good at each storets j^: time of ith (r) visit to each storej^: time intervals between purchasesD„, Dp^: consumption rate of each goodAte^: time between purchase and expiry of perishable goods : instantaneous storage cost of nonperishable goodet , cs^: trip cost to each storeC t^: long-run average trip costsCs^: long-run average storage costsCp^: long-run average price costsC : total long-run average costsDecision Variables: The consumer's problem is to decide when to shop at whichstore, and how much to buy. This is formalized by identifying shopping patterns, orsequences of visits to the stores (for example the sequence "large, small, small, small,Chapter 2: Grocery Shopping Behaviour^ 23large"), and optimizing over purchase quantities Q„, 1 , Q„,„ Qp , 1 , Qp,„ and associated purchaseoccasions tti and t, J . Through a series of lemmas and propositions, the huge number ofpossible shopping patterns are reduced to a manageably small set of possibly optimalpatterns, for which analytic expressions for the cost function are derived. These costfunctions can then be minimized over Q(. ,.) , and the smallest selected as the optimal pattern.Domain: All parameters are positive real. Quantities are non-negative real.Notation is summarized in Table I.Chapter 2: Grocery Shopping Behaviour^ 242.4 Results for Some Simple CasesBefore addressing the full problem of two goods at two stores, results for some restrictedcases are presented. This helps to clarify the issues involved, and some of the results willbe used later.2.4.1 Non-perishable goods and one store onlyFor a single good available at a single store, we may ignore price, and the well-knownEconomic Order Quantity (EOQ) result from inventory theory applies (e.g., Clark andHowe, 1962). Equal quantities are ordered at equal time intervals, with the inventory justat zero at each repurchase occasion ("zero inventory rule"). Omitting the price the customerpays for the goods ("price costs") for now, because they only have an impact when shoppingfrom two sources with different prices, the optimal long-run average quantities, times, andcosts are:Q*2cDnSA ti*^2csDnC * = \i2scDn(3)Now suppose there are two non-perishable goods, differing only on storage costs, s 1and s2 , and purchased at a single store. Note that even if the goods have different prices,Chapter 2: Grocery Shopping Behaviour^ 25price will not affect the solution, because goods are available at only one store and theconsumer must always have some on hand (and we are not considering the time value ofmoney). It is necessary to consider the possibilities that only one or the other good ispurchased on a trip, or that both goods are purchased on a trip. Consider first a fixedhorizon, during which time f1 purchases of good 1, and f2 purchases of good 2 are made.Let the number of trips to purchase only good 1 be m 1 ; the number of trips to purchase onlygood 2 be m2 ; and the number of trips to purchase both goods be m 12 . We note that m 1 +M12 = f1, m2 + m12 = f2, m1 + m2 + m12 = f1^f2, and that the zero inventory rule stillholds for each good (if any quantity of the good were on hand at repurchase time, storagecosts could always be reduced by purchasing less the time before), so that the integratedstorage costs may be expressed as the summation of terms like (2). The consumer's problemis thenMIN CTMD m2, m12,11,12, QD Q2whereCT = c(n1 ±m2+m12)fl-1 Q .^ f2-1^Q .E^A + E^A t.=o^2^i2=o^2^12(4)and, as in the usual single good case, the no-stockout and zero-inventory rules implyconstraints on the total quantity purchased over the horizon T:Chapter 2: Grocery Shopping Behaviour^ 26fi -1rEQi l =i t =012 _1E Qi2 = Q2i2= D i T= D2 T(5)With only one store, and two non-perishable goods with different storage costs, thefollowing proposition states that the customer purchases both goods each time he shops.Proposition 1: For s 1 , s 2 > 0, (4) above is minimized only when m 1 = m2 = 0.Proof: Suppose not. Suppose that for an optimal CT* , m1 '^0. Consider CT ',where m 1 ' =^- 1, and m12' = m12' + 1, that is we replace one of the "good-one-only"trips with a "both-goods" trip. The first and second terms of equation (4), namely the tripcosts and the summed storage costs for good 1, are unaffected. The third term, the summedstorage costs for the second good, has one of its summands (say, the j th) replaced with twosummands. It suffices to examine this summand, as all others are unaffected.For the second good (dropping the goods subscript), we have that the j th time interval,and the j th quantity purchased are now divided: At; = et; ' + et'i+1 , and IQ; = + Q'i+1 .Let the affected term be IS* = sQ*Ati . , and the new terms K'; and K'i+i be similarly defined.Then= -S (Q./ + Q./+ 1)(Atj + At'2^ i+1)(6)= K. ++ -S (Q.1At.1 + Q.I At.1)j+1^j^j +1^j+1 j2Chapter 2: Grocery Shopping Behaviour^ 27implying K* > Kj ' +^and hence CT* > CT ' . Thus CT* cannot be optimal, and m 1(and similarly m 2) must be zero D.Proposition 1 gives f1 = f2^f, and identical interpurchase time intervals for bothgoods, Atii * = Ati2 * (although not yet necessarily equal for all i).^Since each pair ofinterpurchase intervals are the same, we also have Qi1 = ( 31/D2)Qi2, so we may write thecost function asD QiCT = cf + E (s,i.0^D1^2(7)which is identical to the single good cost function, with the storage cost replaced by aweighted sum of the different storage costs. Hence the usual EOQ solutions given in Eqn.(3) apply, with s replaced by the weighted sum of s 1 and s2 , and therefore interpurchasetimes are equal for all i. Times, quantities, and long-run average costs for the case oftwo non-perishable goods available at one store are thus:Chapter 2: Grocery Shopping Behaviour^ 28O ti = 2c s 1D1 + s2D22cs1D1 + s2D2 (8)= D ^2c 2\!s1D1 + s2D2C* = V2c(siD i + s2D2)2.4.2 Perishable goods at one store onlyThe optimal shopping policy for one perishable good at one store only is obviously to shopat intervals equal to the expiry time of the good, and purchase just enough to last to expiry.This simple case will be used as an opportunity to demonstrate the roles of the various modelassumptions by deriving the result formally.The no-stockout assumption has two slightly different expressions depending on therelation between the amount purchased on a trip, the consumption rate, and the expiry time:Dp > At^Ati < Ate^(9)Chapter 2: Grocery Shopping Behaviour^ 29Q„,.< Ate^Ati <DpEquations (9) and (10) simply state that if more perishable goods are purchased than can beconsumed before expiry, the repurchase interval must not exceed the expiry time; and if lessare purchased than can be consumed before expiry, the repurchase interval must not exceedthe purchased quantity divided by the consumption rate.To minimize inventory costs, which for perishable goods are assumed to be the priceof any expired unconsumed goods, the consumer must avoid purchasing more than she canconsume in the expiry time:Qp,,^.Dps ateHence, only (10) above is allowed. Finally, minimizing the long-run average travel costsimplies that, since each trip cost is fixed, the interpurchase interval At ; must be maximized.In the right hand side of (10), then, equality must obtain. Maximizing the interpurchaseinterval implies maximizing the quantity purchased, so that equality also obtains in (11).The optimal policy is (ignoring price costs as in the non-perishable case):(10)DpChapter 2: Grocery Shopping Behaviour^ 30At: = Ate= DpAt: = DpAteCT — CAte(12)Minimum long run average costs are just the averaged trip costs, which occur at intervalsequal to the lifetime of the good.2.4.3 Perishable and non-perishable goodsA well-established result in the perishable inventory management literature is that if a goodhas both holding costs and perishability costs, the optimal order quantity is (Nahmias, 1982):2cD , DAte )SQ = min( (13)In other words, the optimum is the amount considering only the storage costs, unless thatis more than can be consumed before expiry, in which case the consumer purchases onlyenough to last to expiry.The case here is similar, in that two goods, each of which has only one type of cost,are purchased at a single store. The results parallel Equation (13). To clarify the modelstructure and assumptions, first we will consider the total costs that would be incurred fortwo goods if each had both quantity-dependent storage costs and perishability costs, for theChapter 2: Grocery Shopping Behaviour^ 31finite horizon. The formulation will then be explicitly restricted so that there is one goodof each type, and, using previous results, solved for the number of trips of each type overthe finite horizon. Finally, the optimal timing of trips for the infinite horizon will be found.The general formulation for trip and inventory costs over a finite horizon T, for twoperishable goods, (identified by subscripts 1 and 2) with storage costs and expiry times isQ^f2-1 Q^CT = c(m 1 + m2 + m12) E s1-1Ati^E s2_-211/.. (14)i1 =0^2^1^12 =0^2^12such that the sum of the quantities purchased over the horizon T are fixed by the demandrate,'1 _1EQi,i1=0(15)f2 _1E Qi = D271i2=0and repurchase must occur at or before expiry:^Chapter 2: Grocery Shopping Behaviour^ 32^AAt.^t1 1(16)At. s Ate2where ml, m2, m12, f1 , and f2 are defined as in the previous section. Note the expression ofthe integrated storage costs in (14) as summations result, again, from the "zero inventoryrule": it is never optimal to be holding inventory when a shopping trip is made.The above formulation is now specialized as follows. Assume good 1 has negligiblestorage costs (s 1 0), so that the summation in (14) for good 1 is much less than theremaining terms. Good 1 is the perishable good, and as argued before, its quantity-dependent storage costs are ignored because of the assumption that perishability limits thequantity purchased much more severely than any other consideration; for example, it is easyto buy and store more tomatoes than can be eaten before they spoil. Now assume good 2is not perishable (Ate2 00; spaghetti will keep in the cupboard for a long, long time), sothe related constraint in (16) may be ignored.Since s 1 and s2 are both strictly positive, Proposition 1 applies: it is always lessexpensive to buy both goods on each trip: m 1 = m2 = 0,^f2^f, and Atii^etaet, = Q, 1 /D i = Q,2/D2 . This means the timing problem is isomorphic with the problem ofpurchasing one good that has both costs. If the EOQ solution for the non-perishable goodby itself gives an interpurchase time less than the expiry time of the perishable good, thenthat interpurchase time controls the purchase of both goods. The perishability constraint (16)is not binding. On the other hand, if the EOQ interpurchase time exceeds the expiry time,Chapter 2: Grocery Shopping Behaviour^ 33then the perishability constraint is binding, and the expiry time controls the purchase of bothgoods.Formally, the cost function with the above restrictions is now^f-1^Qi^CT = E^sn—,1Ati)i=0 2such thatf-1E Qi = DpTi =0f-1E = DnTi =0 nO ti^towhere the goods subscripts have been relabled n and p. Note that the perishable good doesnot enter the cost function directly; it only operates through constraint (20). Thus, if (20)is not binding, the problem is the same as the single good case with storage costs only.Perishability Non-Binding Case: Assuming the perishability constraint is not binding, and following the standardprocedure for determining the EOQ in inventory theory, we note that the purchase quantitiesare equal, and equally spaced, for a convex cost function (Clark and Howe, 1962).Alternatively, explictly minimizing (17) over Qin subject to (18) gives the finite horizon(17)(18)(19)(20)Chapter 2: Grocery Shopping Behaviour^ 34solutions. With the occasion subscript, "i", dropped on the purchase quantities because theyare all identical, but retained on the interpurchase time At ; to distinguish it from the expirytime Ate , the optimal quantities and times, and minimum cost, are easily shown to beQn = DnA ti*A t:C; = fcTsDnTz2f(21)To extend to the infinite horizon case, consider the long-run average cost CT*^SnDnAt^2A :CT* = (22)and minimize over the interpurchase interval At:. This gives the usual EOQ solutions(where we now take the " * " to mean optimal over quantity and time, over the long run) asin equation (3):Qn* = DnA\I  2c snDn^ (23)T*C = 1 2sDncChapter 2: Grocery Shopping Behaviour^ 35and invoking Proposition 1,Q; = DpAti*^(24)Once again, we note that since there is only one store, it is not necessary to consider price,and hence the perishable good has no costs uniquely associated with it.Perishability Binding Case: Assume now that the perishability constraint (20) is binding. Then (dropping thesubscript "p" on the perishability constraint),Qn = DnAt,Q; = DpAte^ (25)C^snDn A I-,A to^2From (20) and (23), the condition for the perishability constraint to be binding is2c sipn Ate(26) Finally, to establish that this is the complete set of solutions, we note that the Kuhn-Tucker first-order conditions are sufficient for the minimization problem, because the costfunction is convex and the inequality constraint, (16), linear. We may now write, in analogywith (13),Chapter 2: Grocery Shopping Behaviour^ 36Qn* = min( 2cDn, DnAte )Sn= min( D P 2cN1 snDn , DpAte )^(27)Ati* = min( 2c , At ) .snDn^eShopping Pattern PossibilitiesChapter 2: Grocery Shopping Behaviour^ 372.5 Two Goods at Two Stores: Feasible Normative BehaviourWith the introduction of two stores, each with its own prices, the problem is complicated bythe huge variety of qualitatively different shopping behaviours. Given a sequence of storevisits (e.g., small, large, small, small...) and an associated sequence specifying which goodsare purchased, one could in principle minimize the total cost over the quantities purchased,and purchase timing. However, with an infinite horizon, there are an infinity of thesesequences of shopping patterns. The same problem is encountered in the JRP literature,where the analysis is simplified by restricting the problem to periodic policies, which allowsthe analysis of only one period. That a periodic policy is in fact generally optimal has onlybeen demonstrated for a few situations (Andres and Emmons, 1975; for a review, seeIyogun, 1987), all of which are simpler than that considered here.Figure 2: A complete and mutually exclusive classification of shopping behaviour.First, consider the three-way classification of shopping only at the small store, onlyat the large store, and at both (Figure 2). The minimal costs (over quantities) for each ofthese cases must be compared to see which one obtains. For the single store cases, theresults have been found in the previous sections. For the mixed case, there remains anChapter 2: Grocery Shopping Behaviour^ 38infinity of possibilities. In this section, a series of results restricting the optimal patterns ofshopping,in the case where both stores are visited, will be proven.In the previous sections the "zero inventory rule" and the associated requirement thatboth goods be purchased on each trip greatly facilitated the solution by defining the onlypossible shopping pattern. When there are two stores, it may well be optimal to visit onestore, i.e., the more expensive (small) store, while still holding goods purchased at the cheap(large) store. The zero inventory rule thus requires careful consideration.The next result states that the "zero inventory rule" still applies to the large store.Lemma 1: Any trip to the large store that is part of an optimal policy will only occur whenboth goods have just been depleted.Proof: First note that it is never optimal to make a trip without purchasing at least one ofthe goods, say good a. Let the amount purchased at the large store on trip i be qa , i . Theno-stockout assumption implies that inventory on hand any time before the trip is made mustbe non-zero. This means that the inventory on hand immediately after the last trip wheregood a was purchased must be:Da(ti — ti -1)^ (28)Suppose the above inequality is strict, and the amount on hand at time t i is84(2a; =^— Da(ti — ti-1) > 0^(29)The total amount on hand immediately after the trip at time t i is then Qa; +Chapter 2: Grocery Shopping Behaviour^ 39The total costs consist of inventory costs, trips costs, and price. If Q ai_ i t (and henceOQa,i) is reduced, and Qa; increased by the same amount, and no other changes made to thepolicy, then1. Total storage costs are decreased.2. Trip costs remain the same.3. Price costs either decrease (if some of the inventory SQ.; was purchasedat the small expensive store) or remains the same (if all the inventory OQ a , i*was purchased at the large store).4. All goods are stored for the same, or shorter, time, so that perishabilitycosts do not increase.Therefore total costs decrease, and the policy cannot be optimal. Equation (28) must be anequality, and hence from (29), SQ.; = 0.Now consider the second good, say good b. If it is optimal to purchase any amountof good b, the same argument as for the first good applies. If not then some amount SQ, ;*must be on hand from the previous purchase. But if an amount 6() b , ;* is purchased at thelarge store at time i, and the purchase of good b at time is decreased by the same amount,then:1. Since the customer is already at the store to purchase one good, no extratrip costs are incurred by purchasing the second good.2. Since the store is the cheapest, no extra price costs are incurred.3. Storage costs are decreased, as above.4. Perishability costs are not increased, as above.Chapter 2: Grocery Shopping Behaviour^ 40Therefore it cannot be optimal to have any of the second good on hand at time t i ^ .Lemma 1 establishes the "zero-inventory rule" for visits to the large store. It alsoshows explicitly why such a lemma doesn't hold for the small store, in that condition (3)regarding the price costs doesn't hold. Note also that lemma 1, like Proposition 1, relies onthe perishable good having some nonzero storage cost. Lemma 1 also leads immediately tothe following result.Corollary 1-1: Both goods are purchased on each trip to the large store in any optimalpolicy.Proof: Suppose not. From Lemma 1, both goods must be just depleted when the trip attime 4 occurs. The no-stockout rule therefore requires that both goods must be purchasedsomewhere at time 4. If either good (or both) is not purchased on the trip to the large storeat ti , another trip must immediately be made, incurring extra trip costs, and possibly (if thetrip is to the small store) extra price costs, with no savings. Therefore the policy cannot beoptimal ^ .Lemma 2: If the optimal policy has three successive occasions where both goods are at zeroinventory, then the time-average cost between the first and second occasion is equal to thetime average cost between the second and third.Proof: Suppose not. Then replacing the higher cost interval with a copy of the lower costinterval will produce an overall lower long-run cost policy, and the original policy cannotbe optimal ^ .The equality of costs required by lemma 2 can always be achieved, for any arbitraryset of parameters, by having identical shopping patterns, or policies, in the two intervals.Chapter 2: Grocery Shopping Behaviour^ 41It is possible for some special sets of parameters to allow different policies with the sameminimal costs in the two intervals. Such parameters could be found by first finding the costfunctions C 1 '(0) and C2*(0) defined over the n-dimensional parameter space, where thesubscripts correspond to two different shopping patterns, and the optimization is over thequantities purchased. Imposing the constraint C 1* = C2* defines a lower-dimensionalsubspace of the parameter space where the two cost functions are equal. (This, in fact, is theprocedure that will be used later on to find the transition, or crossover points, in trip costparameter space, between different optimal policies). Except for such special cases, lemma2 requires that the detailed policies in the two intervals be identical. Since an arbitrarilyselected set of parameters will lie in a lower-dimensional subspace of parameter space withvanishingly small probability, we will say that the detailed policies in the two intervals are"almost always" identical. On the basis of this heuristic argument, we state, without proof:Corollary 2-1: If the optimal policy has three successive occasions where both goods are atzero inventory, then the policy between the first and second occasion is (almost always)identical to the policy between the second and third.Proposition 2: The optimal policy for the two good, two store scenario is periodic (almostalways), with period defined by the interval between trips to the large store.Proof: Follows immediately from lemma 1, corollary 1-1, and successive application ofcorollary 2-1.Proposition 2 is a major restriction on the variety of shopping patterns that need tobe examined. It restricts the possibilities not only to periodic policies, but to a specificsubset of periodic policies, and allows application of finite horizon results (although not fixedChapter 2: Grocery Shopping Behaviour^ 42horizon, so some care must still be taken). From here on, it is only necessary to considerthe interval between two trips to the large store, with the inventory level of both goods atzero at the beginning and end of the interval, and both goods purchased at the large store atthe beginning of the interval.Note also that, while a nonperiodic policy may (rarely) be optimal, it occurs whenthe long-run average costs of two or more different shopping patterns are identical, so thatthe long run pattern can be any sequence of the individual cycles. Replacing each cycle withjust one of the possible cycles will give exactly the same costs, and be periodic. Therefore,even if there is a nonperiodic optimal policy, there will also be a periodic optimal policy.Since we are examining the mixed shopping case, we also know there is at least onetrip to the small store. Note, however, that nothing has been said about the the purchasepatterns at the small store. One cycle, with notation, for m visits to the small store, isdepicted below:large^small^small^...^small^largeI I I I I<^ Ato ^ > < -- ets3 --- > < ---^-- > < ---- Ats ,o, ---- >to^to^ts,2 t8,„,^tm+1^  At/ ^One of the two goods purchased at the large store, either the perishable or the non-perishable, must last the longest. The following results are designated with an "n" or a "p"depending on which good lasts longest.Non-perishable <^  A t 1 >^Quantities^0 n Ipurchased atlarge storetrips tosmall storet o^1^Q n,1 i1^D n. .t s,i-1^t 8, 1Chapter 2: Grocery Shopping Behaviour^ 43Case n: Refer to Figure 3 for the following lemma and corollary.Figure 3: Timing of first purchase of non-perishable at small store. Diagram refers onlyto lemma 3n. Subsequent results will further restrict this general picture.Lemma 3n: Consider the optimal policies where the non-perishable good purchased at thelarge store lasts at least as long as the perishable good, i.e.,12„,1^ CpiD'D,^P(30)The first purchase of the non-perishable at the small store is on the trip immediately before,or at, the depletion of the non-perishable purchased at the large store, i.e., the trip definedbyChapter 2: Grocery Shopping Behaviour^ 44{ i I max ts:iProof: The no-stockout assumption and lemma 1 (zero inventory at t o and tni, i) give the totalamount of the non-perishable purchased at the small store asQn^ *=^— Qnj (32)The no stockout constraint implies that t.; is the latest the first purchase can occur.Purchasing earlier increases storage costs without affecting any other costs, so that anyearlier purchase than to* cannot be optimal ^ .Corollary 3n-1: Any trips to the small store before occasion to* defined in (31) must be forthe perishable good only.Proof: Follows immediately from lemma 3n and the fact that something must be purchasedon each trip ^ .The same results, with parallel proofs, apply to the case where the depletion time forthe perishable good purchased at the large store is longest. Since the results are essentiallyidentical, they will not be restated, but will be referred to as Lemma 3p and Corollary 3p-1.Lemma 4n: Again consider the case given by (30). The only time the non-perishable willbe purchased at the small store, if at all, is the last trip in the period,Proof: Suppose not. Then between time Q,,, s/D. and ts ,,,. +1* (interval a in Figure 4) thecustomer must optimize his purchases with respect to only one store (he has no inventory onhand from the large store, and doesn't visit the large store). Thus he will buy bothQ,:j^ (31)DnChapter 2: Grocery Shopping Behaviour^ 45perishables and non-perishables simultaneously, according to the single store policy; and attime to+1 *, his inventory is zero. But by corollary 2-1, this can only be optimal in the knife-edge case that the average costs over the intervals before and after t o+i* are equal.Generally, either one or the other interval will have lower costs, and the policy cannot beoptimal.If trips to both stores are optimal, then the first interval must have lower costs andbe the repeated interval. Hence t s , 1, 1  = ts,..„ 1*, or i = m D .Figure 4: Visits to the small store to purchase both goods more than once per periodcannot be optimal.Again, the same results apply to the case where the perishable good from the largestore lasts longer, and will be referred to as Lemma 4p, but not explicitly stated. It is, ofcourse, possible, and more economical, to combine the two cases for arbitrary goods forLemmas 3 and 4, as the argument is the same to this point. Henceforth, however, thesituation at the end of the interval is slightly different between the two cases. Also, later on,Shopping Pattern PossibilitiesChapter 2: Grocery Shopping Behaviour^ 46it will be necessary to distinguish the two cases. Therefore, economy has been sacrificedin the hope that clarity is gained by distinguishing the cases early on.Note that lemmas 3n and 3p specify the earliest (the trip just before both goodspurchased at the large store have run out) that a trip to the small store to purchase bothgoods can occur, and 4n and 4p specify the latest (when the inventory of both goodspurchased at large store has just run out) such a trip can occur. Together this may berestated as:Proposition 3: If both goods are purchased at the small store in an optimal policy thatincludes shopping at both stores, then the second good will only be purchased once perperiod, and on the last trip to the small store in the period.Proof: Follows immediately from lemmas 3n, 3p, 4n, and 4p D.Figure 5: Shopping Behaviour PossibilitiesChapter 2: Grocery Shopping Behaviour^ 47Figure 5 indicates the possibilities so far. Next, we will examine the non-perishablecase ( referred to as the "4n case" in Figure 5, or simply the "n" case later on) in moredetail, and restrict the possible policies further.Proposition 4n: The amount of perishable good purchased at the large store, in the "n"mixed trip shopping case, is just enough for the lifetime 4 of the good, which must be lessthan the consumption time Q,,, s/Dn of the non-perishable good purchased at the large store:'2;1 — Ate <Dp^D.(33)Proof: Consider the three possibilities relating the lifetime of the perishable good to theconsumption time of the non-perishable purchased at the large store.(a) Lifetime less than consumption time: Ate < %,'/D.. (See Figure 6).Any value of Qp3 less than DpAte would increase the price cost by requiring the purchase ofmore at the small store, and have either no effect on, or an increase in, trip costs.(b) Lifetime equal to consumption time: et, =As in (a), Qp; = Dp Ate. But that implies a zero inventory point for both goods at the timeAte . By Corollary 2-1, that implies that, almost always, shopping would occur only at thelarge store, or only at the small store, which violates the assumption of mixed trip shopping.(c) Lifetime greater than consumption time: At e > Q,,'/Dn .By Corollary 2-1, as in (b), and the assumption of restriction to the "n" case, theconsumption time for the perishable must be less than the nonperishable. But costs cannonperishablelarge Q *pperishable Q *0—1521p .° 16' tQ *n,1smallD ntime I IN E 513 B le 1 Pi Mk LS -Chapter 2: Grocery Shopping Behaviour^ 48Figure 6: The amount of perishable purchased at the large store in the "n" case (diagramshows case (a) of Proposition 4n).always be reduced by increasing the quantity of the perishable purchased, so this cannot beoptimal ^ .Proposition 5n: The amount of perishable goods purchased on each trip to the small store,and hence the interpurchase intervals, is identical, except possibly for the last trip andinterval.Proof: For any number of trips to the small store, only one good is being purchased on allbut possibly the last trip. Since the perishable good has a small but nonzero storage costassociated with it, the standard result from perishable inventory theory applies (eg.,Nahmias, 1982): equally spaced trips, with equal purchase quantities on each trip,minimizes costs E.Corollary 5n: If the non-perishable is never purchased at the small store, proposition 5napplies to all trips in the interval.Chapter 2: Grocery Shopping Behaviour^ 49Proof: As for Proposition 5n ^ .Lemma 5n: If some non-perishable is purchased at the small store, then the depletion of thenonperishable from the large store cannot coincide with depletion of the perishable purchasedat the small store.Proof: When there is zero inventory of both goods, Corollary 2-1 applies, and since mixed-store shopping is optimal, the cycle would be repeated without ever purchasing non-perishable goods at the small store O.At the end of the period, the following proposition shows that there are only twopossibilities to consider:Lenuna 6n: If there is a purchase of the non-perishable good at the small store, at the endof the period, either it is purchased by itself when the nonperishable purchased at the largestore runs out, i.e., the last trip to the small store in the period, at time t a , a, or it ispurchased with the perishable good on the last trip to the small store before the non-perishable runs out.Proof: First note that lemma 5n eliminates the possibility of the two cases coinciding.The purchase of the extra non-perishable must occur between the time t s ,m* of the lasttrip to the small store and the depletion of the existing stock (in the interval Ata in Figure 7).Purchase at time ts ,m* incurs extra storage costs A through the interval At a . If it is optimalto delay the purchase, and save some of the storage costs A at the expense of an extra trip,then the total costs can be minimized by delaying the trip until the goods on hand from thelarge store are depleted, thereby reducing A to zero O.Chapter 2: Grocery Shopping Behaviour^ 50Figure 7: End-of-period effects in "n" case.Next, these two possibilities are examined in detail, and it is found that one can beeliminated.Lemma 7n: An optimal policy in the "n" case will almost never include the purchase of anyof the non-perishable good at the small store at the time of perishable good purchase.Proof: Proceeding by contradiction, assume some Q„ , s* ( 0 :5. Q.;^Dn(Ata + tb) )ispurchased at time t, ^Let the total amount of non-perishable purchased in the period beQ.* = Q.; + Q. For any given shopping pattern, then, the only costs which vary withQ„,s are the storage costs and price costs associated with different allocations of the non-perishable purchases between the large and small store (Q* held fixed). Storage costs forthe non-perishable areChapter 2: Grocery Shopping Behaviour^ 51CssQajAtai sQa4Ata sQa4Atb(34)2At1^At1^2At1 ^Let T be the time from the beginning of the period to^the last trip to the small store, tobuy perishables. Noting thatQnj = Qn Qn, ; Attu = (Qn Qn,y)IDn :Ata = (Qa — Qa,^)IDa — T ; Atb = (2a1D aequation (34) becomesCs = A te^n. 2D.^'s^D.S r  (Qn Qn,^)2^Q Qn Qn„,^Q„,,( ^T)s (2,, - 2Q nQ„,, Qn,s2^Qn,sQn — (2:4Q 7' + Qn's IA ti^2Dn^ n,s 2DnQ n2=^— Qn,sTA ti 2Dnwhich implies that as Q,. increases (and Q, 1 decreases) storage costs decrease in proportion.Now consider price costs of the non-perishable.cp^Qn 11)^(2n,sPn,sAt /^A t1^(37)= —Qn + Q„,At/^A ti^n75 Pn)(35)(36)Chapter 2: Grocery Shopping Behaviour^ 52Price costs also vary linearly with Q., s . Hence, we have the following boundary solutionsfor^if price costs dominate, i.e.,^P ^P n,1^sT^ (38)At,^Atthen Q„, s. = 0. If storage costs dominate, then^(2,45, = D.(Ata + Atb) =^ (39)DpIn the case (which almost never occurs) that (38) is an equality, any value of Q a,, between0 and the value given by (39) will be optimal.But in the case that storage costs dominate, (39) implies that the non-perishablepurchased at the large store runs out at time ts,.. Hence both goods are at zero inventoryand by corollary 2-1, that is almost never optimal. Therefore Q, s = 0 almost always D.Combining the above lemmas into a summary proposition:Proposition 6n: For an optimal policy in the "n" case, if any non-perishable is purchasedat the small store, it is purchased only once, and by itself, on the last trip to the small storein the period, at the the time when the non-perishable purchased at the large store is justdepleted (almost always).Proof: Follows immediately from lemmas 5n, 6n, and 7n 0.The "n" case, then, has only two shopping pattern possibilities, that vary by whetheror not any non-perishable is ever purchased at the small store. Even if some is purchased,Chapter 2: Grocery Shopping Behaviour^ 53the shopping pattern differs from the never-purchase case only by an extra "fill in" trip tothe small store to top up only nonperishables, at the end of the period.P-case: Now consider the "p" case: shopping still occurs at both stores, but the depletion timeof the perishable purchased at the large store is longer than the time for the non-perishablegood puchased at the large store. This case has questionable face validity--the implicationis that a trip is made to the the large, inexpensive store for tomatoes and spaghetti, but thespaghetti runs out before the tomatoes, and is replenished, by itself, at the small store.Given that the tomatoes expire (rather quickly), spaghetti trips must be quite frequent. Theparameters necessary for such behaviour to be optimal (e.g., a large price cost differentialon tomatoes, a small differential on spaghetti, large storage costs, and/or small trip costs tothe small store) are possible within the model structure, but somewhat pathological. Whileacknowledging that this is slightly bizarre behaviour, the "p" case will still be examined herefor completeness. Refer to Figure 8.Lemma 5p: If any perishable good is purchased at the small store, then the depletion of theperishable good purchased at the large store cannot coincide with depletion of the non-perishable purchased at the small store.Proof: As for Lemma 5n U.Lemma 6p: If any perishable good is purchased at the small store, it must be purchasedsimulataneously with the last trip to purchase non-perishables.Chapter 2: Grocery Shopping Behaviour^ 54Qlarge^,perishablenon-perishablet „ , 1small^t pjQ p,.t s,mtFigure 8: Notation for the "p" case.Proof: By Proposition 3, the good must be purchased on the last trip in the period to thesmall store, so it cannot be purchased any earlier than the last trip to purchase non-perishables. Delaying the purchase would incur an extra trip, with no savings (storage costsfor the perishable good are negligible compared to other costs), so the last non-perishabletrip is also the last perishable trip O.Lemma 7p: If any perishable is purchased at the small store then just enough perishablemust be purchased at the large store to last to expiry, and the period must be longer than theexpiry time:_= At *1 = Ate < At,Dn(40)Chapter 2: Grocery Shopping Behaviour^ 55Proof: First recall that the stock of perishable good must be depleted at the beginning andend of the period (Proposition 2). Increasing the amount Q p; decreases price costs withoutincreasing any other costs, up to the limit imposed by either the expiry time or the end ofthe period. If the end of the period imposes the limit by being less than or equal to theexpiry time, then no perishable good will be purchased at the small store. Therefore theexpiry time must be the limit ^ .Proposition 7p: If any perishable good is purchased at the small store in the "p" case, thenit is purchased simulataneously with the non-perishable good on the last trip to the smallstore in the period, before the perishable good at the large store expires, which in turn isbefore the end of the period.Proof: Follows immediately from Lemmas 5p, 6p, and '7p 0.Proposition 8p: In the "p" case, the non-perishable purchased is purchased at the smallstore in equal quantities, at equal intervals.Proof: Any other distribution of purchases increases storage costs, as in standard EOQresults O.Chapter 2: Grocery Shopping Behaviour 56Figure 9: The complete set of shopping possibilities.Chapter 2: Grocery Shopping Behaviour^ 572.6 Optimal Shopping Behaviour: Solution to a Special CaseThe previous section reduced the possible shopping behaviours to six patterns, each of whichmay be optimized over the quantities purchased, which then immediately induces the optimaltiming. In the most general case, for a given set of parameters, the six optimizations couldbe done, and the minimal of the six would determine the unrestricted optimal shoppingpattern. Then, in principle, the parameters could be varied to determine where the shoppingpatterns change. With 6 possible patterns, and a 10-dimensional parameter space, the resultswould be difficult to interpret. In this section, a subset consisting of three important casesis examined and solved.Two of the cases to be considered are when the customer shops at one store or theother exclusively. For the mixed-store shopping, only the "n" case will be considered,because the "p" case (as noted previously) has little face validity. Of the two "n" casepossibilities, only the case where the non-perishables are purchased exclusively at the largestore will be considered. The modified version of the "n" case, where some non-perishablemay be purchased on the last trip to the small store, will be left for future investigation.2.6.1 Shopping at the small store onlyThis problem was solved in section 2.4.3, without price costs. If the perishability constraint(26) is binding, the long-run average cost with prices, from (25), issDn A teCT,s,b+ D P +DPA te^2^P^n n,s(41)When the perishability constraint is not binding, the cost from (23) isChapter 2: Grocery Shopping Behaviour^ 58CT:s,nb = 112C sD nS + DpPp4 + D nP 14s^(42)where the cost has been subscripted to indicate "binding" and "not binding", and "small".2.6.2 Shopping at the large store onlyCosts for the large-store only case are as above with the appropriate subscripts: Cl^ sD At+P P'e DPl DnPn'iAte CT,* 1,b(43)CT, 1,nb V2c1Dns DpPp,1 D nP nj^(44)From here on, upper case "C" will mean total long-run average costs, and the bar and thesubscript "T" will be dropped from the cost symbol.2.6.3 Shopping at Both StoresThe shopping pattern being investigated is the periodic policy with period defined by the timebetween trips to the large store, and where non-perishables are purchased only at the largestore (Figure 10).The problem is MIN CmQn,l , Qp ,1Qp , sin(where the "m" subscript on the cost refers to the mixed case described above and in Figure10, and minimization is over quantities, and m, the number of trips per period to the smalln,1 = tD nQ n,quantiQlargestorenonperishable"--t equantitsmall^Qstorem tripstimeperishableChapter 2: Grocery Shopping BehaviourI59Figurere 10: Mixed Store Shopping Patternstore) subject to the perishability constraints,At s Ate^ (45)At 1 s Ate^ (46)Note (Figure 10) that Proposition 4n requires the quantity of the perishable good purchasedat the large store to be exactly enough to last to expiry, so that the perishability constraint(46) is always binding on Qp ,„ so it is unnecessary to minimize over that quantity. Corollary5n also fixes the total amount purchased at the small store during the period as the numberof trips, m, to the small store, times Qp ,s , which can be expressed in terms of Qn,1:Chapter 2: Grocery Shopping Behaviour^ 60mQp,s  ( Atnj  Ate )DPQP,s = (Consequently, the minimization need only be done over Q, 1 and m.The cost function isAte)) PM(47)C C l2+ MCs + SQnj+2DnQn,1Pn,1 +(48)QPiPPJ + MQP,sPp, ^)Substituting for Q,,,, and Q. from above, and letting Po - Pp , s = Op ,2Dn Cl1 + mcs + s(4 1C  ^' + 0, n, n,i — DpAteAPp + DPPP 'sQnj )o'<ii,1 2Dn^ Dn(49)The effect of the binding perishability constraint on Qp3 has already been incorporated intoequation (49), but the constraint is not necessarily binding on Qp , s . Since Qp , s has beeneliminated from the cost function, the constraint is rewritten as a minimum on the numberof trips that must be taken to the small store in the period:m > ^Atli'/ - AteAt(50)or, in terms of Q,, 1 :Chapter 2: Grocery Shopping Behaviourm + 1^Qn r ' >_ 0AteDnFigure 11: Long-run average trip costs to the small store as a function of length of theperiodic interval.A brief discussion of how the perishability constraint (51) affects optimal shoppingpatterns at the small store, and how that effects the length of the period, may be helpful.Since the only costs associated with changing the trip pattern to the small store are trip costs,an optimal policy will minimize the number of small store trips for any given period length.However, the perishability constraint and the no-stockout rule puts a lower bound on thenumber of trips for any given period length. Now consider what happens if the parametersof the model are varied in such a way that the length of the period At o increases; refer toFigure 11, which shows just the (time-average) optimal (small store) trip costs as the periodincreases. As the period increases from once to twice the expiry time, one trip must bemade to the small store, and the fixed cost of that trip translates to a decreasing average tripcost. If the period exceeds twice the expiry time, and extra trip must be made, whichincreases the trip costs discontinuously. The pattern repeats itself, with a fixed upper bound61(51)Chapter 2: Grocery Shopping Behaviour^ 62c s/Ate on average trip costs. At each discontinuity, the lower bound is a factor (m - 1)/mtimes the upper bound. Since the remaining costs (storage and price) are continuous in themodel parameters, what must happen as the parameters change is that the optimal periodlength increases to an integer multiple of the expiry time At e , and then locks there: addinga continuous function to the discontinuous function of Figure 11 cannot produce a minimumtotal cost at the upper bound on the trip cost. As the parameters continue to change in sucha way as to force the period length to increase, the period will eventually jumpdiscontinuously to a length near, or at the next integer multiple of the expiry time.When the period is locked at an integer multiple of at e, the perishability constraint,(45) or (51), is binding. The subsequent jump in period length, as parameters change, maybe to a value less than the next integer multiple, at which point the perishability constraintis not binding. As the parameters change to increase optimal period length continuouslyfrom this point, the period will eventually hit the next integer multiple, at which point theconstraint is again binding. It is also possible that the period will jump directly from thebinding m th to the binding m th + 1 case. However, the nonbinding case must always becomebinding before an extra trip is added. These results will be demonstrated formally later.In the following, the subscripts n,1 will be dropped from Q n,„ as that is the onlyquantity being dealt with in the optimization.The cost function (49) is convex in Q and the constraint (51) is linear. Therefore theKuhn-Tucker first-order conditions are sufficient for minimization with respect to Q. TheLagrangian isChapter 2: Grocery Shopping Behaviour^ 63L=c_ µ ( in-Fi-  QnjA teDn f+ M Cs +SQnj2Dn+ n .P-c- n,r nj — DpAteAPp +D P 0p p,s — ,l)Dnin + 1 (4,1 A teD)(52)where 1.1 is the Kuhn-Tucker multiplier. The first order condition with respect to Q isal, _ ac 1_ ^p. aQ aQ A teDn_ 0^(53)The other FOC is the perishability constraint (51).If the constraint is binding, (51) gives12b:in = D nAte( m + 1 )^(54)where the subscripts on Q indicate "binding" and dependence on the trip parameter "m".The binding case implies IL > 0; differentiating the cost and substituting in (53),sQb,m + 2Dn( APpAteDp — c1 — csm ) < 0^(55)substituting (51) into (55) and rearranging gives the parameter range where the constraintbinds:Chapter 2: Grocery Shopping Behaviour^ 64C1 + MC sA te2Dn( 1 + m )2s^ + APpAteDp (56)2 Finally, the minimal (binding) cost is*cb,m^ + DP— 1 + /tics^A P D^ sAt D (1 + m)P P^niz) + DpPAs + ^e nA te(1 + m)^1 + m 2(57)When the constraint is not binding, it = 0, andQ < ( m + 1 )Dn A te^(58)The first-order condition (53) becomes, after differentiation and simplifying,Qn*b,m2Dn( CI + MC, — APpAteD) (59)Substituting into the constraint gives2Dn(c 1 + MC, — APpAteDp) < (m + 1)213,2 At (60)Sor+ 1)2DnAte2S2CI + MCs < ^ + APpAteDpwhich is the complement of the set defined in (56). The (not binding) cost is(61)Chapter 2: Grocery Shopping Behaviour^ 65Cn*on = D,,P„J + DpPp  + V2Dns(ci + mcs — APpAteDp) (62)Let us consider how the minimum cost varies with the travel cost per period, c l + mcs .From equations (57) and (62), the minimum cost is linear with travel cost in the bindingcase, and varies with the square root of travel cost in the nonbinding case. At the pointwhere the constraint just binds, i.e., equality in (56), the two cost functions (substituting (56)at equality) and their first derivatives are equal:Cb or nb,m = DnPn,1 DpPp,s AteDn8( 1 +^)a (c, ± MC) .nblb^Ate( 1 + m )where the notation is intended to indicate that the derivatives are evaluated at the point whereequality obtains in (56).So, ceterus paribus, the perishability constraint binds when the trip costs are large,as indicated in Figure 12. As can be readily seen in Figure 12, the binding of the m thperishability constraint leads to a higher cost (C b* > Ciib* *)Also note that if trip costs are some positive monotonically increasing function ofdistance, and the stores are separated, so that there is a minimum total trip cost, that it isquite possible for the constraint to be binding for all customers that are homogeneous onother parameters. This would be the case where the period At„,, only takes values equal tointeger multiples of the expiry time Ate.acb* or nb,m 1(63)(64)Chapter 2: Grocery Shopping Behaviour^ 66Figure 12: Perishability constraint binds as trip costs increase (fixed m).Figure 12 is for a fixed value of m. How does it change with different values of m?Note that in (56) or (61) that the RHS increases as m 2 , while the LHS as m, implying thatfor larger values of m, there is greater chance of the constraint being nonbinding. The pointof tangency in Figure 12, defined by equality in (56), moves to the right. So, as trip costsincrease for fixed m, the optimal cost function will switch from nonbinding to binding (ifnonbinding is ever feasible); but for larger values of m, the switch will come at higher tripcosts. One useful application is the following: suppose one were working with a particularset of parameter values, and distance related trip costs, and found that, because of fixed storeseparation, the constraint was always binding for the m = 3 case. Then one would knowthat the constraint was always binding for m = 1 and m = 2.The final step is to determine the optimal number of trips to the small store, m,where m is restricted to integers greater than or equal to one. One approach is to assumem is continuous, and solve the first order conditions. Then for any particular numericalvalues of the parameters, take the value of m rounded up or down as appropriate.Chapter 2: Grocery Shopping Behaviour^ 67If one is interested in a particular parameter (or parameters), an alternate approachis to take the cost functions (57) and (62), and determine at what parameter value the costwith m trips equals the cost with m+ 1 trips. For a sequence of values for m, this will givethe points where the shopping behaviour changes by addition of an extra trip to the smallstore, as a function of the parameter. Because the next section addresses the change inshopping behaviour as a function of household location (which translates to trip costs), thelatter approach will be taken here. The method can easily be adapted for other parameters.We want to find when C.* = C.."*• Because there are different cost functions forthe binding and non-binding cases, there are four possibilities for the transition region:1) Cb • = C- b,m+1 *2) Cb,m* = Cnb,m+1 *^*^*3) Cnb,m^L'nb,m+ 14) C * = C^nb,m^b,m+ 1.A brief geometric discussion of these transitions in the context of Figure 11, whichshows trip costs, follows. As noted before, the remaining costs are convex functions ofAdding a convex function to the trip costs, and varying model parameters so that theminimum in the total cost function moves to the right (increasing period length) in Figure11, can produce transitions from binding to binding, as in 1) above, and from binding tononbinding, as in 2) above. These possibilities are displayed graphically in Figures 13 (a)and (b) respectively, which shows in a stylized fashion the total cost as a function of periodlength. These costs are the sum of the trip costs of Figure 11 and the convex remainingcosts. The minimum total cost occurs at a period length which increases as the parameterschange; and the increase in optimal period length must change discontinuously. However,transitionC* =C*b,m^b,m+1binding, m trips(a)binding, m+1 tripstransitionbinding, m trips not binding, m+1 tripsC* n b,m+1increasing period length(b)Chapter 2: Grocery Shopping Behaviour^ 68Figure 13: Possible transitions from m trips to m+1 trips as period At,,,,* increases: (a)perishability constraint binding for both m and m+ 1 (b) constraint binding for m and notbinding for m + is also apparent from Figure 13 that transitions given by 3) and 4) above cannot occur.As the minimum cost moves to the right, the non-binding, m trip case must first switch tothe binding case, which then jumps discontinuously to either the binding or nonbinding,m + 1 trip case.Proposition 9: For optimal mixed-store shopping, as parameters vary, direct transition fromthe non-binding case, to a case with one extra trip to the small store, will not occur.Chapter 2: Grocery Shopping Behaviour^ 69Proof: Recall c s and ci are assumed positive real. The proposition states that 3) and 4)above will not occur. First consider 3), substituting in the cost functions, and eliminatingcommon terms:\12Dn( c i + mcs ) — 2DnAteAPpDp^(65)= V2Dn( c1 + mcs + cs ) — 2Dn1teAPpDpwhich implies cs = 0, a contradiction.Next consider 4) above, substituting the cost functions and eliminating commonterms:V2Dns( ci + csm — APpAteDp )c 1 + cs( m + 1)^AP DP^sAteDn( m + 2 )P  ^+ Ate( m + 2 )^( m + 2 ) 2Squaring equation (66) gives a quadratic in c, , which may be solved forc1 = APp Dp Ate^e+ At 2Dn s( m + 2 )2 — cs( m + 1 )Ate( 2 + m ) ^2 ^—8cp.which also has no real solution for positive cs El.Referring to Figure 13, we can see that the c. = 0 condition which arises in theabove proof corresponds the disappearance of the discontinuity in the cost function; this isconsistent with the intuitive argument that it is the discontinuity in trip cost that forces the(66)(67)Chapter 2: Grocery Shopping Behaviour^ 70discontinuity in the optimal period length, At„,, (from the binding case, m trips, to the m+1trips case).Proposition 9 eliminates two of the four possibilities for transitions. For thenumerical optimization to be carried out below, it now remains to find the relation betweenthe parameters that defines the region of parameter space where the remaining two allowedtransitions occur. Because the examples later in this chapter focus on the effects of tripcosts, the relation between the parameters are solved for c l , the trip cost to the large store.The first case, binding m trips, to binding m+ 1 trips isc1 + mcs^APpDp sDnAte( 1 + m )^ + ^Ate( 1 + m )^1 + m^2c1 + ( 1 + m )cs^APpDp 4. sDnAt-e( 2Ate( 2 + m )^2 + m^2+ m )(68) which simplifies toc I = cs + Ate [ APp Dp + 'IDn Ate ( 1 + m )( 2 + m )]^(69)2 The last case, binding m trips to non-binding m+ 1 trips, is\12Dns( c1 + cs( m + 1) — APpAteDpc1 + csm^APpDp  + sAteDn( in+1^ _Ate( m + 1 )^( m + 1 )^2Squaring and solving for c l simplifies (70) to)^ (70)Chapter 2: Grocery Shopping Behaviour^ 71c1 = APpDpAte — csm + 2Ate2Dn( m + 1 )2A te( 12^m ) ±^Ocp.sThe above transition equations (69) and (71) are bounds of the areas where various valuesof m are optimal in the mixed shopping cases. With the constraint equations (56) and (61),which give the regions in parameter space where, for each value of m, the constraint isbinding or not binding, the transition equations are helpful in the task of comparing theoptimal cost surfaces for the mixed and single store shopping behaviours.(71)Chapter 2: Grocery Shopping Behaviour^ 722.7 A Numerical Application: Two-Dimensional Trading AreasThe model asssumes that customers shop deterministically, with the shopping behaviour thatminimizes long-run average costs. It is natural to then ask what this sort of shoppingbehaviour implies for the market areas of the grocery stores. This, of course, will dependon the specific parameters describing each customer in the competitive market space. Thetwo parameters which are immediately related to customer location, and hence market area,are the trip costs to the large and small store. In the absence of information on othercustomer-related parameter values, trip costs based on distance can provide a firstapproximation of market areas, assuming customers are homogeneous on other parameters.In this section, the shopping behaviour of spatially heterogeneous customers, and theresulting market areas of the two stores, will be derived using numerical and graphicalmethods.In particular, assume trips costs are directly proportional to the Euclidean distancebetween the customer's household and the store. This is consistent with many theoreticalmodels (Hotelling 1929; Ingene and Ghosh 1990) as well as being a special case of the manyempirical Huff-type spatial interaction models (Huff, 1962). For the small store:Cs = T(X — Xs )2 + ( y — ys )2 (72)where r is the sensitivity to travel in dollars per kilometre, the customer is located at (x,y),and the small store is located at (x„y s). The expression for the large store trip cost issimilar. Table 2 shows the arbitrary, but reasonable, parameter values used in the numericaland graphical calculations.Chapter 2: Grocery Shopping Behaviour^ 73Table 2: Parameter values used in numerical calculations.Locations:Large store:^(x,y) = (-5,0)Small store: (x,y) = (5,0)(store separation = 10 km.)Prices:Perishable, large store: = $40.00/unitPerishable, small store: Pp , s = $50.00Perishable difference: APp = $10.00Non-perishable, large store: Po = $40.00Non-perishable, small store: = $50.00Demand rates:Perishable:Non-perishable:Travel cost:Storage cost:Expiry time:Dp = 1 unit/weekD. = 1r = $4.00/kms = $2.00/unit/weekAte = 1 weekThe households represented by these parameters would spend $100.00 per week ifthey shopped exclusively at the small store, and $80.00 per week if they shopped exclusivelyat the large store. Travel costs can be compared to this savings: a difference in 1 km. tothe stores means a savings of $4.00. Storage costs of $2 per unit per week appear relativelysmall by comparison, but since they are quadratic in quantity purchased, if (for example) thecustomer stocks up on 4 weeks supply, the storage cost will be $16.00 for that purchase.A lifetime of 1 week seems reasonable for many vegetables and dairy products. Theconsumption rates are fixed at one unit per week, and are essentially scaling parameters.Alternatively, one could think of the above parameters as representing some average of allperishables and all nonperishables purchased by the household. The point is to illustrate thenature of shopping behaviour and trading areas that emerge from the model.Chapter 2: Grocery Shopping Behaviour 74The parameter values are used to evaluate the optimal (over quantity) cost functionsderived in the last section. The smallest of these functions at each point in the planedetermines the shopping pattern at that point. First, for the single-store cases, we determinethe region in the plane where the constraints are binding or nonbinding, using equation (26part 1).10.5.^.r---.^. ..0C.$,..^.,..)-0.5 --14 4.5 5 5.5 6Figure 14: Perishability constraint bindsoutside circular region (small-store onlyshopping behaviour).0.50r----c,-0.5-6^-5.5 -5 -4.5Figure 15: Perishability constraint isbinding outside circle (large-store-onlyshopping behaviour).Figures 14 and 15 show these regions: note that the horizontal axis is shifted in the twodiagrams, as the stores are located at different points in the plane. Also note that the bindingcase dominates except for a small region of about one-quarter kilometer radius around eachstore. Outside this region, the perishability constraint binds. For the majority of the plane,then, we need to consider only the binding cost functions, (41) and (43). (While the size ofthe region that is nonbinding depends on the parameters, in the typical case there will beboth binding and nonbinding areas in a similar geometric relation).AsP.:,.„4r4174! 1,4 0 wteop,,,...,..c....,.,..0_,_,.,...-4..4-"flINOWitfilr4g., ...-Ttatle Vt.'44,,,,,,,r. 7.....431,...,.• ;44,44 iz-.444.-.1' t, ItttlIzr 7Chapter 2: Grocery Shopping Behaviour^ 75Figure 16 shows the cost surfaces with perishability constraint binding for the single-store only shopping behaviours. If multi-store shopping were not available to the customer,the intersection of these surfaces would determine the trading areas of the two stores.(a): Large store (b): Small store(c): (a) and (b) superimposed.Figure 16: Single-store-only, binding constraint, optimal cost surfaces.Note that the large store's shopping area would be larger than the small store's, andthat the lowest cost at the large store is less than the lowest cost at the small store.Chapter 2: Grocery Shopping Behaviour2, ,^. r......\Ii\7 .-- ---„,,el^\^\0 .i (^...----...I.;^(^1N.. ) /^....,. ^i^i\ \.\^"--......„^,...--/^I1 /'-2 .3 4^5^6 7Figure 17: Progressively larger regionsaround the small store where theperishability constraint is not binding, form = 5, 6, and 776The interesting aspect of the modelexamined in this research, however, is that itallows shopping at both stores. This means wemust superimpose the mixed-trip cost surfacesas well. Moreover, we need to consider bothbinding and non-binding optimal costs for eachvalue of m, the number of trips to the smallstore per period. As with the single storecases, first we determine where the mixed tripcases switch from binding to non-binding. Forthe parameters here, it turns out that for m less than or equal to 4, the perishabilityconstraint is binding for all locations in the plane. This occurs because the stores areseparated by a fixed distance, so that there is a non-zero minimum on the travel cost, c i +mcs , and this minimimum is greater than the transition value (binding to nonbinding... thepoint of tangency in Figure 12) as long as m is less than 5. To explicitly show this, considerthe following. If m = 1, (trips to the stores alternate between large and small), theminimum travel cost in the plane lies on any point on the line joining the two stores. (Sincetrips alternate between the stores, total travel cost is the same for any household on thisline). If, on the other hand, m > 1, the minimum travel cost is the for the customer wholives in (on?) the small store, i.e., c, = 0. Therefore, for any fixed value of m, no customerengaging in mixed trip shopping will have lower costs than the customer at this point.Therefore, (again referring to Figure 12), if this customer is in the binding region, allcustomers in the plane will be in the binding region. At this point, c1 = 10 km. SubstitutingChapter 2: Grocery Shopping Behaviour^ 77these values and the numerical parameter values above into the condition (61) for theconstraint to be nonbinding gives40 < ( m + 1 )2 + 10^(73)4.47 < mi.e., m must be 5 or more for the perishability constraint to be nonbinding anywhere in theplane.For m = 5, 6, and 7, there is a progressively larger region around the small storewhere the constraint is not binding (Figure 17). Referring to Figure 12, the change in thenonbinding area as m changes can be seen as the change in the relation between theminimum cost and the point of tangency. For each different m, the actual cost curves differ,and so does the point where the minimum cost occurs. For m less than 5, the point oftangency is to the left of the minimum cost. As m increases, the point of tangency movesto the right relative to the minimum cost.In summary, then, we need only consider the nonbinding mixed-trip cost as long asm is less than five. At this stage, however, we haven't eliminated the possibility that highervalues of m may be optimal, and if m exceeds 4, we will have to consider both binding andnonbinding. However, from Figure 17, we need only examine limited regions around thesmall store when comparing the nonbinding, mixed-trip cost surfaces, with the other costsurfaces.Figure 18 shows the mixed-trip, constraint binding, cost surfaces for m =--- 1 and m= 3. Note that for m = 1, the optimal cost is constant and minimal between the two stores,Chapter 2: Grocery Shopping Behaviour^ 78and for m = 3, it is linear between the stores and minimal at the small store, as required bythe preceding argument. Also, the smallest cost for m = 3 is less than for m = 1.(a): m = 1. (b): m = 3.(c): (a) and (b) superimposed.Figure 18: Optimal cost surfaces for mixed-store, bindingconstraint, shopping behaviour.The next step is to determine where the intersections between the surfaces fordifferent values of m are. These will indicate where the transition occurs from one type ofdiscrete shopping behaviour (m trips per period) to the next (m + 1 trips per period). Usingthe transition formula (69) for binding, m, to binding, m+1, the regions of the plane whereeach shopping pattern is minimal can be found. Figure 19 shows the regions where m =0 (which is equivalent to the large-store-only shopping pattern), m = 1, m= 2, m = 3, andm = 4.79There is no intersection between m =4 and m = 5, indicating that 4 trips to thesmall store, between trips to the large store, isthe maximum number that can be optimal (forthe binding case) in the market. In otherwords, we have now eliminated nonbindingmixed-trip costs for m less than 5, and binding,mixed-trip costs for m greater than 4, fromfurther consideration.A note regarding the resolution of theChapter 2: Grocery Shopping Behaviour10 • / /^j1 / /ii /f/,o .^'( CM \t\ 1 \I\ \^N.\ \ \^\\ \ \ \-10 I1-10 -5 0 5 10Figure 19: Comparison of binding, mixed-trip costs for various values of m giveregions where each is minimal. From leftto right the regions are for m = 0, 1, 2,3, and 4.graphical approach is in order here. Considerthe line y = 0 for x > 5, i.e., to the right of the small store. On this half-line, the distanceto the large store is 10 km. more than the distance (say, d s) to the small store, so that c l +mc, = 40 + 4(m + 1)d,. Substituting this into (57) gives the the mixed, binding optimalcost along the half-line for any value of m:Cb*,in   + 90 + 1 + m (74)When either m = 4, or m = 5, this reduces toCI; = 101 + 4ds^ (75)That is, along this half-line, Cbm=4 * = Cbm= 5 * : the two cost surfaces are osculating. Thismeans there should be a degenerate contour in Figure 19 extending from (5,0) in the positive40 + 4( m + 1 )d,^101 + m^1 + mChapter 2: Grocery Shopping Behaviour^ 80direction along the x axis, indicating this equality. The graphics software, however, cannotpick up this degenerate contour, and it appears that there is no possibility of the m = 5 case.All of the intersection contours represent subspaces where either of the equal-cost shoppingpatterns can occur. They represent the vanishingly small regions of the market, referred toin Corollory 2-1, where customers "almost never" live. In the case of the degeneratecontour, we can say, further, that the m = 5 behaviour almost never occurs. 1050-5-10-10^-5^0^5^1 0Figure 20: Contours of intersection ofthe small-store only cost surface with themixed trip (from left to right: m = 0,1,2,and 3) surfaces.-5^0^5^10 Superposition of Figure 20 onThe Figure 20 intersectionsoccur to the right of the correspondingFigure 19 regions.101 -5005-10Figure 21:Figure 19.Figure 20 shows where the small-store only, constraint binding cost surface intersectseach of the cost surfaces in Figure 19. (The small store surface is less than the mixed-storesurfaces on the right of each of the contours). By superimposing Figure 19 on Figure 20,(Figure 21) it can be seen that each of the contours in Figure 20 is to the right of thecorresponding region in Figure 19. This means that none of the Figure 20 intersectionsoccur in the region of Figure 19 where that mixed-trip surface is smallest. In other words,moving from the large store towards the small store (left to right), the customer wouldChapter 2: Grocery Shopping Behaviour^ 81switch from the m th to mth + 1 surface before encountering the small-store-only surface. Sofor these parameter values, the optimal binding, small-store-only shopping pattern is alwaysmore costly than the lowest binding, mixed-trip pattern.Next consider the small regions around each store where the single-store cost functionis not binding (Figures 14 and 15). Since that region for the large store is already in thelarge-store only shopping region, the pattern won't change...just the form of the optimalsolution. For the small store, however, we need to check to see if the non-binding, small-store only cost is less than the mixed trip costs in Figure 19, inside the region where thesmall-store cost is the non-binding cost. From Figures 14 and 19, this specifically meanscomparison of the small store non-binding cost with mixed trip binding cost for m = 4.First we can look at the location of the small store, (x,y) = (5,0), or (c l ,cs) = (40,0). At thispoint the optimal costs are, from equations (42) and (57),^Cs :14 4) = 100^ (76), ^Con .41c,.0 = 101^ (77)There is, then, some region right around the store where the nonbinding, small-store onlycost is less than the binding, mixed, m=4 cost. The intersection of these two cost surfacesis shown in Figure 22. However, the small store non-binding surface here has not beentruncated by the curve that indicates where the constraint must be binding (Figure 14).Superimposing Figure 14 on Figure 22 gives the region where the nonbinding, smallonly, cost function occurs, and the region where it is less than the binding, m =4, mixedChapter 2: Grocery Shopping Behaviour^ 82function. The result, shown in Figure 23 as the smallest intersection region inside the circle,is where shoppers prefer patronizing only the small store, to the m=4, mixed trip behaviour.0.5-0.54 4.5 5 5.5 6 Figure 22: Intersection of mixed, m =4,binding optimal cost surface with small-only, nonbinding optimal cost surface.-----„...,.'L ..,0.5 %. -1 : .0---------..(I,^r____„..., ,.:„.6.... ,....___.,f,, .-0.5' / -;z..-- ...-----"• -4 4.5 5 5.5 6Figure 23: Perishability constraint forsmall-only cost function is nonbindinginside the circle.The remaining consideration is the non-binding mixed trip cases, which are feasiblefor some values of m greater than 4 (Figure 17). First we note from equation (62) that theoptimal cost for this case, regardless of the value of c, and c 1 (and hence of customer position(x,y) in this numerical example), is increasing in m. Therefore it suffices to consider thecase m = 5, which is the smallest cost that is feasible for this case. First note that at thelocation of the small store, (x,y) = (5,0), the cost for the mixed, m = 5, nonbinding caseis, from (62),C,21,„,=51, . 0 = 90 +^101S(78)Comparison with (76) indicates that at this point, the single store behaviour is optimal.Next, where, if at all, do these surfaces intersect?• . . .---•-1...,.... .\ .0.5 1•\.....- .^../r. ...—.—^,^.-^\ •1 / ....... s.,0 .;.,1.1- ...___.."^,^1.E^-4'1.. . (....'.i. •,—0.5 /r•of....•...--•—••—1 •4 4.5 5 5.5 6Figure 26: Perishabiltynonbinding on the mixed,inside the small circle.constraint ism = 5, caseChapter 2: Grocery Shopping Behaviour^ 831 7 -•--...,...0.5 11.k; .o (\-..----1.1..(1—0.5 ,f.....'...'...--...'—1 — . ..._.,-.14 4.5 5 5.5 6Figure 24: Intersection of nonbinding costsurfaces for small-store only, and mixed,m = 5.---.^ •^.1..... .-.......,-..0.51\%...'tf'^.....•--.._.:^! .0 ' .44i^f\ ‘%.^---_—.111."............,....."^i1—0.5 I .•/.,....'— 1 •.........."-'.......•4 4.5^5 5.5 6Figure 25: Perishability constraint isnonbinding, for small-only shopping,inside the small circle.The intersection contour, shown inFigure 24, encloses the region where the singlestore cost is greater than the mixed, m =5, cost(both nonbinding). As before, it is necessaryto restrict the surfaces to the region where theconstraints, for both surfaces, are actuallynonbinding. Superimposing the constraintcontours for the small-only case (Figure 14),and the mixed m = 5 case (the inner contourin Figure 17) on the intersection (Figure 24)gives the results shown in Figures 25 and 26. The central portion of the intersection contourlies inside the non-binding region of both cost functions, so that portion indicates a transition10.50-0.5-14 4.5 5 5.5 6Figure 27:^The intersection of thenonbinding, mixed, m = 6 contour withthe nonbinding, small only, contour.Chapter 2: Grocery Shopping Behaviour^ 84in minimum costs from the small store only to the mixed, m = 5 case, as the customer'sposition moves away from the store.It was asserted that it was only necessary to inspect the case when m = 5 becausethe optimal costs, for the nonbinding mixed cases, increase with m regardless of thecustomer's location. As a check on this assertion, the region in Figure 24, where the m =5, nonbinding case has lower cost than the small- only, nonbinding case, should be largerthan the corresponding region with m = 6. The latter region is shown in Figure 27, and isin fact much smaller.The graphical approach to determiningthe regions where different shoppingbehaviours occur will be summarized shortly,but first a note regarding the resolution of thepreceding few figures is in order. Twodifferent contours are nearly identical in thepreceding: the intersection of the small onlynonbinding cost with the mixed, binding, m =4 contour (Figure 22); and the intersection ofthe small only nonbinding cost with mixed,nonbinding, m = 5 contour (Figure 24). This indicates that the latter cost surfaces are asnearly identical as the graphical approach can discern here. This is consistent with theprevious observation that the binding costs for m = 4 and m = 5 are identical to the rightof the small store. Furthermore, the transition from binding to nonbinding for both thesmall, and the mixed m = 5 costs (the circles in Figures 25 and 26) occur very close to theChapter 2: Grocery Shopping Behaviour 85small store. Hence, in the region around the small store, all six cost functions are nearlyidentical, so that in the immediate vicinity of the small store, and particularly to the right,the precise shopping behaviour cannot be determined from the preceding diagrams.Since this is an artifact of the particular parameters arbitrarily chosen here, it will notbe explored further. We can nonetheless describe the shopping behaviour in the plane inFigure 28: Market areas of the large and small store, located at (-5,0), and (5,0), forthe parameters given in the text.some detail (see Figure 28). There is a tiny region immediately around the small store thatChapter 2: Grocery Shopping Behaviour 86is its own market area exclusively. At about one-quarter of a kilometre away, shoppingbehaviour switches to a mixed, m = 4, or m = 5 type of behaviour. Moving strictly to theright along the x axis, this behaviour doesn't change. In any other direction, however, itchanges to the mixed, m = 4 behaviour. In directions towards the large store, the behaviourgoes through progressively fewer fill-in trips to the small store, until eventually shopping isdone only at the large store.From these market areas, market shares can be calculated from a given populationdensity distribution. For purposes of illustration, assume that the city is the twenty bytwenty kilometer square area that has been plotted, and that the population is uniformlydistributed up to the boundaries of the city, and zero outside the boundaries. Everyoneexcept for those living inside the small store's exclusive region purchases all of theirnonperishables at the large store. Assuming this area is a circle of radius 0.2 km, the smallstore's share of nonperishable good is 0.03 %, and the large store's share is 99.97%.Share of the perishables varies with the different regions. The detailed calculationsmay be found in Appendix A. For the entire region, the large store captures 71.86% of theperishables, and the small store has a 28.14% share.Smaller Price Differences: The model has two parameters under the control of management:price and store location. Price, of course, is a much shorter term strategic variable, and itis interesting to see what happens to the market areas if the price difference is less than the20% postulated in the previous numerical example. Let us reduce the price difference to10% by increasing the price of both goods at the large store from $40 to $45. Leaving allother parameters the same gives the trading areas shown in Figure 29. (The procedurefollowed is the same as in the preceding example, but not duplicated here). For comparison,10 ,-10-10 -5 0 5 10Figure 29: Market areas with a 10% pricedifference between stores. From left toright, the regions are exclusively the largestore's; mixed with m = 1 and m= 2; andexclusively the small store's.Chapter 2: Grocery Shopping Behaviour 87refer to Figures 19 or 28. Unlike the casewith 20% price difference, the small store nowhas a large market area exclusively its own.The mixed behaviour is confined to a narrowregion between the two exclusive areas. Themarket shares that this shopping patterncorresponds to (using the same approach as forthe 10% price difference) are 36.6% of thenonperishable for the small store and 63.4%for the large store; and 40.5% of theperishables for the small store and 59.5% ofperishables for the large store. Changing the price difference between stores from 20% to10% affects the share of non-perishables much more than perishables. As price differencesincrease, it becomes worthwhile to travel further, but less frequently, for goods that can bestored. Where trip frequency is governed by perishability, the effect of price difference isnot so dramatic.Increased Consumption Rate: One would expect that larger households would be moresensitive to price differences, if only from income effects. However, even if all sensitivityparameters are kept the same, the increased consumption rate will make it worthwhile fora larger household to stockpile. Figure 30 shows the trading areas for the same prices (10%difference between stores), and all other parameters the same, except for doubling theconsumption rates: Do = Dp = 2. The regions of mixed shopping are increased again,although not to the same extent as in Figure 19. Market shares for the nonperishable are1050-5-10-10^-5^0^5^10Chapter 2: Grocery Shopping BehaviourFigure 30: A greater consumption rateincreases the tendency to stockpile. Fromleft to right market areas are exclusivelythe large store's; mixed with m = 1 and m= 2; and exclusively the small store's.8818.5% and 81.5% for the small and large storerespectively; and, for the perishable, 29.1 %and 70.9%. The small store still has asubstantial region of exclusivity. The marketareas are, not surprisingly, sensitive to theabsolute differences in consumer expenditurerates at the two stores. Another way of sayingthis is that people with large families will gofurther for a deal, even if they have the sameprice, travel, and stockpiling sensitivities as asingle person.The small store's market shares for the three preceding sets of parameters aresummarized in Table 3.Table 3: Small store shares under the three scenarios.20% pricedifference10% pricedifference10% price difference;2x consumption ratenonperishableshares0.03% 36.6% 18.5%perishableshares28.14% 40.5% 29.1%Chapter 2: Grocery Shopping Behaviour^ 892.8 DiscussionThere are several interesting results from the preceding model. Perhaps the two mostinteresting are that, first, consideration of perishable and nonperishable goods allows multi-store shopping to be the outcome of a purely deterministic rational consumer choice process;and second, consideration of consumer stockpiling allows a grocer's share response to priceto be dramatically nonlinear.Multi-store shopping parallels the multi-purpose shopping results (Ingene and Ghosh,1990). It differs in that the stores in the model here each offer both goods. Themultipurpose models, on the other hand, seek to explain how stores offering frequentlypurchased (low-order) goods will agglomerate with stores offering infrequently purchased(high order) goods, due to savings associated with multipurpose shopping, thereby creatinga high order centre. It seems less surprising that consumers will shop at several stores whenthey offer different goods, than it does when they offer the same goods. Similarly, whilethe retailing and promotions literature recognize that multi-store shopping strategies are thenorm, and that loyalty to one store is low (Uncles and Ehrenberg, 1990), the possibility ofsuch a strategy resulting from the combination of a power retailer (who has a large priceadvantage, but few locations) with the fact that some groceries are perishable and othersnonperishable, has not been considered.Nonlinear response, the second interesting result, is the way the market area of theprice leader can expand dramatically as the price difference between stores increases. In thenumerical example, dropping the price from $45 to $40, against the small store's $50,produced almost complete domination of the nonperishables market by the large store. This"increasing returns" effect is due to the consumers ability to reduce long-run average tripLARGE STORE'SSHARE RESPONSE to PRICE DIFFERENCEfor perishables and nonperishables0^2^4^6^8^10^12^14^18^18^20100so802.)coco3 7o'g8a so5040Percentage Price DifferenceChapter 2: Grocery Shopping Behaviour^ 90costs by stockpiling and shopping less frequently at the large store, and thus being able totravel longer distances to take advantage of low prices.Figure 31: Share response of the large store to its own price reductions, using thenumerical example of the last section. At small price differences, no mixed shoppingoccurs, and the two goods have the same share.Figure 31 shows this effect, using the shares calculated in the preceding section, aswell as intermediate points calculated at 5% and 15% price differences. Note that theperishable goods are much less sensitive to price differences than nonperishables; and thatthe shares of the two goods are the same when the optimal shopping pattern for everyone inthe region is exclusively one store or the other, which occurs when price differences aresmall.Chapter 2: Grocery Shopping Behaviour 91One of the questions posed in the introduction was "Why was Safeway taken off-guard by the entry of the Real Canadian Superstore?" One possibility lies in the highlynonlinear response to the price differences that arises when stockpiling is considered.The retailing literature has many examples of location and trading area models thatdo not consider stockpiling. These usually trade off store attractiveness (which may be afunction of several different variables such as price or image) against distance, and can betraced from Reilly's (1931) deterministic gravity model, through Huff's (1962) probabilisticmodel, and subsequent spatial interaction models. (For a review, see Ghosh and McLafferty(1987). These models are discussed in more detail in Chapter 3). When the "attraction" ofthese models is due to price, then there is some tradeoff between price and distance.Typically these models are empirically tested, and are calibrated to determine some modelparameters; these parameters can then be used to evaluate the sales potential of a proposedoutlet, considering the existing market and the competition. The models have been quitesuccessful, and also have a strong intuitive appeal. In fact, even without a formal model,a manager can readily look at trading areas and create a reasonable model of the price-distance tradeoff.For purposes of comparison with the trading areas implied by the stockpilingshopping model, let us take a simple gravity-type model. Assume that consumers trade offprices and distances in such a way that the boundary of the trading area between stores isgiven by the line where the ratio of distances to the stores is equal to the inverse of the ratioof prices:where d s , d 1 are the distances from the consumer to the small and large store, and p s and piare the expected prices at the small and large store. In an empirical application, X would-10^-5^0^5^101050-5-10^Chapter 2: Grocery Shopping Behaviour^ 92^ds^(PIT^ (79)c11^PSbe estimated. Here, for purposes of illustration, it is simply assumed to be unity.10 / -(5' I10-5 1 -.\-10-10 -5 0 5 10Figure 32: Trading area defined by thegravity model, (42), with a 10% pricedifference; cf. Figure 32.Figure 33: Trade areas with 10% pricedifference using the stockpiling model(from Figure 29).For the two stores located at (-5,0) and (5,0), and the 10% price difference ($45 and$50 at the large and small store respectively), the trading area boundary is shown in Figure32. For comparison, the stockpiling model, with the same parameters, is shown again inFigure 33. If Figure 33 represented actual data, the much simpler gravity model would bea good approximation of the trading areas. An analyst, or manager operating intuitively,would be justified in applying Occam's razor, and accepting equation (79) as a fair modelof shopping behaviour.Chapter 2: Grocery Shopping Behaviour10 45 ...10‘-5 ‘ _-10' \-10 -5 0 5 10Figure 34: Gravity model prediction ofmarket areas with a 20% price difference.931050i /fi //,7, ,^i0i\\ ,,\,\ \\\ \N -\ \ \-10 \\^\-10 -5 0^5^10Figure 35: Trade areas from stockpilingmodel with 20% price difference (fromFigure 19).If the competition was always within a 10% price difference, the analyst, or manager,would have a reasonable tool for predicting trading areas. But what if there is a new marketentrant with 20% price reduction? Applying the model (79) to predict trading areas wouldgive the map in Figure 34. The incumbent would see the threat of some erosion of marketshare, but perhaps not enough to elicit a dramatic response. This would be even more trueif the incumbent was a chain with many stores, located every few kilometres in the plane,and the entrant was known to have a strategy of widely dispersed stores. Unfortunately, forthe established retailer, consumers can stockpile, and the resulting trade areas turn out to bethose shown in Figure 35. The market area, and hence share of the established store, hasdeclined dramatically, from 41% to 28% for perishables, and from 37% to near zero fornonperishables. "Safeway" has been taken completely off-guard.If this argument is reasonable, then we should expect to see substantially greater pricedifferences between RCS and the supermarkets, than between supermarkets. A western+ IGA ^ A SafewayEDMONTON GROCERY PRICES, 1992IGA and Safeway relative to RCS YIY1 4 1V1 1 13 1 16 1 17 122Vi ltMtteijett111 143 1 45WEEK1.381.981.941.321.31.281261. 2: Grocery Shopping Behaviour^ 94Figure 36: Weekly prices of a food basket of approximately $100 value, at twosupermarkets, relative to the price of the same basket purchased at RCS, the powerretailer, in Edmonton, Alberta, in 1992. (Week 1 = Jan 6)Canadian supermarket that is similar to Safeway is IGA. Both stores predate the entry ofRCS. A weekly survey of grocery prices in Edmonton, Alberta (Satanove, unpublished)shows that RCS is indeed dramatically less expensive than either Safeway or IGA. Eachweek, a "shopping basket" (different each week), of approximately $100 value was comparedat the stores. In 1989, IGA was 16% more expensive than RCS on average, and Safewaywas 19% more expensive. In 1992, IGA was 18% higher, and Safeway was, again, 19%higher. Figure 36 shows the relative value of the IGA and Safeway baskets compared to theRCS basket for each week in 1992. Although data is not available on prices before the entryChapter 2: Grocery Shopping Behaviour^ 95of RCS, if the relative prices between IGA and Safeway were similar before the entry ofRCS, we have a scenario very similar to the one described above.Chapter 2: Grocery Shopping Behaviour^ 962.9 Contributions and Research OpportunitiesThis research addresses the problem of consumer response to power retailers in thecontext of the grocery industry. While power retailing has received much attention in thepopular media, it has as yet received little attention from marketing academics, presumablybecause of the recency of the phenomenon. This research contributes to the retailingliterature by showing how dramatic and unexpected changes may occur in consumershopping behaviour when a power retailer enters the market.To capture the unique characteristics of the retail food industry, three diverse researchstreams are drawn from: the promotions literature, to assist in understanding consumerstockpiling behaviour; the inventory management literature, to understand the structure ofjoint replenishment problems; and the multipurpose shopping behaviour, to structure the costminimization problem. The model developed here contributes indirectly to each of theseliteratures. It shows how multi-store shopping, so well documented in the promotionsliterature, may arise from planned behaviour of rational consumers making a purelydeterministic choice. It provides another example of a joint-replenishment problem that hasperiodic optimal policy. The model also extends the multipurpose shopping models byincluding goods differentiated by perishability and stores differentiated by prices. It showsthat with these characteristics, multi-store shopping doesn't require that the stores selldifferent goods, or have temporary price reductions.There are at least four general directions that future research could take. First andforemost, there is the need for empirical validation of the model. While actual calibrationwould be ideal, it would also be a major project. The model, however, makes predictionsthat could be much more easily tested. On the consumer side, this includes the relationChapter 2: Grocery Shopping Behaviour 97between trip patterns to the different stores, distances travelled, and types of goodspurchased. From the stores' perspectives, it suggests the power retailer should have a largershare of nonperishables than perishables.A second direction is strategic. What are the theoretical implications for equilibriumprices and locations? From the large literature on spatial competition (Hotelling, 1929; dePalma et. al., 1985; Vandenbosch and Weinberg, 1992), it is known that increasedseparation on spatial dimensions (either geographic space, or product attribute space)decreases price competition. In this model, consumers travel further for nonperishables thanperishables, which suggests that price competition should be fiercer for nonperishables.However, the fact that each store carries both goods may create interactions that affect thisconclusion. Other strategic issues include how perishables and nonperishables might betreated differently, and how advertising might augment this treatment.A third direction is model extensions. Additional variables, such as advertising,service, or product quality may be considered. The dynamic effects of short termpromotions could be incorporated. This could lead to consumer uncertainty about prices,and require a stochastic approach. Consumers could also be uncertain about their ownsensitivities to the various costs in the model; or, perhaps more interestingly from amanagerial point of view, these sensitivites might be susceptible to manipulation; in fact,one of Safeway's responses to RCS has been to emphasize convenience of location in theiradvertising, presumably to increase the consumer's sensitivity to travel costs. For any modelextension, an overriding issue is the robustness of the increasing returns effect.An interesting issue is the structure of the trip costs. In the numerical section, tripcosts were assumed to be only travel costs. However, it has long been acknowledged thatChapter 2: Grocery Shopping Behaviour 98there is a cost associated with being in a store. Baumol and Ide (1957) suggest that thelarger a store becomes, the more time consuming, and hence costly, shopping becomes.Ingene and Ghosh (1990) explicitly state that their trip costs include a fixed store-specificcost. This is intuitively reasonable, and of interest to management because, once locationis set, the only possible short term control over trip costs is the in-store costs. The RCS,for example, is not only large, but often has long check-out lines. Is it worth their while toincrease the number of checkouts? One step they have taken is to introduce expresscheckouts, but that only assists the small-purchase consumer. The effect of making the tripcost to the large store greater by a fixed amount for each customer would be interesting toknow. It would certainly reduce the large store's exclusive area, but would it have a greatereffect on the mixed-trip region, or the small store's exclusive area?The final issue relates to the dynamic and unpredictable nature of retailing. In thischapter, a specific example is given of how consumers following a relatively simpleoptimizing model can dramatically alter their shopping behaviour in response to a relativelysmall change in the price structure of the market, and that while this change may seemperfectly sensible in hindsight, it would be asking much of managers to predict it before thefact. There are many rapid changes in competitive environments, store formats, consumerneeds, and regulatory environments that are even less predictable. While this is true of anyindustry in a market economy, retailing is perhaps more dynamic and uncertain than most.Nonetheless, even if management is unable to exercise much foresight in this environment,it can be very clever in its response. Retailers typically engage in experimentation andimitation of successful strategies, and respond quickly to adverse situations. The questionis, then, if static equilibria are unlikely to occur, are there other patterns that may arise?Chapter 2: Grocery Shopping Behaviour 99Many authors have long felt that there are patterns in retailing, and have sought to describethem; however, they tend to not be empirically verifiable. The issue of dynamic patternsarising from many spatially competing agents with limited foresight, but the ability to reacteffectively to adversity, is addressed in the next chapter.100CHAPTER THREESPATIAL COMPETITIONAND SELF ORGANIZED CRITICALITY3.1 IntroductionThe preceding chapter demonstrated how retail market share can exhibit increasing returnsto price reductions when the price differential between competitors becomes large enough.This effect is offered as an explanation for incumbents being surprised by the success of anew retailing format in the particular case of food retailing in western Canada. In thischapter, I consider the implications of unexpected adversity more generally for long runindustry structure.The model of competitive dynamics is again inspired by the Canadian food retailingindustry: while the attack of the superstores may have been unexpectedly successful,incumbents have generally not been exterminated. Rather, they are fighting back, and withsome success. To quote a recent (April, 1993) Financial Times of Canada article,Safeway, too, has been struggling with declining market share--although itstill holds the top spot in the West with nearly 24% of the region's sales in1992. Sales throughout the company were down an unhealthy 2.7% last year--and 3.7% in the fourth quarter. Safeway, in fact, used these numbers toconvince its Alberta workers to accept wage rollbacks in March as part of itseffort to try to match the warehousers' costs of doing business. But thecompany is not scrimping on store remodelling. It's halfway through a five-year, $3.2-billion (U.S.) capital investment plan-- double that spent during theprevious cycle.Supermarkets have responded to the entry of power retailers in a wide variety ofways: cost cutting, dramatically increasing service, adding high-margin delicatessens, andintroducing their own superstores. Safeway's television advertising now promotes theChapter Three: Spatial Competition and Self Organized Criticality^101locational convenience of their outlets (the message is that you can walk to Safeway, incontrast to the Superstores), in a clear attempt to increase customer's sensitivity to travelcosts. As the Financial Times article concludes,Little wonder, then, that Price Co.'s [a "superstore" company] earningsdipped last year for the first time. Or that investors have sent Price Co. andCostco share prices tumbling in the past year. Even so, Loblaws, Loeb andthe rest have no room for complacency. Loblaws' Gilles Potvin [a storemanager], for instance, survived the first wave of the warehouse invasion byscrambling astutely [emphasis added] to put his store on a sound footing.He'll survive the next wave because he's discovered the warehousers can't beall things to all people.The need to "scramble astutely" is not confined to supermarket managers faced withnew competitive forms. Corstjens and Doyle introduce a recent (1989) Marketing Sciencearticle as follows:A central facet of modern retailing management is repositioning--adapting thebusiness to a changing retail environment. A retailer's existing positioningbase is continually being eroded by maturing markets and aggressivecompetitors seeking opportunities for profit and growth. Often therepositioning required is small and gradual...Sometimes, however, therepositioning has to be more radical--a switch into new types of stores, achange into major new merchandise areas or a total re-presentation of thestores.This notion of continual erosion and repositioning is consistent with historical views ofretailing as a very dynamic, continually changing industry. McNair's(1958) "wheel ofretailing", and associated notions like "the accordion of retailing" (Mason and Mayer, 1981),suggest that retailers cycle through various formats.The objectives of this chapter are:1) to develop a model of competitive retail markets that recognizes theinterdependency among the outlets, and captures the micro-level behaviour ofChapter Three: Spatial Competition and Self Organized Criticality^102smart management in the face of unexpected adversity ("scrambling astutely");and2) to show that such markets reach a steady state, known as self-organizedcriticality (SOC), and to thereby introduce a novel equilibrium concept to themarketing literature.The model described in the first objective has the following main features. Manyfirms located in a two dimensional plane compete for market share through a probabilisticshare attraction, or spatial interaction, model. In each period, customers allocate theirexpenditures to firms according to distance and intrinsic attractiveness of the firms. Eachperiod, a randomly selected firm experiences an exogenous shock in the form of a decrementto its market share. Firms monitor their own revenues, and react when they fall below athreshold level. The reaction is "astute" in the sense that it increases revenues, at theexpense of the competition. The model is implemented numerically, and a series ofcomputer experiments conducted to investigate dynamic behaviour.The self-organized critical state of the second objective is discussed in detail in thenext section. The remainder of the chapter is organized as follows. The relevantcompetition literature is reviewed and related to the model components in section three.Section four develops the model, and section five describes the results of numericalexperiments. Results are summarized in section six, and the final section identifiescontributions and research opportunities.Chapter Three: Spatial Competition and Self Organized Criticality^1033.2 Self-Organized CriticalityThe self-organized critical state has been introduced to the economics literature by Bak,Chen, Scheinkman, and Woodford (BCSW) (1992). It has its origins in several independentdevelopments in such diverse fields as theoretical biology (Kauffman and Johnson, 1992),solid state physics (Bak, Tang, and Weisenfeld, 1988) and computer simulations of artificiallife (Langton, 1989,1992), although the term originated with Bak et. al. (1988).BCSW (1992) consider a model of production and inventory dynamics for an artificialeconomy with a large number of firms. The highly stylized model consists of a twodimensional network of producers on a cylinder, each of whom buy supplies from two oftheir neighbours at a higher level and sell goods to two other neighbours at a lower level.At one end of the cylinder, final goods are demanded randomly, from the last row ofproducers, by consumers. This demand creates a flow of goods from progressively higherlevels in the supply network. The system converges to a state known as self-organizedcriticality (SOC). SOC will be discussed in more detail shortly, but first consider BCSW'smain point: the law of large numbers does not apply in this situation'. The central limittheorem states that the sum of independent random variables with finite mean and varianceconverges to a normal distribution, and that the variance of the distribution of the averageof the independent random variables approaches zero as the number of random variablesincreases; roughly speaking, in the large number limit, aggregated independent shocks shouldtend to cancel out. In BCSW's SOC state, however, in the limit of a large number of firms,the (appropriately scaled) aggregate response (production) to the independent exogenous'See Judge, Griffiths, Hill, Lutkepohl, and Lee, (1985), page 156 for a disscusion of variouscentral limit theorems and assumptions.Chapter Three: Spatial Competition and Self Organized Criticality 104shocks (consumer demand) does not converge to a distribution with zero variance, but to aPareto-Levy distribution with nonzero variance. In other words, for large but finiteeconomies, the probability of large shocks decreases according to a power law distribution;that is, much more slowly than the exponential decrease that would be expected if the centrallimit theorem applied.' To quote the introduction to the BCSW article,Explaining the observed instability of economic aggregates is a long-standingpuzzle for economic theory. A number of possible reasons for variation inthe pace of production are easily given, such as stochastic variation in thetiming of households' desired consumption of produced goods, or stochasticvariation in the costs of production. But it is hard to see why there should belarge variations in those factors that are synchronized across the entireeconomy--why most households should want to consume less at exactly thesame time, or why most firms should find it an especially opportune momentto produce at the same time. Instead it seems more likely to suppose thatvariations in demand or in production costs in different parts of the economyshould be largely independent. Thus, one might ask, should one not expectthese local variations to cancel out, for the most part , in their effects on theaggregate economy, due to the law of large numbers? Fluctuations in activityof macroeconomic significance, it might be thought, should occur only whenmany independent shocks happen by coincidence to have the same sign, andthis should be an extremely unlikely event (with the probability of occurrencedecreasing exponentially with the square of the size of the event, by thecentral limit theorem).2 For a discussion of Pareto-Levy, or "scaling", or "stable" distributions, and an empiricalexample of Pareto-Levy fluctuations in the price of cotton see Mandelbrot (1982) pp 338-340.It should be noted that the evidence for stable infinite-variance distributions in many financialand commodity markets ( Mandelbrot, 1963 a and b, 1966, 1967; Fama 1963, 1965, 1970) hasbeen challenged. Blattberg and Gonedes (1974) suggest that Fama's 1965 data could be fit bya finite-variance t-distribution. Others (e.g., Westerfield 1977) have shown that if time isredefined so that the analysis is based on prices per transaction, rather than per calendar time,the distribution is normal.The generalized central limit theorem and the resulting stable distributions are discussedin Levy, (1925,1954).Chapter Three: Spatial Competition and Self Organized Criticality 105There are a number of ways in which the law of large numbers can be made to fail 3 , andmore ways still that aggregate instability can arise. BCSW offer SOC as one way in whichthe law of large numbers may fail.In fields other than economics, SOC has been shown to arise in an amazing varietyof very different models and circumstances. While it would be possible to describe SOC inthe context of BCSW's model, I will use the prototype "sandpile" model introduced by Bak,Tang, and Weisenfeld (1988) in statistical physics, which is mathematically isomorphic tothe subsequent economic model of BCSW. This prototype is also the simplest model thatdisplays SOC, and has a strong and concrete intuition associated with it that gives itsubstantial face validity, and also makes it easier to describe and understand. For moredetail in relatively painless prose, the reader is referred to Bak and Chen's (1991) ScientificAmerican article.Consider a flat tabletop on which grains of sand are individually dropped.Eventually, the sand will pile up and start falling off the edge. At some point, the pile willreach a maximum height, with constant slope in all directions to the edge of the table. Atthis point in time, the addition of a grain of sand may have no effect, or it may trigger asmall avalanche of sand, or, occasionally, it may trigger a large avalanche, on the order ofthe size of the whole sandpile. The distribution of avalanches follows a power law: theprobability of an avalanche involving N grains of sand occurring is proportional to 1\1— ,3 These include Shleifer's (1986) innovations, to be discussed in the next section; see Jovanic(1987) for more discussion and examples.'For example, periodicities or deterministic chaos arising from low dimensional nonlinearequations involving relations between macro variables; see Frank and Stengos (1988); Boldrinand Woodford (1990). A little known marketing example is the Bass diffusion equation, whichis theoretically capable of producing chaotic behaviour.Chapter Three: Spatial Competition and Self Organized Criticality 106where a is a constant. This state is arrived at whether the grains of sand are dropped atrandom locations, or at a single location. It can also be arrived at from the other direction--by putting barriers around the edge of the table, filling the resulting box with sand, and thenremoving the barriers, allowing the pile to relax to its natural slope. The resulting state ofthe sandpile will be "critical".The notion of criticality comes from condensed matter physics, which has described"critical states" in a variety of circumstances, usually associated with phase transitions'. Inthe subcritical state, local disturbances have only a weak effect on neighbouring parts of thesystem, and die out in a finite distance. Correlations between different parts of the systemapproach zero exponentially as the distance between the parts increases. By varying a"tuning parameter" such as temperature, however, the fluctuations can be made to propagatefurther. At a certain critical value of the tuning parameter, a state is reached wheredisturbances can just barely propagate to infinity. At this point, correlations no longer falloff exponentially, but with a power law. As the tuning parameter is changed further, a"phase transition" typically occurs, a new structure forms, and correlations again fall offexponentially. Disturbances, or shocks, can propagate to infinity only at the critical point'.'For example, from a solid to liquid phase, or from a magnetized to an unmagnetized phase.The "phases" in the sandpile are a stable phase when the slope is below the critical value, anda turbulent high energy dissipation phase, as the sand everywhere flows downwards, when theslope is above the critical value.'The classic example is spontaneous magnetization of a ferromagnetic material in thepresence of a magnetic field as the temperature drops below the Curie point. On either side ofthe Curie point, the external field has no macroscopic effect. However, at the Curie point,macroscopic fluctuations of the aggregate magnetization are possible, and the weak field candetermine the final aggregate magnetization. This effect was key in establishing plate tectonics,by "freezing" the orientation of the earth's magnetic field, in cooling minerals, at various timesin geologic history.Chapter Three: Spatial Competition and Self Organized Criticality^107The critical state is a region where spontaneous macroscopic instability may occur. BCSWstate,The problem with this as a model of spontaneous macroeconomic instabilityis that, traditionally, critical states were thought to be associated with certain"critical" parameter values (such as temperature), that would almost certainlynot occur in any existing system unless they were "tuned" to be at the criticalvalue in a laboratory experiment. (Jovanovic's (1987) examples of economicmodels in which independent sectoral shocks produce aggregate fluctuationsno matter how large the number of sectors...are special in exactly thissense). But [the sandpile model provides a mechanism whereby] largeinteractive dynamical systems can "self-organize" into a critical state. Thatis, the critical state can actually be an attractor for the dynamical system,toward which the system naturally evolves, and to which it returns after beingperturbed by some large external shock.The power law distribution of the sizes of disturbances, or "avalanches", is mixed news fora competitor living in a SOC state. The bad news is that a competitor may be affected bya shock happening anywhere in the system. The good news is that the probability of beingaffected by the shock decreases as the distance from the shock increases. This "mixed news"has substantial intuitive appeal.There is another interesting way in which the state is "critical": in terms of howsensitive the evolution of the system is to small changes in initial conditions (that is, "small"relative to the range of the system variables normally observed in the system over a longperiod of time). If, at a fixed point in time, we take two realizations of any dynamic systemthat are separated by a "small" distance in state space, and watch their evolution over time,three general kinds of behaviour are possible. The two systems may converge or divergein state space, or they may remain at a constant separation, either absolutely or on theaverage. The steady state achieved in the model developed here is in the constant-separationcategory. A constant separation in a nonlinear system seems surprising--it appears to be aChapter Three: Spatial Competition and Self Organized Criticality^108knife-edge effect. In fact, since divergent systems are commonly referred to as chaotic',the SOC state is often said to be "on the edge of chaos"'.Another feature of this complex extended system is that the degrees of freedomremain high, even after the self-organized state is reached. As BCSW point out, "This isin contrast to a macroscopic description of economics in terms of a few global variables,where micro economic fluctuations are assumed to average out in the final analysis".SOC models have proliferated at an amazing rate since introduced by Bak, Tang andWeisenfeld in 1988. Perhaps the most successful empirical application has been as anexplanation of the Gutenberg-Richter power law distribution of the magnitude of earthquakes.(Sornette and Sornette, 1989) Another class of models in theoretical biology, which havean entirely different structure, also show SOC (Kauffman and Johnson, 1992). Thebiological models will not be discussed here, as the model developed in this research is moreakin to the models of Bak et. al. It should be noted, however, that the biological modelshave substantial promise for economic applications because of their explicitly optimizing'This is the "butterfly effect" due to Lorenz (1969), and the rational behind empiricalmethods to detect chaos. See Grassberger and Procaccia(1983); Sugihara and May (1990). Thebutterfly effect states that the difference between a butterfly fluttering its wings or not in Beijingcan make the difference between whether or not a thunderstorm occurs in New York a monthlater. Erickson (1993) provides a theoretical example of the possibility of chaotic behaviour ina marketing context: closed-loop Nash equilibrium strategies of duopolistic competitors in aLanchester advertising model may be chaotic. While this is a micro-level effect, Erickson'sconcluding statement is relevant: "Chaos theory can be useful in marketing, because marketresponses to marketing activities are dynamic and nonlinear. It is also the case that rarely aremarkets observed to be in steady state, or in a state of repeating cycles, so that exclusivelyempirical approaches are not likely to be sufficient in the study of dynamical behavior inmarketing settings."'For an accessible historical/biographical introduction, see Complexity: The EmergingScience at the Edge of Order and Chaos, by M. Mitchell Waldrop (1992).Chapter Three: Spatial Competition and Self Organized Criticality^109nature. Routledge(1993), in a finance context, has shown behaviour similar to Kauffman andJohnson's in a model of many agents playing a repeated prisoner's dilemma game.In summary, the appeal of SOC is, first that it provides a mechanism for generatingaggregate fluctuations from independent random shocks that do not have vanishing variancein the large limit; in particular, these fluctuations follow power-law (or Pareto-Levy)distributions, which have been shown to occur in economic time series (Mandelbrot, 1982).Second, the self-organized state is generally robust to model details. This means that it isnot necessary to have very special conditions. Third, it has been shown to arise in verydifferent contexts. It remains unclear what criteria are necessary for SOC to arise, but onecommon feature seems to be that systems have many degrees of freedom, with dynamicallyinteracting elements: not unlike the many agents in an economy.In conclusion, the research presented in this chapter can be considered in the contextof Moorthy's (1993) article on the role of theoretical modelling in marketing research.Moorthy persuasively suggests that this type of research can be considered "logicalexperimentation"; progress occurs by a series of "treatments", consisting of substantivelydifferent model assumptions, "very likely [made] by different researchers", which eventuallybuild a picture of which assumptions are responsible for which outcomes. The research inthis chapter may be considered, first, as a second "treatment" of economic SOC. Theadvance here is that, unlike BCSW's model, the model is not just an economicreinterpretation of an existing model from physics. Rather, it is drawn directly frommarketing models of spatial competition, with a dynamic inspired by the apparent micro-dynamics in retailing. On a second level, this research may be considered as a novel"treatment" of micro-level decision dynamics in the context of spatial competition inChapter Three: Spatial Competition and Self Organized Criticality^110retailing. The resulting macro-level behaviour is unusual, but contains much of the spirit ofverbal descriptions of the volatile and cyclic nature of retailing, as in the wheel of retailing.Chapter Three: Spatial Competition and Self Organized Criticality^1113.3 Literature ReviewEliashberg and Chatterjee (1985) provide a classification framework and review, from amarketing perspective, of analytical models of competition. They describe their view of amarketing perspective as follows:For marketing scholars and managers, the principal focus is on the conductof the competing firms in the market, recognizing that the activities of onefirm affect the performance of other firms in the market. Thus, marketers areprimarily interested in competitive models based on an oligopolistic marketstructure, where the interdependence among the competing firms is recognizedexplicitly.This section relates the model in this chapter to the relevant literature on theoreticalmodels of competition and to Eliashberg and Chatterjee's classification. Table IV givestheir scheme, and positions the model of this chapter according to those categories.The first classification is the objective of the model: basic understanding versusdecision-oriented. "Models directed at basic understanding study the industry as a wholerather than one specific company. The industry analysis involves such questions as, 'Undervarious scenarios, what is the nature of the dynamic evolution of the industry... '". Themodel developed in this research is solidly in this category. In the concluding section of thesame paper, Eliashberg and Chatterjee address the issue of "informational, motivational, andbehavioral assumptions". They state, "From the perspective of basic understanding ofcompetitive markets (i.e., from a "detached" viewpoint), we need a descriptive model basedon how competitors believe the others act and how the competitors actually act (moreprecisely, the modeller's "best judgement of how they act)." While I don't propose that thebehaviour as modelled here captures the entire dynamics of real decision processes in spatialcompetition, the introduction to this chapter suggests that "scrambling astutely" is anChapter Three: Spatial Competition and Self Organized Criticality^112Table 4: Model Characteristics in Chatterjee and Eliashberg's (1985) Classification.I. ProblemII. Objective of model: basic understandingvs decision-orientedIII. Basic AssumptionsIII.1 Demand CharacteristicsNumber of segments and the natureof their interrelationshipFactors affecting primary demandFactors affecting market shareUncertainty111.2 Supply CharacteristicsNumber of productsProduct differentiationBathers to entryCost structureUncertainty111.3 Competitive Activity andDecision Making ProcessNumber of competitorsDecision variablesCompetitive behavioral modeDecision makers' objectivesDecision makers' attitudes to riskUncertaintyIV. Mode of AnalysisLevel of aggregationStatic vs dynamicEquilibrium conditionsAnalytic vs simulationV. Basic ResultsEffect of limited information andforesight in adverse environmentsBasic understanding1Intrinsic attractivenessIntrinsic attractivenessRandom exogenous shocksN/A (store choice)ExplicitNot consideredlinear f(attraction)Not consideredMany (64 - 100)Intrinsic attractivenessNoncooperativeSatisfice: maintain profitsNot consideredNot consideredIndividualDynamicDerived SOCSimulationSystem evolves to SOCimportant component. Just as game theoretic models with full information and perfectChapter Three: Spatial Competition and Self Organized Criticality 113foresight can provide substantial insight into the nature of spatial competition, in spite ofbeing an idealization, so should this model provide insight by examining a neglectedcomponent of dynamic competitive processes.The remainder of this section is divided into two parts. The first discusses the marketcontext, namely probabilistic customer choice in spatial competition. The second deals withthe decision environment and competitive dynamics.Spatial Competition: Structure of Supply and DemandThe model assumptions regarding supply and demand are in the context of spatialcompetition. Hotelling (1929) is generally credited with the original work on spatialcompetition, in his analysis of two firms competing for market share in a linear market.This work has been expanded in many directions. Much of it assumes demand arises froma deterministic customer choice process, and is not directly relevant to the research in thischapter, which uses probabilistic choice. A notable exception is dePalma, Ginsburgh,Papageorgiou, and Thisse (1985), where a logit model is used to account for unknownheterogeneity in consumer tastes. The probabilistic model removes discontinuities thatplague Hotelling first-choice models, and has equilibrium configurations where firmsminimally differentiate, as originally proposed by Hotelling. Probabilistic customer choiceis intuitively appealing; however, it is analytically more difficult. Choi, DeSarbo, andHarker (1990;1992) also use a logit choice model in a product-attribute space, and use anumerical solution method to find simultaneous product-price Nash equilibria for manyfirms. The model is demonstrated with residential telecommunications equipment. Thesemodels are generally concerned with strategic issues of location, pricing, and other marketingChapter Three: Spatial Competition and Self Organized Criticality^114mix variables. In contrast, the model developed here is concerned with the dynamicevolution of the industry.A second stream of spatial competition literature, which is more relevant to thisresearch, is the location literature in marketing, geography, and regional science. This isgenerally a much more empirically oriented literature. A major branch is concerned withoptimally locating retail outlets. An early version of this problem was addressed by Reilly(1931), who used a "gravitational" model to determine intermetropolitan trading areaboundaries. Customers are assumed to make a deterministic choice based on a tradeoffbetween the size of the centre and their distance from it. Christaller's (1933) Central Placestheory places primary importance on distance to the centre.Unlike the theoretical spatial competition literature, the spatial location literatureadopted probabilistic choice early. Huff (1962) is generally credited with the originalformulation, although the probabilistic content is Luce's choice axiom (Luce, 1959). A richempirical modelling literature, known as location-allocation modelling, has followed thegeneral Huff formulation. Customers trade off travel related costs against intrinsic storeattractiveness, which may be a function of any number of marketing mix variables. Thetradeoff function gives a utility or attractiveness for each store. However, rather thanallocating all of their expenditures deterministically to the winner of the tradeoff, it isallocated probabilistically according to share of attraction. The allocation is also ofteninterpreted in the aggregate, rather than strictly probabilistically, so that each customeractually allocates shares of expenditure according to share of attraction. The relationbetween this work and the model here is examined in more detail in the model developmentsection. See Ghosh and McLafferty (1987) for a review of location-allocation models.Chapter Three: Spatial Competition and Self Organized Criticality 115The likely reason that probabilistic customer choice is usually assumed in thesemodels is because of their empirical and decision-focused nature, which must captureuncertain heterogeneity in customers. A primary justification for using first choice intheoretical work, even though empirical work uses probabilistic choice, is tractability. If thetheoretical analysis is numerical, rather than analytic, tractability is less of a concern.Consequently, in the model developed in this chapter, the more realistic probabilistic choicemodel is used.It is most commonly assumed in models of spatial competition that the industry-wideprimary demand is fixed, and that firms compete for market share. This is also the morecommon assumption in the representative set of models reviewed by Eliashberg andChatterjee (1985), except when the marketing mix variable of concern is advertising. In thatcase, it is unreasonable to assume inelastic total demand (e.g., Erickson,1985).In the probabilistic spatial competition models, one way to make primary demandelastic is to include an option for the customer that is outside the market. Choi, DeSarbo,and Harker (1990, 1992) use this device, and refer to it as either a generic choice, or a "nopurchase" option. Customers still have a fixed total expenditure to allocate each period, butsome of it may not go to any of the competitors. In this way, the total expenditures withinthe industry varies with the total industry attractiveness, relative to the total externalattractiveness. The model in this chapter also uses this device, and thus has elasticity of totaldemand. However, the main reason here for the external option is to capture dynamicadversity, which will be discussed shortly.Demand is most often expressed as a function of some quite specific decisionvariables, such as price, advertising, or location. In the model developed here, demand isChapter Three: Spatial Competition and Self Organized Criticality 116affected (through the attraction function) by store location relative to the customers, and thefirm's intrinsic attractiveness. Because we want to capture the ability of firms to makeinnovative changes--that is, discontinuous and qualitatively different changes--the intrinsicattractiveness itself is the variable under control of management. It is not specified as afunction of more specific marketing mix elements.The industry structure consists of a fixed number of firms (typically 64 to 100) atfixed locations9 . Eliashberg and Chatterjee (1985), in their review of competiton models,consider three categories:1. Deterministic models addressing competition among incumbent firms.2. Deterministic models addressing competitive entry issues.3. Models addressing competitive decision making under uncertainty.In this scheme,the model is partially in category 1--deterministic models addressingcompetition among incumbent firms. The model might be considered in category three--models addressing competitive decision making under uncertainty--because of the exogenousrandom shocks. However, the decision process itself does not take uncertainty intoconsideration, because the decision makers do not have explicit expectations. The dynamicsand competitive activities are discussed in the next section.A feature which occasionally appears in both the theoretical economic modellingliterature (Carpenter, 1989) and the applied location-allocation modelling literature (Ghoshand Craig, 1991) is "reservation distance". Analogous to reservation price, this representsa distance beyond which customers will not travel to patronize the firm. In many situations,9As the number of firms is increased, the statistics improve but the strain on computerresources also increases. At 64 firms, the main results could be demonstrated; At 100 firms,resource limitations began to be severe.Chapter Three: Spatial Competition and Self Organized Criticality 117this has substantial appeal. For example, it would not seem reasonable for a customer toallocate some portion of his dry cleaning, however small, to every dry cleaner in a city.Aside from that appeal to intuition, a major reason for introducing a distance cutoff is fortractability in analytic models, and to reduce the pressure on computing resources innumerical models. The latter is the dominant reason for the cutoff in my model.Competitive Dynamics As briefly described in the introduction, the approach in this model is to assume a rule-based decision dynamic, and allow the system to evolve, rather than assume a particularequilibrium concept. This is not a common approach to competitive dynamics, but neitheris it highly unusual. The dynamics may be divided into four main features. First, thedecision maker operates in a low information environment; second, the heuristic involvessatisficing behaviour; third, the challenged firm has the ability to react with a successfulinnovation; and fourth, the entire system is driven by exogenous shocks in the form ofindirect outside competition hitting individual firms randomly. First, rule-based heuristicsgenerally will be reviewed, and then these four specific features.The game-theoretic equilibrium approach has well-known advantages andshortcomings, which have been addressed eloquently by many, including Gould and Sen(1984) and Kreps (1990). Regarding the Nash equilibrium, Gould and Sen state that "thedefinition is essentially ad hoc in the sense that it is not endogenously motivated by themodel itself. Among other things, this means that in a comparative static model themechanism which assures equilibrium is unspecified". Gould and Sen conclude thatChapter Three: Spatial Competition and Self Organized Criticality 118It is not yet clear whether the ambiguity will be resolved empirically ortheoretically (most likely a combination of both), but it is clear that aresolution of these issues is important to marketing and economics. Indeedone hopes that some of the questions marketers are working on will be themeans by which more is learned about this topic.Kreps (1990, p405) introduces the Nash equilibrium [author's emphasis] answer to the question: If there is an obvious wayto play the game, what properties must that "solution" possess? ... But thisis a very weak question, and it is clear that having the answer "Nashequilibrium" is pretty thin gruel if what we are after is a way to solve games.All we have is a test of solutions derived by some other means."The dynamic modelling approach which uses only recursive decision rules, in theform of sensible heuristics, directly deals with this problem of "how do we get where we'regoing". Examples are Baumol and Quandt (1964); Day (1967); Day and Tinney (1968); andCohen and Axelrod (1984). These models, however, usually involve some explicit form ofadaptation or learning behaviour. An example where learning is via Bayesian updating isEliashberg (1981), who distinguishes circumstances where the competition may evolvecyclically, or it may converge to the game-theoretic static equilibrium. In contrast, myconcern is not with the explicit modelling of the adaptation process; rather, the model simplyassumes that firms can adapt in response to adversity.Much of the work on decision rules is designed to show how heuristics can arrive atstates that are close, or identical to, the game-theoretic equilibria implied by optimalstrategies of fully informed managers with perfect foresight. A recent marketing exampleis Jeck (1991). He examines three heuristic price setting strategies, in a competitive channelcontext involving two manufacturers and two retailers. Jeck's work is also quite relevantto my model because of the "low information environment" assumed. The information isChapter Three: Spatial Competition and Self Organized Criticality^119low in the sense that each decision maker observes only his own cost, his own decision, andhis own quantity demanded. Two of Jeck's decision rules lead to states that closelyapproximate a Bertrand-Nash equilibrium. The third, which involves more "memory" thanthe other two, leads (approximately) to a collusive equilibrium.Turning now to low information environments, Jeck contends that, in spite of beingunder-researched, low information environments are important:There is considerable evidence that managers have great difficulty deducingthe form and parameters of the consumer demand curve even under fullinformation about the elements of the marketing mix of all competitors. It istrue that the firm can employ very sophisticated statistical analysis to developestimates for the firm's demand function, but only a few such analyses havebeen performed. Perhaps more importantly, perfect information aboutcompetitors' actions even when it is available often is not taken into accountwhen decisions are made. For example The Marketing WorkbenchLaboratory at Duke University has found that store by store reports of prices,which can be obtained by the decision makers, have not been used by manyfirms even though it is felt that many consumer purchase decisions are basedon available stimuli at the point of purchase (Russo, 1977; Aaker and Ford,1983). The decisions arrived at by many managers arise in decision makingenvironments that do not match those used to derive high information closed-form equilibrium descriptions of markets. Given managers' apparentpredilection to make decisions in environments that do not match those usedto derive Bertrand-Nash equilibrium, it would appear beneficial to study the"equilibrium" conditions for markets when the decision makers are ill-informed about their competitors and their reactions.My model has this low information condition, in that decision makers observe, and respondto, only their own revenue levels.In the quote in the previous section, Corstjens and Doyle characterize as a "centralfacet" of retailing the need to respond to "continual erosion" of the retailer's position. TheFinancial Times of Canada article, in reference to the retail grocery industry, speaks of"scrambling". To capture this phenomenon, the model assumes that the players in theChapter Three: Spatial Competition and Self Organized Criticality 120market are under continual external pressure, from both exogenous shocks and endogenouscompetition. Furthermore, decision makers react when their revenues fall below a fixedthreshold. This is a "satisficing" (Simon, 1965) type of behaviour, rather than optimizing.An example of recursive decision rules which involve satisficing can be found in Day(1967). In Day's model, however, the decision maker keeps attempting to improve his lotuntil the incremental improvement (in profit) falls below some satisfactory level. As this"satisficing level" is set smaller, the outcome approaches optimal "marginal costs equalsmarginal revenues" solution. In my model, the satisficing level is a fixed revenue level,rather than a differential level; in the turbulent environment, the retailer simply scramblesto keep his head above the threshold. This dynamic is novel, and perhaps unpalatable, inthat it precludes any possibility of producing outcomes that could be described as optimal.It must be borne in mind, however, that description, rather than prescription, is the objectivehere.The third feature of the model is that the decision maker can make a good decision,once prodded, in that the decision results in improved revenues. In the context of the spatialcompetition model, this means that the store attractiveness can be increased, drawing in morerevenue'. Thus we imagine, rather than a specific setting of some marketing mix variable,a qualitative change in the nature of an innovation. For a food retailer, this might mean achange in advertising strategy (perhaps to increase consumer's sensitivity to travel costs), oran introduction of new high-margin departments, or the extraction of wage concessions from'The implication, of course, is that the revenue more than offsets the cost of implementingthe decision, so that profits increase. The model can be implemented with profits, rather thanrevenues, and an example is provided in the results section. The issue of costs, however, isoutside the main focus of this work.Chapter Three: Spatial Competition and Self Organized Criticality 121unions followed by price reductions. It might also be an imitation of a successful strategy--in some Western Canadian and U.S. cities, Safeway introduced its own low-cost superstorechain, Food-For-Less. A key assumption here is that there is always some revenue-increasing "innovation" available to the firm. It simply requires pressure on revenues forthe innovation to be implemented. So the "innovations" could be referred to not only as"imitations" but as "implementations" (Shleifer, 1986). In the interest of verbal economy,and in keeping with common marketing terminology, however, the decision will be referredto as an innovation. The possibility that innovations are available to firms, but notimplemented, has been addressed by Judd (1985) and Shleifer (1986). In both articles,"innovation cycles" (implementation cycles, in Shleifer's case) occur. However, the cyclesare driven by almost opposite mechanisms. In Judd's model, the introduction of too manynew products within a short time causes competition for consumer resources, and reducesprofits for each one. Subsequent imitation puts even more price pressure on the products.Thus, after a period of innovation, entrepreneurs hold off on introducing new products untilintroduction is again profitable. Shleifer's model, on the other hand, is driven byexpectations of large profits. Entrepreneurs would like to release their inventions when theycan get the most profits, namely when the economy is booming. But a boom is driven byinvestment in the release of innovations. Hence, if firms share beliefs about the timing ofa boom, they can make the boom a reality. Foresight leads to cyclical equilibria.As in the above articles, a central issue of my research is to explore how innovativereactions sweep across a market. Both the driving mechanisms and the nature of theresponse, however, are entirely different. My driving mechanism, to be contrasted with theabove mechanisms, is innovative response by firms to unexpected adversity in an industryChapter Three: Spatial Competition and Self Organized Criticality^122that competes through spatial interaction. My response is waves, or avalanches, that occuron all scales, and with no periodicity.My model shares one important characteristic with Shleifer's. To quote, " Thepossibility of a cyclical equilibrium sheds doubt on a frequently articulated view that amarket economy smooths exogenous shocks. Inventions here can be interpreted as shockshitting the economy, which are essentially identical each period. But these shocks can be"saved" ...The economy follows a cyclical path when a smoother path is available (p. 1165)."In my model, the exogenous shocks come in the form of a decrease in revenues for anindividual firm, modelled as the appearance of competition outside the industry. Theresponse of a firm, by innovating, can then become an endogenous shock for it's nearestcompetitors. The resulting SOC state is characterized by responses that have a power lawdistribution; that is, the distributions' tails fall off as a power law.This leads to the fourth feature: the nature of the shocks. In a share attraction model,the total market demand may be constant, allowing the modeller to focus on the distributionof share among firms". Alternately, the customers may be given a choice that is outsidethe industry, in the form of an additional term in the denominator of the share expression.This device is used by Choi, DeSarbo, and Harker (1990) to introduce price elasticity intotheir logit model of spatial competition. Choi et. al. refer to this as a "no purchase option"and use it to preclude the possibility of an equilibrium with infinite prices and infiniteprofits. In my model, the additional term is used to model the effect of "continual erosion",as described by Corstjens and Doyle, by incrementing the term each period at a randomly"See Bell, Keeney, and Little (1975) for an axiomatic development of the share attractionmodel and a discussion of its features and limitations.Chapter Three: Spatial Competition and Self Organized Criticality^123chosen store. Although this "erosive shock" enters the model as another option for thecustomer, and hence may be easily thought of as either "no purchase", or as indirectcompetition, its main purpose is to create revenue erosion.Chapter Three: Self-Organized Criticality^ 1243.4 Model DevelopmentIn this section, the components of the model and numerical analysis are introduced. Twoimportant points regarding the research strategy guiding model development will first bebriefly summarized.The first point is that the focus is the long-run behaviour of a model industrycharacterized by features of the retailing industry that have been often commented on: it isdynamic, intensely competitive, and innovative. The model assumes that successful,innovative responses to adversity will be made. These may arise from some evolutionaryor learning process, as for example Eliashberg's (1981) Bayesian learning; the details of theorigin of the successful response, however, is not the issue being addressed here. Nor is theprecise nature of the innovation. It is simply assumed that an effective response can bemade. Modelling at this level is necessary to capture the long-run behaviour, because theretail industry changes dramatically and qualitatively in the long run. It is similar in spiritto Shleifer (1986), who also examines innovations.The second point relates to the outcome state, self-organized criticality. Marketinghas a long tradition of borrowing from other fields, and this research is no exception. Inborrowing concepts, however, there is always an issue of appropriateness. Unlike BCSW(1992), the approach here was not to find marketing labels to place on a model fromstatistical physics. Rather, the issue was whether the SOC state could arise in an establishedmarketing context. In particular, the model for the interactions between the elements is thespatial interaction model which has been used in many empirical settings, and which involvessubstantially more complex computations that the sandpile and related models.Chapter Three: Self-Organized Criticality^ 1253.4.1 Customer ChoiceThe geography and retail location literature use a probabilistic customer choice mechanism(eg., Huff 1962), where the market allocates its expenditures to stores according to relativeattractiveness. The economic literature in the Hotelling tradition, in contrast, almostexclusively uses a first choice model, where the entire expenditure is allocated to the mostattractive alternative. One exception is dePalma, Ginsburgh, Papageorgiou, and Thisse,(1985), where a logit model is used to account for unknown heterogeneity in consumertastes. The probabilistic model removes discontinuities that plague Hotelling first-choicemodels, and is intuitively appealing; however, it is analytically more difficult. Because themajority of this work is simulation, and to maintain contact with the retail location literature,this model uses probabilistic customer choice.In general, the utility of store j to customer i is assumed to take the formK^LU = TT [f (A. M ak TT [g (D)]A^k jk^AL 1 ulk=1 1=1The utility--often referred to as the attraction in share attraction models (Cooper andNakanishi, 1988)--is a function of K characteristics A i, intrinsic to store j; the ak are usuallyassumed positive, so that utility increases with increasing Ajk. Ghosh and McLafferty(1987) state, "The [intrinsic] attractiveness of a store results from a number of factors,including its size (which is often a surrogate for breadth and assortment of goods carried),its relative prices, and consumer perceptions of quality of merchandise and service" (p.63).Utility is also a function of L travel cost components, D 11 , usually measured as distance or(80)^Chapter Three: Self-Organized Criticality^ 126travel time between the it' customer and the j th store; the B l are assumed negative to capturedisutility for travel. If the time-honoured assumption is made that either aggregate marketshares or individual choice probabilities (Luce's choice axiom) are proportional to the shareof utility (or attraction). The j th store's share of jth customer's purchases is given byu„^M -^, E UmhIn this relation, total demand is inelastic. Elasticity is introduced by including anextra term, KJ , in the denominator, representing a generic choice, or a no-purchase option.(See, for example, Choi, DeSarbo, and Harker,1990). This operates like indirectcompetition, in the sense of being outside the market. Further, this term is store specific,so that each store may have a different level of "indirect competition" to deal with. It isthrough this term that adversity in the environment is modelled. The customer's allocationthen takes the formviiim., -( E Uih ) + KJh(82)In the marketing retail location literature, the functions fk and gi are almost alwaysthe identity function, giving the multiplicative competitive interaction (MCI) model. Earlyversions were limited to K = L = 1, with size as the surrogate for intrinsic attractiveness,and Euclidean distance for D 11 . If the absolute value of beta is much greater than alpha, themodel approaches "nearest centre" models, as in Christaller's (1933) Central Places(81)Chapter Three: Self-Organized Criticality^ 127formulation. In Reilly's 1931 "Law of Retail Gravitation", which focuses onintermetropolitan trading area boundaries, alpha is one and beta is -2. Huff (1962) assumesalpha is one, and estimates beta to be 2.1 to 3.7 for various types of outlets in Los Angeles.Later authors consider more attributes, such as sales area, number of checkout counters, andwhether credit cards are taken for the Ak (Jain and Mahajan, 1979); and auto travel time,transit travel time, and travel cost per unit income for the Dig (Weisbrod, Parcells, and Kern,1984). Typical parameter values (for a and B) are between one and two, with extremes of0.1 to 3.7. Fixing B = 0 gives a more usual MCI market share model, as, for example, inHansen and Weinberg's (1979) analysis of retail banking outlets.Exponential fk and g i result in the multinomial logit (MNL) share model. An examplein the economics theoretical-equilibrium literature is dePalma, Ginsburgh, Papageorgiou, andThisse (1985), who consider the intrinsic attractiveness to have two components, productvaluation and price; and D i; to be the linear distance on a Hotelling "beach". The associatedexponents are a l = 1/s, a 2 = -1/14, and B = -c/A, where is a population heterogeneityparameter and c is travel costs.The above discussion suggests that the minimal requirement for capturing spatialcompetition is one "intrinsic attractiveness" parameter and one distance parameter. Forexpositional intuition, the attractiveness parameter will be referred to as "size", as suggestedby Ghosh and McLafferty (1987), although it may be equally well thought of as any numberof other quantities (such as product valuation minus price). Because the focus here is ongeographic space, distance will be taken as the usual Euclidean distance on geographiccoordinates.Chapter Three: Self-Organized Criticality 128There is relatively little reason to prefer either the identity or the exponential formsfor the functions f and g. The greater analytic tractability of the exponential form inoptimization and equilibrium analysis is not relevant for this research, which uses simulation.The marketing retail location literature commonly uses the identity, and the dynamicgeographic literature uses the identity for f and the exponential for g, the distance function.For this research, f and g are taken to be the identity function, to be consistent with themarketing retail location literature.As modelled so far, each customer considers all possible retail outlets, which is notonly unrealistic and ignores the notion of consideration sets, but is computationallycumbersome. In a product space context, Carpenter (1989) introduces a "reservationdistance" to limit the customer's consideration set. Similarly, Ghosh and Craig (1991) usea reservation distance in a location model for franchises. Applied to the retail locationproblem, from the firm's point of view, the reservation distance determines the outlet'strading area.In the basic spatial interaction model, then, customer i allocates a proportion mi; ofhis expenditures to store j in each period:RR >(83)where R is the reservation distance, and^Chapter Three: Self-Organized Criticality^ 129a^0,B 5 Firm RevenuesIn each period, each firm's revenues are the sum of attracted customer's shares ofexpenditures:^R1 =^ (84)3.4.3 Market ConfigurationCustomers are uniformly distributed on a bounded plane. For the simulation the plane isrectangular, with customer-origin points in a regular grid. It is slightly more intuitive tothink of the origin points as city blocks, rather than individual customers, in this setup.Stores are located in a coarser grid in the market--for example, every third block in onedirection and every fourth block in the other. In each period, each customer-origin spendsone unit ( e.g., dollar), allocating the unit to all the stores within the customer-origin'sreservation distance according the share of attraction.3.4.4 DynamicsAt time t = 0, the system is initialized by assigning store sizes and unique advantagerandomly to all the stores in the plane. This ensures that the results do not depend on a0 0 0 0 0 0 0 0 00a0 0130 0g30° TeadeWed0 0 0 0 0 00 0 0 0Chapter Three: Self-Organized Criticality^ 130Market Configuration: Small circles represent customer origin points, squares are stores,and the trading area of one store is the area inside the large circle.uniform distribution of store sizes. Revenues are then calculated for each store. A revenuethreshold is initialized at some fraction of this initial revenue. The actual value of thefraction doesn't matter--it is only set different from unity in order to investigate the transientbehaviour of the system. Because of the random initial store sizes, each store starts withdifferent revenues, and will have a different threshold. Again, this guards against resultsarising from uniformity in the model.Stores are shocked by the addition of an increment bk, which remains constant overstores and time, to K. Stores innovate, when their revenues drop below their individualChapter Three: Self-Organized Criticality^ 131revenue thresholds, by the addition of an increment OS, fixed over stores and time, to S i .After initialization, the following algorithm is implemented.ALGORITHM: SCRAMBLING ASTUTELY1. Shock a store chosen at random.2. Calculate revenues of all stores.3. If all stores have revenues above their threshold, increment time andreturn to 1.4. All stores whose revenues have dropped below their threshold innovate.5. Increment time and return to SoftwareThe details of the software used to carry out the numerical experiments described in the nextsection is now briefly discussed.Each experiment consists of a run of the simulation routine with a pariticular set ofparameters. The parameters for each run are set in a parameter file, which is read by thesimulation routine. These parameters describe the market configuration, the size anddistance exponents a and (3, the initial store sizes, the reservation distance, the number ofiterations, and the size of the exogenous shocks and innovation responses. Parameters whichvary across stores are store location, external environment K i and sizes Si . The latter two,of course, also vary with time. All other parameters are held constant for each run, but canbe changed from one run to the next.Chapter Three: Self-Organized Criticality 132Once the parameters are set, the simulation is run. The routine reads in theparameters and initializes the simulated market. If one is not interested in the transientbehaviour, there is a method for getting the system to the steady state more quickly, whichis described in the next section. The system is allowed to run until a fixed number of shockshave been delivered, determined in the input parameter file. Typically, 64 stores are used,and the simulation is run for 1000 to 2000 periods. The size of each innovation response,or avalanche, is measured as the number of stores which innovate after each shock. Thisnumber is recorded, and a histogram of response sizes is output at the end of the run. Aswell, matrices of the revenue level and innovation level are recorded for each innovationcycle. This data can then be input to an animation routine, which is used to observe thedynamics of the system as it evolves.All software is written in C. The code for the simulation may be found in AppendixB. Simulations were run on both a 486 PC, and an IBM RS/6000 560 mainframe.Animations were run on 386 and 486 PC's with VGA.Chapter Three: Spatial Competition and Self-Organized Criticality^1333.5 Results of Numerical ExperimentsIn this section, the characteristics of the model are examined. First, the transient behaviouris examined in terms of total system revenues to demonstrate convergence to a fixed value.Table 5: Parameters Investigated in Sensitivity TestsParameter^Meaning Rangea^size exponent 0.5 - 3.0(3 distance exponent 0.5 - 3.0R reservation distance 2.9 - 6.5CommentsDetermines customer sensitivity to "size"or "intrinsic attractiveness" of store.Empirical estimates of a are in this range.Determines customer sesntivity todistance. A larger value means storeattractiveness drops more rapidly withdistance, implying weaker competitiveinteractions between stores. Range isrepresentative of published estimates.Maximum distance customers will travelto a store. Units are related to "customerorigin points", which are separated by 1unit of distance, and store locations,which are separated by 3 units. R alsois the radius of a store's trading area Like/3, R affects the strength of competitiveinteraction. At the low value (2.9), nostore is within any other's trading area,and each has a monopoly in a small area(of radius 0.1). If R dropped to 1.5, thesystem would be entirely decoupled, andeach store would be a monopolist. WhenR is 6.5, the trade area is 5 stores indiameter, which approaches the size of the8 x 8 system.Chapter Three: Spatial Competition and Self-Organized Criticality 134Then the size distribution of avalanches in the steady state is examined, and found to followa power law. The robustness of the power law distribution to model changes is theninvestigated. Table 5 gives the parameters and ranges that were investigated. Finally, thesensitivity of the system to initial conditions is examined by introducing a small perturbationat a single point in time, and then tracking the subsequent evolution in state space.In most of this chapter, the concern is with the behaviour of the system as timeapproaches infinity, that is, the steady state of the system. Since the system is being driveat all times by shocks delivered to stores chosen at random, the steady state is stochastic.This means that some care must be taken to ensure that any initial transients have decayedbefore starting to count responses, as it is not apparent simply by examining the responsesof the sytem in terms of the innovation avalanches. The issue of transient behaviour isexamined in the next subsection.3.5.1 Transient BehaviourThe nature of the environment and the decision process ensure that each firm's revenues willeventually be close to its threshold level, in particular within a distance determined by thesize of the exogenous shocks and the size of the innovative response. To demonstrate thatthe system will actually converge on this general region, the threshold level is set away fromthe initialization level, and the total revenues in the system monitored over time. In thefollowing, the threshold is set at 80% of the initial revenues. The parameters used in thisrun are shown in Table 6.Chapter Three: Spatial Competition and Self-Organized Criticality^135Table 6: Parameter values for transient test.3, 3^STORE SEPARATION IN x & y. These are the number of customerorigin points between each store in the x and y directions.23, 23^MARKET GRID SIZE. The number of customer origin points in themarket in x an y directions. This gives a store grid of 8 x 8, or 64stores.1, 1^EXPONENTS of DIST & SIZE. Alpha and beta in the choice model.30^SEED. Initializes the random number generator.3.0, .0001^SIZE INITIALIZATION. Size = a + b*ran; a, b are the parametersand ran a random number between 1 and 32,767.4.5^RESERVATION DISTANCE. In units of customer origin points.1000^# OF ITERATIONS. Total number of exogenous shocks delivered.0.8 0.2^MINREVENUE and INNOVATION. The threshold level and the sizeof ai .0.05 1.^ADVERSITY. Initial value of KJ ; and size of shock.Figure 37 shows the sum of the revenues of all 64 stores over 1000 iterations. Notethat it does indeed converge, and to 80% of the starting value, as expected by the setting ofthe threshold.The break in the slope around period 150 indicates the point where enough firms havebeen driven below their revenue threshold, and consequently innovate, to noticeably slowthe decline of total industry revenues. Up to that point, only the external shocks have anyimpact on revenues, and so they are continually being driven down. As more and morefirms become involved in innovation, the total revenue levels out at its steady state value.Figure 38 repeats the environment of Figure 37, but now with much larger shocks.The size of the innovation increment is increased from 0.2 to 10.0, and the size of theChapter Three: Spatial Competition and Self-Organized Criticality^136Figure 37: Transient behaviour: convergence of total profits in 64 store system.Parameters as in Table 5.external shock from 1 to 50. This fifty-fold increase in the magnitude of the adjustmentdynamics causes a faster initial approach to the steady state, and has slightly largerfluctuations, particularly noticeable in the transient region. One might expect a moredramatic change in the behaviour of the revenues, when shocks to market share and resultingresponses are fifty time larger. Recall, however, that once stores' revenues have crossed thethreshold once, the numerator in the share allocation function will thereafter increase inresponse to the increase in the denominator, regardless of the size of the shock andinnovation increments. The share allocations and the revenues will remain in the sameneighbourhood as time progresses. Furthermore, the initial store sizes have numerical values 580570560550540530520510500490480470460450TRANSIENT BEHAVIOURMedium Shocks and Innovations0^100^200^300^400^500 600^700^800^900 1000Time IncrementsChapter Three: Spatial Competition and Self-Organized Criticality^137Figure 38: Convergence of system profits with shock and innovation magnitude 50times those in Figure the range of 3 to 6. With innovation increments to store size of 0.2, the initial valueshave an effect for some time; with increments of 10, they very quickly become washed out.In both cases, once a few hundred shocks have been delivered, the magnitudes of the shocksand innovations relative to the attraction functions is the same. To check this, the incrementswere increase by a factor of 500 over the initial values. The results, shown in Figure 39,are essentially identical to Figure 38, confirming the explanation.Chapter Three: Spatial Competition and Self-Organized Criticality^138 580570560550540530520510500490480470460450TRANSIENT BEHAVIOURLarge Shocks and Innovations0^100 200 300 400 500 600 700 800 900 1000Time IncrementsFigure 39: Transient behaviour with increments 500 times those in Figure 37 is the sameas with increments 50 times those in Figure 37, indicating that large shocks andinnovations quickly remove the effect of the initial conditions.3.5.2 Size Distribution of Avalanches of Innovation in the Steady StateA shock to an individual firm's market share through the attraction function reduces thatfirm's revenues. This reduction may or may not drive the firm to react, depending onwhether or not its minimum revenue threshold is crossed. If it doesn't react, another shockis delivered to another random site. If it does react, however, it increases its intrinsicattractiveness, and captures share (and hence revenues) from not only the extrinsic sources(19, but from any competitors with whom it shares customers--that is, any stores that haveoverlapping trading areas with its own trading area. This causes a reduction in revenues ofChapter Three: Spatial Competition and Self-Organized Criticality^139those stores, some of which may be also be driven to respond. In this way, it is possiblefor innovative responses to cascade across the market through overlapping trade areas.Once the system is in steady state, we would like examine its behaviour. To do this,we keep track of the size of the avalanche produced by each shock, and plot the frequencydistribution of avalanche sizes.Two technical details to briefly note have to do with initialization, and the relativesizes of the shocks and innovative responses. To get the system quickly to the steady state,all stores are initialized above their threshold; then shocks are delivered to the entire systemsimultaneously, rather than to individual stores, so as to drive down the revenues quickly.Once the first store is forced to innovate, the shocks are stopped, and all stores allowed toinnovate until all are just above their own threshold, which they would be in the steady state.The random shock algorithm is then implemented, and run for 100 shocks, before startingto record avalanche sizes.The second issue has to do with the relative sizes of the shocks. Since each store isonly allowed to innovate once for each shock, the shocks must not be too large relative tothe innovation. More precisely, the effect on revenue reduction of the shock must be lessthan the effect of revenue increase, on average, of innovations; if not the revenues in thesystem will eventually all go to the external source; revenues in the system will approachzero. If the shocks are relatively small, on the other hand, it will take many shocks to driverevenues down after an innovation. Since we are only interested in counting innovationavalanches, small shocks with no response place extra demands on computer resources. Inpractice, the relative sizes of the shocks are set so that about half of them produce at least76543210DISTRIBUTION OF AVALANCHE SIZES1^2^3^4Log Number of firms responding to shock0 0Chapter Three: Spatial Competition and Self-Organized Criticalityone innovation. This is a compromise between keeping computer runs from beingexpensive, and allowing the system to recover between shocks.140Figure 40: Size Distribution of avalanches in the steady state, for the base case, withapproximately 60% of shocks producing avalanchesSize Distribution--Initial ResultsFigure 40 is a log-log 10 plot of the size distribution of the avalanches for the casegiven in Table 5, except that the innovation size is 1.0; this gives avalanches for 776 of the1300 shocks. The striking feature is the apparent power law behaviour. The probability oflarge responses does not fall off exponentially, but rather with an exponent of about -1.6 (the'Logarithms are to base e, that is, natural.Chapter Three: Spatial Competition and Self-Organized Criticality^141slope of the regression line, superimposed on the plot"). This is the footprint of self-organized criticality.Since the relative sizes of the innovation and shocks are set to conserve computerresources in the remaining tests, we should ensure that the power-law distribution is nothighly sensitive to this adjustment. Figures 41 and 42 show the distribution with theinnovation increment at 0.8 and 3.0 respectively. The change in the size of the innovationincrement relative to the shock increment changes the proportion of shocks that cause some(at least one innovation) effect, from 80% (for 0.8) down to 20% (for 3.0). There is littleeffect on the shape of the distribution, at least in this range. In all that follows, theincrements are set so that about half the shocks produce a response'.The size of the system, of course, places an absolute limit on the maximum size ofthe avalanches. There is a further effect on an avalanche of any size, however. Whenevera propagating cascade of innovations encounters a boundary, it must stop. Had the boundarynot been there, the size of that avalanche may have been larger. For example an avalanchethat started near a boundary and involves only 10 stores may well have involved 20 or 30had it started in the centre. The result is that the distribution of avalanches tends to shiftedtoward the small size of the distribution. This is a well-known "finite-size effect" insimulations of critical phenomena (Bak, Tang, and Weisenfeld, 1988). The problem is ageneric one of trying to infer system behaviour in the large limit, using only a finite system."The regression line and reported slope on this, and subsequent plots, are intended only asreference points to help with interpretation of the data.12man_y of the following plots have a code, such as "4L62", on them. This is an identifierfor cross-referencing to data files and may be ignored.1^2^3^4Log Number of firms responding to shock0 0765O 4U.•^3021DISTRIBUTION OF AVALANCHE SIZESTest of Relative Increment Sizes 4L62080% of shocks cause responseslope = -1.7Test of Relative Increment Sizes 4L6a7620% of shocks cause responea5slope - -1.8O 4al:^30211^2^3^4Log Number of firms responding to shock0 0Chapter Three: Spatial Competition and Self-Organized Criticality^142Figure 41: Avalanche size distribution when shocks are relatively large.DISTRIBUTION OF AVALANCHE SIZESFigure 42: Size distribution when shocks are relatively small.The plots here show finite size effects to varying degrees. It appears as a reductionChapter Three: Spatial Competition and Self-Organized Criticality 143in the frequency of avalanches at the large end. The effect can be seen most dramaticallyin the plots of tests of sensitivity to the distance exponents, and in that section it will bediscussed further.3.5.3 Robustness to model parametersOne of the characteristics of the self-organized critical state is that the power law distributionis relatively insensitive to model details. In this section, the size distribution of theavalanches will be investigated for various values of the size and distance exponents, and thereservation distance.Empirical estimates of the exponents of the spatial interaction model have been madein a variety of contexts. In Huff's (1962) original work, the size exponent a was assumedto be unity, and the distance exponent estimated. In suburban Los Angeles, a beta value of2.6 to 3.7 was found for clothing stores, and 2.1 to 3.2 for furniture stores. Haines, Simon,and Alexis (1972) estimated beta, again assuming alpha fixed, for grocery stores in varioussuburban and inner city neighbourhoods in a U.S. city. They found values between 0.5 and1.8. Jain and Mahajan (1979) estimated a multiattribute model for supermarkets in a "largenortheastern metropolitan area", and found alpha values between .02 and .56 for their fourintrinsic attractiveness attributes (sales area, number of checkout counters, credit cardsaccepted, and intersection location), and a beta value of 0.3, for the distance exponent.The range of exponents estimated empirically is roughly from 0 to 3. Since the casewhere both are equal to one has already been described, the values of 0.5, 2.0 and 3.0 willexamined in the following, first for a (size), and then for 13.8765432100-JO0 1^2^3^4Log Number of firms responding to shockDISTRIBUTION OF AVALANCHE SIZESTest of Size Exponent SensitivityChapter Three: Spatial Competition and Self-Organized Criticality^144Size ExponentFigures 43, 44, and 45 show the avalanche size distribution for values of alpha equal to 0.5,2.0 and 3.0. For the high and low values, we see more deviation from a straight line thanbefore. In both cases, the falloff appears faster than a power law.Figure 43: The distribution appears to fall off faster than a power law when the sizeexponent is 0.5.Consider the dynamics of incrementing the intrinsic attraction and the externalattraction (S ; and KJ respectively) in Equation (4). When alpha is greater than one, the effectof constant increments of innovation on S becomes progressively greater as S grows.However, the effect of the increments to K remain linear. As the system evolves, there isa systematic increase in the effects of the innovation on revenues, relative to the effects ofthe exogenous shocks on revenues. A similar argument (in the opposite direction) applieswhen alpha is less than one. This systematic effect in the dynamics may well keep the3CD00 1^2^3^4Log Number of firms responding to shock0^4210765DISTRIBUTION OF AVALANCHE SIZESTest of Size Exponent SensitivityTest of Size Exponent 4L820DISTRIBUTION OF AVALANCHE SIZES0^1^2^3^4Log Number of firms responding to shock7650^430210Chapter Three: Spatial Competition and Self-Organized Criticality^145Figure 44: Avalanche size distribution when size exponent is 2.0Figure 45: Avalanche size distribution when size exponent is 3.0.system away from the steady state. To eliminate this effect, we would like to keep theChapter Three: Spatial Competition and Self-Organized Criticality 146absolute magnitude of the intrinsic attractiveness, S, relatively constant as the systemevolves. To accomplish this, the magnitude of S can be adjusted downwards across theentire system after each avalanche. It is important, however, that this adjustment doesn'taffect any store's revenues; therefore, the value of K is adjusted as well. The followingadjustment is used. Equation (4) is repeated below for convenience.(6)Once the initialization of the system is complete, calculate the average initial value of S:Nk)0 =^1..d k0^ (7)k=1After each avalanche is complete (step 5 in the "scrambling astutely" algorithm) calculatethe new average value of S:NSt = Eskt1 ‘ k=1Reset each value of S by reducing it by the ratio of the initial to the new means:sly =^—St(8)(9)Finally, reduce each K value by the following amount:Chapter Three: Spatial Competition and Self-Organized Criticality^147SKio = K,,°—St )"(10)With these new values, the next random shock may be delivered, and the process repeated.This adjustment keeps the average value of Sm over the system at its initial value, foreach shock. Furthermore, the customers' expenditure allocations, given by (6), areunaffected by the adjustment given in (9) and (10); hence the revenues of each store remainunaffected, as required. The only effect on revenues is the exogenous shocks and theinnovative response, as before.Figures 46 and 47 show the avalanche size distributions using this adjustmentalgorithm, using the same parameters as in Figures 43 and 45 (the size exponents of 0.5 and3.0). The avalanche size distribution is now much closer to a power law, particularly foralpha = 0.5 (Figure 46), indicating that the deviation was due to the systematic change overtime of the dynamics, as described above, rather than the sensitivity of the steady state tothe magnitude of the exponent. There remains some systematic deviation in the case of alpha= 3.0. It is unclear how much of this is due to finite size effects, and how much is a dueto the value of the exponent. One might well ask if the curvature in Figure 47 could becaptured by an exponential decay of the distribution. Figure 48 shows a log-linear plot ofthe same data, which, if the decay were exponential, would fall on a straight line. It appearsthat, at the smaller avalanche sizes, the distribution is not exponential; however, theincreased variance at the larger sizes, (due to fewer samples) plus finite size effects, makesit difficult to say much about the shape of the distribution at the large limit tail.DISTRIBUTION OF AVALANCHE SIZESTest of Size Exponent 4182N7605C.'0^4211^2^3^4Log Number of firms responding to shock0 0Chapter Three: Spatial Competition and Self-Organized Criticality^1488DISTRIBUTION OF AVALANCHE SIZESTest of Distance Exponent 4L8076C.^50=0- 42u_3 3210 1^2^3^4Log Number of firms responding to shockFigure 46: The adjustment algorithm restores the power law (cf Fig. 43).Figure 47: Distribution with adjustment algorithm for alpha = 3.0 (cf Fig 45).In summary, the SOC state appears quite robust to changes in the value of the size exponentDISTRIBUTION OF AVALANCHE SIZES7Test of Size Exponent 41_826o5 oo a No 3.00 4 ^^i ^ oit 3 ^12)o_J^^0,3 ° 0^^2o^ ^1 ^ ^ o^ ^ ^0 o'o 0060 oo'0 10 20 30 40Number of firms responding to shockChapter Three: Spatial Competition and Self-Organized Criticality^149Figure 48: Log-linear plot of the data in Figure 47. Note the curvature at small sizes.over the range of values that have been reported in the empirical literature. At larger values,around a = 3, the conclusion is not as strong; the SOC state may be breaking down. Thisissue will be discussed again in the section on research opportunities.Distance ExponentThe distance exponent, 0, determines how rapidly the attraction falls off with distance. Itreflects the relative sensitivity of customers to travel costs. In the context of the largesystem, it reflects how strongly the stores are coupled together. As 0 increases, the couplingstrength decreases.Figures 49, 50, and 51 show results for /3 = 0.5, 2.0 and 3.0 respectively, with aheld at one. As in the case of a, this is representative of the range of empiricallydetermined values for the distance exponent. In these diagrams the break from a straight lineis particularly pronounced in Figures 49 and 51, the cases where 0 is 0.5 and 3.0,DISTRIBUTION OF AVALANCHE SIZES1^2^3^4Log Number of firms responding to shock08765C4u_3210Test of Distance Exponent 4171Chapter Three: Spatial Competition and Self-Organized Criticality^1508DISTRIBUTION OF AVALANCHE SIZESTest of Distance Exponent 4L776 — 0.5C 5C slope = -1.32 43 • • •02 0(3031^^^0 'cdi^coo0^1^2 3 4Log Number of firms responding to shockFigure 49: Size distribution, / = 0.5.Figure 50: Size distribution when (3= 2.0.respectively. To emphasize this break, the reference line has been fit only to the first 12DISTRIBUTION OF AVALANCHE SIZES43Test of Distance Exponent 4L728763211^2Log Number ef *me responding lo shod(00Chapter Three: Spatial Competition and Self-Organized Criticality^151Figure 51: Size distribution when = points in the first case, and the first 9 in the second case.At smaller avalanche sizes, where there are many samples of each size and thevariance is low, the distribution follows an extremely precise power law. Since the breakat larger sizes, which is noticeable in all the plots, is pronounced here, the opportunity willbe taken to briefly explore the finite size effect. Figures 52 to 55 repeat Figure 49, withsmaller and larger markets. Figure 52 is the results of the run on 36 stores in a 6 x 6 array;Figure 53 is for 64 stores in an 8 x 8 array; Figure 54 is for 121 stores in an 11 x 11 array;and Figure 55 is 324 stores in an 18 x 18 array. The change in the location of the break canbe seen between Figures 52 and 53. With the smaller array, the deviation from a powerlaw occurs at smaller sizes, consistent with a finite size effect. A similar increase between64 and 121 stores (8 x 8 to 11 x 11) is not noticeable, but with 324 stores in an 18 x 180 1^2^3^4Log Number of firms responding to shock7650m0- 4E321DISTRIBUTION OF AVALANCHE SIZESFinite Size Effect03 ^800 1^2^3^4Log Number of firms responding to shockDISTRIBUTION OF AVALANCHE SIZESFinite Size Effectot■^876543210Chapter Three: Spatial Competition and Self-Organized Criticality^152Figure 52: A small array: The distribution departs from a power law at around 8 firms.Figure 53: With 64 stores, distribution breaks from a power law at around 12 firms.array, (Figure 55) the increase is again apparent.^ ^ ^CIM3^ ^o' ®o1^2^3^4Log Number of firms responding to shock08763 3210a avow ad765›...c()a)zcr4Eu_c_i3210DISTRIBUTION OF AVALANCHE SIZESFinite Size Effect0^1^2^3^4Ln avalanche sizeChapter Three: Spatial Competition and Self-Organized Criticality^153DISTRIBUTION OF AVALANCHE SIZESFinite Size EffectFigure 54: The break in the power law for the large array is also around 12 stores.Figure 55: For an 18 x 18 array, the break is around 16 stores.While the conclusion that the break is a finite size effect is tentative, it remains aChapter Three: Spatial Competition and Self-Organized Criticality^154likely candidate explanation. A complete investigation of finite size effects is beyond thescope of this work. It would involve investigating a wide range of array sizes, includingthose much larger than investigated here. Bak, Tang, and Weisenfeld (1988) demonstratethe effect for their sandpile model, which is a cellular automaton which involves relativelysimple and fast code. The model here is much more complex in its interactions and hencecalculations, and the load placed on computer resources correspondingly greater, particularlyas array sizes become large. The resources necessary to examine large arrays and manyiterations were prohibitive for this model'. Within these limitations, however, the effectappears to be identical to that described by Bak et. al.'For example, Figure 55 required 10 hours on the IBM RS/6000 560.Chapter Three: Spatial Competition and Self-Organized Criticality^155Reservation Distance The reservation distance is a limit on how far the customer is willing to travel. It determinesthe number of stores to which the customer allocates his expenditures. From the stores'point of view, it represents the radius of the trading area. The larger the reservationdistance, the more stores compete directly with each other. Conversely, the smaller thereservation distance, the more monopolistic each store can become. As the reservationdistance decreases, the whole system will eventually decouple.Recall that the store separation is 3 (measured in terms of customer origin units) andthe runs so far have used a reservation distance of 4.5. (This puts eight stores in the tradingarea of every store not on the boundary of the system). Figures 56 and 57 show results withreservation distances of 6.5 (with 20 competitor stores in a trade area), and 2.9. The latteris small enough that, even though stores share customers, no store is actually in any other'strade area.The distribution remains essentially the same for both the larger and smallerreservation distances. For the small reservation distance, the slope of the reference line issteeper; however, as before, one could fit a shallower line to the smaller avalanches.At some point, the reservation distance will become small enough that the system willbecome entirely decoupled. An interesting technical issue (beyond the scope of this research)is the system behaviour as the stores become independent monopolists. Does the SOC statehold as long as there is at least one customer with divided loyalties? It would undoubtedlyrequire very long computer runs to answer this question; and in any case, the rational formonopolists "scrambling astutely" is unclear.Test of Reservation Distance 4L907654_F:210Chapter Three: Spatial Competition and Self-Organized Criticality^156DISTRIBUTION OF AVALANCHE SIZES0^1^2^3^4Log Number of firms responding to shock876543210Figure 56: Distribution of avalanche sizes with a larger reservation distance.DISTRIBUTION OF AVALANCHE SIZESTest of Reservation Distance 41910^1^2^3^4Log Number of firms responding to shockFigure 57: Distribution of sizes with a smaller reservation distance.Chapter Three: Spatial Competition and Self-Organized Criticality^157The results of the tests of sensitivity to parameter values are summarized in Table 7.Table 7: Summary of parameter sensitivity tests.a 8 R.D.Base Case 1 1 4.5Alpha tests 0.5 1 4.52.0 1 4.53.0 1 4.5Beta tests 1 0.5 4.51 2.0 4.51 3.0 4.5R.D. tests 1 1 2.91 1 6.5RESULTSpower law size distributionsimilar to base casesimilar to base caseslight deviation at large sizespronounced finite size effectsimilar to base casedeviation from power law at large sizessteeper slope, deviation from power lawsimilar to base case3.5.4 Robustness to Dynamic StructureHow dependent is the steady state on the particular dynamics used in the preceding? Toinvestigate this question, a variation on the innovation dynamics was implemented. Ratherthan incrementing the size, or intrinsic attraction only, the increment was added to thecomplete attraction function. This represents an innovation that uniformly increases thestore's attractiveness to all its customers. The customer now allocates expendituresaccording toChapter Three: Spatial Competition and Self-Organized Criticality^158a. + VDP.1^if (ak + 574) + K.Dv.. R0^ > Rwhere the increment is now added toThe model structure was changed to introduce a cost linear with S i , and the decisionto innovate based on profits rather than revenues. (Profits are not dependent on ai). Thisis a relatively minor change. Since the store sizes are all initialized at different values, thecosts, profits, and profit thresholds will all be different across the system. The costs andprofit thresholds remain constant with time; revenues and profits themselves change withtime. The shocking dynamic remains the same, as does the response dynamic, except ofcourse for being triggered by the profit threshold and being added toFigure 58 is the usual size distribution for the base case parameters ( ci,( = 1;reservation distance = 4.5 ). The model was also run with a variety of other parameters.The distribution only departs substantially from the power law when the reservation distancebecomes small, that is when the stores are only weakly coupled through competition in themarket. In all other cases, the distribution of avalanche sizes followed a reasonable powerlaw. These results parallel the original model. The steady state is robust to these changesin system structure and dynamics.3 410 2Ln Size07654321Alternate Model DynamicsOo bon ous'Chapter Three: Spatial Competition and Self-Organized Criticality^159AVALANCHE SIZE DISTRIBUTIONFigure 58: Changing the form of the innovation dynamic and the threshold from profitsto revenues does not affect the avalanche size distribution.3.5.5 Sensitivity to Initial ConditionsWe would like to know what happens in this market when the butterfly flaps its wings.When two realizations of the system start out very close to each other in state space, do theyconverge or diverge? To test this, the system was initialized and run for 200 iterations(shocks). At this point, three copies were made of the system. The first was perturbed byreducing the external attraction of one store (19 by an amount equivalent one quarter of oneshock ( the small change). The second had reduction in Is equivalent to one shock (themedium change) and the third had a reduction equivalent to one shock applied to three stores(the large change). The four systems, including the base case, were then allowed to run foranother 500 iterations. The state space examined was the 64 dimensional space of storeSENSITIVITY TO INITIAL CONDITIONSSmall Initial Separation=DChapter Three: Spatial Competition and Self-Organized Criticality^160sizes. The Euclidean distance between each of the three perturbed systems and the originalsystem was calculated each period. The distance is given by64E (s, - sk)2k=1(12)where the prime indicates the perturbed system.The distances are plotted in Figures 59, 60, and 61. The average separation over the500 shocks is 2.3, 7.8, and 21.2 respectively. Note that the average separation increaseswith the initial separation across the three cases.Figure 59: Separation in store size space after one store is perturbed.For the small initial separation (Figure 59), the two systems repeatedly return to thesame point in state space, that is, zero separation in Figure 59. They don't diverge toseparations that are seen in Figures 60 and 61. In order to achieve those larger separations,1SENSITIVITY TO INITIAL CONDITIONSMedium Initial Se • arationChapter Three: Spatial Competition and Self-Organized Criticality^161the initial separation must be greater. This means that, at least by this measure, theaggregate system is not chaotic in the steady state. Another way of saying this is that at anaggregate level, the evolution of the system is predictable within the accuracy of initialmeasurements, and the variance induced by the random shocks.Figure 60: As in Figure 59, with initial perturbation 4 times greater.Neither do the systems show evidence of permanent convergence within the timeframe examined here. Once apart, even if they return to the same point in state space, theywill separate again. If one considers the constant exogenous shock rate and the adjustmentChapter Three: Spatial Competition and Self-Organized Criticality^162mechamism, it is perhaps not surprising that there is no point attractor within the store sizestate space for the system."88CSENSITIVITY TO INITIAL CONDITIONSLarge Initial Separationa 70--V 60-EOcoag5C-4C-30r2C- 110-0 0^ 500Time (number of shocks)Figure 61: Separation in store size space after the larger perturbation in Figure 59 isapplied to 3 stores."One could also examine sensitivity to initial conditions in the store revenues state space.Since an attractor does exist in this state space, the maximum separation would be bounded, andthe issue of divergence/convergence less obvious. This question is left for future research.Chapter Three: Spatial Competition and Self-Organized Criticality^1633.6 Summary of Numerical ExperimentsThe retail interaction model investigated here converges to a stochastic steady state,characterized by power law avalanche distributions, and is quite robust to changes in modelparameters and structure. Changing the customers' sensitivities to travel costs and intrinsicstore attractivenesses over the ranges where these values have been empirically determineddoesn't affect the power law distributions until the sensitivities become high, i.e. theattraction varies with the cube of the size or distance attributes.The steady state is sensitive to systematic changes in the relative sizes of the effectsof the shocks and the innovation responses. When the shocks cause the external attractionto grow linearly, but the size, or internal attractiveness, to grow nonlinearly, the power lawdistribution breaks down. A normalizing adjustment that keeps both external and internalattractions from growing, without affecting revenues for any store, restores the power lawdistribution, for the values of the size exponent tested.The steady state appears insensitive to changes in the reservation distance until itbecomes small enough that firms become monopolistic, and the system decouples.In most of the tests, there appears to be a distinct break in the power law curve inthe large-avalanche limit. It is quite likely that the deviation is due to the finite-size of themarket simulated. A full exploration of this effect has been precluded by the limitation oncomputer resources, however, it appears identical to that described in the literature.The steady state is also robust to a different specification of the dynamic, withinnovation being triggered by profit levels rather than revenues, and the innovative responseadded to the entire attraction, rather than the intrinsic attraction.Chapter Three: Spatial Competition and Self-Organized Criticality 164Finally, a comparison of two systems with initially small separations in the store size(intrinsic attraction) state space indicates that they evolve along approximately parallel paths,neither converging nor diverging.These characteristics are those of "self-organized criticality", a phenomenadescribed, to the best of my knowledge, only once previously in an economic model, butwith many applications in a variety of other fields, including theoretical biology andstatistical physics.Chapter Three: Spatial Competition and Self-Organized Criticality^1653.7 Contributions, Limitations, and Research OpportunitiesContributionsThis chapter's two main contributions are, first, the modelling of a facet of retaildynamics often discussed, but not modelled; and second the demonstration of steady statemodel behaviour known as self-organized criticality in a marketing context.The conventional wisdom that much of competitive retailing involves repositioningin response to continual erosion, or scrambling astutely, is captured by a shock-and-innovation dynamic embedded in a spatial competition model. The resulting stochasticsteady state is characterized by a continually innovating industry, with firms innovating inresponse to the shocks, in waves, or avalanches, of various sizes. These responses areinteresting generally because of the similarity in spirit to long-standing notions that there issome underlying order to the apparent turbulence in retailing, as exemplified by traditionaltheories of wheels and accordions in retailing. More specifically they are interesting becausethe dynamic allows a small shock to have a large response. The response size falls of as apower law, with exponent typically around -1.5, rather than exponentially, as would beexpected from a simple linear aggregate of independent shocks.The state the system evolves to has the characteristics of self-organized criticality(SOC). This state has been described once previously in an economic context by Bak, Chen,Scheinkman, and Woodford (1992). This chapter is the first application in a marketingcontext, and is an advance over BCSW in that it uses established marketing models as thecontext. BCSW, on the other hand, impose an economic interpretation on the prototypesandpile cellular automaton introduced by Bak, Tang, and Weisenfeld (1988).Chapter Three: Spatial Competition and Self-Organized Criticality 166One reason why SOC is potentially important to marketing is because of itsrobustness at several levels. At the lowest level, SOC is the end state over a range ofparameter values investigated in this model. At the next level, model structural details canbe changed without affecting the final state. At the highest level, SOC arises in entirelydifferent fields. This robustness increases the probability that the phenomena may occur ina marketing context. Furthermore, although the necessary requirements for SOC have yetto be spelled out, the models in all the varied fields have in common many dynamicallyinteracting degrees of freedom, a situation that describes many marketing contexts.Limitations to this research, and implications for future work, will be discussed inthree areas: model structure, method of analysis, and empirical evidence.Model StructureThe model structure involves shocks of adversity and effective response; the possibility ofpositive shocks, and ineffective response, have been abstracted away. Closely related toineffective response is the possibility of exit, and, of course entry. These are certainlyfeatures of retailing not captured in this model, and represent possible model variations orextensions. Incorporating such features in a simulation is relatively easy; however, the extralayers of complexity only increase the difficulty of exploring the model and generalizingresults. Furthermore, the gain is likely only incremental.A more important extension is to investigate optimizing behaviour explicitly.Although the model assumes the ability of management to make good decisions, it doesn'taddress the specifics of the decision. This is an important next step for this line of researchto lead to normative results. The models of theoretical biology, which incorporate short-term optimizing, will be helpful. The likely approach is to consider agents who operate onChapter Three: Spatial Competition and Self-Organized Criticality 167the basis short-term profit maximizing heuristics, such as those described by Jeck (1991),and to compare the various steady states that can arise (static Nash equilibrium, cyclic, SOC,etc. ) on the basis of system-wide profits. The hypothesis one would infer from thebiological literature is that the SOC state, with its very dynamic nature, provides the highestsystem profits.Method of AnalysisThe numerical method of analysis, which allows the investigation of complex models, lacksthe generality of analytic methods. Analytic approaches to SOC is an active area of researchin statistical physics, and BCSW borrow from some of this work to demonstrate the Pareto-Levy distribution of avalanche sizes in their producer-suppler network model. The analyticwork to date is in the context of relatively simple discrete cellular-automaton models. It isnot at all clear that it will be applicable to the more complex model described here, and itis therefore likely that numerical approaches will dominate for the foreseeable future.An important limitation of this research is the inability to completely investigate thefinite size effect, which itself is a limitation of the numerical method of analysis. There aretwo possible approaches here. One is simply brute force--let the model run on large arraysfor a long time. The other is to simplify the model, attempting to retain the basic features,but making the computations much less intensive. The former is probably the moredesirable, as the latter still leaves open the question of whether or not the deviations frompower law behaviour in this particular model are actually finite size effects.Empirical IssuesThe final issue is empirical. A crucial problem in detecting the kind of behaviour impliedby this research, whether in the context of retailing or elsewhere, is that of countingChapter Three: Spatial Competition and Self-Organized Criticality 168avalanches. In this, and other SOC models, each shock triggers a single avalanche, whichcomes completely to rest, before another avalanche is triggered. This is unlikely to occurin any real setting. If shocks are occurring more rapidly, avalanches may run into eachother. Even if they don't physically interfere with each other, there is the likelihood thatseveral avalanches in any large system are occurring simultaneously and econometric datasets will only record the aggregate response. The general question of aggregation ofresponses is also an active area of research in statistical physics. It is well known that whenevents, or pulses, of various shapes and sizes in time, are superimposed at random startingtimes, to produce a time series, the resulting series is highly correlated. The correlation ismost often expressed in the frequency domain in terms of the power spectra, which can beshown to be a power law (A. van der Ziel, 1950). The power law behaviour of powerspectra arising from aggregates of the pulses produced by systems in the SOC state has beenan important part of SOC research from the beginning. Exponents between one and twowere suggested originally by Bak, Tang, and Weisenfeld (1988), depending on thedimensionality of the system. Chau and Cheng (1992) suggest that whether the exponent isone or two depends on whether the model is continuous or discrete.'Relating power law power spectra in economic time series to SOC models seems tobe the best hope for empirical support for the model. The difficulties inherent in sorting outthe values of the exponents are compounded by the possibilities of other explanations.15Some initial work on aggregate traces produced by superposition of the pulses generatedby the model in this chapter gives a power spectra of 1/f, that is an exponent of 1. Bycomparison, a time series of monthly Canadian bankruptcy data for retail businesses for 12 yearshas a power spectra at low frequencies that goes as 1/f2 . The hypothesized connection is thatif some fixed proportion of the firms involved in each avalanche are unable to effectivelyinnovate, and exit the system, they will show up in bankruptcy data.Chapter Three: Spatial Competition and Self-Organized Criticality^169Random walks, for example have power spectra that obey power laws, with the simpleGaussian random walk having an exponent of two (Mandelbrot, 1982). This implies that theSOC explanation for power law behaviour will, at least, have to be contrasted with existingexplanations of the origin of random walks.170CHAPTER FOURCONCLUSIONS AND CONTRIBUTIONS,LIMITATIONS, AND FUTURE RESEARCHThis dissertation is concerned with spatial competition in the currently prevailing dynamicretail environment in North America. Chapter 2 shows the possibility of nonlinearincreasing returns to scale to price reductions in grocery retailing, and suggests that theeffect would have been difficult to detect, let alone predict, before the appearance ofsuperstores. Chapter 3 builds on this result, and the conventional wisdom that strugglingwith unexpected adversity is the norm in retailing, to analyze the long-run behaviour of amodel of the industry that captures the elements of successful reaction to unexpectedadversity. Though the chapters are linked, they differ in terms of specifics, and so nooverall conclusions are offered. Instead, the purpose of this chapter is to summarize theresults of the previous two chapters in turn. First, the conclusions and contributions of theresearch are highlighted. This is followed by a discussion of limitations and of relevantfuture research.Chapter Five: Contributions, Limitations and Future Research^ 1714.1 Grocery Shopping Behaviour in the Presence of a Power Retailer4.1.1 Conclusions and contributionsIn Chapter 2, a normative model of individual consumer choice in grocery shopping isdeveloped. The model assumes long-run cost minimizing consumers, who trade offinventory costs, travel costs, and price of goods purchased. Customers are heterogeneousin household location, and make a deterministic choice between two stores. The stores aredifferentiated by location and price; in addition, each store sells two goods, one perishableand one nonperishable. Except for their price, the goods are identical between stores. Thebroad objective of this model structure is to investigate shopping behaviour in the presenceof a power retailer, an area not previously addressed in the literature.Multistore shopping strategiesThe first interesting finding is a new explanation for multi-store shopping byconsumers among direct competitors in the grocery trade. While it is well documented thatloyalty to any one store is low (Uncles and Ehrenberg, 1990; Bucklin and Lattin, 1992), themulti-store shopping strategy is either treated as an exogenous fact, or explained in terms ofresponses to promotions or search for deals. In contrast, the multi-store strategy in Chapter2 is an endogenous result of rational optimizing in the face of long-term expectations aboutaverage store prices. Applied to the context of the entry of extremely price-competitiveretailers, i.e., superstores, to the market, this may be more than a new explanation--it maybe a reason for multi-store shopping that did not exist before the superstores. Thispossibility is supported by the numerical analysis that showed very small "mixed-store"shopping regions until price differences were on the order of 20%.Chapter Five: Contributions, Limitations and Future Research 172Another established reason for mixed store shopping arises from uncertainty in thechoice process. The spatial interaction models originating with Huff (1962), as well as sometheoretical positioning models (dePalma et. al., 1985; Choi et. al. 1990) consider a stochasticchoice process that allocates expenditures probabilistically to more than one firm. Incontrast, the model developed here involves a purely deterministic choice process; and,again, the multistore shopping arises endogenously.The structure and results of the model have a strong parallel with the multipurposeshopping literature (Ghosh and McLafferty, 1987; Ingene and Ghosh, 1990). In both cases,deterministic choice and rational long-term planning in the presence of known parametersleads to the resulting behaviour. However, the emphasis in the multipurpose shoppingliterature is on different categories of retailers, rather than on direct competitors, and howthese categories may agglomerate as the result of efficiencies achieved by multipurposeshopping. The research reported in Chapter 2 extends this literature by considering storesdifferentiated by price, rather than category, and by introducing two goods--perishables andnonperishables--with very different dynamic characteristics.The fact that consumers in the model treat these two goods very differently hasimportant implications, and raises some interesting strategic questions. These questions willbe addressed in the "future research" section.Increasing returnsIt was shown that consideration of the time dimension in spatial competition modelsthrough consumer stockpiling leads to the possibility of high sensitivity of trade areas toprice differences between competitors. Furthermore, this sensitivity may not be apparentwhen price differences are small. In other words, market share may exhibit an "increasingChapter Five: Contributions, Limitations and Future Research 173returns" effect to price reductions. This effect is offered as a candidate explanation for why,according to several independent industry sources, Safeway was surprised by the success ofthe Real Canadian Superstore in western Canada.Periodic policies and the joint replenishment problem (JRPIFrom the theoretical standpoint of the JRP literature in inventory management, thisresearch contributes another example to the limited set of JRP's where periodicreplenishment policies are optimal. A periodic replenishment policy is one where aparticular fixed pattern of replenishment in a finite time interval is repeated indefinitely. Itreduces the infinite horizon case to a finite horizon problem.From this result, a series of propositions further constrains the possible shoppingpatterns to six. Three of these are argued to be either unlikely to occur or minor in impact.Analytic solutions are obtained for the optimal shopping policy for the remaining threepatterns, and a method for determining which of the three will give minimal costs as afunction of household location is developed.The setting is unusual in that it involves consumer stockpiling, rather than firminventory management. From the practical standpoint of inventory management, however,it is not clear that the properties of this setting translate to any firm-based inventorymanagement problem. The insights it provides for consumer shopping behaviour, however,are interesting, as purchase timing issues are increasing in importance.In summary, the model and results of Chapter 2 should be of interest to marketingacademics and practitioners, in that it provides a theoretical basis, rich with implications, forexamining an important current phenomenon in retailing.Chapter Five: Contributions, Limitations and Future Research^ 1744.1.2 LimitationsThe store choice model developed here combines and extends existing models in severalways. However, the results need to viewed in the light of a number limitations, discussedbelow.1. The numerical portion of the analysis has the usual limitation of lack ofgenerality. The results must be considered as special numerical cases, if quiteplausible ones. In particular, even though the analytic choice results are onan individual basis, for the aggregation analysis to determine market shares,customers are assumed to be homogeneous except for their spatial origin, orhousehold location. This is a common assumption in theoretical work inspatial competition, usually made for reasons of tractability. The assumption,however, is obviously not tenable in any real setting. Consumers will varyon parameters such as price and distance sensitivity. The aggregate resultsshould therefore be viewed as the behaviour of a particular segment ofcustomers, who have homogeneous parameters. It would be possible,although cumbersome, to solve the numerical problem for differenthomogeneous segments, and combine the results.2. The model has only two marketing mix variables--the long-run variable oflocation, and the short-run variable of price. While these two variables arethe most obvious determinants of grocery store choice, other mix elements arealso important. For incumbent firms in particular, who may be unable tomatch the power retailers' prices, strategies based on attributes such as qualityand service may be viable options.Chapter Five: Contributions, Limitations and Future Research^ 1753. The analysis is limited to three of the six possible optimal shopping patternsidentified. One pattern not examined, namely shopping for nonperishablesmore frequently than perishables, lacks face validity. The other pattern notinvestigated, however, is more plausible. The possibility of special trips tothe small store for nonperishables at the end of each periodic interval, and itsimpact on market share, remains to be investigated. It is expected, however,that this is likely to be a minor effect.4. In any urban setting there will be many directly competing retail outlets,whereas this model only considers two independent outlets. Increasing thenumber of outlets is well within the framework of the model, although itwould be numerically cumbersome. A more important issue is that much ofretailing is dominated by chains, which qualitatively changes the nature of thestrategic implications. For example, incumbent chain stores with manylocations have the option of varying prices across locations to respond moreeffectively to the entry of a single power retailer.5. The analysis is confined to existing firms. One of the important issues withsuperstore entry is that they can drive some stores out of business. Thisinteracts with the issue of chains, and the possible strategic reponse ofincumbents to superstore entry of closing some outlets and opening new ones.4.1.3 Future ResearchFour directions for future research are identified, some of which arise from the limitationsof the previous section.Chapter Five: Contributions, Limitations and Future Research 1761. Empirical: Several model predictions could be tested fairly easily. Examplesare a) consumers at large distances from superstores make less frequent tripsand purchase larger quantities per trip than closer consumers b) customers atlarge distances from superstores make more fill in trips to supermarkets thancloser consumers c) average purchase quantities at supermarkets are smallerthan at superstores d) average trip frequency to supermarkets is greater thanto superstores e) superstores share of nonperishables is larger than their shareof perishables.2. Strategic: Theoretical research in spatial competition suggests thatequilibrium configurations involve the tradeoff between market sharemaximizing forces, which lead to minimal differentiation and maximum pricecompetition, and a strategic force to reduce price competition by maximaldifferentiation. Since this model results in consumers willing to travel furtherfor perishables than nonperishables, one would hypothesize that pricecompetition should be more intense on nonperishables. The fact that bothstores carry both goods would make this an interesting, if difficult, setting foran equilibrium analysis. Other strategic issues include how perishables andnonperishables might be treated differently by the different stores, and howadvertising might augment this treatment.3. Extensions:  Several of the results here were derived by graphical andnumerical methods, because the model is already complex enough to makeanalytic solutions impractical. Relaxing the requirement of discrete trips infavour of continuous trips would allow taking analytics further, and that stepChapter Five: Contributions, Limitations and Future Research^177may be necessary to incorporate extensions without getting bogged down inhighly specialized numerical analyses.Possible extensions include consideration of additional variables, suchas advertising, service, and product quality. Short term dynamics in the formof promotions are another possibility. This could lead to consumeruncertainty about prices and require a stochastic approach. A stochasticapproach could also be used to address the issue of consumer heterogeneityin sensitivities to various costs, as in dePalma et. al. (1985). The possibilityof customer sensitivities being susceptible to marketing activities is also a veryinteresting avenue of research.Different cost structures are also possible. Perhaps the most obviouschange is to include a fixed in-store cost that is different across stores, in thetrip cost. This captures the effect of the very large stores being more time-consuming to shop in.In all model extensions, an issue to investigate is whether or not theincreasing returns effect holds.4.^Another research direction is that taken in Chapter 3, namely the long termeffect on industry structure of a competitive dynamic involving effectiveresponse to unexpected adversity.Chapter Five: Contributions, Limitations and Future Research^ 1784.2 Spatial Competition and Self-Organized Criticality4.2.1 Conclusions and ContributionsThe academic, trade, and popular literature have often referred to the dynamic nature ofretailing. Phrases such as "scrambling astutely" (Financial Times of Canada. April 3, 1993) and "repositioning in response to continual erosion" (Corstjens and Doyle, 1989) refer tothe fact that much of retail management is a matter of making major creative changes inresponse to adversity. Chapter 3 formalizes this notion as a shock-and-innovation dynamicset in an oligopolistic spatial competition model. To the best of my knowledge, this is thefirst attempt to formally capture this type of dynamic behaviour.The model's steady state has some interesting characteristics. The exogenous shocksdriving the system are delivered at random to firms in the model industry, so it is naturalto look at distributions of responses. The probability of a large response, where the size ismeasured as the number of firms responding to a single shock, declines according to a powerlaw. The model therefore contributes by describing a mechanism where the law of largenumbers does not apply, as one would expect an exponential decline in that case. In otherwords, the likelihood of a system-wide wave of innovation in response to a small shock isrelatively high.This parallels the observation that retailing as whole tends to undergo dramatic shifts,such as described in the December 21, 1992 Business Week article on power retailers. Thefact that a micro-level dynamic motivated by the conventional wisdom that retailers must"scramble astutely," leads to a stochastic steady state involving sweeping changes at themacro-level, which also parallels conventional wisdom, is intuitively appealing.Chapter Five: Contributions, Limitations and Future Research 179The steady state has the characteristics of self-organized criticality, and the researchtherefore contributes by providing, to the best of my knowledge, the first example of SOCin marketing, and the second in economics. Furthermore, SOC arises here from a purelymarketing model, whereas the previous economic example (Bak et. al., 1992) ismathematically isomorphic to the prototype sandpile model of statistical physics, with aneconomic interpretation imposed. The research thus contributes to the SOC literature byproviding an example in marketing, and a purely economic model.SOC is of interest not only because of its rather unusual characteristics, but becauseit is quite robust to parameter values and model details. Furthermore, it appears to be ageneral organizing principle that has been applied to models in many different areas of thephysical and life sciences, as well as economics. While the models vary widely, onecommon feature is the dynamic interaction of entities with many degrees of freedom, acharacteristic of many marketing situations. This, plus the robustness and generality ofSOC, increase the probability of it occurring in marketing situations.4.2.2 LimitationsA number of relevant limitations of Chapter 3 will now be identified.1. Like the majority of work on SOC, this research involves simulation, which,while it allows the investigation of complex systems, limits generality incomparison with analytic approaches. While analytic work on systemsdisplaying SOC is an active area of research, it is based on simple cellularautomaton models. Whether the methods can be applied to the more complexmodel here is an open question.Chapter Five: Contributions, Limitations and Future Research^ 1802. The simulation is based on a finite number of stores, and so the maximumsize an avalanche can attain is limited. Although the strong power lawbehaviour at smaller avalanche sizes and general nature of the deviation atlarger sizes is entirely consistent with the "finite size effect," it remains tobe conclusively demonstrated that the effect is the reason for the deviation.This, again, is a limitation of the numerical approach, combined with thecomplexity of the interactions in this particular model.3. The negative exogenous shock and effective response captures a subset ofpossible behaviours. The question of positive shocks, and/or the possibilityof failure to respond effectively, is not addressed. The latter in particular isimportant, as exit is a common feature in retailing. Whether models thatincorporate these features will still exhibit SOC is not dealt with here.4. The deliberate focus on low information environments, surprise and satisficingignores the more usual approach to modelling competitive activity, wheresome form of optimizing and foresight is generally considered. To the extentthat retailers do more than "scramble astutely," this model is limited.4.2.3 Future ResearchSeveral research possibilities present themselves. Two of the more important directions willbe discussed in this section.^1.^Incorporating positive shocks, entry, and exit adds realism, but increasescomplexity. While it would be interesting to further test the robustness ofSOC on these dimensions, it is believed that it would be more productive toChapter Five: Contributions, Limitations and Future Research 181construct a new model that involves at least short-term optimizing behaviour.This is a potentially interesting direction because of results in theoreticalbiology (Kauffman and Johnson, 1992) that relate SOC to Nash equilibriawith short-term optimizing entities. They show that a SOC state occurs whenNash equilibria are just barely stable, and tenuously extend across a system.The competing species make single period adjustments which increase theircompetitive advantage, or "fitness." Furthermore, average fitness of thesystem is maximal in the SOC state. It would be very interesting toinvestigate circumstances under which industry-wide, or even economy-wideprofits were maximal in an SOC state; and to investigate further the relationbetween Nash equilibria and self-organized criticality.2. The footprint of SOC is power law distributions, and power laws are commonin economics. For example, Pareto-Levy distributions (with power-law tails)which Bak, Chen, Scheinkman and Woodford (1992) have shown to be aconsequence of SOC, have also been identified in various economic series(see Mandelbrot, 1982). SOC also gives rise to power spectra that havepower law dependency on frequency, as do any time series that can bemodelled as a random walk. 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In the mixed regions, since the constraint isbinding, on each trip exactly enough perishable to last to expiry time is purchased. Thus inthe m = 1 region, trips alternate between stores, and the large store has half the share. Inthe m = 2 region, the large store has on third the share; and so on. (This may also beeasily shown formally by substituting the solution for the optimal quantities of the non-perishable purchased at the large store in the binding, mixed case, equation (54), into theexpression for the perishable quantity purchased at the small store, equation (47), andcomparing with constraint (46) when binding).To determine the areas in Figure 28, first express the boundary contours, fromequation (69) and the parameter values given in Table 2, Section 2.7, as x = f(y,m):x(y,m) = ±-8 [ 144 + 72m + 33m 2 + 6m 3 + m 4 + 64y 2(1)102400y 2 ^1/21456 - 72m - 33m 2 - 6m 3 - m 4 -ITo find the area to the right of each of the contours, transform the x coordinate by shiftingthe origin to the right edge of the city and reflecting: x' = 10 - x(y,m). Now the integralAppendix A: Calculation of Market Shares^ 194w.r.t y will give the area between the contour and the right edge of the city. For the m =0 to m = 1 transition, the area to the right isA1= f [10 - x(y,0)]cly-io= [10y - 3y\/91^43' 2401= 154.525- 2 81§1 Arcsinh[—k8 0110-10(2)Similar integrations give the areas to the right of the remaining three contours. From theseareas, and the total area of the city of 400 km2 , the areas of each of the regions can befound, and the areal shares calculated. The large store's exclusive region is 61.37%. Themixed shopping areas, for m = 1, 2, 3, and 4 respectively are 4.12%, 6.97%, 12.18%, and15.33%. The last region has the .03% market share of the small store's exclusive regionremoved. Therefore, the large store's share of the perishables is61.37 + 4 . 12^6.97^12.18^15.33 2^3^4^5^(3)= 71.86%The small store's perishables share is 28.14%.Appendix BC CodeCl. Simulation Routine/* SIM4LB.0 descendent of sim4fb.cSize incremented directly with inoadvSize and ghat normalized after each avalanchefixed mkt radiidescendent of sim4c.c; threshold = minprof*refprofany firm can innovate--no restriction on availability ofinnovationElastic total demand through ghostatt, which increasesuniformly by ghostinc across marketStore advantage added rather than multipliedNash only found once; after, Size remains fixed duringinnovationsThen ghost attraction increases uniformly until avalancheOccursThen ghost attraction (ghat[][]) incremented at random sites(oloop to ninov now counts number of these)Avalanche size distribution recorded in histsimultaneous movessimulation program */#include < stdio.h >#include < string. h >#include < stdlib . h >#include < math. h >int igridx, igridy, inszf, inadvf, intrup, iseed, ninov;double storsepx, storsepy, alpha, beta, gc, gama, ghostatt, ghostinc;double arn,brn,crn,drn, attmin,dsize,damp,nashtol, minprof, inoadv;char c[13][50];char *chat[] = { &c[0][0], &c[1][0], &c[2][0], &c[3][0], &c[4][0],&c[5][0], &c[6][0], &c[7][0], &c[8][0], &c[9][0],&c[10][0], &c[11][0], &c[12][0], &c[13][0] };195Appendix B: C Code^ 196int parfilin( char *parfnm );void sizfilin( char *sizfnm, double storsiz[32][20], int maxnumx, intmaxnumy );void advfilin( char *advfnm, double storadv[32][20], int maxnumx, intmaxnumy );void sizout( int maxnumx, int maxnumy, double siz[32][20] );void revenue( int maxnumx, int maxnumy, double storlocx[32], doublestorlocy[20],double storadv[32][20], double storsiz[32][20], doublerev[32][20],double distmax[32][20], double ghat[32][20] );void profit( int maxnumx, int maxnumy, double storsiz[32][20], doublerev[32][20],double prof[32][20] );/*void tograph(int maxnumx, int maxnumy, double storsiz[32][20],double prof[32][20], int storino[32][20], FILE *bf );*/int innovate(double prof[32][20], double refprof[32][20], doublestorsiz[32][20],int storino[32][20], int maxnumx, int maxnumy );void adjust( double storsiz[32][20], double ghat[32][20],int maxnumx, int maxnumy, double initmsize );main ( int argc, char *argv[] ){FILE *bf, *lfp;double dummy[32][20], storadv[32][20], prof[32][20], xstart,ystart, distmax[32][20];double refprof[32][20], ghat[32][20], initmsize;double storsiz[32][20], storlocx[32], storlocy[20], rev[32][20];int retin, ntol, storino[32][20], oloop, immflag, immmflag;int maxnumx, maxnumy, i, j, k, 1, iloop, aysize, hist[640];char temp;/* printf( "\n PARFILE:^%s\n SIZEFILE:^%s\n ADVANTAGEFILE: %s\n",argv[1], argv[2], argv[3] );printf( "\nGRAPH OUT FILE: %s\n", argv[4] );printf( "\n If this is OK, Hit Any key to continue \n" );printf( " Otherwise, Hit 0 to break out\n" );temp = getcharO;if( temp = = '0' )Appendix B: C Code^ 197goto Break;*/retin = parfilin( argv[1] );if (retin = = 1 ){/* ******CALCULATE SOME STORE PARAMETERS****** */maxnumx = ((igridx + storsepx/2)/storsepx) - 1;maxnumy = ((igridy + storsepy/2)/storsepy) - 1;/*^printf ( "\nmaxnumx = %i maxnumy = %i\n", maxnumx, maxnumy);xstart = ( igridx - ((maxnumx ) * storsepx)) / 2;ystart = ( igridy - ((maxnumy ) * storsepy)) / 2;for(i = 0; i < = maxnumx; i+ +)storlocx[i] = i*storsepx + xstart;for(j = 0; j < = maxnumy; j+ +)storlocy[j] = j*storsepy + ystart;/*^for(i = 0; i < = maxnumx; i+ +)printf( "\nstorlocx[%i] = %f", i , storlocx[i] );printf( "\n" );for(j = 0; j < = maxnumy; j+ +)printf( "\nstorlocy[%i] = %f', j, storlocy[j] );printf( "\n Any key to continue \n" );printf( "0 to break out\n" );temp=getchar();if( temp = = '0' )goto Break;*1/* ************************************************************* *1/* ****ASSIGN INITIAL STORE SIZE AND DIFFERENTIAL ADVANTAGE****if( inszf = = 0 ){srand( (unsigned)iseed );for( i = 0; i < = maxnumx; i+ + ){for( j = 0; j < = maxnumy; j+ + ){storsiz[i][j] = am + brn * (double)rand();Appendix B: C Code^ 198elsesizfilin( argv[2], storsiz, maxnumx, maxnumy );printf( "\n^Initial Store Sizes\n\n" );sizout( maxnumx, maxnumy, storsiz );printf( "\n Any key to continue \n" );printf( "0 means break out\n" );temp = getcharO;if( temp == '0' )goto Break;if( inadvf = = 0 ){srand( (unsigned)(iseed+1) );for( i = 0; i < = maxnumx; i+ + ){for( j = 0; j < = maxnumy; j+ + ){storadv[i][j] = crn + drn*randO;}}}elseadvfilin( argv[3], storadv, maxnumx, maxnumy );/*^printf( "\n^Initial Store Advantages \n\n" );sizout( maxnumx, maxnumy, storadv );*//*^printf( "\n Any key to continue \n" );printf( "0 means Break out\n" );temp = getcharO;if( temp = = '0' )goto Break;*1/* ****INITIALIZE INNOVATION & GHOST ATTRACTION ARRAY****************** *1for( i = 0; i < = maxnumx; i+ + ){ for( j = 0; j < = maxnumy; j + + ){^storino[i][j] = 0;ghat[i][j] = ghostatt;}Appendix B: C Code^ 199}/******* OPEN LOGFILE ********************************/if( (lfp = fopen(argv[5], "at") ) == NULL )printf("\nUnable to open Logfile" );goto Break;/*printf("\n Opening binary graphics file" );* *************** OPEN AND START BINARY GRAPHICS FILE ***if( argv[4] == NULL ){if( (bf = fopen( "simtograph", "w" ) ) == NULL ){printf("\nUnable to open graphics file" );goto Break;}}else{ if( (bf = fopen( argv[4], "w" ) ) = = NULL ){printf("\nUnable to open graphics file" );goto Break;}}printf("\n writing graphics header");fprintf( bf, "%If %lf %i %i %i %i \n",storsepx, storsepy, maxnumx, maxnumy, igridx, igridy );for( i = 0; i < = maxnumx; i++ ){fprintf( bf, "%lf ",storlocx[i] );fprintf( bf, "%lf \n",storlocy[i] );fprintf( bf, "%lf %lf %lf %lf %i %i \n",alpha, beta, gc, gama, inszf, inadvf );fprintf( bf, "%i %lf %lf %lf %lf \n", iseed, am, brn, crn, drn);fprintf( bf, "%lf %lf %lf %i %i \n", attmin, dsize, damp,ninov, intrup );fprintf( bf, "%lf %lf %lf %lf %lf \n",nashtol,minprof,inoadv,ghostatt,ghostinc );*/Appendix B: C Code^ 200/**** INITIALIZE REFERENCE PROFIT ****/revenue( maxnumx, maxnumy, storlocx, storlocy, storadv, storsiz,rev, distmax, ghat );printf("\n\n INITIAL^REVENUE MATRIX " );sizout( maxnumx, maxnumy, rev );profit( maxnumx, maxnumy, storsiz, rev, prof );printg" \ n \ n^PROFIT MATRIX " );sizout( maxnumx, maxnumy, prof );for( i = 0; i < = maxnumx; i++ ){ for( j = 0; j < = maxnumy; j+ + ){refprof[i][j] = prof[i][j];I}temp=getcharO;if( temp = = '0' )goto Break;printf( "\n^Initial Store Sizes\n\n" );sizout( maxnumx, maxnumy, storsiz );printf( "\n Any key to continue \n" );printf( "0 means break out\n" );temp = getchar();if( temp = = '0' )goto Break;/******* SOC SECTION, IHOPE ***************************/immflag = 0;/**^*****Push down all the profits*******/while(immflag = = 0){for( i = 0; i < = maxnumx; i++ ){ for( j = 0; j < = maxnumy; j + + )ghat[i][j] = ghat[i][j] + ghostinc;}printf("\n\n pushing down profits ");Appendix B: C Code^ 201revenue( maxnumx, maxnumy, storlocx, storlocy, storadv, storsiz,rev, distmax, ghat );/* printf(" \n\n^REVENUE MATRIX " );sizout( maxnumx, maxnumy, rev );*/^profit( maxnumx, maxnumy, storsiz, rev, prof );/* printf("\n\n^PROFIT MATRIX " );sizout( maxnumx, maxnumy, prof );tograph( maxnumx, maxnumy, storsiz, prof, storino, bf );*1/**** Check store's profit threshold *//**** If below, they innovate */immflag = innovate( prof, refprof, storsiz, storino, maxnumx,maxnumy );Iprintf("\n\n Profit threshold exceeded \n Avalanche commences");/* ****Let the avalanche go*****/while( immflag != 0 ){revenue( maxnumx, maxnumy, storlocx, storlocy, storadv, storsiz,rev, distmax, ghat);/* printf(" \ n \ n^REVENUE MATRIX " );sizout( maxnumx, maxnumy, rev );*/^profit( maxnumx, maxnumy, storsiz, rev, prof );printf(" \ n \ n^PROFIT MATRIX " );sizout( maxnumx, maxnumy, prof );printf( "\n Any key to continue \n" );printf( "0 means break out\n" );temp = getcharO;if( temp = = '0' )goto Break;/*^tograph( maxnumx, maxnumy, storsiz, prof, storino, bf );*1/**** Check store's profit threshold *//**** If below, they innovate */immflag = innovate( prof, refprof, storsiz, storino, maxnumx,maxnumy );Iprintf("\n\nAvalanche Complete\n Commence Random incrementing");202Appendix B: C Code/** ****Increment ghosts at random sites****//* *** First init'z hist array and mean size */for( i = 0; i < = (maxnumx +1)*(maxnumy +1)-1; i+ + )hist[i] = 0;initmsize = 0;for( i = 0; i < = maxnumx; i++ ){ for( j = 0; j < = maxnumy; j + + )initmsize = initmsize + storsiz[i][j];}initmsize = initmsize/( (maxnumx +1)*(maxnumy + 1) );printf("\n initial average storsize %fln",initmsize );temp = getchar0;if( temp = = '0' )goto Break;/************************** Loop **********************************/for( oloop = 0; oloop < = ninov; oloop++ ){i = (int)( (float)rand()*(float)(maxnumx+1)/(float)32768 );j = (int)( (float)rand0*(float)(maxnumy+1)/(float)32768);ghat[i][j] = ghat[i][j] + ghostinc;immmflag = 1;aysize = 0;while(immmflag != 0){revenue( maxnumx, maxnumy, storlocx, storlocy, storadv,storsiz,rev, distmax, ghat);/*^printf("\n\n^REVENUE MATRIX " );sizout( maxnumx, maxnumy, rev );*1^profit( maxnumx, maxnumy, storsiz, rev, prof );/*^printf("\n\n^PROFIT MATRIX " );sizout( maxnumx, maxnumy, prof );tograph( maxnumx, maxnumy, storsiz, prof,*/^/**** Check store's profit threshold/**** If below, they innovate */immmflag = innovate( prof, refprof, storsiz,maxnumx, maxnumy );/*^printf("\n immmflag on return from innovate:storino, bf );*/storino,%i",immmflag);Appendix B: C Code^ 203*/^aysize = aysize + immmflag;/*^printf("\n aysize on incrementing by immflag: %i", aysize);*/^}hist[aysize] = hist[aysize] + 1;printf("\n aval. size = %i",aysize);printf("\n PRE ADJUSTED REVENUES \n" );sizout( maxnumx, maxnumy, rev );printf( "0 means break out\n" );temp = getcharO;adjust( storsiz,ghat,maxnumx,maxnumy,initmsize );printf("\n POST ADJUSTED REVENUES \n" );sizout( maxnumx, maxnumy, rev );printf( "0 means break out\n" );temp = getchar();if(temp = = '0' )goto Break;for( 1 = 0; 1 < = maxnumy; 1++ ){fprintf(lfp, "\n %i ", 1*(maxnumx+1));for( k = 0; k < = maxnumx; k++ ){fprintf(lfp,"[%i] %i ", 1*(maxnumx+1) +k,hist[1*(maxnumx+ 1) +k]);}}printf("\n\n Probe at location %i %i complete", i, j );printf("\n oloop = %i, ninov = %i\n", oloop, ninov );temp = getcharO;if(temp = = '0' )goto Break;*1Ifprintf( lfp, "\n\n avalanche size histogram" );for( j = 0; j < = maxnumy; j++ ){fprintf(lfp, "\n %i ", j*(maxnumx+1));for( i = 0; i < = maxnumx; i++ ){fprintf(lfp,"[%i] %i ", j*(maxnumx+l) +i, hist[j*(maxnumx +1)Appendix B: C Code^ 204}}}Break:fclose( bf );fclose( lfp );return( 0 );}/* ***function INPUT PARAMETER FILE *** *1int parfilin( char *parfnm ){FILE *pfp;int result;char temp;if( pfp = fopen( parfnm, "r" ) ){fscanf( pfp, "%lf %lf', &storsepx, &storsepy );fgets( chat[0], 50, pfp );fscanf( pfp, "%i %i", &igridx, &igridy );fgets( chat[1], 50, pfp );fscanf( pfp, "%lf %lf', &alpha, &beta );fgets( chat[2], 50, pfp );fscanf( pfp, "%lf %lf', &gc, &gama );fgets( chat[3], 50, pfp );fscanf( pfp, "%i %i", &inszf, &inadvf );fgets( chat[4], 50, pfp );fscanf( pfp, "%i", &iseed );fgets( chat[5], 50, pfp );fscanf( pfp, "%lf %lf', &arn, &brn );fgets( chat[6], 50, pfp );fscanf( pfp, "%lf %if', &crn, &drn );fgets( chat[7], 50, pfp );fscanf( pfp, "%If', &attmin );fgets( chat[8], 50, pfp );fscanf( pfp, "%lf %lf', &dsize, &damp );fgets( chat[9], 50, pfp );fscanf( pfp, "%i %i", &ninov, &intrup );fgets( chat[10], 50, pfp );fscanf( pfp, "%lf %lf %if', &nashtol, &minprof, &inoadv );Appendix B: C Code^ 205fgets( chat[11], 50, pfp );fscanf( pfp, "%If^&ghostatt, &ghostinc );fgets( chat[12], 50, pfp );fclose( pfp );result = 1;return result;}else{perror("Couldnt open parameter file");result = 0;return result;void sizout( int maxnumx, int maxnumy, double siz[32][20] ){int i, j, temp;for( j = 0; j < = maxnumy; j+ + ){printf( "\n%i: ", j );for( i = 0; i < = maxnumx; i+ + ){printf( "%2.4f ", siz[i][j] );}}/* *****function**REVENUE CALCULATION FOR ALL STORES ONGRm************* *1/* Attractions to stores at a distance greater than attmin are set= 0 *//* For each customer grid point, one unit of revenue is apportionedaccording to attractions*/void revenue( int maxnumx, int maxnumy, double storlocx[32], doublestorlocy[20],double storadv[32][20], double storsiz[32][20], doublerev[32][20],double distmax[32][20], double ghat[32][20] ){Appendix B: C Code^ 206double bufatt[32][20];double dist, bufsav, hold;int i, j, k, 1, kpref, 1pref, temp;/* printf("\n\nCalculating Attractions for each point on %i by %igrid\n",igridx,igridy );printf("to each of %i stores,\n",(maxnumx+1)*(maxnumy+1) );printf("and assigning 1 unit of revenue proportionately.\n");printf( "\nThis may take a while so RELAX...\n");temp = getcharO;*/for( k = 0; k < = maxnumx; k+ + ){ for( 1 = 0; 1 < = maxnumy; 1++ ){rev[k][1] = 0;/* hold = storadvv[k][1] * pow( storsiz[k][1], beta ) / attmin;distmax[k][1] = pow( hold, 1/alpha ); */distmax[k][1] = attmin;}}for( i = 0; i < = igridx; i+ + ){ for( j = 0; j < = igridy; j+ + ){bufsav = 0;for( k = 0; k < = maxnumx; k+ + ){ for( 1 = 0; 1 < = maxnumy; 1++ ){dist = sqrt( pow((i - storlocx[k]), 2) + pow((j -storlocy[1]), 2) );if( dist < = attmin ){ bufatt[k][1] = pow(storsiz[k][1],beta )* pow( dist + 1, -alpha );/*^printf("\n\n i,j = %i, %i",i,j);printf("\n stlocx[%i] stlocy[%i] = %f %f' ,k,l,storlocx[k],storlocy[1] );printf("\n\nSBeta: %f tothe %f = %f' ,storsiz[k][1],beta,pow(storsiz[k][1],beta ) );printf("\nD+lAlpha: %f tothe %f = %f", dist + 1, -alpha,pow(dist + 1, -alpha ) );printf("\n attraction = %f', bufatt[k][1] );temp = getchO;*/^bufsav = bufsav + bufatt[k][1];Appendix B: C Code^ 207}elsebufatt[k][1] = 0;}}if( bufsav ! = 0 ){ for( k = 0; k < = maxnumx; k+ + ){ for( 1 = 0; 1 < = maxnumy; 1++ )rev[k][1] = rev[k][1] + bufatt[k][1] / (bufsav +ghat[k] [1]);}}}}}/* ***************PROFIT FUNCTION********************************* *1void profit( int maxnumx, int maxnumy, double storsiz[32][20], doublerev[32][20],double prof[32][20] ){int i, j;for( i = 0; i < = maxnumx; i+ + ){ for(j = 0; j < = maxnumy; j+ + ){prof[i][j] = rev[i][j] - gc*pow( storsiz[i][j], gama );}/* *********INPUT INITIAL SIZE FILE *********************** */void sizfilin( char *sizfnm, double storsiz[32][20], int maxnumx, intmaxnumy ){FILE *sfp;int i, j;if( sfp = fopen( sizfnm, "r" ) ){for( i = 0; i < = maxnumx; i+ + ){ for(j = 0; j < = maxnumy; j+ + ){fscanf( sfp, "%lf', &storsiz[i][j] );Appendix B: C Code^ 208}}fclose( sfp );}elseperror("Could not open inital store size file" );}void advfilin( char *advfnm, double storadv[32][20], int maxnumx, intmaxnumy )FILE *afp;int i, j;if( afp = fopen( advfnm, "r" ) ){for( i = 0; i < = maxnumx; i+ + ){ for(j = 0; j < = maxnumy; j+ + ){fscanf( afp, "%lf", &storadv[i][j] );}fclose( afp );}elseperror("Could not open inital advantage file" );}/* ******** TO BINARY FILE FOR GRAPHICS ********* *1/*void tograph(int maxnumx, int maxnumy, double storsiz[32][20],double prof[32][20], int storino[32][20], FILE *bf ){int i,j;for( i = 0; i < = maxnumx; i++ ){ for( j = 0; j < = maxnumy; j+ + )fprintf( bf, "%lf %lf %i \n", storsiz[i][j], prof[i][j],storino[i][j] );}*/1* **** INNOVATION ************************************* *1Appendix B: C Code^ 209int innovate(double prof[32][20], double refprof[32][20], doublestorsiz[32][20],int storino[32][20], int maxnumx, int maxnumy ){int immflag = 0, i , j;for( i = 0; i < = maxnumx; i+ + ){ for( j = 0; j < = maxnumy; j+ + ){if(prof[i][j] < (minprof*refprof[i][j]) ){storsiz[i][j] = storsiz[i][j] + inoadv;storino[i][j] = storino[i][j] + 1;immflag = immflag +1;/*^printf("\nStore at %i, %i innovates",i,j);}}}return immflag;/* **************** ADJUST Readjustment of dynamic parametersvoid adjust( double storsiz[32][20], double ghat[32][20],int maxnumx, int maxnumy, double initmsize ){int i,j;char temp;double msize = 0;for( i = 0; i < = maxnumx; i+ + ){ for( j = 0; j < = maxnumy; j+ + ){msize = msize + storsiz[i][j];}}msize = msize/( ( maxnumx + 1 )*( maxnumy + 1) );for( i = 0; i < = maxnumx; i+ + ){ for( j = 0; j < = maxnumy; j+ + ){storsiz[i][j] = storsiz[i][j]*(initmsize/msize);ghat[i][j] = ghat[i][j]*(pow( (initmsize/msize), beta) );}***********1}Appendix B: C Code^ 210printf("\n current mean storsize = %f',msize);printf("\n ADJUSTED SIZES \n");sizout( maxnumx, maxnumy, storsiz );printf( "0 means break out\n" );temp = getcharO;printf("\n ADJUSTED GH ATT\n" );sizout( maxnumx, maxnumy, ghat );printf( "0 means break out\n" );temp = getcharO;}C2. Parameter File Set Routine/* SET4D.0 INITIALIZATON program */#include < stdio.h >#include < string.h >#include <stdlib.h>int igridx, igridy, inszf, inadvf, intrup, ninov, iseed;double storsepx, storsepy, alpha, beta, gc, gamma, am, brn, attmin;double dsize, damp, crn, drn, nashtol, minprof, inoadv, ghostatt, ghostinc;char c[13][50];char *chat[] = { &c[0][0], &c[1][0], &c[2][0], &c[3][0], &c[4][0],&c[5][0], &c[6][0], &c[7][0], &c[8][0], &c[9][0],&c[10][0], &c[11][0], &c[12][0], &c[13][0] };int parfilin( char *parfnm );void parfilch( char *parfnm );main ( int argc, char *argv[] ){FILE *bf;int retin;char temp;printf( "\n PARFILE:^%s\n SIZEFILE:^%s\n ADVANTAGE FILE: %s\n",argv[1], argv[2], argv[3] );printf( "\n If this is OK, Hit Any key to continue \n" );printf( " Otherwise, Hit 0 to break out\n" );Appendix B: C Code^ 211retin = parfilin( argv[1] );if (retin = = 1 ){printf(" \n Hit any key to change any of these parameters.\n" );printf(" Hit ENTER for none. \n");if( ( temp = getch() ) != '\r'){parfilch( argv[1] );}temp = getch();}elseperror( "couldn't open parameter file " );}/* ***function INPUT PARAMETER FILE *** */int parfilin( char *parfnm ){FILE *pfp;int result;if( pfp = fopen( parfnm, "r" ) ){fscanf( pfp, "%lf %lf", &storsepx, &storsepy );fgets( chat[0], 50, pfp );fscanf( pfp, "%i %i", &igridx, &igridy );fgets( chat[1], 50, pfp );fscanf( pfp, "%lf %lf', &alpha, &beta );fgets( chat[2], 50, pfp );fscanf( pfp, "%lf %lf', &gc, &gamma );fgets( chat[3], 50, pfp );fscanf( pfp, "%i %i", &inszf, &inadvf );fgets( chat[4], 50, pfp );fscanf( pfp, "%i", &iseed );fgets( chat[5], 50, pfp );fscanf( pfp, "%If %lf", &arn, &brn );fgets( chat[6], 50, pfp );fscanf( pfp, "%lf %lf", &crn, &drn );fgets( chat[7], 50, pfp );fscanf( pfp, "Mr, &attmin );fgets( chat[8], 50, pfp );fscanf( pfp, "%lf %lf", &dsize, &damp );fgets( chat[9], 50, pfp );fscanf( pfp, "%i %i", &ninov, &intrup );Appendix B: C Code^ 212fgets( chat[10], 50, pfp );fscanf( pfp, "%lf %lf %lf', &nashtol, &minprof, &inoadv );fgets( chat[11], 50, pfp );fscanf( pfp, "%lf %lf", &ghostatt, &ghostinc );fgets( chat[12], 50, pfp );printf( "\n\n^INPUT FILE PARAMETERS^COMMENTS \n\n" );printf( "storsepx: %5.1f \t storsepy: %5.1f ", storsepx, storsepy );printf( chat[0] );printf( "igridx: %0.3i \t igridy: %0.3i ", igridx, igridy );printf( chat[1] );printf( "alpha: %5.3f \t beta: %5.3f ", alpha, beta );printf( chat[2] );printf( "gc: %5.3f \t gamma: %5.3f ", gc, gamma );printf( chat[3] );printf( "inszf: \t inadvf: ", inszf, inadvf );printf( chat[4] );printf( "iseed: %i^", iseed );printf( chat[5] );printf( "arn: %5.3f brn: %5.5f', arn, brn );printf( chat[6] );printf( "crn: %5.3f drn: %5.5f", crn, drn );printf( chat[7] );printf( "attmin: %5.3f^", attmin );printf( chat[8] );printf( "dsize: %5.3f \t damp: %5.3f ", dsize, damp );printf( chat[9] );printf( "ninov: \t intrup: ", ninov, intrup );printf( chat[10] );printf( "nashtol: %5.3f \t minprof: %5.3f \t inoadv: %5.3fln ", nashtol,minprof,inoadv);printf( chat[11] );printf( "ghostatt: %5.3f \t ghostinc: %5.3f ", ghostatt, ghostinc );printf( chat[12] );fclose( pfp );result = 1;return result;}else{perror("Couldnt open parameter file");Appendix B: C Code^ 213result = 0;return result;}}/* ******function CHANGE PARAMETER FILE ***** */void parfilch( char *parfnm ){FILE *pfp;int num;do{printf( "\n\nType the number of the parameter from the list below, \n" );printf( "followed by a space and the parameter's new value. \n\n" );printf( "0 MEANS NO MORE CHANGES\n\n" );printf( "1 storsepx: %5.1f \n2 storsepy: %5.1f ", storsepx, storsepy );printf( "\n3 igridx: %0.3i \n4 igridy: %0.3i ", igridx, igridy );printf( "\n5 alpha: %5.3f \n6 beta: %5.3f ", alpha, beta );printf( "\n7 gc: %5.3f \n8 gamma: %5.3f ", gc, gamma );printf( "\n9 inszf: \n10 inadvf: ", inszf, inadvf );printf( "\n11 iseed: %i \n", iseed );printf( "12 arn: %5.3f \n13 brn: %5.7f ", arn, brn );printf( "\n14 crn: %5.3f \n15 drn: %5.7f ", crn, drn );printf( "\n16 attmin: %5.3f\n", attmin );printf( "17 dsize: %5.3f \n18 damp: %5.3f ", dsize, damp );printf( "\n19 ninov: \n20 intrup: ", ninov, intrup );printf( "\n21 nashtol: %5.4f \n22 minprof: %6.3f \n23 inoadv: %5.3f",nashtol,minprof,inoadv );printf( "\n24 ghostatt: %5.3f \n25 ghostinc: %5.3f :", ghostatt, ghostinc );scanf( "%i", &num );switch( num ){case 1:scanf( "%lf", &storsepx );break;case 2:scanf( "%lf', &storsepy );break;case 3:Appendix B: C Code^ 214scanf( "%i", &igridx );break;case 4:scanf( "%i", &igridy );break;case 5:scanf( "%lf", &alpha);break;case 6:scanf( "%If', &beta );break;case 7:scanf( "%lf", &gc );break;case 8:scanf( "%lf', &gamma );break;case 9:scanf( "%i", &inszf );break;case 10:scanf( "%i", &inadvf );break;case 11:scanf( "%i", &iseed);break;case 12:scanf( " %If', &arn );break;case 13:scanf( "%If', &brn );break;case 14:scanf( "%lf", &crn );break;case 15:scanf( "%lf', &drn );break;case 16:scanf( " %If', &attmin );break;case 17:scanf( "%If", &dsize);Appendix B: C Code^ 215break;case 18:scanf( "%lf", &damp );break;case 19:scanf( "%i", &ninov );break;case 20:scanf( "%i", &intrup );break;case 21:scanf( "%lf", &nashtol );break;case 22:scanf( "%lf', &minprof );break;case 23:scanf( "%lf", &inoadv );break;case 24:scanf( "%lf", &ghostatt );break;case 25:scanf( "%lf", &ghostinc );break;default:printf( "\nNo such parameter---execution continues\n" );}}while( num != 0 );if( pfp = fopen( parfnm, "w" ) ){fprintf( pfp, "%f %f%s\n", storsepx, storsepy, chat[0] );fprintf( pfp, "%i %i%s\n", igridy, igridy, chat[1] );fprintf( pfp, "%f %f%s\n", alpha, beta, chat[2] );fprintf( pfp, "%f %f%s\n", gc, gamma, chat[3] );fprintf( pfp, "%i %i%s\n", inszf, inadvf, chat[4] );fprintf( pfp, "%i%s\n", iseed, chat[5] );fprintf( pfp, "%f %5.7f%s\n", am, brn, chat[6] );fprintf( pfp, "%f %5.7f%s\n", crn, drn, chat[7] );fprintf( pfp, "%f%s\n", attmin, chat[8] );fprintf( pfp, "%f %f%s\n", dsize, damp, chat[9] );fprintf( pfp, "%i %i%s\n", ninov, intrup, chat[10] );fprintf( pfp, "%5.4f %6.3f %5.3f%s\n", nashtol, minprof, inoadv, chat[11] );Appendix B: C Code^ 216fprintf( pfp, "%f %f%s\n", ghostatt, ghostinc, chat[12] );printf( "\nNew parameters written to file\n" );fclose( pfp );}elseperror("\nCouldn't open parameter file for write" );}C3. Example Parameter File3.000000 3.000000 STORE SEPARATION IN x & y (float)23 23^MARKET GRID SIZE (max 320, 200) (int)0.000000 1.000000^EXPONENTS of DIST & SIZE (float)0.000000 0.000000^CONST. AND EXP. OF COST FUNC. (float)0 0^1= = > INITIALIZE FROM FILES (int)30^IF inszf=0, USE THIS SEED AND3.000000 0.0001000 TO INIT. SIZE: size= am + brn*ran0.001000 0.0000100 TO INIT. ADV: adv = cm + drn*ran4.500000 MINIMUM ATTR. FOR PURCHASE (float)0.100000 0.010000^GROWTH DSIZE & DAMPING100 2^# OF INNOVATIONS; MAX # ITERATIONS0.2000 0.950 0.600 TOLERANCE; MINPROFIT; INNOVATION1.000000 1.000000 GHOST ATTRACTION & INCREMENTC4. Animation Routine#include < graph.h >Appendix B: C Code^ 217#include <stdlib.h>#include < conio. h >#include < stdio.h >#include < math. h >#include < errno. h >main( int argc, char *argv[]){FILE *lfp;char temp;double prof[32][20], storlocx[32], storlocy[20],sizmax, am, brn, crn, drn, maxdistmax, dsize, damp;/* double histdens[20], histsiz[20], histmaxsiz, histminsiz,histmaxdens, histmindens;*/double nashtol,minprof,inoadv,ghostatt,ghostinc ;int ihist = 12, ninov, ihit, jhit, isavhit, jsavhit, first, last;double storsepx, storsepy, scrsepx, scrsepy, scale, alpha, beta,gc, gamma, pbc, rho, attmin;int igridy, igridy, maxnumx, maxnumy, iseed, intrup, hitflag[32][20], aggflag;int i,j, k, 1, loop, iattminx, iattminy, radx[32][20],rady[32][20], inszf, inadvf, storino[32][20], savino[32][20];unsigned char diagmask[8] =0x93, OxC9, 0x64, OxB2, 0x59, Ox2C, 0x96, Ox4B };unsigned char solid[8] =OxFF, OxFF, OxFF, OxFF, OxFF, OxFF, OxFF, OxFF };int istorlocx[32], istorlocy[20];short oldvpage, oldapage, vpage, apage;oldapage = _getactivepageO;oldvpage = _getvisualpageO;printf( "\n Current visual page: %d, active page: %d", oldvpage, oldapage );1* ***************** GET DATA *********************** *1if( (lfp = fopen( argv[1], "r" ) ) = = NULL ){printf("\nUnable to open file" );goto Break;}elseAppendix B: C Code^ 218printf("\n File Opened " );fscanf( lfp, "%lf %lf %i %i %i %i ",&storsepx, &storsepy, &maxnumx, &maxnumy, &igridx, &igridy );for( i = 0; i < = maxnumx; i++ ){fscanf( lfp, "%lf ",&storlocx[i] );fscanf( lfp, "%lf \n",&storlocy[i] );}fscanf( lfp, "%lf %lf %lf %lf %i %i \n",&alpha, &beta, &gc, &gamma, &inszf, &inadvf );fscanf( lfp, "%i %lf %lf %lf %lf \n", &iseed, &arn, &brn, &crn, &drn );fscanf( lfp, "%lf %lf %lf %i %i \n", &attmin, &dsize, &damp, &ninov, &intrup );fscanf( lfp, "%lf %lf %lf %lf %lf \n", &nashtol,&minprof,&inoadv,&ghostatt,&ghostincprintf( "\n^SIMTOGRAPH FILE PARAMETERS\n\n" );printf( "storsep x and y: %f %An", storsepx, storsepy );printf( "maxnum x and y: %i^%i\n", maxnumx, maxnumy);printf( "igrid x and y: %i^%i\n\n", igridx, igridy );printf( "STORLOCATIONS X:\n " );for( i = 0; i < = maxnumx; i+ + )printf( " %f", storlocx[i] );printf( "\n\nSTORLOCATIONS Y:\n " );for( j = 0; j < = maxnumy; j+ + )printf( " %f", storlocy[j] );printf( "\n\nattraction exps alpha & beta: %f %An", alpha, beta );printf( "cost parameters gc & gamma: %f %An", gc, gamma );printf( "input switches inszf & inadvf: %i %i\n", inszf, inadvf );printf( "random params iseed, ant & brn: %i %f %An", iseed, arn, brn );printf( "attraction distance limit attmin: %An", attmin );printf( "Nash convergence parameters dsize & damp: %f %fln", dsize, damp );printf( "iterations & innovation counts: %i %i\n", intrup, ninov );printf( "nashtol,minprof,inoadv,ghostatt,ghostinc: %f %f %f %f %An",nashtol,minprof,inoadv,ghostatt,ghostinc );printf( " \n\n'ENTER' to continue, \n'n' to continue without histogram, \n'0' to break out\n");temp = getch();switch( temp ));Appendix B: C Code^ 219{case '0':goto Break;break;case 'n':ihist = 0;break;case '\r':ihist =12;break;case '1':ihist = 1;break;case '2':ihist = 2;break;case '3':ihist = 3;break;case '4':ihist = 4;break;case '5':ihist = 5;break;case '6':ihist = 6;break;case '7':ihist = 7;break;case '8':ihist = 8;break;case '9':ihist = 9;break;case 'q':ihist = 10;break;case 'w':ihist = 11;break;Appendix B: C Code^ 220case 'e':ihist = 13;break;case 'r':ihist = 14;break;case 't':ihist = 15;break;}I*** *****^GET FIRST SIZE MATRIX *************** *1fscanf( lfp, "%i %i ", &ihit, &jhit );isavhit = ihit;jsavhit = jhit;for( j = 0; j < = maxnumy; j+ + ){for( i = 0; i < = maxnumx; i+ + ){fscanf(lfp," %lf ", &prof[i][j]);fscanf(lfp," %i ", &storino[i][j]);savino[i][j] = storino[i][j];printf( "\n\n PROFIT MATRIX\n" );sizmax = 0;for( j = 0; j < = maxnumy; j+ + ){printf( " \n%i: ", j );for( i = 0; i < = maxnumx; i+ + ){if( prof[i][j] > sizmax )sizmax = prof[i][j];printf( " %f", prof[i][j] );IIprintf( "\n\nsizmax = %f", sizmax );printf( "\nEnter Profit Scale Factor--1 is OK\n" );scanf( "%lr, &scale );Appendix B: C Code^ 221printf( " \n\n Enter first and last iterations to plot");scanf( "%i %i",&first,&last );if( last > ninov ){printf( " \n Error: Only %i iterations in file", ninov );goto Break;}/* *** ******************* GRAPHICS ********************* *1if(setvideomode(_ERESCOLOR ) = = 0 ){printf("VIDEOMODE NOT AVAILABLE\n" );getchO;exit( 0 );}vpage = 0;apage = 1;setactivepage(0);_setvisualpage(1);scrsepx = storsepx*(640/igridx);scrsepy = storsepy*(350/igridy);for(i = 0; i < = maxnumx; i+ + )istorlocx[i] = (int)(storlocx[i] *640/igridx);for(j = 0; j < = maxnumy; j+ + )istorlocy[j] = (int)(storlocy[j] *350/igridy);iattminx = (int)(attmin*640/igridx);iattminy = (int)(attmin*350/igridy);/********* ****** ****** LOOP ****** ********** *********/for( loop = 1; loop < = last; loop+ + ){if( loop > = first ){temp = getchO;apage = _getvisualpageO;_setvisualpage( _getactivepage0 );_setactivepage( apage );Appendix B: C Code^ 222printf( "%i", loop - 1 );/* **set flag for hit */for(i = 0; i < = maxnumx; i++ ){ for(j = 0; j < = maxnumy; j+ + )hitflag[i][j] = 0;}aggflag = 0;if( (isavhit != ihit) 11 (jsavhit != jhit) ){hitflag[ihit][jhit] = 1;aggflag = 1;}isavhit = ihit;jsavhit = jhit;for(i = 0; i < = maxnumx; i+ + ){ for(j = 0; j < = maxnumy; j+ + ){radx[i][j] = (int)(scale * (prof[i][j]/sizmax)*scrsepx );rady[i][j] = (int)(scale * (prof[i][j]/sizmax)*scrsepy );}}setviewport(0,0,640,350);_clearscreen(_GCLEARSCREEN);_setbkcolorLBLUE);_setfillmask(solid);_setcolor(11),for(i = 0; i < = maxnumx; i++ ){ for(j = 0; j < = maxnumy; j + + ){if(savino[i][j] != storino[i][j] ){_rectangle( _GFILLINTERIOR, istorlocx[i] - radx[i][j]+10,istorlocy[j] - rady[i][j]+10,istorlocx[i] + radx[i][j]+10,istorlocy[j] + rady[i][j]+10 );Appendix B: C Code^ 223savino[i][j] = storino[i][j];}}_setcolor(14);for(i = 0; i < = maxnumx; i+ + ){ for(j = 0; j < = maxnumy; j+ + ){_rectangle( _GFILLINTERIOR, istorlocx[i] - radx[i][j],istorlocy[j] - rady[i][j],istorlocx[i] + radx[i][j],istorlocy[j] + rady[i][j] );}if(aggflag != 0 ){for(i = 0; i < = maxnumx; i+ + ){ for(j = 0; j < = maxnumy; j+ + ){if( hitflag[i][j] = = 1 ){_setcolor(12);_rectangle( _GFILLINTERIOR, istorlocx[i] - radx[i][j]-6,istorlocy[j] - rady[i][j]-6,istorlocx[i] + radx[i][j]-6,istorlocy[j] + rady[i][j]-6 );}}_setcolor(13 );for( k = 0; k < = 1; k+ + ){ for( j = 2; j < = 3; j++ ){_ellipse( _GBORDER, istorlocx[2] - iattminx - k,istorlocy[j] - iattminy - k,istorlocx[2] + iattminx + k,Appendix B: C Code^ 224istorlocy[j] + iattminy + k);_ellipse( _GFILLINTERIOR, istorlocx[2] - (int)(radx[2][j]/3),istorlocy[j] - (int)(rady[2][j]/3),istorlocx[2] + (int)(radx[2][j]/3),istorlocy[j] + (int)(rady[2][j]/3) );II/***************** Calculate and Draw histogram ***************Count number in each logarithmic intervalif(ihist != 0){for( k = 0; k < = 19; k++)histdens[k] = 0;for(i = 0; i < = maxnumx; i+ + ){ for(j = 0; j < = maxnumy; j+ + ){k = 0;while( (k < = 19) & ( log(storsiz[i][j]) > = histsiz[k] ) )k++;if( k < = 19 )histdens[k] + +;Turn frequency "histdens[k]" (which is now just the numberof stores in each interval) into logarithm of densityof stores in each intervalif( histdens[0] != 0 )histdens[0] = log( histdens[0] ) - histsiz[0];elsehistdens[0] = histmindens;for( k = 1; k< =19; k++ ){ if( histdens[k] != 0 )histdens[k] = log( histdens[k]) - log( exp( histsiz[k] )- exp( histsiz[k-1] ) );elsehistdens[k] = histmindens;}}}Appendix B: C Code^ 225_setviewport(320,20,610,150);_^.setwmdow( 1, histminsiz, histmindens, histmaxsiz, histmaxdens );_setcolor( ihist );_rectangle_w(_GBORDER,histminsiz,histdens[0],histsiz[0],histmindens );for( k = 1; k < = 19; k+ + )rectangle_w(_GBORDER,histsiz[k-1],histdens[k],histsiz[k],histmindens );*1}/****************Get next frame of data*******************/fscanf( lfp, " %i %i ", &ihit, &jhit );for( j = 0; j < = maxnumy; j+ + ){for( i = 0; i < = maxnumx; i+ + ){fscanf(lfp," %lf ", &prof[i][j]);fscanf(lfp," %i ", &storino[i][j]);***** ******** END LOOP ****** ******** *************1apage = _getvisualpage();_setvisualpage( _getactivepage() );_setactivepage( apage );getch();_setvideomode(DEFAULTMODE);_setactivepage( oldapage );_setvisualpage( oldvpage );iireak:fclose( lfp );}


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