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Regional land use allocation models and their application to planning Fricker, Urs Josef 1969

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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 • ••• 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 . • 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 - ^ § - 0 . 0 5 • t ) p^(t) = 20 - 1 .5 • 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 • 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 -•- 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 • k + k -f* • • • • 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. •. 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 ¥/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: • - 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•Corporation, 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<JT^ ~ " 8 August Losch, Die raumliche Ordnung der Wirt-' s c h a f t (Jena: 1940); see a l s o h i s a r t i c l e "The Nature of Economic Regions", i n R e g i o n a l  Development and P l a n n i n g , ed. by John Friedmann and W i l l i a m Alonso" (Cambridge, Massachusetts: The MIT P r e s s , 1964), p. 107 - 115. 9 Walter I s a r d , Op. c i t . 10 Walter I s a r d , Op. c i t . , p. 200. 11 David R i c a r d o , On the P r i n c i p l e of P o l i t i c a l  Economy and T a x a t i o n , 1817• ~ 12 Johann H e i n r i c h von Thunen,. Op. c i t . 13 Walter I s a r d , Op. c i t . 14 Edgar S. Dunn, J r . , The L o c a t i o n of A g r i c u l t u r a l  P r o d u c t i o n ( G a i n e s v i l l e : U n i v e r s i t y of F l o r i d a P r e s s , 1954). 15 W i l l i a m Alonso, L o c a t i o n and Land Use (Cambridge Massachusetts: Harvard U n i v e r s i t y P r e s s , 1964). - 30 -16 Hugh 0 . Bourse, Op. c i t . 17 William Alonso, Op. c i t . , P. 37 - 4-2. 18 Walter Isard, Op. c i t . , p . 2 0 0 . 19 William Alonso, Op. c i t . , P. 4-4-. 20 Hugh 0. Nourse, Op. c i t . , P • 1 0 5 . 21 Hugh 0 . Nourse, Op. c i t . , P- 1 0 1 . 22 Hugh 0. Nourse, Op. c i t . , P. 1 0 7 . 23 Hugh 0 . Nourse, Op. c i t . , P. 108. 24- Hugh 0 . Nourse, Op. c i t . , P- 108. 25 Harland Bartholomew, Land Uses i n American'Cities p. 4-6. 26 William Alonso, Op. c i t . , p. 5 9 . 27 William Alonso, Op. c i t . , p. 59 - 7 4 . 28 Hugh 0 . Nourse, Op• c i t . , p. 117 . 29 See f o r instance Chauncy D. Harris and Edward L. Ullman, "The Nature of C i t i e s " , i n Paul K. Hatt and Albert J . Reiss, J r . (eds.), C i t i e s and  Society (New York: The Free Press of Glencoe, 1 9 5 7 ), p. 237 - 24-7; the concentric zone theory originates i n the work by Ernest W-. Burgess, _ "The Growth of the C i t y " i n R.E."Park ed.,;The  C i t y (Chicago: U n i v e r s i t y of Chicago Press, 1 9 2 5 ) . He found t h i s theory i n studying the e c o l o g i c a l processes within the c i t y of Chicago. 30 This hierarchy i s the concern of the Central Place Theory. The o r i g i n a l work is.Walter C h r i s t a l l e r , Die zentralen Orte i n Suddeutschland (Jena: Gustav Fischer.Verlag,• 1 9 3 3 ) ; see also, Brian J.L. .Berry and A l l a n Pred, Central Place Studies: A B i b l i o - graphy of Theory and A p p l i c a t i o n (Philadelphia: Regional Science Research I n s t i t u t e , 1 9 6 1 ) ; the concepts of the c e n t r a l place theories are' also applied to the formation of subcenters within a c i t y : Hans Carol, "The : Hierarchy of ..Central.Func-tions within the City","Annals of the Association of American Geographers, V o l . 50 ( I 9 6 0 ) , p. 4-19 -4-38. - 3 1 -31 G e r a l d A.F. C a r r o t h e r s , "An H i s t o r i c a l Review of the G r a v i t y and P o t e n t i a l Concepts of Human I n t e r a c t i o n " , J o u r n a l of the American I n s t i t u t e  of P l a n n e r s , V o l . 22 ( S p r i n g 1956), p. 94 - 102. 32 Walter I s a r d , Methods of R e g i o n a l " A n a l y s i s (New York: John Wiley & Sons, I n c . , I 9 6 0 ) , p. 493 - 568. 3 3 Gerald A.P. C a r r o t h e r s , Op. c i t . , p. 94. 3 4 H.C. Carey, P r i n c i p l e s of S o c i a l Science ( P h i l a d e l p h i a : J.B. L i p p i n c o t t and Co., 1858). 35 E.G. Ravenstein, "The"Law of M i g r a t i o n " , J o u r n a l  of the Royal S t a t i s t i c a l S o c i e t y , 48 (6/13853 p. 167 - 235, and 52 (6/1889) p. 241- - 305-36 E.C. Young, The Movement of Farm P o p u l a t i o n ( I t h a c a : C o r n e l l A g r i c u l t u r a l Experiment S t a t i o n , B u l l e t i n 426; 1924). 3 7 W.J. R e i l l y , The Law of R e t a i l G r a v i t a t i o n '(New York: W.J. R e i l l y Co. , 1 9 3 1 ) . : : ~ 38 George K. Z i p f , Human Behavior and the P r i n c i p l e  of Least E f f o r t (Reading, Mass.: Addison-Wesley P r e s s , 1949), and "The P, Pp/D Hypothesis on the I n t e r c i t y Movement of Persons", American Socio-" l o g i c a l Review, V o l . 11 (Oct. 1946) p. 677 - 86. 3 9 John Q. Stewart "Demographic G r a v i t a t i o n s Evidence and A p p l i c a t i o n " , Sociometry, V o l . 11 (Febr. and May 1948). 40 C a r r o t h e r , G e r a l d A.P., Op. c i t . , p. 9 6 . 41 Theodore Anderson, " P o t e n t i a l Models and ..Spatial D i s t r i b u t i o n of P o p u l a t i o n " , Papers and.Pro-' ceedings of the R e g i o n a l Science A s s o c i a t i o n , V o l . 2 (1956), p. 178. 42 Walter I s a r d , Op. c i t . , p. 505. 43 I b i d . , p. 508. 44 B r i a n V. M a r t i n , F r e d e r i c k W.: Memmpt.and Alexander J . Bone, P r i n c i p l e s and Techniques of P r e d i c t i n g  Future Demand f o r Urban Area T r a n s p o r t a t i o n (Cambridge: The MIT-Press, 1966), p. 1 3 9 . 45 Walter I s a r d , Op. c i t . , p. 506. - 3 2 -46 Brian Y. Martin, Frederick W. Memmot and Alexander J . Bone, Op. c i t . , p. 139. 47 Gerald A.P. Carrothers, Op. c i t . , p. 98. 48 Walter Isard, Op. c i t . , p. 566. 49 Brian Y. Martin, Frederick W. Memmot and Alexander J . Bone, Op. c i t . , p. 145. 50 Walter Isard, Op. c i t . , p. 513-3. ELEMENTS OP MODEL BUILDING In t h i s chapter we s h a l l deal with a few important " r u l e s " or "standards" which should be considered by a model-b u i l d e r i n order to b u i l d or evaluate a model. F i r s t of a l l , mention w i l l be made of the d i f f e r e n t a v a i l a b l e types of models. This w i l l be followed by a discussion of the basic steps i n design and c a l i b r a t i o n of a model. With the help of these r u l e s i t i s possible to b u i l d better models by s p e c i f y i n g t h e i r l i m i t a t i o n s and d e f i c i e n c i e s because i t w i l l never be possible to overcome a l l d i f f i c u l t i e s ; i t w i l l always be necessary to make compromises between the ava i l a b l e resources. 3.1 Typology of Models A great v a r i e t y of models has been discussed and b u i l t i n recent years and very often i t i s extremely d i f f i c u l t to get a systematic overview. Therefore, selected attempts toward a general typology of models w i l l be summarized. In a "rough c h a r a c t e r i z a t i o n " Churchman, Ackoff and Arnoff d i s t i n g u i s h 3 types of models, which are: - yv -- i c o n i c models which are a p i c t o r a l or a v i s u a l representation of c e r t a i n aspects of a system, e.g. photographic or a r c h i t e c t u r a l models - analogue models which employ one set of proper-t i e s to represent some other set of properties which the system i n study possesses, e.g. the flow of e l e c t r i c i t y i n a wire can "be studied i n considering an analogue, i n the form of the flow of water i n a pipe; maps and graphs belong also to these types - symbolic models which employ symbols to designate properties of the -system under study i n the form of mathematical equations. This general c l a s s i f i c a t i o n i ndicates the wide range of appl i c a t i o n s of the term "model" i n a great number of s c i e n t i f i c d i s c i p l i n e s . The land use models which are considered i n t h i s study are symbolic models. A more d e t a i l e d attempt toward a typology was undertaken p by H a r r i s . He distinguishes models by defi n i n g " f i v e or s i x dimensions of d i f f e r e n c e s " i n a "sequence of dichoto-mies and antinomies" as follows: d e s c r i p t i v e h o l i s t i c macro s t a t i c d e t e r m i n i s t i c simultaneous versus versus versus a n a l y t i c p a r t i a l micro versus versus versus dynamic p r o b a b i l i s t i c sequential Descriptive versus A n a l y t i c a l Models The d e s c r i p t i v e models are only an exploratory i n v e s t i -gation into r e l a t i o n s h i p s , whereas a n a l y t i c a l models make - 35 -statements about cause and e f f e c t of the r e l a t i o n s h i p i n the r e a l world. In applying the discussion of the previous chapter concerning the approaches to the a l l o c a t i o n of economic a c t i v i t i e s , we would c l a s s i f y models based on -A economic l o c a t i o n theory^ as a n a l y t i c a l and on s o c i a l 4 physics as d e s c r i p t i v e models. In most of the models which have been developed i n the l a s t decade i t has been found that a n a l y t i c a l models are much more d i f f i c u l t to b u i l d because of the lack of knowledge about human behav-i o r a l aspects. H o l i s t i c versus P a r t i a l Models The planner i s forced to consider the implications of h i s p o l i c i e s on the " t o t a l environment"^ which means that he i s n a t u r a l l y i n c l i n e d toward a h o l i s t i c approach. On the other hand, a s o c i a l researcher such as the economist, i s more concerned with p a r t i a l aspects. Hence, he b u i l d s p a r t i a l models and the model-builder i n the planning profession has a tendency to use them and to construct h o l i s t i c models out of a number of p a r t i a l models. In a systems approach t h i s can be expressed as an i d e n t i f i c a t i o n of subsystems. H a r r i s ^ sees three inherent problems i n t h i s approach and urges a c a r e f u l design of such models. The f i r s t problem - 3 6 -concerns the complexity of communication between the sub-models because each submodel often has a large number of 7 d i f f e r e n t v a r i a b l e s . The second . d i f f i c u l t y stems from the fa c t that " p a r t i a l models are apt to use var i a b l e s not o r d i n a r i l y predicted by any other p a r t i a l models." The t h i r d problem a r i s e s from the need to ensure that the d i v i s i o n of the problem in t o p a r t i a l problems i s r e a l i s t i c not only i n regard to the subsystems, but also i n regard to the o v e r a l l or t o t a l system. I t i s conceivable that p a r t i a l land use models combined do not represent the t o t a l land use pattern of an area because the i n t e r a c t i o n s between the submodels are not adequately taken into account. Nevertheless, i t seems that i n proceeding from p a r t i a l to h o l i s t i c models by "expanding the number of variables and processes that are endogenous to our model system and reduc-ing the number that are exogenous, we s h a l l wind up with a h o l i s t i c model that represents the t o t a l i t y of human s o c i a l o development." Although such a model would represent an i d e a l i t has to be pointed out that the present stage of model b u i l d i n g i s f a r from reaching i t . But i n t h i s study we s h a l l deal with h o l i s t i c models - i n a narrower sense -by di s c u s s i n g models which a l l o c a t e a l l land using a c t i v i t i e s within a region. - 37 -Macro- versus Micro Models Generally, planners and public decision-makers are i n t e r -ested i n the aggregated, or macro aspects of t h e i r problems, e.g. t o t a l population or population groups. But on the opposite side a region i s a sum of micro u n i t s , i n c l u d i n g i n d i v i d u a l s , f a m i l i e s and organizations, and these u n i t s make decisions i n regard to t h e i r welfare. These two aspects are not always recognized and most of the aggregated macro models more or l e s s make use of theories, concepts, i n t r o -spection, and observation regarding the behavior at the q micro l e v e l . The opinions i n regard to the usefulness of both types of models d i f f e r . L o w r y f i n d s that macro-models are more s a t i s f a c t o r y whereas Harris favors micro-models. S t a t i c versus Dynamic Models Planning f o r c i t i e s and regions and the involved d e c i s i o n -making process works within a continuously changing system, which means that a model ought to be dynamic. Time can be b u i l t into a model by using d i f f e r e n t i a l equations or d i f f e r e n c e equations. But there are d i f f i c u l t i e s involved i n formulating dynamic models, e s p e c i a l l y i n regard to the observation and analysis of the "time-dependent b e h a v i o r " 1 1 of the d i f f e r e n t parts of the system. The a v a i l a b i l i t y of time-series data i s l i m i t e d and observations during several - 38 -years can hardly he made. In H a r r i s ' opinion manufac-tu r i n g l o cations should he observed t h i r t y or more years, r e t a i l trade f i v e to ten years, and r e s i d e n t i a l l o c a t i o n s between f i v e to f i f t y years, depending on the purpose and on the view of the processes involved. Apart from data there are d i f f i c u l t i e s i n formulating 15 optimization problems; ^ l i n e a r programming i s s t a t i c i n character and the r e s u l t i s an equilibrium. But i n the r e a l world the conditions are dynamic and there i s only a strong tendency toward an equilibrium. Nevertheless, s t a t i c models provide i n a great number of cases a s a t i s -f a c t o r y abstraction and are widely used because they are easier to formulate and to use than are dynamic models. Deterministic versus P r o b a b i l i s t i c Models Human behavior i s p r o b a b i l i s t i c rather than deterministic i n character; therefore we have u n c e r t a i n i t y i n decisions r e l a t i n g to the development of an urban region. In addition, there are probable changes i n technology and taste which make p r e d i c t i o n extremely d i f f i c u l t . How can we, f o r instance, p r e d i c t the r e l o c a t i o n of f a c t o r i e s or the amount of public investment i n transportation? Such features are s t i m u l i to b u i l d p r o b a b i l i s t i c models which - 39 -i n t e r n a l l y generate random events, thus enabling them to simulate the u n c e r t a i n i t y of the behavior of the r e a l world, and to determine c e n t r a l tendencies and t h e i r v a r i a -t i o n s (Monte Carlo Simulation). 1' 4" Although t h i s nature of the r e a l world can be conceptualized, i t i s very d i f f i c u l t to b u i l d p r o b a b i l i s t i c models and only a few have been b u i l t , mostly i n regard to r e s i d e n t i a l development. 1^ On a re g i o n a l scale there e x i s t s no proba-b i l i s t i c model which locates a l l land-using a c t i v i t i e s . Simultaneous versus Sequential Models The treatment of the locators can be c l a s s i f i e d i n the above manner. Har r i s points out that the d i s t i n c t i o n i s not very c l e a r but .given mainly "for the sake of complete-ness and c l a r i t y " . 1 ^ This d i s t i n c t i o n r e f e r s to the s o l u t i o n method of a set of equations and the choice between the two i s " l a r g e l y one of convenience." The attempts by Churchman et a l . and by Harris to set f o r t h a typology of models are i n d i c a t o r s of a great v a r i e t y of models. But these types are not necessarly r e l a t e d to community and regional planning. I t would therefore be quite h e l p f u l to c l a s s i f y models i n r e l a t i o n to the planning 17 process. Such an attempt was undertaken by Lowry when he distinguished between d e s c r i p t i v e models, p r e d i c t i v e models - 4 - 0 -and planning models. This c l a s s i f i c a t i o n i s mainly based on the purpose of the models. Descriptive models have the li m i t e d objective of r e p l i c a -t i n g or.simulating the relevant features of the urban environment. They are of value i n planning because they reveal much about the structure and mechanism of the urban environment. P r e d i c t i v e models do not only r e p l i c a t e ; they also "specify 18 a causal sequence", e.g. one might postulate that a one-unit change of a variable x w i l l cause a change i n variable y by three u n i t s . These models are of great s i g n i f i c a n c e i n planning because planning i s future oriented and one of i t s main tasks i s p r e d i c t i o n . The t h i r d type, planning models, are not s u f f i c i e n t l y developed yet, but they could be of utmost value f o r planners. Such models can be used not only f o r p r o j e c t i o n purposes but also f o r evaluation of the outputs i n terms of the goals which are intended to be achieved. In t h i s context Steger and Lakshmanan state that i f "a shorthand d e s c r i p t i o n of the emphasis i n the.planning process i n the 1950's was p r o j e c t i o n , i n the 1960's the corresponding term would be evaluation. -.19 The development of game theory and i t s a p p l i c a t i o n to - 41 -planning could be viewed as a step i n the d i r e c t i o n toward planning models. Dresner describes the main features of the game theory as follows: The theory of games of strategy may be described as a mathematical theory of decision-making by p a r t i c i p a n t s i n a competitive environment. In a t y p i c a l problem to which the theory i s appl i c a -ble, each p a r t i c i p a n t can bring some influence to bear upon the outcome of a c e r t a i n event; no single p a r t i c i p a n t by himself nor chance alone can determine the outcome completely. The theory i s then concerned with the problem of choosing an optimal course of action which takes into account the possible actions of the p a r t i c i p a n t s and the chance events.22 This d e s c r i p t i o n gives evidence that game theory can also help the planners to f i n d optimal st r a t e g i e s toward the achievment of t h e i r goals. This discussion of attempts toward a typology of models ind i c a t e s a wide v a r i e t y of types. In the following part of t h i s study we s h a l l leave t h i s broad spectrum and focus on the main aspects which have to be considered i n an actual model-building process. 3.2 Design of a Model When we say that a model i s designed i t implies mainly that - we decide which f a c t o r s or variables are relevant - 4-2 -to the problem which requires a sol u t i o n , - from the relevant f a c t o r s , those which can be described q u a n t i t a t i v e l y are selected, - the qua n t i f i a b l e f a c t o r s are then cut down to size by aggregation - f i n a l l y , the r e l a t i o n between the elements are expressed q u a n t i t a t i v e l y i n the form of mathema-23 t i c a l equations. ^  These steps are s i m i l a r to those which one takes i n a systematic study of a problem. The p r i n c i p a l d i f f e r e n c e i s that the model-builder's main concern i s q u a n t i f i c a t i o n . Nevertheless, c a r e f u l formulation of the problem i s a basic requirement f o r a successful model. This aspect i s emphasized by d i f f e r e n t s c i e n t i s t s . Ackoff, f o r example, quotes an old saying that "a problem well put i s h a l f 24-solved", while Lowry argues that "the art of model b u i l d -ing i s above a l l the art of s i m p l i f y i n g complicated prob-25 lems." A s i m i l a r opinion i s expressed by Bellman, who has extensive experience i n mathematical research and model-b u i l d i n g . He writes: I t often comes as a b i t of a shock to the young s c i e n t i s t when he r e a l i z e s that the basic problem Is more to f i n d the r i g h t question than the r i g h t answer."26 Following these remarks which stress the importance of the formulation of the problem i t i s now necessary to proceed - 45 -to the main steps mentioned above regarding model design. $.2.1 The Variables and t h e i r Relevance In studying the problem an attempt should be made to draw up a l i s t of a l l the elements which might be of influence. In most cases a long l i s t of variables w i l l r e s u l t . As an example, consider r e s i d e n t i a l l o c a t i o n . Influencing va r i a b l e s are income, size and structure of family, kind of employment, place of work, ethnic background, p r i c e of housing, socio-economic preferences (tastes),, a v a i l a b i l i t y of services and u t i l i t i e s , and housing p o l i c i e s . With such a l i s t the relevance of each of these variables i s extreme-l y d i f f i c u l t to evaluate. Nevertheless, f i r s t considera-tions can r e s u l t i n a s e l e c t i o n of high p r i o r i t y variables such as income, p r i c e of housing, socio-economic preferences and place of work. Now the possible q u a n t i f i c a t i o n of the variables should be considered. For several of these variables numbers -will be a v a i l a b l e ; i n some cases, such f i g u r e s w i l l even be quite d e t a i l e d . But f o r others, numbers w i l l not be a v a i l -able. In the above example of r e s i d e n t i a l l o c a t i o n , i t w i l l be extremely d i f f i c u l t to measure socio-economic preferences or the influence of housing p o l i c i e s . Such circumstances w i l l always force the model-builder to leave out some of the - 4-4- -elements which may he relevant. There are mainly two reasons which influence the omission of v a r i a b l e s : the nature of the variable - they are often not suited to numer-i c a l measures; or the l i m i t e d knowledge and a b i l i t y of the 27 analyst. ' It should be emphasized, however, that decisions concerning omissions should not be based on convenience. Rather," a l l possible e f f o r t s should be made to get quantitative meas-ures f o r relevant v a r i a b l e s and the model buil d e r " w i l l do well to understand what i s l e f t out as well what i s l e f t • „28 3.2.2 The Level of Aggregation Important decisions have to be made i n regard to the l e v e l of aggregation. They should be determined by the purpose of the model and the e n t i t i e s which the model i s presumed to r e p l i c a t e . ^ But above a l l , the features of ava i l a b l e data are key f a c t o r s , ^ 0 and they should be c a r e f u l l y e v a l -uated together with the purpose and e n t i t i e s of the model. The data are i n most cases cross-section data which means that they deal with e n t i t y to e n t i t y contrasts. I f e n t i t i e s are aggregates made up a r t i f i c i a l l y , e.g. census t r a c t s , then contrasts of behavior of variables w i l l occur between d i f f e r e n t l e v e l s of aggregation. This means that problems - 4-5 -occur by studying, f o r instance, the l o c a t i o n a l behavior of households. There w i l l be d ifferences i n model-building between neighborhood planning by the disaggregation of census t r a c t data and regional planning by aggregating census t r a c t data to l a r g e r zones. In model-building there are often c o n f l i c t s between the p a r t i c i p a t i n g groups of people i n r e l a t i o n to the l e v e l of aggregation. Wolfe and Ernst d i s t i n g u i s h between three groups: the planners, the operations researchers or mathe-maticians, and representatives of the planning a c t i v i t y (e.g. p o l i t i c i a n s , i n t e r e s t g r o u p s ) . ^ The f i r s t basis of c o n f l i c t i s that these groups have never worked together before and agreement on the l e v e l of aggregation i s d i f f i -c u l t . The planner and the representatives of the planning a c t i v i t y favor d e t a i l while the operations researcher wishes to r e t a i n s i m p l i c i t y e s p e c i a l l y i n the i n i t i a l phase, even at the cost of possible loss of v a l i d i t y and u t i l i t y of the r e s u l t s . At the beginning of model-design there i s i n most cases an over-ambition i n the l e v e l of d e t a i l sought. This has been the primary reason f o r unanticipated cost and time. Therefore, one should always s t a r t on a r e l a t i v e l y simple and not too ambitious scale, incorporating greater d e t a i l only as i t s p o t e n t i a l u t i l i t y i s c l e a r l y recognized. - 46 -5*2.3 Formulation of the Mathematical Relationship In b u i l d i n g a model we abstract from the r e a l world and use data of d i f f e r e n t q u a l i t y . This means that Wo types 33 of errors o c c u r : " 1. erx^ors of s p e c i f i c a t i o n which r e s u l t from abstraction, e.g. representation of a nonlinear r e l a t i o n s h i p by a l i n e a r equation or the omission of vari a b l e s of les s relevance; and 2. measurement errors which Include errors of data c o l l e c t i o n and sampling. I t i s the objective of the model bu i l d e r to minimize errors as much as po s s i b l e . Hence, we s h a l l discuss the implica-tions of errors which r e s u l t i n rules f o r the structure of the mathematical r e l a t i o n s h i p of a model. The model i s a mathematical equation or a system of equa-t i o n s . There are errors i n the inputs (measurement errors) and the question i s how w i l l these a f f e c t the r e s u l t or the 34 output. The output i s influenced as follows:^ Model: z = f . (x-, ; x ? . . . . x ) E r r o r : . e x where error of function z (output) e measurement errors of the input v a r i a b l e s i d c o r r e l a t i o n between x. and x. - 4-7 -This equation, which i s exact f o r l i n e a r functions and a good approximation f o r nonlinear f u n c t i o n s , ^ shows that f i r s t of a l l the q u a l i t y of data (e ) influences the x i e r ror of the output. But, i n addition, the mathematical dz structure (JJJ—) of the model influences the degree of accumulation of e r r o r s . This equation can he applied to the d i f f e r e n t mathematical operations (see Appendix 1); t h i s leads to the following "rules of thumb" f o r b u i l d i n g or evaluating a model: 1. Avoid i n t e r c o r r e l a t e d v a r i a b l e s . 2. Add where p o s s i b l e . 3. I f you cannot add, multiply or d i v i d e . 4-. Avoid as f a r as possible taking differences or r a i s i n g v ariables to powers. We can also discuss s p e c i f i c a t i o n errors which, i n turn, leads to a consideration of simple and complex models. Assume, f o r example, a simple l i n e a r model with a c e r t a i n s p e c i f i c a t i o n e r r or. A good model b u i l d e r t r i e s to improve h i s model i n order to reduce the s p e c i f i c a t i o n e r r o r . This means that the model i s becoming more complex. But i n -creasing complexity implies a greater number of mathema-t i c a l operations and therefore, as Alonso argues, the model i s more "explosive with regard to the compounding of [measurement] e r r o r s . " 37 The sum of s p e c i f i c a t i o n and measurement errors ( t o t a l - 4-8 -errors) i s equal to E. tot + e m The r e l a t i o n s h i p between the t o t a l error and the complexity •58 of a model has been postulated by Alonso^ as i n Figure 5 (see p. 4-9). The best point f o r p r e d i c t i o n i s the bottom of the t o t a l e r r o r curve. Under Alonso's assumptions i t i s not possible to gain a reduction of the errors through increasing the complexity of the model. This concept i s also applied by Alonso to two models whose s p e c i f i c a t i o n errors are i d e n t i -c a l but where the q u a l i t y of data varies between the two. He f i n d s that with l e s s accurate data the bottom of the t o t a l e r r or curve moves to the l e f t . This implies that when 39 data are poor, simpler models should be used. y I f errors i n a simple model are a d d i t i v e , i t can be expected that they w i l l also accumulate more r a p i d l y i n recursive and chain models. For example, i n a recursive system of equations the errors accumulate at each time period and, by the time period t+2, the model may be inaccurate. t+2 - 4-9 -Figure 5 : Cumulation of errors E r r o r s t o t a l Complexity of a model Figure 6 : Structure of a chain-model Submodel I Submodel I I Submodel I I I FINAL MODEL Figure 7 : Structure of an improved chain-model Submodel I . Submodel rv Submodel Submodel II FINAL MODEL - 50 -S i m i l a r l y , long chains of models are studied by Alonso and he f i n d s that chains of models i n which the output of 41 one submodel i s the input to the next should be avoided. The structure of a chain model can be seen i n f i g u r e 6 (P- 49). In taking t h i s chain-effect into account, Colenutt proposes checks i n the form of a serie s of models which pr e d i c t the same a c t i v i t y feeding into one submodel as shown i n figur e 7 (p. 4 9 ) . 4 2 As a f i n a l note on model b u i l d i n g i t i s well to consider 45 Alonso's v advice. He concludes that one should d i s t i n g u i s h between models f o r research and models f o r applied work. Research i n s t i t u t i o n s should t r y to obtain data of high q u a l i t y and b u i l d complex models i n order to advance and extend the f i e l d of model b u i l d i n g . On the other side the planning agencies should work with simpler and therefore safer models. 3 . 3 C a l i b r a t i o n and Testing of a Model The r e s u l t of the design process i s the mathematical structure of the model. The next phase i s c a l i b r a t i o n which involves mainly two steps: 1. The variables mentioned i n the model must be - 51 -given precise empirical d e f i n i t i o n , and 2. numerical values must be provided f o r the / | / | model's parameters. In the case of a simple l i n e a r equation model i n form of y = a + bx the f i r s t step r e l a t e s to the variables x and y and the second step to the parameters a and b. The variables and t h e i r q u a n t i f i c a t i o n i n the design of a model have already-been discussed i n the preceding part of t h i s chapter. Therefore, we s h a l l focus only on the estimation of the parameters. The parameters are constants and determine the r e l a t i o n -ship between the exogenous variable x and the endogenous variable y. Their estimation i s an extensively developed part of s t a t i s t i c a l t h e o r y . ^ The best known technique i s regression a n a l y s i s . In t h i s approach the parameters are estimated by applying the method of the l e a s t squares which means that the squared errors f o r sample estimates are minimized. For single equations with one or more independent variables ordinary l e a s t squares methods are applied. However, i n land use models there i s more than one single equation; we have often a system of simultaneous equations. For the estimation of parameters f o r such systems d i f f e r e n t and more complicated s t a t i s t i c a l methods are a p p l i e d . ^ I t i s not intended to give p a r t i c u l a r s about - 5 2 -these s t a t i s t i c a l procedures since the scope of t h i s study only includes the main phases i n model-building. A f t e r the c a l i b r a t i o n of a model the f i n a l question i s : w i l l i t r e a l l y work and represent the features of the r e a l world i n which one i s inte r e s t e d . This means that the model has to be tested. The importance of t h i s phase i s stressed by Branch when he states that: The -validity of the model must be r e g u l a r l y tested by comparing i t s representative and p r e d i c t i v e accuracy with the actual ^behavior of the organism i t depicts; otherwise, the decision-maker w i l l not or should not accept and use i t as a base f o r h i s conclusion.47 The main method of t e s t i n g i s to check i t s a b i l i t y to r e p l i c a t e the features of the r e a l world. This can be done f o r d e s c r i p t i v e models but f o r p r e d i c t i o n and planning 4-8 models i t i s extremely d i f f i c u l t . Boyce and Lote i n d i c a t e two main reasons f o r the d i f f i c u l t i e s i n t e s t i n g : 1. The models cannot be v e r i f i e d i n a s t r i c t sense because t h e i r formulations do not provide a confidence statement about the r e l a t i o n s h i p between the observed and predicted values. 2. There are often not adequate data avai l a b l e f o r t e s t i n g . ^ - 53 -Nevertheless, the model-builder should consider the t e s t i n g phase with equal importance to the other phases because there are models which run on the computer but "the output which they produce i s often lac k i n g i n realism SO and accuracy." Although t e s t i n g has often been neglected i n the past, i t i s i n c r e a s i n g l y emphasized i n recent writings and statements. Footnotes 1 C. West Churchman, Russel L. Ackoff and Leonard E. Arnoff, Introduction to Operations" Research (Hew York: John Wiley & Sons, Inc., 1956), p. 151. 2 B r i t t o n H a r r i s , "Quantitative Models of Urban Development: Their Role i n Metropolitan P o l i c y -Making", i n Issues i n Urban Economics, ed. by Harvey S. P e r l o f f and Lowdon Wingo, J r . ( B a l t i -more, Maryland: The John Hopkins Press, 1968), p. 366 - 380. 3 As examples r e l a t e d to land use see William Alonso, Location and Land Use - Toward a General  Theory of Land Rent (Cambridge, Mass.: Harvard U n i v e r s i t y Press, 1964-) ; Lowdon Wingo, J r . , Transportation and Urban Land (Wahington: Resources f o r the Future, 1961). 4- See f o r instance Walter Isard, Methods of Regional' Analysis (New York: John Wiley & Sons, I n c . , 1 9 6 0 ) , p. 4-93 - 568. 5 B r i t t o n H a r r i s , Op. c i t . , p. 371. 6 I b i d . , p. 371. 7 See Center f o r Real Estate & Urban Economics, Jobs,  People and Land: Bay Area Simulation Study (Berkeley, C a l . : The Center f o r Real Estate and Urban Economics, 1968). 8 B r i t t o n H a r r i s , Op. c i t . , p. 372. 9 B r i t t o n H a r r i s , I b i d . , p. 373• 10 I r a S. Lowry, "A Short Course i n Model Design", Journal of the American I n s t i t u t e of Planners, V o l . 31 No. 2 (May 1965), P- 160. 11 B r i t t o n H a r r i s , Op. c i t . , p. 377. 12 B r i t t o n H a r r i s , I b i d . , p. 377. 13 Linear programming applied to a land use model: Kenneth J . Schlager, "A Land Use Plan Design Model", Journal of the American I n s t i t u t e of Planners, V o l . 31 No. 2 (May 1965), P- 103 - H I -14- B r i t t o n H a r r i s , I b i d . , p. 379. - 55 -15 Thomas G. Donnelly, F. Stuart Chapin J r . , and S h i r l e y F. Weiss, A P r o b a b i l i s t i c Model f o r  R e s i d e n t i a l Growth (Chapel H i l l : Center f o r •Urban and Regional Studies U n i v e r s i t y of Northv Carolina, 1964); C u r t i s C. H a r r i s , "A Stochastic Process Model of R e s i d e n t i a l Development", Journal  of Regional Science, V o l . 8 No. 1 (1968). 16 B r i t t o n H a r r i s , Op. c i t . , p. 3 7 9 . 17 I r a S. Lowry, Op. c i t . , p. 1 5 9 ' 18 I b i d . , p. 159-19 Wilbur A. Steger and T.R. Lakshmanan, "Plan Evalu-a t i o n Methodologies: Some Aspects of Decision. Requirements and A n a l y t i c a l Response", i n Urban  Development Models, Highway Research Board (Sp e c i a l Report 97, Washington, D.C. 1968), p. 3 8 . 20 The fundamental work of game theory i s : John von Neumann and Oskar Morgenstern, Theory of Games and  Economic Behavior (Princeton: U n i v e r s i t y Press, 1944). 21 See f o r example, Richard L. Meier, "The Gaming Simulation i n Urban Planning", Journal of the Ameri- c a r r I n s t i t u t e of Planners, V o l . 32 No. 1 (January 1966), p. 3 - 1 6 ; A l l a n G. F e l d t , "Operational Gaming i n Planning Education", I b i d . , p. 17 - 3 2 ; A l l a n G. F e l d t , The C o r n e l l Land Use Game (New York: C o r n e l l U n i v e r s i t y , Center f o r Housing and Environ-mental Studies, D i v i s i o n of Urban Studies, 1964), Miscellaneous Papers, No. 3« 22 Melvin Dresner, Games of Strategy: Theory and  Ap p l i c a t i o n (Englewood C l i f f s , N.J.: P r e n t i c e - H a l l , Inc., 1961), p. 1. 23 R.D. Specht,"The Why and How of Model Building", i n Analysis f o r M i l i t a r y Decisions, ed. E.S. Quade, The RAND Corporation- (Chicago: Rand McNally & Company, 1964), p. 68. 24 Russel L. Ackoff, The Design of S o c i a l Research (Chicago: The U n i v e r s i t y of Chicago Press, 1 9 5 3 ) , p. 14. 25 I r a S. Lowry, "Seven Models "of Urban Development" i n Urban Development-Models, Highway Research Board (Special Report 97, Washington, D.C. 1968), p. 122. - 56 -26 Richard Bellman, Mathematical Optimization  Techniques (Berkeley: U n i v e r s i t y of C a l i f o r n i a Press, 1963), p. 335. 27 R.D. Specht, Op. c i t . , p. 69. 28 I r a S. Lowry, Op. c i t . , p. 122. 29 W.L. Garrison, " D i f f i c u l t Decisions i n Land Use Model Construction", Highway Research Record, No. 126 (1966), p. 22. 30 This aspect is'" emphasized "by several authors i n Highway Research Board, Urban Development Models (Special"Report 97, Washington, D.C., 1968)," p. 3 - 1 7 , 20, 2 3 . An e x p l i c i t statement about the l e v e l of aggregation \¥hich could also be applied, i n t h i s context: W. Miernyk, The Elements of Input - Output Analysis (New York: Random House, 1965), p. 16. 31 Harry B. Wolfe and Martin C . Ernst, "Simulation Models and Urban Planning", i n Operations Research  f o r Public Systems, P h i l i p M. Morse, ed. (Cambridge, Mass.: The MIT Press, 1967), p. 4-9 - 81. 32 I b i d . , p. 56. 33 This part of the study r e l i e s h e a vily on William Alonso, "Predicting with. Imperfect Data", Journal  of the American I n s t i t u t e of Planners, Vol" 35 No. 3 (July 1968), p. 24-8 - 2 5 5 ; i n addition there i s knowledge included which r e s u l t s from the back-ground education of the author i n error theory as applied to surveying. 34- For a more d e t a i l e d ' explanation of t h i s formula see A. de Forest Palmer, The Theory of Measurements (New York: McGraw-Hill Book Company, 1912), p. .95 -104-; or E. Bright Wilson, J r . , An Introduction to S c i e n t i f i c Research (New York: McGraw-Hill Book Company, 1952), p. 2 7 2 . 35 William Alonso, Op. c i t . , p. 24-9. 36 I b i d . , p. 24-9. 37 I b i d . , p. 2 5 1 . 38 I b i d . , p. 2 5 1 . 39 I b i d . , p. 251. - 57 -4-0 R.J. Colenutt, "Building Linear P r e d i c t i v e Models f o r Urban Planning", Regional Studies, V o l . 2 No. 1 (Sept. 1968), p. 139 - 14-3. 4-1 William Alonso, Op. c i t . , p. 2 5 2 . 4-2 R.J. Colenutt, Op. c i t . , p. 14-2. 4-3 William Alonso, Op. c i t . , p. 254. 4-4- I r a S. Lowry, Op. c i t . , p. 1 6 3 . 4-5 See the general s t a t i s t i c a l l i t e r a t u r e , f o r instance G.W. Snedecor, S t a t i s t i c a l Methods (Ames: Iowa State College Press, 1 9 5 0 ) ; or A.L. Edwards, S t a t i s t i c a l Methods f o r the Behavioral Sciences (New York: Rinehart and Co., 1 9 5 4 ) . 4-6 For review of methods see Donald M. H i l l and Daniel Brand, "Methodology f o r Developing A c t i v i t y D i s t r i -bution Models by Linear Regression Analysis", Highway Research Record, No. 126 (1966),. p. 66 - 78. 4-7 M e l v i l l e C. Branch, Planning: Aspects and Applications (New York: John Wiley & Sons, Inc., 1 9 6 6 ) , p. 153• 4-8 I r a S. Lowry, "A Short Course i n Model Design", Journal of the American I n s t i t u t e of Planners, Vol. 31 No. 2 (May 1 9 6 5 ) , p. 164-. 4-9 David E. Boyce and Roger W. Lote, " V e r i f i c a t i o n of Land Use Forecasting Models: Procedures and Data Requirements", Highway Research Record, No. 126 ( 1 9 6 6 ) , p. 60. 50 Michael A. Goldberg, "The Bay Area Simulation Study: It s Use f o r Comprehensive Urban Transportation Planning", U n i v e r s i t y of B r i t i s h Columbia, Commerce 510 Lecture, 1 9 6 9 . - 58 -4. SELECTED REGIONAL LAND USE MODELS Models are of recent o r i g i n i n planning. 1 Land use models developed, f i r s t of a l l , i n connection with transportation planning because the transportation studies had both a need and the resources f o r the preparation of such models. 2 More recently, the progress i n computer technology and the growing recognition of the complexity of urban problems have promoted even greater i n t e r e s t i n land use model b u i l d -ing. The f i e l d has expanded extremely f a s t ^ and today there i s a great number of land use models which are eith e r i n actual use or under study. The purpose of t h i s chapter i s to discuss three selected regional land use models. The Pittsburgh Model was the f i r s t operational model on a regional l e v e l and i t s inge-n u i t y influenced numerous model-builders. The Connecticut Model deals with a region as large as a State and i s there-fore of great i n t e r e s t as a macro-approach. The Bay Area. Simulation Study i s one of the most recent models and introduces a high l e v e l of disaggregation on a regional s c a l e . - 5 9 -4.1 The Model of the Pittsburgh Region by Lowry The b u i l d i n g of t h i s model was started by Lowry while he worked f o r the Pittsburgh Regional Planning Association 4 which sponsored an Economic Study of the Pittsburgh Region. The model was l a t e r completed as a part of the RAND Corpo-r a t i o n ' s research program i n urban transportation. The Pittsburgh Region covers an area of 420 square miles centering on the C i t y of Pittsburgh. The area i s defined by the Pittsburgh Area Transportation Study (PATS) as the probable "commutershed" of t r a v e l into the c e n t r a l c i t y as 5 f a r into the future as 1980 and encompasses about 1 . 5 m i l l i o n inhabitants and 550,000 jobs. Included i n t h i s area are 225 square miles of usable vacant or a g r i c u l t u r a l land which i s enough to accommodate the growth of the region f o r several decades. 4.1.1 The Concept of the Model The model locates urban a c t i v i t i e s i n sub-areas of the region; i t i s not designed f o r a p r o j e c t i o n of regional growth variables such as t o t a l population or employment. It describes the s p a t i a l organization and i s mainly intended as 1. a device f o r evaluating the impact of public decisions (e.g. concerning urban renewal, - 60 -tax p o l i c i e s , land-use controls, transpor-t a t i o n investment) on metropolitan form; and 2. a device f o r p r e d i c t i n g changes i n metro-p o l i t a n form which w i l l follow over time as a consequence of c u r r e n t l y v i s i b l e or a n t i c i p a t e d changes i n key variables such as the pattern of "basic" employment, the e f f i c i e n c y of the transportation system, or the growth of population.6 The land use a c t i v i t i e s are divided into three groups: 1. Basic sector, i n c l u d i n g manufacturing; wholesale and heavy commercial; public u t i l i t i e s , communication, transportation; h o s p i t a l s , colleges, i n s t i t u t i o n s ; outdoor p u b l i c services; mining and a g r i c u l t u r e . These a c t i v i t i e s are " r e l a t i v e l y unconstrained i n l o c a l s i t e -7 s e l e c t i o n by problems of access to l o c a l markets. Therefore they have been treated as exogenous va r i a b l e s ; t h e i r l o cations and number of employees are assumed as "given". 2. R e t a i l sector i n c l u d i n g r e t a i l trade, personal and business services, l o c a l i n s t i t u t i o n s and schools and other establishments which serve the l o c a l r e s i d e n t i a l population. I t was assumed that these establishments are bounded i n s i t e s e l e c t i o n because they have to be accessible to the l o c a l residents. 3. Household sector or r e s i d e n t i a l population; i n t h i s case s i t e s e l e c t i o n i s powerfully influenced by the residents' journey to work. - 61 -The main feature of the model i s that the basic sector i s given exogenously and only the remaining two sectors are a l l o c a t e d to subzones of one square mile. It d i s t r i -butes these a c t i v i t i e s by means of algebraic p r o b a b i l i t y functions which were developed from the analysis of t r i p data of the region's transportation study. The algorithm of the computer s t a r t s with the given basic workplaces and d i s t r i b u t e s that r e s i d e n t i a l population g which i s able to "supply an appropriate labor force." This r e s i d e n t i a l population i s then a basis f o r the l o c a t i o n of a c t i v i t i e s of the r e t a i l sector. The market p o t e n t i a l of each l o c a t i o n i s evaluated and r e t a i l employment i s d i s t r i -buted i n proportion to these p o t e n t i a l s . In the next phase the residences of the r e t a i l sector are located. This r e s u l t s not only i n a change of r e s i d e n t i a l population but also i n market p o t e n t i a l s . This i t e r a t i o n proceeds u n t i l a stable d i s t r i b u t i o n i s achieved within the constraints of a v a i l a b l e land, e f f i c i e n t s i z e of r e t a i l establishment and maximum r e s i d e n t i a l d e n s i t i e s . This procedure can be seen i n a flow diagram (see f i g u r e 8, p. 62). 4-. 1.2 The Structure of the Model The formal model i s a system of 9 simultaneous equations and three i n e q u a l i t i e s . TRACT STATUS VARIABLES ( E V A L U A T E D S E P A R A T E L Y F O R E A C H T R A C T ) i<3— — — • : T O T A L E M P L O Y M E N T : S T A R T ^ S S i t e E M P L O Y M E N T v R E T A I L 1 : E M P L O Y M E N T : L J O T A L R E S I D E N J J = H O U S E H O L D S = | | | T O T A L L A N D A R E A ' I j U N U S A B L E L A N D ? B A S I C U S E S I : R E T A I L U S E S : <3—=— : R E S I D E N T I A L U S E S : = ( R E S I D U A L ) = RETAIL EMPLOYMENT LOOP ( R E P E A T E D F O R E A C H R E T A I L G R O U P ) RESIDENTIAL POPULATION LOOP ( P R E C E D E S R E T A I L E M P L O Y M E N T L O O P S ) R E T A I L E M P L O Y M E N T P E R H O U S E H O L D ; T O T A L R E T A I L ! E M P L O Y M E N T R E Q U I R E D 1 M A R K E T P O T E N T I A L F U N C T I O N ' Y E S 1. : R E T A I L E M P L O Y M E N T = P E R T R A C T E M P L O Y M E N T D E N S I T Y C O E F F I C I E N T ( L A B O R F O R C E P A R T I C I P A T I O N . r a ; e • + : T O T A L H O U S E H O L D S : = R E Q U I R E D = P O P U L A T I O N P O T E N T I A L F U N C T I O N : H O U S E H O L D S : = P E R T R A C T : : N E T R E S I D E N T I A L 1 ( ^ D E N S I T Y P E R T R A C T : t T- : Y E S F i g . 8 — I n f o r m a t i o n F l o w s i n t h e P i t t s b u r g h M o d e l - 63 -Notations: A = area of land (1000 sq. feet) E = employment (number of persons) N = population (number of households) T = index of t r i p d i s t r i b u t i o n ( a i r distance) Z = constraints U = unusable land B = basic sector R = r e t a i l sector H = household sector k = category of establishment within the r e t a i l sector m = number of classes of r e t a i l establishments (k = l,...m); i n t h i s model m = 3 i , j = sub-areas of the region of about 1 sq. mile, c a l l e d t r a c t s n = number of t r a c t s ( i = 1 , . . . n; j = 1 , . . . n) Model: Land Use R e t a i l Sector A^ + A? + A? + A1* 0 0 0 0 a kN • = b k n 1=1 c k N i I T i a + d ks; ( i ) (2) (3) E = A? = n _k 3^1 E a i i ^ w (5) (6) n Household Sector N = Ko • N = s i = l E. T. -10 ( 7 ) (8) (9) Constraints £ Z k, or else E k = 0 N. < z M 0 u u (10) (11) (12) - 64 -The nine equations express the following r e l a t i o n s h i p : Land Use and Basic Sector: (1) T o t a l a v a i l a b l e land i n each subarea equals the sum "R TT of the d i f f e r e n t land uses. A., A. and A. are exogen-3 0 D ously determined. R e t a i l Sector: (2) T o t a l r e t a i l employment i n each category i s a function of the number of households i n the region. (3) R e t a i l employment by category i n each t r a c t i s propor-t i o n a l to the strength of the market i n the t r a c t which i s expressed as a p o t e n t i a l derived from shopping t r i p s . I t i s assumed that shopping t r i p s originate from the households i n a l l t r a c t s and from workplaces only with-i n the t r a c t . The t r i p s from workplaces are pedestrian t r i p s , while those from home are vehicular t r i p s which diminish with distance (gravity p r i n c i p l e ) . (4) T o t a l r e t a i l employment by category equals the sum of r e t a i l employment by category i n each t r a c t . (5) T o t a l employment i n each t r a c t i s the sum of the exogen-ously determined basic employment and the endogenously determined r e t a i l employment f o r that t r a c t . (6) Land use f o r r e t a i l a c t i v i t i e s i n each t r a c t i s the sum of the uses i n each category. The r e t a i l employment-density r a t i o e k i s determined exogenously f o r each r e t a i l category. Household Sector: (7) T o t a l population i s a function of t o t a l employment. - B S -CS) Number of households i n each t r a c t i s a function of that t r a c t ' s a c c e s s i b i l i t y to employment ( g r a v i t y model).. The c o e f f i c i e n t g Is a scale f a c t o r so as to f u l f i l l equation (7) (9) T o t a l population i s the sum of population i n a l l t r a c t s . Constraints: (10) In order to avoid d i s p e r s i o n of r e t a i l employment there i s a minimum-size constraint expressed as a minimum number of r e t a i l employment. (11) I t i s possible that highly accessible t r a c t s get an excessive population and therefore maximum d e n s i t i e s (Z.) have to be given (number of households per 1000 sq. f t . of r e s i d e n t i a l space). These d e n s i t i e s vary from t r a c t to t r a c t and can be determined exogen-ously (e.g. from zoning ordinances). (12) F i n a l l y , the land f o r r e t a i l establishments must not exceed the a v a i l a b l e land. This constraint and equation (1) prevent negative values of r e s i d e n t i a l land. The model without the constraints contains 4-n + mn + 2m + 2 unknowns (see Appendix 2) and the same number of equations which i s "a necessary but not s u f f i c i e n t condition of s o l u t i o n . " ^ Lowry does not s p e c i f y under which conditions a s o l u t i o n e x i s t s . For the s o l u t i o n of t h i s equation system Lowry developed an i t e r a t i v e method. - 6 6 -4-. 1.3 In t e r p r e t a t i o n of the Model The Pittsburgh Model i s mainly a s o c i a l p h y s i c a l model. The choice between the two theories, as mentioned i n part 2 of t h i s study, was " p a r t l y a matter of circumstance. The avai l a b l e data were c o l l e c t e d f o r the regional trans-p o r t a t i o n study and seemed adequate to f i t gravity type models. There were no data c o l l e c t e d i n regard to the " l o c a t o r s ' preference functions" which would be necessary f o r b u i l d i n g an economic l o c a t i o n model. Apart from these l i m i t a t i o n s of ava i l a b l e data, Lowry points out that the g r a v i t y concept i s much eas i e r to apply and also cheaper to operate. For the l o c a t i o n of r e t a i l a c t i v i t i e s profit-maximation i s the main objective of the entrepreneur and he locates where he can a t t r a c t maximum patronage. According to our poten-t i a l formula (equation 3 ) , patronage attracted to any given t r a c t j depends on the " d i s t r i b u t i o n of residence and employment with respect to t r a c t j , and also on t h i s same d i s t r i b u t i o n with respect to a l l other t r a c t s . . . . The model thus incorporates both competition and d i s t a n c e " 1 as determining f a c t o r s . For the l o c a t i o n of households the journey to work i s con-sidered as a main determinant. The proximity to place of work and the resultant minimization of commuting costs - 67 -influence the choice of l o c a t i o n of households. This r e l a t i o n s h i p , i n turn, can be expressed by the c l a s s i c a l g r a v i t y function with T.. = r x which indicates the dimin-i s h i n g employment r e s u l t i n g from increasing distance from the work-place. Although the parameter x varies with socio-economic status and the kind of occupation, i t was not possible f o r t h i s model to elaborate on that aspect. Rather, only one single t r i p - d i s t r i b u t i o n function was used to cover a l l occupations and i n d u s t r i e s . With t h i s p r o b a b i l i t y function the a l l o c a t i o n algorithm works within the constraints of available land. Households with no working members are d i s t r i b u t e d i n the same way and house-holds with more than one worker are only " i n d i r e c t l y and poorly taken into account i n the empirical evaluation of 12 the access variable (T..) f o r a l l households." 4-. 1.4- C a l i b r a t i o n of the Model The data c o l l e c t e d by the Transportation Study used the c i t y block as the smallest u n i t . These c i t y blocks were then aggregated in t o 4-56 larger units which were f i t t e d to a coordination g r i d of one square mile. The land-use data were c o l l e c t e d on an i n d i v i d u a l parcel basis. The f o r t y - f i v e categories were reduced to f i v e categories: basic, r e t a i l , r e s i d e n t i a l , unusable, and a g r i c u l t u r a l or vacant. - 68 -Information concerning household and t r i p c h a r a c t e r i s t i c s were c o l l e c t e d through an area-wide sampling of households. The sample siz e was approximately 5 percent and the universe was estimated at about 448,000 households. In regard to employment there were some d i f f i c u l t i e s . In the f i r s t instance, the U.S. Census only records employment by the employee's place of residence and not by place of work. Therefore an "employment surface" 15 w a s developed by u t i l i z i n g the work-trip data of the transportation study. In t h i s study, however, employed persons who l i v e outside the study area were not included i n the home interviews. As a r e s u l t , several adjustments were necessary i n order to get t o t a l employment and i t s s p a t i a l d i s t r i b u t i o n . A second problem concerned c l a s s i f i c a t i o n into basic and service sectors. For example, ho s p i t a l s and colleges were s h i f t e d back and f o r t h between the two sectors. I n i t i a l l y , the model was designed to treat ten r e t a i l ' c a t e g o r i e s but Lowry found that t h i s was too expensive i n terms of computer 14 time. Therefore, categories "with s i m i l a r market-patterns" were combined. The r e s u l t was three d i s t i n c t types of r e t a i l c l u s t e r s : Number of Employees Neighborhood f a c i l i t i e s : Food stores; 5 0 , 0 0 0 drug stores; gasoline service s t a t i o n s ; . personal s e r v i c e s ( p a r t ) ; elementary and secondary schools; domestic services. - 69 -Local f a c i l i t i e s : Parts of the following: 85 , 0 0 0 Eating and drinking places; medical and health services; welfare and r e l i g i o u s services; personal services; finance, insurance and r e a l estate services; auto-motive dealers and r e p a i r services; depart-ment, general merchandise, and v a r i e t y stores; amusement and re c r e a t i o n f a c i l i t i e s ; p u b l i c administration; miscellaneous r e t a i l and service trades not l i s t e d above. Metropolitan f a c i l i t i e s : Parts of most groups 56 , 7 0 0 l i s t e d under " l o c a l f a c i l i t i e s " , with large shares of department stores, f i n a n c i a l services and public lodgings, business services, and public administration. T o t a l 1 9 1 , 7 0 0 Each of these c l u s t e r s i s represented by one r e t a i l sub-equation (that means the same form but d i f f e r e n t parameters). However, at t h i s point, i t should be emphasized that one t r a c t can receive employment from more than one of these three c l u s t e r types. Estimation of Parameters The parameters have been estimated independently of each other and outside the context of the model. I t was only a f t e r t h e i r estimation that they were then Inserted into the model. In t h i s section, however, i t i s proposed to discuss only the estimation of the t r i p - d i s t r i b u t i o n i n d i c e s (T. . ) . For work t r i p s a sample of nearly 4,000 t r i p - r e c o r d s , - 70 -representing 3 2 , 0 0 0 f i r s t work t r i p s , has been f i t t e d to a negative power function: ^ = T i d _ 1 * a r - x This gives the r e l a t i v e frequency of t r i p s i n r e l a t i o n to a i r l i n e distance from residence. This function has the following values f o r a l l occupations: y w = 4-3.90 r - 1 ' 3 3 The work t r i p s have also been s t r a t i f i e d into four employ-ment classes and Lowry found that "upper income f a m i l i e s 15 have a more dispersed r e s i d e n t i a l pattern." But f o r t h i s model the t r i p d i s t r i b u t i o n function f o r a l l • occupations, as mentioned above, has been used. For shopping t r i p s a sample of about 5 , 0 0 0 t r i p records, representing an estimated 3 9 , 0 0 0 t r i p s , was found to be best approximated by a r e c i p r o c a l quadratic function of the following form: y x = T i o _ 1 = (a - br + c r 2 ) " 1 This function has been f i t t e d to the three types of r e t a i l c l u s t e r s : neighborhood f a c i l i t i e s ; l o c a l f a c i l i t i e s and metropolitan f a c i l i t i e s . The values of these and other parameters of t h i s model can be found i n Appendix 3 ' - 71 -4-.1.5 Testing of the Model A t o t a l of three experimental runs were conducted: 1. Given basic employment and r e l a t e d land uses, the model generated the d i s t r i b u t i o n of r e s i d e n t i a l population and r e t a i l employment. This experiment represented the model i n f u l l use. 2. Given basic and service employment, the model generated r e s i d e n t i a l population. 3. Given basic employment and population, the model generated r e t a i l employment. The f i r s t t e s t showed mainly that the d i s t r i b u t i o n of population i s more symmetrical than the actual development of the region. This i s due to the assumption of the gravi t y p r i n c i p l e which d i s t r i b u t e s a c t i v i t i e s i n pro-portion to distance but without regard f o r d i r e c t i o n . This means that the costs of transportation are equal i n each d i r e c t i o n which does not correspond to the r e a l world. The second test i ndicated that i n d i s t r i b u t i n g households the model i s not very s e n s i t i v e to the l o c a t i o n of r e t a i l employment. But i n experiment three, It became evident that the d i s t r i b u t i o n of r e t a i l employment i s s e n s i t i v e to the l o c a t i o n of the population. - 7 2 -4.1.6 Evaluation In appraising h i s work Lowry states that t h i s model i s "a 16 prototype with a promising future." I t i s the f i r s t land use a l l o c a t i o n model which provides a s a t i s f a c t o r y d i s t r i -bution of a l l land use a c t i v i t i e s on a regional l e v e l . Perhaps the most s a l i e n t f i n d i n g , however, 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 i n t e r a c t i o n s of a 17 v a r i e t y of l o c a t o r s . " ' This model gives evidence that r e l a t i v e l y simple a l l o c a t i o n r u l e s could be e f f i c i e n t i n describing the i n t e r a c t i n g mechanism of l o c a t i o n and function. 18 Nevertheless, the model i s an "instant metropolis." I t represents a l o c a t i o n a l equilibrium which i s only con-strained by the a v a i l a b i l i t y of space. But the r e a l world i s dynamic and lias a number of constraints. I t appears questionable, therefore, whether a dynamic system can be simulated by giv i n g a sequence of equilibrium conditions. Although there i s a great deal of accomplishment i t has to be emphasized that many problems remain unsolved and need consideration. The dynamic features of an urban region must e s p e c i a l l y be given intensive x-esearch. - 73 -4.2 The Connecticut Model The State of Connecticut has heen a pioneer i n planning. For example, the Connecticut Interregional Planning Program (CIPP), a j o i n t e f f o r t of the Connecticut Development Commission, Department of Agriculture and Natural Resources, and Connecticut Highway Department, i s c u r r e n t l y preparing a State-wide comprehensive development plan which w i l l encompass the economy, land pattern, transportation f a c i l i -19 t i e s , open space and outdoor r e c r e a t i o n . J In keeping with 20 t h i s oDjective a mathematical model has developed f o r the a l l o c a t i o n of the d i f f e r e n t land-use a c t i v i t i e s (population and n o n - a g r i c u l t u r a l employment) throughout the en t i r e State. This model i s mainly indended to: Provide estimates of the l e v e l and structure of the economy of the towns of the State. As such, i t provides a t o o l f o r measuring the scale and l o c a t i o n of demand f o r transporta-t i o n , as well as f o r other services and f a c i l i t i e s ; and Determine, to the extent f e a s i b l e , the modi-f i c a t i o n s i n these trends by"various p h y s i c a l planning p o l i c i e s a v a i l a b l e to the State of Connecticut. As s\ich, i t can help i n estima-t i n g impacts of p o l i c i e s implied i n alternate urban land development plans being prepared by the Connecticut Development Commission.21 The, a c t i v i t i e s are a l l o c a t e d i n two steps: f i r s t , from the State to the towns and, second, from the towns to zones. Only the f i r s t model, however w i l l be discussed, namely the "town model" which d i s t r i b u t e s the growth of the State to - 74- -the 169 towns. The reason f o r t h i s l i m i t a t i o n stems from the purpose of t h i s study which i s to consider regional models. In t h i s case, the region under study i s very large, p a r t i c u l a r l y i n comparison to the preceding Pittsburgh Model. 4-.2.1 Formulation of the Model Economic growth f o r the State of Connecticut varies widely between the d i f f e r e n t sub-areas. Hence, i t was necessary to search f o r a measure of economic growth which was se n s i t i v e to changes i n subareas i n r e l a t i o n to the State as a whole. 22 It was found that s h i f t analysis represented an appropriate framework. In s h i f t analysis the change of an a c t i v i t y (population or employment) of a subarea over a s p e c i f i c time period has tv<ro components: a proportional share (P.S.) which means that i t has the same rate of change as the State as a whole and, secondly, a d i f f e r e n t i a l s h i f t (D.S.) r e s u l t i n g from a f a s t e r or slower rate of growth of a subarea i n comparison with the State as a whole. The d i f f e r e n t i a l s h i f t i s also often c a l l e d the competitive s h i f t because i t i s an i n d i c a t o r of "competition" of growth between subareas. This concept i s i l l u s t r a t e d i n the following f i g u r e : - 75 -Change i n A c t i v i t y E^ o State Subarea j Time Figure 9 : D i f f e r e n t i a l S h i f t and Proportional Share Change of a c t i v i t y i i n subarea j = E . - E.. = (P.S.) + (D. i j ^ i d 0 where P.S. = E-l i 1 0 ( ^ t r ^ o ) i o D ' S * = E i j o ( E i j t • E ± J ° - E±t^~ E ± o ) = E. . E. ZLJO I O D.S. . E l j t - B l j 0 i o The d i f f e r e n t i a l s h i f t can be p o s i t i v e or negative; i t i s p o s i t i v e i f the subarea grows f a s t e r than the State and negative i f the subarea grows slower. I f a l l d i f f e r e n t i a l s h i f t s of an a c t i v i t y are summed, we receive the change of the State and t h i s means that the sum of d i f f e r e n t i a l s h i f t s of an a c t i v i t y over a l l subareas i s zero. - 76 -H (D.S.),, = 0 The model i s now designed to pre d i c t the d i f f e r e n t i a l s h i f t s (endogenous v a r i a b l e s ) of each a c t i v i t y i n each subarea. I t includes nine economic a c t i v i t i e s : s i x employ-ment categories and three groups of population. Employment i s broken down into manufacturing; r e t a i l and wholesale; personal services; business and pro f e s s i o n a l services; construction; and others. This d i v i s i o n shows that basic employment i s aggregated i n one sector and non-basic employment i n f i v e sectors. In t h i s regard we can observe s i m i l a r i t i e s with the Pittsburgh Model i n which one basic 23 sector and three service sectors were distinguished. y The three population sub-groups were formed with income as a c r i t e r i o n . The population of the State was divided into t e r t i l e s and the income ranges of these t e r t i l e s were determined f o r 1950 and I960. These t e r t i l e l i m i t s of i n -come were then applied to the formation of the income groups i n the subareas. 4-.2.2 The Structure of the Model The a l l o c a t i o n of the a c t i v i t i e s i s performed by a set of simultaneous equations f o r the d i f f e r e n t i a l s h i f t s of each a c t i v i t y . The simultaneous equation approach d i f f e r s from - 77 -the m u l t i p l e r e g r e s s i o n approach i n the f o l l o w i n g manner. In m u l t i p l e r e g r e s s i o n only one v a r i a b l e i n each equation i s the dependent v a r i a b l e and i t i s e x p l a i n e d by a number of independent v a r i a b l e s . I n the r e a l world there are i n t e r -dependencies of a c t i v i t i e s (dependent v a r i a b l e s y) which should a l s o be expressed i n the mathematical f o r m u l a t i o n of a model. This o b j e c t i v e can be achieved by the s i m u l t a -neous equation approach because i t takes interdependencies i n t o c o n s i d e r a t i o n i n the form of a system of i n t e r r e l a t e d 24 equations: y l + a 1 2 ^ 2 + ' * ' * a l n y n = b l l x l + b 1 2 x 2 + ''' ' b l k 3 C k + -10 b-a 2 1 y l + J2+ a 2 n y n = b 2 1 x l + b 2 2 x 2 + b 2 k x k + b 2 0 ^l*l+**2?2+ •••• ^n = b n l x l + b n 2 x 2 + •••• b n k x k + b n 0 On t h i s b a s i s the Connecticut Model was b u i l t i n the f o l l o w -i n g way: (For n o t a t i o n see f o l l o w i n g page) Model: Manufacturing: D!? = a-. I DS-H-D^E? + b ^ E ^ . +b7.A^+bz,H? +b ° o l k k j l j o 2 g k g o 3 0° 4- o o S e r v i c e s : D ? . = a-, + a~ XD{?. + b , E . +b 0A^ + b^ k j 1 o 2 k ^ 3 D P o p u l a t i o n : D ? = a 1D?.+a 0D M+aJa S.+ b-.APl + b 0 H P .4g 1 2j 2 .j 3^  kg 1 go 2 o 4j = a l D f j + S2 D d I + a3pkd + a4l I ) f j + b l A d o + b2 H 0 4i • a l D ! + a 2 p k a + a ? p e d r t l A 3 o + b 2 H o Notation: E = Level of a c t i v i t y (number of employees, population) D = D i f f e r e n t i a l s h i f t A = P o t e n t i a l or a c c e s s i b i l i t y of a town to an a f f e c t i n g a c t i v i t y •H = Holding capacity f o r a c t i v i t i e s (maximum l e v e l r e s t r i c t e d by p o l i c i e s ) In conjunction with the above symbols, the following sub-s c r i p t s and superscripts are used: M = Manufacturing sector S = Service sectors (k= 1,...5) B = Business and p r o f e s s i o n a l service sector P = Population e = T e r t i l e of income d i s t r i b u t i o n (e= 1 , 2 , 3 ) k = Industry group i n the employment sector j = Towns i n the State o = Beginning of the growth period a,b = Parameters This mathematical equation system expresses the following r e l a t i o n s h i p s : The d i f f e r e n t i a l s h i f t s of the manufacturing sector (D^) u during a time period i n town j Is a function of: - sum of the d i f f e r e n t i a l s h i f t s i n a l l service sectors i n town j - t o t a l employment In manufacturing i n that town at time t (lagged employment) - employment i n a l l service sectors at time t i n town j - a c c e s s i b i l i t y of town j to employment i n business and pro f e s s i o n a l services at time t - holding capacity f o r manufacturing employment i n town at time t The d i f f e r e n t i a l s h i f t s (D, .) i n the d i f f e r e n t service - 79 -sectors are a function of: - d i f f e r e n t i a l s h i f t i n manufacturing i n town j - sum of a l l d i f f e r e n t i a l s h i f t s i n the d i f f e r e n t service sectors - employment i n the sector at time t i n town j (lagged employment) - a c c e s s i b i l i t y of town j to a l l population at time t The d i f f e r e n t i a l s h i f t i n population of an income t e r t i l e i s a function of: - d i f f e r e n t i a l s h i f t i n the next higher population t e r t i l e - d i f f e r e n t i a l s h i f t i n manufacturing i n town j - sum of a l l d i f f e r e n t i a l s h i f t s i n service sectors i n town j - sum of d i f f e r e n t i a l s h i f t i n a l l 3 population t e r t i l e s i n town j - a c c e s s i b i l i t y to population i n that income t e r t i l e i n town j - holding capacity f o r a d d i t i o n a l population i n town j . 4.2.3 Inte r p r e t a t i o n of the Model I t i s evident that i n the simultaneous equation approach a dependent variable i n one equation i s "independent" i n another. This approach allows interdependencies i n the l o c a t i o n of economic a c t i v i t i e s to be simultaneously treated.25 The model also takes i n t o account i n t e r n a l economies of scale - 80 -by introducing the time-lagged l e v e l s of a c t i v i t i e s (at the beginning of the time period). M u l t i p l i e r e f f e c t s are treated by i n c l u d i n g d i f f e r e n t i a l s h i f t s In other a c t i v i t i e s as "independent" v a r i a b l e s . But beyond these concepts the s p a t i a l d i s t r i b u t i o n of the a c t i v i t i e s i s mainly af f e c t e d by a c c e s s i b i l i t y . A c c e s s i b i l i t y i s c a l c u l a t e d using a gravity-type model: a c c e s s i b i l i t y of town j to an a f f e c t i n g a c t i v i t y a c t i v i t y i n town, k which determines a c c e s s i b i l i t y t r a v e l time between j and k exponent where A. . E i k d i k The exponent of t h i s function i s obtained from the g r a v i t y model run of appropriate t r i p interchanges. A c c e s s i b i l i t y i s s e n s i t i v e to changes i n the transportation network. For manufacturing, a c c e s s i b i l i t y to business and profes-s i o n a l services i s relevant as a "measure of the 'spawning' p o t e n t i a l a town o f f e r s f o r small manufacturing plants that u t i l i z e external economies of scale by sharing a set 27 of business and p r o f e s s i o n a l services." ' For the service sector, a c c e s s i b i l i t y to population i n d i c a t e s the a t t r a c t i o n - 81 -of a town as a market and as a source f o r a labor f o r c e . The existence of income group preferences and t h e i r c l u s t e r i n g i s taken into account by introducing, as a v a r i a b l e , a c c e s s i b i l i t y to population of the same income group. 4-.2.4- C a l i b r a t i o n and Testing of the Model The employment data were obtained from d i f f e r e n t sources; the main sources were the State Department of Employment and the Connecticut Labor Department. The income t e r t i l e d i s t r i b u t i o n was received from Census records and the changes were calculated by a s p e c i a l computer program. The a c c e s s i b i l i t y indices were calculated through a program conducted by the Connecticut Highway Department. The holding c a p a c i t i e s f o r a d d i t i o n a l population and employment "were given by the Connecticut Development Commission which based them on development p o l i c i e s . A discussion of the estimation of the parameters i s beyond the scope of t h i s study because systems of simultaneous equations require the a p p l i c a t i o n of advanced s t a t i s t i c a l p o methods. The standard errors of the parameters in d i c a t e PQ "a high order of r e l i a b i l i t y . " J Therefore the model's performance seems to be encouraging. The model has been applied i n pr o j e c t i n g the future a l l o c a -- 82 -t i o n of economic a c t i v i t i e s . The forecasts of•employment and population by the Connecticut Interregional Planning Program 5 0 f o r the years 1 9 7 0 , 1 9 8 0 , 1990 and 2 0 0 0 were d i s t r i b u t e d to the towns of the State. The projections were based on two d i f f e r e n t assumptions: f i r s t , growth was projected on the basis that land development density p o l i -c i e s w i l l also be i n existence i n the p r o j e c t i o n years. Secondly, the e x i s t i n g highway network w i l l also be i n operation i n the future, and i n addition, the committed network of highways, as well as the set of a d d i t i o n a l f r e e -ways as given by the Highway Department are assumed. The above discussion shows that t h i s model can be applied f o r studying the consequences of d i f f e r e n t p o l i c y assump-tio n s within a state wide area. The structure of the model i s r e l a t i v e l y simple, macro-oriented, and the data require-ments are not i n t e n s i v e . Hence, i t seems that such a s h i f t -analysis framework i n which the interdependencies are treated by a simultaneous equation system could serve as a basis f o r the construction of a regional model f o r other areas. - 83 -4 . 3 The Bay Area S i m u l a t i o n Study (BASS-Model) The Bay Area S i m u l a t i o n S t u d y 5 1 was i n i t i a t e d i n 1962 at the Center f o r Real E s t a t e and Urban Economics of the U n i v e r s i t y of C a l i f o r n i a , Berkeley. I t was supported by the A s s o c i a t i o n of Bay Area Governments under a c o n t r a c t w i t h the Department of Housing and Urban Development i n 1964- and by the C a l i f o r n i a S t ate Water Q u a l i t y C o n t r o l Board, under a c o n t r a c t w i t h K a i s e r Engineers i n 1967. I n i t i a l l y the r e s e a r c h focused mainly on r e v i e w i n g the model b u i l d i n g l i t e r a t u r e , but by 1964 a p i l o t model had been s t a r t e d f o r the Santa C l a r a County (BASS I ) . T h i s e a r l y v e r s i o n p r o v i d e d many u s e f u l i n s i g h t s i n t o the problems of model b u i l d i n g , e s p e c i a l l y i n regard t o data c o l l e c t i o n , programming, and i n t e r p r e t a t i o n . ^ The next step was the development of a model f o r 9 coun t i e s of the San F r a n c i s c o Bay Area (BASS I I ) . 5 5 T h i s model was f u r t h e r m o d i f i e d and expanded t o the f i n a l BASS I I I model c o v e r i n g 13 counties 5' 4' and the time h o r i z o n was extended to 2020. The model i s intended to serve as an e l a b o r a t e a n a l y t i c a l device which permits a l t e r n a t i v e economic p r o j e c t i o n s to be "fed i n t o " i t i n order t o produce as an output the r e s u l t a n t i n cremental e f f e c t s on land absorp-t i o n . . . . i t i s designed to measure the im-pact of changing assumptions w i t h respect to employment; incomes, and household t r a v e l - 84 -and spending behavior; public and private i n -vestments; and other variables a f f e c t i n g land absorption and u t i l i z a t i o n . 3 5 4 . 3 . 1 Formulation of the Model The basic structure of the model i s a system of d i f f e r e n t submodels and has two d i s t i n c t parts: 1. forecast of growth expressed as changes i n population and employment by f i v e year periods. 2. a l l o c a t i o n of forecasted growth to subareas of the region. The forecast gives aggregates of population and employment f o r 21 d i f f e r e n t groups of i n d u s t r i e s . These are then used as inputs f o r the l o c a t i o n a l submodels which d i s t r i b u t e the a c t i v i t i e s to 777 subareas (census t r a c t s ) of the .13 counties of the San Francisco Bay Area. The located a c t i v i -t i e s are l a t e r converted to land use f i g u r e s (acreages) using land absorption c o e f f i c i e n t s f o r each type of a c t i v i -ty. In the forecasting phase the employment and population growth models are separate submodels and only connected i n regard to the estimation of migration.'Employment i s f o r e -casted with two submodels: a s t r u c t u r a l model which uses regression techniques and a s h i f t model using s i m i l a r tech-- 85 -niques as described i n the previous model of the State 36 of Connecticut. F i n a l l y , population forecasts are based on b i r t h rates, death rates, and estimates of migration to the Bay Area. These fo r e c a s t i n g models w i l l not be described i n t h i s study which i s mainly concerned with the l o c a t i o n of a c t i v i t i e s . The s p a t i a l a l l o c a t i o n models are divided into two groups of submodels which are shown i n figu r e 9 P» 86: 1. employment l o c a t i o n submodels which appear . i n the f i r s t h eavily dotted black bos, and 2. r e s i d e n t i a l l o c a t i o n submodels which appear i n the second heavily dotted black box. We s h a l l f i r s t of a l l r e f e r to the f i r s t group of submodels which w i l l then provide a basis f o r a discussion of the r e s i d e n t i a l l o c a t i o n submodel. 4.3.2 Employment Location Submodels In discussing these submodels we s h a l l follow the sequence indicated i n the flow diagram (see f i g u r e 9 )« It does not correspond with the importance of the employment sectors; i t i s mainly determined by the SIC Employment Code. This code and i t s subdivision into employment groups can be seen i n App.endix 4. BAY A R E A SIMULATION MODEL ( B A S S ) EMPLOYMENT LOCATION .& RESIDENTIAL LOCATION SUBMODELS - 87 -A f i r s t set of submodels locates employment i n a g r i c u l t u r e , mining, transportation and communication, and m i l i t a r y (employment groups 1, 2, 12 and 21). These employment sectors represent a minor proportion of t o t a l employment 57 and grow les s r a p i d l y than employment as a whole. ' In the case of mining and a g r i c u l t u r e i t was even found that these sectors are a c t u a l l y d e c l i n i n g . M i l i t a r y employment w i l l probably locate at already e x i s t i n g m i l i t a r y bases. I t was therefore decided to d i s t r i b u t e these a c t i v i t i e s simple i n proportion to t h e i r present l e v e l s i n each subarea. This approach seems to be j u s t i f i e d by the small share of t h i s employment sector "so that any d i s t o r t i o n introduced i s 58 probably unimportant." y Construction employment (group 3) was a l l o c a t e d to a sub-area i n proportion to new employment and new houses i n that subarea: New employment .and new Percentage of construction _ houses i n subarea j employment i n subarea n ~ m~4--,n v,~,„ -« i * ° T o t a l new employment and t o t a l new houses This percentage i s then m u l t i p l i e d by the t o t a l amount of new employment i n a l l counties, which i s an output from the growth forecast, i n order to obtain the number of employees i n the subarea j . The t h i r d set of submodels which i s one of the most important ones r e l a t e s to the l o c a t i o n of i n d u s t r i a l employment (groups - 8 8 -4- through. 11) and includes employment i n manufacturing, trucking, warehousing and wholesaling. In t h i s case a f i r s t step toward the a l l o c a t i o n process involved the i d e n t i f i c a -t i o n of the relevant v a r i a b l e s which influence the r a t i o n a l choice of a l o c a t i o n . Two sources served as a b a s i s : a survey of i n d u s t r i a l r e a l t o r s and a regression analysis which supplied the important f a c t o r s and the r e l a t i v e im-portance of each. Out of these variables the " e s s e n t i a l " f a c t o r s f o r each group were selected. With these " e s s e n t i a l " f a c t o r s i t was then possible to t e s t to see whether a sub-39 area possesses the e s s e n t i a l f a c t o r s . I f not, i t was eliminated from fu r t h e r consideration and thus the number of computations f o r the model was reduced. A f t e r these t e s t s the f e a s i b l e areas were av a i l a b l e f o r the a l l o c a t i o n of a c t i v i t i e s . In each area the l o c a t i o n a l 4-0 f a c t o r s were measured and combined, using weights to y i e l d scores which express the attractiveness f o r each of the eight i n d u s t r i a l groups. Employment was a l l o c a t e d on the basis of these scores. A f t e r the a l l o c a t i o n of the a c t i v i -t i e s , land use patterns were determined by applying the land use absorption c o e f f i c i e n t s . R e t a i l trade employment (group 13) i s located by a modified market p o t e n t i a l model (modified to consider more behavioral f a c t o r s ) . ^ 1 The p o t e n t i a l model i s of the g r a v i t y type. The a l l o c a t i o n process i s shown i n a flow diagram (Figure 11,p.89). Piprure 11 i The R e c a l l Development Model Concur Tor Real Estate and Urban Economics, 1967. - 90 -It begins with, new r e t a i l demand which i s an output from the growth model and measured i n terms of r e t a i l employees. Next, a part of new employment i s a l l o c a t e d to planned stores and establishments; t h i s information, c a l l e d "inten-42 t i o n data" was obtained from b u i l d i n g permits and news-paper reports and i s one of the behavioral modifications.! T o t a l r e t a i l demand i s the sum of present plus new demand. In a fu r t h e r step t h i s demand i s s p l i t into worksite and home site...demand ^ on a percentage ba s i s , assuming that 76 44-percent i s homesite demand. Homesite demand i s all o c a t e d i n proportion to population i n the subarea and worksite demand i n proportion to employment. In a next step the homesite demands are d i s t r i b u t e d to a l l areas by a gravity-type model which ca l c u l a t e s the probabi-l i t y that a person i n subarea i w i l l t r a v e l to an area j f o r shopping. For t h i s gravity model the f r i c t i o n of distance was expressed by T.. (T.. = t r a v e l time). This p r o b a b i l i t y i s then m u l t i p l i e d by the homesite demand i n i i n order to f i n d the demand i n j . This i s repeated f o r a l l subareas i and the r e s u l t i s the expected homesite demand i n j . By adding worksite demand i n j t o t a l expected demand i n j was obtained. This expected demand minus the actual demand measures the demand p o t e n t i a l i n subarea j . F i n a l l y , a commercial s i t e s u i t a b i l i t y was calcul a t e d by multiple regression analysis and employment was then - 91 -d i s t r i b u t e d i n proportion to a combination of t h i s a t t r a c -tiveness index and the demand p o t e n t i a l ( r e l a t i v e a t t r a c -tiveness i n d e x ) . ^ A f i f t h set of submodels locates employment i n groups 14, 16, 17. and 18 which include several services as eating and drinking f a c i l i t i e s ; personal services; miscellaneous business services; and medical se r v i c e s . These a c t i v i t i e s have been located by using multiple regression equations. Out of 40 possible variables (the same as i n manufacturing employment) the most relevant ones have been selected and. represented by a regression equation f o r each employment group. The important independent variables i n these equations were a c c e s s i b i l i t y , density of development, and r e l a t e d groups of employment. These equations measured the a t t r a c -tiveness of a subarea and employment was located on t h i s b a s i s . The f i n a l set of employment models includes employment i n finance, insurance, r e a l estate and government (groups 15 and 20), and employment i n education (group 1 9 ) - These employment groups are al l o c a t e d by applying d i f f e r e n t per-centages. The present percentages In groups 15 and 20 have been estimated f o r each county-and a change over time was introduced assuming d e c e n t r a l i z a t i o n forces (e.g. San Francisco 1966: 51% and 2010: 46%). The a c t i v i t i e s were al l o c a t e d i n each county i n accordance with these percentages. - 92 -Employment i n education was allocated by assuming that i t i s a function of the population i n the subarea. 4-. 3 . 3 R e s i d e n t i a l Location Submodel The r e s i d e n t i a l submodel i s based on the assumption "that households can be all o c a t e d to places of residence using the j o b s i t e l o c a t i o n s of e x i s t i n g and new jobholders as the 4-7 only s p a t i a l determinants." ' The approach of the a l l o c a t i o n algorithm may be seen as an attempt at "an e x p l i c i t r e p l i -4-8 cati o n of the market process." The main operation of the model involves the estimation of the new demand f o r housing which i s then matched with supply. The general structure of the r e s i d e n t i a l a l l o c a t i o n process can be seen i n the flow diagram (figure 10 p. 86). The model includes s i x d i f f e r e n t categories of household u n i t s , formed by three income l e v e l s (high, middle, low), and two housing types (sing l e family and mul t i - f a m i l y ) . In the flow diagram there are three i n i t i a l stages of the sub-model: supply, f i l t r a t i o n , and demand. The submodel s t a r t s each i t e r a t i o n period with a f i l t r a t i o n . I t includes the demolition of houses and the s h i f t of houses from high i n -come to middle and from middle income to low income residents. For demolition the model uses exogenous forecasts based on demolition rates ( t o t a l demolition rates and demolition . 4.0. rates f o r the s i x housing types). - 93 -The housing supply i n a subarea depends on the slope of the land, the attractiveness of the t r a c t f o r r e s i d e n t i a l development, the income cl a s s d i s t r i b u t i o n of e x i s t i n g u n i t s , the proportion of sing l e - f a m i l y and multi-family dwellings, the density of development, and the p o t e n t i a l land supply f o r new units.^° The slope i n each t r a c t was c l a s s i f i e d into l e v e l , r o l l i n g and h i l l y . Attractiveness Includes f a c t o r s such as available services and microclimate. The measurement of density takes i n t o account population and employment. The p o t e n t i a l proportion of dwelling types, i s based on the average of two r a t i o s . The f i r s t r a t i o expresses the e x i s t i n g proportion and the second r a t i o , which i s weighted twice as heavily i s calculated as a function of density.' The t o t a l demand f o r new housing i n the Bay Area i s the sum of housing u n i t s removed from the stock by the f i l t r a t i o n process and the new f a m i l i e s projected by the growth model. This demand i s then divided into s i n g l e - f a m i l y and mu l t i -family u n i t s . This d i v i s i o n changes over time and ranges 51 between 65% s i n g l e - f a m i l y u n i t s i n 1965 and 44-% i n 2015• In a next step, t o t a l demand i s p a r t i t i o n e d into three value c l a s s e s . This i s done by averaging three estimates: 1. the e x i s t i n g d i v i s i o n by income, 2. an estimate based on the assumption that the percentage of high income uni t s increases with higher density, and - 94- -3 . an estimate based on the assumption that the percentage of high income u n i t s increases with increasing slope. The next and f i n a l phase involves the s p a t i a l a l l o c a t i o n of the estimated demands of the s i x categories which i s done on the basis of a c c e s s i b i l i t y to employment as the deter-mining f a c t o r . However, i t was pointed out that there has to be a search f o r a more se n s i t i v e measure of a c c e s s i b i l i t y which should include a c c e s s i b i l i t y to other a c t i v i t i e s as well as to employment. The r e s i d e n t i a l submodel a l l o c a t e s 30 percent of the new housing u n i t s according to a c c e s s i -b i l i t y to e x i s t i n g employment i n order to replace stock re-52 moved and 70 percent to new employment. The f i n a l output of the BASS Model gives employment by 21 groups, population and housing u n i t s by s i x categories, and land use f o r 777 subareas of the 13 counties i n the San Francisco Bay Area Region by f i v e year periods between 1965 and 2020. 4 . 3 . 4 Appraisal The basic concepts of t h i s model i n regard to the a l l o c a -t i o n of a c t i v i t i e s are based on the working mechanism of the market process. The model has been tested under a v a r i e t y of assumptions and has produced outputs which are - 9 5 -"consistent with the l o c a t i o n a l trends under way i n the 53 Bay Area f o r the past two decades or more." y y Projections into the future have also been compared with estimates by other agencies concerned with t h i s region and i t has been observed that the r e s u l t s are quite s i m i l a r . The main assets of t h i s model are i t s l e v e l of disaggregation and i t s f l e x i b i l i t y to adapt to changing conditions. In comparison with the Pittsburgh and the Connecticut Models, the disaggregation i n t h i s model into 21 employment groups, 6 housing types and 777 subareas, as well as the i n c l u s i o n of micro-economic behavior f o r a r a t i o n a l choice of a l o c a -t i o n , i s remarkable. The f l e x i b i l i t y of the model can be viewed i n regard to the change i n the model's parameters and i n the p o s s i b i l i t y of 53 changing d i f f e r e n t p o l i c y assumptions. The model includes a great number of parameters which can be adapted and improved. For example, parameters such as the land use absorption c o e f f i c i e n t s or the average f i r m size can be a l t e r e d over time i f new data and i n s i g h t s become a v a i l a b l e . In regard to new p o l i c y assumptions i t i s conceivable that d i f f e r e n t redevelopment p o l i c i e s could be introduced, each of which w i l l have a d i f f e r e n t e f f e c t on the demolition sub-model. But to introduce such changes into the model a great deal of research i s necessary. Much of t h i s new i n s i g h t , - 9 6 -however can only be gained through a p p l i c a t i o n of the model. This idea has been expanded by the authors who have stated that the model "must be used to be u s e f u l . " 5 ^ Conclusions The discussion of these three models gives evidence that promising and powerful t o o l s e x i s t f o r the s p a t i a l a l l o c a -t i o n of land uses within a region. The progress which has been made i n a r e l a t i v e l y short period of about ten years of research and a p p l i c a t i o n i s an i n d i c a t i o n of the strength of the model b u i l d i n g f i e l d . A main feature of models i s that they give us a better understanding of the growth processes i n a region; they show the relevant f a c t o r s behind growth and the i n t e r a c t i o n s between them. Such an under-standing then provides a sound basis f o r the improvement of p o l i c i e s and plans f o r the future development of urban regions. Footnotes 1 F. Stuart Chapin, J r . , Urban Land Use Planning (Urbana: U n i v e r s i t y of I l l i n o i s Press, 1965), P- 4-75. 2 B r i t t o n H a r r i s , "Conference Summary and Recommen-dations", Urban Development Models, Highway Research -Board (Special Report 97? Washington, D.C. 1968), p. 3.' 3 A f i r s t review of models appeared i n a s p e c i a l issue of the Journal of the American I n s t i t u t e of' Planners, Vol" 25 Ho. 2 (May 1959); a comprehensive evaluation of 14- models appeared i n 1963: T r a f f i c Research Corporation, Review of E x i s t i n g Land Use  Forecasting Techniques, Boston Regional Planning.. .. Project, also published i n Highway Research Record No. 88, 1965- (includes extensive' bibliography). 4- See the following studies: The Pittsburgh Regional Planning Association, Region i n T r a n s i t i o n , Vol.1 P o r t r a i t of a Region, Vol. 2 Region with a Future, Vol. 3 (Pittsburgh, Pennsylvania: U n i v e r s i t y of Pittsburgh Press, 1963); the Pittsburgh Model i s described i n I r a S. Lowry, A Model of Metropolis, Memorandum RM-4-035-RC (Santa Monica, C a l . : The RAND Corpora-t i o n , 1954-).-5 I r a S Lowry, 6 I b i d . P. V. 7 I b i d . P • 5-8 Ib i d . P- 4-. 9 I b i d . P. 14-. 10 I b i d . p. 23. 11 I b i d . P- 24-. 12 I b i d . P- 35. 13 I b i d . P- 61. 14- I b i d . p.. 63. 15 I b i d . P- 66. - 98 -16 I b i d . , p. 128. 17 I b i d . , p. 1 2 9 . 18 I b i d . , p. 3 9 . 19 The Connecticut; Interregional Planning Program, Goals f o r Connecticut  The Economy-The Green Land  Urban Development  Transportation Connecticut: Choices f o r Action. (State of Connecticut, 1966). 20 T.R. Lakshmanan, "A Model f o r A l l o c a t i n g Urban.. A c t i v i t i e s i n a State" i n Socio-Ecdnomic- Planning  Sciences, V o l . 1 No. 3 (July 1968), p. 283 - 2 9 5 . 21 I b i d . , p. 284-. 22 For a d e t a i l e d d e s c r i p t i o n , see Harvey S. P e r l o f f , E.S. Dunn, E.E. Lampard and R.F. Muth, Regions, Resources and Economic Growth (Baltimore: The John Hopkins U n i v e r s i t y Press, 1 9 6 1 ) , part I I . 23 For the Pittsburgh Model see p. 59 of t h i s study. 24- Linear equation system models are discussed by Donald M. H i l l and Daniel Brand, "Methodology f o r Developing A c t i v i t y D i s t r i b u t i o n Models by Linear Regression Analysis", Highway Research Record, No. 126 01966), p. 66 - 78. 25 T.R. Lakshmanan, Op. c i t . , p. 290. 26 I b i d . , p. 2 9 0 . 27 I b i d . , p. 2 9 1 . 28 For the c a l i b r a t i o n of t h i s model the two stage l e a s t square method was applied. This and other possible methods'are discussed by Donald M . H i l l and Daniel Brand, Op. c i t . , p. 73 - 7 6 . 29 T.R. Lakshmanan, Op. c i t . , p. 2 9 2 . 30 Employment was forecasted with the use of an input-output model; see Dr. Charles Leven, The cut Socio-Economic Growth Model, CIPP S t a f f Paper 1 9 6 5 . _ 99 -31 Center f o r Real Estate and Urban Economics, Jobs,  People and Land: Bay Area Simulation Study (Berkeley, C a l i f o r n i a : The Center f o r Real Estate and Urban Economics, 1968.) 32 I b i d . , p. 13. 33 BASS II includes the following counties: San Francisco, Marin, SOnoma, Napa, Sclano, Contra Costa, Alameda, Santa Clara, and Santa Mateo 34- The extension to 13 counties includes Sacramento, San Joaquin, Yolo, and Santa Cruz. 35 Center for-Real Estate and Urban Economics, Op.  c i t . , p. 16. 36 See p. 74 - 76 of t h i s study. 37 Center f o r Real Estate and Urban Economic, Op. c i t . , p. 189-38 I b i d . , p. 1 9 0 . 39 I b i d . , p. 114. 40 These weights v/ere based on judgment. 41 I b i d . , p. 1 9 9 . 42 I b i d . , p. 2 0 3 . 43 The same assumption was made by I r a S. Lowry i n the Pittsburgh Model; see p. 64 of t h i s study. 44- Center f o r Real Estate and Urban Economics, Op.' c i t . , p. 180 and 210; t h i s percentage i s based on studies f o r Pittsburgh and Washington. 45 I b i d . 1 P- 208. 46 I b i d . P • 220. 47 I b i d . P- 2 3 5 . 48 I b i d . P • 2 3 7 . 49 I b i d . 1 P- 245. 50 I b i d . t P- 2 5 2 . 51 I b i d . 1 P- 258. - 100 -52 I b i d . , p. 21 . 53 Michael A. Goldberg, "The Bay Area Simulation Study: I t s Use f o r Comprehensive Urban Transportation Study", U n i v e r s i t y of B r i t i s h Columbia, Commerce 510 Lecture, 1969, p. 10 . 54 Center f o r Real Estate and Urban Economics, Op. c i t . , p. 322. 55 I b i d . , p. 28. 5. REGIONAL PLANNING AND LAND USE ALLOCATION MODELS In previous parts of t h i s study, methods f o r model-building and the ways i n which they are a c t u a l l y b u i l t have been reviewed. I t i s now necessary i n t h i s part of the study to r e l a t e these findings to the regional planning process i n order to v e r i f y our e a r l i e r hypothesis. In so doing, the importance of l o c a t i o n i n regional planning w i l l be d i s -cussed i n order to emphasize the need f o r applying land use a l l o c a t i o n models. In addition, an attempt w i l l be made to outline not only the problems of model a p p l i c a t i o n but also recommendations and prospects f o r the model-building f i e l d . 5.1 Regional Planning and the Importance of Land Use A l l o c a t i o n Models Regional Planning i s concerned "with the ordering of a c t i v i -t i e s and f a c i l i t i e s i n space at a scale greater than a single community and l e s s than a n a t i o n . " 1 It i s an extension of l o c a l planning. The need f o r t h i s extension may be argued i n several ways. The strongest reason i s the regional character of human l i f e which i s p r i m a r i l y due to the development of transportation and" communication. There are - 1 0 2 -several studies, f o r example, which show the influence of p c x t i e s over t h e i r hinterlands. In f a c t , more frequent i n t e r a c t i o n , r e s u l t i n g , f o r example, from commuting, i s now becoming such an established f a c t within metropolitan regional areas that the region can i n c r e a s i n g l y be viewed as a new form'of human settlement. 3 In e f f e c t , there i s a greater r e a l i z a t i o n that the study of human behavior i n the urban center i t s e l f i s inseparable from behavior and s o c i a l organisation within the urban region as a whole. Another reason f o r the r e g i o n a l approach to planning stems from the f a c t that the n a t i o n a l economy has more or l e s s homogenous subsystems (regions) with common features (income, unemploy-ment) and each of these regions requires s p e c i a l goal formulation and s o l u t i o n . Regional planning has to order human a c t i v i t i e s . In other words, there must be an a l l o c a t i o n of a c t i v i t i e s "so that they w i l l help rather than hinder each other." y In t h i s regard, the main a c t i v i t i e s are l i v i n g and working which can be expressed by the volume of economic growth: population and employment. The a l l o c a t i o n of growth should be done i n such a way that the system works optimally i n respect to scarce resources. This goal i s seldom achieved. According to H a r r i s , f o r example, "the present tendencies of develop-ment i n human settlement are f a r from optimal and i f i t w i l l be allowed continuously i t w i l l produce unacceptable 6 conditions", as f o r example urban sprawl and p o l l u t i o n . The - 103 -importance of l o c a t i o n i n regional planning i s also expressed e x p l i c i t l y by Friedmann when he states that: Eegional planning must be thought of as a s c i e n t i f i c undertaking of a s p e c i a l kind. P r i m a r i l y oriented to the future, i t looks to the r e l a t i o n between s o c i a l purpose and s p a t i a l arrangement.7 Eegional planning deals with the "supra - urban space" and "common to both c i t y and regional planning i s a c e n t r a l concern with the organization of space."^ The basic question therefore i s always: How are a c t i v i t i e s to be d i s t r i b u t e d so as to meet s o c i a l o b j e c t i v e s ? 1 0 The preceeding remarks suggest that the regional planner i s mainly concerned with the a l l o c a t i o n of a c t i v i t i e s . He must always f i n d the best locations f o r d i f f e r e n t land uses, i n c l u d i n g industry, r e s i d e n t i a l development, commercial establishments, services and u t i l i t i e s . Hence his basic question i s always: where? The next step will'now be to show how land use models can be applied to f i n d solutions f o r such problems. 5 . 2 Advantages of Land Use A l l o c a t i o n Models In an e a r l i e r statement of the problem of t h i s study r a t i o n a l decision-making c r i t e r i a were mentioned. It was pointed out that i n order to a r r i v e at a r a t i o n a l d e c i s i o n a l l a l t e r n a -- 104 -t i v e s and t h e i r possible consequences have to be considered. However, t h i s requirement i s extremely d i f f i c u l t to f u l f i l l i n such a complex system as an urban region because a great number of variables and i n t e r a c t i o n s are involved. This point i s also emphasised by Czamanski when he states that "without a quantitative model the number of a l t e r n a t i v e s which planners are able to develop i s severely l i m i t e d by the vast amount of work necessary i n order to assess the implications of e a c h . " 1 1 This statement also coincides with the hypothesis of t h i s study i n which the d e s i r a b i l i t y of models f o r r a t i o n a l decision-making i s postulated. In order to f u r t h e r v e r i f y our hypothesis we s h a l l discuss below the a p p l i c a t i o n of models i n planning agencies. I t w i l l be of utmost importance to obtain information about the advantages of models f o r actual, p r a c t i c a l a p p l i c a t i o n . 12 Hemmens has undertaken a survey i n order to get information about the use of models i n planning agencies. A question-naire was sent to 34 planning agencies. A response was received from 26 agencies, i n c l u d i n g 16 metropolitan or r e g i o n a l agencies, 6 c i t y planning agencies, 2 state agen-c i e s , and 1 consulting : f i r m . I t was discovered that nineteen agencies apply models, but f o r three agencies the models are not yet highly developed. The remaining'seven agencies have' no plans f o r applying models. I t has to be pointed out, however, that these seven agencies are b a s i c a l l y c i t y - 105 -agencies; only one was a metropolitan/regional agency. Therefore, i t can be concluded that the a p p l i c a t i o n of models i n re g i o n a l planning agencies i s probably more advanced than i n c i t y or l o c a l planning agencies. These agencies viewed models p r i m a r i l y as t o o l s f o r analysing and evaluating p o l i c y a l t e r n a t i v e s . Some selected comments by agencies are l i s t e d below: Models should be used "to simulate the conse-quences of s e l e c t i n g actions, and to dimension' a general plan and make i t i n t e r n a l l y consistent." Models should be used "to predict the e f f e c t s of varying p o l i c y sets on c e r t a i n f a c t i o n s of the urban system considered to be s i g n i f i c a n t and pre d i c t a b l e . . . . " Models should be used "to forecast the e f f e c t of al t e r n a t i v e courses of action on land develop-ment, and the effectiveness of urban systems such as water and sewer." Models should be used "when and where they can sharpen up or i l l u s t r a t e consequences of follow-ing c e r t a i n development p o l i c i e s more r a p i d l y and/or more o b j e c t i v e l y than other procedures." These comments have been summarized by Hemmens who also emphasized that a l l agencies see the e s s e n t i a l function of models to be to "improve the r a t i o n a l i t y of planning In-d e c i s i o n s . " I t i s now evident that models are judged as to be extremely u s e f u l and f l e x i b l e t o o l s f o r the planner. Nevertheless, a f t e r such an op t i m i s t i c p i c t u r e , there should also be some - 106 -discussion concerning the problems inherent i n model b u i l d -ing. This w i l l be followed by some concluding recommenda-tio n s and prospects f o r 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 n the f i e l d of regional planning. 5.3 D i f f i c u l t i e s of A p p l i c a t i o n of Land Use Models Models can be used to evaluate a great many a l t e r n a t i v e s i n order to reach a r a t i o n a l d e c i s i o n . But i t has to be pointed out that there are not very many decision-makers who can be convinced of the value of highly abstract and t e c h n i c a l d escriptions of the urban processes by means of models. It i s obvious that t e c h n i c a l i t i e s are only convincing to people who can understand them. Therefore, model-builders should not only have the a b i l i t y to b u i l d abstractions of the r e a l world; they must also develop the a b i l i t y to convince those people who must accept the r e s u l t s of models as to the soundness and the value of t h e i r work. Such decision-makers w i l l not be convinced unless the d e s c r i p t i o n of the model can be presented to them i n understandable 15 language. ^ This d i f f i c u l t y of communication lias not always been given adequate attention i n the past. But r e c e n t l y i t 16 has been emphasized by several authors, and i t i s hoped that t h i s problem w i l l be a l l e v i a t e d . Nevertheless, i t should be kept i n mind that t h i s task i s extremely d i f f i c u l t . A second d i f f i c u l t y concerns the data which have to be used. - 107 -The model buil d e r must consider the features of the available data. There w i l l always be a desire f o r more and better data. C e r t a i n l y , models are r e s t r i c t e d by data a v a i l a b i l i t y , but t h i s does not mean that no models can be b u i l t i f there are only l i m i t e d data. It i s conceivable that relevant and high q u a l i t y sampling can serve as a sound' basis f o r the b u i l d -ing of a model and that the r e s u l t s which such a model produces w i l l not be of les s value to the decision-maker. This discussion about data leads to another r e l a t e d d i f f i -c u l t y , namely the processing problem, which becomes more s i g n i f i c a n t as the amount of data Increases. Therefore, with a p o s s i b i l i t y of greater q u a l i t y i n the model, there should be more profound e f f o r t s i n regard to the processing of data. It has also been said that a primary problem i n the model-buil d e r ' s dilemma i s choosing between a model which i s ' t h e o r e t i c a l l y "elegant" and one which i s operati o n a l l y 17 " f e a s i b l e " . Thus a land use model can be c r i t i c i z e d i n two ways: f i r s t , the model may be too simple i n regard to a t h e o r e t i c a l base or second, the model may be so complex as to be non-operational. In t h i s regard the model-builder has to make a d i f f i c u l t d e c i s i o n and, i n most cases, he would do well to choose between the two extremes. C r i t e r i a f o r t h i s choice are given i n part three of t h i s study where the cumulation of errors i s discussed (see f i g u r e 5, p. 4-9). - 1 0 8 -A f i n a l problem r e l a t e s to available resources such as time, finances and s t a f f of a planning agency involved i n model b u i l d i n g . Wolfe and Ernst point out that the f i n a l costs of models have almost always been much higher than i n i t i a l estimates. This i s a r e s u l t of the intensive e f f o r t s necessary f o r the development of a successful model. Very often the conceptualization phase i n model b u i l d i n g seems to i n d i c a t e a very prospective achievment. But i n the phase of c a l i b r a t i o n and t e s t i n g d i f f i c u l t i e s occur which delay the production of a r e a l i s t i c output. In t h i s respect, the following advice may be of advantage. It i s important, Ernst and Wolfe argue, that a working model be available when no P IP more than /3 of the time and budget have been spent. The remaining time i s needed to tes t and prepare the use of the model i n a r e l a t e d planning program. 5.4- Cone lusions This study has suggested that the model b u i l d i n g f i e l d has developed very quickly since 1950. With valuable s t i m u l i coming from d i f f e r e n t s o c i a l and p h y s i c a l sciences, i t has been possible to develop quantitative models which r e p l i c a t e the s p a t i a l features of an urban region. In regard to land use i t has even been possible to integrate a l l land uses and t h e i r p r i n c i p a l i n f l u e n c i n g forces on a regional l e v e l . This i n d i c a t e s a greater leaning i n the d i r e c t i o n of a - 109 -system's view of problems. This has enabled planners to study the e f f e c t s of a single p o l i c y action upon the whole reg i o n a l system. The land use model b u i l d i n g f i e l d i s based on a s o l i d t h e o r e t i c a l ground: economic l o c a t i o n theory and s o c i a l physics. Although model builders have applied mainly s o c i a l p h y s i c a l concepts by using the g r a v i t y p r i n c i p l e , there have been recent attempts to integrate the findings of economic l o c a t i o n theory into regional land use a l l o c a -t i o n models. This seems to provide a sounder basis f o r the r a t i o n a l choice of l o c a t i o n than the gravity p r i n c i p l e which represents the a p p l i c a t i o n of a p h y s i c a l law to socio-economic behavior. The d e t a i l e d discussions of the three selected models have . demonstrated the a v a i l a b i l i t y of promising too l s which are able to t e s t a great number of a l t e r n a t i v e p o l i c y assump-t i o n s . We have also discussed the importance of l o c a t i o n i n the context of regional planning and found that the a l l o c a t i o n of a c t i v i t i e s i s i t s basic concern. Therefore, i t can be assumed that the hypothesis of t h i s study has been examined and can be accepted. Hence, I t can be concluded that the regional complexities which face decision-makers are f o r c i n g them away from - 110 -s i m p l i s t i c , i n t u i t i v e judgement. I t has become desirable to apply more comprehensive and quantitative techniques i n order to make r a t i o n a l decisions f o r the development of urban regions. - I l l -Footnotes 1 H.S. P e r l o f f , "Key Features of Regional Planning", Journal of the American I n s t i t u t e of Planners, V o l . 34- No. 2 (May 1968), p. 153. 2 See f o r instance Raymund E. Murphy, The American  C i t y : An Urban Geography (New York: McGraw-Hill Book Company, 1966), p. 51 - 71. 3 Hans Blumenfeld, The Modern Metropolis (Cambridge Mass.: MIT Press, 1969), p. 235-4- A. Boskoff, The Sociology of Urban Regions (New York: Appleton - C e n t u r y - Cr o f t s , 1962), p. 6. 5 Hans Blumenfeld, "Regional Planning", Plan, V o l . 1 No. 2 (1961), p. 122. 6 B r i t t o n H a r r i s , "Quantitative Models of Urban Development" i n Issues i n Urban Economics, Harvey S. P e r l o f f & lowdon Wingo, ed., (Baltimore, Maryland: The John Hopkins Press, 1968), p. 367. 7 John Friedmann and William Alonso, Regional Develop- ment and Planning (Cambridge, Massachusetts: The MIT Press, 1964-), p. 63. 8 I b i d . , p. 63. 9 I b i d . , p. 6 3 . 10 I b i d . , p. 64-. 11 Stanislaw Czamanski, An Econometric Model of Nova Sco t i a (Halifax,•Canada: I n s t i t u t e of Public A f f a i r s , Dalhousie U n i v e r s i t y , 1968), p. 15• 12 George C. Hemmens, "Survey of Planning Agency .. Experience .'with Urban Development Models", i n Urban  Development Models, Highway Research Board (Special Report 97, Washington, D.C, 1968), p. 219 - 230. 13 I b i d . , p. 221. 14- I b i d . , p. 221. 15 Edward H. 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A u s t r a l i a n Planning I n s t i t u t e Journal, (Jan. 1966), p. 10 - 15-- 119 -APPENDIX 1 Cumulation of Err o r s Function: z = f (x-^Xp. . . .x n) p V~ dz p p ' r- r- dz dz where e : measurement errors of the input v a r i a b l e s * i r . . : c o r r e l a t i o n between x. and x. 10 1 0 The basic algebraic operations can be examined assuming that the independent v a r i a b l e s are not i n t e r c o r r e l a t e d . The function w i l l be z = f (x.;y), where x = 10 e = - 1 (10%) -A. 7 = 8 e y = + 1 (12.5%) 1. Addition: z = x + y = 1 0 + 8 = 1 8 dz _ n dz _ dx _ 1 dy ~ x e 2 = i . e 2 + l . e 2 = 1 + 1 = 2 e z = 1.4 (7-8%) The absolute error of z i s greater; but the percentage error i s smaller. This means that addition reduces the r e l a t i v e e r r o r . - 120 -2 . Substraction: z = x - y = 1 0 - 8 = 2 e 2 = 1 • e 2 + 1 . e 2 = 1 + 1 = 2 e z = 1.4 (70%) There i s a r e l a t i v e error of 70 percent which means that substraction i s explosive to the cumulation of errors, e s p e c i a l l y i f the diffe r e n c e z i s small r e l a t i v e to x and y. 3. M u l t i p l i c a t i o n and D i v i s i o n : z = x « y = 1 0 « 8 = 8 0 dx J dy ~ • e 2 = y 2 • e 2 + x 2 • e 2 = 64. • 1 + 100 • 1 = 164 e z = 13.3 (16.7%) M u l t i p l i c a t i o n increases absolute and r e l a t i v e e r r o r s ; but the r e l a t i v e e r r or of z increased only to 16 .7%. D i v i s i o n behaves exactly l i k e m u l t i p l i c a t i o n . 4. Raising to a Power: z = x 2 = 100 £2- = 2x dx - 121 -e 2 = ( 2 x ) 2 • e 2 = 400 • 1 = 400 e = 20 (20%) Absolute and r e l a t i v e errors cumulate; cumulation i s higher than i n m u l t i p l i c a t i o n , e s p e c i a l l y absolute e r r o r . This operation can also be seen as a m u l t i p l i c a t i o n of p e r f e c t l y i n t e r c o r r e l a t e d v a r i a b l e s . Therefore the second term of the above general error - equation comes into play which means that there i s a high cumulation of errors i n a function of i n t e r c o r r e l a t e d v a r i a b l e s . - 122 -APPENDIX 2 Variables and Parameters of the Pittsburgh. Model Type Symbol Number i n Number Expanded Exogenously System Determined Variables Land Use A^ n n A U A? n n u A^ n A H n -o Employment E. n -E"? n n 4 E^ mn E k m Population N. n <] N 1 (one) 2 2 T r i p - d i s t r i b u t i o n indices T. . n n i j nk" .2 2 T. . mn mn I D S t r u c t u r a l Parameters R e t a i l employment c o e f f i - ^ cients a m m R e t a i l employment scale f a c t o r b^ m k k Shopping t r i p weight f a c t o r s c ,d 2m 2m R e t a i l employment density r a t i o Labor force p a r t i c i p a t i o n rate Population scale f a c t o r - 124 APPENDIX 3 Control Totals and S t r u c t u r a l Parameters of the Pittsburgh Model Land Use: Total bounded area Basic land use Unusable land Residual f o r r e s i d e n t i a l and r e t a i l use Thousands of Sq. Feet 11,698,786 2,615,813 1,931,236 7,151,737 Employment: Basic Sector R e t a i l Sector Neighborhood f a c i l i t i e s Local f a c i l i t i e s Metropolitan f a c i l i t i e s Population and Labor Force: Number of Employees 360,948 1 9 1 , 7 0 0 50,000 85,000 56,700 Number of households Employed Residents Labor force numbers per household 447,734 526,346 1,176 - 125 -Parameters f o r R e t a i l F a c i l i t i e s Neighborhood Local Metropolitan F a c i l i t i e s F a c i l i t i e s F a c i l i t i e s Minimum employment per c l u s t e r (Tract) 50 Number of households necessary to support one employee 9.4-0 Square feet of site-space per r e t a i l employee 1 , 9 0 0 Per cent of shopping t r i p s o r i g i n a t i n g from home 90 200 5 . 5 3 1 , 3 0 0 70 20,000 8.29 80 50 T r i p D i s t r i b u t i o n Parameters Type of T r i p & O r i g i n D i s t r i b u t i o n " o f Trip-Ends by A i r l i n e Distance ( r ) from Origi n - Work t r i p s , a l l o.ccupations: Prom workplace to. home -Neighborhood shopping t r i p s : From home to r e t a i l e s t a b l i s h -ment From workplace to r e t a i l establishment - Local shopping-trips: From home to r e t a i l e s t a b l i s h -ment From workplace to r e t a i l establishment - Metropolitan shopping t r i p s : From home to r e t a i l e s t a b l i s h -ment From workplace to r e t a i l establishment - 1 . 3 3 ( . 5 1 0 7 - .7400 r +.2699 r 2 ) " 1 A l l t r i p s terminate i n work-place t r a c t . (.0116 - .0012 r + .0202r 2)" 1 A l l trip's terminate i n work-place t r a c t . (.0664 - .0442 r +.0156 r 2 ) " 1 A l l t r i p s terminate i n work-place t r a c t . - 126 -APPENDIX 4-Employment Groups f o r the BASS Model Group 1 A g r i c u l t u r e , Forestry and F i s h e r i e s 01 Commercial farms 02 Noncommercial farms 07 A g r i c u l t u r a l services and hunting and trapping 08 Forestry 09 F i s h e r i e s Group 2 Mining 10 Metal mining 11 Anthracite mining 12 Bituminous coal and l i g n i t e mining 13 Crude petroleum and natural gas 14- Mining and quarrying of nonmetallic minerals except f u e l s Group 3 Construction 15 B u i l d i n g construction — general contractors 16 Construction other than b u i l d i n g construction — general contractors 17 Construction — s p e c i a l trade contractors Group 4-20 Food and kindred products Group 5 23 Apparel and other f i n i s h e d f a b r i c products 27 P r i n t i n g , publishing and a l l i e d i n d u s t r i e s Group 6 19 Ordnance and accessories 24 Lumber and wood products, except f u r n i t u r e 25 Furniture and f i x t u r e s 26 Paper and a l l i e d products 29 Petroleum, r e f i n i n g and other r e l a t e d i n d u s t r i e s 30 Rubber "and p l a s t i c s 31 Leather and leather products 32 Stone, clay and glass products 39 Miscellaneous manufacturing i n d u s t r i e s Group 7 28 Chemicals Group 8 3 3 3 4 7 Primary metals Coating, engraving and a l l i e d services Group 9 3 4 Fabricated'metals (except coating, engraving and a l l i e d services) 3 5 Mac nine" ry except e l e c t r i c a l • 3 7 Transportation equipment Group 10. 3 6 E l e c t r i c a l machinery 3 8 P r o f e s s i o n a l s c i e n t i f i c and c o n t r o l l i n g instruments Group 1 1 42 Motor f r e i g h t transportation and warehousing 5 0 Wholesale trade Group 1 2 Transporation, Communication and Public U t i l i t i e s 40 Railroad transportation 41 Local and suburban t r a n s i t and inter-urban passenger transportation 44 Water transportation 4.5. Transportation by a i r 46 Pipe l i n e transportation 4 7 Transportation' services 48 Communication 4 9 E l e c t r i c , gas and sanitary services Group 1 3 R e t a i l Trade 5 2 Building materials, hardware and farm, equipment 5 3 General merchandise 5 4 Food 5 5 Automotive dealers 5 6 Apparel and accessories 5 7 Furniture, home furnishings.and equipment 5 9 Miscellaneous r e t a i l stores Group 14 5 8 Eating and drinking places 7 0 Hotels, rooming houses, camps and other lodging places - 128 -Group 15 Finance, insurance and Real Estate 60 Banking 61 Credit agencies other than banks 62 Security and commodity brokers, dealers, exchanges and services 65 Insurance c a r r i e r s 64- Insurance agents, brokers and service 65 Real estate 66 Combination of r e a l estate, insurance, loan and law o f f i c e s • • -67 Holding and other investment companies Group 16 22 Personal services Group 17 73 Miscellaneous business services 75 Auto r e p a i r , auto services, and garages 76 Miscellaneous repair services 78 Motion pi c t u r e s 79 Amusement and rec r e a t i o n services except motion pictures 81 Legal services 84- Museums, g a l l e r i e s , botanical and zo o l o g i c a l gardens 86 Nonprofit membership organizations 88 Private households 89 Miscellaneous services Group 18 80 Medical and other health services Group 19 82 Educational services Group 20 Government 91 Federal government 92 State government 93 Local government Group 21 M i l i t a r y 

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