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UBC Theses and Dissertations

Market preemption as a barrier to entry in a growing, spatially extended market West, Douglas Scott 1979

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MARKET PREEMPTION AS A BARRIER TO ENTRY IN A GROWING, SPATIALLY EXTENDED MARKET by DOUGLAS SCOTT WEST B.A. , U n i v e r s i t y of Michigan, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of ECONOMICS We accept t h i s t h e s i s as conforming to the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1979 (£) Douglas Scott West, 1979 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers ity of B r i t i s h Columbia, I agree that the Library shal l make i t f ree ly avai lable for reference and study. I further agree that permission for extensive copying of th i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publ icat ion of th i s thesis for f inanc ia l gain shal l not be allowed without my written permission. Department of ^ ^ o ^ ^ — ^ a - The Univers ity of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D E - 6 B P 7 5 - 5 1 1 E i i MARKET PREEMPTION AS A BARRIER TO ENTRY IN A GROWING, SPATIALLY EXTENDED MARKET Research Supervisor: Professor B. C u r t i s Eaton ABSTRACT In recent years, a number of economists have become i n t e r e s t e d i n e x p l o r i n g a type of e n t r y - d e t e r r i n g f i r m behavior known as preemptive entry. This type of behavior has been as s o c i a t e d w i t h e s t a b l i s h e d f i r m s expanding t h e i r c a p a c i t y i n the neighborhood of e x i s t i n g c a p a c i t y , e i t h e r i n the form of p r o l i f e r a t i n g brands or s i m i l a r products or opening new p l a n t s , i n order to secure the custom d e r i v a t i v e from e x i s t i n g or a n t i c i p a t e d f u t u r e demands i n that neighborhood. The goal of such behavior i s to deter ent r y , hence securing p r o t e c t i o n f o r monopoly p r o f i t s . The major t h e o r e t i c a l r e s u l t which may be derived from a model of preemption i s that i f growth of the market i s foreseen, an e s t a b l i s h e d f i r m w i l l always have an i n c e n t i v e to preempt the market at a poi n t i n time when i t would not pay a p o t e n t i a l entrant to enter. We d e r i v e t h i s r e s u l t using a one-dimensional,, l i n e a r s p a t i a l model, and we demonstrate that the r e s u l t does not depend upon the assumption of a l a r g e number of p o t e n t i a l e n trants or on whether the market i s one-dimensional or two-dimensional . The t h e s i s i s devoted to t e s t i n g the i m p l i c a t i o n s of the theory of preemption e m p i r i c a l l y . The f i r s t i m p l i c a t i o n which we examine i s a p r o f i t s i m p l i c a t i o n . Using cost and revenue data from the supermarket i i i i n d u s t r y , we search f o r i n d i c a t i v e evidence f o r or against the f o l l o w i n g n u l l hypothesis which i s a s s o c i a t e d w i t h the p r o f i t s i m p l i c a t i o n : The p r o f i t s of a r e p r e s e n t a t i v e new supermarket are l e s s than zero i n i t s f i r s t twelve months of ope r a t i o n , where the supermarket i s r e p r e s e n t a t i v e i n the sense that i t s p r o f i t s are c a l c u l a t e d using average cost and average revenue data, and the average i s over supermarkets. We perform a s e r i e s of annual net p r o f i t c a l c u l a t i o n s f o r the.years 1970-1976 i n c l u s i v e , and f i n d that i n any one of these years,, the annual net p r o f i t s of a r e p r e s e n t a t i v e new supermarket are negative and l e s s than the annual net p r o f i t s of a r e p r e s e n t a t i v e e s t a b l i s h e d supermarket. The second i m p l i c a t i o n of the theory of preemption i s a l o c a t i o n a l one. Using supermarket l o c a t i o n data from the province of B r i t i s h Columbia, we construct, two types of t e s t s i n order to a s c e r t a i n the nature and extent of preemption i n the Greater Vancouver Regional D i s t r i c t (GVRD.) of B r i t i s h Columbia. F i r s t we use cross s e c t i o n data on store, ownership and the neighbor r e l a t i o n s between s t o r e s i n each of the four sub-markets of the GVRD i n order to determine i f our observations are co n s i s t e n t w i t h a random process based on a set of p r o b a b i l i t i e s which i s independent of the neighbor r e l a t i o n s between s t o r e s . We f i n d that t h i s n u l l hypothesis of randomness may be r e j e c t e d f o r the GVRD as a whole and i t s Vancouver sub-market, but not f o r the three other sub-markets which comprise the GVRD. Next, we use time s e r i e s data on the date at which each s t o r e was e s t a b l i s h e d i n the Vancouver sub-market, where that st o r e was l o c a t e d , and which f i r m owned, i t i n order to determine i f our observations are c o n s i s t e n t w i t h a s t a t e dependent p r o b a b i l i s t i c process i n which the p r o b a b i l i t y that any given s t o r e i s owned by a given f i r m depends upon the neighbor r e l a t i o n s that that s t o r e i v had w i t h other stores i n the sub-market at the time when i t was e s t a b l i s h e d . We f i n d that we may accept the hypothesis of s t a t e dependence: f o r the Vancouver sub-market. F i n a l l y , we conduct an a n a l y s i s of the p r o b a b i l i t i e s u n d e r l y i n g the s t a t e dependent process and we o b t a i n the r e s u l t that preemptive l o c a t i o n behavior has taken place i n the Vancouver sub-market of the GVRD. V TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS;:- v LIST OF TABLES i x LIST OF FIGURES x i ACKNOWLEDGEMENTS x i i i CHAPTER 1: INTRODUCTION 1 Footnotes to Chapter 1 . 6 CHAPTER 2 EXCESS CAPACITY, SEQUENTIAL ENTRY, AND MARKET PREEMPTION 7 2.1 I n t r o d u c t i o n ; 7 2.2 Excess Capacity as a B a r r i e r to Entry 7 2.3 Models of Sequential Entry 11 2.4 Brand P r o l i f e r a t i o n and Preemptive D i v e r s i f i c a t i o n 16 2.5 The Eaton and Lipsey Model of Market Preemption 19 Footnotes to Chapter 2 22 CHAPTER 3 THE THEORY OF MARKET PREEMPTION 25 3.1 I n t r o d u c t i o n 25 3.2 Preemption i n a Spaceless Market 25 3.3 A S p a t i a l Model of Preemption 27a 3.4 An A l t e r n a t i v e Model of Preemption 46 3.5 Extension of the Model to Two Dimensions 53 v i CHAPTER 3 THE THEORY OF MARKET PREEMPTION 3.6 Anomalies 61 Footnotes to Chapter 3. . 65 CHAPTER 4 THE PROFITS TEST./ 70 4.1 I n t r o d u c t i o n 70 4.2 Statement of the N u l l Hypothesis 71 4.3 D e s c r i p t i o n of Data Sources 72 4.4 Procedures f o r Estimating Net P r o f i t s 74 4.5 Test Results of the N u l l Hypothesis of Negative I n i t i a l P r o f i t s 84 4.6 I n t e r p r e t a t i o n of Results.and Concluding Remarks 93 Footnotes to Chapter 4 96 CHAPTER 5 THE LOCATION TEST OF PREEMPTION 102 5.1 I n t r o d u c t i o n ' 102 5.2 The Market 104 5.2.1 Reasons f o r S e l e c t i n g the Greater Vancouver Regional D i s t r i c t as the Basis f o r the E m p i r i c a l Work 104 5.2.2 D e f i n i n g a Supermarket and D i s t i n g u i s h i n g Between P o t e n t i a l Preemptors and Competitive Fringe Firms 105 5.2.3 Supermarket Firms Operating i n the GVRD 107 5.3 The Test of Random Firm Ownership 112 5.3.1 M o t i v a t i o n f o r the Test 112 5.3.2 Statement of the N u l l Hypothesis of Random Firm Ownership of Stores 114 5.3.3 Testing Procedure f o r the N u l l Hypothesis When the Number of Stores i n the Sub-Market I s Small 115 v i i CHAPTER 5 THE LOCATION TEST OF PREEMPTION 5.3.4 Testing Procedure and Test S t a t i s t i c s , f o r the N u l l Hypothesis When the Number of Stores i n the Sub-market Is Large ........... 119 5.3.5 Test Results of the N u l l Hypothesis f o r the GVRD and Constituent Sub-markets 128 5.4 The Test of Random Neighbor R e l a t i o n s 133 5.4.1 M o t i v a t i o n f o r the Test 133 5.4.2 Statement of the N u l l Hypothesis of Random Neighbor R e l a t i o n s 134 5.4.3 Test i n g Procedure f o r the N u l l Hypothesis When the Number of Stores i n the Sub-market Is Small 135 5.4.4 Te s t i n g Procedure and Test S t a t i s t i c s f o r the N u l l Hypothesis When the Number of Stores i n the Sub-market Is Large 141 5.4.5 Test Results of the N u l l Hypothesis of Random Neighbor R e l a t i o n s f o r the GVRD and Constituent Sub-markets. 151 -5.5 I n t e r p r e t a t i o n of the Test Results of Randomness . 156 5.6 The Test of State Dependence 158 5.6.1 M o t i v a t i o n f o r the Test 158 5.6.2 Statement of the N u l l Hypothesis of State Dependence 160 5.6.3 Testing Procedure f o r State Dependence 162 5.6.4 Comparative.Analysis of State Dependent P r o b a b i l i t i e s and R e l a t i v e Frequencies 170 5.6.5 Test i n g f o r State Dependence i n the Vancouver Sub-market 177 5.6.6 Comparative A n a l y s i s of State Dependent P r o b a b i l i t i e s and R e l a t i v e Frequencies f o r the Vancouver Sub-market 183 v i i i CHAPTER 5 THE LOCATION TEST OF PREEMPTION 5.6.7 I n t e r p r e t a t i o n of the Comparative A n a l y s i s of State Dependent P r o b a b i l i t i e s and R e l a t i v e Frequencies 187 5.7 An Extension 191 5.7.1 Redefining a Supermarket '.' 191 5.7.2 The Test of Random Firm Ownership 193 5.7.3 The Test of Random Neighbor R e l a t i o n s 200 5.7.4 I n t e r p r e t a t i o n of the Test R e s u l t s of Randomness 205 5.7.5 The Test of State Dependence 208 5.7.6 Comparative A n a l y s i s of State Dependent P r o b a b i l i t i e s and R e l a t i v e Frequencies f o r the Vancouver Sub-market 212 5.7.7 E v a l u a t i o n of Test R e s u l t s Based on the Revised Sample '.' 216 5.8 Concluding Remarks 218 Footnotes to Chapter 5 223 CHAPTER.6 SUMMARY AND CONCLUSIONS 228 BIBLIOGRAPHY 231 APPENDIX 236 i x LIST OF TABLES I :Net P r o f i t C a l c u l a t i o n s : 1970-1976 -. .. 85 I I Member M u n i c i p a l i t i e s of the GVRD by Sub-Market 105 I I I Store Ownership by Firm - B r i t i s h . Columbia (B.C.) 110 IV Store Ownership by Firm - GVRD Sub-markets I l l V Permutations of Firm Ownership 117 VI Summary of P r o b a b i l i t i e s 118 VI I Three Random Permutations of Firm Ownership 123 2 V I I I Results of the Chi-square Test and the X Test f o r •the GVRD - Permutations = 250, 500, 750, 1000 126 IX R e l a t i v e Frequencies of Each Firm's Stores i n B.C 128 X Marginal D i s t r i b u t i o n D e s c r i p t i v e Measures by Firm f o r the GVRD and Constituent Sub-markets 129 2 XI Results of the X Tests of the N u l l Hypothesis of Random Firm Ownership by GVRD and Sub-markets 130 X I I J o i n t Common Boundary D i s t r i b u t i o n 139 X I I I Marginal Common Boundary D i s t r i b u t i o n s 140 XIV Cumulative Frequencies f o r the Marginal D i s t r i b u t i o n s of the GVRD - Permutations.= 250, 500, 750, 1000 146 2 XV Re s u l t s of the X Tests f o r the GVRD - Permutations = 250, 500, 750, 1000 148 XVI Common Boundaries and Boundaries by Firm - GVRD and Sub-markets 151 XVII Marginal Common Boundary D i s t r i b u t i o n D e s c r i p t i v e Measures by Firm f o r the GVRD and Sub-markets 152 2 XVIII R e s u l t s of the X Tests of the N u l l Hypothesis of Random Neighbor R e l a t i o n s by GVRD and Sub-markets 153 XIX Contingency Table of State Dependent Frequencies 164 XX Contingency Table of State Dependent Frequencies 179 X XXI Revised Contingency Table of State Dependent Frequencies ...... ±......... 181 XXII R e l a t i v e Frequencies of 'Each Firm's Stores i n B.C 182 XXI I I Annual Comparisons of TT . and f. f o r the Vancouver Sub-market , 185 XXIV Store Ownership by Firm - B r i t i s h Columbia (Store S i z e > 12,000 Square Feet) 194 XXV Store Ownership by Firm - GVRD Sub-markets (Store Size > 12,000 Square. Feet) 195 XXVI R e l a t i v e Frequencies of Each Firm's Stores i n B.C. (Store S i z e > 12,000 Square Feet) 196 XXVII Marginal D i s t r i b u t i o n D e s c r i p t i v e Measures by Firm f o r the GVRD and Constituent Sub-markets (Store Size > 12,000 Square Feet) 197 XXVIII R e s u l t s of the X Tests of the N u l l Hypothesis of Random Firm Ownership by GVRD and Sub-markets (Store S i z e > 12,000 Square Feet) 198 XXIX Common Boundaries and Boundaries by Firm - GVRD and Sub-markets (Store S i z e > 12,000 Square Feet) 201 XXX Marginal Common Boundary D i s t r i b u t i o n D e s c r i p t i v e Measures by Firm f o r the GVRD and Sub-markets (Store S i z e > 12,000 Square Feet) 202 2 XXXI R e s u l t s of the X Tests of the N u l l Hypothesis of Random Neighbor R e l a t i o n s by GVRD and Sub-markets (Store S i z e > 12,000 Square Feet) 203 XXXII Contingency Table of State Dependent Frequencies (Store S i z e > 12,000 Square Feet) 208 XXXIII Revised Contingency Table of State Dependent Frequencies (Store Size > 12,000 Square Feet) 210 XXXIV R e l a t i v e Frequencies of Each Firm's Stores i n B.C. (Store Size > 12,000 Square Feet) 211 XXXV Annual Comparisons of TT . and f.. f o r the Vancouver r mi I U Sub^-market (Store Size > 12,000 Square Feet) 214 x i LIST OF FIGURES Figure 1 . . 19 Figure 2 • • • 27a Figure 3 32 Figure 4 47 Figure 5 54 Fi g u r e 6 58 Figure 7 63 Figure 8 116 Figure 9 136 Figure 10 137 Figure 11 163 Figure 12 . 188 Figure A . l Richmond,Sub-market .Cross,Section 238 Figure. A.2 North Shore Sub-market Cross Sect i o n 239 Figure A.3 Delta-Surrey Sub-market Cross Section 240 Figure A.4 Vancouver.Sub-market Western Sector Cross Sect i o n 241 Figure A.5 Vancouver Sub-market Eastern Sector Cross Sect i o n 242 Figure A.6 Richmond Sub-market Cross Section (Store Size > 12,000 Square Feet) ••• 243 Figure A.7 North Shore Sub-market Cross Section (Store Size > 12,000 Square Feet) 244 Figure A.8 Delta-Surrey Sub-market Cross Sect i o n . (Store Size.> 12,000 Square Feet) 245 Figure A.9 Vancouver Sub-market Western Sector Cross Section (Store Size > 12,000 Square Feet) ... 246 x i i F igure A.10 Vancouver Sub^-market Eastern Sector Cross Secti o n (Store Size > 12,000 Square Feet) ... 247 Figure A.11 Vancouver Sub-market Western Sector Time Series . 248 Figure. A.12 Vancouver Sub-market Eastern Sector Time S e r i e s . .. 249 Figure A.13 Vancouver Sub-market- Western Sector Time Se r i e s (Store S i z e > 12,000 Square Feet) 251 Figure A. 14 Vancouver Sub-onarket Eastern Sector Time Ser i e s (Store S i z e > 12,000 Square Feet) 252 x i i i ACKNOWLEDGEMENTS I would l i k e to acknowledge the a s s i s t a n c e of a number of people who took the time and e f f o r t to make i n v a l u a b l e suggestions during the course of my research. F i r s t and foremost, I wish to thank Professor B. C u r t i s Eaton, my research s u p e r v i s o r . His i n s i g h t f u l and pe r c e p t i v e comments on previous d r a f t s of the t h e s i s enabled the research to reach i t s present stage of completion. He was a l s o a constant source of encouragement and enthusiasm during those times when the i n e v i t a b l e snags i n a t h e s i s occur. I would a l s o l i k e to thank the other members of my d i s s e r t a t i o n committee, Professors G.C. A r c h i b a l d and Terry. Wales, f o r t h e i r important comments and suggestions at v a r i o u s stages of my research. At one time or another, I have had u s e f u l d i s c u s s i o n s about v a r i o u s aspects of the t h e s i s w i t h P r o f e s s o r s Charles Blackorby, R u s s e l l Uhler, James Forbes, and R u s s e l l Davidson. I owe s p e c i a l thanks to Pro f e s s o r Yoshitsugu Kanemoto, who provided me w i t h many pe n e t r a t i n g comments on e a r l i e r d r a f t s of t h e l t h e s i s . I would a l s o l i k e to acknowledge the generous computer programming a s s i s t a n c e provided by Lewis James, Programmer Analyst f o r A r t s Computing. May McKee and Janet Clark a l s o deserve mention f o r t h e i r superb t y p i n g of the manuscript. I wish to thank the U n i v e r s i t y of B r i t i s h Columbia f o r f i n a n c i a l support provided by a K i l l a m E r e - d o c t o r a l Fellowship during the .duration of the research. F i n a l l y , I owe a s p e c i a l debt of g r a t i t u d e to Kathy West, who provided s i n c e r e and a f f e c t i o n a t e support and encouragement during my long months of research. Chapter 1 INTRODUCTION In recent years, a number of firms have been charged w i t h prematurely p r o l i f e r a t i n g products or p l a n t s i n order to deter entry i n a growing market. On October 11, 1972, an Information was l a i d charging that Canada Safeway, L t d . , " . . . was a party to a monopoly i n the grocery r e t a i l i n g i n d u s t r y between January 1, 1965, and October 10, 1972. The f i r s t count r e l a t e d to the C i t y of Calgary and the second r e l a t e d to the C i t y of Edmonton.""'" On September 17, 1973, the Attorney General of Canada a p p l i e d f o r (and was granted), i n the Supreme Court of A l b e r t a , Orders of p r o h i b i t i o n pursuant to s e c t i o n 30(2) of the Combines I n v e s t i -g a t i o n A c t . (The Information was t h e r e f o r e withdrawn.) One of the Orders r e q u i r e d that " . . . Canada Safeway L i m i t e d w i l l not s i g n i f i c a n t l y over the next three and one-half years i n c r e a s e the t o t a l square footage which i t operates as r e t a i l o u t l e t s i n each of the two c i t i e s , and i s r e s t r i c t e d to opening only one new o u t l e t i n each market during t h i s p e r i o d . In a d d i t i o n , Canada Safeway i s r e s t r i c t e d from a c q u i r i n g new s i t e s f o r r e t a i l o u t l e t s during the f i r s t two and one-half years of the period i n question and i s f u r t h e r r e s t r i c t e d from opening more than two r e t a i l o u t l e t s d u r ing the year f o l l o w i n g the e x p i r a t i o n of the three and one-half year p e r i o d . The i n t e n t of t h i s p r o h i b i t i o n i s to a l l o w f o r the development of competition i n the r e t a i l grocery trade i n Edmonton and Calgary. Fur t h e r , i t i s intended,to prevent Canada Safeway from pre-empting prime s i t e s f o r r e t a i l o u t l e t s i n each of the two 2 markets as these s i t e s become a v a i l a b l e . " (Emphasis added.) At about the same time that an Information was being ladld. against Canada Safeway, the United States Federal Trade Commission i s s u e d a complaint " . . . charging v i o l a t i o n s of Se c t i o n 5 of the Federal Trade Commission Act against the four l a r g e s t U.S. manufacturers of ready-to-eat c e r e a l ( h e r e i n a f t e r simply RTE c e r e a l ) : K e l l o g g , General M i l l s , General Foods, and Quaker Oats. I n a s e c t i o n headed 'Brand P r o l i f e r a t i o n , Product D i f f e r e n t i a t i o n and Trademark Promotion', the complaint discussed the brand i n t r o d u c t i o n and s a l e s promotion a c t i v i t i e s of these firms and charged that 'these p r a c t i c e s of p r o l i f e r a t i n g brands, d i f f e r e n t i a t i n g s i m i l a r products and promoting trademarks through i n t e n s i v e a d v e r t i s i n g 3 r e s u l t i n h i g h b a r r i e r s to entry i n t o the RTE c e r e a l market.'" The type of e n t r y - d e t e r r i n g behavior which the above cases i d e n t i f y has been v a r i o u s l y c a l l e d preemptive ent r y , preemptive d i v e r s i f i c a t i o n , 4 or brand p r o l i f e r a t i o n . I t has been as s o c i a t e d w i t h e s t a b l i s h e d firms expanding t h e i r c a p a c i t y i n the neighborhood of e x i s t i n g c a p a c i t y , e i t h e r i n the form of p r o l i f e r a t i n g brands or s i m i l a r products or opening new p l a n t s , i n order to secure the custom d e r i v a t i v e from e x i s t i n g or a n t i c i p a t e d f u t u r e demands i n th a t neighborhood. The goal of such behavior i s to deter e n t r y , hence s e c u r i n g p r o t e c t i o n f o r monopoly p r o f i t s I t i s apparent from the cases c i t e d above that there i s a number of i n d u s t r i e s i n which preemption might be an important e x p l a n a t i o n of f i r m e n t r y - d e t e r r i n g behavior and, consequently, of f i r m c o n c e n t r a t i o n and ca p a c i t y i n an i n d u s t r y . We have already mentioned that the FTC b e l i e v e s that preemptive d i v e r s i f i c a t i o n or brand p r o l i f e r a t i o n has been p a r t l y r e s p o n s i b l e f o r the c o n c e n t r a t i o n i n the RTE c e r e a l market. This - 3 -possibility has been discussed by Schmalensee [1978]. While the problem was really one for analysis in characteristics space, Schmalensee chose to cast the analysis in terms of location (see section 2.4). One implica-tion of Schmalensee's analysis is that preemptive diversification may be essential to understanding the market structure of those industries where product diversification i s of paramount importance (such as the automobile industry, soap industry, cigarette industry, etc.). Indeed, as early as 1975, Archibald and Rosenbluth [1975] discovered that preemptive diversif-ication was a logical implication of their analysis of monopolistic competition in Lancaster [l966]-Baumol [1967] characteristics space (see section 2.4). The possibility of preemptive strategies i s also important in promoting our understanding of the market structure and performance of those r e t a i l industries which consist of multi-plant firms. In the past, industrial organization economists have paid scant attention to the r e t a i l sector of the economy as a whole, let alone attempting to uncover the different forms of firm behavior which may have been responsible for giving rise to the particular market structures of various r e t a i l indus-tries . For example, a cursory examination of the most prominent industrial organization textbooks would indicate that economists have engaged in l i t t l e substantive analysis of retailing. Bain [1968] limits his discussion of retailing to an examination of the structural evolution in the distributive trades. Scherer's [1970] widely used text contains very l i t t l e discussion of retailing. What discussion there is revolves around Adelman's study of the anti-trust action brought against the A & P Company in the 1940's. Finally, we could find almost no reference to retailing i n Shepherd [1970]. This dearth of analytical and empirical - 4 -examination i s d i f f i c u l t to understand given the important r o l e that t h i s s e c t o r plays i n the economy. In the pages which f o l l o w , we s h a l l explore the c o n d i t i o n s under which fi r m s w i l l engage i n preemption as a b a r r i e r to entry i n a growing, s p a t i a l l y extended market. We r e v i s e and extend the theory of preemption due to Eaton and Lipsey [1979]. Their theory of preemption was developed i n geographic space and c o n s t i t u t e s a new dimension to the l i t e r a t u r e of s p a t i a l competition. We s h a l l focus our a t t e n t i o n on d e r i v i n g the t e s t a b l e i m p l i c a t i o n s of the theory and we s h a l l conduct r i g o r o u s t e s t s of a number of hypotheses which w i l l a l l o w us to e m p i r i c a l l y confirm or r e j e c t these i m p l i c a t i o n s f o r a p a r t i c u l a r i n d u s t r y . The theory of preemption does not lend i t s e l f to e m p i r i c a l t e s t s based on standard econometric techniques. We have ther e f o r e had to design t e s t i n g procedures which r e l y h e a v i l y on nonparametric techniques. F i n a l l y , our e m p i r i c a l research has focused on the supermarket i n d u s t r y . We b e l i e v e that our research makes an important c o n t r i b u t i o n towards our understanding of the market s t r u c t u r e of the supermarket i n d u s t r y , and of other i n d u s t r i e s as w e l l . The theory of preemption has i t s r o o t s i n the l i t e r a t u r e of excess c a p a c i t y as a b a r r i e r to entry and i n s e v e r a l papers on s e q u e n t i a l entry i n s p a t i a l markets. I t i s a l s o r e l a t e d to the concepts of brand p r o l i f -eration: and preemptive d i v e r s i f i c a t i o n . In the next chapter, we s h a l l survey the l i t e r a t u r e l e a d i n g up to and a s s o c i a t e d w i t h the theory of preemption. In Chapter 3, we c o n s t r u c t a model of preemption i n space and we demonstrate that the e s t a b l i s h e d f i r m always has the i n c e n t i v e to preempt the market at a p o i n t i n time j u s t e a r l i e r than the e a r l i e s t date a t which a new entrant would find i t profitable to do so. We also extend this model by relaxing one of i t s restrictive assumptions. Next, the two-dimensional analogue of the one-dimensional spatial model i s discussed and, f i n a l l y , we indicate reasons why we might not expect to observe perfect preemption in the real world. In Chapter 4, we analyze average cost and average revenue data from the supermarket industry in order to investigate the profits implica-tion of the theory of preemption. We examine the null hypothesis that the profits of a representative new supermarket are negative i n i t s f i r s t twelve months of operation. In Chapter 5, we test the locational implications of the theory of preemption. Using supermarket location data from the province of British Columbia, we f i r s t devise tests i n order to determine i f our observations on firm ownership of stores and the neighbor relations between stores are consistent with a random process based on a particular set of state independent probabilities. If we reject the hypotheses of randomness, we proceed to test the hypothesis that our observations are consistent with a state dependent probabilistic process. If we accept the hypothesis of state dependence, we then analyze probabilities underlying the process i n order to determine i f they are consistent with one or more firms having preempted in the market. In Chapter 6, we summarize our analysis and empirical results, and we make some concluding remarks. - 6 -FOOTNOTES TO CHAPTER 1 1. Consumer and Corporate A f f a i r s [1974 $ 32-33]. 2. I b i d . 3. Schmalensee [1978 ; 305]. The case r e f e r r e d to i s F.T.C. v. K e l l o g g et a l . , Docket No. 8883. 4. Eaton and Lipsey [1979], A r c h i b a l d and Rosenbluth [1975], and Schmalensee [1978]. 5. Other forms of preemption are al s o p o s s i b l e . Southey [1978 J 553-557] b e l i e v e s that the development of the wheat i n d u s t r y on the Canadian p r a i r i e s can be modelled as a f r e e access i n d u s t r y i n which rents are d i s s i p a t e d . Southey considers two cases. I n the f i r s t case, an i n d i v i d u a l must a c t u a l l y work the land a f t e r having homesteaded, w h i l e i n the second case the homesteader i s f r e e to leave the land or s e l l i t once the expenditure on b r e a k i n g - i n the land has occurred. He f i n d s that "In both cases homesteading i s premature, as i s a c t i v e farming. A l l e l s e remaining the same, farming commences e a r l i e r i n the f i r s t case than i n the second. In each case no net, c a p i t a l i z e d rents are made." The d r i v i n g f o r c e of these r e s u l t s i s that as long as the present value of expected rents i s p o s i t i v e and s u f f i c i e n t to cover set-up c o s t s , homesteading w i l l occur. Competition among home-steaders f o r the best land pushes back the date of homesteading to the po i n t where the net present value i s equal to zero. - 7 -Chapter 2 EXCESS CAPACITY, SEQUENTIAL ENTRY AND MARKET PREEMPTION 2.1 Introduction In this chapter, we shall examine the major theoretical developments which led to the theory of preemption. We shall begin by focusing on a body of literature which grew out of a dissatisfaction with the theory of limit pricing, and which concerns the possibility of using excess capacity as a barrier to entry. This literature i s closely related to the theory of preemption in that the theory of preemption concerns i t s e l f with the possibility of an established firm constructing excess capacity in the market i n the form of new plants i n order to prevent new entry from taking place. Next, we discuss some recent literature on sequential entry in spatially extended markets. While the models discussed in this literature do not explicitly deal with the excess capacity issue, they may be regarded as the precursors of spatial models of preemption. These models generally make the important assumption that capital, once establish-ed, i s fixed i n location, and then proceed to analyze the entry deterring strategies of single plant firms. Had these models relaxed their assump-tion that each firm is only allowed to own one plant, a theory of preemption could have been derived. Finally, we summarize the theoretical literature which has expli c i t l y discussed preemption. 2.2 Excess Capacity as a Barrier to Entry An area of economics known as "barriers to entry" has preoccupied the minds of many economists for years (see Bain [1965])."'' Among the most frequently cited entry barriers are increasing returns to scale, patents, - 8 -a d v e r t i s i n g , imperfect c a p i t a l markets, t y i n g up raw m a t e r i a l s , and government l i c e n s i n g and r e g u l a t i o n . However, l i m i t p r i c i n g has probably received the most extensive t h e o r e t i c a l treatment, of any of the entry 2 b a r r i e r s . I t i s . not my i n t e n t i o n to. summarize the l i m i t p r i c i n g l i t e r a t u r e here, but rather to note that d i s s a t i s f a c t i o n w i t h l i m i t p r i c i n g theory l e d some economists to speculate upon the p o s s i b i l i t y of using excess c a p a c i t y as a b a r r i e r to entry. For example, P a s h i g i a n J1968] was uneasy over the conventional assumption of l i m i t , p r i c e t h e o r i s t s that, the monopolist cannot charge the monopoly p r i c e and yet block entry by threatening"to lower p r i c e and expand output. Such a s t r a t e g y would, r e q u i r e that' the monopolist "... be prepared to produce the l a r g e r output r e q u i r e d to meet demand at the l i m i t p r i c e w i t h a p l a n t p r i m a r i l y designed f o r e f f i c i e n t production of the smaller monopoly output. To achieve t h i s output f l e x i b i l i t y , the monopolist w i l l e i t h e r - i / s a c r i f i c e some p l a n t e f f i c i e n c y f o r greater p l a n t f l e x i b i l i t y than i s otherwise r e q u i r e d j or i n c u r higher short-run cost i n t r y i n g to produce the l a r g e r output with a s p e c i a l i z e d p l a n t designed f o r the monopoly output, or carry higher i n v e n t o r i e s , 3 again at a d d i t i o n a l c o s t . " Pashigian, however, dismissed t h i s s t r a t e g y w i t h the argument that the entrant would then have a cost advantage over the monopolist. Wenders' [1971] a r t i c l e was i n s p i r e d by Pashigian's suggestion that a monopolist may.wish to pursue a s t r a t e g y of b l o c k i n g entry by threatening to lower p r i c e and expand output, w h i l e s t i l l charging the monopoly p r i c e . . Wenders disagreed w i t h Pashigian's d i s m i s s a l of t h i s - 9 -s t r a t e g y , n o t i n g that ". . . j u s t because t h i s use of excess c a p a c i t y i n v o l v e s higher c o s t s , t h i s does not mean that i t w i l l be u n p r o f i t a b l e ; i t merely means that excess c a p a c i t y should be increased only up to that p o i n t where the incremental b e n e f i t s are matched by the incremental c o s t s . Wanders then proceeded to consider how a monopolist could block entry by b u i l d i n g a p l a n t which i s l a r g e enough to produce e f f i c i e n t l y at the l i m i t p r i c e , w h i l e charging the monopoly p r i c e and producing the monopoly q u a n t i t y when the t h r e a t of entry i s s m a l l . Such a s t r a t e g y would r e s u l t i n lower monopoly p r o f i t s r e l a t i v e to what the monopolist could earn i f i t had b u i l t the most e f f i c i e n t p l a n t f o r the monopoly output and entry were barred, but p o t e n t i a l l y higher p r o f i t s than could be earned by usin g a l i m i t p r i c i n g s t r a t e g y . Esposito and Esposito [1974], w h i l e not foc u s i n g upon excess ca p a c i t y as a b a r r i e r to entry , d i d suggest reasons why excess c a p a c i t y might appear i n d i f f e r e n t market s t r u c t u r e s . Excess c a p a c i t y might a r i s e i n o l i g o p o l i e s c h a r a c t e r i z e d by s u b s t a n t i a l s e l l e r c o n c e n t r a t i o n , s u b s t a n t i a l b a r r i e r s , to entry and a s i g n i f i c a n t competitive f r i n g e ( " p a r t i a l o l i g o p o l i e s " ) i f the l a r g e s t f i r m s f a i l to act c o l l e c t i v e l y and al s o f a i l to increase t h e i r market share a f t e r attempting to do so i n response to a permanent increase i n demand.^ (Esposito and Esposito found evidence to support the hypothesis of excess c a p a c i t y i n p a r t i a l o l i g o p o l i e s . ) Excess c a p a c i t y could a l s o a r i s e i n a t i g h t o l i g o p o l y i f - 10 -". . .at l e a s t one of the o l i g o p o l i s t s views an increase i n i n d u s t r y demand as c r e a t i n g a good opp o r t u n i t y to i n c r e a s e h i s market share . . . . An o l i g o p o l i s t may a l s o create excess ca p a c i t y i n order to r e t a i n h i s own buyers and s e r v i c e h i s r i v a l s ' customers unexpected needs i n case u n a n t i c i p a t e d f u t u r e i n c r e a s e s i n demand occur. . . . In t u r n , fear of f u t u r e l o s s of market share may impel r i v a l o l i g o p o l i s t s to i n c r e a s e c a p a c i t y beyond what i s needed to meet t h e i r current demand."^ This l a s t reason f o r excess c a p a c i t y comes c l o s e s t to approximating an argument i n support of preemption as a cause of excess c a p a c i t y , although they do not expand upon t h i s p o s s i b i l i t y . Spence [1977], f o l l o w i n g Wenders, a l s o considers the p o s s i b i l i t y of an i n d u s t r y c a r r y i n g excess c a p a c i t y i n order to deter new entry. He e s t a b l i s h e d that a r e s u l t of such a s t r a t e g y would be a p r i c e i n excess of the l i m i t p r i c e and i n e f f i c i e n t p r oduction i n the sense that the output produced would be l e s s than c a p a c i t y output and costs would not be minimized f o r t h i s output. U n l i k e Wenders, Spence focused more e x p l i c i t l y on comparing the i m p l i c a t i o n s of an i n d u s t r y using excess c a p a c i t y as a b a r r i e r to entry w i t h the I m p l i c a t i o n s of an i n d u s t r y u s i n g a l i m i t p r i c i n g s t r a t e g y . The d e s i r a b i l i t y of such excess c a p a c i t y i s shown to depend upon the l e v e l of r e s i d u a l demand and the extent to which such a p o l i c y w i l l i n f a c t deter entry. There are at l e a s t two important features of Spence's and Wenders' models of excess c a p a c i t y as a b a r r i e r to e n t r y which d i f f e r e n t i a t e them from models of preemption. F i r s t , t h e i r models are e s s e n t i a l l y s t a t i c . That i s , the monopolist i s confronted w i t h a o n c e - a n d - f o r - a l l d e c i s i o n of - 11 -what s i z e p l a n t to b u i l d given that the l e v e l of r e s i d u a l demand might induce entry. Models of preemption consider the p l a n t c o n s t r u c t i o n or c a p a c i t y expansion s t r a t e g i e s of e s t a b l i s h e d f i r m s and p o t e n t i a l entrants given that they are confronted w i t h an expected i n c r e a s e i n demand i n the f u t u r e . The date of entry i s thus an important aspect of preemption a n a l y s i s . Second, and most important, t h e i r models are s p a c e l e s s , w h i l e models of preemption have been cast i n a s p a t i a l framework. By a n a l y z i n g the d e c i s i o n to preempt i n a s p a t i a l framework, i n s i g h t s are gained i n t o the i n c e n t i v e s which fi r m s have to engage i n t h i s type of behavior as opposed to some other form of entry d e t e r r i n g behavior. 2.3 Models of S e q u e n t i a l Entry S p a t i a l models of s e q u e n t i a l entry are the precursors of s p a t i a l models of preemption. As mentioned i n the i n t r o d u c t i o n to t h i s chapter, these models g e n e r a l l y make the important assumption that c a p i t a l , once e s t a b l i s h e d , i s f i x e d i n l o c a t i o n , and then proceed to analyze the entry d e t e r r i n g l o c a t i o n s t r a t e g i e s of s i n g l e p l a n t f i r m s . In a d d i t i o n , these models are g e n e r a l l y incapable of a n a l y z i n g a preemptive s t r a t e g y s i n c e they assume that firms enter the market one at a time and do not compete w i t h each other f o r the opportunity to e s t a b l i s h new p l a n t s i n the market. The f i r s t model of s e q u e n t i a l entry which we s h a l l comment upon i n t h i s s e c t i o n i s one constructed by Peles [1974]. Peles shows that excess p r o f i t s may be a property of both short-run and long-run e q u i l i b r i u m i n h i s model (given that c a p i t a l i s f i x e d i n l o c a t i o n or immobile). While not p r i m a r i l y i n t e r e s t e d i n a n a l y z i n g entry d e t e r r i n g s t r a t e g i e s , Peles d i d note that producers could use p a r t of t h e i r excess p r o f i t s i n order to deter e n t r y , w h i l e s t i l l r e t a i n i n g a monopoly p r o f i t . " I f producers expect a new e n t r y , they might l o c a t e themselves c l o s e r together from the - 12 -beginning, reducing the market area and the maximum p r o f i t an i n t r u d e r can get. The cost of t h i s s t r a t e g y f o r the o l d producer i s a lower monopoly p r o f i t . " ^ Peles d i d not go on to explore the c o n d i t i o n s under which such an entry d e t e r r i n g s t r a t e g y would be i n the best i n t e r e s t s of the f i r m . A more d e t a i l e d model of s e q u e n t i a l entry was constructed by R o t h s c h i l d [1976]. R o t h s c h i l d was p r i m a r i l y i n t e r e s t e d i n i n v e s t i g a t i n g the e q u i l i b r i u m c o n f i g u r a t i o n s which would be generated by N firms l o c a t i n g s e q u e n t i a l l y on a l i n e segment bounded by two e s t a b l i s h e d f i r m s , given that c a p i t a l i s immobile once set i n p l a c e . He d i d not concern himself w i t h the question of how competition among firm s f o r the opportun-i t y to e s t a b l i s h a new p l a n t i n the market might a f f e c t the time and place of new p l a n t c o n s t r u c t i o n . R o t h s c h i l d assumes that a l l l o c a t i o n s are occupied by d i f f e r e n t f i r m s and that s i n c e each f i r m i s s o l e l y concerned w i t h maximizing the minimum s a l e s which may accrue to i t when a l l have entered, assuming that any successors have s i m i l a r o b j e c t i v e s , each f i r m l o c a t e s so that the worst p o s s i b l e outcome f o r i t on these assumptions i s as f a v o r a b l e as p o s s i b l e . He thus r e f r a i n s from a n a l y z i n g the case i n which an e s t a b l i s h e d f i r m might consider opening a new p l a n t i n the market as opposed to a new f i r m opening a p l a n t . Hay's a n a l y s i s [1976] appeared at approximately the same time as R o t h s c h i l d ' s , and may have been i n f l u e n c e d by i t . Hay, l i k e R o t h s c h i l d , focused on the s e q u e n t i a l entry of new f i r m s given that p l a n t s are f i x e d i n l o c a t i o n once set i n p l a c e . Also l i k e R o t h s c h i l d , Hay considers only the s e q u e n t i a l entry of new f i r m s i n t o a l i n e a r market which i s l a r g e enough to support some u n s p e c i f i e d number of f i r m s , given that there i s already one f i r m l o c a t e d i n the market. (Imperfection i n i n f o r m a t i o n i s assumed to lead some firm s to enter before others.) Hay e s t a b l i s h e s the r e s u l t that under v a r i o u s assumptions, spacing of f i r m s w i l l be r e g u l a r , i . e . market areas w i l l be j u s t l e s s than twice the minimum market necessary f o r p r o f i t a b l e new entry. This type of spacing, under Hay's f o r e s i g h t assumption, i s the best entry d e t e r r i n g l o c a t i o n a l p a t t e r n from the p o i n t of view of i n d i v i d u a l f i r m s . (The gap i n the market i s i n s u f f i c i e n t to a l l o w new entry to take place.) Hay's a n a l y s i s i s e s s e n t i a l l y c o r r e c t i n the case i n which an e n t e r i n g f i r m foresees the p o s s i b i l i t y of new e n t r y , but does nothing i t s e l f , once e s t a b l i s h e d , to deter e n t r y . Once again, the p o s s i b i l i t y of an e s t a b l i s h e d f i r m d e t e r r i n g entry by opening new p l a n t s i n the market p r i o r to the p o i n t i n time when i t would pay a new entrant to enter was not examined. P r e s c o t t and V i s s c h e r [1977] a l s o consider a s e q u e n t i a l entry model i n which each f i r m ' s l o c a t i o n d e c i s i o n i s o n c e - a n d - f o r - a l l . Firms are assumed to enter i n sequence because " . . . some entrants become aware of a p r o f i t a b l e market before others or r e q u i r e longer periods of time i n g which to ' t o o l up.'" They a l s o assumed that the " . . . expectations of the f i r m about the response of other firms to i t s own d e c i s i o n s are r a t i o n a l i n the sense that the expectations are c o n s i s t e n t w i t h the 9 p r e d i c t i o n s of the model." P r e s c o t t and V i s s c h e r are i n t e r e s t e d i n using t h e i r " s o l u t i o n concept" ( i . e . , f i r m s e n t e r i n g s e q u e n t i a l l y and o n c e - a n d - f o r - a l l , w i t h each e n t e r i n g f i r m c o r r e c t l y a n t i c i p a t i n g the d e c i s i o n s of the remaining f i r m s i n the sequence of entrants) i n order to analyze the H o t e l l i n g problem and m o d i f i c a t i o n s of i t . They discus s a number of examples which, f o r the most p a r t , have l i t t l e relevance f o r the present study. However, t h e i r f o u r t h example i s of some i n t e r e s t to us s i n c e i t suggests how the r e l a x a t i o n of the one f i r m - one p l a n t - 14 -assumption i n a s e q u e n t i a l entry model can generate a preemption r e s u l t . For t h i s example, they assume that l o c a t i o n s are f i x e d when chosen and based on the observed l o c a t i o n s of e s t a b l i s h e d f i r m s and c o r r e c t expecta-t i o n s of where new entrant f i r m s w i l l enter i n the f u t u r e . They a l s o assume that f i r m s know that p r i c e s w i l l be determined noncooperatively i n Nash fa s h i o n once l o c a t i o n s are chosen. Using r e c u r s i v e and numerical methods, they compute e q u i l i b r i a under d i f f e r e n t assumptions about the s i z e of the f i x e d costs of entry and demand, and when the e q u i l i b r i u m number of firms i s l e s s than or equal to three. They f i n d that the e q u i l i b r i u m sequence i s c h a r a c t e r i z e d by successive entrants l o c a t i n g f u r t h e r apart and no f i r m choosing a l o c a t i o n a r b i t r a r i l y c l o s e to the l o c a t i o n of any other f i r m . I n a d d i t i o n , " P r o f i t s and market share are l a r g e r the e a r l i e r i n the sequence that a f i r m enters . . .". However, P r e s c o t t and V i s s c h e r note that there may be cases i n which i t pays a f i r m to enter l a t e r r a t h e r than e a r l i e r . In commenting on how to r e s o l v e t h i s indeterminacy, P r e s c o t t and V i s s c h e r came up w i t h the f o l l o w i n g i n s i g h t : I n p r a c t i c e , the indeterminacy i n such a s i t u a t i o n might be reso l v e d by a s i n g l e f i r m ' s o b t a i n i n g s u f f i c i e n t venture c a p i t a l to l o c a t e at m u l t i p l e p o s i t i o n s such that no remaining p o t e n t i a l p o s i t i o n o f f e r s p r o f i t s . The r e s u l t i n t h i s case i s complete monopoly. Indeed, s e q u e n t i a l f o r e -s i g h t e d entry r e s u l t s i n monopoly anytime-the number of l o c a t i o n s any one f i r m can choose i s not r e s t r i c t e d because a l l e q u i l i b r i u m l o c a t i o n s are p r o f i t a b l e , and we expect the f i r s t f i r m i n the sequence to choose a l l p r o f i t a b l e l o c a t i o n s i f p o s s i b l e . - 15 -Unfort u n a t e l y , P r e s c o t t and Vis s c h e r d i d not provide a more rig o r o u s a n a l y s i s of t h i s p o s s i b i l i t y . P r e s c o t t and V i s s c h e r , i n t h e i r f i f t h example, provide f u r t h e r i n t u i t i v e d i s c u s s i o n of t h e i r contention that " . . . making entry i n t o the i n d u s t r y endogenous i s c r u c i a l and . . . that without a c o n s t r a i n t on the number of ' l o c a t i o n s ' any one f i r m may occupy, the r e s u l t i n g i n d u s t r y s t r u c t u r e i s monopoly." In t h i s a n a l y s i s , they i n t e r p r e t l o c a t i o n as corresponding to a choice of p h y s i c a l p l a n t c a p a c i t y . They assume that f i r m s produce a homogeneous product, that market p r i c e i s determined by t o t a l p l a n t c a p a c i t y i n the i n d u s t r y , that marginal revenue i s a decreasing f u n c t i o n of i n d u s t r y c a p a c i t y , that f i x e d costs are p o s i t i v e and marginal costs are constant. They argue that the f i r s t f i r m i n the market " . . . c l e a r l y p r o f i t s more by extending c a p a c i t y to the u l t i m a t e i n d u s t r y size."''"''' P r e s c o t t and Vi s s c h e r support t h i s c o n c l u s i o n by n o t i n g that i f the f i r m stopped short of b u i l d i n g the " l i m i t c a p a c i t y " , a d d i t i o n a l entry would occur and the market p r i c e would be the demand p r i c e corre^-. spondingjt.o t o t a l i n d u s t r y c a p a c i t y . "Had the f i r s t f i r m chosen the e n t i r e i n d u s t r i a l c a p a c i t y , however, market p r i c e would be no d i f f e r e n t , f u r t h e r entry would s t i l l be f o r e s t a l l e d , y e t the f i r s t f i r m would s e l l 12 more than had i t chosen smaller c a p a c i t y . " This a n a l y s i s , w h i l e suggestive, f a l l s short of the a n a l y s i s of excess c a p a c i t y as a b a r r i e r to entry provided by Spence [1977], which appears i n the same i s s u e of the B e l l J o u r n a l . F i n a l l y , P r e s c o t t and V i s s c h e r note s e v e r a l reasons why a f i r m may f a i l to occupy a l l the a v a i l a b l e l o c a t i o n s . There may be f i n a n c i a l c o n s t r a i n t s on expansion, diseconomies of s c a l e to m u l t i - p l a n t expansion, u n c e r t a i n t y regarding the extent of the market, or s i g n i f i c a n t c o s t s of - 16 -r a p i d expansion of f i r m s i z e p o s s i b l y due to the costs of screening new personnel. Thus, P r e s c o t t and V i s s c h e r have a r t i c u l a t e d the i n c e n t i v e which an e s t a b l i s h e d f i r m has to preempt the market, and have a l s o pointed out reasons why we would: not expect to observe p e r f e c t preemption i n the r e a l world. In the next s e c t i o n , we d i s c u s s two papers which f o r m a l l y recognize the p o s s i b i l i t i e s of preemption. 2.4 Brand P r o l i f e r a t i o n and Preemptive D i v e r s i f i c a t i o n In the previous two s e c t i o n s , we have examined two bodies of l i t e r a t u r e which l a i d the basis f o r the theory of preemption. The f i r s t e x p l i c i t reference to preemption as a b a r r i e r to entry i n a s p a t i a l context appears to have been made by A r c h i b a l d and Rosenbluth [1975]. A r c h i b a l d and Rosenbluth were p r i n c i p a l l y concerned w i t h r e c a s t i n g the theory of m o n o p o l i s t i c competition i n terms of c h a r a c t e r i s t i c s space. In doing so, i t was hoped that some of the .weaknesses' of the Chamberlinian theory, such as the d e f i n i t i o n of a group and the e f f e c t s of i n t r o d u c i n g new d i f f e r e n t i a t e d products i n t o the group, could be remedied. I n d i s c u s s i n g the i m p l i c a t i o n s of t h e i r model, A r c h i b a l d and Rosenbluth note that l i m i t p r i c i n g w i l l not be an e f f e c t i v e entry d e t e r r i n g s t r a t e g y , but preemptive d i v e r s i f i c a t i o n w i l l . That i s , i t w i l l pay e s t a b l i s h e d firms to occupy neighboring p o s i t i o n s i n c h a r a c t e r i s t i c s space, provided the market i s "dense" enough to support the i n t r o d u c t i o n of a new product. I f the p o i n t of expected entry i s bounded by two e x i s t i n g f i r m s , then i t w i l l pay e i t h e r of the e x i s t i n g f i r m s to preempt the market from a new ent r a n t , mainly because the e x i s t i n g f i r m s would be able to c o n t r o l the p r i c e of the new product, although i t i s not p o s s i b l e to say which f i r m w i l l a c t - 17 -upon the incentive to preempt f i r s t . "We may have here an explanation of the proliferation of product variants and brand names in which so many 13 firms engage, apparently even at the expense of economies of scale." Schmalensee [1978] has also focused on the possibilities of using preemptive diversification or brand proliferation as a barrier to entry. As mentioned i n Chapter 1, the lack of significant new entry over a long period of time into the ready-to-eat breakfast cereal industry was believed to be due i n part to frequent introductions of new brands by established firms. Schmalensee analyzed the plaus i b i l i t y of such an entry deterring strategy i n this context. Schmalensee makes the following assumptions: (i) " . . . for individual brands, at least at low levels of output, the unit cost of production and marketing f a l l s with increases i n output." (For " i l l u s t r a t i v e purposes", Schmalensee specifies that long run total cost of producing and marketing a typical brand is given by C(q) = F + vq, where F and v are positive constants and q i s output of the brand. He does not specify a capacity constraint.)"' ( i i ) Localized rivalry among brands. Since Schmalensee uses a one-dimensional, linear spatial model for his analysis, localized rivalry i s ". . . present in an extreme (and thus tractable) form: normally only the two brands between which an entrant locates would be affected by changes i n , for instance, i t s price.""^ ( i i i ) Brand locations cannot be changed. He then shows that the optimal entry deterring strategies of established firms are to charge prices which maximize the unconstrained profits of the established brands, and to choose a number of brands which exceeds the unconstrained profit maximizing number. (Schmalensee i s able to obtain - 18 -a determinate solution to the industry's profit maximizing strategy by assuming that established firms collude to deter entry at minimum cost to themselves. Thus, in effect, the oligopoly is treated as a monopoly, and Schmalensee finds that i t i s in the monopolist's interests to protect i t s monopoly position by preempting the market.) Schmalensee also argues that brand proliferation i s superior to limit pricing i n i t s a b i l i t y to actually forestall entry. If firms attempt to use limit pricing as a barrier to entry, a potential entrant may nevertheless enter. Given that the new entrant's costs are now sunk, a l l firms w i l l see that i t i s to their mutual advantage to raise price. Thus, limit pricing may lack credi b i l i t y as an entry barrier. On the other hand, ". . . i f the established firms can crowd economic space with brands before the threat of entry appears, as we have been assuming, the entry-deterring threat is that the brands w i l l not be moved i f entry occurs. Since repositioning brands i s assumed to involve substantial costs, such a threat i s quite credible. Finally, Schmalensee argues that the structure and performance of the ready-to-eat cereal industry are consistent with established firms actually behaving i n a preemptive fashion. After analyzing the welfare implications of entry deterring behavior, he suggests policies which should improve the performance of the industry. Schmalensee's analysis, while persuasive, does not rigorously test for the existence of preemptive behavior. In a later chapter of this thesis, we shall draw out the empirically testable implications of the theory of preemption and test them using location data and cost and revenue data drawn from the supermarket industry. - 19 -2.5 The Eaton and. L i p s e y Model of Market Preemption Eaton and Lipsey [1979] have provided us w i t h the most t h e o r e t i c a l l y complete treatment of preemption as a b a r r i e r to entry i n a growing, s p a t i a l l y extended market, Since t h i s model w i l l form the b a s i s f o r the t h e o r e t i c a l a n a l y s i s i n the next chapter, I s h a l l summarize i t i n some d e t a i l , Eaton and Lipsey consider a one-dimensional market, two u n i t s i n len g t h , w i t h a uniform d i s t r i b u t i o n of customers of density 6, This market i s represented by Figure 1. Figure 1 "1 I —| 1 1 ; 1 -1 -2/3 0 „ 2/3 1 They assumed that each f i r m has the same cost curve and that production i s subject to i n c r e a s i n g returns to s c a l e at the p l a n t l e v e l over some range of output. Firms, maximize p r o f i t s , and t h e i r c a p i t a l i s immobile once set i n place due to l o c a t i o n s p e c i f i c sunk c o s t s . Consumers are assumed to have i d e n t i c a l , demand curves, and demand i s a f u n c t i o n of d e l i v e r e d p r i c e , m i l l p r i c e plus transport costs (which are an i n c r e a s i n g f u n c t i o n of d i s t a n c e ) . Each, consumer buys from.the p l a n t w i t h the lowest d e l i v e r e d p r i c e . An e s t a b l i s h e d f i r m , F^,, i n i t i a l l y l o c a t e d at the o r i g i n , and an i n f i n i t e number of p o t e n t i a l entrants are assumed, capable of a c c u r a t e l y c a l c u l a t i n g the flows of costs and revenues that w i l l be a s s o c i a t e d w i t h - 20 -any p l a n t . (Eaton and Lipsey c a l l t h i s expectations assumption a form of co n s i s t e n t e x p e c t a t i o n s , or those expectations that are c o n s i s t e n t w i t h r e a l i z a t i o n s . ) A l l f i r m s are a l s o assumed to perceive c o r r e c t l y that the market w i l l grow at a time t2 such that one new p l a n t e s t a b l i s h e d i n each of the i n t e r v a l s (-1:0) and (0:1) would earn pure p r o f i t s i f e s t a b l i s h e d at time F i n a l l y * Eaton and Lipsey assume that d e n s i t y of consumers i s i n i t i a l l y such that the e s t a b l i s h e d f i r m i s earning pure p r o f i t s , but that entry of new fi r m s w i l l not o c c u r . ^ The f i r s t p r o p o s i t i o n which they e s t a b l i s h i s c a l l e d preemption by new e n t r a n t s : " i f the e x i s t i n g monopolist does not e s t a b l i s h new ca p a c i t y to meet the increased demand, competition among p o t e n t i a l new entrants w i l l l e a d to the establishment of new ca p a c i t y some time before the date at which demand i n c r e a s e s . " R e c a l l that the market i s expected to grow d i s c r e t e l y a t time and assume that the e s t a b l i s h e d f i r m does nothing to block entry. Given t h a t there i s a l a r g e number of p o t e n t i a l entrants competing f o r the opportunity to e s t a b l i s h a new p l a n t i n the market, the time of entry w i l l be pushed back to a time t ^ < t2 such that the present value of each of the p l a n t s e s t a b l i s h e d i n the i n t e r v a l s (-1:0) and (0:1) w i l l be zero. The second p r o p o s i t i o n e s t a b l i s h e d by Eaton and Lipsey i s c a l l e d monopoly preemption: "the e x i s t i n g monopolist w i l l always f i n d i t p r o f i t a b l e to preempt the market by e s t a b l i s h i n g new cap a c i t y at a time j u s t e a r l i e r than the e a r l i e s t date at which any p o t e n t i a l new entrant would f i n d i t p r o f i t a b l e to do so." Suppose that the e s t a b l i s h e d f i r m , F-j , foresees the p o s s i b i l i t y of new - 21 -f i r m entry. I f the e s t a b l i s h e d f i r m ' s p r i c e s and l o c a t i o n s were p r e c i s e l y the same as those of new en t r a n t s , then the opportunity to e s t a b l i s h new pl a n t s i n the market would be worth as much to the e s t a b l i s h e d f i r m as they would be to the new entrants. However, the e s t a b l i s h e d f i r m would not adopt the same p r i c e s and l o c a t i o n s as new entr a n t s . F i r s t , the e s t a b l i s h e d f i r m would l o c a t e i t s p l a n t s at the j o i n t p r o f i t maximizing l o c a t i o n s (-2/3 and 2/3), w h i l e the new entrants would l o c a t e new p l a n t s to the r i g h t of -2/3 and to the l e f t of 2/3 f o r the reasons advanced by H o t e l l i n g [1929] i n h i s pion e e r i n g a r t i c l e on s p a t i a l competition. Second, the e s t a b l i s h e d firm.would s e l e c t the j o i n t p r o f i t maximizing p r i c e s , w h i l e entry by a new f i r m would be expected to r e s u l t i n p r i c e s which are e i t h e r permanently or temporarily depressed below the monopoly p r i c e s which the e s t a b l i s h e d . f i r m would charge i f i t owned the new p l a n t s . Eaton.and Lipsey then s t a t e that "The value to F^ of monopoly preemption depends on the d i f f e r e n c e between the p r o f i t a b i l i t y of the market when three p l a n t s are owned by F^ and when two of the pl a n t s are owned by new en t r a n t s , " and the preceeding argument c l e a r l y demonstrates that t h i s d i f f e r e n c e i s p o s i t i v e . Thus f i r m F^ w i l l have an i n c e n t i v e to e s t a b l i s h new p l a n t s i n the market j u s t p r i o r to. t ^ , which i s the e a r l i e s t date at which a new entrant would f i n d i t p r o f i t a b l e to enter. In the remainder of t h e i r paper, Eaton and Lipsey consider extensions of the a n a l y s i s , d i s c u s s the r o l e that expectations p l a y i n the model, and als o compare t h e i r s p a t i a l model of preemption w i t h spaceless models. However, it> i s the model presented above which we w i l l f i n d most u s e f u l as the basis, f o r our t h e o r e t i c a l d i s c u s s i o n i n the next chapter. - 22 -FOOTNOTES TO CHAPTER 2 1. Perhaps s u r p r i s i n g l y , . many w e l l known entry b a r r i e r s , have yet to be s c r u t i n i z e d . b y marketing researchers and r e t a i l geographers. Marketing researchers seem to have been preoccupied w i t h the c a l c u l a t i o n of trade areas, to the exclusion, of. the a n a l y s i s of f i r m s ' s t r a t e g i c a l t e r n a t i v e s w h i c h : a f f e c t these trade.areas, (see Applebaum and Cohen [1960, 1961], B u c k l i n [1967], Cohen and Lewis [1967], and Huff [1963]). R e t a i l geographers have been p r i n c i p a l l y concerned w i t h extending and r e v i s i n g Berry's r e f o r m u l a t i o n of C h r i s t a l l e r ' s theory of c e n t r a l , p l a c e s (see Berry and G a r r i s o n [1958a, 1958b, 1958c], Berry [1958, 1963], and Berry, Barnum and Tennant [1962]). A good d e a l of t h i s e f f o r t has. gone i n t o r e f i n i n g v a r i o u s i n d i c e s of c e n t r a l i t y and i d e n t i f y i n g the nature and existence of h i e r a r c h i e s of centers (or c e n t r a l p l a c e s ) . (For a r e p r e s e n t a t i v e sample of t h i s l i t e r a t u r e , see Garner [1966], Simmons [1964, 1966], Brush and Gauthier [1968], Nystuen: [1959];, and M a r s h a l l [1969].) Hence, most of the l i t e r a t u r e i s , e m p i r i c a l , w i t h a very weak underlying t h e o r e t i c a l base. Indeed, most c e n t r a l place t h e o r i s t s do not have a theory of f i r m behavior, as they g e n e r a l l y take the economic landscape as already given. 2. On the subject of l i m i t p r i c i n g , see Baron. [1972, 1973], Bhagwati [1970], Kamien and Schwartz [1971, 1972], M o d i g l i a n i [1958], Osborne 11964, 1973], Pashigian 1 1 9 6 8 ] , P y a t t 1 1 9 7 1 ] , Sherman and W i l l e t t [1967], Shubik [1959], and Bain [1965] ? L a t s i s [1976] has echoed the sentiments of many c r i t i c s of l i m i t p r i c i n g .theory:in the f o l l o w i n g passage: "Thus the theory does not e x p l a i n why a p o t e n t i a l entrant - 23 -should not c a l l the bluff of the established firms by entering the industry when the limit price is being charged,. The theory can give no reason why the established firms would continue to produce the limit quantity'in the changed circumstances, since their profits might well be larger i f they reduced their output and co-existed peacefully with the new entrant, particularly i f i t were clear that the new entrant was prepared to endure a long period of unprofitable operations. If i t were to be inferred from the theory that the established firms would be prepared to suffer losses i n order to force out the new entrant rather than accommodate him in this industry, a further question is raised. Without setting the price at the level of the limit price, the established firms could communicate the threat that they would deal with the problem of new entry by enforcing losses on any new entrant who enters their industry. The theory does not explain why the communication of such a threat should be ruled out, nor why the only effective communication of a threat is that implicit in the adoption of the limit price." 3. Pashigian [1968; 166]. 4. Wenders [1971; 15.1. 5. Esposito and Esposito [1974; 189]. 6. "High seller concentration, high barriers to entry and an insignificant competitive fringe characterize tight oligopolistic markets." Esposito and Esposito [1974; 188-189]. 7. Peles 11974; 628], - 24 r-8. P r e s c o t t and.Visscher [1977; 379J. 9. I b i d . , pp. 379-380. 10. I b i d . , p. 390. 11. I b i d . , pp. 390-391. 12. I b i d . , p. 390. 13. A r c h i b a l d and Rosenbluth [1975; 589]. 14. Schmalensee [1978; 307-308]. 15. I b i d . , p. 310. 16. I b i d . , p. 314. 17. For a. thorough d i s c u s s i o n of the p o s s i b i l i t y of excess p r o f i t s i n a f r e e entry e q u i l i b r i u m , see Eaton and Lipsey [1978]. Chapter 3 THE THEORY OF MARKET PREEMPTION 3.1 I n t r o d u c t i o n In the previous chapter, we discussed: the Eaton and Lipsey model of preemption i n some d e t a i l . This model w i l l form the b a s i s f o r our a n a l y s i s of preemption i n t h i s chapter. The a n a l y s i s commences w i t h a di s c u s s i o n : o f the p o s s i b i l i t y of preemption i n spaceless models. We argue that w h i l e f i r m s do not have any i n c e n t i v e to preempt i n a p e r f e c t l y competitive world, i n c e n t i v e s : to preempt e x i s t i n a market c h a r a c t e r i z e d by n a t u r a l monopoly. In s e c t i o n 3.2', we develop, a model of preemption i n space which uses the same b a s i c s p e c i f i c a t i o n as the Eaton and Lipsey model. We demonstrate that the e s t a b l i s h e d f i r m w i l l always have an i n c e n t i v e to preempt the market. In s e c t i o n 3.3, we develop a model of preemption which r e l a x e s the i n f i n i t e competitive f r i n g e assumption of the model constructed i n s e c t i o n 3.2. In s e c t i o n 3.4, we discuss the two-dimensional analogue of the one-dimensional s p a t i a l model. F i n a l l y , i n s e c t i o n 3.5, we i n d i c a t e reasons why we might not expect to observe p e r f e c t preemption i n the r e a l world. 3.2 Preemption i n a Spaceless Market Preemption of the. market i s a l o g i c a l p o s s i b i l i t y i n a spaceless world. I t w i l l be argued below that, w h i l e preemption can occur i n the n e o c l a s s i c a l p e r f e c t l y competitive w orld, there i s no i n c e n t i v e f o r f i r m s to engage i n t h i s type of behavior? Other types of market s t r u c t u r e are necessary i n order to provide firms w i t h such an i n c e n t i v e . I t i s u s e f u l to consider the problem of preemption i n a spaceless market i n t u i t i v e l y , -IS-as i t w i l l provide some background to the spatial models developed in the later sections of this chapter. In the neoclassical theory of perfect competition, i t is assumed that there are many firms, each producing a homogeneous product. Each firm believes that i t is too small, relative to the market, to have any influence on price. In the short run, firms may be making above normal profits. However, these pure profits attract entry into the market by other firms. Quantity supplied i s increased and price decreases u n t i l the familiar long run equilibrium with no incentive for entry or exit is achieved. In the scenario outlined above, we may think of each firm as being characterized by constant returns to scale or returns to scale which are insignificant with respect to the market. Suppose that a l l firms (existing firms and potential entrants) anticipate that at some time in the future, t^, there w i l l be a discrete increase in demand such that new capacity could be profitably established in the market to meet that anticipated increase in demand."'" (We consider a discrete rather than continuous increase in demand in order to simplify the exposition. We should also note that even though the increase in demand is discrete, the present value of profits to a firm w i l l s t i l l be a continuous function of time.) It is clear that capacity expansion would occur at time t^, and not before. If any firm contemplated establishing enough new capacity prior to time such that i t would supply a substantial amount of the anticipated discrete increase in demand, then i t would expect i t s flow profits (the difference between i t s current revenues and current costs inclusive of normal profits) to be negative. When the increase in demand occurs at time t , capacity - 27 -expansion would take place such that above normal p r o f i t s would be e l i m i n a t e d . Thus, the f i r m that e s t a b l i s h e d new c a p a c i t y p r i o r to t ^ could not expect to balance o f f l o s s e s p r i o r to a g a i n s t above normal p r o f i t s a f t e r t^. This i m p l i e s that there i s no i n c e n t i v e i n the p e r f e c t l y competitive model f o r any f i r m to preempt the market. Each f i r m would be i n d i f f e r e n t as to whether or not any other f i r m preempted the market s i n c e market preemption cannot guarantee an above normal r a t e of r e t u r n to a f i r m i n a p e r f e c t l y competitive world. Now, consider a market i n which each f i r m i s c h a r a c t e r i z e d by i n c r e a s i n g returns to s c a l e at the p l a n t l e v e l . For s i m p l i c i t y , i t i s assumed that at time t ^ (the i n i t i a l c o n d i t i o n ) , there i s only one f i r m serving the market. As i n the case discussed above, a d i s c r e t e increase i n demand at time i s a n t i c i p a t e d by a l l f i r m s such that p o s i t i v e flow p r o f i t s could be earned by e s t a b l i s h i n g a new p l a n t i n the market at time t^. E i t h e r the e x i s t i n g f i r m or a new entrant could e s t a b l i s h a new plant i n the market p r i o r to t^. Since there are many p o t e n t i a l e n t r a n t s , the a c t u a l time of new f i r m entry would occur at t^< such that the present value of p r o f i t s to the new entrant i s equal to zero. In the above model, preemption can occur and f i r m s have the i n c e n t i v e to preempt the market. The monopolist's i n c e n t i v e to preempt derive s from the prospect of o b t a i n i n g a higher present value by preempting the market than i t would o b t a i n i f i t allowed a new f i r m to enter. P o t e n t i a l entrants have an i n c e n t i v e to preempt due to the prospect of o b t a i n i n g a present value greater than or equal to zero by doing so. In c o n t r a s t to the competitive case, the i n c e n t i v e to preempt the market would lead the e x i s t i n g f i r m or a new entrant to e s t a b l i s h a - 27a -new plant in the market at a time prior to the actual increase in density. 3.3 A Spatial Model of Preemption When space is introduced into an economic model of firm behavior, the decision variables of the firm and set of strategic alternatives are augmented. This expansion of the choice set makes i t possible to explain some firm behavior which might otherwise appear to be inexplicable or perverse i n the context of a spaceless model. Perhaps the most fundamental characteristic of a l l location problems is the recognition of i n d i v i s i b i l i t i e s . If i n d i v i s i b i l i t i e s were nonexistent and i f transportation were costly, a l l production would take place at the point of consumption. This implies that "... without recognizing i n d i v i s i b i l i t i e s - in the human person, in residences, plants, equipment, and in transportation - urban location problems, down to those of the n2 smallest village cannot be understood. Capital i n d i v i s i b i l i t i e s in a spatial world also permit firms to make pure profits without attracting 3 additional entry into the market. It w i l l be argued that firms have an incentive to protect their pure profits by preempting a growing spatially extended market. Attention w i l l be confined in this section to a one-dimensional market, two units in length, as described by Figure 2. Figure 2  X l I 1 1 1 \ -1 -2/3 0 2/3 1 - 28 -I t i s a n a l y t i c a l l y convenient to conduct the a n a l y s i s using a l i n e a r market. However, the r e s u l t s do g e n e r a l i z e to a two-dimensional landscape, as w i l l become apparent i n a l a t e r s e c t i o n . The a n a l y s i s of t h i s s e c t i o n w i l l be based upon the f o l l o w i n g assumptions: ( i ) The f i r m maximizes i t s present value at each moment i n time. I t does so by choosing a l o c a t i o n f o r the establishment of a new p l a n t , and the p r i c e s to be charged by a l l of i t s p l a n t s i n the market. ( i i ) Demand i s a f u n c t i o n of d e l i v e r e d p r i c e : Q = y(q) , where q = p+zX, p = an index of the f i r m ' s p r i c e , z = t r a n s p o r t costs which are constant per u n i t of d i s t a n c e per u n i t of the good, and X i s the d i s t a n c e the consumer t r a v e l s to the p l a n t . This assumption i m p l i e s that a l l consumers have i d e n t i c a l t a s t e s . ( i i i ) Assumption ( i i ) i m p l i e s that u t i l i t y maximizing consumers w i l l p a t r o n i z e the p l a n t o f f e r i n g the goods they d e s i r e to purchase at the lowest d e l i v e r e d p r i c e . ( i v ) The cost f u n c t i o n faced by each of the firm's p l a n t s i s c h a r a c t e r i z e d by i n c r e a s i n g returns to s c a l e over some range 4 of output. (v) The f i r m ' s p l a n t s , once e s t a b l i s h e d , are f i x e d i n l o c a t i o n due to l o c a t i o n s p e c i f i c sunk cost s . The f i r m ' s c a p i t a l i s i n d i v i s i b l e and immobile once set i n place.~* ( v i ) Firms are able to foresee the outcome of the competitive process, and are thus able to c a l c u l a t e t h e i r own and t h e i r competitors' returns contingent upon the p u r s u i t of v a r i o u s - 29 -strategies. Thus, we follow Eaton and Lipsey [1979] and assume that firms have what they c a l l "consistent expectations": expectations that are consistent with realizations. An important implication of the above assumptions in the context of a spatial model is that i n i t i a l conditions do not wash out. In a spatial model, the assumptions of i n d i v i s i b i l i t i e s and sunk costs imply that the i n i t i a l distribution and size of firms have an impact upon the equilibrium configuration which is eventually attained. We now specify a set of i n i t i a l conditions. One firm, which w i l l be referred to as F^, i n i t i a l l y has a plant located at the origin at time tg. At time t^, the customer density, 6^  (which is assumed uniform), i s such as only to allow the profitable operation of one plant at the origin, and even though the firm may be making above normal (or pure) profits, no potential entrant would find i t profitable to enter. It i s assumed that F^ and an in f i n i t e number of potential entrants correctly perceive that at some time in the future, t^, customer density w i l l increase to 5^ such that two new plants, one in the interval (-1:0) and the other in the interval (0:1), would earn pure profits i f established at t^. Since the market i s symmetric about the origin, the following analysis w i l l be confined to examining the po s s i b i l i t i e s for entry on the right hand side of the market. Let us now begin the argument by inquiring as to when a potential entrant would establish a new plant in the market described by Figure 2. It i s clear that i f the opportunity were available, a new entrant would like to enter the market at time t ? , when the present value of profits - 30 -is positive and pure profits can be made. However, since the increase in density at i s foreseen, and since there are many new entrants competing for the opportunity to set up a new plant in the market, the potential entrant that actually enters the market w i l l do so at time t^, such that the present value of it s profits i s equal to zero, and i t earns a normal rate of return. In other words, the only way for the successful new entrant to enter the market at time t^ i s for i t to preempt the market from i t s competitors by constructing a plant in the market at that date. The potential entrant or competitor, would thus enter the market at time t^ and choose i t s price and location in order to maximize the present value of i t s profits: (3.1) max (p 2 2, x 2 2) where t n n 2 ( p22' X22 ; P l l ' Xu> e _ r t d t = v 2> Xi£ = t* i e l° c a t :*- o n °^ plant Z when i t is owned by firm F_^  (Z. = 1 refers to the established plant, while £ = 2 refers to the new plant) I?i£ = the mi l l price charged at plant t when i t is owned by firm F_^  r = the firm's discount rate 2 II = F2's profit function. F2 maximizes the present value of i t s profits, taking the i n i t i a l location of F^(X^) as given. If F2 had successfully entered the market at time t^, then F^ would simply maximize the present value of i t s profits: (3.2) max <pu> J t - 31 -1 - r t 1 n (P 1 ; l; x l ; L, p 2 2, x 2 2) e dt = V . 1 F^ maximizes i t s present value, tak i n g as given the l o c a t i o n of i t s p l a n t ( X - Q ) » and the l o c a t i o n of f i r m F 2's p l a n t ( X 2 2 ) . Suppose F^ had e s t a b l i s h e d the new p l a n t i n the market at time t ^ r a t h e r than F 2 < Then F^ would wish to maximize the present value of i t s p r o f i t s over the e n t i r e market: (3.3) max ^ L ( p n , p 1 2 , x 1 2) 1=1 fc l[t < P i r p i 2 > x i 2 > xn)e"rtdt = v 1 , where II ^ = F^'s p r o f i t f u n c t i o n f o r i t s ttb. p l a n t . That i s , F^ would choose the l o c a t i o n of i t s new p l a n t and p r i c e s p ^ and p^ 2 i n order to maximize the present value of p r o f i t s of both p l a n t s . (We p l a c e a t i l d e over the V to d i s t i n g u i s h j o i n t present value maximization on the p a r t of F^ from i n d i v i d u a l present value maximization on the p a r t of and F 2 > ) We now enquire as to whether or not F^ has an i n c e n t i v e to preempt the market, i . e . to e s t a b l i s h a new p l a n t i n the market at time t ^ -• e (where e i s a r b i t r a r i l y small) before the p o t e n t i a l entrant would f i n d i t p r o f i t a b l e to enter. F i r s t , we know that i t must be the case that (3.4) V 1 ( p n , p 1 2 , X 1 2 ; X n ) > V 1 ( p n ; X n , p 2 2 , X 2 2 ) + V 2 ( p 2 2 , X 2 2 ; p n , x u ) - 32 or (3.5) V 1 ( p l l S p 1 2 , X 1 2 ; X n ) - V 1 ( p u ; X u > v l V *22> * 0 2 since V (P22' X 2 2 ' p l l ' X l l ^ = ^' T ^ a t 1 S> t n e j o i n t l y maximized present value of p r o f i t s of F^ must be greater than or equal to the sum of the independently maximized present value of p r o f i t s of F^ and Y^. However, we s h a l l argue t h a t , i n g e n e r a l , (3.5) would hold as a s t r i c t i n e q u a l i t y . This means that the present value of p r o f i t s to F^ obtained by preempting the market at time t ^ i s s t r i c t l y greater than the present value of p r o f i t s which i t could o b t a i n i f i t d i d not preempt the market at time t ^ . (In other words, i s the opportunity cost of pursuing the preemptive strategy.) Thus, we may conclude from t h i s that i f (3.5) holds as a s t r i c t i n e q u a l i t y , F^ w i l l have an i n c e n t i v e to preempt the market at time t ^ - e. Consider the market described by Figure 3. Figure 3 X For expository convenience, we have suppressed a l l s u b s c r i p t s on p and X which r e f e r to the f i r m . In a d d i t i o n , we s h a l l l e t X represent the l o c a t i o n of the new p l a n t . A c t u a l m i l l p r i c e s are thus represented by - 33 p^, w h i l e q represent expected d e l i v e r e d p r i c e s at the market boundaries, We denote the market boundary between F-^'s p l a n t l o c a t e d at the o r i g i n and the p l a n t l o c a t e d at X as b. The d e l i v e r e d p r i c e s at b and 1 are (3.6) (3.7) q1 = 1/2 ( P ; L + P 2 + zX) q 2 = p 2 + z ( l - X ) r e s p e c t i v e l y , where z = transport costs which are constant per u n i t of distance per u n i t of the good.'' We may c a l c u l a t e the q u a n t i t y demanded at any p o i n t i n time from F-^'s P l a n t a t t n e o r i g i n > given that there i s another p l a n t at l o c a t i o n X, as f o l l o w s : (3.8) Q 1 = 6 0 y(p^ + zb)db = 6/z y(p)dp = 6/z[Y( q ; L) - Y ( ? 1 ) J , where y ( p 1 + zb) i s the demand f u n c t i o n and [ Y ^ ) - Y ( p 1 ) ] i s the d e f i n i t e g i n t e g r a l from p^ to q . The quantity demanded from the p l a n t at X i s then (3.9) Q 2 = 6/z[ *1 y(p)dp + y(p)dp] = 6 / z [ Y ( q i ) + Y(q 2) - 2 Y ( p 2 ) ] . The cost f u n c t i o n of a given p l a n t i s obtained as f o l l o w s : Assume that input p r i c e s are constant. Then the f i r m must choose c a p i t a l (K) and labor (L) i n order to (3.10) min irK + wL (K, L) subject to the c o n s t r a i n t that Q = Q(K, L ) , where r = the i n t e r e s t r a t e - 34 -and w = the wage rate. Cost minimization yields the minimum total cost function for each plant C(Q), where we have suppressed input prices since they are assumed constant. We assume that the cost function i s continuous, twice differentiable, non-decreasing in Q, and that i t satisfies the following conditions: (3.11) C > 0, C" < 0 for any output less than Q C > 0, C" = 0 for output equal to Q C > 0, C" > 0 for any output in the interval (Q:Q) C > 0, C" = 0 for any output greater than or equal to Q. These conditions imply that the average cost function is decreasing in output for any output less than Q, and becomes horizontal for any output ~ 9 greater than or equal to Q. The empirical phenomenon which we wish to capture with this particular specification of the cost function i s increasing returns to scale at the plant level over some range of output. Now suppose that establishes a new plant in the market at time t^. F^ would then choose i t s prices, p^ and p^, and the locat ion of the new plant, X, in order to maximize the present value of profits: t, (3.12) V 1 = 2 [P1~Q1 + P 2 ~ Q 2 - C 1 ( V - C 2 ( V ] ^ d t  t l + [ P A + P 2 ~ Q 2 - C 1 ( V - C 2 ( V ] e _ r t d t t2 P l 6 0 P2 60 & [ Y ( q i ) - Y ( P l ) ] + - ^ [ Y ( q i ) + Y ( q 2 ) - 2 Y(p 2)] - 35 -- C 1(6 0/ Z[Y(q 1) - Y ( P l ) ] ) - C 2 ( 6 0 / Z [ Y ( q i ) + Y ( q 2 ) - 2Y(p 2)J) '2 - r t f P1 61 e r t d t + { [Y(qJ - Y(p_)] z 1 l p 2 6 l + [Y( q i) + Y(q 2) - 2Y(p 2)] - ^ ( S ^ z t Y C q ^ - Y ( P l ) ] ) - C^ j / z tYCq , ) +Y(q 0) - 2Y(p0)]) 1 _ r t ^ e dt, where 6^  = the customer density in the market between t^ and t 2 and 6^  = the customer density in the market at t 2 and after. The f i r s t order conditions for joint profit maximization by are as follows: (3.13) 3V1 = 3X p^yCqJ y ( q J 3C y(q x) 3Cn ^ 9X y ( q i ) - y(q 2 )] 60/z e r t d t + i y ( q j 3c, y(q x) 3Cr ^ 3X y(q\> [—2^- - y ( q 2 ) l i 00 - 36 -(3.14) 3 P i + _ [___ _ y ( ^ } ] + 30^  3P^  y ( q j V z [ _ 2 " y ( p l ) ] ~ 3C, 3 P i 6 0 y ( V 2z e-r tdt + + ¥ 1 y ( ^ [ — 2 ^ - y ( P l ) ] + 6 i p 2 y ( V 2z 3^ 3p. y(q,) 8JZ\—~ - yCpJ] - . 61 1 2 1 3 p l 9C2, 6 i y ( q i ) 2z - r t e dt = 0 (3.15) 3V 3p, P2 60 y ( ^ l } 6 0 P l y ( V &Q  + ~1T~ [ + y < V - 2y<p2>] + ^ - b - ^ 5 3p, 6 0 y( q i ) 3C2 2z 3p, y(q,) V z [ ~ 2 ~ + y ( V " 2 y ( P 2 ) ] 1 - r t , e dt Po«n yCqJ 2 + 7(q 2) s 1p ny(q 1) 2 y ( V ^ + 2z 3 P . 6 i y ( q i ) 3C2 6 1 2 2 3 p2 y ( q j V z [ _ 2 + y ( V " 2 y ( P 2 ) ] 1 e- r tdt = 0. - 37 -We use hats over variables to represent optimal values. In addition, we 9C 3C use the notation Q -r— o <3p and „ 6 3X to represent the fact that output and 6 the partial derivatives of the cost function with respect to price and location are dependent on the particular density in the market. It can be shown that F^ would select a location of 2/3 for i t s second plant, and both plants would charge the same price. This solution would maximize the joint present value of profits of F^.^ We now suppose that F^, instead of F^, is able to enter the market characterized by Figure 3 at time t^. We may capture the oligopolistic interdependence by making the price F^ would charge a function of F 2's price and location, (3.16) p± = g(p 2, X), and the price would charge a function of F^'s price, (3.17) P 2 = h ( P ; L ) . (Note that F^'s plant location does not appear in (3.17) since i t is fixed by assumption.) We shall c a l l the function g(') F-^ 's price response function and the function h(«) F 2's price response function. Conditions on ?^'s a n d ^2* S Pr^"ce r e s P o n s e functions w i l l be derived such that i f these conditions hold, F^ would have no incentive to preempt the market. We then ask i f these conditions are reasonable. F 2 would choose a location and price to maximize the present value of it s profits: - 38 -(3.18) V = [P 2Q 2 - C 2(Q 2)] e - r t d t + [P 2Q 2 - C 2(Q 2)] e" r tdt P2 60 [Y( q ; L) +Y(q 2) - 2Y(p 2)] -C2(60/Z[Y(q±) +Y(q 2) 2Y(p2)]) - r t e " d t + ^p[Y( q i)+Y(q 2) - 2Y(p 2)] C 2(6 1/z[Y(q 1) +Y(q 2) - 2Y( ? 2)]) e dt. Maximizing (3.18) with respect to X, we obtain (3.19) cTVj 3X y < V P 2y(q 1)g 2(p 2,x) sc 2 P2[~2 - y ( V ] + 2z " W~ I — y ( q 2 ) ] Jo 3C, 3X y(q 1)g 2(p 2,x) 1 2z 60/z e r t d t + < y(q x) v2i—— - y(^ 2 }] + p 2y(q 1)g 2(p 2,x) ac 2j y( q i) 2z 3X 2 y(q 2)] 3C, 3X y(^ 1)g 2(p 2,x) 6 ^  2z 61/z - r t e dt = 0. We now ask what must be true of F^'s price response function in order that the optimal location which satisfies the joint present value maximization - 39 -condition (3.13) also satisfies (3.19). In other words, the value of X which satisfies (3.19) must also satisfy (3.13), evaluated at optimal prices, in order to rule out the preemptive entry strategy. Solving (3.13) and (3.19) jointly for g^, we obtain (3.20) g 2(p 2,X) = - r t , e dt V ( V 8 C2 ax y(q x) V z e- r tdt + P 2y( q i) 3c 2 3X y(q x) S 1/z e dt. A second f i r s t order condition of F ^ s present value maximization i s derived by maximizing F ^ s present value with respect to price: (3.21) 3V 3P, Vo y (V « 0P 2y(q 1)g 1(p 2 >x) i 2z 3C, 3p. y ( q j 3C, 3P, 6 0y(q 1)g 1(P 2,X) ) 2z - r t e dt + + y(q 2) - 2y(p 2)] + 6 l P 2 y ( V s l ( p 2 ' X ) 8 C2 2z 3p, y ( q j 6 1 / z [ - 2 J -3C, 4y(q 2)-2y(p 2)] fi1y(q1)81(p2,X) 2z - r t e dt = 0. - 40 -We now derive a second condition on F^'s price response function such that the optimal price which satisfies the joint present value maximization condition (3.15) also satisfies (3.21). Solving (3.15) and (3.21) jointly for g^, we obtain (3.22) g 1(p 2,X) = «oV(V 8 C i f C2 e" r tdt+-J t i W ( V 9 c i ^ y C ^ ) • CO e- r tdt J t2 2z 3p 2 50 2z 3p 2 x. 2z 61 6 0P 2y( q i) 3C 2 60y(V e r t d t + -J t l 6l P2 y (V 9 C2 ^ y C ^ ) • CO e- r tdt J t 2 2z 3p 2 - 2z 0 2z 3p 2 - 2z 61 F^ would choose i t s price in order to maximize the present value of i t s profits, given that F 2 has entered the market at time t^ and given equation (3.17). (3.23) V = [p 1Q 1-C 1(Q 1)]e r t d t + [p 1Q 1-C 1(Q 1)]e r t d t P l 6 0 [Y( q i) -Y(p 1)]-C 1(6 0/z[Y(q 1) - Y ( P l ) ] ) f - r t , e dt + \ P1 61 [Y( q ; L) -Y( P ; L)] -C 1(6 1/ztY(q 1) -Y( P ; L)]) e- r tdt. - 41 -(3.24) + hS0 y ( q j I _ i _ _ y ( & i ) ] + 2z 3 C i y(q-|) 60 2z - r t , e dt + i 2 y(p x)] + 2z 5 3 P i y ( q j 3C. 6 i y ( q i ) h ' ( P l ) 2z - r t , e dt = 0. We now derive a condition on F 2' s P r i c e response function such that the optimal price which satisfies the joint present value maximization condition (3.14) also satisfies (3.24). Solving (3.14) and (3.24) jointly for h' yields (3.25) h'( P ; L) 2z 3Pn 5 0 y ( q i ) 2z "e- r tdt + 51 p 2 y ( q 1 ) 3 C , 2z 9p1 1 2z - r t , e dt J t. j 2z 9 C i l V ( V 3 P 1 2z - r t e dt + 6 l V ( V 3 C 1 2z 3 P i 1 2z - r t , e dt It was noted above that the joint present value maximizing solution - 42 -for involves choosing a location at 2/3 and the charging of common prices at both plants. If (3.20), (3.22), and (3.25) are evaluated at the joint present value maximizing prices and location, we obtain Equation (3,27) states that F^ w i l l match a l l price decreases or price increases by Y^. Equation (3.28) states that W 1 H match a l l price increases or price decreases by F^. Equation (3.26) states that i f F^ locates marginally closer to ¥^ (to the le f t of 2/3), w i l l lower i t s price enough such that ^2*s n^^et area remains unchanged (i.e. a small change in F^'s location has the same qualitative effect as a price change). We shall now argue that there are reasons to believe that conditions (3.26), (3.27), and (3.28) w i l l not be simultaneously satisfied. Let us focus on the plausibility of condition (3.26). According to Hotelling's principle of minimum differentiation, under certain assumptions, F^ w i l l have an incentive to locate i t s store to the l e f t of the joint profit maximizing location because, by doing so, i t can expand i t s market area. If, however, the optimal values which satisfy the joint present value maximization conditions are also to satisfy the independent present value maximization conditions, condition (3.26) must hold, thus leaving F^ with no incentive to move. That i s , a marginal change in F2's location from the joint present value maximizing one would induce F^ to lower i t s price such that F2*s market boundary with F^ does not change. To see this more clearly, consider the derivation of the (3.26) (3.27) (3.28) h' = 1. - 43 -delivered price, q . We know that at the market boundary, b, the following equality holds: (3.29) P 2 + z(X -b) = P l + zb. Solving for b we obtain (3.30) b = ^  ( p 2 - p 1 + zX). Evaluating how b w i l l change given a small change in X yields (3.31) db dX _1_ 2z ( - + z) . 3X If this expression i s evaluated at joint present value maximizing prices, 9 p l db r — = g = z, and - j — • = 0. Thus, a small change in X w i l l not result i n oA A QX a change in the market boundary. Condition (3.26) may be interpreted as a threat which is communicated to F 2 > but we shall argue that F^ would not regard this threat as credible. Recall that assumption (v) above stated that the firm's plants, once established, are fixed in location due to location specific sunk costs. The firm's capital i s indivisible and immobile once set in place. Thus, once F^ picks a location, and constructs i t s plant, i t s capital costs are sunk. If F 2 had chosen to locate to the l e f t of the joint present value maximizing location, then F^ could retaliate against F^ by lowering i t s price. However, such retaliatory action on the part of F^ would have an adverse effect on F^'s profits. F^ must eventually recognize that price-cutting behavior on i t s part would not alter the location already chosen by F^, since F^ would be irrevocably committed."'"''' Thus, while F^ may i n i t i a l l y engage in price-cutting behavior, i t must soon learn that i t - 44 -cannot increase i t s profits by doing so, since F w i l l not change i t s location, thereby increasing the market area of F . Once a new entrant's plant i s i n place, i t would be mutually advantageous for both F^ and F^ to increase their prices i f these prices were driven below the present 12 value maximizing levels by the new entrant's entry. Hence, condition (3.26) would be violated, since F^ would not i n general behave according to this condition on i t s price response function. Since the conditions which are necessary for the joint present value maximizing solution to be equivalent to the independent present value maximizing solution would not be expected to hold simultaneously, and since joint present value maximization over a l l plants i n the market w i l l yield the maximum profits, we may conclude that the joint present value maximizing solution yields profits which are s t r i c t l y greater than a summation of the profits obtained from F^ and F^ independently maximizing their present value of profits. There are two further comments which are relevant to the above argument. F i r s t , we have argued that condition (3.26) is unreasonable, and the logic supporting this conclusion is very similar to that employed by the c r i t i c s of Sylos' postulate. Sylos' postulate states that "Established sellers think potential entrants w i l l expect them to maintain their output in the face of new entry, letting the price f a l l 13 and the market be ruined for a l l . " Critics have argued that i f new entry does in fact take place, the established sellers cannot reasonably 14 be expected to adhere to Sylos' postulate. Second, the argument advanced in this section resembles the one put forth by Thomas Schelling [1956] in his classic essay on bargaining. In his discussion of the - 45 -concept that irrevocable commitments reduce the cr e d i b i l i t y of threats, Schelling has stated, "Similar techniques may be available to the one threatened. His best defense, of course, i s to carry out the act before the threat is made; in that case there is neither incentive nor commitment for retaliation. If he cannot hasten the act i t s e l f , he may commit himself to i t ; i f the person to be threatened i s already committed, the one who would threaten cannot deter with his threat, he can only make certain the mutually disastrous consequences that he threatens.""'"^ I shall now conclude this section by reviewing the procedure which has been pursued. The purpose of this section has been to show that the present value of profits to F^ obtained by preempting the market at time t^ i s s t r i c t l y greater than the present value of profits which i t could obtain i f i t did not preempt the market at time t^. It was argued that i f this was the case, then F^ would have an incentive to preempt the market at time t^ - E . The validity of this proposition was demonstrated as follows: F i r s t , the joint present value maximization conditions were derived for F^, given that i t was able to open a new store i n the market prior to F^. Then, the independent present value maximization conditions for and F^ were derived, given that was able to enter the market prior to F^. Necessary conditions on the price response functions of F^ and F^ were then derived such that the optimal values of prices and location which satisfied the joint present value maximization conditions would also satisfy the independent present value maximization conditions. It was then argued that one would not, in general, expect a l l of these conditions on F^'s and F2's price response functions to be simultaneously satisfied, and thus that the joint present value maximizing solution - 46 -yields profits which are s t r i c t l y greater than a summation of the profits obtained from and F^ independently maximizing their present value of profits. By implication, then, (3.5) would i n general hold as a s t r i c t inequality, and F^ would have an incentive to preempt the market. Intuitively, this i s the solution one would expect. That i s , i f entry by F^ would lead to some price competition, and i f F^ would adopt a location different from that of F^, then F^ would find i t profitable to preempt the market at time t^ - e. 16 3.4 An Alternative Model of Preemption In the Eaton and Lipsey model of preemption and in the model developed in the previous section, i t was assumed that there exists a large number of firms which would compete for the opportunity to enter the market i f entry were profitable. In this section, we relax this restrictive assumption by analyzing the preemption decision in the context of a two-person, non-constant sum, non-cooperative game. Our reason for focusing upon the "inf i n i t e competitive fringe" assumption is as follows: Casual empiricism informs us that many r e t a i l markets are oligopolistic in nature. In particular, the r e t a i l grocery trade, which is the industry to be examined for supportive evidence of the preemption hypothesis, does tend to be highly oligopolistic i n many urban markets. It i s thus of practical importance to investigate whether or not the monopolistic preemption result survives the relaxation of the inf i n i t e competitive fringe assumption. In the following analysis, we shall continue to maintain assumptions (i) through (vi) of section 3.3. It i s assumed that the market i s one-dimensional, two units i n length, with a uniform density of customers, 6 n. - 47 -This market i s described by Figure 2, which we reproduce below as Figure 4. Figure 4 -1 We depart from our e a r l i e r model s p e c i f i c a t i o n by assuming that there are only two f i r m s , F^ and F^, competing f o r the opportunity to serve the market. We a l s o assume that each f i r m makes i t s d e c i s i o n s on the b a s i s of a maximin s t r a t e g y . I n i t i a l l y , at time t ^ , F^ i s assumed to have a p l a n t l o c a t e d a t the center of the market (at 0 ) , and the density of customers, 6^, i s l a r g e enough so that F^ earns pure p r o f i t s on i t s one p l a n t , but s m a l l enough so that another p l a n t could not be p r o f i t a b l y operated i n the market. We are i n t e r e s t e d i n e x p l o r i n g the p o t e n t i a l entry s t r a t e g i e s of the e s t a b l i s h e d f i r m , F^, and a competitor, F^, given that the d e n s i t y i n the market i s c o r r e c t l y a n t i c i p a t e d to increase from 6Q to 6^ at some time i n the f u t u r e t ^ . The in c r e a s e i n den s i t y i s l a r g e enough so that ¥^ could earn pure p r o f i t s on a p l a n t e s t a b l i s h e d i n each of the i n t e r v a l s (-1:0) and (0:1) at time t^, provided that F^ has not already opened new p l a n t s i n these i n t e r v a l s . In a d d i t i o n , the increase i n d e n s i t y i s not l a r g e enough to support two new p l a n t s i n each i n t e r v a l a t time t ^ . Since the market i s symmetric about the o r i g i n , the f o l l o w i n g a n a l y s i s w i l l be confined to examining the p o s s i b i l i t i e s f o r entry on the r i g h t hand s i d e of the market. 4 l - 48 -2 Let us define V ( t ) to be the present value to T?^ °f e s t a b l i s h i n g a p l a n t i n the market a t time t when does not e s t a b l i s h a new p l a n t . Since the p o t e n t i a l e n t r a n t , F^, cannot p r o f i t a b l y enter the market when density i s 6Q between t ^ and t^, and sin c e ¥^ can earn pure p r o f i t s on a pl a n t e s t a b l i s h e d i n the market when density i s 5^, there must be some time t ^ , tg < t ^ < t ^ , such that could o b t a i n a present value equal to zero i f i t entered the market at time t ^ . That i s , (3.32) V 2 ( t 2 ) = 0 V 2 ( t ) < 0 f o r t < t± V 2 ( t ) > 0 f o r t> t±. Given that t ^ i s the e a r l i e s t date a t which would consider e n t e r i n g the market, we s h a l l confine our a t t e n t i o n to a n a l y z i n g a game i n which the only p o s s i b l e times of entry are t and t such that (3.33) t Q < t < t t = t -e t 1 < t < t 2 t = t 1 + e. Consider the f o l l o w i n g payoff matrix: (3.34) - 49 -(B,B) (B,N) (N,B) (N,N) (B,B) * A A A (B,N) * A A A (N,B) A \ 1 v 0 ^ X . 1 > s j 2 V 2 \ \_ 1 2 \ v 3 V 3 \ (N,N) A \ 1 \ 1 "N. 1 ><* Each c e l l of the payoff matrix contains e n t r i e s r e p r e s e n t i n g the present values of F^ and F^, contingent upon the s e l e c t i o n of p a r t i c u l a r s t r a t e g i e s at both t and t . The columns represent p o s s i b l e s t r a t e g i e s f o r F^, w h i l e the rows represent p o s s i b l e s t r a t e g i e s f o r f^- S t r a t e g i e s are described by ordered p a i r s i n which the f i r s t element i s an a c t i o n taken at time t and the second element an a c t i o n taken at time t . We l e t B represent the a c t i o n of b u i l d i n g a p l a n t and N represent the a c t i o n of not b u i l d i n g a p l a n t . Thus the s t r a t e g y (B,N) c o n s i s t s of b u i l d i n g a p l a n t at t and not b u i l d i n g a p l a n t at t . We wish to reduce the d i m e n s i o n a l i t y of the game and we do so by r u l i n g out any s t r a t e g y which promises at best a negative present value to the f i r m adopting that s t r a t e g y . F i r s t , r e c a l l that t ^ represents the date at which F^ could c o n s t r u c t a new p l a n t i n the market w i t h a present value equal to zero. Since t < t ^ , F^ would never consider choosing a s t r a t e g y which r e q u i r e d i t to b u i l d a p l a n t at t , and thus we have placed a s t e r i s k s i n the c e l l s c o n t a i n i n g the payoffs to F 0 and F 1 which r e s u l t - 50 -from s e l e c t i n g i t s f i r s t or second s t r a t e g y . Second, r e c a l l our assumption that the market w i l l grow i n density at time t ^ such that at most one new p l a n t could earn pure p r o f i t s i f e s t a b l i s h e d i n the i n t e r v a l (0:1) at time t^. This i m p l i e s that F^ would never s e l e c t i t s f i r s t s t r a t e g y s i n c e t h i s s t r a t e g y would e n t a i l b u i l d i n g two new p l a n t s , one at time t and one at t . Hence, we have placed a s t e r i s k s i n the c e l l s c o n t a i n i n g payoffs to F^ and F^ which r e s u l t from F^ s e l e c t i n g i t s f i r s t s t r a t e g y . The s e q u e n t i a l nature of the game a l s o permits us to r u l e out the outcome i n the t h i r d row and second column of the payoff matrix. Suppose that F^ has taken the a c t i o n B at time t and has taken a c t i o n N at time t . Then F^ would not adopt the s t r a t e g y of b u i l d i n g a p l a n t at time t since the market can only p r o f i t a b l y support the a d d i t i o n of one new p l a n t , 2 and since F^ has already b u i l t a new p l a n t , v^ must be negative. We thus place an a s t e r i s k i n the a p p r o p r i a t e c e l l of the reduced payoff matrix Let us f i r s t consider the p o s s i b l e payoffs f o r F^. In row two, a l l of the payoffs to ¥^ are zero s i n c e row two represents the a c t i o n s of not b u i l d i n g at t and not b u i l d i n g at t . I f F^ s e l e c t e d i t s f i r s t s t r a t e g y and b u i l t a new p l a n t at t , and i f F^ s e l e c t e d i t s second s t r a t e g y and a l s o b u i l t a p l a n t at t , then F ? must r e c e i v e a negative present value (3.35) - 51 -because the den s i t y at i s not large enough to support two new p l a n t s . I f F^ chooses the a c t i o n N at times t and t ( i t s t h i r d s t r a t e g y ) , and i f F^ chooses N at t and B at t , then F^ would obtain a present val u e equal 2 to v^. Since the worst that F^ can do by choosing i t s f i r s t s t r a t e g y i s to r e c e i v e a negative present value, w h i l e the worst i t can do by choosing i t s second s t r a t e g y i s to rece i v e nothing, I 1 ^ ' 8 m a x i i n i n s t r a t e g y must be i t s second. We now consider p o s s i b l e payoffs f o r F^. I f F^ adopts s t r a t e g y (B,N), the worst and only p o s s i b l e outcome i s v^. I f F^ chooses s t r a t e g y (N,B), the worst p o s s i b l e outcome i s v^ s i n c e the d e n s i t y a t t ^ i s not la r g e enough to support two new p l a n t s . I f F^ chooses s t r a t e g y (N,N), the worst p o s s i b l e outcome i s v^ sin c e F^'s p r o f i t s would f a l l a b r u p t l y a t the point i n time when F^ opens a new p l a n t . R e c a l l i n g that we have assumed maximin behavior on the p a r t of both F^ and F^, we wish to compare v^, v^, and v^ i n order to f i n d the best of the worst p o s s i b l e outcomes f o r F^. F i r s t , we note that v^ > v^ sin c e the density a t i s l a r g e enough to support one, but not two, new p l a n t s . Next, we w i l l show that there e x i s t s some e > 0 such that v^ > v^ and v^ > 0 f o r e < e. We do so as f o l l o w s : F i r s t r e c a l l from (3.33) that >s ~ 2 2 2 t = - e and t = t ^ + e. Then v^ = V ( t ^ + e ) , where V i s def i n e d i n (3.32). Define V^(t) to be the present value of F^ when i t e s t a b l i s h e s a new p l a n t at time t and F^ does not e s t a b l i s h a new p l a n t . Then v^ = V ^ C t ^ - e ) . Define V ( t ) to be the present value of both fi r m s when F^ e s t a b l i s h e s a new p l a n t at t and F^ does nothing. Then v^ = V ( t ^ + e) 2 - V (t1+ e ) . Consider - 52 (3.36) lim Cv^-vJ) = lim ( V 1 ^ - e ) - V ( t 1 + e ) + V 2 (t^ +e ) ) = v 1 ( t 1 ) - v ( t x ) > 0. The inequality (3.36) is s t r i c t for the reasons advanced in the previous section. That i s , w i l l choose the joint present value maximizing location and prices while F^ in general would select a location to the left of the joint present value maximizing location and would be expected to engage i n some price competition i f i t entered the market. Given 1 2 (3.36) and our definitions of V , V , and V, there must exist some e > e such that (3.37) V 1 ( t 1 - e ) - V ( t x + £) + V 2 ( t L + E) = 0. We have thus shown that there exists some arb i t r a r i l y small e such that v^ > v^ and v^ > 0. This implies that F^'s maximin strategy i s (B,N). Since we have already shown that F2's maximin strategy i s (N,N), the outcome of the game is for to preempt the market at time t^-£. The monopolistic preemption result that the existing firm has an incentive to preempt the market just prior to the earliest date at which a new entrant would find i t profitable to enter survives the relaxation of the i n f i n i t e competitive fringe assumption. In the model discussed above, attention was confined to the situation where an existing firm and a potential entrant consider building a new plant in a market segment bounded by a plant owned by the established firm and the market boundary. Our results would have been unaltered i f we had considered the possibilities of entry in a market which was - 53 -bounded by two p l a n t s owned by an e s t a b l i s h e d f i r m . I t a l s o would have been p o s s i b l e to analyze the s i t u a t i o n where F^ and consider b u i l d i n g a new p l a n t i n a market bounded by one p l a n t owned by F^ and one p l a n t owned by F^. The i m p l i c a t i o n which such an i n i t i a l c o n f i g u r a t i o n of p l a n t s would have f o r the preemption a n a l y s i s w i l l be discussed i n the next s e c t i o n i n the context of a two-dimensional model. 3.5 Extension of the Model to Two Dimensions In t h i s s e c t i o n , the two-dimensional analogue of the one-dimensional model i s discussed. The p o s s i b i l i t i e s of market preemption are explored, given d i f f e r e n t assumptions about the l o c a t i o n a l c o n f i g u r a t i o n of e x i s t i n g f i r m s . This a n a l y s i s w i l l a s s i s t us i n drawing out e m p i r i c a l l y t e s t a b l e hypotheses. In order to s i m p l i f y the e x p o s i t i o n , i t i s assumed that the market i s a c i r c l e w i t h diameter equal to one. ( I t w i l l soon be evident that t h i s assumption i n no way a f f e c t s the r e s u l t s . ) Assumptions ( i ) through ( v i ) of s e c t i o n 3.3 are r e t a i n e d , w h i l e noting that d i s t a n c e must be s u i t a b l y redefined to r e f l e c t the f a c t that the model of t h i s s e c t i o n i s two-dimensional i n s t e a d of one-dimensional. In a d d i t i o n , i t i s assumed that at time t Q , there are two m u l t i - p l a n t firms e s t a b l i s h e d i n the market and an i n f i n i t e competitive f r i n g e which would compete w i t h each other f o r the opportunity to enter the market i f they expected a present value greater than or equal to zero by doing so. (We s h a l l r e f e r to the competitive f r i n g e c o l l e c t i v e l y as F .) As i n the previous models, customer d e n s i t y i s expected to increase at some time i n the f u t u r e such that a new p l a n t could p r o f i t a b l y be e s t a b l i s h e d i n the market at that time i n the 'neighborhood' of X (see Figure 5 ) . - 54 -There are two general types of l o c a t i o n a l c o n f i g u r a t i o n s which w i l l be considered i n the ensuing a n a l y s i s . Case I represents the type i n which a competitive f r i n g e f i r m would have as i t s neighbors the p l a n t s of only one other f i r m . Case I I represents the type i n which a competitive f r i n g e f i r m would have as i t s neighbors the p l a n t s of at l e a s t two other f i r m s . We s h a l l consider the Case I type of l o c a t i o n a l c o n f i g u r a t i o n f i r s t . Case I Consider F i g u r e 5. Figure 5 The i n i t i a l l o c a t i o n s of firms F^ and F^ i n the market of Figure 5 are denoted by and X^^ r e s p e c t i v e l y , where X_^ represents the l o c a t i o n of p l a n t £ when i t i s owned by F . These l o c a t i o n s c o n s t i t u t e the i n i t i a l c o n d i t i o n s . - 55 -We now i n q u i r e as to when a competitive f r i n g e f i r m would l i k e to e s t a b l i s h a new p l a n t i n the market of Figure 5. As i n the one-dimensional model, competition among competitive f r i n g e f i r m s would push back the time of entry to t ^ , such that the present value of p r o f i t s to the s u c c e s s f u l new entrant would be equal to zero. The competitive f r i n g e f i r m F would thus enter the market a t time t ^ and choose i t s p r i c e and l o c a t i o n i n order to maximize the present value of i t s p r o f i t s : (3.38) max ( P j r x ) nJ(P j l> x j l;P 2i'•••»P 2 n > X21'•'•' X2n *pll» X.^,...^^ ) e r t d t = V J, where X_.^  = F j ' s l o c a t i o n , defined by a p a i r of p o l a r c o o r d i n a t e s , R and 9 (where R i s the length of a radius and 6 i s the measure of an angle) X2£ = t^lG l ° c a t i o n °f a P x a n t owned by F^, d e f i n e d by a p a i r of p o l a r coordinates (£ = 1, 2, n ) = the £th l o c a t i o n of a p l a n t owned by F^, defined by a p a i r of p o l a r coordinates (£ = 1, 2, n^) p.. = F.'s m i l l p r i c e P2£ = n ^ P j , X j »Pn' ••'» P i n ) = a f u n c t i o n d e s c r i b i n g F2's p r i c e response at i t s £th plant to the s e l e c t i o n of a p r i c e and l o c a t i o n by F and the s e l e c t i o n of p r i c e s by F^ Pjj? = g(Pj>Xj>P2 i» •••> P2 n ) = a f u n c t i o n d e s c r i b i n g F^'s p r i c e response a t i t s £th p l a n t to the s e l e c t i o n of a p r i c e and l o c a t i o n by F^ and the s e l e c t i o n of p r i c e s by F2 - 56 -II^(') = F.'s profit function r = the firm's discount rate n_^  = the number of plants owned by F^. If F. had entered the market at time t.. , then F„ would simply maximize J 1 2 the present value of i t s profits: (3.39) max I n 2 2 N »-., j V P21' P2n 0 ; X21' ••" X2n ' ( P21' P 2 n 2 } 1 = 1 h 2 2 P j l , X j l , p u , .... P l , X n , .... X ) e" r tdt = V 2, where p., = d(p„.. , ..., p„ , p , p . ) = a function describing F.'s r j l * z l 2n 2 r l l 1 j price response to the selection of prices by F 2 and F^. F^ maximizes the present value of the sum of profits of i t s n 2 plants i n i t i a l l y located in the market. Suppose that F^ had established the new plant i n the market at time t.. rather than F.. Then F~ would wish to maximize the present value of 1 J 2 its profits over i t s entire operation: n 2 + l (3.40) max 2, ( p 2 1 , P 2 ( n 2 + 1 ) > X 2(n 2+1) ) 1 = 1 J t ~2, V P 2 1 ' P2(n 2+1)' 1 X2(n 2+1) ; X21' X2n 2' P l l ' ^ n ^ X l l ' X, ) e r t d t = V 2, l n ^ where p ^ = g<P21, P 2(n 2+1)' X2(n 2+1) } = a f u n c t i o n describing F ^ s p r i c e response at i t s tth p l a n t to the s e l e c t i o n of a l o c a t i o n and p r i c e s by F 2. We now i n q u i r e as to whether or not F^ has any i n c e n t i v e to preempt the market at time t ^ - e . F i r s t , we know that i t must be true that (3.41) V 2 > V 2 + V j. That i s , the present v a l u e of F^'s j o i n t l y maximized p r o f i t s must be greater than or equal to the sum of F 's and F.'s independently maximized present value of p r o f i t s . Equation (3.41) i s e x a c t l y analogous to equation (3.4) i n the one-dimensional model. In the one-dimensional case, i t was argued that Y^ would p i c k the l o c a t i o n of i t s new p l a n t and the p r i c e s of a l l of i t s p l a n t s i n order to maximize the present value of i t s p r o f i t s over i t s e n t i r e o p e r a t i o n , w h i l e F would s e l e c t i t s p r i c e and l o c a t i o n i n order to maximize the present value of i t s one p l a n t . Since the e q u i l i b r i u m p r i c e s and l o c a t i o n which s a t i s f i e d F^'s j o i n t present value maximization c o n d i t i o n s would not be expected simultaneously to s a t i s f y the independent present value maximization c o n d i t i o n s of F and F., the i n e q u a l i t y (3.4) would be s t r i c t . A s t r i c t i n e q u a l i t y i m p l i e s that F^ would have an i n c e n t i v e to preempt the market at time t ^ - e • Case I I The second l o c a t i o n a l c o n f i g u r a t i o n which i s considered i s described by Figure 6. - 58 -Figure 6 Figure 6 i s the same as Figure 5, except that we now examine the case where a competitive f r i n g e f i r m l o c a t e d i n the neighborhood of X would have as i t s neighbors the pl a n t s of two f i r m s . Consider the competitive f r i n g e entry s t r a t e g i e s . As i n Case I above, competition among competitive f r i n g e firms would push the time of entry back to t ^ such that the s u c c e s s f u l competitive f r i n g e entrant would earn a zero present value by maximizing (3.38) w i t h respect to p r i c e and l o c a t i o n . We now ask whether F^ would have an i n c e n t i v e to preempt the market at time t ^ - e . F i r s t , i f F^ had entered the market at time t ^ , then F^ would simply maximize the present value of i t s p r o f i t s w i t h respect to the n^ p r i c e s of i t s p l a n t s . That i s , F^ would maximize (3.39). I f F^ had entered the market w i t h a new p l a n t at time t ^ , then i t would maximize the present value of i t s p r o f i t s over i t s e n t i r e o p e ration. In other words, - 59 -i t would s e l e c t i t s l o c a t i o n X and p r i c e s p 21' • • • , P i n order to maximize (3.40). In order f o r to have an i n c e n t i v e to preempt the market, i n e q u a l i t y (3.41) must be s t r i c t . The i n e q u a l i t y (3.41) would be s t r i c t as long as F_. would s e l e c t a p r i c e and l o c a t i o n which d i f f e r e d from the j o i n t present value maximizing l o c a t i o n and p r i c e which F^ would s e l e c t . I t w i l l be argued i n t u i t i v e l y below t h a t , i n general, t h i s c o n d i t i o n w i l l h o l d . The argument proceeds as f o l l o w s : F i r s t , i t i s e s t a b l i s h e d that i f F^ s e l e c t e d the same l o c a t i o n as that which maximized F.'s present value, F„ would not be expected to charge the same p r i c e as F . Then, i t i s argued that J ^ ' 3 P r e s e n t v a l u e maximizing l o c a t i o n would d i f f e r from ^..'s. For t n e reasons advanced i n s e c t i o n 3,3, we would then expect the j o i n t present value maximization to y i e l d greater p r o f i t s to F^ than the p r o f i t s which could be obtained i f F. and F independently maximized t h e i r present value of p r o f i t s . This then i m p l i e s that (3.41) would hold as a s t r i c t i n e q u a l i t y and F^ would have an i n c e n t i v e to preempt the market. In order to s i m p l i f y the argument, i t i s assumed that the p l a n t l o c a t e d at X would have one F^ p l a n t and one F^ p l a n t as i t s neighbors. I f F^ e s t a b l i s h e d a p l a n t at the l o c a t i o n that maximized F^'s present value, F would not i n general charge the same p r i c e as F.. F. would ^ J J attempt to charge a p r i c e lower than that which F^ would charge i n the hope of a t t r a c t i n g customers who p r e v i o u s l y p a t r o n i z e d the p l a n t s owned by F^ and F 2 , F^ would charge a higher p r i c e than F because i f F^, inst e a d of F , l o c a t e d at X at time t ^ , i t would have one of i t s own pla n t s as one of i t s neighbors, and customers a t t r a c t e d from i t s o l d pl a n t to i t s new p l a n t by lower p r i c e s could not be counted as net a d d i t i o n s to F 0's c l i e n t e l e . With respect to l o c a t i o n , the same s o r t of - 60 -l o g i c would lead us to expect F^'s j o i n t present value maximizing l o c a t i o n to be' d i f f e r e n t from F^'s independent present value maximizing l o c a t i o n . In p a r t i c u l a r , F^ would be expected to l o c a t e c l o s e r to F^ than would F_.. -F_.. would p i c k i t s l o c a t i o n to maximize i t s present value of p r o f i t s , t a k i n g i n t o account that i t would have as i t s neighbors p l a n t s owned by F^ and F^. F^ would s e l e c t i t s l o c a t i o n to maximize the present value of i t s e n t i r e o p e r a t i o n / t a k i n g i n t o account that one of the neighbors of i t s new p l a n t would be one of i t s . o l d p l a n t s . Thus, F^ would not s e l e c t a l o c a t i o n that would e x c e s s i v e l y encroach upon the market area o f . i t s o l d p l a n t , w h i l e F. would s e l e c t i t s l o c a t i o n to encroach upon the market areas of both F^ and F^. From these arguments, we may conclude that i n e q u a l i t y (3.41) would be s t r i c t and that F^ would have an i n c e n t i v e to preempt the market at time t ^ - e. I n an e x a c t l y analogous manner, the above argument could have been cast i n terms of F^ and F_. . By i m p l i c a t i o n , then, both F^ and F^ have an i n c e n t i v e to preempt the market at time. t^. - e . However, the assumptions underlying the model do not permit the i d e n t i f i c a t i o n of which f i r m w i l l a c t u a l l y preempt. Let us now b r i e f l y summarize the discussion: of t h i s s e c t i o n . F i r s t , i f there i s an a n t i c i p a t e d . i n c r e a s e i n de n s i t y i n the: market such that a new p l a n t could be p r o f i t a b l y e s t a b l i s h e d i n that market, and i f the new p l a n t would have as i t s neighbors the p l a n t s of one e x i s t i n g f i r m , then the e x i s t i n g f i r m w i l l have an i n c e n t i v e to preempt the market. Second, i f t h e r e . i s an a n t i c i p a t e d i n c r e a s e : i n density i n the market such that a new p l a n t could be p r o f i t a b l y e s t a b l i s h e d i n that market, and i f the new plant, would have as i t s neighbors the p l a n t s of more than one e x i s t i n g - 61 -f i r m , then any of the e x i s t i n g f i r m s w i l l have an i n c e n t i v e to preempt the market, 3.6 Anomalies Perhaps the strongest i m p l i c a t i o n of the theory developed i n s e c t i o n 3.3 i s that one f i r m w i l l tend.to dominate each s p a t i a l l y separated market, and that as the market grows, that f i r m w i l l have an i n c e n t i v e to strengthen i t s g r i p over that market by preempting i t . However, casu a l empiricism informs us that s p a t i a l l y separated markets are not always dominated by one f i r m . One i s t h e r e f o r e l e d to ask how "anomalies" may come about. An anomaly i s defined as e x i s t i n g when a m u l t i - p l a n t f i r m has e s t a b l i s h e d , a new p l a n t i n the market such that the new p l a n t has as i t s neighbors only p l a n t s owned by other f i r m s . There are three important sources of anomalies which may be i d e n t i f i e d . The f i r s t source of anomalies i s unexpected inc r e a s e s i n consumer d e n s i t y i n the market., I t w i l l be r e c a l l e d that a l l " . o f the models discussed:above were constructed on the assumption that a l l f i r m s ( e s t a b l i s h e d f i r m s as w e l l as p o t e n t i a l entrants) c o r r e c t l y a n t i c i p a t e that there w i l l be a . d i s c r e t e increase i n d e n s i t y at time t^. Each f i r m then s e l e c t s i t s present value maximizing strategy', given t h i s a n t i c i p a t e d growth.in the market. I f an increase i n density occurred at a time t < which was not a n t i c i p a t e d , then the l o g i c of previous arguments i s s h o r t - c i r c u i t e d i n that the f i r m which enters the market w i l l be the one which f i r s t perceives the p r o f i t a b i l i t y of entry and acts upon that perception. That i s , i n the case of unexpected increases i n consumer de n s i t y , i t i s impossible to say which f i r m w i l l preempt the market. The second source of anomalies i s " c a p i t a l c o n s t r a i n t s " which b i n d . Consider a market which at time t ^ i s dominated by the p l a n t s of only one f i r m . I t i s assumed that there are an i n f i n i t e number of p o t e n t i a l entrants which, would compete w i t h each other f o r the opportunity to e s t a b l i s h a p l a n t i n the market.if they could o b t a i n a present value greater than or equal to zero by doing so. Suppose de n s i t y i s a n t i c i p a t e d to increase at a time such that two new p l a n t s could be p r o f i t a b l y e s t a b l i s h e d i n the market. The theory of.market preemption: suggests that the e s t a b l i s h e d f i r m w i l l have an i n c e n t i v e to preempt the market at a time t ^ - e=by c o n s t r u c t i n g two new p l a n t s . However, the a b i l i t y of the established, f i r m to preempt the market i m p l i e s that there i s no c a p i t a l c o n s t r a i n t which might prevent i t from opening two new p l a n t s at t ^ - e . I f such a c a p i t a l c o n s t r a i n t was b i n d i n g on the e s t a b l i s h e d f i r m such that i t could only open one new p l a n t a t time t ^ - e, then one of the p o t e n t i a l entrants would have an opportunity to open up the second p l a n t at time t ^ . Thus c a p i t a l c o n s t r a i n t s which bind may prevent the e f f e c t i v e implementation of a preemptive s t r a t e g y . The above argument has been cast i n terms of c a p i t a l c o n s t r a i n t s . However, the second type of anomaly could a l s o a r i s e from c o n s t r a i n t s on the growth of the managerial team ("managerial growth costs") or any other c o n s t r a i n t which would e f f e c t i v e l y l i m i t the r a t e of growth of the f i r m . A t h i r d source of apparent anomalies i s r e l a t e d to the c o n s o l i d a t i o n o f . s p a t i a l l y separate markets. Consider F i g u r e 7: F i g u r e 7 Suppose that at time t ^ , there e x i s t two s p a t i a l l y separated markets, and A^, which are dominated by the p l a n t s of F^ and F^ r e s p e c t i v e l y . Suppose that F^ and F^ c o r r e c t l y a n t i c i p a t e that market A^ w i l l grow i n extent a t time t ^ such that the two markets w i l l no longer be s p a t i a l l y separated and such that a new p l a n t could be p r o f i t a b l y e s t a b l i s h e d i n the market at.time t^. Our d i s c u s s i o n of the two-dimensional model i n s e c t i o n 3.5 leads us to conclude that both F^ and F^ w i l l have an i n c e n t i v e to preempt the market, although i t i s impossible to say which f i r m w i l l a c t u a l l y preempt. However, depending upon, when our observations are made, i t may be p o s s i b l e to observe what looks l i k e an outside entrant preempting a market which was p r e v i o u s l y dominated by one f i r m . Hence, t h i s t h i r d , source of anomalies i s r e a l l y a source of pseudo-anomalies. That i s , the l o c a t i o n a l c o n f i g u r a t i o n may appear anomalous i f observed a t one point i n time, but would not be anomalous i f the h i s t o r y of growth of the market and of the p l a n t s s e r v i n g the market were known. - 64 -F i n a l l y , a f o u r t h p o s s i b l e source of anomalies may be the f e a r on the part of one e s t a b l i s h e d f i r m of r e t a l i a t o r y responses on the par t of another e s t a b l i s h e d f i r m . Consider Figure 6. In s e c t i o n 3.5, i t was e s t a b l i s h e d t h a t , i n general, i f there i s an a n t i c i p a t e d i n c r e a s e i n de n s i t y i n the market such that a new p l a n t could be p r o f i t a b l y e s t a b l i s h e d i n that market, and i f . the new p l a n t would have as i t s neighbors the p l a n t s of more than one e x i s t i n g f i r m , then any of the e x i s t i n g f i r m s w i l l have an i n c e n t i v e to preempt the market. Thus, i n the market described by Fi g u r e 6, we should observe. F^ and F^ preempting the market. However, i f F^ and F^ b e l i e v e that preemption w i l l generate a r e t a l i a t o r y , p r i c e response from the preempted f i r m , they may be r e l u c t a n t to take, the i n i t i a t i v e . Thus, some other f±rm,.'F ., might be allowed to enter the market. This i s only one of many a l t e r n a t i v e scenarios which could a r i s e depending upon the assumptions made regarding the e s t a b l i s h e d f i r m s ' expectations. We s h a l l f i n d i t u s e f u l , however, to maintain our s i m p l i f y i n g assumption that f i r m s are able to c o r r e c t l y foresee the outcome of the competitive process. Hence, we s h a l l not consider the f o u r t h p o s s i b l e source of anomalies i n any greater d e t a i l i n the l a t e r stages, of t h i s work. FOOTNOTES TO CHAPTER 3 I t i s assumed, i n t h i s s e c t i o n , and i n the remainder, of t h i s chapter, that firms are able.to foresee the outcome of the competitive process, and are thus able to c a l c u l a t e t h e i r own and t h e i r competitor's returns contingent upon the p u r s u i t of v a r i o u s s t r a t e g i e s . See s e c t i o n 3.3. Koopmans [1957] prefaced t h i s statement w i t h the f o l l o w i n g remarks: " I f we imagine a l l land to be of the same q u a l i t y , both a g r i c u l t u r a l l y and i n amount and a c c e s s i b i l i t y of m i n e r a l resources, then an a c t i v i t y a n a l y s i s model of production that includes the p r o p o r t i o n a l i t y p o s t u l a t e would show a p e r f e c t l y even d i s t r i b u t i o n of a c t i v i t i e s to be most economical. Each square inch of area would, produce the same bundle of a c t i v i t i e s , and a l l t r a n s p o r t a t i o n would.then be avoided. I f t h i s model i s modified so as to r e f l e c t continuous d i s t r i b u t i o n of s o i l f e r t i l i t y and of mineral content., t r a n s p o r t a t i o n does become economical i f i t s resource requirements are not too high i n r e l a t i o n to the advantage to be gained by t r a n s p o r t a t i o n . However, even then there w i l l be no reason, f o r having concentrated c i t i e s unless mineral.deposits (or p o s s i b l y s o i l f e r t i l i t y ) were to be h i g h l y concentrated.. T h i s suggests that without r e c o g n i z i n g i n d i v i s i b i l i t i e s - i n the human person, i n residences, p l a n t s , equipment,, and i n t r a n s p o r t a t i o n - urban l o c a t i o n problems, down to those of the s m a l l e s t v i l l a g e , ..cannot be understood." This p r o p o s i t i o n has been c a r e f u l l y analyzed by Eaton and Lipsey [1978]. In the context of a one-dimensional s p a t i a l model, they demonstrate that pure p r o f i t s w i l l not n e c e s s a r i l y be d r i v e n to zero - 66 -by p r i c e competition and/or f r e e entry of new f i r m s . Our p a r t i c u l a r choice of a cost f u n c t i o n w i l l be s p e c i f i e d at a l a t e r p o i n t i n the a n a l y s i s . One type of c o s t - f u n c t i o n f r e q u e n t l y used i n models of s p a t i a l competition i s of the f o l l o w i n g form: (1) ATC = K/Q + c Q < Q, where K = f i x e d c o s t s , Q = output, c = a constant marginal c o s t , and Q < Q i s a capacity c o n s t r a i n t . (The c a p a c i t y c o n s t r a i n t i s not always imposed, however.) The reasons f o r using t h i s type of cost f u n c t i o n have been stated, by Eaton and Lipsey [1976]: "The e m p i r i c a l phenomenon which i s . o f i n t e r e s t to us i s that average t o t a l costs of production d e c l i n e over some range of output. We know that n e a r l y a l l production involving.machines i s c h a r a c t e r i z e d by. i n d i v i s i b i l i t i e s which give f a l l i n g ATC at l e a s t at low l e v e l s of output. More important f o r the r e t a i l i n g s e c t o r are economies of s c a l e i n the c o n s t r u c t i o n of b u i l d i n g s and i n the management of i n v e n t o r i e s . The assumptions r e f l e c t e d i n the cost f u n c t i o n of equation (1) are the simplest and most manageable way of i n c o r p o r a t i n g economies of s c a l e up to some minimum output l e v e l . " The reasons f o r making t h i s p a r t i c u l a r assumption have been s t a t e d by Eaton and Lipsey [1978], Consider a number of f i r m s e q u a l l y spaced on a l i n e o f / f i n i t e l e n g t h . I f there are no sunk c o s t s , and a new f i r m enters, a l l other f i r m s woud move u n t i l they were again e q u a l l y spaced. That i s , they can a l l increase the q u a n t i t y demanded by moving to the mid-point of t h e i r market areas. However, i f there, are sunk c o s t s , - 67 -and i f the p r o f i t s of e x i s t i n g f i r m s i n t h e i r o r i g i n a l l o c a t i o n s are greater, than they would be i f these f i r m s moved, then the new entrant cannot expect e x i s t i n g firms to move. That i s the r a t i o n a l e f o r the zero c o n j e c t u r a l v a r i a t i o n w i t h respect to l o c a t i o n assumption. I t depends on the exi s t e n c e of l o c a t i o n s p e c i f i c sunk cos t s and may y i e l d the r e s u l t of excess p r o f i t s i n e q u i l i b r i u m . 6. The p o s s i b i l i t y of a f i r m making pure p r o f i t s without inducing f u r t h e r entry i n t o the market has been discussed by Eaton and .Lipsey [1978]. This p o s s i b i l i t y was a l s o discussed i n t h e i r 1976 paper, "The Theory of Market Preemption: B a r r i e r s to Entry i n a Growing S p a t i a l Market": "This geographical d i s p e r s i o n of f i r m s e f f e c t i v e l y segments the market and confers on each f i r m an element of monopoly power over segments of the market that are c l o s e r to that f i r m than to any other f i r m . The d i s p e r s i o n of firms a l s o means that any new entrants must f i t i n t o a space between e x i s t i n g f i r m s and as a r e s u l t w i l l s e l l s i g n i f i c a n t l y l e s s at any p o i n t than would an e x i s t i n g f i r m have s o l d a t the same p r i c e before entry occurred. This r e s u l t , i n combination w i t h c h a r a c t e r i s t i c ( i ) , decreasing LRATC over some range, i m p l i e s that i t i s q u i t e p o s s i b l e f o r p o t e n t i a l entrants to a n t i c i p a t e negative p r o f i t s w h i l e . f i r m s already i n the market enjoy p o s i t i v e pure p r o f i t s . " 7. Equation (3.6) i s obtained as f o l l o w s : P 2 + z ( X - b ) = p + zb zb = p 2/2 - p /2 + zx/2 V± + zb = p 2/2 + P l / 2 + zx/2 q1 = 1/2(p 2 + p + zx) . - 68 -8. Equation (3.8) i s obtained as f o l l o w s : V = 5 y (p + zb) db I f we set u = P x + zb, then du = z db , or db = 1/z du , When b = 0, u = p^. When b = t>, u = p^ + zb = q . A f t e r making the appropriate s u b s t i t u t i o n s , we o b t a i n Q± = 57 z q l y(p) dp p l 9. I t i s w e l l known that i f the production f u n c t i o n i s homothetic, then the m i n i m i z a t i o n problem min -j rK + wL | Q = Q(K,L) (K,L) y i e l d s a minimum t o t a l cost f u n c t i o n of the f o l l o w i n g form: C = c(Q) $ (r,w). This cost f u n c t i o n i s such that i t may.embody the p r o p e r t i e s and s a t i s f y the conditions l i s t e d i n the t e x t . 10. The proof that j o i n t p r o f i t maximization by F^ i n v o l v e s choosing a l o c a t i o n at 2/3 and the charging of common p r i c e s at both of i t s p l a n t s may be found i n Eaton and Lipsey [1976]. 11. This argument i s i n the s p i r i t of Eaton and Lipsey's no m i l l - p r i c e u n d e rcutting assumption, which i s discussed i n Eaton and Lipsey [1978] and an e x p l i c i t assumption i n t h e i r preemption.model. Eaton and Lip s e y argue that no p o t e n t i a l e n t r a n t , p r i o r to entry , can expect to charge a p r i c e low enough to d r i v e the e s t a b l i s h e d f i r m out of the - 69 -market and expect, to earn non-negative pure p r o f i t s charging such a p r i c e . This argument r e l i e s on the f a c t that the e s t a b l i s h e d f i r m has sunk c o s t s , w h i l e the p o t e n t i a l entrant's costs are a l l avoidable p r i o r to entry. 12. A s i m i l a r argument has been advanced by Schmalensee [1978; 313]. "Suppose, f o r in s t a n c e , that e s t a b l i s h e d firms attempt to deter entry by some v a r i a n t of l i m i t p r i c i n g , h o l d i n g p r i c e s below the short-run p r o f i t maximizing l e v e l so that the expected p r o f i t of an entrant brand that takes those p r i c e s as f i x e d would be negative. Suppose f u r t h e r that entry nevertheless occurs.. Once the entrant i s i n pl a c e , i t i s r e l a t i v e l y immobile. Both i t s p r o f i t s and those of i t s immediate r i v a l s can then g e n e r a l l y be r a i s e d by. i n c r e a s i n g p r i c e . As only a s m a l l group of f i r m s i s i n v o l v e d j such mutually b e n e f i c i a l p r i c e increases are not i m p l a u s i b l e . But i f p o t e n t i a l entrants come to recognize t h i s p o s s i b i l i t y , l i m i t p r i c i n g ceases to be an e f f e c t i v e d e t e r r e n t , s i n c e low preentry p r i c e s cease to convey a c r e d i b l e t h r e a t of low postentry p r i c e s . " 13. Osborne [1973]. 14. See L a t s i s [1976], P a s h i g i a n [1968], and Wenders [1971]. 15. See S c h e l l i n g [1956; 294-295]. 16. The model presented i n t h i s s e c t i o n represents a r e v i s e d v e r s i o n of one which was o r i g i n a l l y developed as part of the author's d i s s e r t a t i o n prospectus.,.. The r e v i s i o n was undertaken by B. C u r t i s Eaton and myself. - 70 -Chapter 4 THE PROFITS TEST 4.1 I n t r o d u c t i o n In the previous chapter, the f o l l o w i n g general r e s u l t of preemptive f i r m behavior was deri v e d : i f an e x i s t i n g f i r m and p o t e n t i a l entrants a n t i c i p a t e that the market w i l l grow at some time i n the f u t u r e such that a new p l a n t could be p r o f i t a b l y e s t a b l i s h e d i n that market, and I f the e x i s t i n g firm; does nothing to bl o c k entry, then competition among p o t e n t i a l entrants w i l l lead to a new p l a n t being e s t a b l i s h e d i n the market at a poi n t i n time when the present value of that p l a n t i s equal to zero. This i m p l i e s that the negative p r o f i t s which would accrue to the new entrant's p l a n t p r i o r to the increase i n density would be balanced o f f by the p o s i t i v e pure p r o f i t s earned by the new entrant's p l a n t a f t e r the in c r e a s e i n density takes place. I f an e x i s t i n g f i r m preempted the market instead of a p o t e n t i a l entrant, we know that the present value of p r o f i t s of the new p l a n t would be l e s s than the present value of p r o f i t s of a p l a n t e s t a b l i s h e d at the date when the in c r e a s e i n de n s i t y occurs. However, we are unable to i n f e r whether the p r o f i t s of the e x i s t i n g f i r m ' s new:plant would be negative or p o s i t i v e p r i o r to the increase i n d e n s i t y . In t h i s chapter, we s h a l l perform a s e r i e s of annual.net p r o f i t c a l c u l a t i o n s f o r stores belonging to.the supermarket i n d u s t r y . We s h a l l d e r i v e estimates of the annual p r o f i t s of a " r e p r e s e n t a t i v e new super-market"..and a " r e p r e s e n t a t i v e e s t a b l i s h e d supermarket". Given the general i m p l i c a t i o n of preemptive behavior, as s t a t e d above, we would regard negative estimates of new supermarket p r o f i t s as being evidence which i s co n s i s t e n t w i t h preemptive behavior on the par t of some, f i r m or firms i n the i n d u s t r y . T h i s , then, would provide us w i t h a b a s i s f o r engaging i n a more.detailed examination of the l o c a t i o n a l i m p l i c a t i o n s of preemption, with the goal of determining whether preemption has occurred i n some s p a t i a l l y extended market, and, i f so, which supermarket f i r m or f i r m s have pursued t h i s .type of behavior.. On the other hand,, p o s i t i v e estimates of new supermarket p r o f i t s , w h i l e p o s s i b l y c o n s i s t e n t w i t h preemptive behavior on the par t of e x i s t i n g f i r m s , would not provide us w i t h s u f f i c i e n t i n c e n t i v e to pursue the l o c a t i o n a l a n a l y s i s of preemption f o r the supermarket i n d u s t r y . In s e c t i o n 4.2, we s t a t e the n u l l hypothesis to be examined i n t h i s chapter, as w e l l as our. reasons f o r s e l e c t i n g i t . In s e c t i o n 4.3, we discuss the data sources to be used i n our net p r o f i t c a l c u l a t i o n s . Section^4.4 contains a d e t a i l e d d e s c r i p t i o n of how the p r o f i t s of a re p r e s e n t a t i v e new supermarket and a r e p r e s e n t a t i v e e s t a b l i s h e d super-market may be estimated using the a v a i l a b l e data. In s e c t i o n 4.5, we present the r e s u l t s , of. our p r o f i t c a l c u l a t i o n s , w h i l e i n s e c t i o n 4.6, we i n t e r p r e t our r e s u l t s and make some concluding remarks. 4.2 Statement of the N u l l Hypothesis The i n d u s t r y we have:chosen as the b a s i s f o r our examination of the p r o f i t s i m p l i c a t i o n of the theory of preemption i s the supermarket i n d u s t r y . I d e a l l y we would l i k e to have cost and revenue data f o r i n d i v i d u a l supermarkets covering t h e i r f i r s t twelve months of op e r a t i o n . (The s e l e c t i o n of a year as the p e r i o d of time over which the i n i t i a l p r o f i t s of new supermarkets would be ca l c u l a t e d : i s a r b i t r a r y , but - 72 -suggested by the nature of the.data.) Such data would enable us to t e s t d i r e c t l y the n u l l hypothesis that the I n i t i a l p r o f i t s of new supermarkets are negative i n t h e i r f i r s t twelve months of operation. However, as might be expected, we.do not have access to such a body of data, and the r e f o r e some approximation to the i d e a l t e s t becomes necessary. The approximation we have i n mind i s to examine the f o l l o w i n g , n u l l h ypothesis: For any given year, the "average" p r o f i t s of new supermarkets i n the United; States and Canada are negative i n the f i r s t twelve months of operation of these supermarkets. By average p r o f i t s we mean average t o t a l revenue minus average t o t a l , c o s t s , where the a v e r a g e i i s over supermarkets. The a l t e r n a t i v e hypothesis i s that f o r any given year, the average p r o f i t s of new supermarkets i n the United States and Canada are non-negative i n the f i r s t twelve months of operation of these supermarkets. Acceptance of the n u l l hypothesis would be c o n s i s t e n t w i t h the p r o p o s i t i o n that a l a r g e percentage (although not n e c e s s a r i l y the maj o r i t y ) of new supermarkets established, i n the year f o r which the t e s t i s conducted could not cover t h e i r costs i n t h e i r f i r s t twelve months of operation. This p r o p o s i t i o n i s , i n t u r n , c o n s i s t e n t w i t h the p r o f i t s i m p l i c a t i o n of the theory of preemption. In the. next s e c t i o n , we describe the data sources to.be used f o r our net p r o f i t c a l c u l a t i o n s . 4.3 D e s c r i p t i o n of Data Sources The data which we use f o r our net p r o f i t c a l c u l a t i o n s are drawn from three sources. Two of these sources are published by-the Food Marketing I n s t i t u t e , a United States based supermarket trade a s s o c i a t i o n . The f i r s t one i s called.The Super Market. Industry Speaks -(SMIS) and i s published - 73 -annually.. The data contained i n t h i s p u b l i c a t i o n are.based on i n d u s t r y surveys of the I n s t i t u t e ' s member companies (which may a l s o be regarded as a sample of supermarkets taken from the po p u l a t i o n of a l l supermarkets i n operation i n a:given y e a r ) . We are p r i m a r i l y i n t e r e s t e d i n the I n s t i t u t e ' s c a l c u l a t i o n s of average operating r e s u l t s of: supermarkets, where a supermarket i s defined i n SMIS as a departmentalized food s t o r e doing at l e a s t one m i l l i o n d o l l a r s a year i n s a l e s , or about twenty thousand d o l l a r s weekly.^ The second p u b l i c a t i o n of the Food Marketing I n s t i t u t e of i n t e r e s t to us i s c a l l e d : Facts About New Super Markets Opened i n (year) (FANSM), which i s published annually. Again, the data contained i n t h i s p u b l i c a t i o n are based on a survey of member companies, which have opened 2 new stores during the year f o r which the survey was conducted. The type of data reported i n FANSM v a r i e s from year to year, but s a l e s i n f o r m a t i o n i s always reported. The problem which we confront i n the next s e c t i o n of t h i s chapter i s how to estimate the average p r o f i t s of new supermarkets given the incomplete nature of the cost data reported i n FANSM. The t h i r d p u b l i c a t i o n which we make use of i s c a l l e d Operating Results of Food Chains (ORFC), published annually between 1956 and 1961 3 by Harvard U n i v e r s i t y , and s i n c e 1962 by C o r n e l l U n i v e r s i t y . This p u b l i c a t i o n reports d e t a i l e d breakdowns of costs as a percentage of s a l e s f o r a sample of. supermarkets.taken from the po p u l a t i o n of a l l super-4 markets i n operation i n a given year, For given years,: we s h a l l f i n d i t u s e f u l to derive, estimates of the net p r o f i t s , o f new supermarkets and the..net p r o f i t s of e s t a b l i s h e d super-markets. ~* The bulk of the data contained i n a l l three of the aforementioned p u b l i c a t i o n s i s i n the form of averages, where an average i s defined to be a mean i n ORFC and a -median i n FANSMand SMIS, and where the average i s over a l l supermarkets i n the r e s p e c t i v e sample. Since our data are i n the form of averages,.our net p r o f i t c a l c u l a t i o n s f o r a given year w i l l , be regarded as c o n s t i t u t i n g estimates of the net p r o f i t s of a re p r e s e n t a t i v e new supermarket and a r e p r e s e n t a t i v e e s t a b l i s h e d super-market i n that year. Our. net p r o f i t c a l c u l a t i o n s w i l l be. made under the f o l l o w i n g assumptions: ( i ) the ORFC and SMIS data are based on random samples of supermarkets taken from the same, popu l a t i o n of supermarkets i n operation i n . a given year; ( i i ) the FANSM data are based on a random sample of supermarkets taken from the po p u l a t i o n of new supermarkets opened i n . a given year; ( i i i ) a l l of the independent v a r i a b l e s comprising the net p r o f i t f u n c t i o n are random v a r i a b l e s which are independently d i s t r i b u t e d . In the next s e c t i o n , we d e f i n e the p r o f i t f u n c t i o n which we use to c a l c u l a t e theunet p r o f i t s of a r e p r e s e n t a t i v e new supermarket and the net p r o f i t s of a r e p r e s e n t a t i v e e s t a b l i s h e d supermarket.. We a l s o d i s c u s s the assumptions that are necessary i n order to ob t a i n an estimate of the net p r o f i t s of a r e p r e s e n t a t i v e new supermarket. 4.4- Procedures f o r Estimating Net P r o f i t s In t h i s s e c t i o n , we s h a l l d i s c u s s the procedures which we use to estimate the net p r o f i t s of a representative, new supermarket, and the net p r o f i t s of a r e p r e s e n t a t i v e e s t a b l i s h e d supermarket. The general form of the p r o f i t f u n c t i o n to be c a l c u l a t e d i n a l l cases i s as f o l l o w s : (4.1) n; = A S - o - K - P C - ic, where II - p r o f i t s per square fo o t of s e l l i n g area per week, AS = average s a l e s per square foot of s e l l i n g area per week, 0 = operating costs per square fo o t of s e l l i n g area per week, K = c a p i t a l costs per square f o o t of s e l l i n g area per week, PC = product costs per square fo o t of s e l l i n g area per week, and IC = inventory costs per square fo o t of s e l l i n g area per week, A l l ; v a r i a b l e s are defined i n u n i t s of d o l l a r s per square f o o t of s e l l i n g area per week sin c e our s a l e s data are i n those u n i t s . .(Ideally we would l i k e to be able to c a l c u l a t e (4.1) f o r d i f f e r e n t s t o r e s i z e c a t e g o r i e s since we would not expect costs and revenues to be independent.of s t o r e s i z e . However, as s t a t e d i n s e c t i o n 4.3, our data are i n the form of averages, where the averages are over a l l super-markets i n the sample i r r e s p e c t i v e of s t o r e s i z e . ) In what f o l l o w s , we use s u b s c r i p t N to r e f e r to new supermarkets and s u b s c r i p t E to r e f e r to e s t a b l i s h e d supermarkets. We begin our d i s c u s s i o n by examining the procedure that i s used to obta i n estimates; of each component of equation (4.1) f o r a r e p r e s e n t a t i v e e s t a b l i s h e d supermarket. Data on the f i r s t component of equation (4.1), average, s a l e s per square fo o t of s e l l i n g area per, week, are obtained d i r e c t l y from SMIS. The second component of equation (4.1), operating costs per square foot o f . s e l l i n g area per week, i s defined to i n c l u d e d i r e c t operating c o s t s , DOi and i n d i r e c t operating c o s t s , 10. Data on d i r e c t operating costs as a percentage of average annual sales per e s t a b l i s h e d s t o r e , do , are obtained d i r e c t l y from ORFC, and i n c l u d e labor costs of employees assigned to s t o r e s , s u p p l i e s and s e r v i c e s consumed at. the s t o r e s , and s i m i l a r d i r e c t store c o s t s , covering r e c e i p t , h a n d l i n g , p r e p a r a t i o n f o r - 76 -s a l e , d i s p l a y and s a l e of merchandise and the r e l a t e d customer s e r v i c e s . ^ Data on i n d i r e c t operating costs as a percentage of average annual s a l e s per e s t a b l i s h e d s t o r e , i o , are a l s o obtained from ORFC, and i n c l u d e the . 7 . . 8 costs of. warehouse operations, t r a n s p o r t a t i o n operations, 9 10 merchandising and buying, a d v e r t i s i n g and s a l e s promotion, accounting 11 12 13 and o f f i c e services,. general a d m i n i s t r a t i o n , f i e l d s u p e r v i s i o n , 14 15 employee b e n e f i t s , and non-store occupancy. In order to o b t a i n an estimate of 0 using ORFG data, we must perform the f o l l o w i n g E c a l c u l a t i o n : (4.2) 0 E = D0 E + I 0 E = AS E ( d o E + i o E ) , where do„= d i r e c t operating costs as a percentage of s a l e s and i o = E Ji i n d i r e c t operating costs as a percentage of s a l e s . We. w i l l assume that a l l of the costs included i n 0 are v a r i a b l e c o s t s . That i s , a l l of E these costs w i l l be assumed to. vary p r o p o r t i o n a l l y with.the opening of a new s t o r e by a f i r m . The t h i r d component of equation (4.1) i s c a p i t a l costs per square foot of s e l l i n g area per week. Data on c a p i t a l costs as a percentage of average' annual sales per e s t a b l i s h e d s t o r e , k , are a v a i l a b l e from E ORFC. ORFC defines c a p i t a l costs to i n c l u d e a l l of the company's non-c a p i t a l i z e d costs r e l a t i n g to s t o r e r e a l estate., b u i l d i n g s , f i x t u r e s and equipment ( i n c l u d i n g u t i l i t i e s , insurance, taxes, l i c e n s e s , property and equipment rentals., d e p r e c i a t i o n and a m o r t i z a t i o n , r e p a i r s , and c r e d i t s and a l l o w a n c e s ) . ^ To put these costs on a per square fo o t of s e l l i n g area per week b a s i s , we must perform the f o l l o w i n g simple c a l c u l a t i o n : (.4.3) K E = AS E ( k E ) . The f o u r t h component of equation (4,1), product costs, per square fo o t of s e l l i n g area per week, may be c a l c u l a t e d as f o l l o w s : (.4.4) PC E = ( l - g n i g ) AS E , where gm^ , = gross margin as a percentage of s a l e s . ORFC rep o r t s data on the gross margin as a percentage of average, annual s a l e s per e s t a b l i s h e d s t o r e and defines the gross margin as the amount remaining a f t e r the 18 deduction of net cost of merchandise s o l d from net s a l e s . In our n o t a t i o n , (.4.5) AS - PC = GM, where GM = gross margin per square fo o t of s e l l i n g area per week. I f we assume that P'Q = AS and c Q = PC, where E = p r i c e , Q = q u a n t i t y s o l d , and c = the :unit cost of goods s o l d , then we could r e w r i t e (4.5) as f o l l o w s : (4.6) P p ° = gm. We should p o i n t out. that our d e f i n i t i o n , of gross margin d i f f e r s from the standard d e f i n i t i o n of the term, which u s u a l l y defines: the margin as a mark-up on cos t s . The f i n a l component of equation (4.1) i s inventory costs per square foot of s e l l i n g area per week,. and represents the cost to the f i r m of t y i n g up c a p i t a l i n the form of unsold inventory. Inventory costs may - 78 -be c a l c u l a t e d using the f o l l o w i n g equation: AS (.4.7) I C E = E U - S V ' ITR E 19 where ITR = inventory turnover r a t e (or stockturns).. Data on ITR may be obtained d i r e c t l y from ORFC. The i n t e r e s t r a t e , r , to be used i n our c a l c u l a t i o n of IC> i s the U.S. prime l e n d i n g r a t e plus one percent. The E p a r t i c u l a r choice of i n t e r e s t r a t e was a r b i t r a r y , but i t w i l l have very l i t t l e impact on our net p r o f i t c a l c u l a t i o n s , since IC c o n s t i t u t e s a very small component of t o t a l costs per square foot of s e l l i n g area per week. Using equations (4.2) - (4.7), we may now r e w r i t e equation (4.1) as f o l l o w s : (4.8) AS ( l - g m ) r IT = AS E - AS E(.do E + i o E ) - AS. E(k E) - ( l - g m E ) A S E  ITR E ( l - g m E ) r = AS E(gm E - d o £ - i o E - k E - - — - ) • E From equation (4.8), we f i n d that whether our c a l c u l a t i o n of the net p r o f i t s of a r e p r e s e n t a t i v e e s t a b l i s h e d supermarket i s p o s i t i v e or negative depends s o l e l y on the cost s as percentage of s a l e s data reported i n ORFC and als o the i n t e r e s t r a t e , and not on the absolute l e v e l of ASg. However, we s h a l l f i n d i t u s e f u l to have an estimate of II i n order to N compare i t wi t h the net p r o f i t s of a r e p r e s e n t a t i v e new supermarket, II . In the remainder of t h i s s e c t i o n , we s h a l l d i s c u s s . t h e procedure that i s used to c a l c u l a t e the net p r o f i t s of a r e p r e s e n t a t i v e new supermarket, N II . This c a l c u l a t i o n w i l l r e q u i r e a number of s i m p l i f y i n g assumptions due to the nature of the cost data which we must use. As befor e , we s h a l l proceed by c o n s i d e r i n g how we might o b t a i n estimates of each component of equation (4.1). Data on the f i r s t component of equation (4,1), average s a l e s per square foot of s e l l i n g area per week, are obtained d i r e c t l y from FANSM. Data on the second component of equation (4.1), operating costs per square fo o t of s e l l i n g area per week, cannot be obtained from FANSM. We do have data on do and i o _ from ORFC, but we cannot use t h i s data h E d i r e c t l y i n order to c a l c u l a t e 0^ f o r the f o l l o w i n g reasons: The ORFC f i g u r e s on operating costs as a percentage of s a l e s are based on the sales, of e s t a b l i s h e d supermarkets.; However, from FANSM and SMIS, we f i n d that the average sa l e s per square fo o t of s e l l i n g area per week f o r new supermarkets are l e s s than- the average s a l e s per square fo o t of s e l l i n g area per week f o r e s t a b l i s h e d supermarkets i n any given year, and s e v e r a l studies have shown that average costs: as a percentage of s a l e s are not constant f o r various l e v e l s of average s a l e s per square fo o t (the u t i l i z a t i o n r a t e ) . In p a r t i c u l a r , Mallen and Haberman [1975] have estimated long run and short, run:cost f u n c t i o n s using cost data from 130 supermarkets owned.by a major Canadian supermarket chain. They found that average costs as a percentage of s a l e s d e c l i n e d s i g n i f i c a n t l y over almost the e n t i r e range of u t i l i z a t i o n r a t e s f o r which they had data (holding s t o r e s i z e constant),-while average costs as a percentage of s a l e s i n i t i a l l y decrease and then increase w i t h s t o r e s i z e f o r , f i x e d u t i l i z a t i o n - 80 -r a t e s . ( I t should be noted that the change i n average costs w i t h s t o r e s i z e i s very small r e l a t i v e to the change i n average costs due to changing u t i l i z a t i o n r a t e s . In f a c t , the long run average cost f u n c t i o n 20 appears to be almost L^-shaped.) S a v i t t [1975], i n an independent study of the cost f u n c t i o n s of a l a r g e number of Canadian supermarkets, a r r i v e d . . ., ' . 21 at s i m i l a r conclusions. Thus, given that average costs as a percentage of s a l e s would be expected to d e c l i n e . w i t h i n c r e a s i n g u t i l i z a t i o n r a t e s (over some range), and given that new supermarkets i n general have lower u t i l i z a t i o n rates than e s t a b l i s h e d supermarkets, we cannot use. ORFC data on, operating costs as a. percentage of sal e s i n order d i r e c t l y to estimate 0^. We can, however, use the ORFC data i n order to construct upper and lower bounds on 0^. F i r s t , s i n c e the a v e r a g e ; u t i l i z a t i o n r a t e s of new supermarkets are, i n general, lower than f o r e s t a b l i s h e d supermarkets, we would expect the operating costs of a r e p r e s e n t a t i v e new supermarket to be a b s o l u t e l y lower than the operating costs of a r e p r e s e n t a t i v e e s t a b l i s h e d supermarket w i t h i t s higher u t i l i z a t i o n r a t e . Thus, (4.9) 0 N < A S E ( d o E + i o E ) . However, i t i s a l s o true that (.4.10) A S N ( d o E + i o E ) < 0 N s i n c e average costs as a percentage of sal e s f o r a r e p r e s e n t a t i v e e s t a b l i s h e d supermarket should be "lower than average costs.as a percentage of s a l e s f o r a r e p r e s e n t a t i v e new supermarket. Having derived these upper and lower bounds on'0 , we can perform.our p r o f i t c a l c u l a t i o n s using these - 81 -N bounds i n order to o b t a i n a range estimate of H t The t h i r d component of equation (4.1), c a p i t a l costs per square f o o t of s e l l i n g area per week, must a l s o be estimated for. a r e p r e s e n t a t i v e new supermarket u s i n g ORFC and SMIS data. I t w i l l be r e c a l l e d that ORFC's f i g u r e s on c a p i t a l costs as a percentage of s a l e s , k„", are defined to i n c l u d e a l l o f . t h e company's n o n - c a p i t a l i z e d costs r e l a t i n g to s t o r e r e a l e s t a t e , b u i l d i n g s , f i x t u r e s and equipment. Now, there does not seem to be any compelling reason to b e l i e v e that the c a p i t a l costs per square foot f o r new supermarkets would d i f f e r s i g n i f i c a n t l y from the c a p i t a l costs per square foot f o r e s t a b l i s h e d supermarkets. Thus, we s h a l l estimate as f o l l o w s : (4.11) K N = A S E ( k E ) . The f o u r t h component of equation (4.1) i s product costs per square foot of s e l l i n g area per.week, or PC = (l-gm)AS. Since FANSM does not report gross margin data, we must once again use ORFC data i n order to obtain an estimate of gm^ - This, r a i s e s the question of whether gm^ i s a good estimator of g 1 0^* and we answer t h i s question i n the f o l l o w i n g way: F i r s t , the u n i t costs of goods s o l d by new supermarkets and e s t a b l i s h e d supermarkets might be d i f f e r e n t . However, given that almost a l l super-markets have access to some warehouse, f a c i l i t i e s , we would not expect the u n i t cost of goods s o l d to d i f f e r s i g n i f i c a n t l y f o r d i f f e r e n t f i r m s . That i s , most supermarkets, can take advantage of lower costs of d i s t r i b u t i o n ( i . e . through'quantity discounts, on the purchase of l a r g e volumes of goods, r a t i o n a l i z a t i o n of inventory management, etc.) even i f they do not own t h e i r own warehouses. Second, i f new supermarkets were p r i m a r i l y being -.82 -opened by e s t a b l i s h e d f i r m s , then the theory of preemption suggests that these st o r e s w i l l be l o c a t e d i n t h e i r j o i n t p r o f i t maximizing l o c a t i o n s and w i l l charge the j o i n t p r o f i t maximizing p r i c e s . We would a l s o expect these p r i c e s to be higher than those charged by new entrants i n t o the market. I f the p r i c e s charged by new supermarkets are high e r , on average, than 7those charged by a l l e s t a b l i s h e d f i r m s , then by equation (4,6): we would expect grn^ > gm^ ,. If-, on the other hand, e s t a b l i s h e d f i r m s have, had the. opportunity to adjust t h e i r p r i c e s to the j o i n t p r o f i t maximizing p r i c e s , and i f new supermarkets are p r i m a r i l y being opened by these f i r m s , then we would not expect much d i f f e r e n c e between e s t a b l i s h e d supermarket and new supermarket p r i c e s i n any, given time p e r i o d . Hence, gm^ would be a good estimator of gm^ - Therefore, under the assumption that e s t a b l i s h e d firms preempt t h e i r markets, we could use ORFC d a t a , i n order to c a l c u l a t e PC„ as f o l l o w s : N (4.12) PC N = A S N ( l - g m E ) . Furthermore,, gm^ , would be a good estimator of grn^ i f new supermarkets are p r i m a r i l y opened, by new entrants and i f p r i c e competition prevents e s t a b l i s h e d f i r m s from, charging the p r o f i t maximizing p r i c e s . ( In other words, new entrants and e s t a b l i s h e d f i r m s charge the same p r i c e s . ) We s h a l l - use (4.12) to estimate PC^ f o r our r e p r e s e n t a t i v e new supermarket, w h i l e r e c o g n i z i n g that depending on a c t u a l circumstances, (4.12) might 22 overestimate or underestimate true PC„. N The l a s t . component of equation (.4.1) i s inventory costs per square foot of s e l l i n g area per week. The c a l c u l a t i o n of IC^ would be a straightforward-matter.were i t not f o r the f a c t that FANSM does not supply us w i t h i n f o r m a t i o n on the inventory turnover r a t e . Data on ITR hi - 83 -may be obtained from ORFC, but we would expect ITR^ to exceed ITR^ due to the higher u t i l i z a t i o n r a t e f o r a r e p r e s e n t a t i v e e s t a b l i s h e d super-market. Thus, the best we can do. i s to construct a lower bound estimate of IC„ as f o l l o w s : N • AS ( l - g m ) r (4.13) . IC > — — . I T R E We may now combine equations (4.9) - (4.13) to ob t a i n upper and N lower bound estimates of n : (4.14) N A S N ( 1 - § m E ) r E ( l - g m ) r - A S N ( ^ E - d ° E - l 0 E " — > ~ AW L IK,., hi (4.15) AS H - gin ) r nLB " A S N " A S E ( d ° E + *»J ~ AW ~ ( 1 " § m E ) A S N " ITR E (1 - gm E)r = A S N ( g m E - — >- ^ E ^ E + ^ E + V I T R E The only d i f f e r e n c e between equation (4.8) and equation (4.14), aside from average s a l e s per square f o o t of s e l l i n g area per week, l i e s i n t h e . f a c t that c a p i t a l costs per square foot of s e l l i n g area per week w i l l be a l a r g e r percentage of AS than of• AS . The d i f f e r e n c e s between equation N E (4.8) and equation (4.15) are, aside from average s a l e s per square f o o t of s e l l i n g area per week, r e f l e c t e d i n the f a c t that operating c o s t s as - 84 -w e l l as c a p i t a l costs w i l l be a l a r g e r percentage of AS M than of AS . N E In the next s e c t i o n , we report the r e s u l t s of " c a l c u l a t i o n s of E N N II , , and I I L B f o r a seven year p e r i o d . 4.5 Test R e s u l t s of the N u l l Hypothesis of Negative I n i t i a l P r o f i t s E N In t h i s s e c t i o n , we report the r e s u l t s , of our c a l c u l a t i o n s of H , I I , LIB N 23 and n^g f o r the years 1970-1976 i n c l u s i v e . Before r e p o r t i n g our r e s u l t s , we should note three conventions which were employed i n a l l of our c a l c u l a t i o n s . F i r s t , SMIS and FANSM report AS„ and AS.T r e s p e c t i v e l y , f o r stores operating i n a given calendar year, whereas ORFC data i s based on a f i s c a l year, May - A p r i l . The only means of r e c o n c i l i n g the data s e t s i s an a r b i t r a r y one, and we therefore, use, f o r example, ORFC data from the 1976-77 f i s c a l year i n order to c a l c u l a t e net p r o f i t s i n 19.76. Second, i n order to get an estimate of the annual prime lending r a t e charged by U.S. banks, we found i t necessary to take the average of average monthly prime le n d i n g r a t e s . The i n t e r e s t r a t e , r , i s then obtained by adding one percent to.our estimate of the U.S. prime l e n d i n g r a t e . F i n a l l y , ORFC reports TTR^ , data f o r three c a t e g o r i e s of fir m s doing d i f f e r e n t l e v e l s of sale s ( f o r example, f o r f i r m s w i t h s a l e s below $100 m i l l i o n , w i t h s a l e s between $100 m i l l i o n and $500 m i l l i o n , and with s a l e s above $500 m i l l i o n ) . Our estimate of the inventory turnover rate: i s the simple unweighted average of these three f i g u r e s . As f o r the i n t e r e s t r a t e , the way i n which we estimate the: inventory turnover r a t e w i l l have very l i t t l e impact on our net p r o f i t c a l c u l a t i o n s s i n c e IC c o n s t i t u t e s a very s m a l l component of t o t a l costs per square f o o t of s e l l i n g area per week. In Table I , we report the r e s u l t s of our net p r o f i t c a l c u l a t i o n s . - 85 -Table I NET PROFIT CALCULATIONS: 1970-1976 19 70 E ( l - g m E ) r n = A S ^ g i ^ - d O g - i O g - k g - — — ) = 4.16 (.2139 - .0939 - .0783 - .0347 - — ' ° 8 3 ^ ) 14.39 = .0101674 + n Eas % of AS^ = .24441% ( l - g m ^ ) r nUB " A S N ( g m E " d°E " l 0 E " I T R E ' > " k E ( A S E > = 3.44 (.2139 - .0939 - . 0 7 8 3 - ( 1 ~ ' j ^ 3 ^ ' 0 8 3 4 ) - .0347(4.16) = - .0165763 - n N as % of AS = .48186% UB' N N ( l - g m j r "LB " A S N ( g m E - I T R e > " A S E ( d o E + l o E + k E > = 3.44C .2139 - ( 1 ~ ' 1 4 3 3 9 ' ° 8 3 4 )• - 4.16( .0939 + .0783 + .0347) = - .1405603 -•nJL as % of AS = 4.08605% LB N - 86 -Table I NET PROFIT CALCULATIONS: 1 9 7 0 - 1 9 76 19 71 n = A S , Xgmg-dOg-iOg-kg -(1 - gn^) r E^-E - E ~E ITR, ' = 4 .55( .2153- .0953 - .0791 - .0347 - ( 1 ~ -2153).0661. 15.01 J = .012487 + n E as % of AS,, = .27444% E i N ( l - g m j r % - ^ N ^ - ^ E - ^ E " I T R e } " kE ( AV = 3.50( .2153 - . 0 9 5 3 - . 0 7 9 1 - i i ^ | ^ | p 6 6 1 ) _ .0347(4.55) = - .0268296 N - n as • % of A S N = .76656% ( l - g m ^ ) r nLB = ^ N ^ " ITR, > " ^ E ^ E + ^ E + V = 3 . 5 0 ( . 2 1 5 3 - ( 1 ~ ' ^ 5 5 Q } ' Q 6 6 1 ) - 4 . 5 5 ( . 0 9 5 3 + . 0 7 9 1 + . 0 3 4 7 ) = - . 2 0 9 9 4 9 6 - nJL as % of AS T = 5.99856% LB N - 87 -Table I NET PROFIT CALCULATIONS: 1970-1976 19 72 [ RE n = A S E ( . g m E - d o E - i o E - k E - — j - — ) = 4.34C.2093- .0968 - .0774- .0341 - ( 1 ~ ' i " 9 ^ - ' 0 5 2 4 ) = - .0089499 - n E as % of AS„ = .20622% N ( l - g r n ^ r "UB = A S N ( g m E - d ° E - i o E " ITR E . > " k E ( A S E ) = 3.48C.2093 - .0968 - .0774 - ( 1 ~ ' ^ P . 9 ^ - 0 5 2 4 ) - .0341(4.34) = - .0365025 - I I ^ B as % of AS N = 1.04892% ( l - g n u ) r nLB = ^ N ^ - ITR } - ^ E ^ E + ^ E + V E = 3.48C.2093 - ( 1 ~ ' 2 3 9 5 3 , Q 5 2 4 ) - 4.34(.0968+.0774+.0341) = - .1863145 - as % of AS = 5.35386 % Ld IN * In 1972, average s a l e s per square fo o t of s e l l i n g area per week were reported by- FANSM f o r conventional supermarkets, combination super-markets, and food departments, and not f o r a l l new supermarkets i n general. To o b t a i n an estimate of AS^, we took the unweighted average of AS f o r these three separate c a t e g o r i e s , -.88 -Table I NET PROFIT CALCULATIONS: 1970-1976 1973 E ( l - g m E ) r H = ASgCgtUg - d o E - i o E - k E - - j — ) E = 4. 71(.2090 - ,0960 - .0790 - ,0329 - ( 1 ~ -^090).0772 ) = - .0157059 - n E as % of AS,, = .33346% N , ( l - g n ^ ) r V = A S N ( g m E ' d°E " l 0 E " I T R e > " k E ( A S E > = 3.75(.2090 - .0960- .0790 - (1 " ' ^ j^ 0 7 7 2)- .0329(4.71) = - .0440888 - n£i as % of AS = 1.1757% UB N N ( l - g m ^ r n L B = A S ^ \ ~ I T h >- A S E ( d 0 E + i 0 E + k E ) = 3.75(.2090 - d - •2090).0772 ) _ 4 . 7 1 ( > 0 9 6 0 + > 0 7 9 0 + > 0 3 2 9 ) = - .2120888 - as % of AS = 5.6557% LB N - 89 -Table I NET PROFIT CALCULATIONS: 1970-1976 1974 ( 1 - gm E)r n " = A S E ( g m E " d°E " ± 0 E " k E ITR ~ } = 5.09C.2115 - .0969 - .0744 - .0339 - ( 1 " ' 1 1 7 5 ) = - .0022294 - n E as % of AS,, = .0438% N ( l - g ^ r nUB " A S N ( g m E - d ° E - i 0 E " IT* ) ~ V AV = 4.22C.2115 - .0969 - .0744 - ( 1 " ' j ^ 1 ^ ' 1 1 7 5 ) - .0339(5.09) = - .0313414 - ILj as % of AS N = .74268% ( l - g m ^ ) r n L B = A S N ( g m E " I T R e > " ^ V^E + ^ E + V = 4.22(.2115 ~ ( 1 ~ 'i\15-/\'1175 > " 5.09(.0969 + .0744+.0339) = - .1803724 - as % of AS„ = 4.27422 % LB N - 90 -Table I NET PROFIT CALCULATIONS: 1970-1976 19 75 n = A S E ( g m E - d o E - i o E - k i (1 - gm E)r IT = 5.33( .2122 - .0965 - .0762 - .0347 - — — - 2 1 2 2 ) - 0 8 8 6 ) 14.63 = .0001551 + n E as % of AS^ = .00291% h. N ( l - g m ^ ) r V - A S N ( ^ E - d ° E - i 0 E " IT^ ) ~ = 4.50(.2122 - .0965 - .0762 - ( 1 ~ - 2 1 2 2 ) - 0 8 8 6 ) _ .0347(5.33) 14.63 = - .0291201 - i i as % of AS = .64711% U;B N (1 - gm E)r "LB = A S N ( ^ " I T R / > " ^ E ^ E + V = 4.50(.2122 - ( 1 " - 2 1 2 2 ) - 0 8 8 6 ) _ 5.33(.0965+ .0762+ .0347) 14.63 = - .172011 - nJL as % of AS A T = 3.82246% LB N - 91 -Table I NET PROFIT CALCULATIONS: 1970-1976 1976 E ( l - g r n ^ r n = A s E ( g m E - d o E - i o E - k E - ) £ = 5.50(.2135 - .10 - .0763-.0335 - ( 1 ~ 2 ^ ' 0 7 8 4 ) 13.66 = - .004477 - " n E as % of ASW = .0814% hi N ( l - g m ^ r % B = A S N ( g m E " d°E " l 0 E " . - i T R ^ > ~ k E ( A S E ) = 4.78(.2135 - .10- .0763 - ( 1 " - 2 1 3 5>-0784 ) _ .0335(5.50) 13 . DO = - .028011 - n§„ as % of ASA1 = .586% UB N ( l - g m j r "LB - AV g mE " ' ITR^ > - ^ V ^ E ^ V V = 4.78(.2135 - ( 1 " ' 2 o 3 ^ ' ° 7 8 4 ) ~ 5.50(.10+.0763+.0335) 13.66 = - .154947 - nJL as % of AS = 3.24156% LB N Looking f i r s t at our c a l c u l a t i o n s of n^j we f i n d that was p o s i t i v e i n 1970 and.1971, negative; i n 1972 and 1973, and very c l o s e to zero i n 1974, 19 75, and 19 76. I t i s . . i n t e r e s t i n g to note that even though e s t a b l i s h e d supermarkets, on average, were making negative or zero p r o f i t s between 19 72 and 1976 i n c l u s i v e , new supermarkets were s t i l l being opened. This r e s u l t might be explained by the f a c t that some l o c a l markets were growing, and new supermarkets were being e s t a b l i s h e d , i n these markets, w h i l e other l o c a l markets were d e c l i n i n g , and e s t a b l i s h e d supermarkets i n these l o c a l markets were being closed due to t h e i r u n p r o f i t a b i l i t y . N E When we compare our c a l c u l a t i o n s of n w i t h our c a l c u l a t i o n s of n , N we f i n d that our upper and lower, bound estimates of n are always negative and l e s s than n . N N N We can only speculate about which estimate of n » IIygx or Il-^gj would be c l o s e r to the true net p r o f i t s of our r e p r e s e n t a t i v e new supermarket. One of the most c r i t i c a l determinants would be the shape of the long run average cost curve. I f both AS and AS appear on a f l a t p o r t i o n of the average cost curve, then do„ and io„ would c o n s t i t u t e good estimators of N do^ and i o ^ . This would then imply that IL^g would be a good estimator of N the t r u e II . On the other hand, suppose that AS^ was on the d e c l i n i n g p o r t i o n of an L-shaped short run average cost curve, w h i l e AS„ was on the hi N f l a t p o r t i o n of that curve. Then, nTT, might be a b e t t e r estimator of the N true n . Support f o r t h i s c o n c l u s i o n i s obtained when one considers that many of the component elements of i o would be i n c u r r e d by a f i r m when operating a 'supermarket, regardless of the absolute l e v e l of AS. This would then imply that AS (io„) would be a f a i r l y accurate estimator of 'hi hi 10^. However, s i n c e DO would be expected to vary w i t h AS, we would expect - 93 -N do„(AS ) to overestimate DO •. Thus, whether or not nT-_ i s a b e t t e r N N estimator of II than n ^ g , depends upon the shape of the short run average cost curve and a l s o the extent to which do (AS ) and i o (AS ) are Hi L Hi Xj accurate estimators of D0.T and I0„,. N N N N N Regardless, however, of whether J[Tr> or I T T r D lis c l o s e r to the true n , N we have shown that our estimates of II are more than j u s t m a r g i n a l l y negative and that they.are s u b s t a n t i a l l y l e s s than II . The d r i v i n g f o r c e of these r e s u l t s i s as f o l l o w s : F i r s t , f o r the years 1970-1976, AS^ i s l e s s than AS . Second, s i n c e we have maintained that the c a p i t a l costs per square foot of s e l l i n g area per week of new supermarkets would not d i f f e r s i g n i f i c a n t l y from the c a p i t a l costs per square f o o t of s e l l i n g area per week of e s t a b l i s h e d supermarkets, then (4.16) ^ = A S E ( k E ) = K E. Thus, i n comparing equations (4.8) and (4.14), we f i n d t h a t AS < AS N h, N E and = IC^ , imply that 1 1 ^ < IT . In comparing equations (4.8) and (4.15) we f i n d that AS^ < ASg, = 1 ^ and 0 N = 0 E imply that 1 1 ^ < n^. < I I E . N N F i n a l l y , AS„ i s small r e l a t i v e to AS_. Thus, not only are I I T _ and I I T T T , J N E LB UB l e s s than II , but they are a l s o l e s s than zero. Hence, our c a l c u l a t i o n s support the n u l l hypothesis . that the average p r o f i t s of new supermarkets i n the United States and Canada are, negative i n the f i r s t twelve months of operation of these supermarkets. In the next s e c t i o n , we i n t e r p r e t the s i g n i f i c a n c e of these r e s u l t s and make some concluding remarks. 4.6 I n t e r p r e t a t i o n of Results and Concluding Remarks In t h i s chapter of the t h e s i s , we have examined the n u l l hypothesis of negative i n i t i a l p r o f i t s . Our c a l c u l a t i o n s of. n N provided evidence i n - 94 -support of the n u l l hypothesis. However, we should note some of the l i m i t a t i o n s of our t e s t i n g procedure, One of the most s e r i o u s l i m i t a t i o n s was our i n a b i l i t y to do a proper s t a t i s t i c a l t e s t of the n u l l hypothesis due to the absence of s u f f i c i e n t i n f o r m a t i o n on the v a r i a n c e and d i s t r i b u t i o n s of cost and revenue v a r i a b l e s . ' A second l i m i t a t i o n was N our i n a b i l i t y to c a l c u l a t e II .using data drawn s t r i c t l y r f r o m a sample of N new supermarkets. That i s , to o b t a i n estimates of II , we had to adapt ORFC cost data using s e v e r a l r e s t r i c t i v e assumptions ( e s p e c i a l l y w i t h respect to c a p i t a l c o s t s , operating costs and the gross margin). We were unable to check the accuracy of these assumptions w i t h the l i m i t e d data at our d i s p o s a l . The most s e r i o u s l i m i t a t i o n of the. a n a l y s i s i s that we were only able to o b t a i n evidence i n support of an hypothesis regarding the average p r o f i t s of new supermarkets, and not an hypothesis regarding the p r o f i t s of i n d i v i d u a l , new supermarkets. Apart from the l i m i t a t i o n s of the a n a l y s i s , we should make a few comments regarding the s i g n i f i c a n c e of our r e s u l t s . In p a r t i c u l a r , are N there other reasons why our estimates of II might be negative apart from the e x p l a n a t i o n that f i r m s have engaged i n preemptive l o c a t i o n s t r a t e g i e s ? One e x p l a n a t i o n that comes to mind i s that s t a r t - u p costs f o r new super-markets might be appreciable. Since, however, we use ORFC cost data i n order to estimate the cost components of II , s t a r t - u p costs have not been inc l u d e d i n ourr.net p r o f i t c a l c u l a t i o n s . Thus s t a r t - u p costs do not provide an explanation f o r negative JITTli-. or u T 1 J . Ut> L i i N One might be tempted to a s s e r t that our estimates, of II are negative because i t takes time f o r consumers to r e a l i z e that a new supermarket has been e s t a b l i s h e d i n t h e i r neighborhood,or because i t takes time f o r - 9 5 -consumers to change t h e i r shopping h a b i t s . We do not a t t a c h much importance to these as explanations f o r our r e s u l t s f o r the f o l l o w i n g reasons: F i r s t , .while some consumers may not r e a l i z e that a new super-market has been es t a b l i s h e d , i n t h e i r neighborhood during the f i r s t few weeks that i t i s i n o p e r a t i o n , we would not expect such a l a c k of awareness to l a s t f o r an extended p e r i o d of time. Second, economic theory and e m p i r i c a l s t u d i e s suggest that consumers are responsive to p r i c e , and we would not expect a l a r g e number of consumers to forego the cost savings to be obtained by shopping at a c l o s e r new supermarket i n order to shop at a more, d i s t a n t , but f a m i l i a r , o l d e r supermarket. However, the r e l a t i v e importance of these a l t e r n a t i v e explanations f o r our estimates of N II are u l t i m a t e l y matters f o r e m p i r i c a l . i n v e s t i g a t i o n . We conclude that i n s u f f i c i e n t consumer de n s i t y provides the best N explanation f o r our estimates of II . That i s , our e m p i r i c a l evidence i s c o n s i s t e n t w i t h firms e s t a b l i s h i n g new supermarkets i n the market such that these new supermarkets represent excess c a p a c i t y at the time i n which they are opened. This explanation,, i n t u r n , i s c o n s i s t e n t w i t h the hypothesis of preemptive f i r m behavior. In the next chapter, we consider the l o c a t i o n a l i m p l i c a t i o n s of preemption. We devise t e s t s to determine simultaneously i f preemption has occurred i n p a r t i c u l a r markets and, i f so, which firm: i s the preemptor. FOOTNOTES TO CHAPTER 4 The Super Market Industry Speaks, 1976, conducted by Research D i v i s i o n , Super Market I n s t i t u t e , Inc. In 1976, 40 percent of the I n s t i t u t e ' s U.S. and Canadian member companies p a r t i c i p a t e d i n the survey, and these companies operated,10,278 food s t o r e s . (We should a l s o note that i n 1977, the Super Market I n s t i t u t e changed i t s name to the Food Marketing I n s t i t u t e . ) For example,, i n 1976, a t o t a l of 384 companies r e p l i e d to the Food Marketing I n s t i t u t e survey, and of these, 103 companies had e i t h e r opened'or close d supermarkets during the previous year. See Facts  About New Supermarkets. Opened i n 1976, conducted by Research D i v i s i o n , Super Market I n s t i t u t e , Inc. In 1976, the operating r e s u l t s data reported i n ORFC were based on surveys of 58 companies operating 5831 s t o r e s , with,aggregate s a l e s of 24.4 b i l l i o n d o l l a r s f o r the f i s c a l year May, 1975, to A p r i l , 1976. While ORFC does re p o r t sales, data, the s a l e s data are not i n a form which.is u s e f u l for. our p r o f i t c a l c u l a t i o n s . Our estimate of the net p r o f i t s of e s t a b l i s h e d supermarkets i s based on ORFC and SMIS data. This data i n c l u d e s the ope r a t i n g r e s u l t s of new supermarkets as w e l l . a s that of ol d e r supermarkets. ORFC, 1975-76, p.93. The costs of warehouse operations i n c l u d e the costs of r e c e i v i n g , checking, s t o r i n g , s e l e c t i n g , and l o a d i n g of merchandise and s u p p l i e s - 97 -f o r d i s t r i b u t i o n 1 to the s t o r e s , and exclude employee b e n e f i t s and occupancy cos t s . ORFC, 1975-76, p.93, 8. The costs of t r a n s p o r t a t i o n operations i n c l u d e the costs of operating a f l e e t of v e h i c l e s f o r the d e l i v e r y of merchandise to s t o r e s . H i r e d h a u l i n g i s inc l u d e d i n t h i s cost category, but employee b e n e f i t s and garage occupancy are excluded.. ORFC, 1975-76, p.93. 9. Merchandising and buying costs i n c l u d e a l l of the costs (excluding employee b e n e f i t s and occupancy) of developing merchandise and p r i c i n g p o l i c i e s and the procurement of a l l items s o l d or: consumed i n the st o r e s . This cost category a l s o i n c l u d e s the costs of a l l buyers, merchandising managers, and c l e r i c a l and a d m i n i s t r a t i v e a s s i s t a n t s . ORFC, 1975-76, p.93. 10. A d v e r t i s i n g and sal e s promotion costs i n c l u d e a l l of the costs (excluding employee b e n e f i t s and occupancy) r e l a t i n g , to a d v e r t i s i n g and d i s p l a y , s a l e s promotion, customer r e l a t i o n s , and p u b l i c r e l a t i o n s , and a l l other costs i n c u r r e d to a t t r a c t and r e t a i n customers. ORFC, 19 75-76, p.93. 11. Accounting and o f f i c e s e r v i c e s costs i n c l u d e a l l costs (excluding employee b e n e f i t s and occupancy) of accounting and bookkeeping a c t i v i t i e s i n c l u d i n g t a b u l a t i n g , i n t e r n a l a u d i t i n g , s t o r e inventory t a k i n g and processing, budgeting, and a l l s i m i l a r a c t i v i t i e s u s u a l l y performed by the c o n t r o l l e r ' s o f f i c e , A l s o included are the costs of o f f i c e s e r v i c e s such as m a i l room, telephone switchboard, general o f f i c e s u p p l i e s , e t c . ORFC, 1975-76, p.93. - 98 -12. General a d m i n i s t r a t i o n , costs i n c l u d e the costs of a l l c e n t r a l o f f i c e a c t i v i t i e s not provided for, i n other cost c a t e g o r i e s , Included here are the costs and expenses of c o r p o r a t e . o f f i c e r s , s t a f f , personnel a d m i n i s t r a t i o n , insurance a d m i n i s t r a t i o n , r e a l e s t a t e management, design and c o n s t r u c t i o n , research and development, l e g a l and f i n a n c i a l , e t c . ORFC, 1975-76, p.93. 13. F i e l d s u p e r v i s i o n costs i n c l u d e a l l of the costs (excluding employee b e n e f i t s and occupancy) of the employees engaged i n the s u p e r v i s i o n and a d m i n i s t r a t i o n of the s t o r e s . Included here are the costs of the superintendent of s t o r e s , r e g i o n a l , d i v i s i o n a l , or d i s t r i c t managers and s u p e r v i s o r s , f i e l d merchandisers and s p e c i a l i s t s , and the c l e r i c a l and. a d m i n i s t r a t i v e a s s i s t a n t s of such employees,.whether s t a t i o n e d i n the f i e l d or at the home o f f i c e . ORFC, 1975-76, p.93. 14. Employee b e n e f i t s i n c l u d e the costs f o r f r i n g e b e n e f i t s of employees which a r i s e from management p o l i c y , from n e g o t i a t i o n s w i t h l a b o r unions, or as a r e s u l t of governmental requirements, i n c l u d i n g v a c a t i o n s , s i c k leaves, p a y r o l l taxes, personnel insurance premiums and s i m i l a r payments. ORFC, 1975-76, p. 93. 15. Non-store occupancy costs i n c l u d e a l l of the .non-capitalized costs r e l a t i n g to r e a l e s t a t e , b u i l d i n g s and f i x t u r e s , and equipment, other than.for s t o r e p r o p e r t i e s . ORFC, 1975-76, p.93, 16. For example, the opening of a new s t o r e by a supermarket chain f i r m would; r e s u l t i n an increase i n the q u a n t i t y of goods which must pass through the fi r m ' s d i s t r i b u t i o n network. This increase i n goods i s - 99 -c e r t a i n to. have some.impact on the costs of warehouse op e r a t i o n s , t r a n s p o r t a t i o n operations, merchandising and.buying, and accounting and o f f i c e s e r v i c e s . The opening of a new st o r e would a l s o n e c e s s i t a t e an increase i n labor costs and f r i n g e b e n e f i t s , and would be l i k e l y to increase the costs, of f i e l d s u p e r v i s i o n and general a d m i n i s t r a t i o n . A d v e r t i s i n g and s a l e s promotion co s t s a l s o would be expected to increase w i t h the opening of a new s t o r e i n a new area, e s p e c i a l l y i f one considers the a d d i t i o n a l costs to the f i r m of sending out weekly advertisements, to r e s i d e n t s i n the neighborhood of the new s t o r e . 17. ORFC, 1975-76, p.93. ORFC. does not. provide us w i t h a d e f i n i t i o n of " n o n - c a p i t a l i z e d : c o s t s . " We presume that each f i r m reports the same f i g u r e s . o n n o n - c a p i t a l i z e d costs of s t o r e r e a l e s t a t e , b u i l d i n g s , f i x t u r e s and equipment to ORFC as i t r e p o r t s f o r tax purposes. 18. "Net cost o f merchandise s o l d i s the b i l l e d or i n v o i c e cost of merchandise s o l d , l e s s trade discounts (except cash discounts earned) and l e s s r e t u r n s and allowances received from manufacturers or whol e s a l e r s , p l u s processing expense ( f o r such operations as produce prepackaging a t the warehouse,: bakery, coffee r o a s t i n g , egg handling, banana r i p e n i n g , e t c . ) , and p l u s t r a n s p o r t a t i o n charges." ORFC, 1975-76, p,92, 19. ORFC defin e s the inventory turnover r a t e , or s t o c k t u r n s , as f o l l o w s : "Stockturns i s the number of times the average merchandise inventory was s o l d during the year. The stockturns f i g u r e s are - 100 -based on beginning and ending i n v e n t o r i e s : ( i n . warehouse as w e l l as i n stores) and are computed by d i v i d i n g net cost of merchandise s o l d (as defined under gross margin): by the average inventory a t c o s t . " ORFC, 1975-76, p.92. 20. M a l l e n and Haberman £1975; 163-166]. 21. "One, s t o r e s i z e has l i t t l e e f f e c t , on s t o r e operating expenses; two, st o r e u t i l i z a t i o n as measured i n s a l e s per square f o o t does have a s i g n i f i c a n t e f f e c t on s t o r e costs i n so f a r as high costs are ass o c i a t e d w i t h low rat e s of u t i l i z a t i o n and as u t i l i z a t i o n r a t e s i n c r e a s e cost l e v e l s begin to d e c l i n e at f i r s t and then appear to l e v e l out. These r e s u l t s are s i m i l a r to other s t u d i e s using much the same techniques ( N a t i o n a l Commission, 1967; Dooley, 1968)." S a v i t t [1975; 227]. Als o see the N a t i o n a l Commission on Food Marketing [1967] and Dooley [1968; 145-150]. 22. I f new supermarkets are p r i m a r i l y opened by e s t a b l i s h e d f i r m s and charge the j o i n t p r o f i t maximizing p r i c e s , w h i l e gm i s based on competitive p r i c e s , then we would expect grn^ > gm^. I f new super-markets were p r i m a r i l y opened by new entrants and i f e s t a b l i s h e d firms charge j o i n t p r o f i t maximizing p r i c e s , then we would expect e gm^ < gro^- We do not regard these cases as l i k e l y s i n c e 1) we would not expect new supermarkets opened by e s t a b l i s h e d firms to charge p r i c e s s i g n i f i c a n t l y d i f f e r e n t from other e s t a b l i s h e d f i r m s , 2) we would not expect most.new supermarkets to be opened by new entrants i n any time p e r i o d . - 101 -P r o f i t , c a l c u l a t i o n s for' years, p r i o r to 19 70 would ,have, been p o s s i b l e given f u r t h e r assumptions. The main d i f f i c u l t y i s that f o r years p r i o r to 1970, ORFC does not c a l c u l a t e costs as percentage of s a l e s f o r a l l supermarket f i r m s , but r a t h e r f o r f i r m s w i t h s a l e s below $20 m i l l i o n , w i t h s a l e s between $20 m i l l i o n and $100 m i l l i o n , and w i t h s a l e s above $100 m i l l i o n . However, given the r e s u l t s of p r o f i t c a l c u l a t i o n s f o r the years 1970-1976, f u r t h e r c a l c u l a t i o n s f o r years p r i o r to 1970 seemed unnecessary. - 102 -Chapter 5 THE LOCATION TEST OF PREEMPTION 5.1 I n t r o d u c t i o n In Chapter 3 of t h i s t h e s i s , we developed the theory of market preemption w i t h i n a nonstochastic framework. We derived the i m p l i c a t i o n that i f there i s an a n t i c i p a t e d increase i n d e n s i t y i n a market such that a new s t o r e (or p l a n t ) could be p r o f i t a b l y e s t a b l i s h e d i n that market, and i f the new s t o r e would have as i t s neighbors other s t o r e s that an e x i s t i n g f i r m owns, then the e x i s t i n g f i r m w i l l have an i n c e n t i v e to preempt the market. In a d d i t i o n , given the assumptions of our theory, the e x i s t i n g f i r m w i l l act on t h a t i n c e n t i v e w i t h p r o b a b i l i t y equal to one. In t h i s chapter of the t h e s i s , we wish to t e s t the l o c a t i o n a l i m p l i c a t i o n s of the theory of preemption. To do so, we must view preemp-t i o n as a p r o b a b i l i s t i c process t a k i n g place i n a s t o c h a s t i c world. That i s , w h i l e an e x i s t i n g f i r m , under the c o n d i t i o n s s p e c i f i e d above, w i l l s t i l l have an i n c e n t i v e to preempt the market, i t may sometimes f a i l to act upon t h i s i n c e n t i v e f o r a v a r i e t y of reasons, i . e . u n a n t i c i p a t e d growth i n the market, management m i s c a l c u l a t i o n , c a p i t a l c o n s t r a i n t s , e t c . I f e x i s t i n g f i r m s f r e q u e n t l y f a i l e d to act upon the i n c e n t i v e to preempt, then i t might appear that the a l l o c a t i o n of f i r m ownership to s t o r e s i n the market i s e s s e n t i a l l y random. I n other words, there may not be a d i s c e r n i b l e p a t t e r n to the s p a t i a l a l l o c a t i o n of f i r m ownership of s t o r e s i n the market, suggesting that the p r o b a b i l i t y that a s t o r e i s owned by a p a r t i c u l a r f i r m does not depend upon neighbor r e l a t i o n s or the sequence of past s t o r e openings. However, a preempting f i r m would not e s t a b l i s h i t s new s t o r e s i n the market i n a random f a s h i o n , but r a t h e r would take - 103 -account of the ex post s t a t e of neighbour r e l a t i o n s that would e x i s t i f i t e s t a b l i s h e d a new s t o r e i n the market. Thus, preemption i n a s t o c h a s t i c world should be viewed as a s t a t e dependent p r o b a b i l i s t i c process, meaning a process over time such that the p r o b a b i l i t y that a given s t o r e i s owned by a given f i r m depends upon the neighbor r e l a t i o n s w i t h other s t o r e s i n the market. There are two types of t e s t s which we s h a l l use i n order to a s c e r t a i n the nature and extent of preemption i n a given market. The f i r s t type of t e s t u t i l i z e s cross s e c t i o n data on s t o r e ownership and the neighbor r e l a t i o n s between sto r e s i n a given market. We devise t e s t s to determine i f our data were generated by an independent s t o c h a s t i c process. The second type of t e s t u t i l i z e s time s e r i e s data on the date at which each s t o r e was e s t a b l i s h e d i n the market, where that s t o r e was l o c a t e d , and which f i r m owned i t . We devise t e s t s to determine i f our data were generated by a s t a t e dependent s t o c h a s t i c process. We s h a l l perform the cross s e c t i o n t e s t s f i r s t i n order to see i f we may r e j e c t the hypothesis that our data were generated by an indepen-dent s t o c h a s t i c process. The cross s e c t i o n data are more r e a d i l y access-i b l e than the time s e r i e s data, and the cross s e c t i o n t e s t w i l l perform a screening f u n c t i o n f o r us. That i s , i f we cannot r e j e c t the hypothesis t h a t our data were generated by an independent s t o c h a s t i c process, then there i s r e a l l y no need to proceed f u r t h e r and t e s t f o r the existence of s t a t e dependence. I f we may r e j e c t the hypothesis of randomness, then we s h a l l proceed and t e s t the hypothesis that our data were generated by a s t a t e dependent s t o c h a s t i c process. The data base f o r a l l of our t e s t s w i l l c o n s i s t of supermarket l o c a t i o n data from the Greater Vancouver Regional D i s t r i c t of the , - 104 -province of B r i t i s h Columbia. In the next s e c t i o n , we s h a l l d i s c u s s our reasons f o r s e l e c t i n g the Greater Vancouver Regional D i s t r i c t as the b a s i s f o r our e m p i r i c a l work. We s h a l l a l s o d e f i n e what we mean by a supermarket, d i s t i n g u i s h between p o t e n t i a l preemptors and competitive f r i n g e f i r m s , and provide d e s c r i p t i o n s of the supermarket fi r m s operating w i t h i n the Greater Vancouver Regional D i s t r i c t . 5.2 The Market 5.2.1 Reasons f o r S e l e c t i n g the Greater Vancouver Regional D i s t r i c t as  the Basis f o r the E m p i r i c a l Work The supermarket i n d u s t r y i n the province of B r i t i s h Columbia has been chosen as the b a s i s f o r t e s t i n g the l o c a t i o n a l i m p l i c a t i o n s of the preemption hypothesis. The choice was not an a r b i t r a r y one, but r a t h e r was based on the need to have easy access to supermarket l o c a t i o n data.^ The province of B r i t i s h Columbia may be broken down i n t o a number of geo g r a p h i c a l l y d i s t i n c t "sub-markets". A sub-market i s assumed to conta i n consumers who mainly p a t r o n i z e s t o r e s i n t h e i r sub-market, e i t h e r because the d i s t a n c e ( t r a n s p o r t a t i o n costs) or time costs would make the patron-i z i n g of sto r e s i n other sub-markets too c o s t l y from the point of view of u t i l i t y maximizing consumers. In order to conduct t e s t s f o r randomness or s t a t e dependence, a set of sub-markets of B r i t i s h Columbia had to be chosen which conformed reasonably w e l l to the c r i t e r i o n of c o n t a i n i n g consumers who mainly p a t r o n i z e s t o r e s i n t h e i r sub.-market. The four sub-markets comprising the Greater Vancouver Regional D i s t r i c t (GVRD) were judged to be s u f f i c i e n t l y g e o g r a p h i c a l l y d i s t i n c t to s a t i s f y our c r i t e r i o n . Each sub-market i s separated from i t s neighboring sub-market by bodies of water, and there i s a small number of bridges which allow only l i m i t e d access between sub-markets. In Table XI> the Greater - 105 -Vancouver Regional D i s t r i c t member m u n i c i p a l i t i e s are l i s t e d by sub-market. TABLE. I I MEMBER MUNICIPALITIES OF THE GVRD BY SUB-MARKET I . Vancouver Sub-market 1) Vancouver 2) New Westminster.; 3) Port Coquitlam 4) Burnaby 5) Coquitlam 6) Port Moody I I . Richmond Sub-market 1) Richmond I I I . North Shore Sub-market 1) North Vancouver 2) West Vancouver 3) Lions Bay IV. Delta-Surrey Sub-market 1) D e l t a 2) Surrey 3) White Rock 5.2.2 D e f i n i n g a Supermarket and D i s t i n g u i s h i n g Between P o t e n t i a l  Preemptors and Competitive Fringe Firms -Our next task i s to e s t a b l i s h c r i t e r i a f o r determining which r e t a i l food s t o r e s belong to the supermarket i n d u s t r y . Not a l l s t o r e s which s e l l food would a u t o m a t i c a l l y be c l a s s i f i e d as supermarkets. Many r e t a i l food s t o r e s would be c l a s s i f i e d as convenience s t o r e s , and hence would be excluded from our a n a l y s i s . There are s e v e r a l competing d e f i n i t i o n s of what c o n s t i t u t e s a supermarket. For example, the Super Market I n s t i t u t e i n 1976 defined a supermarket as "... a departmentalized food s t o r e doing 2 at l e a s t $1 m i l l i o n a year, or about $20,000 weekly." S t a t i s t i c s Canada c a l l s supermarkets "combination s t o r e s " , and uses the f o l l o w i n g d e t a i l e d d e f i n i t i o n : - 106 -R e t a i l businesses i n which the sa l e s of a balanced l i n e of g r o c e r i e s , bakery products, d a i r y products, canned and/or f r o z e n foods, prepared meats, f r e s h meats, f i s h and p o u l t r y , f r e s h f r u i t s and vegetables, beer (Newfoundland and Quebec), and other food l i n e s form the dominant business a c t i v i t y . Fresh meat, f i s h and p o u l t r y must account f o r at l e a s t 20% (but l e s s than 60%) of t o t a l s a l e s . In a d d i t i o n , l i m i t e d l i n e s of newspapers, magazines, paper products, s o f t d r i n k s , tobacco items, h e a l t h and beauty a i d s , housewares and other non-food a r t i c l e s may a l s o be c a r r i e d . However, no one commodity l i n e , excepting beer, can account f o r more than 60% of t o t a l s a l e s . 3 (Note that the S t a t i s t i c s Canada d e f i n i t i o n does not i n s i s t on a given l e v e l of s a l e s i n order f o r a s t o r e to be c l a s s i f i e d as a combination st o r e . ) S t i l l other d e f i n i t i o n s of a supermarket were o f f e r e d by l o c a t i o n a n a l y s t s i n the supermarket i n d u s t r y i t s e l f . Since we do not have access to i n d i v i d u a l s t o r e s a l e s data, any d e f i n i t i o n r e q u i r i n g that a grocery s t o r e be c l a s s i f i e d as a supermarket only i f i t does a c e r t a i n l e v e l of s a l e s cannot be used. We have t h e r e f o r e chosen to proceed by c l a s s i f y i n g r e t a i l food st o r e s as supermarkets on the b a s i s of a d e f i n i t i o n which combines elements of the Super Market I n s t i t u t e and S t a t i s t i c s Canada d e f i n i t i o n s . Since we do have access to i n d i v i d u a l s t o r e s i z e data, and s i n c e s t o r e s i z e w i l l be assumed to be a good proxy f o r s a l e s volume, the f o l l o w i n g d e f i n i t i o n of a supermarket w i l l be used i n i t i a l l y f o r purposes of c l a s s i f y i n g data: D e f i n i t i o n 1. A f i r m ' s s t o r e s w i l l be designated as supermarkets i f the average (mean) ground f l o o r area of a l l s t o r e s owned by that f i r m exceeds 10,000 square f e e t , and i f they are capable of being the d e s t i n a t i o n of a consumer's weekly grocery shopping t r i p i n that they stock the goods 4 l i s t e d i n the S t a t i s t i c s Canada combination s t o r e d e f i n i t i o n . Having defined the c r i t e r i a which w i l l be used i n determining whether a f i r m i s a member of the supermarket i n d u s t r y , we must now - 107 -e s t a b l i s h the b a s i s f o r i d e n t i f y i n g those firms which are the p o t e n t i a l market preemptors. The theory of market preemption suggests that i n order f o r a f i r m to be a preemptor, i t should c o n s t r u c t new stores i n the market at l o c a t i o n s which maximize the j o i n t p r o f i t s of the f i r m , and i t should construct these st o r e s at po i n t s i n time when i t would not be p r o f i t a b l e f o r a new entrant to enter. Thus, p o t e n t i a l market preempting fir m s may be i d e n t i f i e d by the extent to which they have opened new s t o r e s i n the market over the p e r i o d of time when the market was expanding, or by the number of s t o r e s which they own r e l a t i v e to the t o t a l number of store s comprising the i n d u s t r y over a given market or sub-market. On the other hand, f i r m s belonging to the "competitive f r i n g e " may be i d e n t i f i e d by the extent to which they have not opened new stor e s i n the market over the period of time when the market was expanding, by the f a c t that i t seems unreasonable to b e l i e v e that the f i r m s e l e c t e d the l o c a t i o n s of i t s s t o r e s i n order to maximize the j o i n t p r o f i t s over a l l of i t s s t o r e s , or by the number of stor e s which the f i r m owns r e l a t i v e to the t o t a l number of s t o r e s comprising the i n d u s t r y over a given market or sub-market. For example, we would immediately c l a s s i f y a l l supermarkets which are independently owned and c o n t r o l l e d as belonging to the competitive f r i n g e . 5.2.3 Supermarket Firms Operating i n the GVRD On the b a s i s of the aforementioned c r i t e r i a , we may i d e n t i f y three supermarket f i r m s which may be regarded as capable of pursuing a preemptive l o c a t i o n s t r a t e g y i n some sub-market of B r i t i s h Columbia. The f i r s t such f i r m i s Canada Safeway, L t d . , a wholly owned s u b s i d i a r y of Safeway S t o r e s , Inc., which i s based i n the United S t a t e s . Canada Safeway, L t d . , i s - 108 -broken up i n t o geographic d i v i s i o n s and each d i v i s i o n i s r e s p o n s i b l e f o r conducting i t s own l o c a t i o n a n a l y s i s and f o r making recommendations regarding new s t o r e c o n s t r u c t i o n . ^ F i n a l d e c i s i o n s on the c o n s t r u c t i o n of new stores are made by Safeway Stores, I nc., the U.S. parent company. As of March, 1978, Canada Safeway owned and operated 87 stor e s i n B r i t i s h Columbia, 46 of which were i n the four sub-markets comprising the GVRD. Canada Safeway a l s o owns and operates i t s own warehousing and procurement agent, Macdonalds Consolidated. The second f i r m i s Overwaitea, a s u b s i d i a r y of Neonex, L t d . Overwaitea i s a p r o v i n c i a l l y based supermarket chain f i r m , and as of March, 1978, i t owned and operated 40 Overwaitea s t o r e s and 6 Your M a r k - i t Food Stores throughout B r i t i s h Columbia. Of these, 3 Overwaitea s t o r e s and 3 Your Mar k - i t Food Stores were l o c a t e d i n the four sub-markets comprising the GVRD.^ Overwaitea a l s o handles i t s own warehousing and d i s t r i b u t i o n . The t h i r d f i r m capable of pursuing a preemptive l o c a t i o n s t r a t e g y i n B r i t i s h Columbia i s K e l l y Douglas & Company, L t d . , a s u b s i d i a r y of George Weston, L t d . K e l l y Douglas not only owns and operates i t s own supermarkets under the Super V a l u , Shop Easy, and Economart names, but i t a l s o grants f r a n c h i s e s to independent supermarket operators under the Super Valu and Shop Easy names. We have chosen to regard a l l of the K e l l y Douglas owned and fr a n c h i s e d s t o r e s as having been lo c a t e d according to a j o i n t p r o f i t maximizing l o c a t i o n s t r a t e g y . The reasons are as f o l l o w s : Most of the K e l l y Douglas f r a n c h i s e s t o r e s are o f f e r e d to independents on the ba s i s of l o c a t i o n s which are pr e - s e l e c t e d by K e l l y Douglas, although o c c a s i o n a l l y an independent i n possession of an e x i s t i n g s t o r e w i l l approach K e l l y Douglas f o r a f r a n c h i s e . The only major - 109 -d i f f e r e n c e between f r a n c h i s e s t o r e s and corporate owned s t o r e s , besides ownership, i s s i z e . The f r a n c h i s e s t o r e s tend to be s m a l l e r , s i n c e few independent operators can meet the c a p i t a l requirements of owning and operating the l a r g e r s t o r e s . 7 Thus, there i s l i t t l e b a s i s f o r c l a s s i f y i n g the f r a n c h i s e s t o r e s as belonging to firms which are members of the competitive f r i n g e , given that K e l l y Douglas uses the same s i t e s e l e c t i o n procedure f o r most of i t s f r a n c h i s e s t o r e s as i t does f o r i t s corporate owned s t o r e s , and given that the d e c i s i o n to f r a n c h i s e a s t o r e i s made on the b a s i s of i t s s i z e and not i t s l o c a t i o n . K e l l y Douglas s t o r e owner-ship f i g u r e s appear i n Table I I I and Table IV. A l l other supermarket f i r m s which operate i n B r i t i s h Columbia w i l l be designated as belonging to the competitive f r i n g e , and we s h a l l b r i e f l y d e s cribe the competitive f r i n g e firms which have sto r e s l o c a t e d i n the GVRD. The l a r g e s t competitive f r i n g e f i r m i s represented by the H. Y. Louie Company, L t d . H. Y. Louie i t s e l f i s predominantly a s e r v i c e o r g a n i z a t i o n , a c t i n g as d i s t r i b u t o r and consultant to independently owned IGA (Independent Grocers A s s o c i a t i o n ) supermarkets. H. Y. Louie grants IGA f r a n c h i s e s to independent operators i n B r i t i s h Columbia f o r IGA Canada, L t d . , provided the p o t e n t i a l f r a n c h i s e e meets c e r t a i n minimum standards. Some of the IGA stor e s are owned by the H. Y. Louie Company, these st o r e s c o n s i s t i n g mainly of those taken over by the c o r p o r a t i o n from franchisees who could not meet t h e i r c o n t r a c t u a l o b l i g a t i o n s . H. Y. Louie does not view i t s e l f as a supermarket chain f i r m and does not engage i n l o c a t i o n a n a l y s i s . 8 The second l a r g e s t competitive f r i n g e f i r m operating i n the GVRD i s Woodward's. Woodward's i s p r i m a r i l y a department s t o r e chain f i r m , but i t does operate a number of "food f l o o r s " , a l l but one of which are - 110 -attached to i t s department s t o r e s . Hence, i t seems reasonable to presume that Woodward's has lo c a t e d i t s food f l o o r s i n order to increase i t s department s t o r e revenues and p r o f i t s by t a k i n g advantage of the demand e x t e r n a l i t i e s created by having the two operations l o c a t e d next to each other. The remaining two competitive f r i n g e f i r m s are Stong's, which owns f i v e s t o r e s i n the GVRD, and High-Low, which owns f o u r . Both f i r m ' s s t o r e s are wid e l y s c a t t e r e d throughout the GVRD, and hence n e i t h e r f i r m can be regarded as a p o t e n t i a l preemptor of any sub-market of the GVRD. Store ownership f i g u r e s by f i r m f o r B r i t i s h Columbia and the sub-markets comprising the GVRD appear i n Table I I I and Table IV. TABLE I I I STORE OWNERSHIP BY FIRM—BRITISH COLUMBIA (B.C.) Firm Canada Safeway (Firm F g) Overwaitea (Firm F^ ;) K e l l y Douglas (Firm F k) Super Valu (Corporate) Super Valu (Franchise) Shop Easy (Corporate) Shop Easy (Franchise) [ Economart (Corporate) Number of Stores i n B.C. 87 46 35 47 7 22 3 T o t a l : 247 - I l l -TABLE IV STORE OWNERSHIP BY FIRM—GVRD SUB-MARKETS Firm Sub-markets Vancouver Delta-Surrey North Shore Richmond Canada Safeway (Firm F,) Overwaitea (Firm F-) K e l l y Douglas (Firm F k) Competitive F r i n g e (cf) IGA Woodward's Stong's [ High-Low 30 3 22 16 4 2 2 7 2 9 3 1 0 2 4 0 6 2 1 3 0 5 1 2 2 1 0 0 T o t a l : 79 24 16 11 Before proceeding f u r t h e r , i t should be noted that Table I I I does not c o n t a i n a f i g u r e f o r the number of st o r e s comprising the competitive f r i n g e i n B.C. In order to determine the p r e c i s e number of competitive f r i n g e s t o r e s i n B.C., i t would be necessary to v i s i t p e r s o n a l l y every town and c i t y i n B.C. i n order to assess whether or not a given r e t a i l food s t o r e was a supermarket. Such a procedure was deemed i m p r a c t i c a l , and we have t h e r e f o r e chosen to estimate the number of competitive f r i n g e s t o r e s i n B.C. i n the f o l l o w i n g manner: We make the assumption that the number of competitive f r i n g e s t o r e s i n B.C. i s p r o p o r t i o n a l to p o p u l a t i o n . The po p u l a t i o n of the GVRD i n 1976 was 1,085,242, w h i l e the po p u l a t i o n - 112 -of B.C. was 2,466,608. Thus, approximately 44% of the people l i v i n g i n B.C. r e s i d e d i n the GVRD i n 1976, and i t i s assumed that the observed number of competitive f r i n g e s t o r e s i n the GVRD, 39, i s 44% of the t o t a l number of competitive f r i n g e s t o r e s i n B.C. We have th e r e f o r e assumed that there are 89 competitive f r i n g e s t o r e s i n B.C. In the next s e c t i o n , we di s c u s s the t e s t i n g procedure, t e s t s t a t i s t i c s , and t e s t r e s u l t s of the n u l l hypothesis that our observations on s t o r e ownership were generated by an independent s t o c h a s t i c process. We s h a l l conduct the t e s t s f o r each sub-market of the GVRD s e p a r a t e l y , as w e l l as f o r a l l sub-markets combined. Thus, when we r e f e r to the t e s t r e s u l t s f o r the GVRD, we s h a l l be r e f e r r i n g to t e s t s based on our observations of the s p a t i a l a l l o c a t i o n of f i r m ownership throughout the GVRD, d i s r e g a r d i n g sub-market boundaries. 5.3 The Test of Random Firm Ownership 5.3.1 M o t i v a t i o n f o r the Test In t h i s s e c t i o n and the next, we s h a l l devise t e s t s i n order to determine whether our observations on f i r m ownership and neighbor r e l a t i o n s were generated by an independent s t o c h a s t i c process. I n order to f a c i l i t a t e our d i s c u s s i o n , we s h a l l u t i l i z e the f o l l o w i n g framework: Assume that there are j firms which own and operate st o r e s i n some s p a t i a l l y extended market, A. These j firms w i l l be defined as c o n s t i t u t -i n g an i n d u s t r y because they s e l l the same ve c t o r of goods. The market A i s assumed to be made up of U ge o g r a p h i c a l l y d i s t i n c t sub-markets, indexed A^ (u = 1,2,...,U), where (as state d e a r l i e r ) each sub-market i s assumed to con t a i n consumers who mainly p a t r o n i z e s t o r e s i n t h e i r sub-market, e i t h e r because the d i s t a n c e ( t r a n s p o r t a t i o n costs) or time costs would - 113 -make the p a t r o n i z i n g of s t o r e s i n other sub-markets too c o s t l y from the poin t of view of u t i l i t y maximizing consumers. We may now define the r e l a t i v e frequency f_^ as being the number of s t o r e s that f i r m ( i = l , 2 , . . . , j ) owns i n market A (n.) d i v i d e d by the number of s t o r e s i n j 1 market A ( J n.) or n i j (5.1) f. = —: i = l , 2 , . . . , j , I n. = n . J A — 1 I n. 1 - 1 i - 1 1 Using , t h i s framework, we may be a b i t more p r e c i s e about what we mean by random f i r m ownership. Consider f i r s t what might be deemed the a n t i t h e s i s of random f i r m ownership, p e r f e c t preemption. I f some f i r m has p e r f e c t l y preempted some sub-market A^ of A, then unless that f i r m owns a l l of the s t o r e s i n market A, there must n e c e s s a r i l y be some divergence between the r e l a t i v e frequency of the preempting firm's s t o r e s i n A^ and the r e l a t i v e frequency of the preempting f i r m ' s stores i n A. The magnitude of t h i s discrepancy w i l l of course depend upon the extent to which the preempting f i r m has preempted other sub-markets i n A. Now consider a more r e a l i s t i c s i t u a t i o n where market A i s dominated by the stor e s of s e v e r a l l a r g e f i r m s , and no f i r m has p e r f e c t l y preempted any sub-market. I f the s p a t i a l a l l o c a t i o n of f i r m ownership of these s t o r e s was completely random over a l l sub-markets of A, then we would expect the r e l a t i v e frequency of any firm's s t o r e s i n a given sub-market to be i n s i g n i f i c a n t l y d i f f e r e n t from the r e l a t i v e frequency of that f i r m ' s s t o r e s i n A. In other words, we would observe the complete absence of a given f i r m ' s s t o r e s being concentrated i n any p a r t i c u l a r sub-market of A. This i s what we mean by random f i r m ownership of s t o r e s , and the hypothesis of random f i r m ownership that w i l l be t e s t e d i n t h i s s e c t i o n i s fo r m a l i z e d next. - 114 -5.3.2 Statement of the N u l l Hypothesis of Random Firm Ownership of Stores The n u l l hypothesis of random f i r m ownership which we s h a l l t e s t i s as f o l l o w s : Each f i r m ' s s t o r e s i n some market A are randomly d i s t r i b u t e d over any a r b i t r a r i l y chosen, but w e l l d e f i n e d , sub-markets of A. This random d i s t r i b u t i o n of s t o r e ownership was generated by a s t o c h a s t i c process such that the p r o b a b i l i t i e s that any given s t o r e i n A i s owned by f i r m F ^ , F 2 > .. . ,F.. are equal to the r e l a t i v e frequencies f ^ , f 2,.. . , f . Thus, the p r o b a b i l i t i e s that any given s t o r e i s owned by f i r m F^,F 2,...,F^ are constant and i n v a r i a n t w i t h respect to which f i r m or firms own neighboring s t o r e s . Henceforth, these p r o b a b i l i t i e s w i l l be r e f e r r e d to as the set of s t a t e independent p r o b a b i l i t i e s . The n u l l hypothesis i m p l i e s that f i r m s n e i t h e r c o l l u d e i n order to d i v i d e up the market r a t i o n a l l y among themselves or c o n s c i o u s l y pursue a s t r a t e g y of preempting the s t o r e l o c a t i o n s i n any p a r t i c u l a r sub-market. The a l t e r n a t i v e hypothesis i s that the d i s t r i b u t i o n of f i r m ownership of s t o r e s i n any a r b i t r a r i l y chosen sub-market of A was generated by a s t a t e dependent p r o b a b i l i s t i c process such that the p r o b a b i l i t i e s that a given s t o r e i n any sub-market of A i s owned by f i r m F ^ , F 2 F_. depend upon the neighbor r e l a t i o n s of that s t o r e w i t h other s t o r e s i n the sub-market. R e j e c t i o n of the n u l l hypothesis w i l l be i n t e r p r e t e d to imply that some form of s t a t e dependence was r e s p o n s i b l e f o r generating our observa-t i o n s , although the s t r u c t u r e of the n u l l hypothesis prevents us from being p r e c i s e about the s p e c i f i c nature of the dependence. While a preemptive process i s n e c e s s a r i l y a s t a t e dependent process, a s t a t e dependent process need not be preemptive, and thus r e j e c t i o n of the n u l l hypothesis does not permit us to conclude t h a t preemption e x i s t s . - 115 -R e j e c t i o n of the n u l l hypothesis f o r a given sub-market of A a l s o i m p l i e s that the observed r e l a t i v e frequencies of each f i r m ' s s t o r e s i n that sub-market are s i g n i f i c a n t l y d i f f e r e n t from the r e l a t i v e frequencies f. of each firm's s t o r e s i n market A. This must mean that one or more 1 f i r m s are r e l a t i v e l y over-represented i n A^ w h i l e one or more f i r m s are r e l a t i v e l y under-represented i n A^ v i s - a - v i s the d i s t r i b u t i o n of f i r m ownership of s t o r e s that would be generated by a random process based on the f ^ . Thus, r e j e c t i o n of the n u l l hypothesis would a l l o w us to conclude that one or more f i r m s ' s t o r e s are r e l a t i v e l y concentrated i n A^, but i t would not a l l o w us to conclude that a preemptive s t a t e dependent process was r e s p o n s i b l e f o r t h i s c o n c e n t r a t i o n . We s h a l l now proceed to a d i s c u s s i o n of the t e s t i n g procedure of the n u l l hypothesis of random f i r m ownership of s t o r e s . 5.3.3. T e s t i n g Procedure f o r the N u l l Hypothesis When the Number of  Stores i n the Sub-market Is Small I n t h i s s u b - s e c t i o n , we s h a l l use an example i n order to i l l u s t r a t e the concepts and r a t i o n a l e behind the procedures employed to t e s t the n u l l hypothesis of random f i r m ownership of s t o r e s . Our example w i l l u t i l i z e the f o l l o w i n g framework: Assume that there are two f i r m s , F^ and Y^, which own and operate st o r e s i n some s p a t i a l l y extended market A. These two f i r m s w i l l be assumed to s e l l the same vect o r of goods at the same p r i c e s . The market A i s assumed to be made up of U g e o g r a p h i c a l l y d i s t i n c t sub-markets, A . Both f i r m F n and f i r m F„ w i l l be assumed to u 1 2 own one h a l f of the s t o r e s i n market A, and thus f ^ = f.^ ~ I f each firm's s t o r e s are randomly d i s t r i b u t e d over a l l sub-markets of A, then we would expect the r e l a t i v e frequency of s t o r e s owned by f i r m F., and f i r m F 9 i n any given sub-market to be i n s i g n i f i c a n t l y - 116 -d i f f e r e n t from 1/2, On the other hand, i f f i r m F or f i r m F 2 has p r e ^ empted a given sub-market, then we would expect the r e l a t i v e frequency of stores owned by F^ or i n that sub-market to be s i g n i f i c a n t l y d i f f e r e n t from 1/2, and we would expect the preempting f i r m to have sto r e s r e l a t i v e l y concentrated i n the sub^-market v i s - a - v i s the f . . Now l e t us focus on how. we might t e s t the n u l l hypothesis that the d i s t r i b u t i o n of f i r m ownership of st o r e s i n a p a r t i c u l a r sub-market of A was generated by a random process based on the f ^ . Consider the sub-market depicted i n Figure 8, where the represent s t o r e l o c a t i o n s . Figure 8 Our f i r s t step i s to generate the random d i s t r i b u t i o n of f i r m ownership f o r t h i s sub-market that i s i m p l i e d by a random process based on the s t a t e independent' p r o b a b i l i t i e s , f ^ and f ^ . This random d i s t r i b u t i o n i s generated by f i n d i n g a l l of the p o s s i b l e permutations of f i r m ownership of the four st o r e s i n the sub-market and the r e s p e c t i v e p r o b a b i l i t y of - 117 -occurrence of each permutation. Since the p r o b a b i l i t y i s % th a t any given s t o r e w i l l be owned by f i r m F^ or f i r m F^, each permutation w i l l have an equal p r o b a b i l i t y of o c c u r r i n g . There are s i x t e e n p o s s i b l e permutations of f i r m ownership of the four s t o r e s , and these are l i s t e d i n Table V. TABLE V PERMUTATIONS OF FIRM OWNERSHIP f l h f i f i f l f l f 3 f i 1) F l F i F i F l F 2 F l F l F l 2) F l F i F i F 2 10) F 2 F l F 2 F l 3) F l F i F 2 F l 11) F 2 F 2 F l F l 4) F l F 2 F l F l 12) F 2 F l F l F 2 5) F l F l F 2 F 2 13) F 2 F 2 F 2 F l 6) F l F 2 F l F 2 14) F 2 F 2 F l F 2 7) F l F 2 F 2 F l 15) F 2 F l F 2 F 2 8) F l F 2 F 2 F 2 16) F 2 F2 F 2 F 2 In Table V I , we provide a summary of the p r o b a b i l i t i e s that given numbers of st o r e s w i l l be owned by f i r m F^ and f i r m F^- The t a b l e was constructed on the ba s i s of the f a c t that each permutation has a p r o b a b i l i t y .0625 of o c c u r r i n g . - 118 -TABLE VI SUMMARY OF PROBABILITIES P rC r^ = 4 and n 2 = 0) = .0625 P r ( n l = 3 and n 2 = 1) = .2500 P r O ^ = 2 and n 2 = 2) = .3750 P r C ^ = 1 and n 2 = 3) = .2500 Pr ( ^ = 0 and n 2 = 4) = .0625 (Note that n^ i n Table VI represents the number of stores owned by f i r m f\ i n the sub-market of Figure 8. The s u b s c r i p t r e p r e s e n t i n g the sub-market has been suppressed f o r e x p o s i t i o n a l c l a r i t y . ) We now wish to s i n g l e out those observations which would lead us to r e j e c t the n u l l hypothesis. I f our c r i t e r i o n f o r r e j e c t i o n i s that an observation have l e s s than a ten percent p r o b a b i l i t y of o c c u r r i n g , then we f i n d that there are two observations which would lead us to r e j e c t the n u l l hypothesis: a) i f f i r m F^ owns four s t o r e s and f i r m F 2 owns zero s t o r e s ; b) i f f i r m F^ owns zero s t o r e s and f i r m F 2 owns four s t o r e s . Thus, i n t h i s example, only p e r f e c t preemption would lea d us to r e j e c t the n u l l hypothesis that our observations were generated by a random process based on the f . . l Before proceeding f u r t h e r , we should note that the d i s t r i b u t i o n described i n Table VI i s bi n o m i a l . That i s (5.2) - 119 -where n = the number of s t o r e s owned by f i r m F. i n the sub-market of i i F i gure 8. The binomial d i s t r i b u t i o n has the f o l l o w i n g moments: E(n.) = y. = (n + n ) f . (5.3) 1 2 1 V(n.) = a 2 = (n.. + n ) f f . l l 1 I 1 2 I f more than two f i r m s were assumed to own; s t o r e s i n market A, then the random d i s t r i b u t i o n of f i r m ownership f o r the sub-market depicted i n Figure 8 would have been m u l t i n o m i a l . (5.4) n n n. g(n n n.) = [(En.)!/n !n !...n.!] (f ) 1 ( f ) . . . ( f ) 3 . X 2. "1 X X ^ J X ^ J When•there are only two f i r m s which own s t o r e s i n market A, we may use the b i n o m i a l d i s t r i b u t i o n i n order to c a l c u l a t e the p r o b a b i l i t i e s that given numbers of stores w i l l be owned by each f i r m . We may then do a o n e - t a i l e d t e s t of the n u l l hypothesis of random f i r m ownership. This t e s t would i n v o l v e determining i f the p r o b a b i l i t y of observing e i t h e r an > n^ or an > n^ was l e s s than .10. When the number of firms which own stores i n market A i s greater than two, i t would s t i l l be p o s s i b l e to c a l c u l a t e the p r o b a b i l i t y that = n^, = n^,...^ =n f o r d i f f e r e n t values of the n.. However, when the number of s t o r e s l i n a given sub-market i s l a r g e , i t would be i m p r a c t i c a l to c a l c u l a t e the exact multinomial d i s t r i b u t i o n corresponding to the f_^. More im p o r t a n t l y , even i f we only wished to c a l c u l a t e the d e n s i t y i n that part of the multinomial d i s t r i b u t i o n which represented values of the n_^  more "extreme" than our observations, i t i s unclear what the proper c r i t i c a l r e g i o n should be and what s i g n i f i c a n c e l e v e l we should choose. In the next sub-section, we d i s c u s s two procedures f o r t e s t i n g the n u l l hypothesis of random f i r m ownership which do not r e q u i r e knowledge of the exact - 120 -shape of the mul t i n o m i a l d i s t r i b u t i o n . The f i r s t t e s t i n g procedure may be used when the number of sto r e s i n the sub-market i s l a r g e , and only r e q u i r e s that we know the observed n^ and the set of s t a t e independent p r o b a b i l i t i e s . The second t e s t i n g procedure w i l l be based on generating an approximation to the exact random d i s t r i b u t i o n of f i r m ownership which does not r e l y on the c a l c u l a t i o n of a l l of the mul t i n o m i a l p r o b a b i l i t i e s . The second.testing procedure may be used when the number of sto r e s i n the sub-market i s too small to use the f i r s t t e s t i n g procedure. 5.3.4 Test i n g Procedure and Test S t a t i s t i c s f o r the N u l l Hypothesis When  the Number of Stores i n the Sub-market I s Large In the previous sub-section, our procedure f o r f i n d i n g the exact random d i s t r i b u t i o n of f i r m ownership was based on l i s t i n g a l l of the p o s s i b l e permutations of f i r m ownership when there are only two f i r m s which own j stores i n market A. However, as E n. , the number of sto r e s i n sub-• -I xu' x=l market A^, becomes l a r g e , and w i t h more than two f i r m s i n A, the number of p o s s i b l e permutations of f i r m ownership r a p i d l y e s c a l a t e s and i t would not be p r a c t i c a l to f i n d the exact shape of the multinomial d i s t r i b u t i o n corresponding to the n u l l hypothesis. In a d d i t i o n , as pointed out at the end of the l a s t sub-section i t i s unclear what the c r i t i c a l r e g i o n of t h i s d i s t r i b u t i o n should be or what s i g n i f i c a n c e l e v e l we should choose. There are two r e l a t e d approaches we may take i n order to t e s t the j n u l l hypothesis when E n . i s l a r g e and when there are more than two . , xu i = l f i r m s i n the market. The f i r s t approach i s to do a chi-square t e s t f o r goodness of f i t of the h u l l hypothesis. Given the way we have stated the n u l l hypothesis, we are r e a l l y i n t e r e s t e d i n t e s t i n g whether our observa-t i o n s on f i r m ownership of sto r e s i n a given sub-market, n, ,n„ ,...,n. , l u 2u j u are compatible w i t h the expected v a l u e s , ei u» e2 u»••*' eju* ^ e e x P e c t e Q values are formed by t a k i n g the products of the number of stores i n sub-market A^ and the p r o b a b i l i t i e s that a given s t o r e i s owned by f i r m - 120 a -E l ' ^ 2 ' " " " ' ^ j ' ° r (5.5) e. = ( I n. ) f . . xu . 6_ xu i x=l "Theorem. I f n 1 ,n„ ,...,n. and e. , e„ ,...,e l u ' 2u' ' j u l u ' 2u' ' j u represent the observed and expected f r e q u e n c i e s , r e s p e c t -i v e l y , f o r the j p o s s i b l e outcomes of an experiment that i s to be performed n times, then as n becomes i n f i n i t e , the d i s t r i b u t i o n of the random v a r i a b l e (5.6) J (n. - e. ) 2 / e . xu xu xu w i l l approach that of a chi-square v a r i a b l e w i t h j = l degrees of of freedom. Thus, given n. and e. , we need only c a l c u l a t e (5.6) and then xu xu determine whether t h i s value exceeds the c r i t i c a l v alue of chi-square that i s obtained from a t a b l e of c r i t i c a l values of the chi-square d i s t r i b u t i o n . I f i t does, then we would r e j e c t the n u l l hypothesis that - 121 -observations on the f i r m ownership of sto r e s i n sub-market were generated by a random process based on the f ^ . The upper t a i l of the chi-square d i s t r i b u t i o n i s the accepted c r i t i c a l r e g i o n f o r purposes of hypothesis t e s t i n g , and K e n d a l l and Stuart j u s t i f y t h i s choice from the point of 10 view of i t s asymptotic power. According to Hoel and othe r s , as long as j > 5 and e^ u > 5, the c h i -square d i s t r i b u t i o n w i l l provide a s a t i s f a c t o r y approximation to the exact d i s t r i b u t i o n of the q u a n t i t y given by (5.6). I f j < 5, then the e^ u should be s l i g h t l y l a r g e r than 5 . ^ Walker and Lev have sta t e d a d d i t i o n a l g u i d e l i n e s to i n s u r e that the approximation i s good. " I f there are 2 or more degrees of freedom and the expectation i n each c e l l i s more than 5, the chi-square t a b l e assures a good approximation t o the exact p r o b a b i l i t i e s . I f there are 2 or more degrees of freedom and roughly approximate probab-i l i t i e s are acceptable f o r the t e s t of s i g n i f i c a n c e , an expe c t a t i o n of only 2 i n a c e l l i s s u f f i c i e n t . I f there are more than 2 degrees of freedom and the exp e c t a t i o n i n a l l the c e l l s but one i s 5 or more, then an exp e c t a t i o n of only one i n the remaining c e l l i s s u f f i c i e n t to provide a f a i r approximation to the exact p r o b a b i l i t i e s . " 1 2 Looking back at the data i n Table IV, i t i s c l e a r that the expected frequencies f o r the Richmond, North Shore, and Surrey-Delta sub-markets w i l l not a l l exceed 5. In a d d i t i o n , i t seems unreasonable to regard the f^ f o r B.C. as being approximately equal. Thus, we cannot do the c h i -square t e s t of the n u l l hypothesis f o r these sub-markets s i n c e the expected frequencies f o r these sub-markets v i o l a t e the c r i t e r i a which i n s u r e that the chi-square d i s t r i b u t i o n w i l l provide a s a t i s f a c t o r y approximation to the exact d i s t r i b u t i o n of the q u a n t i t y given by (5.6). However, there i s an a l t e r n a t i v e procedure which may be used to t e s t the j n u l l hypothesis when £ n. i s too l a r g e to f i n d the exact shape of the i = l random d i s t r i b u t i o n , yet too s m a l l to do the chi-square test,"and when the number of firms i n the market i s greater than two. This - 122 -a l t e r n a t i v e procedure e n t a i l s generating an approximation to the shape of the true d i s t r i b u t i o n , c o n v e r ting the generated d i s t r i b u t i o n i n t o the d i s t r i b u t i o n of the q u a n t i t y given by (5.6), and then f i n d i n g where our observations l i e w i t h i n t h i s d i s t r i b u t i o n . We s h a l l c a l l t h i s a l t e r n a t i v e 2 t e s t i n g procedure the X t e s t , and we s h a l l now proceed to d i s c u s s i t i n more d e t a i l . Let us define a draw as c o n s i s t i n g of the assignment of f i r m owner-ship to one s t o r e i n some sub-market A^. Define a permutation as c o n s i s t i n g of one complete assignment of f i r m ownership to a l l of the j s t o r e s i n some sub-market A , or the set of draws of the 7 n. s t o r e s u . L n xu i = l i n A^. Using the computer, we may generate a " l a r g e " number of random permutations of f i r m ownership, where the draws f o r each permutation are made on the b a s i s of a set of f i x e d p r o b a b i l i t i e s equal to the f ^ of market A. From the l i s t of permutations so generated, we may e a s i l y c a l c u l a t e the number of stores drawn f o r each f i r m i n each permutation. The j o i n t d i s t r i b u t i o n of the numbers of st o r e s drawn f o r each f i r m i n each permutation w i l l c o n s t i t u t e our estimate of the t r u e d i s t r i b u t i o n . Our next step i s to convert our estimate of the t r u e d i s t r i b u t i o n i n t o the d i s t r i b u t i o n of the q u a n t i t y given by (5.6). We do so by - 2 -c a l c u l a t i n g I (n. - n. ) /n. f o r each randomly generated permutation, i = l where f i ^ u = the randomly generated number of s t o r e s owned by f i r m F^ i n sub-market A , and n. = the mean number of stores generated f o r f i r m F. u I U & 1 i n sub-market A , or u (5.7) n. = ) f i . /number of permutations . i u L . i u permutations That i s , suppose Table VII represents the f i r s t three permutations of f i r m ownership i n some sub-market A . u - 123 -TABLE VII THREE RANDOM PERMUTATIONS OF FIRM OWNERSHIP ii,.. n„ fL n.. l u 2 u 3 u 4u (1) 7 10 8 12 (2) 8 11 13 5 (3) 10 8 6 13 For each row of Table V I I , we may compute the q u a n t i t y £ (fi. - n . )^ /n.„ > i = l u u given that we have already computed n_^u f o r each f i r m . We may then p l o t the d i s t r i b u t i o n of t h i s q u a n t i t y . F i n a l l y , we c a l c u l a t e J - 2 -> (n. - n. ) /n. , where n. = the observed number of st o r e s owned by ± ^ i u i u l u i u f i r m F. i n sub-market A , and we f i n d where t h i s s t a t i s t i c l i e s i n the i " - i-d i s t r i b u t i o n of ) (n. - n.•) /n. . I f , f o r example, the s t a t i s t i c ± t l i u ' i u ' i u * l i e s i n the 10% r i g h t t a i l of the d i s t r i b u t i o n , we would r e j e c t the n u l l hypothesis that our observations were generated by a random process based on the f ^ . By choosing the 1 0 % . r i g h t t a i l of the d i s t r i b u t i o n as the c r i t i c a l r e g i o n , we are choosing to> l e t the type I e r r o r equal 10%. There i s a w e l l known trade o f f between the type I e r r o r , the p r o b a b i l i t y of r e j e c t i n g the n u l l hypothesis when i t i s a c t u a l l y t r u e , and the type I I e r r o r , the p r o b a b i l i t y of accepting the n u l l hypothesis when the a l t e r n a -t i v e hypothesis i s t r u e , such that the s i z e of the type I I e r r o r increases as the s i z e of the type I e r r o r decreases. By t o l e r a t i n g a r e l a t i v e l y s m a ll type I e r r o r , we t h e r e f o r e b i a s the t e s t towards accepting the n u l l hypothesis. From the po i n t of view of our hypothesis - 124 -t e s t i n g , t h i s i s d e s i r a b l e s i n c e we wish to be confident that we are 13 r e j e c t i n g a f a l s e n u l l hypothesis. 2 What i s the i n t u i t i v e i n t e r p r e t a t i o n behind using the X t e s t as a t e s t of the n u l l hypothesis? F i r s t , we note that the chi-square s t a t i s t i c i s e s s e n t i a l l y a measure of the discrepancy between observed and 14 expected values. The l a r g e r i s the discrepancy between observed and expected values i n r e l a t i o n to the expected v a l u e s , the l a r g e r w i l l be 2 the c o n t r i b u t i o n of the term (n. - e. ) /e. to the chi-square i u i u xu 2 s t a t i s t i c . This i n t e r p r e t a t i o n a l s o holds f o r the X t e s t . Thus, we may 2 focus on the i n d i v i d u a l terms making up the observed X s t a t i s t i c , j - 2 -7 ( n . - n. ) /n. , i n order to assess the nature of the d i s c r e p a n c i e s . «, i u xu xu x=l between observed and mean number of stor e s owned by each f i r m which are lea d i n g us to e i t h e r accept or r e j e c t the n u l l hypothesis. This s o r t of a n a l y s i s w i l l g i v e us a f e e l f o r the p o s i t i o n s of fir m s i n sub-market r e l a t i v e to the f. and r e l a t i v e to each other, x 2 One f i n a l comment regarding the X t e s t i s i n order. We have noted that we cannot do the standard chi-square t e s t f o r the Richmond, North Shore, and Delta-Surrey sub-markets because the expected frequencies are too s m a l l . However, the expected frequencies f o r the GVRD and the Vancouver sub-market are l a r g e enough to i n s u r e that the chi-square d i s t r i b u t i o n w i l l provide a good approximation to the q u a n t i t y given by (5.6). We may th e r e f o r e use the r e s u l t s from the chi-square t e s t as a 2 check against the r e s u l t s of the X t e s t i n the Vancouver sub-market and the GVRD i n order to in s u r e that our computer program i s p r o v i d i n g an accurate estimate of the true d i s t r i b u t i o n . In other words, s i n c e the chi-square d i s t r i b u t i o n i s an approximation to the d i s t r i b u t i o n of the X s t a t i s t i c s which we generate, and s i n c e t h i s approximation should be - 125 -a good one provided the number of stores i n the sub-market and the number of randomly generated permutations i s l a r g e , the chi-square t e s t and 2 the X t e s t should y i e l d approximately the same r e s u l t s . F i n a l l y , we 2 should note that w h i l e generating the d i s t r i b u t i o n s of the X s t a t i s t i c s i s not a n e c e s s i t y f o r the GVRD and the Vancouver sub-market i n order to t e s t the n u l l hypothesis of random f i r m ownership, these d i s t r i b u t i o n s w i l l be necessary i n order to t e s t a r e l a t e d hypothesis i n a l a t e r s e c t i o n of t h i s chapter. One question which must be confronted before r e p o r t i n g the r e s u l t s 2 from the X t e s t s i s as f o l l o w s : How many random permutations of f i r m ownership are necessary to ins u r e that the approximation of the generated d i s t r i b u t i o n to the true d i s t r i b u t i o n i s a good one? One way of approaching t h i s question i s to compare the r e s u l t s of the chi-square 2 t e s t and the X t e s t f o r the GVRD f o r d i f f e r e n t numbers of permutations. As st a t e d above, i f the approximation i s good, both t e s t s should y i e l d n e a r l y the same r e s u l t s . In Table V I I I , we rep o r t the r e s u l t s of the 2 standard chi-square t e s t and the r e s u l t s of the X t e s t f o r v a r y i n g numbers of permutations f o r the GVRD. Each sum i n Table V I I I i s over j f i r m s , and, henceforth, the j t h f i r m w i l l represent the competitive f r i n g e . 2 As we can see, the r e s u l t s from the chi-square t e s t and the X t e s t s based on d i f f e r e n t numbers of permutations of f i r m ownership are v i r t u a l l y i d e n t i c a l . We may conclude from t h i s that as few as 250 permutations of f i r m ownership w i l l provide a good approximation to the true shape of the random d i s t r i b u t i o n of f i r m ownership."^ However, we s h a l l use 1000 permutations i n order to generate our random d i s t r i b u t i o n s of f i r m ownership, i n part because the accuracy of the estimate to the - 126 -TABLE V I I I RESULTS OF THE CHI-SQUARE TEST AND THE X 2 TEST FOR THE GVRD—PERMUTATIONS = 250, 500, 750, 1000 Chi-square Test 4 I (n. - e ± ) 2 / e . = (46 - 33.661) "733.661 + (6 - 17.798)z/17.798 1 = 1 2 2 + (39,- 44.107) /44.107 + (39 - 34.434) /34.434 = 13.540549 l i e s i n the 1% t a i l of the chi-square d i s t r i b u t i o n w i t h three degrees of freedom. X 2 Tests 250 Permutations 4 I (n. - n. ) Z / n . = (46 - 34.044)Z/34.044 + (6 - 17.924)^/17.924 . - X X 1 1 = 1 2 2 + (39 - 43.512)z/43.512 + (39 - 34.520) /34.520 = 13.1806266 4 ? _ % of y (fi. - n.) In. d i s t r i b u t i o n to r i g h t of 13.1806266 = .4%. . , 1 1 x i = l 500 Permutations 4 I (n. - n.) 2 / n . = (46 - 33.914)2/33.914 + (6 - 17.606)2/17.606 1 = 1 2 2 + (39 - 44.016)^/44.016 + (39 - 34.464)z/34.464 = 13.1264939 4 2 _ % of 7 (fi. - n.) In. d i s t r i b u t i o n to r i g h t of 13.1264939 = .8% . . , 1 1 1 i = l - 127 -Table V I I I (continued) 750 Permutations 4 I (n. - n . ) 2 / n . = (46 - 33.543) 2/33.543 + (6 - 17.737) 2/17.737 , - X X X 1 = 1 2 2 + (39 - 44.095) /44.095 + (39 - 34.625) /34.625 = 13.5343668 4 - 2 -% of £ ( i i . - n.) /n. d i s t r i b u t i o n to r i g h t of 13.5343668 = .4% , i = l 1000 Permutations 4 I (n. - n.) 2 / n . = (46 - 33.49) 2/33.49 + (6 - 17.761) 2/17.761 . - X X X 1 = 1 2 2 + (39 - 44.433) /44.433 + (39 - 34.316) /34.316 = 13.764613 Av - 2 -% of ) ( i i . - n.) /n. d i s t r i b u t i o n to r i g h t of 13.764613 = .4% . i = l - 128 -true d i s t r i b u t i o n increases as we increase, the number of permutations, and i n part because we. s h a l l need the l a r g e r number of permutations f o r a r e l a t e d t e s t of randomness, to be discussed i n a l a t e r s e c t i o n of t h i s chapter. • ' 5.3.5 Test R e s u l t s of the. N u l l Hypothesis f o r the GVRD and  Constituent Sub-markets In t h i s sub-section, we r e p o r t . t h e r e s u l t s obtained from t e s t i n g the n u l l hypothesis that our observations on f i r m ownership of s t o r e s were generated by a random process based on the f ^ . Gur observations on f i r m ownership i n each, sub-market were reported i n Table IV, and the f used to generate the random d i s t r i b u t i o n s are l i s t e d i n Table IX. TABLE IX RELATIVE FREQUENCIES OF EACH FIRM'S STORES IN B.C. Firm R e l a t i v e Frequency Canada Safeway (Firm F g0 87/336 ~ .2589285 Overwaitea (Firm F ) 46/336 ; .1369048 K e l l y Douglas (Firm Efc,) 114/336 z .3392858 Competitive Fringe ( c f ) 89/336 ~ .2648809 In Table X, we provide d e s c r i p t i v e measures f o r the marginal d i s t r i b u t i o n s of f i r m ownership generated f o r each sub-market and f o r the GVRD. (In the two f i r m case, the marginal d i s t r i b u t i o n , f ( n ^ ) , bears the f o l l o w i n g r e l a t i o n to the j o i n t d i s t r i b u t i o n , f ( n ^ , ^ ) :f (n^) - £ f C n ^ , ^ ) . The n 2 f u n c t i o n f(n..,n ) gives the p r o b a b i l i t i e s that NJ. w i l l assume the value - 129 -16 w h i l e at the same time N 2 w i l l assume the value n 2«) We note once again that the random d i s t r i b u t i o n f o r the GVRD was generated s e p a r a t e l y from'the d i s t r i b u t i o n . o f f i r m ownership f o r the i n d i v i d u a l sub-markets. TABLE X MARGINAL DISTRIBUTION DESCRIPTIVE MEASURES BY FIRM FOR THE GVRD AND CONSTITUENT SUB-MARKETS GVRD Vancouver D e l t a -Surrey n. n. North Shore n^ Richmond Firm F/. Firm F . Firm F, Competitive Fringe s_ o_' k_ 33.490 17.761 44.433 34.316 4.7172 3.9425 5.3977 5.2090 20.363 10.895 26.553 21.189 3.5261 3.0808 4.1649 3.7743 6.111 3.258 8.302 6.329 2.1427 1.6650 2.4014 2.2449 4.058 2.270 5.524 4.148 1.7748 1.3620 1.8770 1.7283 2.858 1.496 3.722 2.924 1.4268 1.1105 1.5294 1.4527 A f t e r t r a n s l a t i n g our randomly generated d i s t r i b u t i o n s of f i r m owner-j - 2 -ship i n t o d i s t r i b u t i o n s of the q u a n t i t y ) (h. - n. ) /n. , and usi n g r . -i i u i u xu i = l the i n f o r m a t i o n contained i n Tables IV and X, we o b t a i n the f o l l o w i n g 2 r e s u l t s from doing the X t e s t of the n u l l hypothesis. - 130 -TABLE XI RESULTS OF THE X 2 TESTS OF THE NULL HYPOTHESIS OF RANDOM FIRM OWNERSHIP BY GVRD AND SUB-MARKETS GVRD Firm F , s Firm F,- Firm F k c f (46 - 33.49)' 33.49 (6 - 17.761)' 17.761 (39 - 44.433)' 44.433 (39 - 34.316)' 34.316 4.6730397 7.7879128 .6643145 .6393477 I (n. - n.) /n. = 13.764613 . , i i i i = l (13.540544)* j - 2 -% of I (n. - n.) /n. d i s t r i b u t i o n to r i g h t of 13.764613 = 1=1 1 1 1 .4% Vancouver Firm F. ' .S Firm F ; 0 Firm R. cf (30 - 20.363) 2 (3 - 10.895) 2 (22 - 26.553) 2 (24 - 21.189) 2 20.363 10.895 26.553 21.189 4.5608091 5.721067 .7806955 .3729161 I (n. - n.) /n, = 11.435487 . i I l l i = l (11.413771) J V — 2 — % of I (n. - n.) /n. d i s t r i b u t i o n to r i g h t of 11.435487 = .9% i = l 1 * The f i g u r e i n parentheses i s the r e s u l t obtained by c a l c u l a t i n g the j 2 chi-square s t a t i s t i c , £ . (n. - e.) /e. . i = l 1 1 - 131 -Table XI (continued) Delta-Surrey Firm F Firm F, Firm F, k cf. (7 - 6.111)' 6.111 (2 - 3.258)' 3.258 (9 - 8.302)' 8.302 (6 - 6.329)' 6.329 .1293276 .485747 .0586851 .0171023 J (n. - n . ) 2 / n . = .690862 x=l (.7129479) % of y (fi. - n . ) 2 / n . d i s t r i b u t i o n to r i g h t of .690862 = 87.3% . , x x x x=l North Shore Firm F Firm F Firm F, cf (4 - 4.058)' 4.058 .0008289 (0 - 2.27)' 2.27 2.27 (6 - 5.524)' 5.524 .0410166 (6 - 4.148)' 4.148 .8268813 I (n. - n . ) 2 / n . = 3.1387268 . — x i x x=l (2.987564) % of y (fi. - n.) 2 / n . d i s t r i b u t i o n to r i g h t of 3.1387268 = 38.6% . ^ , 1 1 1 i = l - 1 3 2 -Table XI (continued) Richmond Firm F Firm F Firm Fv., c f s o k i-(5 - 2.858) 2 (1 - 1.496) 2 (2 - 3.722) 2 (3 - 2.924) 2 2.858 1.496 3.722 2.924 1.6053757 .1644491 .796691 .0019753 I (n. - n . ) 2 / n . = 2.5684911 i = l (2.6024497) % of T (fi. - n.) /n.. d i s t r i b u t i o n to r i g h t of 2.5684911 = 48% . , 1 1 1 i = l - 133 -2 In Table X I , we report the i n d i v i d u a l components of the X s t a t i s t i c s j - 2 -1 (n. - n.) /n. i n order that the reader may e a s i l y a s c e r t a i n the s i g n • i i i l 1 = 1 2 of each discrepancy and i t s r e l a t i v e c o n t r i b u t i o n to the X s t a t i s t i c . We a l s o i n d i c a t e the percentage of the generated random d i s t r i b u t i o n l y i n g 2 to the r i g h t of the X s t a t i s t i c , and the f i g u r e s i n parentheses represent j 2 our c a l c u l a t i o n s of the chi-square s t a t i s t i c , £ (n. - e.) /e.. ( I t w i l l i = l be r e c a l l e d that we report the r e s u l t s of the chi-square t e s t i n order to check the accuracy of the computer program which generates the random d i s t r i b u t i o n s . The reader may e a s i l y v e r i f y t h a t the r e s u l t s from the 2 chi-square t e s t and the X t e s t are i n c l o s e agreement w i t h each other, i n d i c a t i n g that our estimates of the tru e d i s t r i b u t i o n s are accurate.) Looking at the r e s u l t s i n Table X I , we see that we would r e j e c t the n u l l hypothesis that our observations were generated by a random process based on the f. f o r the Vancouver sub-market and the GVRD. These r e s u l t s i are s i g n i f i c a n t at the 1% l e v e l . We cannot r e j e c t the n u l l hypothesis f o r the Richmond, North Shore, or Delta-Surrey sub-markets. We s h a l l delay our i n t e r p r e t a t i o n of these r e s u l t s f o r the time being. We wish f i r s t to t e s t a r e l a t e d hypothesis that our observations on the neighbor r e l a t i o n s i n the GVRD and i t s c o n s t i t u e n t sub-markets were generated by an independent s t o c h a s t i c process. This we do i n the next s e c t i o n . 5.4 The Test of Random Neighbor R e l a t i o n s 5.4.1 M o t i v a t i o n f o r the Test In the previous s e c t i o n , our focus was on t e s t i n g the n u l l hypothesis that our observations on f i r m ownership of stor e s i n the GVRD and con s t i t u e n t sub-markets were generated by an independent s t o c h a s t i c process - 134 -based on the f\ . Note i n p a r t i c u l a r that t h i s hypothesis was concerned w i t h the number of st o r e s that each f i r m owned i n each sub-market, and not w i t h where these stores were l o c a t e d w i t h i n the sub-market i n r e l a t i o n to each other. However, the theory of market preemption has d e f i n i t e i m p l i c a t i o n s about the neighbor r e l a t i o n s which should e x i s t i n a sub-market i f i t has been preempted by one or more f i r m s . Consider once again the case of p e r f e c t preemption. I f some f i r m has p e r f e c t l y preempted some sub-market A^ of A, then we would observe a l l of the st o r e s i n A^ having as t h e i r neighbors only other st o r e s that the preempting f i r m owns. I f we assume that market A i s dominated by the sto r e s of s e v e r a l l a r g e f i r m s , then we would expect a s i g n i f i c a n t divergence between the observed neighbor r e l a t i o n s i n sub-market A^ and the set of neighbor r e l a t i o n s that would be generated by a random d i s t r i b u -t i o n of f i r m ownership of st o r e s based on the f , . However, we would not expect to observe p e r f e c t preemption i n any sub-market, given the s t o c h a s t i c nature of the w o r l d , whence we must devise a formal t e s t to determine the extent to which our observations on neighbor r e l a t i o n s w i t h i n given sub-markets are c o n s i s t e n t w i t h a random process based on the f , • In the next su b - s e c t i o n , we f o r m a l i z e the n u l l hypothesis of random neighbor r e l a t i o n s that w i l l be t e s t e d i n t h i s s e c t i o n . 5.4.2 Statement of the N u l l Hypothesis of Random Neighbor R e l a t i o n s The n u l l hypothesis of random neighbor r e l a t i o n s which we s h a l l t e s t i n t h i s s e c t i o n i s as f o l l o w s : The observed set of neighbor r e l a t i o n s i n a given sub-market was generated by a random d i s t r i b u t i o n of f i r m ownership of st o r e s i n the sub-market. This random d i s t r i b u t i o n was produced by a s t o c h a s t i c process such that the p r o b a b i l i t i e s that any given s t o r e i n A i s owned by f i r m F,,F ?,...,F. are equal to the r e l a t i v e - 135 -frequencies f^,f^> •••»fj• These p r o b a b i l i t i e s are constant and i n v a r i a n t w i t h respect to which f i r m or firms own neighboring s t o r e s . The a l t e r n a t i v e hypothesis i s that the set of observed neighbor r e l a t i o n s i n any a r b i t r a r i l y chosen sub-market of A was generated by a s t a t e dependent p r o b a b i l i s t i c process such that the p r o b a b i l i t i e s that a given s t o r e i n any sub-market of A i s owned by f i r m P ^ J ^ ' *" * ' F j depend upon the neighbor r e l a t i o n s of the s t o r e w i t h other stores i n the sub-market. R e j e c t i o n of the n u l l hypothesis w i l l once again be i n t e r p r e t e d to imply that some form of s t a t e dependence was r e s p o n s i b l e f o r generating our observations, although the s t r u c t u r e of the n u l l hypothesis prevents us from being p r e c i s e about the s p e c i f i c nature of the dependence. R e j e c t i o n of the n u l l hypothesis f o r a given sub-market A^ of A w i l l a l s o be i n t e r p r e t e d to imply that at l e a s t one f i r m ' s s t o r e s are " c l u s t e r e d " i n sub-market A . That i s , at l e a s t one firm's s t o r e s are l o c a t e d more i n u p r o x i m i t y to each other than would be expected on the b a s i s of a random d i s t r i b u t i o n of f i r m ownership of stores produced by a random process based on the f^. This i n t e r p r e t a t i o n w i l l become c l e a r e r a f t e r we have o p e r a t i o n a l i z e d the concept of neighbor r e l a t i o n s i n the next sub-section. 5.4.3 T e s t i n g Procedure f o r the N u l l Hypothesis When the Number of  Stores i n the Sub-market I s Small In t h i s s u b - s e c t i o n , we s h a l l extend the example of sub-section 5.3.3 i n order to i l l u s t r a t e the concepts and r a t i o n a l e behind the procedures employed to t e s t the n u l l hypothesis of random neighbor r e l a t i o n s . However, before doing so, we need to devise a measure of neighbor r e l a t i o n s i n order to f a c i l i t a t e t e s t i n g of the n u l l hypothesis. - 136 -In a one-dimensional market, the concept of neighbor r e l a t i o n s t r a n s l a t e s i n t o the concept of market area boundary. Consider the market described i n Figure 9: Figure 9 I 1 4 X l b X 2 I t i s assumed that the fir m s l o c a t e d at X^ and X^ s e l l the same commodity at a common p r i c e . Then, the market area boundary between the firms l o c a t e d at X^ and X^ w i l l be the perpendicular b i s e c t o r of the market segment, X^,X2» represented by poi n t b. In order to implement our t e s t s of the n u l l hypothesis, we s h a l l g e n e r a l i z e t h i s concept of market area boundary to two dimensions w i t h the f o l l o w i n g d e f i n i t i o n s : D e f i n i t i o n 2. A boundary e x i s t s between two s t o r e s , s t o r e 1 and s t o r e 2, i f some part of the perpendicular b i s e c t o r of a l i n e drawn between the l o c a t i o n s of s t o r e 1 and st o r e 2 l i e s c l o s e r to these l o c a t i o n s than the perpendicular b i s e c t o r of a l i n e drawn between the l o c a t i o n of s t o r e 1 and the l o c a t i o n of any other s t o r e I, and between the l o c a t i o n of s t o r e 2, and the l o c a t i o n of any other s t o r e I. D e f i n i t i o n 3. A common boundary e x i s t s between two stor e s i f a boundary e x i s t s between them, and i f these two stor e s are owned by the same f i r m . In d e f i n i n g market area boundary as we have i n D e f i n i t i o n 2, we have i m p l i c i t l y assumed that a l l firms charge the same vector of p r i c e s f o r the same vect o r of goods. This s i m p l i f y i n g assumption i s necessary i n order to o p e r a t i o n a l i z e the concept of market area boundary f o r the - 137 -purpose of generating the random d i s t r i b u t i o n of market area boundaries. While we recognize that s t o r e s owned by d i f f e r e n t f i r m s probably charge s l i g h t l y d i f f e r e n t p r i c e s , small p r i c e d i f f e r e n c e s would not be expected to s e r i o u s l y a f f e c t the q u a n t i t a t i v e r e s u l t s . Let us now focus on how we might t e s t the n u l l hypothesis that the set of observed neighbor r e l a t i o n s i n a p a r t i c u l a r sub-market of A was generated by an independent s t o c h a s t i c process based on'the f ^ . In Figure 10, we reproduce the sub-market which appeared i n Figure 8. The dashed l i n e s are the market area boundaries of the s t o r e s l o c a t e d at X 1 , X 9, X^, X A, and have been drawn i n accordance w i t h D e f i n i t i o n 2. Figure 10 We s h a l l c a l l t h i s method of f i n d i n g market area boundaries the perpendicular b i s e c t o r - l e a s t d i s t a n c e method. We may summarize the i n f o r m a t i o n about market boundaries contained i n Figure 10 by c o n s t r u c t i n g a boundary ma t r i x : - 138 -- X l X 2 x 3 X 4 x l ~ 0 X 2 0 0 X 3 1 1 0 X 4 1 1 1 0 I f two stores have a boundary w i t h each other, then a 1 w i l l appear i n the boundary matrix. A s t o r e cannot have a boundary w i t h i t s e l f , so 0 e n t r i e s appear along the main d i a g o n a l . Since the matrix i s symmetric around the main d i a g o n a l , we need not f i l l i n the upper h a l f of the matrix. Our next step i s to generate the random d i s t r i b u t i o n of common boundaries f o r t h i s sub-market t h a t i s i m p l i e d by a random process based on the s t a t e independent p r o b a b i l i t i e s , f ^ = - Since we have already l i s t e d a l l the random permutations of f i r m ownership of stor e s 1-4 i n Table V, we need only use the boundary matrix i n conj u n c t i o n w i t h t h i s l i s t of permutations i n order to construct the "common boundary d i s t r i b u t i o n " . The common boundary d i s t r i b u t i o n attaches p r o b a b i l i t i e s to the occurrence of various numbers of common boundaries f o r each f i r m , given the p r o b a b i l i t i e s of occurrence of the d i f f e r e n t permutations of f i r m ownership. For example, suppose we wished to f i n d the p r o b a b i l i t y that f i r m F-^  w i l l have three common boundaries w h i l e f i r m F 2 w i l l have zero. Looking at permutation (4) i n Table V, we see that s t o r e s 1, 2, and 4 are a l l owned by f i r m F^, and l o o k i n g at the common boundary matrix (5.8), we see that stores 1 and 3, 1 and 4, and 3 and 4 have boundaries w i t h each other. Thus, we have found one permutation where f i r m F^ has three common boundaries, w h i l e f i r m F 2 has zero. This r e s u l t , however, als o occurs i n permutation ( 9 ) . Since each permutation has a .0625 - 139 -p r o b a b i l i t y of o c c u r r i n g , the p r o b a b i l i t y of observing three common boundaries f o r f i r m and zero common boundaries f o r f i r m F^ equals .125. In Table X I I , we summarize our c a l c u l a t i o n s of the p r o b a b i l i t i e s that f i r m s F, and F„ w i l l have various numbers of common boundaries: TABLE; X I I JOINT COMMON BOUNDARY DISTRIBUTION P r 7 B l l = 5 and B ^ = 0) = .0625 P r ( B i ; L = 3 and B 2 2 = 0) = .1250 P r ( B 1 ] L = 2 and B 2 2 = 0) = .1250 P r ( B i ; L = 1 and B 2 2 = 1) = .2500 Pr,(B 1 1 = 1 and B 2 2 = 0) = .0625 P r ( B n = 0 and B 2 2 = 1) = .0625 P r ( B l l = 0 a n d B22 = 2 ) = - 1 2 5 0 P r ( B i ; L = 0 and B 2 2 = 3 ) = • 1 2 5 ° P r ( B 1 ; L = 0 and B 2 2 = 5) = .0625 B.. = the number of common boundaries f o r f i r m F. x i x In Table X I I I , we have c a l c u l a t e d the marginal d i s t r i b u t i o n s corresponding to the j o i n t d i s t r i b u t i o n of Table X I I . - 140 -TABLE X I I I MARGINAL COMMON BOUNDARY DISTRIBUTIONS P r ( B n « 5) = .0625 P r ( B 2 2 = 5) = .0625 Pr ( B u = 3) = .1250 Pr.(B 2 2 = 3) = .1250 P ^ B U = 2) = .1250 P r ( B 2 2 - 2) = .1250 P r ( B n = 1) = .3125 P r ( B 2 2 = 1) = .3125 P r ( B n = 0) = .3750 P r ( B 2 2 = 0) = .3750 Again, we wish to s i n g l e out those events which, i f they occurred, would lead us to r e j e c t the n u l l hypothesis. I f our c r i t e r i o n f o r r e j e c t i o n i s that an observation on common boundaries f o r a f i r m have l e s s than a ten percent p r o b a b i l i t y of o c c u r r i n g , then there are two observations which would lead us to r e j e c t the n u l l hypothesis: a) i f f i r m F^ has f i v e common boundaries w h i l e f i r m F 2 has zero common boundaries: b) i f f i r m F 2 has f i v e common boundaries and f i r m F, has zero common boundaries. These r e s u l t s could only occur i f f i r m F^ and f i r m F 2 owned a l l four s t o r e s i n the market, r e s p e c t i v e l y , and we have already seen i n sub-section 5.3.3 that such observations on f i r m ownership would lea d us to r e j e c t the n u l l hypothesis of random f i r m ownership. Thus, we have found that the same set of observations on f i r m ownership of stor e s i n the sub-market depicted i n Figure 10 w i l l .lead us to r e j e c t both the n u l l hypothesis of random f i r m ownership and the n u l l hypothesis of random neighbor r e l a t i o n s . Sometimes i t may be true that the set of observations on f i r m owner-- 141 -ship that w i l l lead to r e j e c t i o n of the n u l l hypothesis of random f i r m ownership w i l l p e r f e c t l y c o i n c i d e w i t h the set of', observations on f i r m ownership that w i l l lead t o . r e j e c t i o n of the n u l l hypothesis of random neighbor r e l a t i o n s , . However, .this w i l l not always be the case,'''7 I t i s important to note that the random v a r i a b l e s and are not independent of each other. The p r o b a b i l i t y that s t o r e s 1 and 4 have a common boundary i s not independent of whether or not stor e s 4 and 2 have a common boundary. Thus, whereas the d i s t r i b u t i o n of f i r m ownership of s t o r e s was binomia l or multinomial because the p r o b a b i l i t i e s that a st o r e i s owned by f i r m F^ or f i r m Y^ are independent, the d i s t r i b u t i o n of common boundaries i s not multinomial because such independence of the B does not e x i s t . This f a c t w i l l become important i n our d i s c u s s i o n of the proper t e s t i n g procedure and t e s t s t a t i s t i c s of the n u l l hypothesis of random neighbor r e l a t i o n s when the number of stores i n a sub-market i s l a r g e , a d i s c u s s i o n to which we now t u r n . 5.4.4 Testing Procedure and Test S t a t i s t i c s f o r the N u l l Hypothesis  When the. Number of Stores i n the Sub-market Is Large I n the previous sub-section, our procedure f o r generating the exact random common boundary d i s t r i b u t i o n e n t a i l e d l i s t i n g a l l of the p o s s i b l e permutations of f i r m ownership o f . s t o r e s , and then f i n d i n g the number of common boundaries f o r each f i r m i n each permutation by using the boundary j matrix. However, i f / n; i s l a r g e , such a procedure would be ' /•, i u 1=1 i m p r a c t i c a l . In a d d i t i o n , since the common boundary d i s t r i b u t i o n i s not mul t i n o m i a l , we cannot use the multinomial d i s t r i b u t i o n to generate the p r o b a b i l i t i e s that, firms: Y , F_, F. w i l l have given numbers of common x - z j boundaries, and we cannot do a chi-square t e s t of the n u l l hypothesis, - 142 -2 Therefore, once again the best t e s t of the n u l l hypothesis i s the X t e s t . 2 The procedure f o r doing the X t e s t of the n u l l hypothesis of random 2 neighbor r e l a t i o n s i s e s s e n t i a l l y the same as that f o r doing the X t e s t of the n u l l hypothesis of random f i r m ownership, and we may b r i e f l y d e s cribe the procedure as f o l l o w s : F i r s t , we take the random permutations of f i r m ownership which we generated i n order to f i n d the random d i s t r i b u t i o n of f i r m ownership, and we use the boundary matr i x to f i n d the number of common boundaries which each f i r m has i n each permutation. The j o i n t d i s t r i b u t i o n of the number of common boundaries which each f i r m has i n each permutation w i l l c o n s t i t u t e our estimate of the tru e common boundary d i s t r i b u t i o n . Next, we convert our estimate of the tru e d i s t r i b u t i o n i n t o the d i s t r i b u t i o n of the f o l l o w i n g q u a n t i t y : (5.9) ^Y1 (B. . - B . . )2 L 1 1 1 1 1 = 1 B.. xx (B. . - B . . ) 2 (B - B ) 2 B. . B'.T J J x i where B..= the number of randomly generated common boundaries which stores xx owned by f i r m F. have w i t h themselves, B.. = the number of randomly i JJ generated boundaries between stores owned by competitive f r i n g e f i r m s , and B. T = E b. T = the t o t a l number of randomly generated boundaries between x i x i x^I s t o r e s owned by d i f f e r e n t f i r m s ( i n c l u d i n g boundaries between p o t e n t i a l preemptor f i r m s and competitive f r i n g e f i r m s ) . Each b^ represents the number of randomly generated boundaries between sto r e s owned by fi r m s F, and F^ ., i £ I . Bars over v a r i a b l e s i n d i c a t e mean generated v a l u e s . To s i m p l i f y our d i s c u s s i o n , we s h a l l c a l l each term i n the sum given by (5.9) a " r e l a t i v e discrepancy". A couple of comments regarding (5.9) are i n order. F i r s t , we i n c l u d e the r e l a t i v e discrepancy between observed and mean generated boundaries - 143 -between competitive f r i n g e firms as a separate component of (5,9) because the s i z e of t h i s discrepancy has i m p l i c a t i o n s f o r the acceptance or r e j e c t i o n of randomness i n a given sub-market. The theory of preemption suggests that i f p o t e n t i a l preemptor fi r m s have been e f f e c t i v e l y preempting the market, then we should expect to observe few boundaries between sto r e s which belong to the competitive, f r i n g e r e l a t i v e to the mean number of competitive f r i n g e boundaries that would be generated by a random process based on the f.. The reasons are as f o l l o w s : I n a market which has been 1 preempted by one or more f i r m s , we might s t i l l observe the existence of competitive f r i n g e s t o r e s due to management m i s c a l c u l a t i o n , u n a n t i c i p a t e d increases i n d e n s i t y , e t c . , on the part of preempting f i r m s . However, we would expect these s t o r e s to.be p r i m a r i l y l o c a t e d such that they are bounded by e s t a b l i s h e d f i r m s t o r e s . I f f i r m ownership were randomly d i s t r i b u t e d on the b a s i s of the f. , there would be no such presumption. This i m p l i e s that i f the r e l a t i v e frequency of the competitive f r i n g e stores i n a given sub-market i s approximately equal to our estimate of the s t a t e independent p r o b a b i l i t y , f , then we would expect a l a r g e r number of boundaries between competitive f r i n g e stores, when f i r m ownership i s randomly d i s t r i b u t e d on. the b a s i s of the f compared to the number of competitive f r i n g e boundaries that would e x i s t i f one or more f i r m s have preempted the market. The l a r g e r the discrepancy between observed and mean generated competitive; f r i n g e boundaries, the l e s s l i k e l y i s i t that a random process based on the.f was r e s p o n s i b l e f o r generating our observations, and the b e t t e r our chances may be of r e j e c t i n g the. n u l l hypothesis, Second, we i n c l u d e the r e l a t i v e discrepancy between, observed and mean - 144 -generated boundaries between.stores owned by d i f f e r e n t firms as a separate component of (5.9) because the s i z e of t h i s r e l a t i v e discrepancy a l s o has i m p l i c a t i o n s f o r the acceptance.or r e j e c t i o n of randomness i n a given sub-market. That i s , i f the d i s t r i b u t i o n of f i r m ownership, of st o r e s i n a given sub-market i s completely random and based on the f , then we would expect fewer common boundaries between stores owned, by any given f i r m , and consequently more boundaries between sto r e s owned by d i f f e r e n t firms than would be the case i f one or more fir m s have preempted the sub-market. The l a r g e r the discrepancy-.between observed and mean generated boundaries between s t o r e s owned by d i f f e r e n t f i r m s , the b e t t e r our chances may be of r e j e c t i n g the n u l l hypothesis. Having converted our estimate of the true d i s t r i b u t i o n i n t o the d i s t r i b u t i o n of the qua n t i t y given by (5.9), we need only f i n d where the sum of the observed, and mean generated r e l a t i v e d i s c r e p a n c i e s , j (B. . - B . . ) Z i i i i I i = l B. . i i ( B i i - y 2 H — , l i e s w i t h i n t h i s d i s t r i b u t i o n . (Note that ' 5 l I the r e l a t i v e discrepancy between observed and mean generated boundaries between competitive f r i n g e stores i s represented by the j - t h term i n the sum.) I f , f o r example, t h i s s t a t i s t i c l i e s , w i t h i n the 10% t a i l of the d i s t r i b u t i o n , we would r e j e c t the n u l l hypothesis that our observations on neighbor r e l a t i o n s were generated from a random d i s t r i b u t i o n of f i r m 18 ownership based on the f.. l ' 2 Before proceeding to report- the r e s u l t s of the X t e s t s of the n u l l hypothesis, we must once again confront the question of how many random permutations of f i r m ownership are necessary i n order to ins u r e that the - 145 -approximation of the generated d i s t r i b u t i o n to the- true, d i s t r i b u t i o n i s a good one. However, u n l i k e the random d i s t r i b u t i o n of f i r m ownership, we do not know the t h e o r e t i c a l form.of the common boundary d i s t r i b u t i o n (and 2 thus we cannot compare t h e . r e s u l t s from the X t e s t w i t h the r e s u l t s from some other test- l i k e the chi-square i n order to determine i f our approximation i s good), Therefore, we propose to take the f o l l o w i n g a l t e r n a t i v e approach; F i r s t , we know that the accuracy of. the estimate of the true d i s t r i b u t i o n should i n c r ease as we in c r e a s e the number of permutations. So, our f i r s t step would be to compare the cumulative frequencies at d i f f e r e n t p o i n t s i n the marginal d i s t r i b u t i o n s of common boundaries f o r d i f f e r e n t numbers of permutations i n order to see i f there are any "major" d i s c r e p a n c i e s . Our second step would be to subject these d i s c r e p a n c i e s to a c r i t e r i o n i n order to determine t h e i r r e l a t i v e 2 xmportance. The c r i t e r i o n we have chosen i s to compare the r e s u l t s of X t e s t s based on the d i f f e r e n t numbers of permutations as a means of determining i f the d i f f e r e n c e s i n cumulative frequencies at d i f f e r e n t p o i n t s i n the marginal d i s t r i b u t i o n s are sm a l l enough such that the 2 r e s u l t s of the X te s t s , would be i n v a r i a n t w i t h respect, to the number of random permutations. In other words, i f the accuracy, of the estimate of the t r u e d i s t r i b u t i o n does not increase enough i n going from 250 to 1000 random permutations of f i r m ownership to generate, any fundamental changes i n r e s u l t s , then we can be confident that any increase i n accuracy of the estimate obtained by doing more than 1000 permutations w i l l a l s o not generate any changes i n the r e s u l t s . I t w i l l be r e c a l l e d that i n sub-section 5,3.4, we generated four random d i s t r i b u t i o n s of: f i r m ownership f o r the GVRD based on 250, 500, 750, - 146 -and 1000 permutations of f i r m ownership. For the experiments discussed above, we have converted these random.distributions of firm.ownership i n t o the corresponding common boundary, d i s t r i b u t i o n s by f i n d i n g the number of common boundaries, f o r each f i r m i n each permutation, In Table XIV, we report the cumulative frequencies at d i f f e r e n t p o i n t s i n the marginal d i s t r i b u t i o n s of common boundaries f o r f i r m s F , F , F, , and the s o k competitive f r i n g e . (The number of boundaries between st o r e s owned by d i f f e r e n t f i r m s i s , of course, the t o t a l number of.boundaries i n the GVRD minus the t o t a l number of common boundaries. In other words, i t i s a r e s i d u a l and not independent of the t o t a l number of common boundaries i n the GVRD.) TABLE XIV CUMULATIVE FREQUENCIES FOR THE MARGINAL DISTRIBUTIONS OF THE GVRD - PERMUTATIONS = 250 , 500, 750, 1000 Firm F s X B \ ss # of X Permu- ^ t a t i o n s 6 10 14 18 ' 22 .26 30 3 4 v' 38 42 250 4.0 15.6 35.6 56,8 74.0 87.6 92.8 98.0 99,2 22.008 500 .6 4.8 18.0 34.4 56.4 73.4 86,0 93.8 9 7.4 98,8 21.970 750 .3 4.4 16,9 36.7 5.7,2 75,9 87,9 94,7 98.5 99.3 21.611 1000 .5 4.7 18.8 39,6 61.2 78,4 88,8 94.9 98,0:. 99,3 21.115 Table XIV (continued) - 147 -Firm F o ^ ^ o o # of \ ^ 2 , 4 . 6 8 10 12 14 16 18 y Permu-t a t i o n s 250 14. 1 40.2 61.4 79.1 90.0 94.8 98. 4 99.6 99.6 5.9600 500 13. 6 39.1 64.3 78.7 89.9 96.6 99. 0 99.6 100.0 5.8060 750 15. 1 39.1 63.1 81.1 90.0 95.7 98. 2 99.2 99.6 5.7853 1000. 13. 7 35.8 58.3 77.7 88.9 94.6 97. 3 98.8 99.5 6.0700 Firm F. V S » k k # o f V Permu-t a t i o n s ,21 25 29 33 37 41 45 49 53 57 y 250 5.2 12.4 25.6 41.6 55.6 71.6 81.6 89.2 93.6 98.0 36. 500 500 4.2 11.2 22.8 38.0 54.2 68.6 80.0 89.4 93.2 95.6 37. 250 750 5.9 12.4 25.9 40.4 55.6. 70.1 80.7 8.7.9 93.2 96.3 36. 877 1000 5.7 12.9 22.9 36.9 53.2 65.4 78.3 86.8 92.5 95.6 37. 569 Competitive Fringe \ ? c f 6 10 14 18 22 26 30 34 38 42 y # o f \ Permu-t a t i o n s 250 .4 6.8 20.4 33.6 52.0 69.6 83.2 92.0 96.4 98.4 22.428 500 .2 3.4 13.8 31.2 55.8 74.8 85.8 93.4 96.8 99.2 22.230 750 .4 4.8 14.0 31.7 52.4 70. 7 82.0 90.5 95.5 98.7 22.955 1000 .6 4.0 16.4 335.2 54.8 770.'7 84.5 -,9200 96.2 '98.4 22.461 - 148 -. L o o k i n g a t T a b l e XIV, we note f i r s t t h a t d i f f e r e n c e s i n c u m u l a t i v e f r e q u e n c i e s a t g i v e n p o i n t s o f the m a r g i n a l d i s t r i b u t i o n s r a r e l y exceed 5%. A p l o t of t h e m a r g i n a l d i s t r i b u t i o n s f o r the v a r i o u s numbers of per m u t a t i o n s would a l s o show t h a t the m a r g i n a l common boundary d i s t r i b u t i o n s f o r a g i v e n f i r m have approximately, the same.shape, Second, we not e t h a t t h e r e i s no c o n s i s t e n t tendency f o r the d i s c r e p a n c i e s i n c u m u l a t i v e f r e q u e n c i e s a t g i v e n p o i n t s i n the m a r g i n a l d i s t r i b u t i o n s to d e c l i n e as we i n c r e a s e the number o f p e r m u t a t i o n s . 2 Our next s t e p i s t o see i f the r e s u l t s of the X t e s t based on the above common boundary d i s t r i b u t i o n s a r e i n v a r i a n t w i t h r e s p e c t t o the number o f p e r m u t a t i o n s . The r e s u l t s o f t h e s e t e s t s a r e r e p o r t e d i n Ta b l e XV. TABLE XV RESULTS OF THE X 2 TESTS FOR THE GVRD PERMUTATIONS = 250, 500, 750, 1000 250 Permutations (B. . - B . . ) ' i i i i 1=1 B, i i > + < B i I - B i I > ' 5 i l (31 - 22.008) 2 (0 - 5.96) 2 22.008 5.96 (25 - 36,500) 2 (.8- 22.428) 2 (244 - 221.1) 2 36.500 22,428 221,1 = 24.9106259 - 149 -Table XV (continued) % of (B. . -B. . ) ' 1 1 1 1 i = l B.. i i + ( B . - B . )' i i i i i i d i s t r i b u t i o n to the r i g h t of 24.9106259 = 6.8% 500 Permutations (B. . -B. .)' 1 1 i i i = l B. i i + i i ( 3 1 - 21.97) 2 + (0 - 5.806) 2 21.97 5,806 (25 - 37.25) 2 (8 - 22.23) 2 (244 - 220.74) 2 37.25 22.23 220.74 = 25.1059537 % of { -1 ( B . . - B . . ) 2 i = l B.. i i + ( B i I " B i l } i i d i s t r i b u t i o n to the r i g h t of 25.1059537 = 4.8% 750 Permutations (B.. - B. .)' l i 1 1 i = l B.. i i + < Bii-V ' i l (31 - 21.611) 2 (0 - 5. 7853) 2 21.611 5.7853 (25 - 36.877) 2 (8 - 22.955) 2 (244 - 220.77) 2 36.877 22.955 220.77 = 25.8770136 % of { I i = l (B. . - B . . ) ' i i i i B. . l i + ( B i r V i i d i s t r i b u t i o n to the r i g h t of 25.8770136 = 5.8666.7% - 150 -Table XV (continued) 1000 Permutations j (B. . - B . . ) 2 1=1 B, i i + ( B i I \ l l (.31- 21.115) 2 , ( 0 - 6 , 0 7 ) 2 "T 21,115 6.07 (.25 - 37.569) 2 (.8 - 22,461) 2 (244 - 220, 7 8 ) 2 37.569 22.461 220.78 = 26.655214 % of j I 1=1 (B. . -B. J " i i i i i i < B±I- B±I>' • i i d i s t r i b u t i o n to the r i g h t of 26.655214 = 5.8% The r e s u l t s appearing i n Table XV i n d i c a t e that no fundamental changes occur as we increase the number of permutations from 250 to 1000. I f 2 our c r i t e r i o n f o r r e j e c t i o n i s that the X s t a t i s t i c l i e w i t h i n the 10% t a i l of the d i s t r i b u t i o n , then we would r e j e c t the n u l l hypothesis regardless of the number of permutations which we had generated f o r the t e s t . However, si n c e the accuracy of the estimate should increase as we increase the number of permutations:,' we s h a l l use 1000 permutations of f i r m ownership as the b a s i s f o r our t e s t s of the n u l l hypothesis of random neighbor r e l a t i o n s . The r e s u l t s of these t e s t s appear i n the next sub-section. - 151 -5.4,5 T e s t R e s u l t s of the N u l l H y p o t h e s i s o f Random Neighbor R e l a t i o n s  f o r the GVRD a n d . C o n s t i t u e n t Sub-markets In t h i s s u b - s e c t i o n , we r e p o r t the r e s u l t s o b t a i n e d from t e s t i n g the n u l l h y p o t h e s i s t h a t our observations,;on n e i g h b o r r e l a t i o n s , were g e n e r a t e d by a random pr o c e s s , based on the f_^ of. B.C. Our o b s e r v a t i o n s on n e i g h b o r r e l a t i o n s were o b t a i n e d by u s i n g the p e r p e n d i c u l a r - b i s e c t o r - l e a s t d i s t a n c e method o f finding:common b o u n d a r i e s , and th e s e o b s e r v a t i o n s appear i n T a b l e XVI. TABLE XVI COMMON BOUNDARIES AND BOUNDARIES BY FIRM - GVRD AND SUB-MARKETS Boundary GVRD Vancouver D e l t a - N o r t h Richmond . :. ' Surrey Shore  B 31 20 4 2 5 s s B 0 0 0 0 0 oo B k k 25 8 13 4 0 B j . 8 7 1 0 0 c f B. 244 162 45 22 15 i i __ . T o t a l 308 19 7 63 28 20 In T a b l e XVII, we p r o v i d e d e s c r i p t i v e measures f o r the m a r g i n a l d i s t r i b u t i o n s o f common b o u n d a r i e s g e n e r a t e d f o r each sub-market and f o r the GVRD, These d i s t r i b u t i o n s : a r e based on the.random d i s t r i b u t i o n s of f i r m ownership which were ge n e r a t e d f o r the t e s t s i n s e c t i o n 5,3 and were o b t a i n e d by u s i n g the boundary m a t r i x f o r the GVRD to f i n d the number of - 152 -common b o u n d a r i e s which: each f i r m has i n each random.permutation o f f i r m ownership. TABLE XVII MARGINAL COMMON BOUNDARY DISTRIBUTION DESCRIPTIVE MEASURES BY FIRM FOR THE GVRD AND SUB-MARKETS B B B, , B B. ss oo kk c f x l S u r r e y GVRD B i j L 21.115 6.07 37.569 22.461 220,78 a 7.3977 3.6175 10.709 8.1837 8.4637 Vancouver B.. 13.145 3.793 22.312 14.057 143.69 xx a 5.5779 2.7647 8.0844 6.1184 6.9433 D e l t a - B.. 4.8820 1.3430 9.1500 5.2490 42.376 xx 4.0470 1.7806 5.8674 4.2411 4.5501 N o r t h B.. 1.8110 .56800 3.2860 1.8960 20.439 Shore a 1.9583 .99517 2.6783. 1.9258 2.4063 Richmond B.. 1.3470. .35900 2.2590 1.4220 14.613 xx a 1.6066 .75279 2.2572 1.7581 2.0336 A f t e r . t r a n s l a t i n g our randomly g e n e r a t e d d i s t r i b u t i o n s o f common b o u n d a r i e s i n t o d i s t r i b u t i o n s o f the q u a n t i t y - 153 -(5.10) I 1=1 (B.. . -B. .)' 1.1 l i E. . 11 r + hi and using the informa t i o n contained i n Table XVI and Table XVII, we o b t a i n the f o l l o w i n g r e s u l t s from doing the X t e s t of the n u l l hypothesis: TABLE XVIII 2 RESULTS OF THE X TESTS OF THE NULL HYPOTHESIS OF RANDOM NEIGHBOR RELATIONS BY GVRD AND SUB-MARKETS GVRD (B -B ) 2 ss ss 2 (B -B ) oo oo ( B k k - \ k ) 2 ( B c f - 5 c f ) 2 < B i I - § i I > 2 B ss B oo B k k B c f § i l ( 3 1 - 21.115) 2 (0 - 6.07) 2 (25 - 37.569) 2 (8 - 22.461) 2 (244 - 220.78) 2 21.115 6.07 37.569 22.461 220.78 4.6276687 6.07 4.2050562 9.3103833 2.442107 ( B . . - B . . y i i i i i = l B. i i )• + — — 26.655214 B i l % of { I 1=1 (B. . - B.. .)' l i i i B. . i i > + ^ i i - V 'ii d i s t r i b u t i o n to r i g h t of 26.655214 = 5.8% - 154 -Table XVIII (continued) Vancouver (B -B ) 2 ss ss • (B -B )2 oo oo ( B k k - \ k ) 2 < a c f - 5 c f > 2 ' X i - * ! / B ss B oo 5kk § i l ( 2 0 - 13.145) 2 (.0 - 3. 793) 2 ( 8 - 2 2 . 3 1 2 ) 2 (7 - 14.057) 2 (162 - 143.69) 2 v 13.145 3.793 22.312 14.057 143.69 3.5748212 3.793 9.1804114 3.5425077 2.3331902 - J (B. . -B. J ' XI XX 1=1 B. % of XX j I 1=1 ( B i I - B i I > + — — — — = 22.424229 B (B. . -B. .)' XX XI B. . XX i i ( B i I - B i i r + d i s t r i b u t i o n to r i g h t of B i i 22.424229 = 7.7% Surrey-Delta (B -B ) 2 ss ss (B -B )2 oo oo ( B k k - B k k ) 2 ( B c f - B c f ) 2 ( B i l - B i l ) 2 B ss B oo \ k . B c f B i l (-4- 4.882) 2 (0 - 1.343) 2 ( 1 3 - 9.15) 2 (1 - 5.249) 2 (45 - 42.376) 2 4.882 1.343 9.15 5.249 42.376 .1593453 1.343 1.6199453 3.4395124 .1624829 J I 1=1 % of (B.. - B. .)' XX XX XX ^ i l - v + — i i — = 6.7242859 B 1 . I • 1=1 (B. . -B. .)' xx 11 B. . xx i i ( B ± I - B + d i s t r i b u t i o n to r i g h t of B i i 6.7242859 = 63% - 155 -Table XVIII (continued) North Shore (B -B ) 2 ss ss (B -B )2 oo oo ^ c f - ^ f ) 2 • • • ^ „ = - v 2 B ss B oo \ k B c f 5 i l (2 - 1.811) 2 (0 - .568) 2 (4 - 3.286) 2 (0 - 1.896) 2 (22 - 20.439) 2 1.811 .0197244 :568 .568 3,286 .1551418 1.896 1.896 20,439 .1192191 (B. . -B. .)' XI 11 1=1 B. . i i ( B i I - B i I > i. + — i i i i — = 2.7580853 B ii % of < (B . . - B . .)' i i i i i = l B. i i ^ i l - 5 ! ^ _ l d i s t r i b u t i o n to r i g h t of B i l 2.7580853 = 79% Richmond (B - B ) 2 ss ss (B - B )2 oo oo ( B k k " B k k ) 2 ( B c f - § c f ) 2 ( B i l - B i l ) 2 B ss B oo B k k • 5 c f . B i l (5 - 1.347) 2 (0 - .359) 2 (0 - 2.259) 2 (0 - 1.422) 2 (15 - 14.613) 2 1.347 .359 2.259 1.422 14.613 9.9067624 .359 2.259 1.422 .0102490 (B. . - B. .)' 11 i l ^ i l ^ i P v, + — i i i i — = 13.9570114 i = l B.. i i B I I % of j ( B . . - B . . ) 2 l i i i 1 ~ - 2 y + d i s t r i b u t i o n to r i g h t of =1 B. . B ii 13.9570114 = 14.1% - 156 -Looking a t the r e s u l t s i n Table X V I I I , we see that we would r e j e c t the n u l l hypothesis that our observations on neighbor r e l a t i o n s were generated by a random process based on the f f o r the Vancouver sub-market and the GVRD. These r e s u l t s are s i g n i f i c a n t at the 10%. l e v e l . We cannot r e j e c t the n u l l hypothesis f o r the. Richmond, Delta-Surrey, and North Shore sub-markets. In the next s e c t i o n , we s h a l l analyze these r e s u l t s more c l o s e l y : 5.5 I n t e r p r e t a t i o n of the Test Results of Randomness In the previous two sections., we found that we could r e j e c t the n u l l hypothesis of random f i r m ownership and the n u l l hypothesis of random neighbor r e l a t i o n s f o r the GVRD and the Vancouver sub-market. We s h a l l now i n t e r p r e t these r e s u l t s . Let us focus on the GVRD r e s u l t s f i r s t . In order to f a c i l i t a t e our d i s c u s s i o n , we s h a l l once, again c a l l each term of the sum of the - 2 - - 2 -(n.-n.) /n. and the (B.. - B . .) /B.. a " r e l a t i v e discrepancy". In Table l l l n n n J XI, we see that the l a r g e s t r e l a t i v e discrepancy between observed and mean generated f i r m ownership occurs f o r f i r m F , w h i l e the second l a r g e s t o r e l a t i v e discrepancy occurs f o r f i r m F g. Given the signs of these d i s c r e p a n c i e s before squaring we may conclude that f i r m F g's stor e s are r e l a t i v e l y concentrated i n the GVRD and f i r m F o's stor e s are r e l a t i v e l y unconcentrated or under represented i n the. GVRD v i s - a - v i s the r e l a t i v e frequencies f , . Firm i s a l s o r e l a t i v e l y under represented i n the GVRD, although the discrepancy between observed and mean generated f i r m ownership i s not l a r g e r e l a t i v e to the s i z e s of the other d i s c r e p a n c i e s . In Table X V I I I , we f i n d that the r e s u l t s f o r common boundaries do not show a p e r f e c t correspondence to those j u s t reported f o r f i r m ownership. In Table X V I I I , - 157 -the l a r g e s t r e l a t i v e discrepancy between observed and mean generated common boundaries occurs f o r competitive f r i n g e s t o r e s , and the sign of t h i s discrepancy before squaring i n d i c a t e s the absence of c l u s t e r i n g of competitive f r i n g e s t o r e s ^ i n . t h e GVRD, I f we exclude t h i s r e l a t i v e discrepancy, we see that the ord e r i n g of r e l a t i v e d i s c r e p a n c i e s by t h e i r magnitude i s the same i n Table X V I I I as i t i s i n Table XI: f i r m F q has the l a r g e s t r e l a t i v e discrepancy between observed and mean generated common boundaries, w h i l e f i r m F has the second l a r g e s t , and f i r m F, has S K. t h e ; t h i r d l a r g e s t . Given the signs of these d i s c r e p a n c i e s before squaring i t appears that f i r m F. 's. stores are r e l a t i v e l y c l u s t e r e d i n the GVRD, wh i l e the stor e s of f i r m F and f i r m F, are not. o k When we look a t the Vancouver r e s u l t s , we f i n d that the ord e r i n g of the r e l a t i v e d i s c r e p a n c i e s i n f i r m ownership i n Vancouver i s the same as that f o r the GVRD. Again, firm-'F ''s stores are r e l a t i v e l y concentrated and f i r m F Q's stor e s are r e l a t i v e l y unconcentrated, and the discrepancy between observed and mean generated f i r m ownership f o r f i r m F ^ i s not la r g e r e l a t i v e to the. s i z e s of the other d i s c r e p a n c i e s . However, the ord e r i n g of the r e l a t i v e d i s c r e p a n c i e s i n observed and mean generated common boundaries by t h e i r magnitude i s , not the same i n Vancouver as i t i s i n the GVRD. In p a r t i c u l a r , the l a r g e s t r e l a t i v e discrepancy occurs f o r f i r m F, , w h i l e the r e l a t i v e d i s c r e p a n c i e s f o r f i r m F , f i r m F , and k s o the competitive f r i n g e are approximately.of equal s i z e . Again, f i r m Fg's stores are the only ones, which are candidates f o r being c l u s t e r e d , w h i l e f i r m F 's and f i r m F, T s are not. o k I t i s i n t e r e s t i n g to note that we r e j e c t e d the n u l l hypothesis of random f i r m ownership and the n u l l hypothesis of random neighbor r e l a t i o n s f o r the GVRD, but not f o r three of- i t s c o n s t i t u e n t sub-markets, Richmond, - 158 -Delta-Surrey, and. the North.Shore. This r e s u l t appears to be due to the f a c t that we r e j e c t e d the n u l l hypothesis f o r the Vancouver sub-market, and the Vancouver sub-market accounts f o r 60.77% of the stor e s i n the GVRD. Thus, i n some sense, the Vancouver sub-market has dominated the GVRD i n terms of the r e s u l t s obtained from-our t e s t s of randomness. R e j e c t i o n of the n u l l hypothesis of randomness f o r the GVRD arid Vancouver sub-market was e a r l i e r said, to imply the exis t e n c e of some s o r t of s t a t e dependence. I t i s th e r e f o r e appropriate at t h i s p o i n t to t e s t f o r the exis t e n c e o f : s t a t e dependence i n the Vancouver sub—market. This we do i n the next s e c t i o n . 5.6 The Test of State Dependence 5.6.1 M o t i v a t i o n f o r the Test In the previous two se c t i o n s of t h i s chapter, we teste d the hypothesis that the s t o c h a s t i c process which was r e s p o n s i b l e f o r generating our observations on f i r m ownership and neighbor r e l a t i o n s was an independent s t o c h a s t i c process based on a set of f i x e d s t a t e independent p r o b a b i l i t i e s . These.state independent p r o b a b i l i t i e s were defined to mean that- the p r o b a b i l i t i e s that a store i s owned by f i r m F^, F^, F.. do not depend on the s t a t e of neighbor r e l a t i o n s p r i o r to the poi n t i n time when the s t o r e i s e s t a b l i s h e d . We found that we could r e j e c t these hypotheses f o r the GVRD and the Vancouver sub-market. However., these t e s t s , based as. they were on a set of cross s e c t i o n observations on s t o r e l o c a t i o n s o n l y , permitted us to r e j e c t the hypotheses of randomness, and they d i d not a l l o w us to be p r e c i s e about the nature of the process which a c t u a l l y gave r i s e to our observations. Such p r e c i s i o n r e q u i r e s knowledge of the process i t s e l f or observations - 159 -on the outcomes of the process over time, In t h i s s e c t i o n , we s h a l l u t i l i z e time s e r i e s data.on the date at which each s t o r e was e s t a b l i s h e d i n the Vancouver sub-market i n order to t e s t the hypothesis that the s t o c h a s t i c process which gave r i s e to our observations on the sequence of- store: openings i s a s t a t e dependent process. A s t a t e w i l l be defined over the set of neighbor r e l a t i o n s that a given s t o r e w i l l have w i t h other s t o r e s i n the' sub-market i f that s t o r e i s e s t a b l i s h e d . The theory of preemption suggests that p o t e n t i a l neighbor r e l a t i o n s are key c o n s i d e r a t i o n s i n a preempting f i r m ' s l o c a t i o n s t r a t e g y . By e s t a b l i s h i n g new stores i n a given market at l o c a t i o n s which are only bounded by other s t o r e s that i t owns, and at p o i n t s i n time when i t would not be p r o f i t a b l e f o r other f i r m s to e s t a b l i s h s t o r e s , a preempting f i r m maximizes the present value of i t s p r o f i t s . I t does so by a v o i d i n g c o s t l y p r i c e competition w i t h other f i r m s , and by s e l e c t i n g the p r i c e s and l o c a t i o n s which maximize the j o i n t p r o f i t s over a l l of i t s s t o r e s . Thus, i n order to i d e n t i f y a p a r t i c u l a r f i r m as having preempted i n a given sub-market, we must f i r s t determine i f the s t a t e s of neighbor r e l a t i o n s have had an impact on the outcomes of the s t o c h a s t i c process. Once we have e s t a b l i s h e d that the process which gave r i s e to our observations i s a s t a t e dependent, one, we may-begin to analyze the process more c l o s e l y w i t h the aim of determining whether the u n d e r l y i n g p r o b a b i l i t i e s which generated the process are c o n s i s t e n t w i t h the existence of preemptive f i r m behavior i n the market. In the next sub-section, we f o r m a l l y s t a t e the n u l l hypothesis of s t a t e dependence, and we d i s c u s s the i m p l i c a t i o n s of r e j e c t i n g - or accepting the - 160 r-hypothesis. In order to t e s t the n u l l hypothesis, of s t a t e dependence, we must f i r s t o b t a i n estimates of s t a t e dependent p r o b a b i l i t i e s , and our procedure f o r c o n s t r u c t i n g these estimates i s explained ,in sub-section 5.6.3. We then go on to d i s c u s s how we may use our estimates of s t a t e dependent and s t a t e independent p r o b a b i l i t i e s i n order to conduct a t e s t of s t a t e dependence. As noted above, even i f we accept.the hypothesis of s t a t e dependence, we must s t i l l determine i f our estimates of the s t a t e dependent p r o b a b i l i t i e s are c o n s i s t e n t w i t h one or more fir m s having preempted.in the market. In sub-section 5.6.4, we e s t a b l i s h two r e l a t e d sets of c r i t e r i a based on comparisons of s t a t e dependent p r o b a b i l i t i e s and r e l a t i v e frequencies.which w i l l enable us to make t h i s determination. In sub-section 5.6.5, we conduct a t e s t f o r s t a t e dependence using Vancouver sub-market data, and i n sub-section 5.6.6, we subject our estimates of s t a t e dependent p r o b a b i l i t i e s to the two sets of c r i t e r i a e s t a b l i s h e d i n sub-section 5.6.4. F i n a l l y , i n sub-section 5.6.7, we provide an i n t e r p r e t a t i o n of our r e s u l t s . 5.6.2 Statement of the N u l l Hypothesis of State Dependence The n u l l hypothesis that we s h a l l t e s t i n t h i s s e c t i o n may be s t a t e d as f o l l o w s : The observed set of neighbor r e l a t i o n s and d i s t r i b u t i o n of f i r m ownership i n a given sub-market A^ of A were generated by a s t a t e dependent p r o b a b i l i s t i c process over time. This process may be c h a r a c t e r i z e d by a set of M x j p r o b a b i l i t i e s , where j equals the number of f i r m s ( i n c l u d i n g the competitive f r i n g e ) which have e s t a b l i s h e d s t o r e s i n the sub-market A ,'and"M equals the number of p o s s i b l e . s t a t e s of neighbor r e l a t i o n s which any given s t o r e might have with other s t o r e s i f i t were e s t a b l i s h e d i n the sub-market. (As i n the previous s e c t i o n , "neighbor - 161 -r e l a t i o n " should be thought of i n the context of a s t o r e owned by a given f i r m having a boundary w i t h a s t o r e owned by i t s e l f or w i t h a s t o r e owned by a d i f f e r e n t firm.) Each p r o b a b i l i t y , ir . (m = 1,2,. . . ,M; i = 1,2,. . ., j ) , mi represents the p r o b a b i l i t y that a p a r t i c u l a r f i r m F^ w i l l e s t a b l i s h , a st o r e i n the sub-market, given that that s t o r e would have as i t s neighbors sto r e s owned by fir m s represented by s t a t e m. The a l t e r n a t i v e hypothesis i s that the observed set of neighbor r e l a t i o n s and: d i s t r i b u t i o n of firm-ownership i n a given sub-market A of A u were generated by a random process based on a set of s t a t e independent p r o b a b i l i t i e s . One i m p l i c a t i o n of t h e ' n u l l hypothesis i s that i f the M x j s t a t e dependent p r o b a b i l i t i e s TT^ (m= 1,2,. . . ,M; i = 1,2, .... , j ) are t r u l y the p r o b a b i l i t i e s of a s t a t e dependent process., then they should be s.v s i g n i f i c a n t l y d i f f e r e n t from the p r o b a b i l i t i e s of a s t a t e independent process. I f these p r o b a b i l i t i e s are not s i g n i f i c a n t l y d i f f e r e n t from the s t a t e independent p r o b a b i l i t i e s , then a l l of the M s t a t e s may be co l l a p s e d i n t o one s t a t e , w i t h the consequence that the p r o b a b i l i t i e s of a s t o r e being owned by given f i r m s do not depend upon the s p e c i f i c set of neighbor r e l a t i o n s that that s t o r e would have w i t h other s t o r e s i n the sub-market i f i t was e s t a b l i s h e d . Another i n t e r p r e t a t i o n would be that i f our estimates of the s t a t e dependent p r o b a b i l i t i e s are i n s i g n i f i c a n t l y d i f f e r e n t from the s t a t e independent p r o b a b i l i t i e s . , then the explanatory power of the n u l l hypothesis of s t a t e dependence would be. n i l s i n c e the process from which estimates of the u . are obtained would be o b s e r v a t i o n a l l y r mi equivalent to an independent s t o c h a s t i c process based on the s t a t e independent p r o b a b i l i t i e s . - 162 -F i n a l l y , we should l i k e , to note once again that w h i l e acceptance of the n u l l hypothesis means that the process which gave r i s e to our observations on f i r m ownership and neighbor r e l a t i o n s , i s c o n s i s t e n t w i t h a s t a t e dependent p r o b a b i l i s t i c process, i t does not n e c e s s a r i l y mean that the process was a preemptive one. Whether or not the process was a preemptive one must be i n f e r r e d from a. c a r e f u l a n a l y s i s of the s t a t e dependent p r o b a b i l i t i e s themselves. * In the next s u b - s e c t i o n , we discus s the procedure which we use to o b t a i n estimates of s t a t e dependent p r o b a b i l i t i e s , and we devise a t e s t of the n u l l hypothesis of s t a t e dependence. 5.6.3 Testing Procedure for'. State Dependence In t h i s sub-section, we devise a t e s t i n g procedure f o r the n u l l hypothesis of s t a t e dependence. As i n the previous s e c t i o n s , we s h a l l use an example to i l l u s t r a t e the procedure f o r o b t a i n i n g estimates of the s t a t e dependent p r o b a b i l i t i e s IT and the t e s t i n g procedure f o r the n u l l hypothesis. Consider once again the sub-market represented by Figure 10, which we reproduce below as Figure 11. - 163 -Figure 11 In s e c t i o n 5.4.3, we assumed that Figure 10 represented a snapshot of the sub-market at one p o i n t i n time and that we d i d not know the sequence i n which the stores were e s t a b l i s h e d . Now, however, l e t us assume that we know the sequence, and that s t o r e 1 l o c a t e d at was e s t a b l i s h e d a t time t ^ , s t o r e 2 l o c a t e d at X^ was e s t a b l i s h e d at time t ^ , and so f o r t h , where t n < t„ < t„ < t . . Since s t o r e 1 located at X, was the f i r s t s t o r e to be 1 2 3 4 1 opened i n t h i s sub-market, we s h a l l d e f i n e i t as re p r e s e n t i n g the i n i t i a l c o n d i t i o n . There are a f i n i t e number of s t a t e s of neighbor r e l a t i o n s which a f i r m might be faced w i t h when i t considers opening e i t h e r s t o r e 2, s t o r e 3, or store 4. i n the sub-market at time t^, t ^ and t ^ r e s p e c t i v e l y . For example, the new store might have one, two.or three boundaries w i t h the stor e s owned by f i r m F^ or f i r m F^ depending on when the new s t o r e i s e s t a b l i s h e d i n the sub-market and whether f i r m F-^  or f i r m F^ owns the stor e s that are e s t a b l i s h e d i n the sub-market p r i o r to the new s t o r e . Or, the new s t o r e might have one or two boundaries w i t h s t o r e s owned by f i r m - 164 -F^ ( f i r m F^) and one boundary w i t h a store owned by f i r m F^ ( f i r m F ^ ) , again depending on when the new s t o r e i s e s t a b l i s h e d and which f i r m or firms owned the e x i s t i n g s t o r e s . As the number of firms:and the number of s t o r e l o c a t i o n s i n a sub-market in c r e a s e , the t o t a l number of p o s s i b l e s t a t e s of neighbor r e l a t i o n s increases r a p i d l y . We s h a l l . f i n d i t necessary to assume that what matters to the f i r m i s whether i t s new st o r e would only have boundaries w i t h other s t o r e s that i t owns or i f i t would have boundaries w i t h stores owned by other f i r m s , and that the f i r m i s not concerned w i t h the absolute number of boundaries which i t s new store would have w i t h i t s own stor e s or the stores of other f i r m s . This assumption means that i n our example, there are only three, p o s s i b l e s t a t e s of neighbor r e l a t i o n s which a f i r m might be faced w i t h when i t considers opening a new store at some time t , and these three s t a t e s are represented by the B. on the l e f t hand side of the f o l l o w i n g contingency t a b l e : -TABLE XIX CONTINGENCY TABLE OF STATE DEPENDENT FREQUENCIES F F *1 2 a l l a l 2 a 2 1 a22 a 3 1 a32 - 165 -Given that there are only two f i r m s which c o n s t i t u t e the i n d u s t r y i n our example., a f i r m which e s t a b l i s h e s a new s t o r e i n the sub-market may f i n d that i t s s t o r e has boundaries only w i t h stores'owned by f i r m F^(B^), boundaries only w i t h stores owned by f i r m F2X.B2) , or boundaries w i t h stores owned by both f i r m F.^  and f i r m F 2 ( B 1 , B 2 ) . The F. at the top of the t a b l e symbolize a store owned by f i r m F_^  being e s t a b l i s h e d i n the sub-market. Thus, contingency t a b l e element a ^ represents the number of times that f i r m F^ e s t a b l i s h e d a store- i n the sub-market from time t2 to time t ^ such that i t . o n l y had boundaries w i t h s t o r e s owned by f i r m F^. Contingency t a b l e element a ^ i represents the number of times that f i r m F^ e s t a b l i s h e d a s t o r e i n the sub-market from time t„ to time t . such that 2 4 i t had at l e a s t one boundary w i t h a s t o r e owned by f i r m F^ and at l e a s t one boundary w i t h a store owned by f i r m F^. The r e s t of the e n t r i e s i n the t a b l e may be i n t e r p r e t e d i n a s i m i l a r f a s h i o n . Next, we convert the contingency t a b l e i n t o a matrix of r e l a t i v e frequencies. (5.11) B r B 2 '11 a l l + a12 21 a 2 1 + a22 l31 a 3 1 + a32 *12 a l l + a12 22 a 2 1 + a22 l32 a 3 1 + a32 - 166 -We s h a l l regard the r e l a t i v e frequencies i n matrix (5,11) as c o n s t i t u t i n g our best estimates of the s t a t e dependent p r o b a b i l i t i e s TT^ ,. Our reasons f o r t h i s c o n c l u s i o n are as f o l l o w s : The s t a t e dependent p r o b a b i l i t i e s are s t a t e dependent i n the sense that the p r o b a b i l i t i e s that a given s t o r e i s owned by- f i r m F , F , , . ., F-. depend upon the s t a t e of neighbor r e l a t i o n s that that s t o r e would have w i t h other s t o r e s i n the market i f i t was e s t a b l i s h e d . Thus, the s t a t e dependent p r o b a b i l i t i e s may be i n t e r p r e t e d as being c o n d i t i o n a l p r o b a b i l i t i e s , where Pr (F. n m) (5.12) TT . = Pr(F. I m) = -mx X n / \ Pr (m) Pr(m) > 0 ( i = 1,2,.. , , j ; m=l,2,...,M) Equation (5.12) says that the p r o b a b i l i t y that f i r m F, w i l l own a given store i n A^, given that that store would have boundaries w i t h s t o r e s owned by the firm s represented by s t a t e m, i s equal to the p r o b a b i l i t y that f i r m F. e s t a b l i s h e s a s t o r e i n A and s t a t e m occurs, d i v i d e d by x u the p r o b a b i l i t y that s t a t e m occurs. These c o n d i t i o n a l p r o b a b i l i t i e s may be estimated as f o l l o w s : j a . mx (5.13) TT . = Pr(F. I m) = — ^- — ^ . mx x M j M j m=l i - 1 m=l i = l a . mx The numerator of (5,13) c o n s t i t u t e s our estimate of Pr(F i-nm), That i s , the p r o b a b i l i t y that f i r m F ± e s t a b l i s h e s a s t o r e i n A^ and s t a t e m occurs may be estimated by .the number of times that f i r m F, e s t a b l i s h e d a s t o r e i n - 167 -A given that the s t o r e had boundaries w i t h stores owned by the f i r m s u represented B.y-'.state m, d i v i d e d by the t o t a l number of s t o r e s which are e s t a b l i s h e d i n the sub-market, The denominator of (5,13) c o n s t i t u t e s our estimate of Pr(m). That i s , the p r o b a b i l i t y that s t a t e m occurs may be estimated.by the number of times when the opportunity to e s t a b l i s h a s t o r e i n A , such that the s t o r e would only have boundaries w i t h s t o r e s u owned by the firms represented by s t a t e m, presented i t s e l f , d i v i d e d by the t o t a l number of s t o r e s which are e s t a b l i s h e d i n the sub-market. We may r e w r i t e (5.13) as f o l l o w s : a . mi TT . = —; mi 3 . L, mi 1=1 Note that i m p l i c i t i n t h i s i n t e r p r e t a t i o n of the s t a t e dependent p r o b a b i l i t i e s i s the assumption that what matters to a f i r m i s simply the s t a t e of neighbor r e l a t i o n s i n the immediate v i c i n i t y of where i t plans to open a new s t o r e , and not, f o r example, the s t a t e of neighbor r e l a t i o n s throughout the sub-market at the time when i t wishes to e s t a b l i s h a new s t o r e . A l s o i m p l i c i t i n t h i s i n t e r p r e t a t i o n of the s t a t e dependent p r o b a b i l i t i e s i s that whenever f i r m s are faced w i t h a given s t a t e of neighbor r e l a t i o n s , the same set of p r o b a b i l i t i e s determine which f i r m w i l l , a c t u a l l y e s t a b l i s h the new s t o r e . (We a l s o wish to note that j j M YTT = Y P r ( F . I m) = 1, but that 7 TT i = l m i 1=1 1 vml would not equal one i n ml general.) Having obtained estimates of the s t a t e dependent p r o b a b i l i t i e s , we - 168 -must determine i f they are s i g n i f i c a n t l y d i f f e r e n t from some set of s t a t e independent p r o b a b i l i t i e s . As i n sections. 5.3 and 5.4, we s h a l l l e t the r e l a t i v e frequencies of each f i r m ' s s t o r e s i n market A be our estimates of the s t a t e independent p r o b a b i l i t i e s , We do not use sub-market r e l a t i v e frequencies as our estimates, of s t a t e independent p r o b a b i l i t i e s i n order to conduct a t e s t of s t a t e dependence, f o r the f o l l o w i n g reasons: Suppose that a given sub-market's r e l a t i v e frequencies f are the r e s u l t of one f i r m having preempted the market. I f our t e s t of s t a t e dependence involved determining i f the f. were s i g n i f i c a n t l y d i f f e r e n t from the xu ir ... then we might f i n d that the f. are not s i g n i f i c a n t l y d i f f e r e n t from mx b xu • the TT . p r e c i s e l y because the f. are the r e s u l t of a s t a t e dependent mx xu process. Consequently, we would r e j e c t the n u l l hypothesis of s t a t e dependence when i t i s i n f a c t t r u e . We t h e r e f o r e regard the f from market A as the most appropriate estimates, of the s t a t e independent p r o b a b i l i t i e s s i n c e the f from market A are based on a l a r g e r number of outcomes of the process (or processes) which gave r i s e to them. Returning now to our example, we s h a l l assume that both f i r m F^ and f i r m F^ own h a l f of the st o r e s i n market A, and thus our. estimates of the s t a t e independent p r o b a b i l i t i e s are f = f ^ = 1/2. Te s t i n g f o r a s i g n i f i c a n t d i f f e r e n c e between our estimates of the s t a t e dependent p r o b a b i l i t i e s and the s t a t e independent p r o b a b i l i t i e s i n v o l v e s t e s t i n g f o r the s i m i l a r i t y between matr i x (5.11) and the f o l l o w i n g m a t r i x of s t a t e independent p r o b a b i l i t i e s ; - 169 -F l F 2 B l f 1/2 1/2 (5.14) H 2 1/2 1/2 B r B 2 1/2 1/2 Once again, the most appropriate t e s t of s i g n i f i c a n c e to perform here i s a chi-square t e s t s i n c e we are i n t e r e s t e d i n the s i g n i f i c a n c e of the di s c r e p a n c i e s between our observations a ^ i n the contingency t a b l e and the number of cases e ^ which would be expected to appear i n the c e l l s of the t a b l e i f our observations were generated by an independent s t o c h a s t i c process based on the f^. In order to conduct a chi-square t e s t , we would f i r s t compute (5.15) M j / ,2 J (a . - e .) I _ mx mx m=l i = l e . mx where e . = f . ( / a . ) . The dxstrxbutxon of the random varxable mx a. . , mx x=l represented by (5.15) w i l l approach that of a chi-square v a r i a b l e w i t h ( M - l ) ( j - l ) degrees of freedom as the number of times the experiment i s 19 performed approaches i n f i n i t y . I f the c a l c u l a t e d value of (5.15) l i e s i n the c r i t i c a l r e g i on of a chi-square d i s t r i b u t i o n w i t h ( M - l ) ( j - l ) degrees of freedom, then we may accept the hypothesis that the a are s i g n i f i c a n t l y d i f f e r e n t from the e .. This i n t u r n i m p l i e s that the rax' process g i v i n g r i s e to our observations a i s c o n s i s t e n t w i t h a s t a t e mx - 170 -dependent s t o c h a s t i c process based on the TT . . mi Since the sub-market of our example i s so small (that i s , only three observations on the process a f t e r we exclude the s t o r e r e p r e s e n t i n g our i n i t i a l c o n d i t i o n ) , i t would be i n a p p r o p r i a t e .to c a l c u l a t e (5.15) f o r h y p o t h e t i c a l values of the a .. However, the example has served to mi , i l l u s t r a t e what:the appropriate t e s t i n g procedure i s when the number of stores i n the sub-market i s l a r g e . In the next sub-section, we proceed to a d i s c u s s i o n of how we.may use our estimates of the s t a t e dependent p r o b a b i l i t i e s i n order to determine i f the s t a t e dependent process i s c o n s i s t e n t w i t h preemptive f i r m behavior. 5.6.4 Comparative A n a l y s i s ' o f State Dependent P r o b a b i l i t i e s and  R e l a t i v e Frequencies I n the previous s u b - s e c t i o n , we discussed how we would o b t a i n estimates of the s t a t e dependent p r o b a b i l i t i e s when there are only two firms operating i n a sub-market. This procedure obviously g e n e r a l i z e s to a sub-market w i t h j f i r m s , where the j th f i r m represents the competitive f r i n g e s t o r e s i n the sub-market. We a l s o explained how we could use our estimates of s t a t e dependent p r o b a b i l i t i e s i n conjunction w i t h estimates of s t a t e independent p r o b a b i l i t i e s i n order to do a chi-square t e s t of the n u l l hypothesis of s t a t e dependence. Given that we might accept the n u l l hypothesis of s t a t e dependence, we must e s t a b l i s h c r i t e r i a f o r determining, whether the process i s a preemptive one. We do so by examining the t h e o r e t i c a l l y , p r e d i c t e d d i f f e r e n c e s between s t a t e dependent p r o b a b i l i t i e s and r e l a t i v e frequencies,, However,; p r i o r to t h i s a n a l y s i s , we s h a l l discuss"ways in.which we may reduce the d i m e n s i o n a l i t y of the matrix of s t a t e dependent p r o b a b i l i t i e s . The chi-square t e s t of s t a t e - 171 -dependence and data l i m i t a t i o n s r e q u i r e that such a r e d u c t i o n be made. In t h i s sub-section, we s h a l l consider a sub-market c o n s i s t i n g of two f i r m s , which are p o t e n t i a l preemptors, and a l s o a competitive f r i n g e . The matri x of s t a t e dependent p r o b a b i l i t i e s f o r t h i s sub-market would be as f o l l o w s : F l F 2 cf B l *11 *12 *13 B 2 U 2 1 *22 ^23 B c f TT3 1 *32 ^33 B r B c f \l ^42 ^43 B 2 ' B c f 1151 ^52 TT53 B1' B2 •*61 u62 ^63 B 2 ' B c f *72 TT73 1 By simply adding the competitive f r i n g e to the sub-market, we have increased the number of s t a t e dependent p r o b a b i l i t i e s which r e q u i r e estimates from s i x to twenty-one. This l a r g e increase i n the number of s t a t e dependent p r o b a b i l i t i e s may cause problems f o r our t e s t of s t a t e dependence-since, i d e a l l y , fewer than 20% of the c e l l s of the contingency t a b l e upon which matrix (5.16).is based should have expected frequencies l e s s than f i v e . Thus, we must ask i f there are ways i n which we may combine, c e r t a i n s t a t e s i n matrix (5.160 i n order to increase the expected frequencies i n as many c e l l s of the contingency t a b l e as... p o s s i b l e , w h i l e at the same time not i m p a i r i n g our a b i l i t y to use the s t a t e dependent p r o b a b i l i t i e s to a s c e r t a i n the existence of preemptive firm.behavior i n the sub-market. - 172 -F i r s t , we may combine s t a t e B, and state,B_,B This i s because ' - 1 1 cf f i r m F i n i t s c a p a c i t y as p o t e n t i a l preemptor, would have roughly the same i n c e n t i v e to preempt the market i f i t faced a s t a t e such that i t could e s t a b l i s h a new store i n the sub-market which, would only have boundaries w i t h other f i r m F^ s t o r e s , or i f i t could e s t a b l i s h a new sto r e i n the sub-market which would have boundaries w i t h competitive f r i n g e s t o r e s as w e l l as f i r m F, s t o r e s . This same reasoning allows us to combine s t a t e s B„ and B„,B ... Next, we may combine s t a t e s B,,B and z z c i 1 z B ,B0,B.\p. The reason i s because according to the theory of preemption, both f i r m F^ and f i r m would have an i n c e n t i v e to preempt the market i f they faced these s t a t e s of neighbor r e l a t i o n s , although the theory does not p r e d i c t which f i r m w i l l a c t u a l l y act upon that i n c e n t i v e f i r s t . F i n a l l y , we s h a l l t r e a t B ^ as a r e s i d u a l and combine i t w i t h s t a t e s B^jB^ and B^fB^B^^. The reason i s because i f a new s t o r e would only have boundaries w i t h s t o r e s belonging to the competitive f r i n g e , the theory does not p r e d i c t which f i r m w i l l e s t a b l i s h that s t o r e . Thus, s t a t e B._ more c l o s e l y resembles s t a t e B, ,B„ and B 1 5B-,B . than i t does cf 1 2 1 2 cf st a t e B^ or s t a t e B^. A f t e r combining s t a t e s , we obtain.the f o l l o w i n g m a t r i x of s t a t e dependent p r o b a b i l i t i e s : B i ' B r B c f (5.17) B 2 / B 2 ' B c f B c f 7 B ^ / B ^ B ^ F l F 2 c l " l l *12 " l 3 *21. TT22 *23 *31 *32 *33 - 173 -Now, l e t us assume that we have conducted a t e s t of the n u l l hypothesis of state.dependence f o r t h i s sub-market, and that we have accepted the n u l l hypothesis. We wish to e s t a b l i s h c r i t e r i a which would permit us to::say that the s t a t e dependent p r o b a b i l i t i e s are c o n s i s t e n t w i t h preemptive f i r m behavior on the part of f i r m F^, f i r m F^j or both f i r m F^ and f i r m F^. Our: f i r s t set of c r i t e r i a w i l l be based upon the t h e o r e t i c a l l y p r e d i c t e d d i f f e r e n c e s between the s t a t e dependent p r o b a b i l i t i e s and the r e l a t i v e frequencies f_^. In previous s e c t i o n s , we t r e a t e d the f ^ as our best estimates of the s t a t e independent p r o b a b i l i t i e s . However, we may no longer t r e a t the f ^ as estimates of the s t a t e independent p r o b a b i l i t i e s , because i f a l l f i r m s ' s t o r e s are not randomly, d i s t r i b u t e d , then the f ^ w i l l not be independent of the s p a t i a l s t r u c t u r e of each f i r m ' s s t o r e s i n the sub-markets comprising market A. Nonetheless i f f i r m F^ has engaged i n preemptive behavior i n a sub-market, we would expect T-^JL > ^± ^ O R T N E f o l l o w i n g reasons: F i r s t , r e c a l l that our estimate of i r ^ i s obtained by f i n d i n g the number of cases from t ^ (the i n i t i a l c o n d i t i o n ) to the present i n which f i r m F^ has e s t a b l i s h e d s t o r e s i n the sub-market such that they only had boundaries w i t h other f i r m F^ s t o r e s . The r e l a t i v e frequency f ^ i s a weighted average of the sub-market r e l a t i v e frequencies of f i r m F^'s s t o r e s , where the weights represent the r e l a t i v e importance of each sub-market i n market A. Provided f i r m F^ has not preempted a l l sub-markets to the same extent, there i s a tendency, f o r ' T-Q to exceed f ^ i f f i r m F^ has preempted the sub-market represented by m a t r i x (5,17). This same reasoning leads us to expect that >- f 2 Af £Arm F^'has preempted .the sub-market, Let us now consider the i m p l i c a t i o n s of d i s c r e p a n c i e s between -n and f 2 and i r ^ 3 and fy. I f f i r m F^ has engaged i n preemptive behavior i n the - 174 -sub-market, then the opportunity f o r f i r m F 2 or the competitive f r i n g e to e s t a b l i s h " a s t o r e i n the sub-market when faced w i t h . s t a t e s B^ or B^,B^f w i l l not o f t e n present i t s e l f . Thus, t h e r e . i s a tendency, which depends upon the weights of sub-markets which f ^ and f ^ are based on, f o r TT^ ^ < f 2 and TT^ 2 ^ £ 3 i n a sub>-market which has been a c t i v e l y preempted by f i r m F^. (I t must be kept i n mind that f 2 and f ^ are weighted averages of a l l of the sub-market r e l a t i v e frequencies. I n p a r t i c u l a r , may be based on some sub-market r e l a t i v e frequencies where f i r m has preempted, and f ^ may be based on some sub-market r e l a t i v e frequencies where n e i t h e r f i r m F^ or f i r m F^ has preempted.) This same reasoning leads us to expect < ^1 a n c^ ^23 ^ "^ 3 ^ n a s u b - I I i a r k e t which has been a c t i v e l y preempted by f i r m F2« F i n a l l y , we should note that the theory of preemption does not lead to any p a r t i c u l a r p r e d i c t i o n s about what the d i s c r e p a n c i e s between TT ^ and f ^ , and f 2> and TT,^  an& £ 3 should be i f e i t h e r f i r m F^, f i r m Y^, or both f i r m F^ and f i r m F 2 have engaged i n preemptive behavior w i t h i n some sub-market. We may summarize the above d i s c u s s i o n . b r i e f l y as f o l l o w s : 1) i f TT^ > f ^ , TT^ 2 < r 2 ' ^13 < f i r m F ^ w i l l be s a i d to have engaged i n the preemption of the sub-market; 2) i f TT > ^21 < ^1' U23 < ^3' ^ r m ^ 2 w"*"^ ^ e s a i a t o have engaged i n the preemption of the sub-market. Note once again that TT . and f. are based upon the complete mx 1 set of outcomes of the process i n the sub-market and, market A, r e s p e c t i v e l y , from t ^ to the present. However, the theory of preemption a l s o makes p r e d i c t i o n s about what the d i f f e r e n c e s between state, dependent p r o b a b i l i t i e s and the r e l a t i v e frequencies of each f i r m ' s s t o r e s i n the sub-market should - 175 -be i n each.time p e r i o d between t ^ and the present i f a given f i r m has preempted i n the market, These p r e d i c t i o n s w i l l c o n s t i t u t e a second set of c r i t e r i a f o r determining i f the s t a t e dependent p r o b a b i l i t i e s are c o n s i s t e n t w i t h one or more f i r m s haying engaged i n preemption i n the sub-market. (We should n o t e . t h a t . t h i s second set of c r i t e r i a i s not independent of the f i r s t set.) We s h a l l regard the second set of c r i t e r i a as p r o v i d i n g us with- an a d d i t i o n a l check on the consistency of our estimates of the s t a t e dependent p r o b a b i l i t i e s w i t h the p r e d i c t i o n s of the theory, and as p r o v i d i n g us w i t h v a l u a b l e i n s i g h t s i n t o the nature of the process over time. Consider f i r s t the i m p l i c a t i o n s of a p o s i t i v e discrepancy between T T ^ ( t ) and f (t) i n each time period between t ^ and the present, where j^^(t) represents our estimate of the s t a t e dependent p r o b a b i l i t y based on the outcomes of the s t a t e dependent process between t ^ ( t h e : i n i t i a l c o n d i t i o n ) and some poin t i n time t , and f - ^ u ( t ) represents the r e l a t i v e frequency of f i r m F,'s s t o r e s i n sub-market A at time t . Let us assume 1 u that our i n i t i a l c o n d i t i o n c o n s i s t s of a f i r m F, s t o r e and a competitive f r i n g e s t o r e . I f f i r m F^ e s t a b l i s h e d a s t o r e i n the sub-market at time t ^ such that s t a t e B^ p r e v a i l e d , then our estimate of a t time t ^ would be 1 and C*^) = Z ^ 3 ' ^ n u s » ^1 1 ^ 2 ^ ^ l u ^ t 2 ^ ' Suppose that at time t ^ , f i r m F^ and f i r m F^ e s t a b l i s h s t o r e s i n the sub-market such that s t a t e B^ p r e v a i l e d . Then, ^ 2 . 1 ^ 3 ^ = 2 / 3 > 3/5 = f ^ C * ^ ) * I f f i r m F, continues to act upon the i n c e n t i v e to preempt by e s t a b l i s h i n g s t o r e s i n the sub-market when a p r o f i t a b l e opportunity e x i s t s and-when i t i s faced w i t h . s t a t e B^, then there w i l l be a tendency f o r Tr^Ct) to exceed f ^ (t) i n every time p e r i o d . Thus, T?_,('t) > f- (t) f o r a l l t w i l l be c o n s i s t e n t w i t h f i r m F. 11 l u 1 having preempted i n the sub-market. However, we w i l l not r e q u i r e t h i s - 176 -i n e q u a l i t y to h o l d f o r ' a l l t i n order t o - c l a i m that the s t a t e dependent p r o b a b i l i t i e s are c o n s i s t e n t w i t h a preemptive process., Due to the small number of- observations on the process i n the f i r s t few time p e r i o d s , the i n e q u a l i t y may i n i t i a l l y be i n the opposite d i r e c t i o n f o r the f i r s t s e v e r a l time p e r i o d s , and then change over' to the t h e o r e t i c a l l y p r e d i c t e d d i r e c t i o n f o r the remaining time p e r i o d s . We s h a l l t h e r e f o r e only r e q u i r e that the i n e q u a l i t y hold f o r "almost a l l t " . For the same reasons, -^22^) > f 2 u ( t ) f o r almost a l l t w i l l be c o n s i s t e n t w i t h f i r m F^ having preempted i n the sub-market. Consider next the i m p l i c a t i o n s of d i s c r e p a n c i e s between -n^(t) and f„ (t) and 7r,~(t) and f„ ( t ) . Due to the f a c t that we have not assumed 2u 13 3u that a l l f i r m s are i d e n t i c a l and due to the unpredictable behavior of competitive f r i n g e f i r m s , our p r e d i c t i o n s regarding the expected signs of these d i s c r e p a n c i e s must be weak. A l l that we are able to say i s that i f f i r m F, i s a strong preemptor. i n the sub-market, then the opportunity f o r f i r m F 2 or the competitive f r i n g e to e s t a b l i s h s t o r e s i n the sub-market, such that they would be faced w i t h s t a t e B- .,. w i l l not o f t e n present i t s e l f . Hence, t r ^ C t ) < f 2 u ( t ) a n d ^ 1 3 ^ < ^ 3 u ^ a l m o s t 3 1 1 fc w o u ± d he c o n s i s t e n t w i t h f i r m F^ having preempted i n the sub-market. However, we w i l l not regard i i ^ 2 ( t ) < > f 2 u ^ o r ^ 1 3 ^ > ^ 3 u ^ a l m o s t a ^ t a s implying that f i r m F^ has not preempted i n the. sub-market., provided that 7?,,(t) > f - ^ u ( t ) f o r almost a l l t . These same c o n s i d e r a t i o n s w i l l apply to our i n t e r p r e t a t i o n of the d i s c r e p a n c i e s between H^tt) and f-j^OO and Tr0-(.t) and f„ ( t ) . 23 3u F i n a l l y , we note that the theory of preemption does not make p r e d i c t i o n s about what the d i s c r e p a n c i e s between TT^,(t) and f ^ ^ ( t ) , ^ 3 2 ^ ^ -177 -and f 2 u ( t ) , and TT^^Ct) and f ^ C t ) should be i f e i t h e r f i r m F^, f i r m F2? °r both f i r m F^ and f i r m Y^ have preempted i n the sub-market. Having now discussed the t e s t i n g procedure f o r s t a t e dependence, and what we s h a l l regard as evidence of preemptive behavior, we may now proceed to a t e s t of s t a t e dependence i n the Vancouver sub-market, 5.6.5 Tes t i n g f o r State Dependence i n the Vancouver Sub-market Our t e s t f o r the ex i s t e n c e of s t a t e dependence i n a given sub-market i s l a r g e l y based upon estimates of the s t a t e dependent p r o b a b i l i t i e s . As pointed out i n the previous sub-sections, i n order to o b t a i n such estimates, i t i s necessary to know the sequence i n which the s t o r e s were e s t a b l i s h e d i n the sub-market. Hence, i n order, to conduct the t e s t f o r s t a t e dependence i n the Vancouver sub-market, we had to r e c o n s t r u c t the s e q u e n t i a l development of the supermarket i n d u s t r y i n . Vancouver i n the f o l l o w i n g way: Our i n i t i a l set of observations c o n s i s t e d of the cross s e c t i o n a l l o c a t i o n pattern, of supermarkets i n the Vancouver sub-market as of the. f i r s t quarter of 1978. Using the Vancouver, Lower. Fraser V a l l e y , and B.C. c i t y d i r e c t o r i e s we were able to look up our l i s t of supermarket addresses, year by year, u n t i l we found the opening date f o r each s t o r e . We were e s s e n t i a l l y i n t e r e s t e d i n p l o t t i n g the h i s t o r y of the sto r e s themselves. Thus, i f a s t o r e was opened at some time t by f i r m F g , and i f that s t o r e changed ownership at some f u t u r e time, then the h i s t o r y of that s t o r e was i n t e r p r e t e d as re p r e s e n t i n g two observations on the s t o c h a s t i c process. ( I t i s f o r t h i s reason that the number of observations on the s t o c h a s t i c process i n Vancouver exceeds the number of s t o r e s i n operation i n Vancouver i n 1978.) We traced the s e q u e n t i a l , development of the supermarket i n d u s t r y as f a r back as .1940. There were two supermarkets - 178 -i n operation i n 1940 which are s t i l l i n operation today, and these two supermarkets w i l l represent our i n i t i a l c o n d i t i o n . A f t e r 1940, 89 s t o r e s were e s t a b l i s h e d , and the r e f o r e we have 89 observations on the s t o c h a s t i c process which gave r i s e to the current c o n f i g u r a t i o n of supermarkets i n the Vancouver sub-market. F i n a l l y , we note that the p o p u l a t i o n i n the Vancouver sub-market i n 1941 was 338,648, which compares to a Vancouver sub-market po p u l a t i o n of 674,731 i n 1976, I t i s important to point'out that our.reconstructed. h i s t o r y of the supermarket i n d u s t r y i n the Vancouver sub-market excludes a l l those s t o r e s which opened and closed p r i o r to the f i r s t quarter of... 1978.. The reason f o r t h i s i s that many of these stores would probably f a l l , i n t o the convenience s t o r e or neighborhood grocer category, even though they bore the name of a f i r m which i s today considered to be a supermarket chain f i r m . Supermarkets as we know them today, and as represented by the S t a t i s t i c s Canada combination s t o r e d e f i n i t i o n , are a f a i r l y recent phenomenon, 20 d a t i n g back to the 1940's. In order to avoid b i a s i n g our s e q u e n t i a l data by i n c l u d i n g the h i s t o r y of a l a r g e number of stores which are not supermarkets, and i n the absence of s i z e data on these s t o r e s , we chose to exclude a l l s t o r e s opened and closed, p r i o r to the f i r s t quarter of 1978 from our a n a l y s i s . A f t e r o b t a i n i n g the opening date i n f o r m a t i o n on each, supermarket i n the Vancouver sub-market, we u t i l i z e , the procedure f o r e s t i m a t i n g the s t a t e dependent p r o b a b i l i t i e s discussed i n sub-section 5.6,3,. That i s , we p l o t each s t o r e i n the sequence i n which i t was opened, n o t i n g as we go along the s t o r e s which the p l o t t e d s t o r e had boundaries w i t h . This i n f o r m a t i o n i s then summarized i n a contingency t a b l e , Table XX, and, i n t u r n , - 179 -t r a n s l a t e d i n t o a matrix of s t a t e d e p e n d e n t . p r o b a b i l i t i e s , m a t r i x (5,18). -TABLE XX CONTINGENCY TABLE OF STATE DEPENDENT FREQUENCIES F s F k cf B s 7 2 1 \ 0 3 1 B c f 1 0 0 V B c f 4 5 4 k cf 1 1 2 B ,B. s k 5 4 4 B , B. , B j. s k cf 14 13 16 Note: Only 88 (instead of 89) observations appear i n Table XX sinc e one f i r m F^ s t o r e was e s t a b l i s h e d such that i t had boundaries w i t h no other f i r m ' s s t o r e s . - 180 -cf (5.18) cf B ,B s c f \ ' B c f B ,B. s k B s ' B k ' B c f .700 .000 1.000 .308 .250 .384 .326 .200 .100 ,750 .000 .384 .250 ,308 ,302 .250 .000 ,308 .500 .308 .372 The f i r s t t h i n g to.note i s that f i r m F^'s stor e s have been placed i n the competitive f r i n g e category.. This d e c i s i o n was made, i n p a r t , on the b a s i s of the very small number of f i r m F 's stor e s which were e s t a b l i s h e d J o i n the Vancouver sub-market ( i . e . t h r e e ) . In a d d i t i o n , the d i m e n s i o n a l i t y of both the contingency t a b l e and s t a t e dependent p r o b a b i l i t y m a t r i x would have been d r a s t i c a l l y increased had we not combined the f i r m F and o competitive f r i n g e c a t e g o r i e s . Second, i t i s c l e a r that we cannot use the contingency t a b l e XX as p r e s e n t l y constructed, i n order to do a c h i -square t e s t of the n u l l hypothesis of s t a t e dependence. The reason i s that only f i v e of the twenty-one c e l l s of t a b l e XX would have expected frequencies greater than f i v e , We must th e r e f o r e combine v a r i o u s s t a t e s i n order to increase the expected frequencies i n as many c e l l s of the contingency t a b l e a s . p o s s i b l e . Since we have already discussed our'reasons f o r combining .; p a r t i c u l a r s t a t e s of neighbor r e l a t i o n s i n sub-section 5.6.4, we s h a l l r e f r a i n from reproducing that d i s c u s s i o n here, and ins t e a d provide - 181 -a summary of. our c a l c u l a t i o n s i n the form of a r e v i s e d contingency t a b l e and r e v i s e d s t a t e dependent p r o b a b i l i t y m a t r i x . -TABLE XXI REVISED CONTINGENCY TABLE OF STATE DEPENDENT FREQUENCIES F s \ cf W c f 11 7 5 W B c f 1 4 3 Bcf / Bs' Bk / Bs'V Bcf 20 17 20 cf .217 .375 .351 We s h a l l now use the in f o r m a t i o n contained i n Table XXI i n order to conduct the chi-square t e s t of the n u l l hypothesis of s t a t e dependence, The expected frequencies f o r each c e l l of Table XXI are c a l c u l a t e d on the b a s i s of the r e l a t i v e frequency, of each f i r m ' s s t o r e s i n B.C. These r e l a t i v e f requencies, which again are our estimates of the s t a t e (5.19) B / B ,B s s cf W c f B / B ,B. / B ,B. ,B cf s k s k c f ,478 .125 .351 .305 ,500 ,298 - 182 r-independent p r o b a b i l i t i e s f o r t h i s t e s t , are l i s t e d i n Table XXII, TABLE XXII RELATIVE FREQUENCIES OF EACH. FIRM'S. STORES IN B.C, Firm R e l a t i v e Frequency Canada Safeway (Firm F g) 87/336 « ,2589285 K e l l y Douglas (Firm F^) 114/336 z .3392858 Competitive Fringe ( c f ) 135/336 Z .4017857 The c a l c u l a t i o n of the chi-square s t a t i s t i c y i e l d s the f o l l o w i n g : M j (a - e ) 2 2 2 2 r c K ml mlJ _ (11 - 5.9554) (7 - 7.8036) (5 - 9.2410) S e . 5.9554 7.8036 9,2410 m=l i = l mi (1 - 2.0714) 2 . (4 - 2.7143) 2 (3 - 3.2143) 2 2.0714 2.7143 3.2143 (20 - 14.7589) 2 (17 - 19.3393) 2 (20 - 22.9018) 2 14.7589 19.3393 22.9018 = 4.2730948 + .0827532 + 1.9463349 + .5541652 + .6090057 + .0142875 + 1.8611908 + .2829639 + .367676 = 9.991472, This r e s u l t i s s i g n i f i c a n t at the 5% l e v e l since i t l i e s i n the 5% t a i l of a chi-square d i s t r i b u t i o n w i t h four degrees of freedom. Hence, we may - 183 -accept the hypothesis that the a ^ are s i g n i f i c a n t l y d i f f e r e n t from the e ^ . This i m p l i e s , that the process g i v i n g r i s e to our observations a . i s c o n s i s t e n t w i t h a s t a t e dependent p r o b a b i l i s t i c process based on mx the TT .. Our next task i s to determine i f our estimates of the s t a t e mx dependent p r o b a b i l i t i e s are c o n s i s t e n t w i t h one or more fir m s having engaged i n preemptive behavior i n the Vancouver sub-market. 5.6.6 Comparative A n a l y s i s of State. Dependent P r o b a b i l i t i e s and  R e l a t i v e Frequencies f o r the Vancouver Sub-market Our f i r s t set of c r i t e r i a f o r determining i f the s t a t e dependent p r o b a b i l i t i e s are c o n s i s t e n t w i t h one or more firms, having engaged i n preemptive behavior, i n the Vancouver sub-market i n v o l v e s comparisons between our estimates of the state, dependent p r o b a b i l i t i e s i n m a t r i x (5.19) and the r e l a t i v e frequencies i n Table XXII. Looking at the f i r s t row of m a t r i x (5.19), we discover the f o l l o w i n g : .478 > .2589285 = f. s .305 < .3392858 = f^, .217 < .4017857 = f ,v . cf A l l three of these i n e q u a l i t i e s are c o n s i s t e n t w i t h f i r m F g having preempted i n the Vancouver sub-market (see sub-section 5.6.4). Looking at .the second row of matrix (5.19), we a l s o f i n d that .500 > .3392858 = f. k .125 < .2589285 - f s .375 < .40.17857 * f _ . cf Again, a l l three of these i n e q u a l i t i e s are c o n s i s t e n t w i t h f i r m F., having 11 = 12 = 13 = - 184 -preempted i n the Vancouver sub-market. Thus, on the b a s i s of these c r i t e r i a , . i t would appear that two.firms have engaged i n preemptive behavior i n the Vancouver sub-market, f i r m F and f i r m F, . We s h a l l s delay the f u r t h e r i n t e r p r e t a t i o n of these r e s u l t s u n t i l a f t e r we have determined i f the s t a t e dependent p r o b a b i l i t i e s s a t i s f y our second set of c r i t e r i a f o r d e s i g n a t i n g a given f i r m as a preemptor. In sub-section 5.6.4, we e s t a b l i s h e d that i f f i r m F i s to be s designated as a preemptor, then TT ( t ) should exceed f (t) f o r almost XX s a l l t , where f (t) represents the r e l a t i v e frequency of f i r m F 's sto r e s s s i n the Vancouver sub-market at some time t . In a d d i t i o n , we noted that •rr 1 0(t) < f, (t) and 7r.._(-t) < f ..(t) f o r almost a l l t would be c o n s i s t e n t ±/ k l j cr w i t h f i r m . F g having preempted i n the sub-market. A s i m i l a r set of c r i t e r i a was e s t a b l i s h e d f o r being able to designate f i r m F^ as a preemptor., most importantly T i ^ C t ) > f^Ct) f ° r almost a l l t . In Table X X I I I , we compare the annual estimates of the s t a t e dependent p r o b a b i l i t i e s w i t h the annual r e l a t i v e f requencies. - 185 -TABLE XX I I I ANNUAL COMPARISONS OF 7? . AND f,. FOR THE VANCOUVER SUB-MARKET ml lu 11 s "12 f k "13 f c f 1941 1/1 > 2/3 0 0 0 < 1/3 1942 2/2 > 3/4 0 0 0 < 1/4 1947 3/3 > 4/5 0 0 0 : 1/5 1949 6/8 > 7/10 2/8 > 2/10 0 < : 1/10 1954 6/8 > 7/11 2/8 > 2/11 0 : 2/11 1955 6/10 > 7/16 3/10 » • 4/16 1/10 < : 5/16 1956 6/10 > 9/19 3/10 > • 4/19 1/10 < 6/19 1957 6/11 > 9/23 4/11 > • 7/23 1/11 <<7/23 1958 6/11 > 11/27 4/11 > • 9/27 1/11 < 7/27 1959 6/11 > 13/33 4/11 > 10/33 1/11 < 10/33 1960 4/12 = 13/39 4/12 . 13/39 2/12 < 13/39 1961 6/12 > 15/43 4/12 > 14/43 2/12 < 14/43 1962 6/12 > 16/47 4/12 < 17/47 2/12 < 14/47 1963 8/15 > 20/55 4/15< 18/55 3/15 < 17/55 1964 9/17 > 21/59 5/17 < 21/59 3/17 < 17/59 1965 9/19 > 23/65 6/19 < 22/65 4/19 < 20/65 1966 9/20 > 24/67 7/20 > 23/67 4/20 < 20/67 1967 9/20> 24/69 7/20 > 23/69 4/20 < 22/69 1968 9/20 > 26/72 7/20 > 24/72 4/20< 22/72 1969 10/21 >: >27/75 7/21 > 25/75 4/21 < 23/75 1970 10/21> 28/78 7/21 > 26/78 4/21 < 24/78 1971 11/22 > 30/81 7/22 < 27/81 4/22 <; .24/81 1973 11/22 > 31/83 . 7/22 < 27/83 4/22^ 25/83 1974 11/22 > 32/85 7/22 < 28/85 4/22 < 25/85 1975 11/22 > 33/87 7/22 < 28/87 4/22 < 26/87 1976 11/22 > 33/88 7/22 = 28/88 4/22 < 27/88 1977 11/22 > 33/89 7/22 < 29/89 4/22 < 27/89 1978 11/23 > 33/91 7/23 < 29/91 5/23< 29/91 T 186 -*22 h *21 f s "23 1941 0 0 0 < 2/3 0 < 1/3 1942 0 0 0 3/4 0 < 1/4 1947 0 Q 0 < 4/5 0 < 1/5 1949 0 < 2/10 0 7/10 0 < 1/10 1954 0 < 2/11 0 < 7/11 0 < 2/11 1955 0/1 < 4/16 0 < 7/16 1/1 > 5/16 1956 0/1 < 4/19 0 < 9/19 1/1 > 6/19 1957 6/1 < 7/23 0 < 9/23 1/1 > 7/23 1958 1/2 > 9/27 0 < 11/2.7 1/2 > 7/27 1959 2/3 > 10/33 0 < 13/33 1/3 > 10/33 1960 3/4 > 13/39 0. < 13/39. 1/4 < 13/39 1961 3/4 > 14/43. 0 < 15/43 1/4 < 14/43 1962 3/4 > 17/47 0 16/47 1/4 < 14/47 1963 3/4 > 18/55 0 < 20/55 1/4 < 17/55 1964 4/5 > 21/59 0 < 21/59 1/5 < 17/59 1965 4/6 > 22/65 0 < 23/65 2/6 > 20/65 1966 4/6 > 23/67 0 < 24/67 2/6 > 20/67 1967 4/7 > 23/69 0 < 24/69 3/7 > 22/69 1968 4/7 > 24/72 0 < 26/72 3/7 > 22/72 1969 4/7 > 25/75 0 < 27/75 3/7 > 23/75 1970 4/7 > 26/78 0 < 28/78 3/7 > 24/78 1971 4/7 > 27/81 0 < 30/81 3/7 > 24/81 1973 4/8 > 27/83 1/8 < 31/83 3/8 > 25/83 1974 4/8 > 28/85 1/8 < 32/85 3/8 > 25/85 1975 4/8 > 28/87 1/8 < 33/87 3/8 > 26/87 1976 4/8 > 28/88 1/8 < 33/88. 3/8 > 27/88 19.77 4/8 > 29/89 1/8 < 33/89 3/8 > 27/89 1978 4/8 > 29/91 1/8 < 33/91 3/8 > 29/91 - 187 -The most, important r e s u l t i n Table XXIII i s that Tr^^(t) exceeds• f ( t ) i n every ::tiroe p e r i o d . This c l e a r l y e s t a b l i s h e s f i r m F g as a. preemptor i n the Vancouver sub^market. We a l s o f i n d that from 1958 to 1978, •n^Ct) > fj ^ C t ) . This r e s u l t supports the conc l u s i o n that f i r m F^ began preempting i n the Vancouver sub-market i n 1958. Also of i n t e r e s t i s the f a c t that ir.^.^Ct.) < f ^ ( t ) and T r 2 i ^ t ) < ^ s^ t) "*"n e v e r Y t i m e p e r i o d . These r e s u l t s are c o n s i s t e n t w i t h f i r m F and f i r m F having engaged i n S K preemptive behavior i n the Vancouver sub-market r e s p e c t i v e l y . F i n a l l y , we note that i n 10 out of the 25 time periods l i s t e d i n the t a b l e i n which f i r m F^ had stor e s i n the Vancouver sub-market, T? 2 < f^> and i n 10 out of the 28 time periods l i s t e d i n the t a b l e , TT„„ < f These l a s t r e s u l t s 23 c f are weak, and given the t e n t a t i v e nature of the i m p l i c a t i o n s which we were able to draw regarding the signs of these d i s c r e p a n c i e s i n sub-s e c t i o n 5.6.4, we cannot regard these r e s u l t s as c o n s t i t u t i n g evidence against the p r o p o s i t i o n that f i r m F and f i r m F,_ have preempted i n the s tc Vancouver sub-market. 5.6.7 I n t e r p r e t a t i o n of the Comparative A n a l y s i s of State Dependent  P r o b a b i l i t i e s and R e l a t i v e Frequencies We have seen i n the previous sub-section that our two sets of c r i t e r i a f o r e v a l u a t i n g the s t a t e dependent p r o b a b i l i t i e s lend strong support to the p r o p o s i t i o n that both f i r m F and f i r m F have preempted S K. i n the Vancouver sub-market. In t h i s s u b - s e c t i o n , we s h a l l consider i n somewhat more d e t a i l when and where i t appears that f i r m F and f i r m F, s & est a b l i s h e d , themselves as preemptors. To f a c i l i t a t e our d i s c u s s i o n , we have sketched a very rough map of,;the Vancouver sub-market, which appears as Figure 12 below. We s h a l l d e f i n e the western sector of the Vancouver sub-market as c o n s i s t i n g Figure 12 - 189 -of the C i t y of Vancouver, while the eastern sector w i l l be defined as c o n s i s t i n g of Burnaby, Port Moody, Port Coquitlam, Coquitlam, and New Westminster. Between the years 1941 and 1 9 4 9 . i n c l u s i v e , firm•F. e s t a b l i s h e d s i x stores i n the Vancouver sub-market, f i v e of which were l o c a t e d i n the western s e c t o r . These, s i x s t o r e s represent s i x cases where f i r m F g e s t a b l i s h e d stores i n the Vancouver sub-market such that each s t o r e only had boundaries w i t h other f i r m F s t o r e s . In a d d i t i o n , i n 1949, s i x of s the seven e x i s t i n g s t o r e s i n the western sector were owned by f i r m F . We a l s o f i n d that the p o p u l a t i o n i n the western sec t o r had increased from 275,353 to 344,833 between 1941 and 1951, or by 25.23%. Between 1950 and 1962 i n c l u s i v e , f i r m F e s t a b l i s h e d nine new st o r e s i n the Vancouver s sub-market, but none of these s t o r e s had boundaries only w i t h f i r m F g s t o r e s or w i t h f i r m F g and competitive f r i n g e s t o r e s at the time when they were opened, and only four of these n i n e . s t o r e s were opened i n the western s e c t o r . (The population of the western sector increased from 344,833 to 384,522 between 1951.and 1961., or by 11.51%, w h i l e the popu l a t i o n of the eastern, sector increased from 108,190 to 175,764, or by 62.46%.) Then, between 1963 and 1971 i n c l u s i v e , f i r m F e s t a b l i s h e d fourteen new st o r e s i n the Vancouver sub-market, and eleven of these sto r e s were i n the western s e c t o r . In a d d i t i o n , of these fourteen new s t o r e s , f i v e s t o r e s had boundaries only w i t h f i r m F st o r e s or w i t h f i r m • s. F s and competitive f r i n g e s t o r e s , and a l l : f i v e were, lo c a t e d i n the western sector... The po p u l a t i o n of the. western sector increased: from 384,522 to 426,256 between 1961 and 1971, or by 10,85%, F i n a l l y , from 1972 to 1978, f i r m Fs. only opened three new st o r e s i n the Vancouver sub-market, and a l l - 190 -three had at l e a s t one boundary w i t h a s t o r e owned by f i r m F -.. Between the years 1958 and 1964 i n c l u s i v e , f i r m F^ opened f i v e of the seven sto r e s i n the Coquitlam-Port Coquitlam-Port Moody s e c t i o n of the eastern s e c t o r . Three of these f i v e s t o r e s were e s t a b l i s h e d such that each s t o r e only had boundaries w i t h other f i r m F^ s t o r e s , w h i l e one of the f i v e s t o r e s was e s t a b l i s h e d such that i t had boundaries w i t h f i r m F^ and competitive f r i n g e s t o r e s . These f i v e s t o r e s represented twenty-four percent of the t o t a l number of f i r m F, sto r e s i n the Vancouver sub-k market i n 1964, w h i l e the po p u l a t i o n of Coquitlam-Port Coquitlam-Port Moody represented only 9.53% of the t o t a l Vancouver sub-market pop u l a t i o n . In a d d i t i o n , between the years 1956 and 1966, the po p u l a t i o n of Coquitlam-Port Coquitlam-Port Moody had increased from 28,145 to 59,058, 21 or by 109.83%. From 1965 to 1978, f i r m F f c e s t a b l i s h e d an a d d i t i o n a l eight s t o r e s i n the Vancouver sub-market., but none, of these st o r e s had boundaries only w i t h stores owned by f i r m F^ or w i t h s t o r e s owned by f i r m F and the competitive f r i n g e . A l l of t h i s suggests that f i r m F g preempted i n the western sector of the Vancouver sub-market between 1941 and 1949 and between 1963 and 1971, w h i l e f i r m F^ preempted i n the eastern s e c t o r of the Vancouver sub-market between 1958 and 1964. Further support f o r t h i s c o n c l u s i o n i s obtained when one considers that by 1971, 21 out of 30 (or 70%) of f i r m F_'s sto r e s were l o c a t e d i n the western s e c t o r , while 12 out of 21 (or 57%) of f i r m F^'s stor e s were lo c a t e d i n the eastern sector by 1964. This breakdown gains added s i g n i f i c a n c e when i t i s r e a l i z e d that the eastern s e c t o r represented only 33.76% of the Vancouver sub-market p o p u l a t i o n i n 1966, while the western sector represented 62.85% of the Vancouver sub-market pop u l a t i o n i n 1971. - 191 -In summary, the past: three s e c t i o n s of t h i s chapter have shown the f o l l o w i n g : 1) we may r e j e c t the n u l l hypothesis of random f i r m ownership and random neighbor r e l a t i o n s f o r the Vancouver sub-market; 2) we may accept the n u l l hypothesis of s t a t e dependence f o r the Vancouver sub-market; 3) we accept the p r o p o s i t i o n that both f i r m F and f i r m F, have S RC preempted i n the Vancouver sub-market. In the next s e c t i o n of t h i s chapter, we consider.an extension of t h i s a n a l y s i s . 5.7 An Extension 5.7.1 Redefining a Supermarket In s e c t i o n s 5.3 - 5.6, our a n a l y s i s was based on. d e f i n i n g a l l of a given f i r m ' s s t o r e s as supermarkets i f the mean ground f l o o r area of a l l of the st o r e s owned, by that f i r m exceeds 10,000 square.feet, and i f they are capable of being the d e s t i n a t i o n of a consumer's weekly grocery shopping t r i p i n that they stock the goods l i s t e d i n the S t a t i s t i c s Canada combination store d e f i n i t i o n . This d e f i n i t i o n allowed f o r the p o s s i b i l i t y of s i g n i f i c a n t v a r i a n c e of s t o r e s i z e around the mean f o r a given f i r m . That i s , some of a given f i r m ' s s t o r e s may be i n the neighborhood of f i v e or s i x thousand square f e e t i n s i z e , w h i l e others may be i n the neighborhood of t h i r t y - t w o thousand square f e e t i n s i z e . Should both, extremes of the s i z e d i s t r i b u t i o n of stores owned by a given f i r m be c l a s s i f i e d as supermarkets? There are.two ways'in which to approach,this question, The f i r s t approach i s to argue that the st o r e s in'the. extremes of the s i z e d i s t r i b u t i o n of supermarkets do.not r e a l l y belong to the same i n d u s t r y and hence do not serve the same market. The st o r e s which belong to the - 192 -supermarket industry- would p r i m a r i l y serve a market c o n s i s t i n g of the weekly grocery shopping t r i p s of consumers.- The stores,which belong to what might be c a l l e d the convenience s t o r e or neighborhood grocer i n d u s t r y would p r i m a r i l y serve a market c o n s i s t i n g of the mid-week grocery shopping t r i p s of consumers. These mid-week shopping t r i p s might be occasioned by consumers exhausting t h e i r i n v e n t o r i e s of p a r t i c u l a r goods before the time of the: weekly shopping t r i p . Supermarkets and neighborhood grocers are f u r t h e r d i f f e r e n t i a t e d by the vector of goods which are s o l d , w i t h supermarkets u s u a l l y s t o c k i n g a wider assortment of goods. By p r o v i d i n g a wider assortment of.goods, supermarkets hope to convey the image of being able to s a t i s f y a m u l t i p l i c i t y of consumer needs (both food and nonfood), thus p r o v i d i n g an i n c e n t i v e f o r consumers to bypass neighborhood 22 grocers on t h e i r way to do t h e i r weekly grocery shopping. A second approach i s to argue that the st o r e s which belong to the neighborhood grocer i n d u s t r y are apt to charge higher p r i c e s than super-markets. For example, neighborhood grocers may have a lower volume of s a l e s or lower turnover r a t e than supermarkets, and might charge higher p r i c e s i n order to compensate.for t h e i r lower p r o f i t a b i l i t y . Some neighborhood grocers might be faced w i t h higher wholesale costs than supermarkets due to t h e i r i n a b i l i t y to buy i n volume or maintain t h e i r own warehouse f a c i l i t i e s . This.would be the r e s u l t of the f a c t that most neighborhood grocers are independently owned. I f we have included some neighborhood grocers i n our a n a l y s i s of the p r e v i o u s . s e c t i o n s , and i f they charge markedly higher p r i c e s than supermarkets, then.we w i l l haye v i o l a t e d the assumption that allowed us to f i n d the market area boundaries by the perpendicular b i s e c t o r - l e a s t d i s t a n c e method.' The simplest: way of - 193 -c o r r e c t i n g f o r t h i s d i s t o r t i o n would be to e l i m i n a t e the smaller s t o r e s from the analysis,- (However., w e would continue to maintain our assumption that the remaining st o r e s charge the same v e c t o r of p r i c e s . ) I t seems, then, that our a n a l y s i s based on our e a r l i e r d e f i n i t i o n of a supermarket, may be subject, to a degree of measurement e r r o r . However, these e r r o r s may be of l i t t l e s i g n i f i c a n c e i f our e m p i r i c a l r e s u l t s remain f a i r l y r obust.with respect to a l t e r n a t i v e d e f i n i t i o n s : of a supermarket. In order to see i f t h i s i s . t h e case, we s h a l l conduct t e s t s of the n u l l hypotheses.of randomness and s t a t e dependence on the b a s i s of a r e v i s e d sample which conforms to the f o l l o w i n g d e f i n i t i o n of a supermarket: D e f i n i t i o n 4. A r e t a i l food store, w i l l be designated a supermarket i f i t has a ground f l o o r area i n excess, of 12,000 square, f e e t , and i f i t i s capable of being the d e s t i n a t i o n of a consumer's weekly grocery shopping t r i p i n that i t stocks the goods l i s t e d i n the S t a t i s t i c s Canada combination store d e f i n i t i o n . The a n a l y s i s of t h i s s e c t i o n w i l l exclude the extreme t a i l of the s i z e d i s t r i b u t i o n of supermarkets c o n s i s t i n g of st o r e s l e s s than 12,000 square f e e t i n s i z e . Again, we use the c o n s t i t u e n t sub-markets of the GVRD as the b a s i s f o r our t e s t s , and the procedure f o r t e s t i n g , the n u l l hypotheses of random f i r m ownership, random neighbor r e l a t i o n s , and s t a t e dependence o u t l i n e d i n s e c t i o n s 5.3 - 5.6 w i l l be fo l l o w e d . 5.7.2 The Test of Random: Firm Ownership In t h i s s u b-section, we s h a l l r e t e s t the n u l l ' hypothesis of random f i r m ownership f o r the GVRD and sub-markets us i n g a r e v i s e d data s e t . We begin by p r o v i d i n g l i s t s of the r e v i s e d s t o r e ownership f i g u r e s f o r B r i t i s h - 194 -Columbia and the GVRD sub-markets in-Tables'XXIV and XXV. TABLE XXIV STORE OWNERSHIP BY FIRM - BRITISH COLUMBIA (STORE SIZE > 12,000 SQUARE FEET) Firm Number of Stores i n B.C Canada Safeway (Firm F ) Overwaitea (Firm F ) o K e l l y Douglas (Firm F ) Super Valu . (Corporate) Super Valu (Franchise) Shop Easy (Corporate) Shop Easy (Franchise) Economart (Corporate) 87 34 34 21 4 1 3 [87] [46] [35] [47] [ 7 ] [22] [ 3] To t a l 184 [247] - 195 -TABLE XXV STORE OWNERSHIP BY FIRM - GVRD SUB-MARKETS J. • (STORE SIZE > 12,000 SQUARE FEET) Sub-markets Firm Vancouver •Delta-Surrey North Shore Richmond Canada Safeway (Firm F ) 30 [30] 7 [ 7] 4 [ 4] 5 [ 5] s Overwaitea (Firm F.) 3 [ 3] 2 [ 2] 0 [ 0] 1 [ 1] o K e l l y Douglas (Firm F.) 14 [22J 4 [ 9] 4 [ 6] 1 [ 2] Competitive Fringe ( c f ) " IGA 8 [16] 2 [ 3] 1 [ 2] 0 [ 2] Woodward's 4 [ 4] 1 [ 1] 1 [ 1] 1 [ 1] \ Stong's 1 [ 2] 0 [ 0] 3 [ 3] 0 [ 0] High-Low 1 [ 2J 2 [ 2] 0 [ 0tl 0 [ 0] T o t a l 61 [79] 18 [24] 13 ;:[.16] .8 [11] The f i g u r e s which appear i n brackets represent the st o r e ownership f i g u r e s based on our e a r l i e r d e f i n i t i o n of a supermarket, D e f i n i t i o n 1. Again, we make the assumption that the number of competitive f r i n g e s t o r e s i n B.C. i s p r o p o r t i o n a l to po p u l a t i o n . Since there are twenty-five competitive f r i n g e s t o r e s i n the GVRD which are greater than 12,000 square f e e t i n s i z e , and since the GVRD represents approximately 44% of the po p u l a t i o n i n B.C., we have assumed that there are 57 competitive f r i n g e s t o r e s i n B.C. - 196 -Looking b r i e f l y at the r e v i s e d s t o r e ownership f i g u r e s , we see that Safeway's r e p r e s e n t a t i o n remains unchanged. A l l of t h e i r s t o r e s are greater than 12,000 square fee t i n s i z e . Overwaitea !s r e p r e s e n t a t i o n i n the GVRD has not?, changed, but i t s r e p r e s e n t a t i o n i n B.C. has been reduced by twelve. The most dramatic change has occurred i n the K e l l y Douglas s t o r e ownership f i g u r e s . Approximately 44.7% of i t s st o r e s i n B.C. are l e s s than or equal to 12,000 square fee t i n s i z e , and 92% of these st o r e s are f r a n c h i s e s . In a d d i t i o n , we note that K e l l y Douglas' r e p r e s e n t a t i o n i n the GVRD has been reduced by 41%. We are now prepared to t e s t the n u l l hypothesis that our r e v i s e d observations on f i r m ownership were generated by an independent s t o c h a s t i c process based on the f . , where the f . are l i s t e d i n Table XXVI. TABLE XXVI RELATIVE FREQUENCIES OF EACH FIRM'S STORES IN B.C. (STORE SIZE > 12,000 SQUARE FEET) Firm R e l a t i v e Frequency Canada Safeway (Firm F g) 87/241 ~ .3609958 Overwaitea (Firm F ) 34/241 z .1410788 o K e l l y Douglas (Firm F f c) 63/241 ~ .2614108 Competitive Fringe ( cf,) 57/241 ~ .2365146 In Table XXVII, we provide d e s c r i p t i v e measures f o r the marginal d i s t r i b u t i o n s of f i r m ownership generated f o r each sub-market and the four sub-markets combined (the GVRD). - 197 -' ' " • ~ TABLE XXVII t--.., -MARGINAL DISTRIBUTION DESCRIPTIVE MEASURES BY FIRM FOR THE. GVRD AND CONSTITUENT SUB-MARKETS (STORE SIZE > 12,000 SQUARE FEET) Firm F g Firm F q Firm F^ Competitive Fringe GVRD n. .1 36.091 4.7313 1.4.165 3.5767 . ,26.333 4.5269 23.411 4.3311 Vancouver n. l 22.159 3.7723 8.5400 2.8511 15.897 3.5416 14.404 3.3514 D e l t a -Surrey 6.5380 2.021 2.4800 1.460 4.6680 1.8163 4.3140 1.7924 North Shore n i ^ 4.683 1.7865 1.924 1.303 3.374 1.5769 3.019 1.5418 ; Richmond n. 2.904 1.3837 1.194 1.0244 1.997 1.2388 1.905 1.2106 A f t e r t r a n s l a t i n g our randomly generated d i s t r i b u t i o n s of f i r m j ownership i n t o d i s t r i b u t i o n s of the q u a n t i t y / (n. - n . ) /n. , and J . , i u i u i u ' 1=1 using the in f o r m a t i o n contained i n Tables XXV and XXVII, we o b t a i n the 2 f o l l o w i n g r e s u l t s from doing the X t e s t of t h e . n u l l hypothesis. - 198 -TABLE XXVIII RESULTS OF THE X 2 TESTS OF THE NULL HYPOTHESIS OF RANDOM FIRM OWNERSHIP BY GVRD AND SUB-MARKETS (STORE SIZE > 12,000 SQUARE FEET) GVRD Firm F (46 - 36.091)' 36.091 2.7205752 Firm F Firm F. k (6 - 14.165)' 14.165 (23 - 26.333)' 26.333 4.7064754 .4218618 cf (25 - 23.411)' 23.411 .1078519 Y (n. - n . ) 2 / n . = 7.9567643 (7.8290561)' i = l % of I ( n \ - n . ) 2 / n . d i s t r i b u t i o n to r i g h t of 7.9567643 = 5.3% i = l Vancouver Firm F (30 - 22.159)' 22.159 2.7745512 Firm F Firm F, (3 - 8.54) 8.54 3.5938641 (14 - 15.897)' 15.897 .2263703 c f (14 - 14.404)' 14.404 .0113312 I (n.-n.) /n. = 6.6060976 (6.7929822) i = l 1 1 1 v ~ - 2 -% of ) (n.-n.) /n. d i s t r i b u t i o n to r i g h t of 6.6060976 = 9.8% i = l The f i g u r e i n parentheses i s the r e s u l t obtained by c a l c u l a t i n g the c h i -j 2 square s t a t i s t i c , I (n - e ) /e i = l I X 1 - 199 -Table XXVIII (continued) Delta-Surrey Firm F s Firm F Firm F. o k cf (7 - 6.538) 2 (2 - 2.48) 2 (4 - 4.668) 2 (5 - 4.314) 2 6.538 2.48 4.668, 4.314 .0326466 .0929033 .0955922 .1090857 j j ( n . - n . ) 2 / n = 1=1 .3302278 (.3889026) j % of Y (n. - n . 1=1 1 1 ) 2 / n ± d i s t r i b u t i o n , t o r i g h t of. .3302278 North Shore = 93.4% Firm F s Firm F Firm F. o k cf ( 4 - 4.683) 2 (0 - 1.924) 2 (4 - 3.374) 2 (5 - 3.019) 2 4.683 1.924 3.374 3.019 .0996133 1.924 .1161458 1.2998877 j I (n - n ) 2 / n 1=1 = 3.4396468 (3.2480664) % of 7 ( n . - n . ) 2 / n . d i s t r i b u t i o n to r i g h t of 3.4396468 = 35.0% i = l Richmond Firm F Firm F o Firm F, cf (5 - 2.904) 2.904 1.5128154 (1-1.194) 1.194 .0315209 (1 - 1.997) .1.997 .4977511 (1 - 1.905) 1.905 .4299343 - 200 -Table XXVIII (continued) j I ( n . - n . ) 2 / n . = 2.4720217 (2.5490303) i = l j r ~ - 2 -% of ) (n. -n.) /n. d i s t r i b u t i o n to r i g h t of 2.4720217 = 55.8% 1=1 1 1 1 Looking at the r e s u l t s i n Table XXVIII, we see that we may once again r e j e c t the n u l l hypothesis that our observations were generated by an independent s t o c h a s t i c process based, on the f f o r the Vancouver sub-market and the GVRD. These r e s u l t s are s i g n i f i c a n t at the 10% l e v e l . We cannot r e j e c t the n u l l h y p o t h e s i s . f o r the Richmond, North Shore, or Delta-Surrey sub-markets. Before i n t e r p r e t i n g these r e s u l t s and comparing them w i t h the r e s u l t s of our e a r l i e r t e s t s , we s h a l l t e s t the n u l l hypothesis that our observations on the neighbor r e l a t i o n s i n the GVRD and i t s c o n s t i t u e n t sub-markets were generated by an independent s t o c h a s t i c process based on the f.. x 5.7.3 The Test of Random Neighbor R e l a t i o n s In t h i s sub-section, we s h a l l r e t e s t the n u l l hypothesis of random neighbor r e l a t i o n s f o r the GVRD and con s t i t u e n t , sub-markets u s i n g data based on our r e v i s e d d e f i n i t i o n of a supermarket. In Table XXIX, we report our c a l c u l a t i o n s of common boundaries and boundaries between stores owned by d i f f e r e n t firms f o r the GVRD and i t s c o n s t i t u e n t sub-markets . - 201 -TABLE XXIX COMMON BOUNDARIES AND BOUNDARIES BY FIRM - GVRD AND SUB-MARKETS (STORE SIZE > 12,000 SQUARE FEET Boundary GVRD. Vancouver D e l t a -Surrey North Shore Richmond B ss 43 [ 31] 31 [ 20] 4 [ 4] 3 [ 2]. 5 [ 5] B oo 0 [ 0] 0 [ 0] 0 [ 0] 0 [ 0] 0 [ 0] B k k 9 [ 25] 5 [ 8] 2 [13] 2 [ 4] 0 [ 0] B c f 15 [ 8] 9 [ 7] 3 [ 1] 3 [ 0] 0 [ 0] B i l 163 [244] 105 [162] 33 [45] 16. [22] 9 [15] T o t a l 230 [308] 150 [197] 42 [63] 24 [28] 14 [20] The f i g u r e s appearing i n brackets i n Table XXIX represent our e a r l i e r common boundary c a l c u l a t i o n s . Some major d i f f e r e n c e s i n the two sets of f i g u r e s are apparent.. F i r s t , l o o k i n g at the common boundary c a l c u l a t i o n s f o r Vancouver, we note that f i r m F 's common boundaries s increase from 20 to 31 (or by 55%) wh i l e f i r m F^.'s common boundaries d e c l i n e from 8 to 5 (or by 37.5%). In Delta-Surrey sub-market, f i r m F 's common boundaries, remain the same, w h i l e f i r m F, 's common boundaries s k d e c l i n e from 13 to 2 (or by 84.6%). In the GVRD as a whole, f i r m F_'s common boundaries increase from 31 to 43 (or by 38.7%), w h i l e f i r m F^'s d e c l i n e from 25 to 9 (or by 64%). These r e s u l t s were not unexpected given that f i r m F g's r e p r e s e n t a t i o n i n the GVRD was not changed by our r e d e f i n i t i o n of a supermarket, while f i r m F^'s r e p r e s e n t a t i o n f e l l by 41%. - 202 -In Table XXX, we provide d e s c r i p t i v e measures f o r the marginal d i s t r i b u t i o n s of common boundaries generated f o r each sub-market and f o r the GVRD. These d i s t r i b u t i o n s are based on the random d i s t r i b u t i o n s of f i r m ownership which were generated f o r the t e s t s i n the previous sub-s e c t i o n . TABLE XXX MARGINAL COMMON BOUNDARY DISTRIBUTION DESCRIPTIVE MEASURES BY FIRM FOR THE GVRD AND SUB-MARKETS (STORE SIZE > 12,000 SQUARE FEET) GVRD 1 1 B ss 29.58 9.0098 B oo 4.506 3.0399 B kk 16.041 6.6629 B cf .12.451 5.7826 B i l 167.42 7.4842 Vancouver B. . i i 19.771 2.971 10.338 8.434 108.49 7.9011 2.5645 5.3485 4.7439 6.3112 D e l t a -Surrey i i 5.583 3.9926 .797 1.198 2.777 2.7769 2.442 2.4394 30.401 3.1116 North Shore B. . i i 3.108 2.7882 .506 .970 1.566 1.914 1.291 1.7015 17.529 2.3522 Richmond i i 1.818 .323 .925 .798 10.136 1.9992 .77283 1.3781 1.2872 1.7597 - 203 -A f t e r t r a n s l a t i n g our randomly generated d i s t r i b u t i o n s of common boundaries i n t o d i s t r i b u t i o n s of the q u a n t i t y J I 1=1 (B. . -B..Y xx 1 1 xx 1 ^ i l - 5 ^ '  B i l and using the in f o r m a t i o n contained i n Table XXIX and Table XXX, we 2 ob t a i n the f o l l o w i n g r e s u l t s from doing the X t e s t of the n u l l h y pothesis: TABLE XXXI RESULTS OF THE X 2 TESTS OF THE NULL HYPOTHESIS OF RANDOM NEIGHBOR RELATIONS BY GVRD AND SUB-MARKETS (STORE SIZE > 12,000 SQUARE FEET) GVRD (B. -B )' ss ss ss (B -B r oo oo oo ( B k k - \ k ) 2 ( B c f - B c f ) 2 B kk B cf < B i I - B i I > ' 'II (43 - 29.58) 29.58 6.0884516 (0 - 4.506)' 4.506 4.506 (9 - 16.041) 2 (15 - 12.451) 2 16.041 12.451 3.0905605 .5218376 (163- 167.42)' 167.42 .1166909 (B.. - B . . ) ' XX XX 1=1 B. % of XX j ( B 1 I - B i l > + — — — — = 14.323538 B (B.. - B . . r XX XX i = l B, XX i i + d i s t r i b u t i o n to r i g h t of B i i 14.323538 = 23.8% - 204 -Table XXXI (continued) Vancouver (B - B ) 2 ss ss (B -B )2 oo oo ( B k k - \ k ) 2 ( B c f - § c f ) 2 < B i I - 5 i I ) 2 B ss 1 B oo \ k 5 c f 5 i l (31 - 19.771) 2 CO - 2.971) 2 (5 - 10.338) 2 (9 - 8.434) 2 (105 - 108.49) 2 19.771 2.971 10.338 8.434 108.49 6.3775448 2.971 2.7562627 .0379838 .1122693 3" I i = l (B.. - B . . ) ' XX XX % of XX j < B i I - B i I > + — — — — = 12.255059 B i l „ (B. . - B . . ) ' y xx. xx (.1=1 B. . XX ^ i l - 5 ! ^ + d i s t r i b u t i o n to r i g h t of B i i 12.255059 = 31.7% Delta-Surrey (B -B ) 2 ss ss (B. —B )2 oo oo ( B k k " \ k ) 2 - ( B c f - 5 c f ) 2 • ' ( B i l - B i l ) 2 B ss B oo B k k B c f B i l (4 - 5.583) 2 (0 - .797) 2 (2 - 2.777) 2 ( 3 - 2.442) 2 (33 - 30.401)' 5.583 .797 2.777 2.442 30.401 .4488427 .797 .2174034 .1275037 .22219 3 I i = l (B. . -B. . ) ' X I XX XX + — — — — = 1.8129398 B i i % of j ( B x . - B . . ) 2 1=1 B.. xx 1 ~. ^ 2 + — — , — d i s t r i b u t i o n to r i g h t of B i i 1.8129398 = 92.4% - 204a -North Shore (B - B ) 2 ss ss (B -B )2 oo oo ( B k k " \ k ) 2 ( B c f - 5 c f ) 2 ( B i l - § i l ) 2 B ss B oo B k k 5 c f § i l (3 - 3.108) 2 ( 0 - i 5 Q 6 ) 2 (2 - 1.566) 2 ( 3 - 1.291) 2 (16 - 17.529) 2 3.108 .506 1.566 1.291 17.529 .0037528 .506 .1202785 2.26235 .1333699 J I i = l (B. . - B . .)' IX XX XX ' • ( B i I - B i I ) + — — — — = 3.0257413 B i i % of j ( B . . - B . . ) 2 i = l B.. xx ( B i I B i I ) -1 : d i s t r i b u t i o n to r i g h t of B i l 3.0257413 = 76.0% - 205 -Table XXXI (continued) Richmond (B - B ) 2 (B. o-B ) 2 ss ss oo oo B ss B oo (5 - 1.818) 2 (0 - .323) 2 1.818 5.5693752 ,323 ,323 "kk (0 - .925)' .925 .925 ( B c f - B c f ) 2 ( B i I - § i I ) 2 ' B cf B i l (0 - .798) 2 (9 - 10.136) 2 ,798 ,798 10.136 .127318 (B... -B. .)' i i i i i = l B.. i i < B i I - B i I > + — — — — = 7.7426932 B i i % of (B. . -B. .) ' i i i i i = l B. i i ( B i I " B i l > •4 d i s t r i b u t i o n to r i g h t of B i l 7.7426932 = 30.8% Looking a t the r e s u l t s i n Table XXXI, we see that we would accept the n u l l hypothesis that our observations on neighbor r e l a t i o n s were generated by an independent s t o c h a s t i c process based on the f ^ f o r the GVRD and a l l of i t s c o n s t i t u e n t sub-markets. Our acceptance of the n u l l hypothesis f o r the GVRD and the Vancouver sub-market comes as somewhat of a s u r p r i s e , and i t s i m p l i c a t i o n s w i l l be explored, i n the next sub-s e c t i o n . 5.7.4 • I n t e r p r e t a t i o n of the Test Results of Randomness In the past two sub-sections, we have shown that the n u l l hypothesis of random f i r m ownership can be r e j e c t e d f o r the GVRD. and the Vancouver sub-market, but that the n u l l hypothesis of random neighbor r e l a t i o n s cannot be r e j e c t e d f o r the GVRD and the Vancouver sub-market. In t h i s - 206 -sub-section, we s h a l l i n t e r p r e t these r e s u l t s and compare them w i t h the r e s u l t s of our e a r l i e r t e s t s of random f i r m ownership and random neighbor r e l a t i o n s . Let us begin by examining the t e s t r e s u l t s f o r the. GVRD. As i n our e a r l i e r t e s t s , the l a r g e s t r e l a t i v e discrepancy between observed and mean generated f i r m ownership occurs f o r f i r m F , wh i l e the second l a r g e s t r e l a t i v e discrepancy occurs f o r f i r m F . Thus, i t appears that f i r m F 's s s stores are r e l a t i v e l y concentrated i n the GVRD, wh i l e f i r m F 's stor e s • o are r e l a t i v e l y unconcentrated, given, the signs of these d i s c r e p a n c i e s before squaring. However, we do not f i n d the ordering of r e l a t i v e d i s c r e p a n c i e s between observed and mean generated common boundaries to be unchanged. From Table XXXI. we see that the l a r g e s t r e l a t i v e discrepancy occurs f o r f i r m F , the second l a r g e s t f o r f i r m F , and the t h i r d l a r g e s t f o r f i r m F^. No longer does the l a r g e s t r e l a t i v e discrepancy occur f o r c o m p e t i t i v e . f r i n g e stores (9.3103833 i n our e a r l i e r c a l c u l a t i o n s and .5218376 i n our c a l c u l a t i o n s based on the re v i s e d sample). When we compare the magnitude of the r e l a t i v e d i s c r e p a n c i e s between observed and mean generated, common boundaries appearing i n Tables XVIII and XXXI, we f i n d that the t o t a l c o n t r i b u t i o n of the r e l a t i v e d i s c r e p a n c i e s f o r f i r m F. , f i r m F , and f i r m F, i s not s o k markedly d i f f e r e n t . I t seems as i f the very small r e l a t i v e discrepancy between observed and mean generated boundaries f o r competitive f r i n g e s t o r e s has been the main cause f o r accepting the n u l l hypothesis of random neighbor r e l a t i o n s . f o r the GVRD when the t e s t i s based on the re v i s e d sample. Let us now examine the t e s t r e s u l t s f o r the Vancouver sub-market. Again, the ord e r i n g of r e l a t i v e d i s c r e p a n c i e s between observed and mean - 2 0 7 -generated f i r m ownership i s the same i n Table XI as i t i s i n Table XXVIII. Again, i t appears that f i r m F^'s stores are r e l a t i v e l y concentrated i n the Vancouver sub-market and that f i r m F 's st o r e s are r e l a t i v e l y o unconcentrated, and these r e s u l t s are s i g n i f i c a n t . However, the ord e r i n g of r e l a t i v e d i s c r e p a n c i e s between observed and mean generated common boundaries has changed con s i d e r a b l y , as has t h e i r magnitude. The l a r g e s t r e l a t i v e discrepancy occurs f o r f i r m F G ( 6 . 3 7 7 5 4 4 8 compared to 3 . 5 7 4 8 2 1 2 ) , the second l a r g e s t f o r f i r m F Q ( 2 . 9 7 1 compared to 3 . 7 9 3 ) , and the t h i r d l a r g e s t f o r f i r m F F C ( 2 . 7 5 6 2 6 2 7 compared to 9 . 1 8 0 4 1 1 4 ) . I t seems that the la r g e r e d u c t i o n i n f i r m F^'s r e p r e s e n t a t i o n i n the Vancouver sub-market caused by our r e d e f i n i t i o n of supermarket has l e d to a s i g n i f i c a n t r e d u c t i o n i n mean generated common boundaries f o r f i r m F^, and hence a reduced r e l a t i v e discrepancy between observed and mean generated common boundaries. What seems c l e a r at t h i s point i s that t e s t s based on our. r e v i s e d supermarket d e f i n i t i o n have not a l t e r e d our conclusions regarding c o n c e n t r a t i o n , but have had some i n f l u e n c e on our conclusions regarding the nature and extent of c l u s t e r i n g . Since we have r e j e c t e d the n u l l hypothesis of random f i r m ownership f o r the Vancouver sub-market, but accepted the n u l l hypothesis of random neighbor r e l a t i o n s f o r the Vancouver sub-market, our conclusions regarding whether our observations were generated by an independent s t o c h a s t i c process based on the f of Table XXVI must n e c e s s a r i l y be ambiguous. I t i s th e r e f o r e appropriate to proceed w i t h the t e s t of s t a t e dependence f o r the Vancouver sub-market. - 208 -5.7.5 The Test of State Dependence In t h i s sub-section, we f o l l o w the procedure discussed i n sub-s e c t i o n 5.6.3 i n order to t e s t the n u l l hypothesis of s t a t e dependence for the Vancouver sub-market. On the b a s i s of t h i s procedure and our r e v i s e d data s e t , we o b t a i n a r e v i s e d contingency t a b l e of s t a t e dependent frequencies and a new matrix of s t a t e dependent p r o b a b i l i t i e s . TABLE XXXII CONTINGENCY TABLE OF STATE DEPENDENT FREQUENCIES (STORE SIZE >.12,000 SQUARE FEET) F s F k •cf B s 8 0 3 B k 0 0 . 1 B c f 1 0 0 B s ' B c f 11 6 6 B. ,B _ k ..cf 1 2 0 B ,B. s k 1 1 0 B , B, , B c s k cf 9 ' 6 6 - 209 -cf (5.20) cf B ,B , s cf V Bcf B ,B. s k B ,B. ,B s k cf ,727 .000 1.000 .478 .333 .500 ,428 ,000 273 .000 1.000 ,000 ,261 ,667 ,500 ,286 .000 ,261 .000 .000 ,286 As f o r our e a r l i e r t e s t of s t a t e dependence, r e s t r i c t i o n s on the c h i -square t e s t make, i t necessary to combine as many s t a t e s as p o s s i b l e . A f t e r combining s t a t e s , we o b t a i n contingency t a b l e XXXIII and s t a t e dependent p r o b a b i l i t y m a t r i x (5.21). - 210 -TABLE XXXIII REVISED CONTINGENCY TABLE OF STATE DEPENDENT FREQUENCIES (STORE SIZE > 12,000 SQUARE FEET) F s F k cf B / B ,B . s s cf 19 6 9 B. / B. , B . k k cf 1 2 1 B / B ,B. / B ,B. ,B cf s k s k cf 11 7 6 (5.21) B / B ,B r s s cf B k / B k ' B c f B £>/B ,B /B ,B',B cf s k s k cf .559 ,250 .458 k ,176 ,500 .292 cf ,265 ,250 ,250 In Table XXXIV, we provide a l i s t of the r e l a t i v e frequencies of each f i r m ' s stores i n B.C. These r e l a t i v e frequencies c o n s t i t u t e our estimates of the s t a t e independent p r o b a b i l i t i e s f o r the t e s t of s t a t e dependence. - 211 -TABLE XXXIV RELATIVE FREQUENCIES OF EACH FIRM'S STORES IN B.C. (STORE SIZE > 12,000 SQUARE FEET) ^ i r m R e l a t i v e Frequency Canada Safeway (Firm F ) 87/241 * .3609958 K e l l y Douglas (Firm F f e) 63/241 ~ .2614108 Competitive Fringe ( c f ) 91/241 * .3775934 Using the i n f o r m a t i o n contained i n contingency t a b l e XXXIII, and using our estimates of the s t a t e independent p r o b a b i l i t i e s i n order to c a l c u l a t e the expected frequencies f o r each c e l l of Table XXXIII, we are able to perform the chi-square t e s t of s t a t e dependence. y I ( a m i " e m i ) = (19 - 12,2739) 2 (6 - 8.888) 2 (9 - 12.8381) 2 12.2739 8.888 12.8381 m=l i = l e . mi + (1 - 1-444) 2 (2 - 1.0456) 2 (1 - 1.5104) 2 1.444 1.0456 1.5104 (11 - 8.6639) 2 (7 - 6.2739) 2 (6 - 9.0622) 2 8.6639 6.2739 9.0622 = 3.6859043 + .938405 + 1.1474447 + .1365207 + .8711546 + .1724762 + .6298968 + .084034 + 1.0347452 = 8.7005815. This r e s u l t i s s i g n i f i c a n t at the 10% l e v e l since i t l i e s i n the 10% t a i l of a chi-square d i s t r i b u t i o n w i t h four degrees of freedom. Thus, we may accept the hypothesis that our observations a are s i g n i f i c a n t l y d i f f e r e n t from the e ^ . This i m p l i e s that the process g i v i n g r i s e to our - 212 -observations i s c o n s i s t e n t w i t h a s t a t e dependent p r o b a b i l i s t i c process based on the -IT-... Our next step i s to. determine i f our estimates of the mx s t a t e dependent p r o b a b i l i t i e s are c o n s i s t e n t w i t h one or more fir m s having preempted i n the Vancouver sub-market. 5.7.6 Comparative A n a l y s i s , of State Dependent P r o b a b i l i t i e s and  R e l a t i v e Frequencies f o r the Vancouver Sub-market I t w i l l be r e c a l l e d that our f i r s t set of c r i t e r i a f o r determining i f the s t a t e dependent p r o b a b i l i t i e s are c o n s i s t e n t w i t h one or more fir m s having preempted i n the Vancouver sub-market i n v o l v e s comparisons between our estimates of the s t a t e dependent p r o b a b i l i t i e s i n matrix (5.21) and the r e l a t i v e frequencies i n Table XXXIV. Looking at the f i r s t row of matrix (5.21), we f i n d that .559 > .3609958 = f s .176 < .2614108 = f. k .265 < .3775934 = f c . cf A l l of these i n e q u a l i t i e s are c o n s i s t e n t w i t h f i r m F g having preempted i n the Vancouver sub-market (see s e c t i o n 5.6.4). Looking at the second row of matrix (5.21), we f i n d that TT„„ = .500 > .2614108 = f, 22 k ir _ - = .250 < .3609958 = f 21 s TT00 = .250 < .3775934 = f ,. 23 cf These three i n e q u a l i t i e s are c o n s i s t e n t w i t h f i r m F7 having preempted i n the Vancouver sub-market. However, given that there i s only a t o t a l of *12 = TT12 = - 213 -four cases i n the three c e l l s of the second row of matrix (5.21), we do not a t t a c h much s i g n i f i c a n c e to t h i s r e s u l t . Indeed, i f we look back at our i n i t i a l contingency t a b l e before we combined s t a t e s , we f i n d that f i r m never e s t a b l i s h e d a s t o r e i n the sub-market such that i t only had boundaries w i t h other f i r m F s t o r e s . Furthermore, only two out of the K. f i f t e e n f i r m F sto r e s were e s t a b l i s h e d i n the sub-market such that they only had boundaries w i t h f i r m F^ and competitive f r i n g e s t o r e s . We cannot regard f i r m F^ as being a preemptor of the Vancouver sub-market on the b a s i s of such weak evidence. Our f i r s t set of c r i t e r i a , t h e r e f o r e , supports the co n c l u s i o n that e l i m i n a t i n g f i r m F^'s smaller s t o r e s from the a n a l y s i s r e s u l t s i n e l i m i n a t i n g f i r m F^ as being one of the preemptors of the Vancouver sub-market. In sub-section 5.6.4, we e s t a b l i s h e d that i f f i r m F i s to be s designated a preemptor, then TT^^(t) should exceed fgCt) f ° r almost a l l t . In a d d i t i o n , we noted that T r ^ C t ) < a n c* ^ 1 3 ^ < ^ c f ^ a ^ i n o s t a l l - t would be c o n s i s t e n t w i t h f i r m F g having preempted i n the sub-market. In Table XXXV, we compare the annual estimates of the s t a t e dependent p r o b a b i l i t i e s w i t h the annual r e l a t i v e frequencies. - 214 -TABLE XXXV ANNUAL COMPARISONS OF TT . AND f. FOR THE VANCOUVER SUB-MARKET mx xu (STORE SIZE.> 12,000 SQUARE FEET) u l l f s TT^ "13 f c f 1942 1/1 > 2/3 0 < 1/3 1947 2/2 > 3/4 0 < 1/4 1949 5/5 > 6/7 0 < 1/7 1954 5/6 > 6/8 1/6 < 2/8 1955 5/7 > 6/9 2/7 < 3/9 1956 7/9 > 8/11 2/9 < 3/11 1957 7/10> 8/12 3/10- 4/12 1958 8/12 = 10/15 1/12 > 1/15 3/12 < 4/15 1959 10/17 < 12/20 1/17 > 1/20 6/17 > 7/20 1960 10/20 > 12/25 2/20 < 3/25 8/20 = 10/25 1961 11/21 > 14/28 2/21 < 4/28 8/21 > 10/28 1962 11/21 > 15/31 2/21 < 6/31 8/21 > 10/31 1963 13/23 > 19/35 2/23 < 6/35 8/23 > 10/35 1964 14/25 > 20/38 3/25 < 8/38 8/25 > 10/38 1965 14/27 > 22/43 4/27 < 9/43 9/27 > 12/43 1966 14/28 < 23/45 5/28 «10/45 9/28 > 12/45 1967 14/28 = 23/46 5/28 < 10/46 9/28 > 13/46 1968 16/31 > 25/49 6/31 < 11/49 9/31 > 13/49 1969 17/32 > 26/51 6/32 < 12/51 9/32 > 13/51 1970 .;' 17/32 > 27/54 6/32 <j 13/54 9/32 > 14/54 1971 18/33 > 29/57 6/33 < 14/57 9/33 > 14/57 1973 18/33 > 30/58 6/33 < 14/58 9/33: > 14/58 1974 19/34 > 31/60 6/34 < 15/60 9/34 > 14/60 1975 19/34 > 32/62 6/34 < 15/62 9/34 > 15/62 1976 19/34 > 32/63 6/34 < 15/63 9/34 > 16/63 1978 19/34 > 32/64 6/34 < 15/64 9/34 < 17/64 - 215 -When.we examine the r e s u l t s i n Table XXXV, we f i n d that TT^Ct) > f ( t ) i n a l l but four of the time periods l i s t e d i n the t a b l e . This r e s u l t confirms our c o n c l u s i o n , based on the f i r s t set of c r i t e r i a , that f i r m F g has preempted i n the Vancouver sub-market. We a l s o f i n d that i n a l l but two. time p e r i o d s , TT-^Ct) < f ^ ( t ) , and t h i s r e s u l t i s co n s i s t e n t w i t h f i r m F_ having preempted i n the sub-market. F i n a l l y , we note that T r ^ ( t ) < f j ( t ) i n nine of the twenty-six time periods l i s t e d i n Table XXXV. This r e s u l t i s weak and we cannot regard i t as c o n s t i t u t i n g evidence f o r or against the p r o p o s i t i o n that f i r m F_ has preempted i n the Vancouver sub-market. When we compare Table XXXV with our e a r l i e r comparisons of -n and f i n Table X X I I I , some i n t e r e s t i n g observations emerge. F i r s t , between 1941 and 1949 i n c l u s i v e , f i r m F_ e s t a b l i s h e d f i v e new st o r e s (as opposed to s i x i n our e a r l i e r a n a l y s i s ) i n the Vancouver sub-market such that a l l of these sto r e s had..boundaries only w i t h s t o r e s owned by f i r m F g or wi t h s t o r e s owned by f i r m F^ and the competitive f r i n g e . Between 1950 and 1962 i n c l u s i v e , f i r m F e s t a b l i s h e d nine new stor e s i n the s Vancouver sub-market, s i x of which had boundaries only w i t h other f i r m F s t o r e s or w i t h f i r m F and competitive f r i n g e s t o r e s . In our e a r l i e r s s a n a l y s i s , ; none of the nine f i r m F g stores opened between 1950 and 1962 had boundaries only w i t h other f i r m F stor e s or w i t h f i r m F and J s s competitive f r i n g e s t o r e s . Between 1963 and 1978 i n c l u s i v e , f i r m F_ opened an . a d d i t i o n a l seventeen sto r e s i n the Vancouver sub-market, eig h t of which had boundaries only w i t h stores owned by f i r m F g or w i t h s t o r e s owned by f i r m F and the competitive f r i n g e . This compares w i t h f i v e out of seventeen i n our e a r l i e r a n a l y s i s . Thus, we see that f i r m F_ es t a b l i s h e d nineteen stores i n the Vancouver sub-market such that each - 216 -store had boundaries only w i t h other f i r m F stor e s or w i t h f i r m F and s s competitive f r i n g e s t o r e s , and f i f t e e n of these nineteen s t o r e s (or 78.95%) were l o c a t e d . i n the western s e c t o r . To put these l a s t f i g u r e s i n p e r s p e c t i v e , we note that the western sector represented only 61.11% of the Vancouver sub-market p o p u l a t i o n i n 1976. A l l of t h i s suggests, that f i r m F g has been engaged i n the continuous preemption of the Vancouver sub-market si n c e 1941, and that i t has p r i m a r i l y focused i t s preemptive l o c a t i o n behavior i n the western sector of the Vancouver sub-market. 5.7.7 Evalua t i o n . o f Test Results Based on. the Revised Sample The m o t i v a t i o n f o r the t e s t s performed i n t h i s s e c t i o n has been to see i f the e m p i r i c a l r e s u l t s based on our f i r s t d e f i n i t i o n of a super-market remain f a i r l y robust w i t h respect to an a l t e r n a t i v e d e f i n i t i o n of a supermarket. By changing the number of stores: which we designate as supermarkets, we not only change the r e l a t i v e frequencies of each fi r m ' s s t o r e s i n B.C. and i t s c o n s t i t u e n t sub-markets, but we a l s o change the set of s p a t i a l r e l a t i o n s h i p s between s t o r e s . These a l t e r e d s p a t i a l r e l a t i o n s h i p s can have s i g n i f i c a n t repercussions on our c a l c u l a t i o n s of boundaries, and hence on t e s t s based on the c a l c u l a t i o n of boundaries. In p a r t i c u l a r , under our broad d e f i n i t i o n of the supermarket i n d u s t r y ( i . e . D e f i n i t i o n 1 ) , we may have included a number of s t o r e s i n our sample which do not e f f e c t i v e l y compete w i t h l a r g e r supermarkets. By i n c l u d i n g these smaller s t o r e s i n our common boundary c a l c u l a t i o n s based on the perpendicular b i s e c t o r - l e a s t d i s t a n c e method, a f i r m which owns p r i m a r i l y l a r g e supermarkets w i l l l i k e l y have i t s common boundaries reduced v i s - a - v i s the number of common boundaries i t would - 217 -have i f the smaller s t o r e s were, excluded from the sample. This r e d u c t i o n i n common boundaries could cause us to f a l s e l y accept randomness or to f a l s e l y r e j e c t s t a t e dependence. .Thus, we would expect that under a broad d e f i n i t i o n of the supermarket i n d u s t r y , we would be l e s s l i k e l y to f i n d evidence of preemptive behavior s i n c e the extent of competition would be a r t i f i c i a l l y (and p o s s i b l y erroneously) increased by the i n c l u s i o n of stor e s which are not i n e f f e c t i v e competition w i t h the l a r g e r supermarkets. That i s , the more broadly we de f i n e the i n d u s t r y , the l e s s l i k e l y i t i s that we w i l l f i n d one or two fi r m s dominating i t . Having completed a s e r i e s of t e s t s based on a r e v i s e d sample, i t i s c l e a r that our r e v i s e d supermarket d e f i n i t i o n has i n f l u e n c e d the outcomes of the t e s t s and the conclusions which may be derived from them. F i r s t , we have seen that the r e j e c t i o n of randomness was not complete f o r the Vancouver sub-market. While we were able to r e j e c t the hypothesis of random f i r m ownership, we could not r e j e c t the hypothesis of random neighbor r e l a t i o n s . This s o r t of ambiguity d i d not e x i s t i n our f i r s t s e r i e s of t e s t r e s u l t s based on D e f i n i t i o n 1 of a supermarket. Second, we found that we could s t i l l accept the hypothesis of s t a t e dependence, but w i t h d i f f e r e n t i m p l i c a t i o n s f o r the nature of preemption i n the Vancouver sub-market. I n . p a r t i c u l a r , w h i l e our o r i g i n a l data set supported the p r o p o s i t i o n that f i r m F and f i r m F, preempted i n the Vancouver sub-market,,our r e v i s e d data set supports the p r o p o s i t i o n that f i r m F g i s the s o l e preemptor i n the Vancouver sub-market. The i n a b i l i t y to designate f i r m F^ as a preemptor.is to be expected i n l i g h t of the la r g e r e d u c t i o n i n f i r m F^'s r e p r e s e n t a t i o n i n B.C. and the GVRD occasioned by our r e d e f i n i t i o n of supermarket. In a d d i t i o n , t h i s change i n r e s u l t s i s con s i s t e n t w i t h our expectations regarding the probable - 218 -e f f e c t s of more narrowly d e f i n i n g the supermarket i n d u s t r y , s i n c e the r e d e f i n i t i o n of supermarket enabled us to designate only one f i r m as the preemptor of the Vancouver sub-market• 5 .8 Concluding Remarks In t h i s chapter, we have sought to determine the nature and extent to which the s p a t i a l c o n f i g u r a t i o n of supermarkets i n the Greater Vancouver Regional D i s t r i c t i s c o n s i s t e n t w i t h one or more f i r m s having preempted i n some sub-market of the GVRD. Our s t r a t e g y has c o n s i s t e d of i n i t i a l l y t e s t i n g two hypotheses on the randomness of our observations on f i r m ownership and neighbor r e l a t i o n s i n p a r t i c u l a r sub-markets. R e j e c t i o n of the n u l l hypotheses that our observations were generated by an independent s t o c h a s t i c process based on the-f was i n t e r p r e t e d to mean that some form of s t a t e dependence was r e s p o n s i b l e f o r generating our observations. However, these t e s t s would not permit us to be more p r e c i s e about the s p e c i f i c nature of the s t a t e dependence which was r e s p o n s i b l e f o r generating our data. In p o i n t of f a c t , there are other explanations f o r why we might r e j e c t the hypothesis of randomness, and these explanations/do not n e c e s s a r i l y imply a preemptive l o c a t i o n s t r a t e g y on the p a r t of any f i r m . One explanation f o r why our observations on f i r m ownership and neighbor r e l a t i o n s may be i n c o n s i s t e n t w i t h an independent s t o c h a s t i c process based on the f . i s the i n c e n t i v e which each f i r m has to economize on wholesaling or d i s t r i b u t i o n c o s t s . One of the major co s t s to any f i r m engaged i n food r e t a i l i n g i s the cost of p r o c u r i n g goods to s e l l i n i t s s t o r e s . As the number of stores which a f i r m owns i n c r e a s e s , the q u a n t i t y of goods which must pass through a d i s t r i b u t i o n network a l s o increases. At some p o i n t , i t may pay f o r a f i r m to v e r t i c a l l y - 219 -i n t e g r a t e i t s wholesale and r e t a i l operations i n order to take advantage of the lower costs which would be i n c u r r e d through such i n t e g r a t i o n . (These lower costs might be due to the f i r m ' s a b i l i t y to o b t a i n q u a n t i t y discounts, on the purchase of l a r g e volumes of goods, to r a t i o n a l i z e i t s inventory management, to coordinate product d i s t r i b u t i o n to minimize transport c o s t s , etc.) Thus, the i n c e n t i v e which a f i r m has to economize on. d i s t r i b u t i o n costs might lead i t to,concentrate i t s r e t a i l o u t l e t s i n p a r t i c u l a r sub-markets.. This c o n c e n t r a t i o n could lead to there being more common boundaries between stores, that that f i r m owns than would be expected on the b a s i s of a random a l l o c a t i o n of f i r m ownership. A second explanation f o r why we might r e j e c t the n u l l hypotheses that our observations on f i r m ownership and neighbor r e l a t i o n s were generated by an independent s t o c h a s t i c process based on the f has to do w i t h the i n f l u e n c e that marketing co s t s might have on a f i r m ' s l o c a t i o n s t r a t e g y . In p a r t i c u l a r , . a v i a b l e marketing s t r a t e g y might r e q u i r e a f i r m to have a s i g n i f i c a n t presence.in a p a r t i c u l a r sub-market. In a d d i t i o n , w i t h only a s m a l l number of s t o r e s i n a given sub-market, i t may not pay f o r a f i r m to engage i n a widely based marketing str a t e g y from the p o i n t of view of the net revenues generated by such a s t r a t e g y . Thus, there may be some economies to be r e a l i z e d i n a f i r m ' s marketing costs i f i t concentrates i t s r e t a i l o u t l e t s i n a given sub-market. Both of the above explanations suggest that firms might have i n c e n t i v e s to concentrate stores i n p a r t i c u l a r sub-markets. However, they do not suggest that neighbor r e l a t i o n s would be of much importance to a f i r m ' s l o c a t i o n s t r a t e g y . In f a c t , marketing c o n s i d e r a t i o n s might - 220 -lead a f i r m to d i s p e r s e i t s s t o r e s throughout the sub-market i n s t e a d of concentrating them i n a corner of i t . F i n a l l y , we might ask i f c o l l u s i o n i s a p o s s i b l e e x p l a n a t i o n f o r observing c o n c e n t r a t i o n or c l u s t e r i n g of a f i r m ' s s t o r e s i n a given sub-market. On the face of i t , i t might seem reasonable to b e l i e v e that f i r m s would c o l l u d e i n order to d i v i d e up sub-markets or p a r t s of sub-markets among themselves, and; even to r e s t r i c t p r i c e competition w i t h i n sub-markets. The normal i n c e n t i v e s f o r such behavior are entry prevention, market s t a b i l i t y , higher p r o f i t s through higher p r i c e s and r e d u c t i o n of competition. A major goal of c o l l u s i v e behavior would a l s o be to postpone the time of entry u n t i l the p r o f i t s to be derived from opening a new s t o r e i n the market are greater than zero. However, due to the r e l a t i v e l y low costs of entry i n t o an i n d u s t r y such as the supermarket i n d u s t r y , i t i s l i k e l y t hat non-colluding f i r m s would compete w i t h each o t h e r . f o r the opportunity to e s t a b l i s h a new s t o r e i n the market.if the present value obtained from doing so. i s greater than or equal to zero. Thus, a c o l l u d i n g f i r m would not be able to postpone the time of entry or new s t o r e c o n s t r u c t i o n u n t i l the time when the p r o f i t s a t t r i b u t a b l e to a new s t o r e are greater than zero. I t would s t i l l have to preempt the market i f i t wished to maintain i t s monopoly p o s i t i o n , and i t would, have an i n c e n t i v e to l o c a t e the s t o r e such that i t only had as i t s neighbors other s t o r e s that i t owned. Apart from the f a c t that a c o l l u d i n g f i r m would a l s o have to be a preempting f i r m i f i t wished to maintain i t s monopoly p o s i t i o n i n a given sub-market, there would be no i n c e n t i v e to c o l l u d e i f the c o l l u s i v e agreement would only permit the f i r m to do what i t would do i n the absence of such an agreement. - 221 -Given that there may be a l t e r n a t i v e explanations f o r the r e j e c t i o n of the n u l l hypotheses of random f i r m ownership and random neighbor r e l a t i o n s , we have chosen to regard t e s t s of these hypotheses as performing a screening f u n c t i o n and as p r o v i d i n g i n s i g h t s i n t o how the observed s p a t i a l c o n f i g u r a t i o n o f . s t o r e s d e v i a t e s from a h y p o t h e t i c a l random one, I f we r e j e c t e d these hypotheses f o r a p a r t i c u l a r sub-market, then we proceeded to t e s t f o r the existence of the p a r t i c u l a r type of s t a t e dependence, that i s i m p l i e d by the theory of preemption. The type of s t a t e dependence that we t e s t e d f o r i s one that emphasizes p o t e n t i a l neighbor r e l a t i o n s as being'a key determinant i n a f i r m ' s l o c a t i o n a l choice behavior. Since the most p l a u s i b l e a l t e r n a t i v e explanations of r e j e c t i o n of randomness do not have the same i m p l i c a t i o n s w i t h respect to neighbor r e l a t i o n s that the theory of preemption has, we can e l i m i n a t e these explanations as being, the primary.causes f o r acceptance of the n u l l hypothesis of s t a t e dependence. Now, acceptance of the n u l l hypothesis of s t a t e dependence does not i n and of i t s e l f imply that;preemptive f i r m behavior was r e s p o n s i b l e f o r generating our observations. The.reason i s . because the d i s c r e p a n c i e s between observed and expected number of s t o r e s that each f i r m e s t a b l i s h e s i n the subr-market given d i f f e r e n t s t a t e s of neighbor r e l a t i o n s may be s i g n i f i c a n t , but i n the wrong d i r e c t i o n or w i t h the wrong s i g n . For example, i f the state.dependent p r o b a b i l i t i e s were such that TT-J^ < f ^ , TT^ 2 > f2» and r f ^ > then we would not say t h a t our estimates of the s t a t e dependent p r o b a b i l i t i e s are c o n s i s t e n t w i t h f i r m F^ having preempted i n the sub-market, even though we may have accepted the n u l l hypothesis of s t a t e dependence. The f a c t that the d i s c r e p a n c i e s between observed and expected number of s t o r e s that each f i r m owns i n the submarket - 222 -given d i f f e r e n t s t a t e s of neighbor r e l a t i o n s have the t h e o r e t i c a l l y p r e d i c t e d s i g n s , and t h e . f a c t that we are able to accept the n u l l hypothesis of s t a t e dependence, together c o n s t i t u t e evidence i n support of the p r o p o s i t i o n that preemptive f i r m behavior i s the best e x p l a n a t i o n of the observed s p a t i a l c o n f i g u r a t i o n , of s t o r e ownership and l o c a t i o n . - 223 -FOOTNOTES TO CHAPTER 5 1. Since we wished to have accurate, up-to-date i n f o r m a t i o n on super-market l o c a t i o n s , we found i t necessary to contact p e r s o n a l l y the l o c a t i o n a n a l y s t s f o r s e v e r a l supermarket chain f i r m s i n order to o b t a i n a l i s t of supermarket l o c a t i o n s throughout the province of B r i t i s h Columbia. Accurate and complete i n f o r m a t i o n on supermarket l o c a t i o n s was not a v a i l a b l e from such obvious sources as telephone and c i t y d i r e c t o r i e s . In a d d i t i o n , even i f i t were a v a i l a b l e from these sources, i t would be q u i t e time consuming to compile the re q u i r e d i n f o r m a t i o n . 2. The Super Market Industry Speaks,. 1976,- conducted by Research D i v i s i o n , Super Market I n s t i t u t e , Inc. (Chicago, I l l i n o i s ) . 3. Census of Canada, 1971. 4. In a l a t e r s e c t i o n of the.chapter, we report the r e s u l t s of a s e r i e s of l o c a t i o n a l t e s t s which u t i l i z e a r e v i s e d d e f i n i t i o n of a super-market. The r e v i s e d d e f i n i t i o n w i l l r e q u i r e that a r e t a i l food s t o r e be at l e a s t 12,000 square f e e t i n t o t a l ground f l o o r area i n order to be c l a s s i f i e d as a supermarket. 5. Information regarding Canada Safeway's s i t e s e l e c t i o n and approval procedure, as w e l l as a l i s t of i t s supermarket l o c a t i o n s i n B r i t i s h Columbia, were obtained i n a personal i n t e r v i e w w i t h ;the property manager of the Vancouver D i v i s i o n of Canada Safeway, L t d . , February, 1978. - 224 -6. A l i s t of supermarket l o c a t i o n s of Overwaitea and Your Mark-It Food Stores i n B r i t i s h Columbia was provided by a l o c a t i o n a n a l y s t f o r Overwaitea, March, 1978. ' 7. We have obtained our informa t i o n regarding the K e l l y Douglas f r a n c h i s e operation, from, a personal i n t e r v i e w w i t h a l o c a t i o n a n a l y s t f o r K e l l y Douglas, April,..1978. A l i s t of the l o c a t i o n s of a l l K e l l y Douglas owned and franehised. supermarkets was a l s o provided. 8. Information regarding the H.Y..Louie f r a n c h i s e o p e r a t i o n , as w e l l as a l i s t of supermarket l o c a t i o n s i n B r i t i s h Columbia, were su p p l i e d i n a personal i n t e r v i e w w i t h an executive of the H.Y. Louie Company, L t d . , March, 1978. 9. Hoel [1971; 228-229]. 10. Kendall and Stuart [1973; 439-,- 453-454]. 11. I b i d . 12. Walker and Lev [1953; 107].. For a more d e t a i l e d d i s c u s s i o n and d e r i v a t i o n of the chi-square t e s t , see K e n d a l l and Stuart [1963; 355-356], and Ke n d a l l and Stuart [1973; 436-440]. 2 13. Hoel [1971; 108-112]. Choice of the upper t a i l of the X d i s t r i b u t i o n as our c r i t i c a l r e g i o n i s a l s o suggested by the f a c t that the choice of the upper t a i l i s accepted p r a c t i c e when 2 performing the chi-square goodness of f i t t e s t , and the X t e s t and 2 the chi-square t e s t are q u i t e s i m i l a r . (In f a c t , the X t e s t and the chi-square t e s t y i e l d almost i d e n t i c a l r e s u l t s when the und e r l y i n g - 225 -d i s t r i b u t i o n i s multinomial.) 14. For a d i s c u s s i o n of t h i s i n t e r p r e t a t i o n , see Walker and Lev [1953; 107]. 15. This r e s u l t was not unexpected. We would expect f a i r l y r a p i d convergence of the generated d i s t r i b u t i o n to the exact d i s t r i b u t i o n as the number of permutations, increases since each assignment of f i r m ownership to a s t o r e i s based on the same set of f i x e d p r o b a b i l i t i e s . 16. Hoel [1971; 33-35]. 17. In f a c t , as we s h a l l discover i n a l a t e r s e c t i o n of t h i s chapter, i t i s q u i t e p o s s i b l e that a given set.of observations on f i r m ownership of s t o r e s w i l l lead to r e j e c t i o n of the n u l l hypothesis of random f i r m ownership, but w i l l not permit the r e j e c t i o n . o f the n u l l hypothesis of random neighbor r e l a t i o n s . 18. U n l i k e our t e s t i n g procedure f o r the n u l l hypothesis of random f i r m 2 ownership, use of the X t e s t as a t e s t of random neighbor r e l a t i o n s cannot be j u s t i f i e d by i t s c l o s e approximation to the:chi-square t e s t . Other t e s t i n g procedures based on the d i s t r i b u t i o n s of 1 } . (B ... - B . . ) • l i l i i = l + <B ± fB ± I.)- or /- 1 i l i l 1=1 + .l B i l - B i l l or I j B... - B . . | / B . . ^ , i i i i i i i = l + B. -B. I / B . T could have been used, il i l 1 il However, we wished to use a c o n s i s t e n t set of t e s t i n g procedures f o r both n u l l hypotheses of randomness. In a d d i t i o n , we d i d conduct t e s t s - 226 -of the n u l l hypothesis or random neighbor r e l a t i o n s f o r the GVRD using the above a l t e r n a t i v e d i s t r i b u t i o n s , and the r e s u l t s were not q u a l i t a t i v e l y a f f e c t e d by the choice of d i s t r i b u t i o n . In f a c t , our observations were found.to l i e even f u r t h e r to the r i g h t i n the generated d i s t r i b u t i o n s when we used the d i s t r i b u t i o n s l i s t e d above 2 as compared to the d i s t r i b u t i o n of the X s t a t i s t i c . 19. S i e g e l [1956; 104-111]. 20. See Markin [1968; 6-17] and Mueller-and Garoian [1961; 8-15] f o r b r i e f h i s t o r i e s of the development of the supermarket i n d u s t r y i n the United States. Another argument f o r excluding from our a n a l y s i s s t o r e s which no longer e x i s t today i s as f o l l o w s : Suppose a given f i r m has preempted some sub-market A^ at some time i n the past. As these stores become economically and p h y s i c a l l y o b s o l e t e , the preempting f i r m w i l l have an i n c e n t i v e to e i t h e r renovate these stores or replace them w i t h new stor e s at nearby l o c a t i o n s . Thus, even i f we ignore the openings and c l o s i n g s of stor e s which no longer e x i s t i n the market today, we should s t i l l be able to a s c e r t a i n the existence of preemption on the b a s i s of a more l i m i t e d data set c o n s i s t i n g of the s e q u e n t i a l openings of stores which c u r r e n t l y operate i n the sub-market. 21. Some of the stor e s e s t a b l i s h e d i n the Coquitlam-Port Coquitlam-Port Moody s e c t i o n of the eastern sector would a l s o serve people r e s i d i n g i n the Burnaby and New Westminster s e c t i o n of the eastern s e c t o r . The p o p u l a t i o n i n Burnaby-New Westminster increased from 115,410 to 150,049 between 1956 and 1966, or by 30.01%. - 227 -I f consumers engage i n multipurpose, one stop shopping, then there i s a high p r o b a b i l i t y that they w i l l bypass s m a l l e r , neighborhood grocers i n order to d o . t h e i r weekly grocery shopping at supermarkets By shopping at the l a r g e r supermarkets w i t h t h e i r wider assortment of goods, consumers may be maximizing the p r o b a b i l i t y of f i n d i n g the goods which they d e s i r e , and t h i s may r e s u l t u l t i m a t e l y i n maximizing the consumer's u t i l i t y . We do not wish to s p e c i f y a formal model here, but f o r more d e t a i l s , see Baumol and Ide [1956]. - 228 -Chapter 6 SUMMARY AND CONCLUSIONS In t h i s t h e s i s , we have explored the c o n d i t i o n s under which firms w i l l engage i n market preemption as a b a r r i e r to-entry i n a growing, s p a t i a l l y extended market. We developed a model of preemption i n one-dimensional space, and derived the r e s u l t that the e s t a b l i s h e d f i r m has an i n c e n t i v e to preempt the market a t a poin t i n time j u s t p r i o r to the e a r l i e s t date at which a new entrant would f i n d i t p r o f i t a b l e to enter. We a l s o found that t h i s r e s u l t does not depend on the i n f i n i t e competitive f r i n g e assumption or on whether the space i s one-dimensional or two-dimensional. The major focus of the t h e s i s has been on d e r i v i n g , the e m p i r i c a l l y t e s t a b l e i m p l i c a t i o n s of the theory, and t e s t i n g the associated hypotheses. F i r s t , we examined the p r o f i t s i m p l i c a t i o n of the theory, which was stat e d as f o l l o w s : i f an e x i s t i n g f i r m and p o t e n t i a l entrants a n t i c i p a t e that the market w i l l grow.at some time i n the f u t u r e such that a new- p l a n t could be p r o f i t a b l y e s t a b l i s h e d i n that market, and i f the e x i s t i n g f i r m does nothing to block e n t r y , then competition among p o t e n t i a l entrants w i l l lead to a new p l a n t being e s t a b l i s h e d i n the market at a poin t i n time when the present value of that p l a n t i s equal to zero. Due to data l i m i t a t i o n s , we were constrained from performing a rigo r o u s s t a t i s t i c a l t e s t . o f the i d e a l n u l l hypothesis a s s o c i a t e d w i t h t h i s i m p l i c a t i o n . However, we were able to obt a i n i n d i c a t i v e evidence i n support of the hypothesis, that the average p r o f i t s of new supermarkets are negative i n the f i r s t twelve months of ope r a t i o n of these super-markets. The d r i v i n g f o r c e of t h i s r e s u l t was that a r e p r e s e n t a t i v e new - 229 -supermarket, with i t s lower average sales, r e l a t i v e to a r e p r e s e n t a t i v e e s t a b l i s h e d supermarket, could not cover i t s c a p i t a l c o s t s i n i t s f i r s t twelve months of operation. We argued that the lower average sales of new supermarkets were most l i k e l y the product of sto r e s being e s t a b l i s h e d i n the market at dates when the customer de n s i t y was i n s u f f i c i e n t to guarantee p o s i t i v e p r o f i t s f o r these s t o r e s . We next, examined the l o c a t i o n a l i m p l i c a t i o n of the theory of preemption. This i m p l i c a t i o n h e l d that i f there i s an a n t i c i p a t e d increase i n de n s i t y i n a market such that a new p l a n t could be p r o f i t a b l y e s t a b l i s h e d i n that market, and i f the new p l a n t would have as i t s neighbors other p l a n t s ' t h a t an e x i s t i n g f i r m owns, then the e x i s t i n g f i r m w i l l have an i n c e n t i v e to preempt the market. In order to t e s t t h i s i m p l i c a t i o n of the theory f o r a p a r t i c u l a r i n d u s t r y and a p a r t i c u l a r s p a t i a l market, we designed two types of t e s t s . F i r s t , we used cross s e c t i o n data on store ownership and neighbor r e l a t i o n s i n order to determine i f our observations were generated by an independent s t o c h a s t i c process. Using a broad d e f i n i t i o n of what c o n s t i t u t e s a supermarket., we were able to r e j e c t the n u l l hypotheses that our observations were generated by an independent s t o c h a s t i c process based on a p a r t i c u l a r set-of s t a t e independent p r o b a b i l i t i e s f o r the Vancouver sub-market and the GVRD. We then u t i l i z e d time s e r i e s data on the date at which each s t o r e was e s t a b l i s h e d i n the Vancouver sub-market, where that s t o r e was lo c a t e d and which f i r m owned i t i n order to determine i f our data were c o n s i s t e n t w i t h a s t a t e dependent s t o c h a s t i c process. We found that we could accept the n u l l hypothesis of s t a t e dependence f o r the Vancouver sub-market, and that the un d e r l y i n g p r o b a b i l i t i e s of t h i s process were c o n s i s t e n t w i t h there being two - 230 -preempting f i r m s . F i n a l l y , we r e v i s e d our d e f i n i t i o n of what c o n s t i t u t e s a supermarket i n order to see how s e n s i t i v e our e m p i r i c a l r e s u l t s would be to a narrower d e f i n i t i o n of the i n d u s t r y . We discovered that we could only p a r t i a l l y r e j e c t the n u l l hypotheses of randomness f o r the Vancouver sub-market and the GVRD, but that our Vancouver sub-market time s e r i e s data were s t i l l c o n s i s t e n t w i t h a s t a t e dependent p r o b a b i l i s t i c process. We a l s o found that an a n a l y s i s of the underlying p r o b a b i l i t i e s of the s t a t e dependent process supported the p r o p o s i t i o n that only one f i r m preempted the Vancouver sub-market. We concluded that how narrowly we d e f i n e the in d u s t r y w i l l have an impact on the e m p i r i c a l r e s u l t s of our t e s t s , and that t h i s c o n c l u s i o n was supported by the theory and to be expected given our t e s t i n g procedure. Our general c o n c l u s i o n , then, i s that the. theory of preemption provides a c o n s i s t e n t and e m p i r i c a l l y acceptable explanation of the s t r u c t u r e of the supermarket i n d u s t r y i n the Vancouver sub-market of the GVRD. The f a c t t h a t we se l e c t e d the GVRD f o r a n a l y s i s on the b a s i s of data a v a i l a b i l i t y and not on the b a s i s of any preconception as to what the l o c a t i o n a l p a t t e r n i n the GVRD. would be suggests that the theory of preemption should provide an exp l a n a t i o n of supermarket i n d u s t r y s t r u c t u r e and performance i n many other markets as w e l l . In a d d i t i o n , we would expect to f i n d the s t r u c t u r e of other r e t a i l i n d u s t r i e s to be c o n s i s t e n t w i t h preemptive behavior: on the part of the dominant f i r m s . - 231 -BIBLIOGRAPHY Applebaum, W, and Cohen, S. (.I960), " E v a l u a t i n g Store S i t e s and Determining Store Rents",: Economic Geography, 36, 1-35. Applebaum, W. and Cohen, S. (1961), "Dynamics of Store Trading Areas and Market E q u i l i b r i u m " , Annals of the A s s o c i a t i o n of American  Geographers, 51, 73-101. A r c h i b a l d , G.C. and Rosenbluth, G. (1975), "The 'New' Theory of Consumer Demand and M o n o p o l i s t i c Competition", Quarterly J o u r n a l of  Economics, 89, 569-590. Bain, J.S, (1965), B a r r i e r s to New Competition, Cambridge: Harvard U n i v e r s i t y Press. Bain, J.S. (1968), I n d u s t r i a l O r g a n i z a t i o n , New York: John Wiley and Sons. Baron, D.P.- (1972), " L i m i t P r i c i n g and Models of P o t e n t i a l E n t r y", Western Economic J o u r n a l , 10, 298-307. Baron, D.P. (1973), " L i m i t P r i c i n g , P o t e n t i a l Entry, and B a r r i e r s to Entry", American Economic Review, 63, 666-674. Baumol, W.J. and Ide, E.A. (1956), " V a r i e t y i n R e t a i l i n g " , Management  Science, 3, 93-101. Baumol, W.J. (1967), " C a l c u l a t i o n of Optimal Product and R e t a i l e r C h a r a c t e r i s t i c s : The A b s t r a c t Product Approach", J o u r n a l of  P o l i t i c a l Economy, 75, 674-685. Berry, B.J.L. (1958), "Shopping Centers and the Geography of Urban Areas", Ph.D. D i s s e r t a t i o n , U n i v e r s i t y of Washington. Berry, B.J.L. (1963), Commercial S t r u c t u r e and Commercial B l i g h t : R e t a i l  Patterns and Processes i n the C i t y of Chicago, Research Paper No. 85, Department of Geography, U n i v e r s i t y of Chicago. Berry, B.J.L. and G a r r i s o n , W.L. (1958a), "The F u n c t i o n a l Bases of the C e n t r a l Place Hierarchy", Economic Geography, 34, 145-154. Berry, B.J.L. and G a r r i s o n , W.L. (1958b), "A Note on C e n t r a l Place Theory and the Range of a Good"., Economic Geography, 34, 304-311. Berry, B.J.L. and G a r r i s o n , W.L. (1958c), "Recent Developments of C e n t r a l P l a c e Theory", Papers and Proceedings of the Regional Science  A s s o c i a t i o n , 4, 107-120. Berry, B.J.L., Barnum, H.G. and Tennant, R.J. (1962), " R e t a i l L o c a t i o n and Consumer Behavior", Papers and Proceedings of the Regional Science  A s s o c i a t i o n , 9, 65-106. - 232 -Bhagwati, J.N. (1970), "Oligopoly Theory, Entry-Prevention, and Growth", Oxford Economic Papers, 22, 297-310. Brush, J.E. and Gauthier, H.L. (1968), S e r v i c e Centers and Consumer T r i p s : S t u d i e s . i n the P h i l a d e l p h i a M e t r o p o l i t a n F r i n g e ; Research Paper No. 113, Department of Geography, U n i v e r s i t y of Chicago. B u c k l i n , L.P. (1967), "The Concept of Mass i n Intra-urban Shopping", J o u r n a l of Marketing, 31, 37-42. Cohen, S.B. and Lewis, G.K. (1967), "Form and Function i n the Geography of R e t a i l i n g . " , Economic Geography, 43, 1-42. Consumer and Corporate A f f a i r s (1974), Report of the D i r e c t o r of I n v e s t i g a t i o n and Research, Combines I n v e s t i g a t i o n A c t , For  the Year Ended March 31, 1974, Ottawa: Information Canada. Dooley, P.C. (1968), R e t a i l O l i g o p o l y : An E m p i r i c a l Study of the S t r u c t u r e , Conduct and Performance of the Grocery Trade on the P r a i r i e s , Supporting Study No. 3, Studies of the Royal Commission on Consumer Problems and I n f l a t i o n , Regina: Queen's P r i n t e r . E a r l e , W. and Hunt, W. (1970-71-1976-77), Operating R e s u l t s of Food  Chains ( y e a r ) , I t h a c a : C o r n e l l U n i v e r s i t y . Eaton, B.C. and Lipsey, R.G. (1975), "The P r i n c i p l e of Minimum D i f f e r e n t i a t i o n Reconsidered: Some New Developments i n the Theory of S p a t i a l Competition", Review of Economic S t u d i e s , 42, 27-49. Eaton, B.C. and Lipsey, R.G. (1976), "The Theory of Market Preemption: B a r r i e r s to Entry i n a Growing S p a t i a l Market", U n i v e r s i t y of B r i t i s h Columbia, (mimeo). Eaton, B.C. and L i p s e y , R.G. (1977), " S p a t i a l Monopoly, N a t u r a l Monopoly, Pure P r o f i t s , and Land Rents", D i s c u s s i o n Paper No. 265, Department of Economics, Queen's U n i v e r s i t y . Eaton, B.C.. and Lipsey, R.G. (1978), "Freedom of Entry and the Existence of Pure P r o f i t " , Economic J o u r n a l , 88 455-469. Eaton, B.C. and L i p s e y , R.G. (1979), "The Theory of Market Preemption: The P e r s i s t e n c e of Excess Capacity and Monopoly i n Growing S p a t i a l Markets", Economica, (forthcoming). E s p o s i t o , F.F. and E s p o s i t o , L. (1974), "Excess.Capacity and Market S t r u c t u r e " , Review.of Economics and S t a t i s t i c s , 56, 188-195. Garner, B.J'. - (1966), The I n t e r n a l S t r u c t u r e of R e t a i l N u c l e a t i o n s , Northwestern U n i v e r s i t y Studies i n Geography No. 12, Evanston: Northwestern U n i v e r s i t y . -233 -Hay, D.A. (1976), "Sequential Entry and En t r y - D e t e r r i n g S t r a t e g i e s i n S p a t i a l Competition", Oxford- Economic Papers;, 28, 240-257. Hoel, P.G. (1971), I n t r o d u c t i o n to Mathematical S t a t i s t i c s , New York: John Wiley, and Sons, Inc. H o t e l l i n g , H. (1929), " S t a b i l i t y i n Competition", Economic J o u r n a l , 29, 41-57. Huff, D.L. (1963), "A P r o b a b i l i s t i c A n a l y s i s of Shopping Center Trade Areas", Land Economics, 39, 81-90. Kamien, M.I, and Schwartz,. N.L. (1971), " L i m i t P r i c i n g and Uncertain Entry", Econometrica, 39, 441-454. Kamien, M.I. and Schwartz, N.L. (1972), "Uncertain Entry and Excess Capacity", American Economic Review, 57, 918-927. K e n d a l l , M.G. and S t u a r t , A. (1963)., The Advanced Theory, of S t a t i s t i c s , V o l . L: D i s t r i b u t i o n Theory, 2nd. ed., London: Charles G r i f f i n and Company, L i m i t e d . K e n d a l l , M.G. and S t u a r t , A. (1973), The Advanced, Theory of S t a t i s t i c s , V o l . 2: Inference and R e l a t i o n s h i p , 3rd ed., London: Charles G r i f f i n and Company, L i m i t e d . Koopmans, T.C..(1957), Three Essays on. the State of Economic Science, New York: McGraw H i l l Book Company, Inc. Lancaster, K.J. (1966), "A New Approach to Consumer Theory", J o u r n a l of  P o l i t i c a l Economy, 74, 132-157. L a t s i s , S.J. (1976), "A Research Programme i n Economics", i n L a t s i s , S.J. (ed.), Method and A p p r a i s a l i n Economics, Cambridge: Cambridge U n i v e r s i t y Press, 1-42. Lee, Y. and Koutsopoulos, K. (1976), "A L o c a t i o n a l A n a l y s i s of Convenience Food Stores i n M e t r o p o l i t a n Denver", Annals of Regional Science, 10, 104-117. Luce, R.D. and R a i f f a , H. (1957), Games and D e c i s i o n s , New York: John Wiley and Sons, Inc. Ma l l e n , B. (1976), "A P r e l i m i n a r y Paper on the L e v e l s , Causes, and E f f e c t s of Economic Concentration i n the Canadian R e t a i l Food Trade: A Study of Supermarket Power", Reference Paper No. 6, Food P r i c e s Review Board. Mallen, B. and Haberman, M. (1975), "Economies of Scale: A Determinant of 'Overstoring and S u p e r s t o r i n g " 1 , i n Proceedings, Third Annual Conference, Canadian A s s o c i a t i o n of A d m i n i s t r a t i v e Sciences, 158-166. - 234 -Markin, R.J. (1968), The Supermarket: An Analysis, of Growth, Development  and Change;, Pullman: Washington S t a t e . U n i v e r s i t y Press. M a r s h a l l , J.U. (1969), The Location of Service Towns:. An Approach to the  A n a l y s i s of C e n t r a l Place Systems, Toronto: U n i v e r s i t y of Toronto Press. Meehan, J.W. (1970), " J o i n t Venture Entry i n P e r s p e c t i v e " , A n t i t r u s t  B u l l e t i n , 15, 693-711. Modigl i a n i , F. (1958), "New Developments on the Oligopoly Front", J o u r n a l of P o l i t i c a l Economy, 66, 215-232. Mu e l l e r , W.F. and.Garoian, L. (1961), Changes i n the Market S t r u c t u r e of  Grocery R e t a i l i n g , Madison: U n i v e r s i t y of Wisconsin Press. N a t i o n a l Commission on Food Marketing (1966), Organization and Competition  i n Food R e t a i l i n g , T e c h n i c a l Study No. 7, Washington, D.C: - Government P r i n t i n g O f f i c e . Nystuen, J.D. (1959), "Geographical.Analysis of Customer Movement and R e t a i l Business L o c a t i o n " , Ph.D. D i s s e r t a t i o n , U n i v e r s i t y of Washington. Osborne, D.K. (1964), "The Role of Entry i n Oligopoly Theory", J o u r n a l  of P o l i t i c a l Economy, 72, 396-402. Osborne, D.K. (1973), "On the R a t i o n a l i t y of L i m i t P r i c i n g " , J o u r n a l of  I n d u s t r i a l Economics, 22, 71-80. Pa s h i g i a n , B.P. (1968), " L i m i t P r i c e and the Market Share of the Leading Firm", J o u r n a l of I n d u s t r i a l Economics, 16, 165-177. Pe l e s , Y. (1974), "A Note on E q u i l i b r i u m i n M o n o p o l i s t i c Competition", J o u r n a l of P o l i t i c a l Economy, 82, 626-630. P y a t t , F.G. (1971), " P r o f i t Maximization and the Threat of New Entry", Economic Journal;, 81, 242-255. R o t h s c h i l d , R. (1976), "A Note on the E f f e c t of Sequential Entry on Choice of Location!', J o u r n a l of I n d u s t r i a l Economics, 24, 313-320. S a v i t t , R. (1975), " E f f i c i e n c y and S i z e i n Canadian Supermarkets", i n Proceedings, Third Annual Conference, Canadian A s s o c i a t i o n of A d m i n i s t r a t i v e Sciences, 225-232. S a v i t t , R. (1977), "Some P r e l i m i n a r y Findings About E f f i c i e n c y i n Canadian Supermarkets", The Canadian Marketer, 9, 22-26. S c h e l l i n g , T.C. (1956), "An Essay on Ba r g a i n i n g " , American Economic Review, 45, 281-306. - 235 -Scherer, F.M. (1970), I n d u s t r i a l Market St r u c t u r e and Economic  Performance, Chicago: Rand McNally and Company, Schmalensee, R. (1978), "Entry Deterrence i n the Ready-to-Eat Breakfast Cereal Industry", B e l l J o u r n a l of Economics, 9, 305-327. Shepherd, W.G. (1970), Market Power and Economic Welfare: An I n t r o d u c t i o n , New York: Random House, Inc. Sherman, R. and W i l l e t t , T.D. (1967)., " P o t e n t i a l Entrants Discourage Entry", J o u r n a l of P o l i t i c a l Economy, 75, 400-403. Shubik, M. (1959)., Strategy.and Market S t r u c t u r e : Competition, O l i g o p o l y , and the Theory of Games, New York: John Wiley and Sons, Inc. S i e g e l , S. (1956), Nonparametric S t a t i s t i c s f o r the B e h a v i o r a l Sciences, New York: McGraw H i l l Book Company. Simmons,. J . (1964)., The Changing P a t t e r n of R e t a i l L o c a t i o n , Research Paper No. 92, Department of Geography, U n i v e r s i t y of Chicago. Simmons, J . (1966), Toronto's Changing R e t a i l Complex: A Study i n Growth  and B l i g h t , Research Paper No. 104, Department of Geography, U n i v e r s i t y of Chicago. Southey, C. (1978), "The Staples Thesis, Common Property and Homesteading", Canadian J o u r n a l of Economics, 11, 547-559. Spence, M. (1977), "Entry, Capacity, Investment and O l i g o p o l i s t i c P r i c i n g " , B e l l Journal of Economics, 8, 534-544. Super Market I n s t i t u t e (1970-1976), Facts About New Super Markets Opened  i n ( y e a r ) , Chicago: Super Market I n s t i t u t e . Super Market I n s t i t u t e (1970-1976), The Super Market Industry Speaks, (y e a r ) , Chicago: Super Market I n s t i t u t e . Walker, H.M. and Lev. J (1953), S t a t i s t i c a l Inference, New York: Holt Rinehart and Winston. Wenders, J.T. (1971), "Excess Capacity as a B a r r i e r to Entry", J o u r n a l  of I n d u s t r i a l Economics, 20, 14-19. - 236 -APPENDIX In t h i s appendix, we present a s e r i e s of diagrams which show the r e l a t i v e l o c a t i o n s of supermarkets i n the c o n s t i t u e n t sub-markets of the GVRD. The f i r s t group of diagrams represents cross s e c t i o n s of f i r m ownership i n the f i r s t quarter of 1978. We have used the f o l l o w i n g symbols to i n d i c a t e which fi r m s own which.stores: pi Canada Safeway ^ Overwaitea ^ K e l l y Douglas Competitive Fringe In order to ensure v i s u a l c l a r i t y . , the Vancouver sub-market has been broken down i n t o a western sector ( C i t y of Vancouver) and an eastern sector (Burnaby, New Westminster, Coquitlam, Port Coquitlam, Port Moody). We have constructed, two sets of cross s e c t i o n diagrams, one of which i s based on D e f i n i t i o n 1 of a supermarket and the other of which i s based on our r e v i s e d d e f i n i t i o n of a supermarket. The second group of diagrams p o r t r a y s the s e q u e n t i a l establishment of supermarkets i n the Vancouver sub-market. Thus, associated w i t h each symbol i s a number representing the date at which that p a r t i c u l a r s t o r e was opened. We have constructed two time s e r i e s diagrams p o r t r a y i n g the s e q u e n t i a l development of the supermarket i n d u s t r y , one of which i s based on D e f i n i t i o n 1 of a supermarket and the other of which i s based on our r e v i s e d supermarket d e f i n i t i o n . Again, f o r the purpose of v i s u a l c l a r i t y , we have broken..down the Vancouver sub-market i n t o a western sector and an eastern s e c t o r . However, we have - 237 -numbered the stor e s i n the western sector and eastern sector diagrams corresponding,to a given supermarket d e f i n i t i o n as i f the two diagrams appeared as one. In a d d i t i o n , i t w i l l be r e c a l l e d that some of the store s i n our sample changed ownership at some p o i n t i n t h e i r h i s t o r y . Due to the l i m i t a t i o n s of the type of diagrams which we have chosen to use, change of ownership i s not d i r e c t l y r e f l e c t e d i n the diagrams. However, a f t e r each set of western and eastern sector diagrams, we provide a d e s c r i p t i o n of which sto r e s changed f i r m ownership and when. For example, 12(F k) 27(F g) should be i n t e r p r e t e d to mean that s t o r e number 12, owned by f i r m F^, was taken over by f i r m F and became s t o r e number 27. • s F i n a l l y , we should note two conventions which were employed i n our c a l c u l a t i o n s of boundaries f o r these sub-markets. F i r s t , i f a perpendicular b i s e c t o r boundary between a s t o r e 1 and a store 2 i s only an a r b i t r a r i l y small d i s t a n c e e more d i s t a n t than the nearest perpendicular b i s e c t o r boundary between s t o r e 1 and another s t o r e , then sto r e s 1 and 2 w i l l be assumed to have a boundary w i t h each other. Second, i f two or more stores are w i t h i n e distance of each other, then i t w i l l be assumed that these two s t o r e s have the same market area boundaries w i t h other s t o r e s . A cursory examination of the diagrams which f o l l o w should convince the reader of the n e c e s s i t y of these conventions,, and a l s o g i v e the reader an idea of. how l a r g e ,e. was, allowed to be ( i n our c a l c u l a t i o n s , one c i t y b l o c k ) . Figure A . l RICHMOND SUB-MARKET CROSS SECTION o in. CM a a . CM CD a . in CD • A • to Co 00 • T 50 1 0 0 150 200 X 250 ~ 1 300 ~ 1 — 350 400 Figure A.2 NORTH SHORE SUB-MARKET CROSS SECTION • m • 6 "i r 50 100 ~1 150 200 X 250 300 350 400 Figure A.3 DELTA-SURREY SUB-MARKET CROSS SECTION <2& • * i CD • 6 1 1 1 1 1 r— 50 100 350 200 X 350 250 300 400 Figure A.4 VANCOUVER SUB-MARKET WESTERN SECTOR CROSS SECTION A • • • A • Q * • • A n * * • * i • A A • * • • 0 i 1 1 1 1 1 1 — r — i 1 — i — i 1 — i 1 — i 1 50 100 150 200 250 300 350 400 X Figure A.5 VANCOUVER SUB-MARKET EASTERN SECTOR CROSS SECTION • A A • • 50 • • A A • 100 150 200 X A 250 • A "1 300 350 400 a a in o a . C M a a . a-4 in a Figure A.6 RICHMOND SUB-MARKET CROSS SECTION (STORE SIZE > 12,000 SQUARE FEET) CD A N3 -P-U) • CD CD • ~ T I I I I 1 1 1 1 1 1 — 1 1 1 1 - r — 0 5 0 1°° 150 200 250 30 0 350 400 X Figure A 7 7 NORTH SHORE SUB-MARKET CROSS SECTION (STORE SIZE > 12.,000 SQUARE FEET) i <=>-! in • -r 50 1 0 0 150 i r 200 X 250 300 - i 1 r— 350 400 Figure A.8 DELTA-SURREY SUB-MARKET CROSS SECTION (STORE SIZE >. 12,000 SQUARE FEET) RH o o . C M • A o 1 a N3 a " l i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r— 0 50 100 150 200 250 300 350 400 X i n — Figure A.9 a a a . C M a Q - l VANCOUVER SUB-MARKET WESTERN SECTOR CROSS SECTION (STORE SIZE > 12,000 SQUARE FEET) 4f A * • rrj A • A • • • a ^ • • • N5 O N a . « • * # m • • a -• A • ~ r i i i 1 1 — i — i 1 — i — | r — | 1 — r -50 100 150 200 250 300 350 400 X Figure A.10 VANCOUVER SUB-MARKET EASTERN SECTOR CROSS SECTION (STORE SIZE > 12.,000 SQUARE FEET) Q I D . C M a a . C M a in. a a . a . IT) • • <!> • • N3 a 0 T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 50 300 150 200 250 300 350 400 X Figure A.11 VANCOUVER SUB-MARKET WESTERN SECTOR TIME SERIES 7 6 jm 58 43 2 5 5 7 A 7 9 • 7 5 CLt 49 A 6 1 CD8o n * G32 _ i3 A 7 3 9 1 • s 5 0 A 2 0 6 6 34 *17 Q 8 1 or 7 2 • *36 A 5 6 • 2k * 3 6 N ^ 6 0 Q ] 2 2 8 x ^ T J 5 2 2 9 2 2 6 2 5 3 A 35 A12 K30 Q 6 A 7 0 a * i ^8 5 • K3 00 1 r 50 ~ i r 1 0 0 150 T — 200 X - T — 250 "1 300 "~T 350 -7— 400 Figure A.12 VANCOUVER SUB-MARKET EASTERN SECTOR TIME SERIES Q I D — , rvi a in. a . • 1 8 A"1 6 7 31 )K 46 AfT ] 87 a . in A 9 • 45 <S8 ^ 42 3 7 1 9 27 64 A 3 8 A 5 9 1 + 3 G f c , ! CD77 89 A 50 100 *16 • 7 i r~ 150 A 2 6 ~~1 200 X 250 a82 6 8 <!>65 3 3 A r—i r 300 350 400 N5 -P-- 250 -Notes to Figures A,11 and A.12 2 ( F s ) 10(F k) 51(cf) 12(F k) 54(cf) 14(cf) 23(F k) 9( F k ) 41(cf) 22(F k) 74(cf) 26(F k) 83(cf) 20(F k) 48(cf) 55(F s) 90(cf) 38(F k) 69(F ) o 5 ( F s ) 88(cf) 16(cf) 39(F k) jn_^  Figure A. 13 VANCOUVER SUB-MARKET WESTERN SECTOR TIME SERIES (STORE SIZE > 12,000 SQUARE FEET) 1 37 ism 36 Ass rrj32 rjj2 Aso o a . a in. Q. • 5 1 A C356 Q ] 3 3 ^ ^ 2 3 * z i ° 5 ? D 3 ° 1 ? i f i f ^ 1 8 2 2 A A 3 5 A 6 0 CD" in o 0 1 — I — I — I — I — I — I — I — I — I — I — I — I — I — I — I — 1 50 100 150 200 250 300 350 400 X Figure A.14 VANCOUVER SUB-MARKET EASTERN SECTOR TIME SERIES (STORE SIZE > 12,000 SQUARE FEET) a i n _ C M a a . C M o VTJ. 10 <+5 A 2 9 • 25 15 42 A 3 8 28 • K 8 • 59 of <!> .4 3 |S5 a a . a . in 19 3^A |TJ62 <!> 54 3 0 • .7. 27 \ i r 2 4 • 20 a 6 0 1 — I — I — I — I — I — I — I — I — I — I — 1 — I — I — I — I — I 50 100 150 200 250 300 350 400 X - 253 -Notes to Figures A.13 and A.14 4(F ) 63(cf) s 25(F k) 46(F o) 35(F ) 64(cf) 

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