"Applied Science, Faculty of"@en . "Community and Regional Planning (SCARP), School of"@en . "DSpace"@en . "UBCV"@en . "Fricker, Urs Josef"@en . "2011-06-07T19:07:41Z"@en . "1969"@en . "Master of Science in Planning - MSc (Plan)"@en . "University of British Columbia"@en . "In the planning profession there is increasing recognition of the complex relationship of variables in an urban region which impede rational decision-making. In order to cope with this problem, quantitative models have been developed in recent years. It is the purpose of this study to investigate and evaluate the present stage of model-building as it applies to regional planning.\r\nIt is hypothesized that the application of land use allocation models is a desirable aid for rational decisionmaking in regional planning.\r\nThe study begins with an outline of the theoretical basis for building land use allocation models: economic location theory and social physics. Economic location theory is mainly concerned with finding criteria for a rational choice of the location for a firm or a household. In this context, the concept of economic rent is discussed. In order to give explanations of the land use patterns within a region the basic notion in respect to agricultural production is developed and then extended to the urban land uses. The second approach to land use allocation models, social physics, is mainly based on statistical regularities in explaining human mass behavior. The most commonly employed concept is the gravity principle, which is an attempt to apply Newton's physical law of gravitation to social, mass behavior. This concept is very often applied in community and regional planning and has yielded acceptable results in a great number of studies.\r\nIn part three the most important elements and steps in the process of model-building are discussed, including rules or standards which should be considered by a model-builder. First of all, a wide range of types of models are examined in order that the proper model may be selected for an actual regional planning problem. The design process is also discussed in some detail and it is shown that there is evidence of fundamental criteria for model building.\r\nPart four is concerned with three selected existing regional land use allocation models. The model of the Pittsburgh Region was the first operational model on a regional level and its ingenuity influenced numerous model-builders. One of the most salient findings of this model, which is mainly based on social physics, relates to the fact that the gravity principle seems to have enough flexibility to comprehend the spatial pattern of land uses within an urban region.\r\nThe model of the State of Connecticut is based on the shift-analysis framework and distributes three population and six employment groups to the 169 towns of the State of Connecticut. Its basic feature is the ability to replicate the structure of a region as large as a state and it is therefore of great interest as a macro-approach. The structure of the model is relatively simple and the data requirements are not intensive. Hence, it seems that such a model framework could serve as a sound basis for models in other study areas.\r\nThe Bay Area Simulation Study is one of the most recent models. It introduces a high level of disaggregation and assumptions which are based, to some extent, on economic location theory. Hence, it can be said that its basic concept relates to the working mechanism of the market process. The structure of the model is based on a number of interrelated submodels, including a set of employment allocation models and a set of residential allocation models.\r\nThe final part of this study relates the findings of the preceding parts to regional planning. It is shown that regional planning is fundamentally a locational problem. In addition, some experiences of model application by planning agencies are discussed. These experiences emphasize the fact that, the essential feature of land use allocation models is to improve the rationality of decision-making. By comparing the advantages of models with the principal difficulties in application it is then possible to derive the final conclusion that land use allocation models are a desirable aid for rational decision-making in regional planning."@en . "https://circle.library.ubc.ca/rest/handle/2429/35163?expand=metadata"@en . "REGIONAL LAND USE ALLOCATION MODELS AND THEIR APPLICATION TO PLANNING by URS JOSEF THICKER D i p l . Ing. ETH, Swiss I n s t i t u t e of Technology, 1 9 6 5 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE. REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the School of Community and Regional P l a n n i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d staadard THE UNIVERSITY OF BRITISH COLUMBIA May, 1 9 6 9 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Community and Regional Planning The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date A p r i l 3 0 , 1969 i i i ABSTRACT In the planning profession there i s increasing recognition of the complex r e l a t i o n s h i p of v a r i a b l e s i n an urban region which impede r a t i o n a l decision-making. In order to cope with t h i s problem, quantitative models have been developed i n recent years. I t i s the purpose of t h i s study to investigate and evaluate the present stage of model-b u i l d i n g as i t applies to regional planning. I t i s hypothesized that the a p p l i c a t i o n of land use a l l o -cation models i s a desirable aid f o r r a t i o n a l d e c i s i o n -making i n regional planning. The study begins with an outline of the t h e o r e t i c a l basis f o r b u i l d i n g land use a l l o c a t i o n models: economic l o c a t i o n theory and s o c i a l physics. Economic l o c a t i o n theory i s mainly concerned with f i n d i n g c r i t e r i a f o r a r a t i o n a l choice of the l o c a t i o n f o r a f i r m or a household. In t h i s context, the concept of economic rent i s discussed. In order to give explanations of the land use patterns within a region the basic notion i n respect to a g r i c u l t u r a l pro-duction i s developed and then extended to the urban land i v uses. The second approach to land use a l l o c a t i o n models, s o c i a l physics, i s mainly based on s t a t i s t i c a l r e g u l a r i -t i e s i n explaining human mass behavior. The most commonly employed concept i s the g r a v i t y p r i n c i p l e , which i s an attempt to apply Newton's p h y s i c a l law of g r a v i t a t i o n to social, mass behavior. This concept i s very often applied i n community and regional planning and has yielded accept-able r e s u l t s i n a great number of studies. In part three the most important elements and steps i n the process of model-building are discussed, i n c l u d i n g r u l e s or standards which should be considered by a model-b u i l d e r . F i r s t of a l l , a 'wide range of types of models are examined i n order that the proper model may be selected f o r an actual regional planning problem. The design process i s also discussed i n some d e t a i l and i t i s shown that there i s evidence of fundamental c r i t e r i a f o r model b u i l d i n g . Part four i s concerned with three selected e x i s t i n g regional land use a l l o c a t i o n models. The model of the Pittsburgh Region was the f i r s t operational model on a regional l e v e l and i t s ingenuity influenced numerous model-builders. One of the most s a l i e n t findings of t h i s model, which i s mainly based on s o c i a l physics, r e l a t e s to the f a c t that the g r a v i t y p r i n c i p l e seems to have enough f l e x i b i l i t y to comprehend the s p a t i a l pattern of land uses within an urban region. V The model of the State of Connecticut i s based on the s h i f t - a n a l y s i s framework and d i s t r i b u t e s three population and s i x employment groups to the 169 towns of-the State of Connecticut. Its basic feature i s the a b i l i t y to r e p l i -cate the structure of a region as large as a state and i t i s therefore of great i n t e r e s t as a macro-approach. The structure of the model i s r e l a t i v e l y simple and the data requirements are not i n t e n s i v e . Hence, i t seems that such a model framework could serve as a sound basis f o r models i n other study areas. The Bay Area Simulation Study i s one of the most recent models. I t introduces a high l e v e l of disaggregation and assumptions which are based, to some extent, on economic l o c a t i o n theory. Hence, i t can be said that i t s basic concept r e l a t e s to the working mechanism of the market process. The structure of the model i s based on a number of i n t e r r e l a t e d submodels, inclu d i n g a set of employment a l l o c a t i o n models and a set of r e s i d e n t i a l a l l o c a t i o n models. The f i n a l part of t h i s study r e l a t e s the findings of the preceding parts to regional planning. I t i s shown that regional planning i s fundamentally a l o c a t i o n a l problem. In addition, some experiences of model a p p l i c a t i o n by planning agencies are discussed. These experiences emphasize the f a c t that, the e s s e n t i a l feature of land use a l l o c a t i o n models i s to improve the r a t i o n a l i t y of decision-making. By v i comparing the advantages of models with the p r i n c i p a l d i f f i c u l t i e s i n a p p l i c a t i o n i t i s then poss ib le to der ive the f i n a l conc lus ion that land use a l l o c a t i o n models are a d e s i r a b l e a id f o r r a t i o n a l decis ion-making i n r e g i o n a l p l a n n i n g . v i i TABLE OF CONTENTS ABSTRACT - i i i LIST OF FIGURES . . . i x LIST OF APPENDICES x ACKNOWLEDGEMENT S x i 1. INTRODUCTION .1 1.1 The Problem 1 1.2 Purpose and Scope of t h i s Study 4 1.3 Hypothesis 6 1.4 D e f i n i t i o n s 6 1.5 O r g a n i z a t i o n of the Remainder 7 2. APPROACHES TO REGIONAL ALLOCATION OF ACTIVITIES 10 2.1 Economic L o c a t i o n Theory............. 11 2.1.1 A g r i c u l t u r a l Rent and Land Use 13 2.1.2 Urban Land Uses . 16 2.1.3 General E q u i l i b r i u m 20 2.2 S o c i a l P h y s i c s 23 3 . ELEMENTS OF MODEL BUILDING . . .' 33 3.1 Typology of Models 33 3.2 Design of a Model 41 3.2.1 The V a r i a b l e s and t h e i r Relevance 43 3.2.2 The L e v e l of Aggregation 44 3.2.3 Fo r m u l a t i o n of the Mathematical R e l a t i o n s h i p 46 v i i i 3- 3 C a l i b r a t i o n and Testing of a Model 50 4. SELECTED REGIONAL LAND USE MODELS 58 4 . 1 The Model of the Pittsburgh Region by Lowry .. 59 4 . 1 . 1 The Concept of the Model 59 4 . 1 . 2 The Structure of the Model 61 4- . 1 . 3 I n t e r p r e t a t i o n of the Model 66 4.1.4 C a l i b r a t i o n of the Model 67 4 . 1 . 5 Testing of the Model '. 71 4.1.6 Evaluation 72 4 . 2 The Connecticut Model 73 4 . 2 . 1 Formulation of the Model 74 4 . 2 . 2 The Structure of the Model 76 4.2.3 I n t e r p r e t a t i o n . .... 79 4 .2.4 C a l i b r a t i o n and Testing 81 4 . 3 The Bay Area Simulation Study 83 4 . 3 . 1 . Formulat ion of the Model 84 4 . 3 . 2 Employment Location Submodels 85 4 . 3 . 3 R e s i d e n t i a l Location Submodel 92 4 .3.4 Appraisal ' 94 4.4 Conclusions 96 5 REGIONAL PLANNING AND LAND USE ALLOCATION MODELS 101 5 . 1 Regional Planning and the Importance of Land Use A l l o c a t i o n Models 101 5 . 2 Advantages of Land Use A l l o c a t i o n Models 103 5 . 3 D i f f i c u l t i e s of Applications 106 5.4 Conclusions 108 BIBLIOGRAPHY 113 APPENDICES 119 i x LIST OF FIGURES Figure Page 1 A g r i c u l t u r a l bid rent function f o r one crop ... 15 2 A g r i c u l t u r a l bid rent functions f o r two crops . 15 3 Bid rent functions f o r urban land uses 21 4 Bid rent functions f o r a hierarchy of centers . 21 5 Cumulation of errors 4-9 6 Structure of a chain model 49 7 Proposed structure of an improved model 49 8 Information flows i n the Pittsburgh Model 62 9 D i f f e r e n t i a l s h i f t and proportional share 75 10 Structure of the Bay Area Simulation Study 85 11 R e t a i l a l l o c a t i o n flow diagram \u00E2\u0080\u00A2 \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 89 X LIST OF APPENDICES Appendix Page 1 Cumulation of Er r o r s 119 2 Variables and Parameters of the Pittsburgh Model 122 3 Control Totals and S t r u c t u r a l Parameters of the Pittsburgh Model- 124 4 Employment Groups f o r the BASS Model 126 x i ACKNOWLEDGEMENTS I wish to express my thanks to the many people who contr ibuted to the completion of t h i s t h e s i s . Dr . H . Peter Oberlander, D i r e c t o r of the School of Community and Regional Planning deserves p a r t i c u l a r thanks f o r h i s i n i t i a l encouragement to inves t iga te t h i s t op ic and h i s continuous i n t e r e s t i n my planning educat ion . G r a t e f u l apprec ia t ion i s also extended to Dr . H . C r a i g Davis and Dr. V . Set ty Pendakur f o r the concern, ad-vice and cons truc t ive c r i t i c i s m they have of fered dur ing the prepara t ion of t h i s t h e s i s . I am also indebted to my f r i e n d and col league Fraser L . Manning who with great pat ience as s i s t ed with the f i n a l E n g l i s h s t y l e . But above a l l , thanks has to be expressed to the Canada C o u n c i l which through a scho larsh ip made my studies here i n Canada p o s s i b l e . F i n a l l y , my greatest indebted-ness i s to my wife Insa who always helped dur ing my s tudies and brought my s c r i b b l e of t h i s thes i s in to f i n a l form .and -gestal t . - 1 -1. INTRODUCTION 1.1 The Problem A c t i v i t i e s by men, s o c i a l groups, communities or entire s o c i e t i e s are determined by a purpose or a number of purposes. We act i n order to achieve goals. But there are always a l t e r n a t i v e ways to achieve them and one has to be chosen. This s e l e c t i v e d e c i s i o n i s fundamental i n human l i f e , and therefore can also be seen i n the context of planning what i s \"designing a course of action to achieve ends.\" 1 In community and r e g i o n a l planning, designing or selec-t i n g a course of action i s impeded by several circumstances. F i r s t of a l l , goals have to be formulated. This i s extremely d i f f i c u l t because groups of human beings are l i m i t e d i n t h e i r a b i l i t y to agree on common goals, to communicate and p to cooperate. Apart from goal s e t t i n g there i s increasing r e c o g n i t i o n that a course of action influences a great number of variables and \"planners are now prisoners of the discovery that i n the c i t y [and i n the region] everything - 2 -af f ec t s everything e l s e . \" J A t h i r d d i f f i c u l t y i s the l i m i t a t i o n of time i n dea l ing with a complex and c o n t i n u -ously changing system such as the c i t y or the r e g i o n . Very of ten the planner i s asked to give recommendations i n a short time and therefore he i s not able to study a l l the necessary aspects of h i s task . These remarks i n d i c a t e severa l d i f f i c u l t i e s and r e s t r i c -t i ons i n s o l v i n g the urban and r e g i o n a l problems, and a statement by Simon i s extremely true f o r the p lanner . He s tates that the capac i ty of human mind f o r formulat ing and s o l v i n g complex problems i s very smal l compared with the s ize of the problem whose s o l u t i o n i s requ ired f o r o b j e c t i v e l y r a t i o n a l behavior i n the r e a l world - or even f o r a reasonable approximation to such objec t ive r a t i o n a l i t y . 2 1 \" Despite these d i f f i c u l t i e s and l i m i t a t i o n s not only a \"good\" a c t i o n , but the \"best\" act ion^ should be found. Meyerson and B a n f i e l d express t h i s as \"e f f i c i en t\" planning which \"under given condi t ions leads to the maximization of the attainment of the re levant ends.\" They assume also \"that a planned course of a c t i o n , which i s se lec ted r a t i o n a l l y i s most l i k e l y to maximize the attainment of the re levant ends and that therefore ' r a t i o n a l ' p lanning and ' e f f i c i e n t ' p lanning are the same.\"^ In a f u r t h e r statement they state that a r a t i o n a l d e c i s i o n has to be made i n the f o l l o w i n g manner: - 3 -The decis ion-maker considers a l l of the a l t e r n a t i v e s (courses of act ion) open to him; i . e . he considers what courses of a c t i o n are poss ib le w i t h i n the condi t ions of the s i t u a t i o n and i n the l i g h t of the end he seeks to a t t a i n . He i d e n t i f i e s and evaluates a l l the conse-quences which would fo l low from the adoption of each a l t e r n a t i v e ; . . . and he s e l ec t s that a l t e r n a t i v e the probable consequences of 'which would be pre ferab le i n terms of h i s most valuable ends . ' Meyerson and Banf ie ld point out that t h i s i s an i d e a l concept and \"no d e c i s i o n can be p e r f e c t l y r a t i o n a l \" s ince a l l a l t e r n a t i v e s and consequences can never be known. Nevertheless decis ion-making i n community and r e g i o n a l p lanning should be based on knowledge of the main a l t e r -nat ives and t h e i r consequences. We have now o u t l i n e d on one side the circumstances which impede decis ion-making and on the other side the c r i t e r i a f o r r a t i o n a l dec is ion-making . We may now pose the d i f f i c u l t quest ion: how can a r a t i o n a l d e c i s i o n i n regard to a complex system such as the c i t y or the reg ion be made? I t i s obvious that the planner can no longer r e l y s o l e l y on i n t u i t i v e judgement or experience. He has to apply too l s and techniques which are able to tes t i n a short time a v a r i e t y of goa ls , a l t e r n a t i v e s , and t h e i r consequences which inf luence a great number of v a r i a b l e s . 1 . 2 Purpose and Scope of t h i s Study Since World War I I there has been i n c r e a s i n g concern about s c i e n t i f i c methods to solve complex problems. Operations 8 9 research and systems ana lys i s are now appl i ed i n indus-t r i a l management, warfare, government, p lanning and many other f i e l d s . These techniques provide a s c i e n t i f i c bas i s f o r s o l v i n g problems which involve i n t e r a c t i o n s of many v a r i a b l e s . Churchman, Ackoff and Arnof f descr ibe operations research i n the fo l lowing way: The concern of O.R. with f i n d i n g an optimum d e c i s i o n , p o l i c y , or design i s one of i t s e s s e n t i a l c h a r a c t e r i s t i c s . I t does not seek merely to f i n d be t ter so lu t ions to a p r o b l e m . . . ; i t seeks the best s o l u t i o n . It may not always f i n d i t . . . . But O . R . r s e f f o r t s are c o n t i n u a l l y d i r e c t e d to ge t t ing to the optimum or as c lose to i t as p o s s i b l e . A main phase i n operations research i s the cons truc t ion of a model to represent the system under study. With the help of a model i t i s poss ib le to run experiments which would otherwise be imposs ib le . The purpose of t h i s study i s to inves t iga te and evaluate the present stage of model -bui ld ing f o r a p p l i c a t i o n i n r e g i o n a l p l a n n i n g . In reviewing the l i t e r a t u r e there i s evidence of a great v a r i e t y of models. The scales range from i n t e r n a t i o n a l trade flow models to the i n t e r a c t i o n between individuals.\"'\"\"'\" - 5 -In t h i s study we s h a l l focus on i n t r a r e g i o n a l growth a l l o -c a t i o n or land use models. These models a l l o c a t e economic a c t i v i t i e s or land uses to subareas w i t h i n a r e g i o n . The reason f o r t h i s focus stems from the importance of the s p a t i a l a l l o c a t i o n i n p lanning - as we s h a l l see i n t h i s study - and t h i s leads to the fac t that 7 0 percent of 1 2 urban development models are l o c a t i o n a l models. The f i e l d of model b u i l d i n g i n the p lanning profes s ion i s expanding extremely f a s t , and often d i f f i c u l t i e s of communication ex i s t between the \"hardl iners\" and the 1^ \" s o f t l i n e r s \" , as f o r instance expressed by Lowry: The Model -bui lders - a group that overlaps but does not co inc ide with the p lanning pro fe s s ion - c la im that t h e i r b r a i n - c h i l d r e n have present or p o t e n t i a l value as p lanning a i d s . One of the f r u s t r a t i o n s of the planner as c l i e n t i s that he does not u s u a l l y f i n d i t easy to judge these claims or to choose among the many a l t e r n a t i v e s now a v a i l a b l e f o r h i s c o n s i d e r a t i o n . ! ^ The author hopes to help to bridge the \"gap\" between these two groups. Therefore the scope of t h i s study w i l l be l i m i t e d to t h i s o b j e c t i v e . With in the great number of land use models only a few w i l l be reviewed and only those aspects thereof w i l l be examined which seem to be of p r i -mary relevance i n regard to t h e i r a p p l i c a t i o n i n r e g i o n a l p l a n n i n g . - 6 -1.3 Hypothesis T h i s study w i l l examine the fo l l owing hypothes is : Given that the major dec i s ions p e r t a i n i n g to r e g i o n a l p lanning r e l a t e to the s p a t i a l a l l o c a t i o n of economic a c t i v i t i e s , the a p p l i c a t i o n of land use a l l o c a t i o n models i s a d e s i r a b l e a i d for r a t i o n a l d e c i s i o n -making i n r e g i o n a l p lann ing . 1.4 D e f i n i t i o n s For the purpose of t h i s study re levant terms are def ind i n the fo l lowing way: Region: A space l a r g e r than a s ing le community and smal ler 15 than a whole country . y Model: A s i m p l i f i e d representa t ion of some subject of i n q u i r y (such as ob jec t s , events, processes , systems) . \" ^ Land Use: Man's a c t i v i t i e s on land which are d i r e c t l y 17 r e l a t e d to l a n d . ' Land Use A l l o c a t i o n Model: A symbolic statement about the a l l o c a t i o n of economic a c t i v i t i e s (populat ion and employment) and land use categor ies which can \"include s t r u c t u r e s , eco-nomic a c t i v i t i e s , f l o o r areas , and genera l ly any items that - 7 -can be used to descr ibe regions and subareas i n q u a n t i t a -t i v e s p a t i a l terms.\" S imulat ion: The operat ion of a model or s imulator . The model i s amenable to manipulations which would be impos-s i b l e , too expensive or i m p r a c t i c a l to perform on the e n t i t y i t p o r t r a y s . The operat ion of the model can be s tudied and, from i t , proper t i e s concerning the behavior 19 of the ac tua l system or i t s subsystems can be i n f e r r e d . 1.5 Organizat ion of the Remainder A f t e r these in troduc tory remarks the second chapter w i l l dea l with the two a l t e r n a t i v e approaches to the formula-t i o n of land use a l l o c a t i o n models. The t h i r d chapter d iscusses the most important r u l e s which have to be con-s idered i f such a model i s to be b u i l t . The f o u r t h chapter expla ins and evaluates three e x i s t i n g r e g i o n a l models. We s h a l l see how the f i n d i n g s of the f i r s t three chapters are inputs to an opera t iona l model. The f i n a l part of t h i s study i s concerned with the a p p l i c a t i o n of models i n r e g i o n a l p lanning programs as t oo l s for improving d e c i s i o n -making . - 8 -Footnotes 1 Mart in Meyerson and Edward C. B a n f i e l d , P o l i t i c s , P lanning and the Publ i c\"Interes t (Glencoe, 1 1 1 . : The Free Pres s , 1955), p . 314. 2 Herbert A . Simon, Models of Man (New York: John Wiley & Sons, I n c . , 1 9 5 7 ) , p . 199-3 I r a S. Lowry,' \"Short Course i n Model Des ign\", ' Journa l of the American I n s t i t u t e of P lanners , V o l . 31 No. 2 (May 1965 ) , p . 158. 4 Herbert A . Simon, Op. c i t . , p . 198. 5 B r i t t o n H a r r i s , \"New Tools f or P l a n n i n g , \" Journal of the American I n s t i t u t e of P lanners , Vol\" 3 l No. 2 (May 1965) , p . 9 1 . ' 6 Mart in Meyerson and Edward C . B a n f i e l d , Op. c i t . , p . 314. 7 I b i d . , p . 314. 8 See f o r instance the. comprehensive and bas ic work by C . West Churchman, Russel L . Ackoff and Leonard E . A r n o f f , In troduct ion to Operations Research (New York: John Wiley and Sons, I n c . , 1956) . 9 See for instance .Claude McMil lan and Richard F . Gonzalez,. Systems A n a l y s i s (Homewood, 1 1 1 . : Richard D. Irwin , I n c ; , 1965 ) . 10 C . West Churchman,\"Russel L . Ackoff and Leonard E . A r n o f f , Op. c i t . , p . 8 . 11 Iowa State U n i v e r s i t y , Center f o r A g r i c u l t u r a l and Economic Development, Research and Educat ion f o r Regional and Area Development\"(Ames,'Iowa: Iowa State U n i v e r s i t y Pres s , 1966) , p . 2 5 5 . 12 G. Hemmens, \"Survey of Planning Agency Experience with Urban Development Models,\" Journa l of the American I n s t i t u t e of P lanners , V o l . 34 No. 5~ (Sept. 1968) . ' 13 W i l l i a m Goodman i n a l e c t u r e at the U n i v e r s i t y of B r i t i s h Columbia, January 1968. - 9 -14 I r a S. Lowry, \"Seven Models of Urban Development,\" i n Urban Development Models by Highway Research-- .' Board (Spec ia l Report 97 ; Washington, D . C . , 1968) , p . 121 . 15 Harvey S. P e r l o f f , \"Key Features of Regional P l a n n i n g , \" Journa l of the Amer ican ' Ins t i tu te of P lanners , V o l . 34- No. 2 (May 1968) , p . 153. 16 G. West Churchman, Russel L . Ackoff and Leonard E . A r n o f f , Op. c i t . , p . 151. 17 Marion Clawson and Charles L . Stewart, Land Use Information (Bal t imore: The John Hopkins Press , 1965) , P . 29 . \u00E2\u0080\u00A2 18 T r a f f i c Research Corpora t ion , Boston Regional Planning P r o j e c t , \"Review of E x i s t i n g Land Use Forecas t ing Techniques,\" i n -Highway Research Record, No. 88 ( 1 9 6 5 ) , p . 183. 19 Mart in Shubik, \"Simulation of the Industry and the F i r m , \" American Economic Review, L , No. 5 (Dec. I 9 6 0 ) , p . 9 0 9 . 2. APPROACHES-TO REGIONAL ALLOCATION OP ACTIVITIES In order to proceed to b u i l d i n g models of the s p a t i a l d i s t r i c u t i o n of a c t i v i t i e s we must f i r s t of a l l have a bas ic understanding of the under ly ing f o r c e s . Priedman and Alonso state that : Human a c t i v i t i e s are d i s t r i b u t e d over the n a t i o n a l t e r r i t o r y i n c e r t a i n rhythms and pat terns that are n e i t h e r a r b i t r a r y nor the working of chance. They r e s u l t ra ther from the interdependencies that give form to economic space. S p a t i a l patterns w i l l change with s h i f t s i n the s tructure of demand and of product ion , i n the l e v e l of technology, and i n the s o c i a l and p o l i t i c a l organ iza t ion of the n a t i o n . The economic and s o c i a l development of the nat ion i s r e f l e c t e d i n i t s patterns of sett lement; i t s systems of f low and exchange of commodities, money, and in format ion; i t s p a t t e r n of commuting and migrat ion; and i t s r e t i c u l a t i o n of areas of urban i n f l u e n c e . l There i s evidence of a complex framework of i n t e r a c t i o n s which makes a systematic i n v e s t i g a t i o n d i f f i c u l t . Neverthe-l e s s , f o l l o w i n g Lowry, there are mainly two a n a l y t i c a l t r a d i t i o n s or theor ies which o f f e r guidance: economic l o c a t i o n theory and s o c i a l p h y s i c s . Two approaches are a lso d i s t i n g u i s h e d by K i l b r i d g e and Carabateas as they descr ibe the organiz ing p r i n c i p l e of a model which i s the - 1 1 -\"essent ia l manifes tat ion of i t s under ly ing t h e o r y . \" 5 They d i s t i n g u i s h m i c r o - a n a l y t i c a l behavior or choice models, and m a c r o - a n a l y t i c a l growth-forces or ind ice s models. Th i s d i s t i n c t i o n co inc ides with Lowry's . The former models are based on economic l o c a t i o n theory, the l a t t e r ones on s o c i a l p h y s i c s . Therefore we s h a l l focus on these two approaches i n the f o l l o w i n g d i s c u s s i o n . 2 . 1 Economic L o c a t i o n Theory The main feature of t h i s theory i s that c r i t e r i a f o r the r a t i o n a l choice of a l o c a t i o n f o r a f i r m or a household are g iven . G e n e r a l l y , households and f irms locate where they ZL can \"gain more than they can elsewhere.\" Businesses locate where.they can obta in the highest p r o f i t , and house-holds where they have the greatest s a t i s f a c t i o n and employment o p p o r t u n i t i e s . Economic l o c a t i o n theory i s an extensive f i e l d i n i t s e l f . We w i l l therefore only out l ine a few fundamental p r i n c i p l e s . The \"father\" of l o c a t i o n theory i s von Thunen who i n h i s 5 study the \"Isolated State\"^ d i d \"progress somewhat toward a general l o c a t i o n a l a n a l y s i s . \" ^ He found that the s p a t i a l arrangement of a g r i c u l t u r a l product ion around a s ing le c i t y takes the form of concentr ic r i n g s . In regard to the l o c a t i o n of i n d u s t r i e s , the f i r s t comprehensive theory - 12 -was developed by Weber' who emphasized three bas ic l o c a -t i o n a l forces: ' transport cost d i f f e r e n t i a l s , labor cost d i f f e r e n t i a l s and agglomeration (deglomeration) economies and diseconomies. Weber's theory i s mainly a microeconomic approach from the point of view of the i n d i v i d u a l p r o -ducer . Hence i t was recognised that a general e q u i l i b r i u m theory was necessary instead of a p a r t i a l l o c a t i o n theory . This attempt was made by Losch i n h i s \"Economics of L o c a -8 t i o n \" i n which he found that the hexagon i s the most economical shape f o r t r a d i n g areas . These bas ic works v/ere s t i m u l i f o r s evera l scholars to study l o c a t i o n problems and to advance the theory . A com-, prehensive review of the l i t e r a t u r e i s contained i n I sard ' s \"Location and Space Economy\".^ Eor the purpose of t h i s study we w i l l focus on the d i s t r i -but ion of economic a c t i v i t i e s and land uses w i th in the sphere of the c i t y and i t s h i n t e r l a n d and o u t l i n e some concepts which seem to be of great help i n understanding the s p a t i a l forces which shape the urban reg ions . I sard s tates that t r a d i t i o n a l l y the t h e o r e t i c a l ana lys i s of the s p a t i a l a l l o c a t i o n of urban land uses.has f a l l e n outs ide the realm of l o c a t i o n theory . But i n regard to such a theory he f inds that \" in many aspects urban land use theory i s a l o g i c a l extension of a g r i c u l t u r a l l o c a t i o n theory.\"\"^ The bas ic concept of a g r i c u l t u r a l l o c a t i o n i s land rent ; - 13 -therefore we s h a l l def ine i t and show how i t can he extended and appl i ed to commercial, i n d u s t r i a l , and r e s i d e n t i a l land uses . 2 . 1 . 1 A g r i c u l t u r a l Rent and Land Use The concept of the land rent was f i r s t mentioned by 11 .. 12 Ricardo and f u l l y developed by von Thunen. Recent 13 and more complete formulat ions are those by Isard , Dunn 1 ^, A l o n s o 1 ^ and N o u r s e 1 6 . We assume that there i s a s ing le market center at which a g r i c u l t u r a l products from the surrounding h i n t e r l a n d can be s o l d . A l l the land i s uniformly f e r t i l e , and t r a n s p o r t -a t i o n costs are equal i n a l l d i r e c t i o n s from the market. The p r i c e s are determined at the market where demand equals supply. 17 We may consider one a c t i v i t y ' which, f or ins tance , p r o -duces 30 bushels of corn per acre at a cost of $ 10 i n labor and machinery. I f the p r i c e at the market i s $ 1 per bushel the product per acre can be so ld f o r # 3 0 . But corn produced at any dis tance from the center has to be transported at l e t us say, a p r i c e of # 0 . 0 5 per bushel per m i l e . In a case where corn i s produced at 10 miles from the market center we would have t r a n s p o r t a t i o n costs of - 14- -$ 1 5 per acre . While revenue remains f> 30 the d i f f erence between revenue and cost would only be # '5- Th i s d i f f erence between revenues and costs i s the economic r e n t . A farmer not i ces that t h i s d i f f erence increases $ 1 . 5 0 per mile and acre . Therefore he w i l l b i d rents up to these amounts for each mile nearer the market. The rent at any l o c a t i o n can be c a l c u l a t e d as fo l lows: p c ( t ) = N ( P c - C - k c . t ) = 3 0 ( l - ^ \u00C2\u00A7 - 0 . 0 5 \u00E2\u0080\u00A2 t ) p^(t) = 20 - 1 .5 \u00E2\u0080\u00A2 t where P c ( t ) : the rent per u n i t of land at a d is tance t from the market center N : number of u n i t s of the crop produced per un i t of land P : p r i c e per un i t of the crop at the market C : cost of producing one u n i t of the crop (farmers \"normal p r o f i t \" i s inc luded as labor cost) k Q : cost of t r a n s p o r t a t i o n per mile of one u n i t of the product t : d is tance from the market This b i d rent func t ion can be presented g r a p h i c a l l y as shown i n f i gure 1 (p. 1 5 ) . At the d is tance t ( i n the above example 13^/3 mi les ) the rent i s zero which means that at a d is tance greater than t , corn can only be pro-duced at a l o s s . - 15 -A g r i c u l t u r a l b i d rent f u n c t i o n f o r one crop Distance t (miles) F i g u r e 2 : A g r i c u l t u r a l b i d rent f u n c t i o n for two crops Rents Distance t (miles) - 16 -I f potatoes are a lso produced we can determine which area w i l l be used f o r each k ind of product ion . We may also c a l c u l a t e the b i d rent func t ion for potatoes and combine both funct ions as i n f i g u r e 2 (p . 15). In the competit ive market f o r land the highest b idder f o r any p a r t i c u l a r s i t e w i l l obta in i t . This means i n our case that farmers i n potato product ion outbid farmers i n corn product ion for a l l land between the market center and the dis tance t^. Between t and t c , corn producers can b i d the highest which means that they rece ive the lands . In t h i s way more crops can be added r e s u l t i n g i n the land use p a t t e r n f o r the a g r i c u l t u r a l produc t ion . 2 .1 .2 Urban Land Uses This s i m p l i f i e d model of the a g r i c u l t u r a l rent can now be appl ied to urban land uses . I sard mentions that the follow-ing f a c t o r s are determining the p r i c e each p o t e n t i a l user i s w i l l i n g to b i d : 1. e f f e c t i v e dis tance from core . 2. a c c e s s i b i l i t y of the s i t e to p o t e n t i a l customers. 3. number of compet i tors , t h e i r l o c a t i o n s , and the i n t e n s i t y with which they v ie f o r sa les ; and 4-. prox imi ty to land devoted to an i n d i v i d u a l use or a set of uses which are complementary - 17 -i n terms of \"both a t t r a c t i n g p o t e n t i a l customers and c u t t i n g cos t s , whether they be product ion , s e r v i c e , a d v e r t i s i n g , or other . 18 In Alonso ' s study only the f i r s t of these fac tors i s cons idered . He states that i n a c e n t r a l i z e d c i t y the second f a c t o r i s a lso impl ied because a c c e s s i b i l i t y of the s i t e to p o t e n t i a l customers w i l l decrease with d is tance from the center and furthermore that the other f a c t o r s , which r e l a t e to the interdependence of business l o c a t i o n s , are too complex f o r an ana lys i s w i th in the 19 scope of h i s work. y Commercial and Service Land Uses Commercial and serv ice a c t i v i t i e s are t r y i n g \"to maximize the volume of business that they transact\" and t h i s can be achieved i f \"they are located near the center of the day-2 0 time popu la t ion .\" Th i s means that a c c e s s i b i l i t y i s the key f a c t o r to a p r o f i t a b l e l o c a t i o n . I f we assume a market center at which most t ransac t ions take p l a c e , such as the C e n t r a l Business D i s t r i c t , we w i l l have fewer t ransac t ions with i n c r e a s i n g d i s tance , and s e l l i n g costs ( e . g . a d v e r t i s i n g ) must increase v/ith d is tance from the center i n order to o f f se t the decrease of a c c e s s i b i l i t y . Another cost i tem which changes with l o c a t i o n i s rent per acre , which leads to the b i d rent func t ion whose slope i s - 18 in f luenced by the fo l l owing f a c t o r s : The c e i l i n g rent per acre would dec l ine f a s t e r with d is tance from the market center , the greater the increased s e l l i n g costs to achieve the same volume of bus iness , the greater the number of t ransac t ions per square foot of f l o o r space and the l e s s poss ib le i t would be to subs t i tu te cheaper land for other i n p u t s . 1 These f a c t o r s are the same as those p e r t a i n i n g to a g r i -c u l t u r a l product ion: s e l l i n g costs are t r a n s p o r t a t i o n cos ts ; t ransac t ions per square foot of f l o o r space r e p r e -sent y i e l d per acre; and s u b s t i t u t i o n of land f o r nonland inputs i s the same. I n d u s t r i a l Land Uses For the l o c a t i o n of manufacturing a c t i v i t i e s the b i d rent func t ion i s in f luenced by f a c t o r s which d i f f e r from a g r i -c u l t u r a l - and urban a c t i v i t i e s depending on a c c e s s i b i l i t y . The p r o f i t of a manufacturing plant i s i n general de ter -mined by sales cost not only to the l o c a l market but p r i m a r i l y to outs ide areas . Therefore \" t o t a l revenues w i l l not n o t i c e a b l y s h i f t as the p lant i s located at d i f f e r e n t d is tances from the c i t y center , s ince we are i n i t i a l l y assuming that transport costs are equal i n every d i r e c t i o n . \" i But t o t a l cost may change with the l o c a t i o n of the p l a n t . With i n c r e a s i n g distance from the daytime popula t ion , higher wages have to be paid because commuting costs have to be - D e -compensated. Very important f o r manufacturing i s the s u b s t i t u t i o n of nonland inputs f o r l a n d . Because rents are d e c l i n i n g with d i s tance , the p lants can use more land which enables them to have more e f f i c i e n t flows i n the assembly process . Therefore greater economies of scale 23 can be achieved. ^ Hence the slope of the b i d rent func t ion depends upon the increase i n the wage r a t e , the decrease i n rent per acre necessary to o f f se t t h i s wage increase and economies of scale made poss ib le by the s u b s t i t u t i o n of more land f o r nonland f a c t o r s of product ion.24 R e s i d e n t i a l Land Uses R e s i d e n t i a l land i s of primary importance because i t covers f o u r - f i f t h s of a l l p r i v a t e l y developed land i n major 25 26 American c i t i e s . y The b i d rent or b i d p r i c e curve of a res ident i s the set of p r i c e s f o r land the i n d i v i d u a l would pay at d i f f e r e n t d is tances while d e r i v i n g a constant l e v e l of s a t i s f a c t i o n . A comprehensive d i s c u s s i o n of the r e s i d e n t i a l b i d p r i c e curve i n diagrammatic and mathematical form was done by Alonso^?. He combines ind i f f erence ana lys i s with the budget or p r i c e - o p p o r t u n i t y theory and a r r i v e s at a mathematical equation f o r the r e s i d e n t i a l b i d p r i c e curve . - 20 -I t i s \"beyond the scope of t h i s study to give, the p a r t i c u -l a r s concerning the d e r i v a t i o n of t h i s equat ion . Therefore only the main f ind ings w i l l be mentioned. Once again the slope of the b i d p r i c e curve i s negative and depends upon the commuting costs and the tastes of the i n d i v i d u a l s . 2 .1 . 3 General E q u i l i b r i u m A f t e r t h i s d i s c u s s i o n of c lasses of land uses we can now superimpose a l l b i d rent curves . We have seen that a l l rents dec l ine with dis tance from the market center o r , as we might also name i t , the center of the daytime popu la t ion . In g e n e r a l i z i n g we can state that the slopes of the b i d rent curves depend on the output of the land us ing a c t i v i t y per acre , t r a n s p o r t a t i o n , s e l l i n g or wage cos t s , and the p o s s i b i l i t y of s u b s t i t u t i o n between land and nonland i n p u t s . The shape of these curves may be as shown i n F igure 3 (p. 21). By r o t a t i n g these curves around the market center we get a concentr ic land use p a t t e r n ; we a r r i v e d at the concentr ic 29 zone theory ^ i n a d i f f e r e n t way. U n t i l now we discussed i d e a l i z e d condi t ions assuming uniform f e r t i l i t y of s o i l , uniform topography, equal t r a n s p o r t a t i o n cost i n a l l d i r e c t i o n s and one market center . In order to represent a r e a l i s t i c s i t u a t i o n , a l l these f a c t o r s have to be modi f i ed . F i g u r e 5 : B i d \"rent f u n c t i o n s f o r u r b a n l a n d u se s Rent \u00E2\u0080\u00A2 Source : Hugh. 0 . N o u r s e , Region9-1 Eccmomics (New Y o r k : Mc G r a w - H i l l , 1968) , p . 115 . F i g u r e 4- : B i d r e n t f u n c t i o n s w i t h a h i e r a r c h y o f c e n t e r s M e t r o p o l i t a n / y R e g i o n a l C e n t e r i Ne ighborhood I C e n t e r 1 ! ! i i 1 1 i i i i i i 1 i i i i ! I j ' i i i i i R e g . Shopping C e n t e r i i ' ^ > s ^ \ I I 1 i i i I 1 1 S a t e l i t e C i t y 1 1 1 D i s t a n c e Source : Hugh 0 . N o u r s e , R e g i o n a l Economics (Hew Y o r k : Mc G r a w - H i l l , 1-968), p . 120. - 22 -I f we have s o i l s with d i f f e r e n t f e r t i l i t y , the p r o f i t -a b i l i t y of a g r i c u l t u r a l product ion w i l l be changed and the b i d rent f u n c t i o n w i l l s h i f t up or down. D i f f e r e n t topo-g r a p h i c a l features r e s t r i c t development because product ion , s e l l i n g or cons truc t ion costs w i l l increase with ascending s lope . A d r a s t i c change occurs when we modify the assump-t i o n of equal t r a n s p o r t a t i o n costs i n a l l d i r e c t i o n s . A major highway, f or example, allows eas i er and f a s t e r t r a v e l . Therefore t r a n s p o r t a t i o n costs w i l l be reduced and the slope of the b i d rent func t ion dec l ines along the highway. The r e s u l t i s that the concentr ic r ings change to a \"star\" p a t t e r n . F i n a l l y , we have not only one market center . There ex i s t s 30 a h i e r a r c h y of centers-' and each subcenter causes a peak i n the b i d rent f u n c t i o n . The r e s u l t i n g shape of the b i d rent f u n c t i o n i n one d i r e c t i o n from the main center might occur as shown i n f i g u r e 4- (p. 21). The d i s c u s s i o n of these modi f icat ions gives a r e a l i s t i c p i c t u r e of land use patterns wi th in a metropol i tan r e g i o n . There i s evidence of s trong economic forces which determine the l o c a t i o n of a c t i v i t i e s . Hence i t i s important f o r the planner and model -bui lder to recognize t h e i r magnitude and to take them into account. - 2 3 -2.2 S o c i a l Physics The findings of s o c i a l physics are mainly based on s t a t i s -t i c a l r e g u l a r i t i e s i n explaining mass behavior. They do not explain the behavior of the i n d i v i d u a l . In the search f o r explanation of the s p a t i a l structure of urban areas and regions, g r a v i t y and p o t e n t i a l concepts have been applied. It was recognized that these p h y s i c a l p r i n c i p l e s could also be applied to s o c i a l phenomena. H i s t o r i c a l and comprehensive reviews of these concepts i n 31 a p p l i c a t i o n to human mass behavior are those by Carrothers^ 32 and Isard.^ In the following part of t h i s study the main features of these concepts w i l l be given i n order to provide i n s i g h t and understanding of the basic concepts which are applied i n b u i l d i n g l o c a t i o n a l models. B a s i c a l l y , the gravity concept of human i n t e r a c t i o n s postu-l a t e s that an a t t r a c t i n g force of i n t e r a c t i o n between two areas of human a c t i v i t y i s created'by the population masses of the two areas, and a f r i c t i o n against i n t e r a c t i o n i s caused by the intervening space over which the i n t e r a c t i o n must take place. 5 3 In mathematical notation, the r e l a t i o n s h i p can be expressed as follows: - 24 -1 0 f < V where I . . : i n t e r a c t i o n between center i and j -'-CI P. ; P. : population of area i and j , r e s p e c t i v e l y -\u00E2\u0080\u00A2- J D. . : distance between the two centers. -*-u The f i r s t statement of t h i s concept was made by C a r e y ^ i n the l a s t century. Later i t was applied to migration by 55 56 57 R a v e n s t e i n ^ and Young^ and to r e t a i l trade by R e i l l y . A f t e r these e a r l y applications Z i p f ^ and Steward^ generalized the concept and formulated the \"force\" of in t e r a c t i o n s which i s where k : a constant of p r o p o r t i o n a l i t y , and, i n analogy to physics, the \"energy\" of i n t e r a c t i o n which r e s u l t s from t h i s force P i P i 3-D Stewart also formulated the \"pot e n t i a l of population\" which i s a measure \" i n d i c a t i n g the i n t e n s i t y of the p o s s i b i l i t y of i n t e r a c t i o n . \" - 25 --V -i a k where : p o t e n t i a l at i of the population of \" area j i n d i c a t e s the p o s s i b i l i t y of i n t e r a c t i o n between an i n d i v i d u a l at i and a population at g. In r e a l i t y there i s more than one mass j which leads to the t o t a l p o t e n t i a l at a point i . It i s the sum of the separate p o t e n t i a l s l T t o t \u00E2\u0080\u00A2 k + k -f* \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 lc ' n n = k D. i l D i 2 D. i n 0 = 1 D 10 To c a l c u l a t e t h i s t o t a l p o t e n t i a l i t has to be pointed out that the mass at i also creates a p o t e n t i a l . But the distance cannot be considered as zero because t h i s would r e s u l t i n an i n f i n i t e p o t e n t i a l . This d i f f i c u l t y may be overcome i n two ways. Carrothers proposes to take the average of the distance from the center of area i to i t s periphery. It i s also possible to express a l l denominators 41 as 1 + D. \u00E2\u0080\u00A2. For a p p l i c a t i o n s of gravity models i t i s Important to discuss some measurements of the two v a r i a b l e s , mass and distance. Not only can population be used f o r measuring mass. The choice of the measurement depends to a great extent on the problem to be studied, a v a i l a b l e data, and 42 r e l a t e d considerations . I f , f o r instance, migration i s . t h e - 26 -focus, employment opportunities seem to be a more adequate measure than population. Yery often i t i s also necessary 45 to assign weights to the mass. J Suppose we are studying consumer behavior. In t h i s case i t i s obvious that income l e v e l also has to be taken into account because an area with higher income w i l l consume more than an area with the same population but a lower per c a p i t a income. In t h i s case population could be m u l t i p l i e d by per c a p i t a income. In other cases weights can be given i n form of sex, education, ethnic composition etc. Distance can i n i t i a l l y be measured along a s t r a i g h t l i n e or along transportation routes. But f o r t r a f f i c studies i t 44 seem that t r a v e l time provides better estimates. Other p o s s i b i l i t i e s are transportation costs, number of stops or 45 even number of gear s h i f t s . ^ The influence of distance i s determined by an exponent which has d i f f e r e n t values depending on the phenomena studied. In t r i p behavior i t 4. was found that the exponent i s a function of trxp purpose. There i s also discussion about an exponent applied to the 47 measure of mass. Carrothers notes that, f o r instance, t h i s may be necessary i n a case \u00C2\u00A5/here \"agglomeration econo-mies\" e x i s t . I f these various modifications of distance and mass are applied, the g r a v i t y concept can be expressed i n i t s most general form: \u00E2\u0080\u00A2 - 27 -k . w. ( P , ) * d a u where in t e r a c t i o n s between i and j constant of p r o p o r t i o n a l i t y w weighting f a c t o r s P mass mass exponent distance exponent The g r a v i t y concept has been applied to a great v a r i e t y of problems of human in t e r a c t i o n s and i n many cases the r e s u l t s indicate that i t \"constitutes a very promising 48 technique f o r regional a n a l y s i s \" . . Nevertheless, i t i s often c r i t i c i z e d as an attempt to apply the p h y s i c a l law of g r a v i t a t i o n of Newton to s o c i a l behavior of men. Although i t i s apparent that s i m i l a r i t i e s do ex i s t between the p h y s i c a l and the s o c i a l world, i t i s necessary to search f o r more fundamental p r i n c i p l e s which determine human behavior. y But m absence of such a theory the gr a v i t y concept can be applied i f i t s l i m i t a t i o n s and features are considered. Since t h i s study focuses on the l o c a t i o n of a c t i v i t i e s within a region, i t means that we are dealing with a macro-scopic scale and aggregated v a r i a b l e s . Hence i t seems that - 28 -the g r a v i t y concept i s applicable to a s i g n i f i c a n t degree because i t s \"fundamental notion pertains to a r e l a t i v e l y 50 huge mass composed of a multitude of i n d i v i d u a l u n i t s . \" Footnotes 1 John Friedmann and W i l l i a m Alonso, Regional Development and P l a n n i n g (Cambridge,\"Massa-c h u s e t t s : The MIT P r e s s , 1964), p. 2. 2 I r a S. Lowry, A Model of M e t r o p o l i s (Santa Monica, C a l i f o r n i a : The Rand\u00E2\u0080\u00A2Corporation, Memorandum RM - 4035 - RO, August 1964), p. 2 0 . 3 M. K i l b r i d g e and S. Carabateas, \"Urban P l a n n i n g Model\", E k i s t i c s , V o l . 24, No. 145 (Dec. 1967), p. 481. 4 Hugh 0. Nourse, R e g i o n a l Economics (New York: McGraw-Hill Book Company, 1968), p. 1. 5 Johann H e i n r i c h von Thiinen, Der i s o l i e r t e Staat i n Beziehung auf L a n d w i r t s c h a f t und N a t i o n a l - okonomie (Hamburg: 1826). ; ' 6 Walter I s a r d , L o c a t i o n and Space Economy (New York: John Wiley & Sons, Inc\"., 1956), p. 27 - 28 7 A l f r e d Weber, Uber den Standort der I n d u s t r i e n (Tubingen: 190 "Thesis/Dissertation"@en . "10.14288/1.0103949"@en . "eng"@en . "Planning"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Regional land use allocation models and their application to planning"@en . "Text"@en . "http://hdl.handle.net/2429/35163"@en .